Lecture Notes in Computer Science Edited by G. Goos, Karlsruhe and J. Hartmanis, Ithaca
10 Computing Methods in Applied Sciences and Engineering Part 1 International Symposium, Versailles, December 1 ?-21, 1973 IRIA LABORIA Institut de Recherche d'lnformatique et d'Automatique
Edited by R. Glowinski and J. L. Lions
Springer-Verlag Berlin-Heidelberg New York 1974
Editorial Board: P. Brinch Hansen • D. Gries C. Moler • G. Seegm011er • N. Wirth Dr. R. Glowinski Dr. J. L. Lions IRIA LABORIA Domaine de Voluceau - Rocquencourt F - 7 8 1 5 0 Le Chesnay/France
AMS Subject Classifications (1970): 65-02, 6 5 K 0 5 , 65Lxx, 65Mxx, 65Nxx, 65P05, ?6-04, 9 3 E 1 0 , 93E25 CR Subject Classifications (1974): 3.1, 3.2, 5.1
ISBN 3-540-06768-X Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06768-X Springer-Verlag New York • Heidelberg - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Library of Congress Catalog Card Number 74-5712. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
textes des communications
Ce colloque est organis6 par I'IRIA sous le patronage de l'International Federation for Information Processing.
This symposium is organised by IRIA under the sponsorship of the International Federation for Information Processing. Organisateurs
Organizers R. Glowinski J.L. Lions
PREFACE
L,e present v o l u m e r a s s e m b l e les travaux l~rEsentEs au Colloque "International sur les MEthodes de Calcul Scientifique et Technique", organisE par I ' I R I A - L A B O R I A du 17 au Zl D 4 c e m b r e 1973, sous le patronage de I'IFIP. C e Colloque a rEuni h Versailles prhs de 400 chercheurs et ingEnieurs de t o u s l e s pays du m o n d e . L'originalitE des travaux pr~sentEs, la qualit4 de l'auditoire et des questions pos4es tout au long du Colloque, attestent de l'extr~me intEr~t scientifique et technique, qui s'attache k l'usage des ordinateurs pour le calcul scientifique. Les par tieuli~r ement
organisateurs
tiennent h r e m e r c i e r
Monsieur Monsieur
Andre
DANZIN,
Directeur de I'IRIA
MichelMONPJ~TIT,
Directeur Adjoint de I'IRIA
L e Service des Relations ExtErieures qui ont p e r m i s
tout
:
de l'IRIA
l'organisation de ce Colloque.
les conf4renciers
N o s r e m e r c i e m e n t s vont, 4galement, h tous et aux diff4rents pr4sidents de seance : MM.
J.H. A R G Y R I S A.V. BALAKRISHNAN P. BROUSSE J. DOUGLAS D. FEINGOLD B. F R A E I J S de V E U B E K E P. MOREL W. PRAGER E. ROUBINE O.C. ZIENKIEWICZ
qui ont anita4 d'int4ressantes
R.
discussions.
GLOWINSKI
et J.L.
~, Institut de R e c h e r c h e d'Informatique et d'Automatique Zaboratoire de R e c h e r c h e de I'IRIA.
LIONS
PREFACE
This book contains the lectures which have been presented during the " International Symposium on Computing Methods in Applied Sciences and Engineering " organised by I R I A - L A B O R I A • u n d e r t h e s p o n s o r s h i p of I F I P . ( D e c e m b e r 17, 2 1 , 1 9 7 3 ) 4 0 0 people~ s c i e n t i s t s a n d e n g i n e e r s c o m i n g f r o m many countries attended this meeting in Versailles. The originality of t h e w o r k p r e s e n t e d , t h e h i g h q u a l i t y of t h e a u d i e n c e a n d t h e pertinent questions raised during the symposium show how important is, at the present time, the scientific and technical interest for the u s e o f c o m p u t e r s in a p p l i e d s c i e n c e a n d e n g i n e e r i n g . The organisers Monsieur
w i s h to e x p r e s s
t h e i r g r a t i t u d e to :
Andr4 DANZIN, Director
Monsieur
of I R I A ,
Michel MO,~,~PETIT, Deputy Director of IRIA
The IRIA Public Relations Office who have contributed to the organisation of this Symposium. They also address a n d to t h e c h a i r m e n of s e s s i o n s : MM.
who have
directed
J.H. A.V. P. J. D. B. P. W. E. O.C.
t h e i r t h a n k s to a l l t h e s p e a k e r s
ARGYRIS BALAKRISHNAN BROUSSE DOUGLAS FEINGOLD F R A E I J S de V E U B E K E MOREL PRAGER ROUBINE ZIENKIEWICZ
interesting discussions.
R.
* Institut de Recherche d'Informatique Laboratoire de Recherche de I'IRIA
GLOWINSKI
et d'Automatique
J.L.
LIONS
TABLE TABLE
DES MATIERES OF CONTENTS
TOME PART
I I
GENERALITES GENERA LITIES Methods of stuctural optimization W. Prager .................................... Optimisation et propulsifs singularit
L.
des par
syst~mes la m4thode
portants des
~s -
Malavard
..................................
Some contributions t o non-linear solid mechanics J.H. Argyris, P.C. Dunne .......................... ELEMENTS FINITE
20
42
FINIS EILEMENTS
One-sided approximation and plate bending G. Strang ..................................
140
Quelques mdthodes d' 414ments finis pour le probl~me d' une plaque encastrde P.G. Ciarlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
Un nouvel 414ment de coques g4ndrales B.M. Irons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
177
Numerical solution of the stationary Navier-Stokes equations by finite element methods P. Jarnet, P.A. Raviart ..........................
193
Finite elements method engineering problems B. Fraeijs de Veubeke
224
in aerospace ...........................
Viseo-plasticity and plasticity An alternative for finite element solution of material nonlinearities O.C. Zienkiewicz ..............................
259
VIII
Som~ superconvergence results for an II-- Galerkin procedure for the heat equation J. Douglas, T. Dupont, M.F. Wheeler
..................
Application de la m6thode des dldments finis - Un procddd de sous-assemblage J.M. Boisserie .................................. PROBLEMES NON-LINEAR
288
312
NON-LINEAIRES PROBLEMS
Formulation and application of certain primal and mixed finite element models of finite deformations of elastic bodies J.T. Oden ..................................
334
M4thodes num4riques pour le projet d'appareillages industriels avane4s S. Albertoni ..................................
366
Etude num4rique du champ magn4tique dans un alternateur t4trapolaire par la m~thode des 414ments finis R. Glowinski, A. Marrocco ..........................
392
Une nouvelle m4thode d' analyse num4rique des probl~mes de filtration dans les mat~riaux poreux C. Baiocchi .....................................
410
CIRCUITS NETWORKS
ET
TRANSISTORS AND SEMI-CONDUCTORS
Numerical methods for stiff systems of differential equations related with transistors, tunnel diodes, etc.W. Miranker, F. Hoppensteadt .......................
416
Conception, simulation, optimisation d' un filtre ~ l' aide d" un ordinateur A. Guerard .....................................
433
Computing methods in semiconductor problems M . Reiser ....................................
441
Simulation num4rique de la fabrication et du comportement des dispositifs semiconducteurs D. Vandorpe ...................................
467
TABLE TABLE
TOME PART
DES MATIERES OF CONTENTS
II II
MECANIQUES DES FLUIDS MECHANICS
FLUIDES
Recent advances in computational fluid dynamics T.D. B u t l e r ................................. Mdthodes et techniques d' int4gration num4rique adapt4es ~ i' 4tude des 4coulements plan4taires R. Sadourny ....................................
22
Flow ,computations with accurate space derivative methods J. Gazdag .........................................
37
Numerical simulation of the TaylorGreen vortex S.A. Orszag .....................................
50
Probl~mes et m~thodes num~riques en physique des plasmas k tr~s haute temperature C. Mercier, J.C. Adam, Soubbaramayer, J.L. Soule ......................................
65
-
Probl~mes de contr~le optimal en physique des plasmas J.P. Boujot, J.P. Morera, R. Temam
...................
107
Probl~mes de stabilit4 num4rique pos4s par les syst~mes hyperboliques avec conditions aux limites J.J. Smolderen ......................................
135
R4solution num4rique des 4quations de Navier-Stokes pour les fluides compressibles R. Peyret ........................................
160
-
PROBLEMES D' ONDES WAVES PROBLEMS Three dimensional flows around airfoils with shocks A. Jameson .....................................
185
Lage amplitude wave propagation in arteries and veins Y. Kivity, R. Collins ..............................
213
Increase of accuracy of projectivedifference schemes G.I. Marchuk, V.V. Shaydourov ......................
240
M4thodes num4riques en 41ectromagn4tisme J. Ch. Bolomey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
261
CONTROLE OPTIMAL
OPTIMAL CONTROL
Time-optimal control synthesis for non-linear systems A flight dynamic example A.V. Balakrishnan ...............................
289
Numerical analysis of problems arising in biochemistry J.P. Kernevez ...................................
312
Sur I' approximation num4rique d" in4quations quasi-variationnelles stationnaires A. Bensoussan, J.L. Lions .......................
325
Gestion optimale des r~servoirs d' une vallde hydraulique A. Breton, F. Falgarone .........................
339
FILTRAGE FILTERING
ET IDENTIFICATION AND IDENTIFICATION
Algorithmes de calcul de modules markoviens pour fonctions al4atoires P. Faurre ..................................
352
Estimation de param~tres distribu4s dans les equations aux d~riv4es partielles O. Chavent ....................................
361
Adapdon de la m4thode du gradient un probl~me d' identification de domaine J. Cea, A. Gioan, J. Michel . . . . . . . . . . . . . . . . . . . . . .
391
Application de la m 4 t h o d e des 414ments finis ~ la r4solution d' un p r o b l ~ m e de d o m a i n e optimal D. B4gis, R. Glowinski ........................
403
GENERALI,TES GENERALITIES METHODS OF STRUCTURAL OPTIMIZATION William Prager Professor Emeritus, Brown University, Providence, R. I., USA ABSTRACT The paper is concerned with methods of optimal structural design.
Typical ingredi-
ents of structural optimization problems are discussed in Sect. i.
The basic prob-
lem is identified as one in mathematical programming, in general nonlinear programming, and the difficulties are indicated that are experienced in the application of standard methods of nonlinear programming.
The following three sections deal with
optimal plastic design, which is the most developed area of structural optimization because linear programming is applicable to it.
Sections 2 and 3 are respectively
concerned with the optimal plastic design of a truss of given layout and the determination of the optimal layout of a truss that has to transmit given loads to a given foundation. Sect. 4.
Optimal plastic design of beams and grillages is discussed in
Section 5 is devoted to optimal design of elastic trusses and beams.
Com-
putational aspects of structural optimization are discussed in Sect. 6, and some new ideas are mentioned in Sect. 7. 1.
INTRODUCTION
To be well-posed, a problem of optimal structural design requires specification of the purpose of the structure, the design constraints, and the design objective. general purpose of a structure is to carry given loads.
The
In general, a structure
will have to carry several alternative sets of loads, but it may happen that the design of the structure is governed by only one of them.
Design constraints may
concern the geometry of the structure or its behavior under the given loads.
Geo-
metric constraints specify at least the space that is available for the s%-~ucture, but may go much further in restricting the shape of the structure and the dimensions of its members.
Behaviomal constraints set bow~nds on quantities that characterize
the response of the structure to the loads for which it is being designed.
Examples
of behavioral constraints are upper bounds on stresses and deflections, and lower bounds on fundamental natural frequencies or on the ratio in which the given loads would have to be increased before they would cause failure by buckling or plastic flow.
The general design objective is minimization of the combined costs of the
manufacture of the structure and its operation over the expected lifetime.
It is
typical for aerospace structures that the cost of the fuel that would be needed to carry additional structural weight and the accompanying reduction in payload are much more important considerations than the manufacturing cost. e Research supported by the U. S. Army Research Office - Durham.
In these ¢ircum-
stances, minimization of structural weight becomes the design objective.
Since
structural optimization is particularly important in the aerospace industry, a vast majority of papers on structural optimization is concerned with design for minimal weight. Naturally discrete or artificially discretized problems of structural optimization are essentially problems of mathematical progra~nlng, usually nonlinear program,ninE. The development of powerful methods of nonlinear programming, and the availability of computers with large immediate-access memories therefore raised hopes for automated optimal design of practical structures by direct application of these programruing methods.
Except for optimal plastic design, which may be treated by linear
programming, these hopes have not so far been fulfilled.
Automated design proce-
dures were in fact developed that solve the basic nonlinear progra~ing problem by gradient or feasible directions methods, or treat it by a sequence of linear programs, or transform it into an unconstTained problem by the intToductlon of penalty f~nctions (for a survey, see Pope and Schmit, 1971). According to Gellatly and Berke, 1971, however, the use of these direct methods entails a prohibitive number of design iterations for structures with more than about 150 design variables.
Even
for much smaller numbers of design variables, the possibility is troublesome that the procedure may converge towards a local optimum.
To get some assurance that a
global optimum has been achieved, it may be necessary to start the procedure from quite different initial designs and choose from the corresponding final designs the one with the smallest weight. Sved and Ginos, 1968, have pointed out another reason why direct search techniques may not be satisfactory. truss for given loads.
Consider, for example, the minimum-weight design of a One way of approaching the problem of optimal layout is to
start with a lattice of possible nodes and consider the truss in which any two of these nodes are connected by a bar.
The bar forces of this highly redundant basic
truss must satisfy conditions of equilibrium and compatibility, which will be incorporated in the formulation of the problem, and the search will be conducted over "feasible" trusses for which these conditions are satisfied.
The optimal truss,
however, may be a subtruss of The basic truss obtained from the latter by The omission of certain bars.
In this case, the bar forces of the optimal truss together
with zero forces in the omitted bars will not satisfy all compatibility conditions of the basic truss.
Accordingly, the optimal truss is not a member of The set over
which the search is extended.
Sved and Ginos concluded that it would he necessary
to supplement the search described above by a similar search of the feasible designs of each statically stable subtruss of the basic truss.
Since each of these subtrus-
ses has a different set of compatibility conditions, this would be an enormous task. Sheu and £chmit, 1972, have shown how it can be reduced to manageable size, at least for the comparatively small example trusses they considered.
(The largest of Their
haslc trusses has eight nodes and Twenty-Two bars.) In recent years, attention has tumned from the direct application of the seamch techniques of nonlinear p~o~Tamming to the derivation of optimality conditions and their use in desiEn procedumes.
Some of this work is surveyed in the following.
Various problems of optimal plastic design and optlmal elastic design ape discussed in Sects. 2-4 and 5, mespectively.
Section 6 treats computational aspects of struc-
tumal optimization, and some new ideas ame mentioned in Sect. 7. 2.
OPTIMAL PLASTIC DESIGN OF TRUSS OF GIVEN LAYOUT
By far the most developed area of structumal optimization is optimal plastic design. Heme~ the structural material is tmeated as rigid , perfectly plastic, and the structure is to use the smallest possible amount of material subject to the condition that a given state of loading should r~pmesent the load-carTying capacity of the structure. It was recognized quite early (Foulkes, 1954) that the optimal plastic design of a structure of given layout may be formulated as a linear proETam.
Consider~ for in-
stance~ the minimum-volume design of a truss with the layout shown in Fig. i.
The
bars of the truss are to be made of a rigid, perfectly plastic material with the
Fig. i:
Optimization of truss of given layout
tensile and compmessive yield limits disPegamded in the analysis. to be rigid.
+a , and the possibility of buckling is %o he
The foundation, which supports the truss, is assumed
To exclude degenel-ate cases, in which some of the four bars are miss-
ing in the optimal truss, we prescribe a minimum cross-sectional area
Aio
for bar
i , (i=l,... ,4) ~ and write the actual c~oss-sectional area of this bar as Aio + Ai, where
A. >. 0 . 1
At the joint
0 , which is taken as the origin of the coordinates
X~ , (u -- 1,2) , a load with components be denoted by
Pa
is to act.
The length of bar
i
will
h i , and the direction cosines of the axis of this ham (oriented from
the foundation towards
0 ) by
c.
The following concepts will be useful in the discussion of this pmoblem.
Fom gener-
ality, they will be defined without reference to the partlcula~ly simple example in Fi E . i.
A system of axial fomces in the bars of a truss that equilibrates given
loads acting at the joints of the tmuss will be called "statically admissible" fom these loads.
Fom specified cross-sectional areas of the bars, a statically admis-
sible system of bar forces will be called "safe" if the cormesponding axial stress in each ham does not have an absolute value in excess of
~ .
A system of mates of
extension of the bars of the truss will be called "kinematically admissible" if it is derived from velocities of the joints of the tmuss that do not violate the kinematic constmaints at the supports. Let us now return to the problem in Fig. i.
According to the static theorem of
limit analysis (see, for instance, PmaEer , 1959), the load
P
will not exceed the
load-car~ying capacity of the tmuss if there exists a safe, statically admissible system of ham forces
F.
for this load.
AccoPdingly, an optimal design is charac-
terized by the following llnear program : Minimize ~
O
Z.£.A. 1
!
(2. la)
1
subject to EiciaF i
=
P
,
(2.1b)
F i + ~A i >. -OAio ,
(2.1c)
-F i + cA i >. -qAio .
(2.1d)
The sum in (2.1a) is the excess of the total volume of the bars of the truss over the minimal volume
?i£iAio .
Equation (2.1b) stipulates that the bar forces
should be statically admissible for ~he load
P
Fi
, and the inequalities (2.1c) and
(2.1d) stipulate that these forces should be safe. If the dual variables corresponding to the constmalnts (2.ib) thmough (2.1d) are denoted by
v
, ~i ' ~i ' the dual program has the following fomm:
Maximize ~aPavc~ - cZiAio(~i + ~i )
(2.2a)
subject to
+ -~i "< ~o £'~ "
(2.2c)
To facilitate the mechanical interpretation of the dual program, the factor ~e o has been introduced in the objective function, where ~ is a reference mate of extension, o
Note that the variables applies in (2.2c) if Fi/(Aio + A i) value
o .
v
are not restricted in sign, and that the equality sign
Ai > 0 .
has the value
Note further that
-c , and that
Accordingly, either
~i
or
Pi
~i > 0
Pi • 0
only if the stress
only if this stress has the
or both vanish.
The dual problem may be given a mechanical interpretation by identifying vs,(e=l,2), as the velocity components of the joint P~
and
~. • 0
~i - ~i
O
in a collapse mechanism under the load
as the corresponding rate of elongation
Xi
of bar
only if the equality sign holds in (2.1c), i.e. if the bar
i
i .
Now,
is at the
l
compressive yield limit; in this case,
"i + 6 1
=
The same r e l a t i o n
Ixil is
Pi = 0 , I i = ~i - ~i < 0 , and
.
(=3)
obtained
for
Pi • 0 , in which
case
~i
= 0
and
X'z • 0
m
Accordingly, the constraint (2.2c) may be written as fell "< So ' with equality sign for
Ai > 0 ,
(2.~)
where
¢. is the axial strain rate of bar i in the considered collapse mechanism. 1 Note that the signs of s i and F i are coupled by the relation F . c . >. 0 . 1 1
(2.5)
If a truss of the given layout is at all capable of carr~ylng the load
P
, there
exists at least one statically admissible system of bar forces, and the cross-sectional areas
A. + A. may be chosen sufficiently large $o fulfill the constraints io 1 (2.1c) and (2.1d). According to the existence theorem of linear programming, this
fact assures the existence of solutions of both the primal and the dual problems. There exists therefore a joint velocity that entails rates of extension satisfying (2.4) and (2.5). Note that the linear programming formulation of our problem is readily adapted to the case where none of several alternative loads load-carrying capacity of the truss.
P'
P"
is to exceed the
For each load, we then have equilibrium equa-
tions of the form (2.1b) and yield constraints of the forms (2.1c) and (2.1d).
The
mechanical interpretation of the dual yields the optimality condition
Iql where P'
~' P"
+ ~':11 + ~" ....
"'"
...
"< So ,
with
equality sign for
A, • 0 ,
(2.6)
are axial strain rates in collapse mechanisms under the loads
If one of these loads is too small to influence the optimal design,
the corresponding rates of extension vanish. 3.
OPTIMAL LAYOUT OF A TRUSS
The preceding discussion of the optimal design of the truss in Fig. 1 obviously remains valid if the joint
O
is connected to the foundation by more than four bars.
This remark enables us to attack the problem of optimal layout by starting from a "basic" layout that comprises, in addition to the joint
0 , a large number of po-
tentlal joints at the foundation, and setting
Aio = 0
for all bars of thls layout
To allow for the possibility that some of them may be omitted in the optimal truss. In The limiting case~ wheme any point of the homizontal foundation is a potential joint, the optimality condition (2.4) calls for a homogeneous strain mate field in the stmlp between the foundation and the horizontal Through mates
¢
0
that has strain
satisfying The following conditions:
Icl
=
0
for the homizontal direction,
(3.1a)
=
c°
for the direction of any bar of the optimal Truss,
(3.]30)
¢ o
fom any othem direction.
(3.1c)
It follows from (3.1) that the bars of the optimal truss are along the principal directions of the considemed stmain mate field.
Depending on whether one or both
principal mates of extension have the absolute value in Figs. 2a and b.
~o ' we have the cases shown
In the first case, the optimal truss is degenemate and consists
P O
A
P
O
// (a)
(b) (a) Degenerate optimal tmuss for load inclined by 45 ° or less against vertical
Fig. 2:
(b) Optimal truss for load inclined by more than 45° against vertical
of a single bam
OA
along the line of action of The load~ which forms an angle of
45° or less with The vertical. the bars
OA
and
OB
In the second case, the optimal truss consists of
that form angles of 45° with the vertical, while the angle
between load and vertical exceeds 45 ° . The conditions (3.//)) and (3.1c) quite generally gover~ the optimal layout of trusses fop a single state of loading (Michell, 1904). truss for the transmission of the load of the limited width
AB.
P
at
0
Figur~ 3a shows the optimal to a rigid, horizontal foundation
The hams of the Truss follow the principal lines of a
strain mate field with pmlncipa! strain rates
±¢
o
; along the foundation, this
field has vanishing strain rate. onal net in the region shown.
In the circular sectors
bars.
The bars of the optimal truss form a dense orthog-
0CDE ; in the figure, only a few of these curved bars are
Along the contour bars
ACD OCA
and and
BDE , there are dense systems of radial 0EB , the axial forces are constant, and
the curvature of these bars causes small axial forces in the bars of the other family.
Accordingly, the interior bars of the truss are light in comparison to the
contour bars.
There is a substantial body of literature on trusses of this kind,
which are known as Miehell trusses (see, for instance, Hemp, 1966).
While Michell
trusses wlth their infinity of bars are not practical, they provide the smallest
C
E
A
B
A
(b)
(a) Fig. 3:
(a) Optimal transmission of load
P
to foundation
(h) Optimal transmission of alternative loads to foundation AB
P'
AB and
p,,
possible structural weight, which is used in assessing the efficiency of more practical designs.
The Michell efficiency of a truss is defined as the inverse ratio
of the weights of this truss and the Michell truss for the same loading. To formulate an optimization problem that leads to more realistic designs, one might enforce a finite number of bars by including the weight of the connections at the joints in the structural weight that is to be minimized, for instance by adding, for each joint, a fixed weight corresponding to the average weight of a connection.
As
is well known, fixed cost problems of this kind create considerable algorithmic dif-
ficulties. An interesting super~ositlon principle that furnlshes the optimal truss for two alternative loadings is due to Hemp, 1968.
The illustrative problem in Fig. 3h con-
cerns the transmission of the alternative loads represented by the vectors OP"
to a rigid foundation of the limited width
segment
P'P" , the given loads are written as
AB .
With
Or'
and
as the center of the
OQ -+ QP' , and the optimal trusses
("component trusses") ape deterlnined for the single loads consists of the single bar
Q
OQ
and
QP' .
The first
0C , while the second has the layout in Fig. 3a.
The
strain rate fields for the collapse mechanisms of the component trusses ~ which satisfy the conditions (3.i), will he called "cor~onent fields".
The line elements of
the plane of the truss may be divided into three groups depending on whether they are along a bar of the first or second component truss or not along a bar of either II truss. If EI and c are the strain rates of the same line element in the two component fields,
¢oe°l
I¢I I "<'< i¢II I
{ group. with equality sign for line elements in the ~second~Ifirst
(3.2)
It follows from (3.2) that leI + eII I * leI - e I I I .< 2e 0 , with equality sign for line elements of first and second groups.
(3.3)
The sum and the difference of the component fields thus satisfy the conditions (3.1) for the truss obtained by the supemposition of the component trusses (with the reference strain rate
2¢ O) , while the sum and the difference of the bar forces of the
component trusses ape in equilibrium with the sum and difference of the loads on the component trusses, i.e. with the loads
OP'
and
0P" .
For fu1~ther examples of the
use of this superposition principle, the reader is meferTed to a paper by Nagtegaal and Prager, 1973.
Unfortunately, it does not appear that a similarly simple super-
position principle is available for more than Two alternative loadings. Retu1~nlng to the optimal design of a truss for a single loading, we mention the use of dynamic programming by Palmer and Sheppapd, 1970. transmission tower that carries a horizontal load height 2y I
h
and
a discrete set of given values, say
Y3
has a given value, say
panel for each possible value of W3 .
2y 3
at the top.
The u n i f o ~
at the base, but the widths
at the bottoms of the Two top panels are to he optimally chosen fmom 0.2h ~ 0.4h , 0.6h , 0.Sh .
the typical panel with the forces acting on it. which
2P
of the panels is given, as is the width 2y 2
Figure ~a shows a symmetric
Figure 4h shows
Starting with the bottom panel, for
Y3 = 0.6h ~ we compute the weight Y2
and select the value
Y2 = Y2
W3
of this
that minimizes
A similar computation is carried out for the next higher panel using Y2 = Y2 ~
and so on.
The optimal tmuss found in this way has y~ = 0.6h , Yl = 0.4h (Fig. 5a).
Although this truss has %he high Michell efficiency of
0.91 , ~he method is open
Y2 ~
Y2 --."
Qi-I
Qi-I :(i-I)Ph/Yi_ ~
P._,,.~ 2 Yi-' ~__.. P
//
Qi
(o) Fig. 4:
f
Qi = i Ph/Yi
(b)
(a) General layout of transmission tower: h and Y3 given, Yl and Y2 ar~ to be optimally chosen
a~e
(b) Equilibrium of typical panel to the criticism that the general layout of the truss has been chosen in advance, whereas the optimal truss is likely to have the general layout in Fig. 5b~ which does not lend itself to optimization by dynamic proETamming.
//
X
"////~ / /
/
(G) Fig. 5:
/ (b)
(a) Optimal layout by dynamic pro~rammin~ (b) Likely optimal layout
10
4.
OPTIMAL PLASTIC DESIGN OF OTHER STRucTuRES
The optimal design of a sandwich beam with a core of uniform rectangular section and identical cover plates of varying thickness may be reduced To a linear programming problem by choosing a sufficiently dense set of nodes along the axis of the beam and replacing the loads on each segment between consecutive nodes by equipollent loads at the ends of the segment.
At each node an equation of equilibrium must be satis-
fied, which is linear in the bending moments at this node and its immediate neighbors.
Furthermore, at each node, the absolute value of the bending moment must not
exceed the yield moment, which is proportional to the Thickness of the cover plates. These yield inequalities and equilibrium equations are the constraints of a linear programming problem; The objective function, which is to be minimized, represents the weight of the cover plates, which is proportional to is half the distance between nodes moment at
node
i , Yio
i - i
and
Zi£i(Yio + Yi) , where
£i
i + 1 , and
Y. + Y. is the yield io 1 being a prescribed minimal value for This yield moment,
and
Y. % 0 . This linear programming problem has the same structure as (2.1), and l its dual may be mechanically interpreted in a similar manner. For a building frame with many floors and bays, this fortaulation obviously results in a linear programming problem of considerable size, particularly when several states of loading m u s t be considered.
To reduce the problem to a manageable size, special methods were
developed (see, for instance, Livesley, 1956, and Heyman and Prager, 1958).
With
the large core memories of medemn computers, the need for such methods, which require special programming, has disappeared. It follows from the analogy between the problems of optimal plastic design of trusses and beams that the superposiTion principle of Hemp, 1968, applies also to beams subject to two alternative loadings. by Nagtegaal, 1973a.
Examples of this supemposition have been given
Thereare no similarly efficient methods of dealing with Three
or more alternative loadings. this kind is straightforward
The linear programming formulation of problems of but results in linear programs of very large size.
An
interesting limiting case has, however, been treated analytically, namely the optlmal plastic design for a "moving load", i.e., a load of fixed intensity that may act at any cross section of the beam (Save and Prager, 1963, and Lambiin and Save,1971). Let us briefly return to the optimal design of a sandwich beam with prismatic core of rectangular cross section for a single loading measured along the beam. for the yield moment
x
denotes distance
We shall assume That no positive lower bound is prescribed
Y(x) .
collapse under the load
p(x) , where
p(x)
For the optimal beam, the bending moment then has the absolute value
M(x)
at
Y(x) , and we have The
following continuous problem of linear programming: Minimize
[I
MI dx
(~.l)
subject to d2M/dx 2 ' =
p •
(4.2)
The integration in (4.1) is extended over the entire beam; (4.2) expresses the fact that The bending moment
M(x)
must be statically admissible for the load
p(x).
In
analogy with the discussion leading to (2.4) and (2.5), it can be shown that a beam is optimal, if there exist statically admissible bending moments ically admissible rates of deflection K
=
- d2v/dx 2 IKI ~ mo ' MK ~ 0
v(x)
M(x)
and kinemat-
such that The rates of curvature
satisfy the conditions with equality for
M # 0 ,
(4.3)
,
(~.4)
where
K is an arbitrary reference rate of curvature. Examples for the use of o L this optimality condition were given by Heyman, 1951. Figure 6a shows a propped cantilever beam carrying a uniform load
p .
The kinematic constraints at the sup-
ports call for the vanishing of the rate of deflection ishing of
dv/dx
at the clamped end.
Since
v(x)
v
at both ends and the van-
must be in class
conditions Together with The optimality condition (4.8) determine has been chosen (Fig. 6b).
It is found that
K(X)
I-
and hence
C 1 , these
v(x)
M(x)
when K O change sign at
P l J i,i i l i i i l ' l , ~
t/l l't I i
(o)
X
(b)
Xo I/v Fig. 6:
,
(a) Uniformly loaded~ propped cantilever beam (b) Rates of deflection at collapse of optimal design
X
:
XO
= £I~
With the conditions
(4.2) uniquely deter~nines
M(x)
M(0) : M(x O) = 0 , the differential equation
and hence the yield moment
Y(x) = IM(x)l
of the
optimal beam. The optimal plastic design of a dense grillage of sandwich beams with uniform core dimensions but variable thickness of cover plates (Rozvany, 1972) leads to a continuous problem of linear programming that is a two-dlmensional
analog of The prob-
12
lem indicated by (~.i) and (4.2). M xy
Denote by
M x , M>
the bending moments, and by
the twisting moment in the g1~illage with mespeot to rectangula~ Cartesian cooP-
dinates
x,y
uted load.
in the median plane of ~he gTillage, and let
p(x,y)
be the distTib-
The beams of the grillage follow the lines of the principal bending mo-
ments
and have the
lMll
yield moments
IM21 . The
optimal
grillage is
characterized
by the following continuous problem of linear programming: Minimize
I(IMll + IM21) dA
(4.6)
subject t o 2Mx/3X 2 + In (4.6),
dA
~2My/~y2÷ 2B2Mxy/~X~y
=
p .
(4.7)
is the a~ea element of the planform of the grillage, the integration
is extended over this planform, and
M1
and
M2
are defined by (4.5).
The con-
straint (4.7) is the equation of equilibrium. At collapse of an optimal grillage, we must have a statically admissible moment field with principal moments deflection field in
M
, (~=1,2)
, and a kinematically admissible rate of
C 1 , with principal mates of cumvature
K
(of the same direc-
tions as the principal moments) that must satisfy the conditions IKal "<
M
# 0 ,
(4.8)
>. 0 .
(4.9)
Rozvany (1972, 1978) has shown that the planfo~m of an optimal grillage is divided into subdomains in each of which the net of lines of principal bending moments consists of a one-parameter family of straight lines and their orthogonal trajectories, with zero principal moment alon E curved trajectories.
He also established a set of
rules for the determination of the boundamies of subdomains.
For a summary of this
work, see Rozvany, 1974. Figure 7a shows an optimal layout of a square grillage that is simply supported alon E the edges and subjected to a distributed load of varying intensity but constant sign.
For a downward load on a horizontal grillage, the directions of posi-
tive or negative principal moments are indicated by full om dashed lines.
The prin-
cipal moments have opposite signs in the corner regions (e.g. OAB) but the same sign in the central square
ABCD .
Figure 7b shows an optimal layout of a square gril-
lage that is clamped along the edges.
Only in the cent1~al square
BCEF
does this
layout correspond to the idea evoked by the term ~Tillage~ outside this squame, one of the principal bending moments vanishes.
A load at a point in the centmal square
13
C
)
:k
:
I F - ' / x,<" \," hl B
D
0
A
O
I
(a) Fig. 7:
A
(b) (a) Optimal grillage simply supported along edge (b) Optimal "grillage" built in along edge
is transmitted to the edge beams of this square and, by them, to cantilevers such as and
AB
or
DC .
A beam such as
GH
is simply supported by the cantilevers
IG
KH .
Hemp's supei~position principle may also be applied to the optimal design of a grillage for i~4o alternative loadings (Rozvany, 1974). 5.
OPTIMAL ELASTIC DESIGN
The compliance of an elastic structure to a given system of loads is defined as the virtual work of these loads on the elastic displacements caused by them.
Alterna-
tively, the compliance may be evaluated as twice the strain energy stored in the stmucture when it is carrying the given loads.
Prescribing an upper bound for the
compliance of an elastic structure thus amounts to introducing an overall constraint on its deformations. As an example of optimal design with a compliance constraint, consider the design of a sandwich beam with constant core dimensions and se~mentwise constant cmosssectional areas of the identical cover plates. compliance
C to a given loading.
optimal beam in the segment
Let
si ~ 0
The beam is to have a prescribed be the bending stiffness of the
(Xi_l, x i) , and denote by
the optimal beam produced by the given loads.
K = <(x)
the curvature of
If a second design with the same
14
compliance is specified by the stiffnesses given loads is denoted by
'
s#l >. 0 , and its curvature under the
~°: = <~(x) , then
xi_ 1
xi_ 1
Since the virtual work of the loads has the given value
C , the principle of mini-
mum potential energy is here reduced to one of minimum strain energy.
Thus,
(5.2) xi_ I
xi_ 1
because the curvature
<
is kinematically admissible for the design
s# . l
Combi-
nation of (5.1) and (5.2) yields
si) i [
(5.3)
,
0
where
112 i
All
(5.4)
xi-I
is the mean square curvattt~e for the segment
(x i_l,xi)
and
A.~ is the length of
this segment. It is possible that some segments of the optimal beam have vanishing stiffnesses. For example, a beam may be conceived as a propped cantilever.
If, however, it only
has to carry a load close to the built-in end, the optimal design will be a cantilever that extends only from this end to the point of application of the load. the segment s~
1
-
s.
l
(Xi_l,Xi)
has vanishing stiffness, we have
~
=
0
,
s.
1
0
.
If
(5.5)
Finally, because the weight per unit length of a sandwich beam of this kind is proportional to its stiffness, the optimality of the design
si
is expressed by the
inequality
Zi(s ~ - si)A i ~ 0 .
(5.6)
According to a theorem of Farkas, 1902, the inequality (5.6) will follow from the inequalities
(5.3) and (5.5) if and only if it is a nonnegative linear combination
of these inequalities.
Let the coefficients of this linear combination be
and
is a reference curvature, and
Bi/K2o ' where
Bi = 0
for
1/<~
s.l > 0 .
Farkas' theorem then yields the optimality condition 2 2 <..< < , with equality sign for l o
s. > 0 . l
(5.7)
15
By the introduction of the mean square bending moments
=
Mi
M2dx " "'i-i
~i !
,
(5.8)
the optimality condition (5.7) may be given the alternative form Mi/s i ~ K °
with equality sign for
,
Note that the compliance C
=
C
si
> 0
.
(5.9)
may be written as
2 Zi£iMi/s i .
(5.10)
It follows from (5.9) and (5.10) that the reference curvature is given by C/Z.Z.M. i 1 i
0
When the compliance is not to exceed a prescribed value
C
for either one of two
alternative loadings, an inequality of the form (5.3) is valid for the mean square curvatures K[ under the first loading as well as for the mean square curvatures l ~'J under the second loading. Application of Farkas' theorem to these inequalities l and (5.5) and (5.6) then yields the optimality condition , ,2 2 2 K i + ~"K~ .< K O , where
~'
and
~"
with equality sign for
s. > 0 , l
(5.12)
are dimensionless, nonnegative Lagrangian multipliers satisfy-
ing
For
U' + W'
:
p' = 0
or
design,
i .
(5.13)
~" = 0 , the first or second loading does not influence the optimal
In tlhis case, (5.12) reduces to (5.7) applied to the other loading, and K
is found from (5.il).
For other values of
of the loadings has the value mean square bending moments
O
~' , ~" , the compliance to either one
C , and Mi
K is again found from (5.11), where the o may be taken from either loading.
Because the optimality condition (5.12) contains the squares of •
K[
i
~
<~ I
it cannot
be stated in "the form of a superposltlon principle similam to Hemp's principle for optimal plastic design for two alternative loadings. As was pointed out by Prager and Taylor, 1968, the way in which the optimality conditions (5.7) and (5.12) were derived may be used for other problems of optimal design~ provided that the behavioral constraints concern quantities that are characterized by minimum principles in the manner in which the compliance is characterized by the principle of minimum potential energy.
Constraints that have been treated in
this manner concern not only static elastic compliance but also dynamic elastic compliance under harmonically varying loads (Icerman, 1969; Mroz, 1970; Plaut, 1971)~
16
fundamental frequency (Taylor, 1967a E 1968; Sheu, 1968), elastic buckling load (Taylor, 1967b; Taylor and Liu, 1968), static deflection at a specified point (Shield and Prager, 1970; Chern, 1971), and rate of deflection in stationary creep (Prager, 1968).
Much earlier, essentially the same technique was used by Drucker
and Shield, 1957, for optimal plastic design. 6.
COMPUTATIONAL ASPECTS
For the problems in Figs. 3, 6, and 7, the optimality condition directly led to the optimal designs.
Problems of this kind are exceptional.
As a rule, optimality con-
ditions are used as guides in the iterative impmovement of an initial design.
To
indicate some basic ideas used in procedu1~es of this kind, we consider a very simple example.
Let a single set of loads be castled by a sandwich beam with segmentwise
constant cross section, which is loads.
to be designed for a given compliance to these
The optimality condition (5.9) remains valid if the total volume of the cov-
er plates is prescribed (or, what amounts to the same, the value of and the compliance is to be minimized.
S = Z 1.£.s.) i l
s'. for 1 ' instance the uniform design s' = S/tit i , we compute its mean square bending mol ments M'. , which will not, in general, satisfy the optimality condition (5.9). The 1 compliance C" of a second design s~ with mean square bending moments M~ satls1 1 lies the relations
C" = ~i~iM[2/s~ ~ ~ihM[2/s~
Starting with an a~bitrary design
,
(6.1)
where the inequality follows from the principle of minimum complemental ~] e n e r ~ because the mean square bending moments M v. are computed from bending moments that l are statically admissible for the second design. Since (6.1) gives an upper bound for
C" , and since a design of minimal compliance is sought, it is natural to de-
ter~nine the stiffnesses dition that
s': that furnish the least upper bound subject to the con1 Z.~.s': = S . These stiffnesses are found to be 1
s': : 1
i
M~/< °
1
where
K
0
:
?.£.M~./S . 1
1
(6.2)
1
Note that the use of the redesign formula (6.2) amounts to using the optimality condition (5.9) with the assumption that the redesign does not change the mean square bending moments.
This is strictly true fop a statically determinate beam.
For an indeterminate beam, we can only assert that the mean square bending moments will not change materially if the stiffness changes are sufficiently small. ingly, the redesign formula (6.2) will have to be used iteratively.
Accord-
Experience with
many problems has shown that this iterative method furnishes excellent results when the number of redesign steps is made to equal the number of segments.
Fop large
numbers of segments, it has been found that the e-algorithm of Wynn, 1961~ may be used advantageously to speed convergence, though no theometlcal justification has
17
as yet been offered for this. Another means of speeding up convergence is as follows. Z.£.s. = S
to find
iii
K
= Zi£iMi/S
Usin E the condition
in (5.9), we write this optimality condition in
0
the fol~m si where
= A
[A + (1-1)MiS/~iZi£iMi)]s i , is an arbitrary constant.
(6.3)
We then transforln (6.3) into an iteration
for~nula by writing the left side as
s~ but using the values of M! and s[ on the 1 1 1 Experience has shown that with, say, i = -0.9 , this kind of iteration
right.
works extremely well in the neighborhood of the optimal design.
For example, for a
beam with five segments and starting from a unlfomm design, a single use of the redesign formula (6.2) followed by a single iteration based on (6.3) typically furnish as good results as five successive uses of (6.2).
On the other hand, the immediate
application of itePations based on (6.3) to the uniform initial desiEn is less satisfactory Than The iterations (6.2). In connection with itePative optimal design, it is impoPtant to note that the analysis of a design fuPnlshes information that m a y b e used to evaluate uppeP and loweP bounds fop the con~liance of the optimal desiEn (Nagtegaal, 1973b).
This enables
the designer to decide whether it is worthwhile to continue the iterations. The treatment of multiple constraints is much mope difficult. cussed the case of prescribed upper bounds native loadlngs.
are lineaP in
Mamtin, 1970, dis-
on the compliance to two alter-
When both constraints are active, the equations expressing
in Terms of the bendinK moments
~' , ~"
C' , C"
C'
C"
M' , M"
but nonlinear in
and the Lagrangian multipliers ~' , ~"
C' , C"
~' , ~"
Martin therefoPe suggests that
be tPeated as the independent variables, and that the mapping of the
~' , ~"-plane on the
C' , C"-plane be explored.
While this approach can be extend-
ed to thPee oP more constraints, it rapidly becomes awkward as the numbeP of constPaints increases. Gellatly and BePke, 1971, discuss the combination of stPess constmaints with several displacement constPaints.
In each Pedesign step, the stress constraints and one dis-
placement constPaint at a time ape tPeated as the constraints of a sepaPate optlmization problem, and, for each membeP, the largest of the Pesulting member sizes is adopted fop the next design.
When the design obtained in this manneP is analyzed,
it will, in general, be found That none of the constPaints is satisfied as an equality.
Before the next Pedesign is initiated, the membeP sizes are uniformly scaled
so that at least one constraint is satisfied as equality.
FoP details of the pPo-
ceduPe~ the PeadeP is referred to the original papeP and its discussion by Kiusalaas, 1972.
48
7.
CONCLUDING REMARKS
There are numerous ideas that could not be taken up in this survey.
A few will be
briefly mentioned in this section. For a sandwich beam with segmentwise constant cross section, the segment boundaries x. may be at the choice of the designer (Sheu and Prager, 1968). At the same time, l the cross-sectional areas A i of the cover plates may be restricted to the members of a discrete set of standard sections.
This kind of condition has long been re-
garded as difficult because it transforms the problem into one of integer programming.
Masur, 1974, has recently shown that the problem is not as difficult as had
been assumed. Pickett, Rubinstein, and Nelson, 1973, have suggested to reduce the dimensionality of the optimization problem by restricting the search to the linear combinations of a number of trial designs, the choice of which m a y b e guided by experience with similar problems.
Moreover, if the trial designs are scaled to have equal structural
weights, the one with the smallest coefficient in the optimal linear combination may be replaced by a new trial design, and a new optimal linear combination may be sought. A somewhat related idea is to simplify the analyses of subsequent designs by a Ritz approach.
This is, of course, done anyhow when finite element methods are used for
the analysis (Dupuis, 1972; Major, Zavelani-Rossi, and Benedetti, 1972), hut nonlocalized coordinate functions might be more advantageous for the optimization problem. Discussing the optimal design of an elastic plate, Mroz, 1973, has used linear combinations of coordinate functions for both the plate thickness and the deflection. If both combinations have the same number of terms, this approach leads to a system of non-linear equations for the deflection coefficients.
After these equations have
been solved, the thickness coefficients are found f~om a system of linear equations. The limited space does not permit the discussion of further problems.
It is hoped,
however, that those that were discussed will have conveyed the idea that optimal structural design is a field that merits the attention of numerical analysts. REFERENCES Chern, J.-M., Int. J. Solids $ Structs. 7, 373 (i971). Dupuis, G., Int. J. Num. Moth. Engg. 4, 331 (1972). Drucker, D. C., $ Shield, R. T., Proc. 9th Int. Congr. Appl. Mech. (Brussels) 5, 212 (1957). Farkas, J., J. reine & angew.
Math. 124, i (1902).
Fouikes, J., Proc. Roy. Soc. (London) A 223, 482 (1954). Gellatly, R. A., & Bemke, L., Wright-Patterson Air Force Base, Tech. Rep. AFFDLTR-165 (1971).
19
Hegemier, G. A., £ Prager, W., Int. J. Mech. Scis. Ii, 209 (1969). Hemp, W. S.~ Proc. Int. Congr. Appl. Mech.(Munich) 621 (1966). , Abstract of Lect. Course "Optimal Structures", Oxford, 15 (1968). Heyman, J., Quart. Appl. Math. 8, 373 (1951). , & Prager, W., J. Franklin Inst. 266, 339 (1958).
-
Icerman, L. J., Int. J. Solids $ Structs. 5, 473 (1969). Kiusalaas, J., NASA Tech. Note D-7115 (1972). Lamblin, D. 0., & Save, M., Meccanica 6, 151 (1971). Livesley, R. K., Quart. J. Mech. & Appl. Math. 9, 257 (1956). Maler, G., Zavelani-Rossi, A., ~ Benedettl, D., Int. J. Num. Meth. Engg. 4, 455 (1973). Martin, J. B., J. Optim. Theory & Appls. 6, 22 (1970). Masur, E. F., Computer Meth. Appl. Mech. & Engg. (1974), to appear. Michell, A. G., Phil. Mag. 8, 859 (1904). Mroz, Z., J. Struct. Mech. l, 371 (1973). Nagtegaal, J. C., Int. J. Solids & Structs. (1973a), to appear. , SIAM J. Appl. Math. 25 (1973b), to appear. , g Prager, W., Int. J. Mech. Sets. 15, 583 (1973).
-
Palmer, A. C., & Sheppard, D., Proc. Instn. Civ. Engrs. 47, 363 (1970). Pickett, R. M., Rubinstein, M. F., g Nelson, R. B., AIAA J. ll, 489 (1973). Plaut, R. H., Quart. Appl. Math. 29, 315 (1971). Pope, G. C., g Schmit, L. A., AGARDograph AG-149 (1971). Prager, W., An Introduction to Plasticity, Reading, Mass., 1959. -
- , J. Appl. Maih.$ Phys. 19, 252 (1968). - , g Taylor, J. E., J. Appl. Mech. 35, 102 (1968).
Rozvany, G. I. N., Computer Meth. Appl. Mech. $ Engg. i, 253 (1972). -
, J. Struct. Mech. 2 (1973), to appear.
- , Proc. IUTAM Sympos. 0prim. Struct. Design (Warsaw) (1974), to appear. Save, M., & Prager, W., J. Mech. & Phys. Solids 20, 255 (1963). Sheu, C. Y., Int. J. Solids & Structs. 4, 953 (1968). , $ Prager, W., J. Opt. Theory ~ Appls. 2, 179 (1968). , & Schmit, L. A., AIAA J. i0, 155 (1972). Shield, R. T., & Prager, W., J. Appl. Math. $ Phys. 21, 513 (1970). Sved, G., & Ginos, Z., Int. J. Mech. Scls. i0, 803 (1968). Taylor, J. E., AIAA J. 5, 1911 (1967a) g 6, 1379 ~1958). -
-
, J. Appl. Mech. 34, 486 (1967b). , ~ Liu, C. Y., AIAA J. 6, 1497 (1968).
Wynn, P., Math. of Computation 15, 151 (1961).
OPTIMISATION DES SYSTEMES PORTANTS ET PROPULSIFS PAR LAMETHODE DES SINGULARITES
L. M A L A V A R D Universit@ Paris Vl et LIMSI e - CNRS
INTRODUCTION
La m~thode des singularit~s ~ r~partition discr~tis~e est utilisEe depuis de nombreuses ann~es pour r~soudre num~riquement des probl~mes de potentiel des vitesses d'Ecoulements aEro-hydrodynamiques. ExploitEe d'abord par une Equipe d'aErodynamiciens de Douglas Aircraf Company, J.L. Hess, A.M. Smith, J.P. Giesing, en utilisant des distributions de sources sur la surface d'un obstacle pour dEfinir le mouvement du fluide qui l'entoure,
elle fut reprise par divers auteurs et en
particulier au LIMSI par T.S. Luu, G. Coulmy, J. Corniglion et al. qui la dEvelopp~rent avec succ~s en faisant intervenir des singularit~s de types tr~s varies et dont la nature pouvait ~tre la mieux appropri~e au probl~me aux limites consid~rE. Cette co~unication a pour objet de montrer que cette mEthode est particuli~rement bien adapt@e ~ la resolution numErique d'un important probl~me d'aErohydrodynamique : celui de la recherche de la r@partition optimale de circulation des @IEments actifs d'un syst~me portant ou propulsif animEs d'un mouvement donnE. Bien qu'il ne s'agisse ici que de la r~solution d'~quations de Laplace, on verra qu'en gEn@ral la complexitE de la gEomEtric des surfaces fronti~res et de certaines conditions aux limites rendrait inop@rantes la plupart des autres m~thodes classiques. La position du probl~me a@ro-hydrodynamique est pr@sent~e dans la premiere partie : module thEorique du fonctionnement des syst~mes portants et/ou propuls&fs et la condition d'optimum qui traduit une perte d'Energie minimum pour un effet portant ou propulsif imposE. Les bases de la m~thode des singularit@s, telle qu'elle est exploit~e par Luu, Coulmy et al., sont reprises darts la seconde partie pour en marquer certains aspects importants : non-unicit~ de la nature des singuiaritEs dans la creation d'un potentiel, leur choix pour rEaliser au mieux le type de conditions aux
LIMSI - CNRS : Laboratoire d'Informatique pour la M~canique et les Sciences de l'Ing~nieur du Centre National de la Recherche Scientifique (France).
21
limites ~ imposer, le cas particulier de distributions p~riodiques de singularit~s, etc.
La derni~re partie est consacr~e aux applications de la m~thode pour le traitement des syst~mes cycliques b i e t
tridimensionnels
: ailes oscillantes,
ailes battantes, voilures tournantes ; d~termination des lois optimales de circulation et recherche des mouvements propres ~ donner aux ~l~ments actifs pour satisfaire la condition d'optimum.
I . POSITION
DU
PROBLEME
I.]. - D~finition et hypotheses de base. Un syst~me portant ou/et propulsif est un ensemble m~canique form~ d'~l~ments actifs, de types ailes ou pales, animus de mouvements appropri~s afin d'assurer ~ l'ensemble une portance ou/et une propulsion. Outre les ailes et h~lices de formes diverses, entrent dans cette d~finition aussi bien les voilures tournantes des h~lieopt~res, des gyroplanes, des autogyres que les ailes battantes e~ les propulseurs ~ voilure oscillante. Pour effectuer l'~tude du fonctionnement a~ro-hydrodynamique de ces syst~mes on consid~re, en premiere ~tape, que le fluide est parfait et incompressible. Dans le cas g~n~ral le mouvement d'un ~l~ment (d'une pale par exemple) dans un fluide illimit~ et au repos ~ l'infini engendre ~ partir de sa ligne du bord de fuite une surface
5-" de
discontinuit~ des vitesses tangentielles, habituellement d~sign~
par "sillage tourbillonnaire". Cette surface ~'~ est caract~ris~e par la valeur de la circulation ~(~'~) en chacun de ses points ~ sur un lacet passant ~ l'ext~rieur de
Z'
: la circulation ~tant prise
et partant du point ~
sur l'une des
faces de ~-' pour y revenir sur l'autre face. La valeur de ~ ( ~ )
est fix~e au
moment de la formation de ~_~ au bord de fuite par les conditions classiques de Kutta-Joukowsky et de la constance de la circulation avec le temps ~ . La forme g~om~trique de
~volue avec le temps ; il est cependant possible
d'en obtenir une premiere approximation en admettant l'hypoth~se usuelle des faibles perturbations : si les ~l~ments du syst~me sont tr~s minces et, par exemple, rgduits ~ des surfaces mat~rielles dont les sections (profils) sont en outre tr~s peu courb~es et inclin~es sur le vent relatif, les vitesses V
des particules
fluides sont tr~s petites vis-a-vis de la vitesse V o qui caract~rise le d~placement
g~n~ral du syst~me,
/lV=l <<~
. Ii est alors loisible de confondre le
sillage ~--~ avec la surface ~--"-engendrge au tours du temps par le d~placement de la ligne du bord de fuite, Cette approche est oonforme aux m~thodes adopt~es dans les theories classiques des surfaces sustentatrices et des h~lices, Le momvement du fluide peut ~tre d~fini, ~ tout i n s t a n t S ,
par un potentiel
22
des vitesses ~
satisfaisant l'~quation de Laplace. Les conditions aux limites sont
les suivantes : - ~ l'infini,
fluide au repos,
- en chaque point ~
~
~
de la surface ~
~
=
0
d'un ~iEment (aile ou pale), dont la vitesse
d'entrainement est ~%/(~), la condition de glissement exige que la d~riv~e norma+ - en c h a q u e p o i n t ~
du s i l l a g e
sur la face
supgrieure
circulation
Pen
~-' l a d i f f e r e n c e
et la valeur
ce point :
et cette circulation ~
~
(4~
entre
sur ta face
-- ~ - - =
la valeur inffirieure
correspond
@ ~ la
C
est celle autour du profil au moment ~
fuite passait au point ~
du p o t e n t i e l
o~ son bord de
. Enfin la d~riv~e normale demeure continue R la tra-
vers~e de ~ . Dans le cas g~n~ral et tel qu'il est pose la resolution pratique de ce probl~me soul~ve encore beaucoup de difficult~s, sauf darts le cas bidimensionnel d'un profil anim~ d'un mouvement arbitrairement donnE. On se bornera donc ici rechercher des conditions d'optimisation des syst~mes R fonctionnement cyclique,
1.2. - Energie communiqu~e au fluide ; effort moyen. Lorsque le fonc£ionnement du systgme est cyclique le sillage ~_ est form~ de surfaces ~ g~om~trie p~riodique. En se pla~ant suffisamment loin ~ l'aval du syst~me on peut considErer que dans chaque tranche de fluide, correspondant ~ une p~riode, l'~coulement pr~sente les m~mes propri~t~s de p~riodicit~ que si le nombre de tranches ~tait infini dans les deux sens. Si l'on d~signe par p volume et par ~
, ~
, ~
la masse volumique du fluide, par ciV'un ~l~ment de les composantes de la vitesse, l'Energie cin~tique E
du fluide contenu dans le domaine
que l'on peue encore ~crire,
]~) d~fini par une telle tranche est donn~e par :
compte-tenu de la d~finition du potentiel des vitesses
0
o~ ~-0 est la portion de surface de
correspondant
une p~riode.
Quand les cordes des ~l~ments (ailes, pales, etc..) du syst~me sont petites par rapport ~ une longueur de r~f~rence de celui-ci, telle que son envergure, ou la longueur de la pale, ou l'amplitude de ~ o ° u
le dEplacement du syst~me pour
une p~rlode To, on peut admettre l'approximation de la ligne portante. La surface sustentatriee fdrmant un gl~ment est ainsi r~duite g une ligne portante confondue avec la ligne A B
du bord de fuite.
_. D~signons par ~
le vecteur unitaire tangent en un point de eette ligne et
par ~&/ la vitesse d'entrainement de ce point. Puisque le d~placement de cette
23
ligne de bard de fuite A B ggngre la surface~
on peut d@finir la normale ~ en prenant le vecteur unitaire d@duit du produit v e c t o r i e l ~ A ~ .
B
Avec les approximations admises on peut alors montrer que l'effort
V~ ~ ~ A
a@rody~amique qu~ supporte I'@I@~
ment ~ ~ ~\
de la ligue p°rta~ie
est donn@ par :
soit encore :
Ainsi la force moyenne support@e par la ligne portante ~ B pendant une p@riode r a de fonctionnement est donn@e par :
I:o+TO~
=
~ f p F ~ . cL~ '
"~a "A 1.3. - C o n d i t i o n o p t i m a l e .
Pour une direction fix@e par le vecteur unitaire ~ projection suivant ~
on peut imposer que la
de la force moyenne ait une valeur constante et rechercher
quelle est la condition ~ respecter pour que l'@nergie ein@tique E
eonnnuniqu@e au
fluide soit minimale. Ce qui revient encore ~ imposer que le travail
j
(o3 V o est la vitesse de r@f@rence) de sorte que
E
=-
£
r~
(~.~)
cL2~
soit minimale. Dans ce but, supposons que l'on donne ~ r , pris eonmle variable indgpendante une variation ~
; on a
;F-o et
~E
-
-~
~r~+r~
~z
partir de la d@finition du potentiel (D il est ais@ de v@rifier que les deux int@grales ~I~P (~' ~ particulier, @crire:
et I [' # ~ ~w~-L~-
cSE - - p
sont @gales ; on pourra donc, en
I Eo~r ~ ' r
24
D'apr~s le calcul des variations l'extremum sera atteint pour los fonctions telles que la difference gE
-
c
=
o
o~ = est une constante arbitraire. Soit encore :
(%+
>a r
=
o
ce qui ne pout avoir lieu q%el que soit ~[~ que si pour tout point de >-o
'
la
con-
dition suivante est satisfaite :
q~
-
-
c
c~.~
"~.~
Vo Cette condition correspond ~ des donnges aux limites de Neumann sur la surface
~
; la r~solution du probl~me permet alors d'atteindre la r~partition 0
sur cette surface de la difference du potentiel ( ~ loi optimale de distribution de la circulation r
~)
et donc, finalement, la
pour le mouvement du syst~me
considerS. La condition(~.~)est classique et son usage est familier dans l'~tude du fonctionnement stationnaire des ailes ordinaires et des h~lices usuelles ; il ~tait bon d'en rappeler le caract~re g~n~ral et notamment pour los applications aux syst~mes g fonctionnement instationnaire ou cyclique.
2. LA M E T H O D E
DES
SINGULARITES
A REPARTITION
DISCRETISEE
2.1. - G~n~ralit~s et choix des singularit~s. Bien que le principe de la m~thode des singularit~s soit classique, certains de ses aspects, importants pour les r~solutions num~riques, restent mal connus. Aussi, avant d'appliquer cette m~thode aux probl~mes ~nonc~s pr~c~den~nent, n'estil pas inutile de revenir sur quelques considerations g~n~rales qui pr~cisent sos possibillt~s pour traitor los champs Laplaciens ou Poissonniens avec une diversit~ portant aussi bien sur la g~om~trie des fronti~res que sur los donn~es aux limites. +
Consid~rons, par exemple, dans un domaine ~D enveloppe ferm~e C
, un potentiel ~
illimit~ ~ l'ext~rieur d'une
solution de l'~quation de Poisson ~ O = ( ~ p
et remplissant los donn~es aux limites sur C •Dans le domaine ~D- int~rieur ~ C~ , le prolongement du potentiel ~O reste arbitraire et on peut consid~rer qu'il est r~gi par l'~quation de Laplace. II est alors bien connu que l'application du th~orgme de Green permet d'exprimer le potentiel ~
C
dans ])+ par :
25
A 4~r
~3
oO
C~(M,~) = GL(P~M ) =
(~
~tant la direction de la normale ext~rieure ~ C ) .
On peut donc considErer que ~
pour
mB
P = I~--M~
est engendrE par une rEpartition de source dans
l'espace ]~+ avec une intensit~ dEfinie par le second membre de l'Equation de Poisson et une rEpartition de source et de doublet sur la fronti~re C , La densit~ de source est donnEe par la discontinuitE de la d~riv~e normale de la f o n c t i o n ~ tandis que la densitE de doublet correspond ~ la discontinuitE de la fonctinn ~ Si l'on consid~re que ~
, .
, dEfini dans ~I~+ correspond ~ une solution d'un
probl~me aux limites bien determine, dans le domaine ~
int~rieur ~ C
la d~fi-
nition du champ harmonique prEsente un degrE d'arbitraire. En effet, si le champ dans T)- est d~fini par la condition de Dirichlet telle que ~
est identifi~
la valeur de ~ t sur la fronti~re, la resolution du probl~me conduit ~ la eonnaissance de
~"
sur C
•Dans l'expression
(~,~) de ~ ( M ~ ,
ne subsiste ~ la fron-
tigre que la rEpartition de source. Mais on peut aussi envisager d'Eliminer cette r~partition de source de la mani~re suivante : si le flux champ dans ~ -
~ =
(/~
~ travers la fronti~re est nul, le
peut ~tre d~fini par la condition de Neumann
resolution du probl~me donne la distribution de ~ reste plus que la rEpartition de doublet sur ~ flux ~ ~
sur ~ .
~ =
~
et la
Darts ce cas il
ne
dans l'expression de ~ M ) .
Si le
n'est pas nul il est toujours loisible d'introduire la fonction
= ~-F~(~,M)
o~ ~ ( ~ M )
au point~m~ ~ D - p o u r
correspond ~ une source ponctuelle plac~e
se ramener au cas precedent. Finalement ~ ( ~ )
est donn~
par :
Ii est encore possible de transformer une rEpartition de simple couche en une rgpartition de double couche et vice-versa. Pour le montrer bornons-nous au cas bidimensionnel
(le raisonnement est le m~me en tridimensionnel) d'une r~par-
tition de source et tourbillon sur une courbe ~
; une integration par parties
donne, en effet, l'identit~ suivante : P
le premier membre repr~sente la rgpartition de source et de tourbillon sur la courbe ~ B
, d'intensitEs respectives ~
et ~
; au second membre le deuxigme
terme repr~sente une r~partition de doublet ~ axe tangentiel d'intensitE ~ et de doublet ~ axe normal d'intensit~
~+ ~
~_
=
telles que :
+ ~) e-~
26
et le premier terme une source et un tourbillon d'intensit~ Q
et ~
places en"~5 .
Les considerations qui pr~cgdent montrent qu'il n'y a pas unicit~ de la r~partition de singularitg dans la creation d'un potentiel. L'int~r~t de cette non-unieit~ est de permettre, dans une certaine mesure, le choix de la nature de la singularitg g r~partir sur les fronti~res de fagon g r~aliser au mieux le type de conditions aux limites impos~es.
2.2. - Formulation de la m~thode. En se limitant au cas de champs harmoniques et lorsque l'on a choisi la nature de singularit~ la plus appropri~e pour les conditions aux limites impos~es ~ ( I =') sur ~
, le probl~me revient ~ d~terminer la densitg ) ~ ,
a priori incon-
nue, de la singularit~ par l'~quation int~grale : .)
Le noyau
K (P,P') , qui d~Cend des donn~es aux limites et de la nature de la sin-
gularitg n'est pas toujours r~gulier lorsque ~ - ~ ~:~ . La m~thode, dite ~ r~partition diser~tis~e, consiste ~ diviser la frontigre C ments
/%6
en un hombre fini de petits gig-
(faeettes ou segments) sur chacun desquels on d~finit ~ p
moyenne ) ~
par sa valeur
consid~r~e comme eonstante sur route l'~tendue de l'~l~ment, son v
numgro d'ordre ~tant ~
. La
condition pr~c~dente
C~ ~) , oO l'int~grale est
remplac~e par une somme, est satisfaite en des points de contr$1e fixes au centre de chacun des ~l~ments. L'~quation int~grale est ainsi remplac~e par un syst~me d'~quations lin~aires d~finissant les ~ 3
:
o~ Fi est ~a valeur de ~O au point de contrBle "P: et o~ A~i repr~sente l'int~grale
.[^K(Im, P ' ) d ~ p
K(~'~)
sur l'gtendue
/~06 pour un point ]Dg impos~. M~me si
est singulier au voisinage de ] m
n~ral de d~finir ~6~
le passage aux limites permet en g~-
en ce point.
Pour faciliter la r~solution pratique de ce probl~me il y a ~videmment int~r~t g choisir la nature de la singularitY, en fonetion des donn~es aux limites
~ traiter,
de mani~re ~ rendre la diagonale p r i n e i p a l e de ta matriee
~:)
ptfi--
pondfirante afin de rendre aisge l'utilisation de mfithodes itgratives, rapides et pr~cises comme celle de Gauss Siedel~pour rgsoudre le syst~me.
2.3. - Distributions p~riodiques de singularit~s. II est ho=s de propos de reprendre ici le calcul des A ~
, c'est-~-dire
des effets produits par des r~partitions de densit~ constante de singularit~s de nature varige en vue de leur utilisation dans les probl~mes aux limites usuels de types Neumann, Dirichlet, Fourier. Un recueil de oes formules existe dans divers articles de synth~se parus ~ ce sujet
( c f: ~e~.).
27
Dans Le problgme posg pr~c~demment il s'agit de dgterminer la r~partition optimale de circulation
~=
~+_~-
sur une surfaee-coupure d'un espace illimit~,
avec des conditions aux limites de Neumann pour le potentiel harmonique ~0. II est ~vident qu'il s'agit ici d'un potentiel de double-couche : la singularit~ la plus adgquate correspond done ~ une distribution de doublet ~ axe normal,
2.3.1. Pour le cas bidimensionnel, rappelons que le potentiel produit par une r~partition de doublet de densitg constante ~ fZ~
sur un segment Z ~
z,
soit
Z~,J //\~ _
/
,
(~
r"2 \
La vitesse induite par cette rgpartition est .
est donn~ par :
~e °
/ e,~
.-
4 2
g
:
z I Soit encore, pour les eomposantes tangentielle et normale rapport~es g u n rep~re local li~ ~ ~.
:
Bien que la fonction ~ et ~ 2 du segment,
et son gradient soient singuliers aux deux extr~mitgs
Zd
leurs valeurs restent finies lorsque Z
tend vers la face + +~ +_, sup~rieure ou inf~rieure de ce segment. En dgsignant par (~- ~ QPf , ~ les
valeurs correspondantes, on a : _+
On n o t e r a , situ~
d'apr~s
au c e n t r e
ces ferules,
+~
+'
la contribution
pr~pond~rante,
du s e g m e n t , que donne une r ~ p a r t i t i o n
au p o i n t de c o n t r $ 1 e
de d o u b l e t p o u r une c o n d i t i o n
de Neumann. Pour le problgme g traiter d'un sillage ~ ggom~trie p~riodique ~
em est o
conduit ~ considgrer une infinite de segments identiques, p~riodiquement espac~s et supportant une m~me densitg constante ~
de doublet,
Z
/ w
28
L'expression correspondante du potentiel @
en un point ~
est donn~e par :
soit encore
-~ et la vitesse
induite
~
rep~r~e suivant ~
et ~
~ C~-~I attaches ~ ~
, par :
-
Au centre d~ segment =, Z~ les valeurs qO: qO%, C ~
sont toujours donn~es par
les expressions de (~,3 Lorsque l'on cherche ~ d~finir la loi du mouvement quail faut donner au profil d'aile (cf. § 4.4.) pour que sa circulation ~ ( ~ I soit effectivement optimale, on a besoin de connaitre l'influence d'un sillage semi-infini, II est done n~eessaire de traiter aussi la contribution au potentiel ~
d'une file semi-
infinie de segments r~guli~rement espac~s porteurs d'une densit~ constante / ~ de doublet.
Soient ~
du segment de rang ~
z~÷~L
et ~ = ~ - - = ~ + ~
(0<~,~).
vitesse induite ~ l'affixe ~
les affixes des extr~mit~s
Le potentiel resultant de cette file et la
sont donn~s par :
) Pour ~
suffisamment grand, on a :
?o~
Z-~o-Jh~.+~"
:
A ~ e ~8°
c'est-g-dire que les segments lointains peuvent ~tre templates par des doublets ponctuels de m~me axe et m~me intensitY,
Ce r~sultat ~vident est important
rappeler car il facilite les calculs num~riques d'autant plus que la s~rie donnant l'expression de ~
_ ~
~tant absolument convergente il est l~gitime
dans la pratique de se limiter g u n nombre limit~ de termeso
2.3.2. Pour le probl~me tridimensionnel on est conduit ~ utiliser la r~partition densit~ constante de doublet appliqu~e sur un quadrilat~re
A~CI3
. Le po-
2g tentiel et la vitesse induite s'~crivent :
Mais il est bien connu que cette r~partition constante de doublet produit un champ identique g celui de l'anneau de tourbillon fQrm~ par les quatre c$t~s du quadrilatgre, la circulation eorrespondante ~tant identique g la densit~ ~ , le calcul de ~ d ~ o n
Pour
a done int~r~t g appliquer la formule de Biot et Savart._~
pour chaeun des cSt~s tourbillons du quadrilat~re ; ainsi la contribution ~" ~¢~d~
donn~e par le eSt~ A B
en un point du champ ~
Pour le ealcul du potentiel
~(M)
s'gcrira :
on peut remplacer le quadrilat~re
en g~n~ral vrill~, par un 61gment plan dont la normale se d~duit du produit vectoriel des deux diagonales
Ac e t ~ D
, En glissant un
z
plan perpendiculaire ~ cette normale ~ mi-distance entre
A
les deux diagonales et en prenant la projection des
M (x,y,z) n x,~ T :~(~.~.,) iI' m (x,y)
quatre sommets sur ce plan on obtient le quadrilat~re plan A'B~ "~'
•
y,~
Cette operation
est avantageuse car le poten-
B
c, % 9
tiel ~ ( M ) r~sultant d'une densit~ unitaire de doublet support~e par cet ~l~ment
plan est identique g la composante suivant ~L de la vitesse induite en M
par une
r~partition de source de densit~ unitaire sur ce m~me ~l~ment. Or la formulation des composantes de cette vitesse a d~j~ ~t~ donn~e par Hess et Smith
[~.] de
sorte qu'il suffit de transposer leurs r~sultats et l'on obtient :
~rM)= _
4;
~ ( 4 ~j.) ao
/geD,
Aq~ (~i~ ~ ~. 6-~)[-A(~-J~-J~-J3~-J~ ] a~
30 o~ : I ~
](
,
.lr
,
I
("~)
h (~) | I
J
"
C~
~=IT
s i / ~ se trouve ~ l'int~rieur de l'~l~ment, nul dans le cas contraire. Les expressions pr~c~dentes qui permettent le calcul pratique de ~ M )
~%ocL~O
et
relatifs g u n quadrilat~re sont ~videmment utilisables pour d~finir la
contribution d'une suite finie de quadrilatgres identiques r~guli~rement espac~s. Mais lorsque la suite devient finie ou semi-infinie, on peut se contenter d'utiliser des formules asymptotiques pour les ~Igments inducteurs places ~ grande distance du point M
consid6r~ ; ce qui revient g remplacer l'~l~ment lointain
par un doublet ponctuel ~quivalent. Ainsi la contribution d'un tel ~l~ment donnera :
pour le potentiel, la valeur
~___~ ~ ~rL
T'~.zYL F~
pourle gradient,la valeur_~
~0 ; ~ )( ~Q -T ~ L~'~ ~
(~)
(I~)
o~
Une s~rie dont les ~l~ments sont form's des expressions
merit eonvergente.
(~'~I ou (=.~) est absolu-
31
3. A P P L I C A T I O N S
Dans les paragraphes qui suivent on pr~sente quelques ~tudes de syst~mes portants ou propulsifs, b i e t
tridimensionnels, en rue d'illustrer, pour chaque
cas, les particularit~s d'application de la m~thode d~crite pr~c~demment.
3.1. Effet propulsif d'un profil en mouvement sinusoldal. On consid~re un profil, form~ d'une plaque plane, dont le bord de fuite d~crit une sinusolde ; la composante horizontale ~/~ de la vitesse de d~placement est constante et dirig~e suivant le sens n~gatif de l'axe des o~. La vitesse de d~placement du bord de fui~e s'~crit :
=
d
On se propose de d~terminer : I. la loi de circulation ~(~) du profil en fonction d~ temps pour que la perte d'gnergie soit minimale, 2. la loi de braquage du profil, par rapport ~ la direction de ~/F ' qu'il faut rgaliser g chaque instant pour que la condition optimale soit satisfaite.
3.1.1.
~E~Em!~E~Em_~_!~_!~_!m_~EEm!!E!~.
courbe sinusoldale d~crite par
Le sillage ~
, rgduit ici g la
F , eat suppos~ en ~quilibre au sein du fluide ;
cet gquilibre des pressions de part et d'autre de ~
exige alors d'apr~s le
th~or~me de Bernoulli et l'hypoth~se des faibles perturbations que ~ ( ~ - ~ ) / ~ soit nul. L'intensitg ~ =
~0+-~
rappelges au § I.I., g is circulation
correspond donc, d'apr~s lea conditions ~ ( £ ) autour du profil au moment o~ son
bord de fuite passe par le point correspondant de ~- . La d~termination de la loi de circulation F ( £ ) sur ~
revient donc g la recherche de la densit~ ~ d e
doublet
pour que la condition d'optimum de l'effet propulsif soit r~alis~e : on
devra doric v~rifier que sur ~ o
,
correspondant ~ une pgriode, on a :
Le probl~me aux limites eat donc traduit par l'~quation int~grale :
La solution n'est dfifinie qu'~ une constante pros car le sillage ~,constituant la frontigre~partage le plan complexe ~ r~partition ~ =
~ ~
en deux parties ind~pendantes; mais une
n'apporte aucun effet sur la d~riv~e normale. La r~solution
de( $.¢I eat effectuge suivant la technique de discr~tisation d~crite au paragraphe prgcgdent 2.3.1. L'indgtermination due g la constante additive eat glimin~e par
32
la discr~tisation car le syst~me d'~quations lin~aires rempla~ant (5.&) est bien conditionn~ et sa r~solution donne une r~partition ~ s u r
le sillage qui corres-
pond sur le sillage ~ une constante additive nulle.
3.1.2. ~ E ~ ! ! ! £ ~ _ ~ _ ! ~ - ! £ ~ - ~ - ~ E ~ £ ~ g ~ " lation
Une fois d~termin~e la loi de circu-
~(~) du profil en fonction du temps, il faut pour d~terminer la loi de
braquage r~soudre un probl~me de champ ~ tout instant ~
en tenant compte cette
lois de la presence du profil dont la circulation est connue ~ l'instant consid~r~ et dont le sillage, pr~c~dente des
semi infini, est caract~ris~ par sa forme et la loi optimale
~.
Consid~rons l'instant ~ point F
o3 le bord de fuite du profil rectiligne passe au
de sa trajectoire sinusoldale et soit o((~) l'angle que fait le profil ~ F
avec la direction de l'axe ~X~
/
Compte-tenu du pivotement du profil autour de 'F , la vitesse d'un de ses points tel que M
~ la distance M F = r
vo
est done :
avec
IYFI=
(cf. ~.I.) exige done qu'en ce point M
IA
v~/oo4~
La condition de glissement
F~t
v~,~- ,~
~
!
f.I
=
•
.~
la d~riv~e normale du
potentiel qO , d~fini ~ cet instant,
soit :
~'(~4 ) =
Vo 4 ~ (~- e ) - &,-
o3, avee les hypoth&ses admises, ~ ( ~ - 0 )
peut ~tre confondu avec l'arc ; la
condition de Neumann pr~c~dente peut aussi ~tre aplatie sur la sinusolde que dgcrira F
, c'est-~-dire sur F M ' ~ ' projection du profil sur celle-ci.
Sur le sillage semi-infini
~[] , qui s'~tend de F
on devra imposer la r~partition optimale de circulation curviligne de ~
compt~e ~ partir de F
jusqu'~ l'infini aval, FC~ ] , ~
~tant l'abscisse
. Cette loi optimale est connue apr~s la
r~solution du probl&me precedent (cf. 3.1.|.) qui donne la distribution des ~ # ~ ) ou encore la densit~
~(~) des tourbillons libres du sillage
T(~)
=
<~C~o)/<~
~ , c'est-g-dire
.
Enfin il conviendra de satisfaire la condition de Joukowsky au bord de fuite, qui exige, tout en imposant la circulation ~ )
connue du profil, de v~rifier que
est un point r~gulier de l'~coulement o3, en particulier, la d~rlv~e normale
~
demeure continue.
33
Pour satisfaire la condition aux l i m i t e s ( 3 , % ~ qui contient l'inconnue ~ ( t ] et sa d~riv~e O~ , il est co,node de d~composer le potentiel des vitesses en trois parties, en ~crivant :
Les conditions aux limites ~ satisfaire pour chacune de ces fonctions sont r~sum~es dans le tableau suivant :
Sur le profil
Sur le sillage
-~
=
d~
Circulation du profil
conou
~F C~C')
connu
7
~Z~
~
--
~
d
A~
:
O
-- __ _~
CL
A~
: O
~
~ d6terminer
V~
~ d6terminer
Compte-tenu de la condition ei-dessus de Joukowsky ~ satisfaire dans les trois cas ces conditions aux limites fixent sans ambiguit~ chacune de ces trois fonctions ~
La d~termination des potentiels C~
et ~ 3
est classique (absence du sillage
des tourbillons libres) et les solutions donnent les valeurs des circulations et ~
, ind~pendantes du temps ~
.
Par contre le potentiel ~4 d~pend du temps ~
, en pratique on effectue sa
d~termination ~ des s~quences cons~cutives s~par~es par un intervalle de temps ~ pris constant sur une p~riode %
. A chaque s~quence la r~solution numgrique est
obtenue en appliquant le proc~d~ du § 2.3.1. pour une file semi-infinie de singularit~s ; on en d~duit ainsi Connaissant
~(t],
~ (~) .
~
et ~
il est possible de calculer la loi de braquage
~ ( ~ ) cherch~e. La m ~ h o d e utilis~e est la suivante. Soit (X~ la valeur de C~ pour la s~quence d'ordre J
, on peut ~crire :
Soit encore, en rempla~ant ~ .j j
par ~ / ~ t
J+(
34
Par ailleurs, la valeur
r~_(45)
,
pour l'instant 2C considerS, doit ~tre respect@e,
d'o~ une seeonde @quation :
Pour chaque s~quence on dispose ainsi de deux @quations (A) et (B) et si la p@riode est divis~e en ~
s@quenees,
on a ~
~quations pour d~terminer les ~5[ ineonnues
En posant s'~crit
et, pour les N
J syst~me :
s@quences on a l e
I C~'
to(~l
--
4
C~
N
-o-
a;i
4.-~ -4
4
'4
%
0
Soit encore
®+®
/C,+ C~1
dl~ ~ z
I
C~+C3I N"
(~1,t
0,. 4
I
-~8 L'~limination des ~ £ ~
~,
I i
4
,o
)
donne :
ii
(~-~B)
4
:i
$5
Posons
~÷~ +~-~
¢xi = -
o-.i
-
~
9.~
C~ + C~+~
--_ a ~ + ~ m ~ _~--~-~ C_.N + (2 .~
q,--
le syst~me pr~cgdent devient
q~ +
o/,t -
~
~ . , o/~
= q~+ ~
d'o~ l'on tire finalement :
Les autres valeurs de ~
se d~terminent par substitution.
Un exemple de r~sultat est donn~ sur la figure ci-contre pour un profil rectiligne dont le bord de fuite d6crit une sinusolde telle que l'amplitude soit ~gale au d~placement ~% pour une p~II
riode ; la corde du profil la
:OjOT~. quage ( 0 - ~ )
loi de braest donn~e pour
'°I kil
trois valeurs de la constante C
de la condition opti-
male(~.~
/\
. La valeur c = O
correspond au mouvement
~\.
"',
I/,'°
a/i/
;
i.
donner au profil pour que la circulation
~{~)
nulle ~ tout instant ~
soit :
dams ce cas le sillage disparait ainsi d'ailleurs que l'effet propulsif.
3.2. Loi de circulation optimale pour des syst~mes tridimensionnels. Des applications ont ~t~ effectu~es pour des syst~mes portants et propulsifs de types ailes oscillantes, ailes battantes, voilures tournantes. Ii s'agit donc de d~terminer la densit~ de doublet ~
sur la surface sillage ~
normale correspondante soit ~gale ~ e V o ~ . ~
pour que la d~riv6e
. Dams le cas g~ngral / ~
est
36
donn~ par la solution de l'~quation int~grale suivante :
Pour les syst~mes precedents ~ mouvement cyclique, le caract~re p~riodique de la nappe permet de remplacer l'int~grale pr~c~dente par une int~grale ~tendue ~ la surface
~ o d'une seule p~riode ; en introduisant ~
= rpM + N ~L, o3 ~L d~signe
le pas g~om~trique ou avance du syst~me par p~riode, on a :
JE o En g&~ral pour les syst~mes ~ ailes oscillantes ou ~ ailes battantes la sym~trie des mouvements des parties droite et gauche de l'envergure entraine une sym~ ~ trie de
m~me caract~re pour le sillage
~
, il est avantageux d'en tenir compte
dens l'~criture de la condition pr~c~dente
~o/~
Dens le cas d'un syst~me ~ voilure tournante, un rotor d'h~licopt~re par exemple, form~ de ~ b
pales identiques r~guli~rement espac~es, le sillage comprend
nappes tourbillonnaires identiques. Le champ du potentiel des vitesses pr~sente alors le m~me caract~re de p~riodicit~ que ces nappes et le cal~ul des ~ p e u t ~tre effectu~ en consid~rant une seule nappe ~-O
engendr~e par le bord de fuite
d'une pale quelconque au tours d'un tour complet de celle-ci. Cette remarque permet d'~crire l'~quation int~grale (~.~) sous la forme suivante : N=+~
~o o~ ~% d~signe l'avance par tour. La r~solution num~rique de ces ~quations int~grales est obtenue par la m~thode de colocation bas~e sur l'usage des r~partitions discr~tis~es du paragraphe pr~c~dent. Elle fournit, dens cheque cas, la loi de circulation optimale qui permet d'abord de caract~riser les performances globales du syst~me par une ~valuation des efforts moyens portant et propulsif et, en seconde ~tape, de d~finir les lois de d~formation en envergure ~ donner ~ l'aile ou g la pale au cours d'une p~riode pour obtenir les r~sultats de l'optimum. Pour les applications pratiques il est important de rechercher des crit~res de qualit~s permettant de comparer les syst~mes entre eux. Ces crit~res se d~duisent des valeurs des efforts moyens de portence et de propulsion et de l'~nergie communiqu~e au fluide au cours d'une p~riode ; or, il suffit de trois coefficients pour
37
caract~riser un syst~me en fonctionnement optimal. Pour ohtenir ces coefficients il faut r~soudre deux probl~mes d'optimum pour la surface-sillage ]o
Sur ~ o
~o
du syst~meo
on impose la condition d'optimum :
" l a s o l u t i o n du p r o b l g m e f o u r n i t
q
~A'-- ~ f
V o ~ % ,-*
l a t o i de r ~ p a r t i t i o n
de c i r c u l a t i o n
optimale :
et done les deux int~grales :
VoL
"
-6~
~
-
J
Eo
(o~ L e s t
une longueur de r~f~rence,
l'envergure par exemple). 2 °.
Sur
~o ' on impose une autre condition
d' optimum
~
--~ - ~
on en d~duit une autre Ioi de circulation optimale, soit
A~
-
r'~
~.L
~
et les int~grales :
~z
VoL
L~
o
-
~
"Vl.. ,1.,
dE
o
Les int~grales
A4%
et ~ I
sont 6gales, comme on le v6rifie immediatement
en appliquant, la formule de Green aux potentiels
94
et ~ %
et leurs d~riv~es
)
normales ~,~,
q~
.
Soit maintenant ~ r~soudre le probl~me d'optimum pour la direction de ~-~ -~ " -- a " " " ~"~
~
+,~,
circulation P
vec la condition ~ -
CVo~A,
r' = c ( % . C L'effort propulsif moyen, ou traction T
o~
l~ solution concerna~t ~a
s'obtient par combinaison
+
~',. ~)
, est alors donn~ par :
T o J~o est la longueur d'une p~riode. De m~me la portance moyenne " ~ est donn~e par
To
~o
38
L'gnergie ~
communiqu~e au fluide se calcule de m~me : P
p
avec pour l'optimum
~
=
-
cVo $ . ~
soit finalement : p_
-
Vol
~
~-
,~
Ii est souvent commode d'interprgter eette gnergie F-- comme le travail d'un effort r~sistant moyenT~
qui s'oppose au mouvement du syst~me, effort analogue ~ la
r~sistanee induite d'une aile ordinaire en translation uniforme. Pour la p ~ r i o d e T o et le d ~ p l a c e m e n t &
on ~crira ainsi ~
Les ealeuls de T ~.
,]=~ , E
our
----~ / ~ , o3 ne figurent que les trois coefficients
, A % ~ , A i~ , permettent d'~tablir d'utiles comparaisons entre les qualit~s
des syst~mes. Ainsi lorsque le syst~me est purement portant sa qualit~ est caract~ris~e par le seul coefficient ~ ( l e s
autres ~tant nuls), sa r~sistance induite
est proportionnelle au carr~ de la portance comme on le v~rifie imm~diatement en ~liminant la constante C
entre " P e t E
(ou~),
on obtient ainsi
Soit encore, en passant ~ des coefficients sans dimension familiers aux a~rodynamiciens :
=
(4"A,,k)
expression qui montre, qu'~ ~galit~ de C z et d'allongement ~
, le syst~me sera
d'autant plus avantageux que le terme entre parenth~se est grand. Pour un syst~me purement propulsif on pourrait de m~me caleuler une r~sistance et la comparer & la traction T
par une formule analogue ~ la pr~c~dente, mais
o3 apparait eette fois le seul coefficient
~44
• Dans le cas g~n~ral d'un systgme
portant et propulsif il est plus indiqu~ de s'int~resser ~ un coefficient rapport de la puissance propulsive T V o
~
~ la somme de la puissance propulsive et
de l'~nergie cin~tique communiqu~e au fluide par unit~ de temps, ~oit
~ / T o.
D'apr~s les formules pr~e~dentes ee coefficient, que l'on peut d~signer par "indice de rendement", est donn~ par :
4 + ~ (.'~T~) pour
un eoeffioient
de portance impes~e, soit
calcul~ en fonction de ~ = T / ~
@=&L~/~eV$L~,
& partir des 3 coefficients
~,
N~ peut ~tre ,
~,
~
.
Les r~sultats montrent qu'il existe, pour chaque valeur de l'amplitude d'une aile oseillante ou bmttante ou de l'inclinaison d'un rotor une valeur mmximale de ~ ( ~ ) p o u r
~
fix~.
Pour illustrer ees considerations nous donnons ci-dessous quelques r~sultats des calculs num~riques :
39
v
Aile oscillante, longueur de r~f~rence : envergure de l'aile ~ longueur ~
pour une p~riode : ~ =
amplitude
/~
/ \
////:I////
.
U
/~%
/ ~ % = /~z~
0
0
0,3
0,3070
0,7854 = T L / ~
0
0,6528
0
0,5
0,6114
0,5188
0
0,8
1,0816
0,3896
0
Aile battante. 0
0
0,7854
0
0,3
0,10434
0,5583
0
0,4
0,18068
0,4278
0
0,5
0,4424
0,1429
0
Darts les deux cas, l'amplitude nulle correspond g l'aile ordinaire en translation uniforme cient
~/o , caract~ris~e par le seul effet portant c'est-~-dire par le coeffi-
A~
qui vaut ici T [ / q
, puisque
L = ~.
Rotor tripale., longueur de r~f~rence : rayon d'une pale L=~=.
et
~/~
: angle du plan de rotation avec
angle ~
A ~
A~
= ~/~o
O~-
A,~
30 °
0,50014
1,4491
0,84398
50 °
1,1880
0,82072
0,97114
70 °
1,8012
0,25521
0,63873
La figure suivante donne, pour ces trois inclinaisons ~
et @ =
O, OS
,
la
40
variation de l'indice de rendement ~
avec
C~ =
-F/~ :::~
1 --
70"
0,9 O,8. q?
i x /y
,:~o~ '
IX
i /,,
O,6
../
../
/
\ ~ l , Ty. \ o,-~-
--PX
,-o.o. \ - .
qs 2
0
~ : T/P
CONCLUSION
Si le choix d'un systgme portant et propulsif peut ~tre effectu@ d'aprgs des performances globales les mieux adapt@es ~ un r@gime de fonctionnement impos~, il reste ensuite ~ d@terminer les lois de braquage et de vrillage & donner, ~ chaque instant ~ circulation
, g l'aile ou aux pales pour satisfaire la r@partition optimale de ~(~)
en envergure. Le probl~me est analogue ~ celui abord@ en 3.1.2.
pour le cas bidimensionnel d'un profil, mais le cas d'un syst~me tridimensionnel exige une miss en oeuvre num@rique beaueoup plus laborieuse. Son exploitation est n~anmoins en eours en utilisant toujours la d~composition du potentiel ~
en @l@-
ments plus simples, puis en satisfaisant les conditions par eombinaisons fin@aires. Bien que bas~ sur des principes consid@r~s comme classiques et @l@m~ntaires la m~thode des singularit~s ~ r~partitions discr~tis~es ne parait pas avoir eu l'audience qu'elle m@ritait. II faut reconnaltre que sa pratique exige souvent une tr~s bonne connaissanee des subtilit@s de la mod~lisation th@orique du probl~me physique analys@. Par cette communication, traitant d'un exemple typique d'a~rohydrodynamique, nous esp@rons avoir attir@ l'attention sur l'int~r~t et la souplesse de ses possibilit@s. Telle qu'elle a @t@ d~velopp@e au LIMSI, dans les travaux de Luu, Coulmy, Corniglion et al, la m@thode peut ~tre utilis~e dans des applications tr~s diverses et nota~ent pour la r@solution d'@quations non lin@aires, de type mixte elliptique-hyperbolique pour la d@termination d'@coulements
41
compressibles autour d'obstacles. Ii faut dire que les d~veloppements donn~s par ces auteurs leur furent souvent sugg~r~s par analogie avec des proc~d~s de calcul at simulation rh~o~lectriques auxquels ils ~taient auparavant rompus. A c e
propos
on peut d'ailleurs noter que le probl~me g~n~ral dont il a ~t~ question ici aurait pu faire l'objet d'une telle simulation : son principe est ais~ ~ concevoir (cf.l~A~), mais le module correspondant, souvent plus difficile ~ r~aliser.
REFERENCES
Hess, J.L. et Smith, A.M.O., Journal of Ship Research, vol. 8, n ° 2, septembre 1964. Hess, J.L. et Smith, A.M.O., Progress in Aeronautical Sciences, vol. 8, Pergamon Press,
1967.
Giesing, J.P., Journal of Bassic Engineering, Trans. ASME, series D, vol. 90 n ° 3, septembre 1968. Luu, T.S., Coulmy, G. et Corniglion, J., Bull. Ass. Tech. Maritime et AKron. (ATMA~, session de 1969. Luu, T.S., ATMA, session de 1970. Luu, T.S.
Coulmy, G. et Corniglion, J., ATMA, session de 1971.
Luu, T.S.
Coulmy, G. et Sagnard, J., ATMA, session de 1971.
Luu, T.S.
C.R. du Colloque du C.N.R.S., Marseille, 2-5 novembre 1971.
Luu, T.S.
Coulmy, G. et Dulieu, A., ATMA, session de 1972.
Luu, ToS.
Corniglion, J., ATMA, session de 1972.
Luu, T.S,
Coulmy, G., Proceeding 3 e Conf. Numerical Methods in Fluid Mechanics,
juillet 1972, Ed. Springer-Verlag. Malavard, L., Agardograph n ° 18, ao~t 1956, Ed. AGARD-NATO, Paris, France.
SOME CONTRIBUTIONS
TO NON-LINEAR SOLID MECHANICS
J.H.Ar.qyrls and P.C.Dunne Imperial College of Science and Technology, University of London Institut fur Stafik und Dynamik der Lufl-und Raumfahrtkonstruktionen,University of Stuttgart
SUMMARY
The finite element method has now developed to the point where practically any linear problem can be solved with adequate accuracy. There is still room for improvement, much dead wood remains to be removed and pracedures have not yet reached the degree of standarization which will eventually be possible.
On the other hand the treatment of non-linear problems By the
finite element method, and indeed by any method on the scale required by engineers, is still at the beginning. Non-linear problems are very difficult to visualize. It seems that there is a built-ln tendency for the human brain to think linearly. Analogies are especially helpful in this respect. A problem often becomes easier to think about when it is translated into the field to which one is accustomed. The present report is mainly concerned with non-llnear problems in structural engineering but some other types of problems are mentioned. Part 1 attempts to make the theory of large strains more accessible to the engineer who has no background in tensor calculus. This is done mainly through extending the idea of natural stresses and strains to the case of large strains. The simplicity of this approach is greatest 'with the family of elements deriving from triangular TRIM3 andtetrahedronal TET4. Part II reports on recent work at ISD on the dynamics of very large structural systems. Two types of integrating algorithms have been developed. The first which requires small Hme steps is applicable to both linear and non-linear systems. The second type enables linear systems to be integrated with very large time steps and is an alternative to modal analysis. The development of a similar method for non-linear systems is discussed. Although the small step algorithms are especially suitable for structural dynamics they are also applicable to other problems. The classic n-body problem is one example and another is the dynamics of an underground explosion with idealized non-ti near material.
43 ACK NOWLEDGEME NTS
The work represented here is the work of a team. Of this team we must mention especially T. Angelopoulos who was responsible for the development and implementation on the computer of the dynamics algorithms of Part II and the shape finding technique of Part II1. A valuable contribution to the theoretical work was made by Maria Haase and a large proportion of the programming was the work of B.Bichat. The problem on elasto-plastic wave propagation was contributed by J. Dol~sinis. Others who must be thanked for their extensive and unselfish help in the preparation of the paper a r e G.Grlmm, I.Hucklenbroich, K .Mai, H.Ries and K .Strauf].
44
I. LARGE STRAIN THEORY FOR FINITE ELEMENTS
1. Introduction The theory of large strains and its application to non-linear elasticity is generally considered by practising engineers to be a subject difficult for them to Follow. The idea is prevalent tha~ a proper
understanding of the subject requires facility in the tensor calculus. Particularly confusing to the non-specialist is the multiplicity of strain and stress tensors such as the Green and Almansi strainss and the Eulerian, Lagrangian and Kirchhoff stresses. Whereas in small strain theory the shear strain is the easily understood change in the angle between two originally orthogonal lines, in large strain theory it depends also on the direc~ strains along these lines. In the book of O.H .Varga [1 ] the behaviour of isotropic material is described almost entirely in terms of the principal strains and stresses. This approach is easy for the engineer to understand. Another relatively simple way of treating the problem is through the idea of natural strains in simplex Finite elements. These latter are the triangular (TRIM) and tetrahedronal (TET) family of finite elements. In the following we particularly develop this method. The natural large strains of the TRIM3 and TET4 elements are constant and are the Green strains along the edges. The associated natural stresses are the component stresses parallel to the sides of the triangle or tetrahedron in its final position. For higher order simplex elements these definitions may be used in relation to the constituting small subelements. The present work is restricted to consideration of TRIM 3~ TET4 and TRIAX 3 with and without axial symmetry. In the latter case it is necessary to make use of cartesian strains in conjunction with circumferential Fourier components. The non-llnear stiffness matrices for the TRIM 3, TET4 and TRIAX3 axial symmetrical case are established For an arbitrary polynomial strain energy Function in terms of the natural strains. For isotroplc materials the strain energy polynomials are expressed in terms of the strain invarlants.
4S 2. Two-Dimensional Strain The strain is supposed to be a pure extension in the direction of the cartesian axes Ox , Oy (Fig.l.1)Apoint
P[x,y)
moves to
P'(x',Z').
Then if
fl .~)
/1. = r ' / r
Z2= ~cos 2e + and
2 1 sin20 v&]
f1.2)
if (I .3)
E" =
E
Thus the
=
Eicos2e + Eiisin2e
strain measures
A2 and E
transform as in the Mohr Circle.
which is related to the ordinary Cauchy strain ¢
:
/1. - 1
or
(I .4)
E
=
E
by the equations 1 2 -=-E Z
E *
The Green shear strain is obtained From £I
E is the Green strain
(1.5)
and ~II by the usual transformation. However, it
has no longer the simple physical meaning that it has in the small strain case. Now consider a triangle 1,2,3 which is uniformly strained and displaced to a
new position
1~,263 ~ . The strains of the sides 1 2, 2 3, 3 I as defined in eq. (I.3) are called the total natural straih; which form the vector
Et
= { Eta: Etp
Eta, }
(1.6)
Then
Eta = $1c°s2VI ~tp
+ ~Zllsin2Vz
= ~Ic°s2(Vt+~') + ~:iIsin2(Vz + ~')
(1.7)
Etz = ~icos2(Vr-/~) + 611sin2(V1-/~) These equations are the same as those For the small strains. It Follows that the device of co=nponenf strains ~'c
may Be used althouuh they are no longer physically superposable.
Thus, with, LEC
:
{ ECCL £Cf2 EC, }
IE t
=
A Ec
or
Ec
=
~ll-lEt
0.8)
46 where, cos2~
1 A
:
cos2p 1
J, c°?l
cos2g,
cos 2p
f1.9)
The strain invariants are,
J1
= el + eII
=eca
= m~
+ ec# ÷ ecz
t -I e c = le3A a" t
0.10)
and
~
=
= 1
~,~,~ 7(~, ~ -,,~"c)
=
1
t
T-c[E~
_
"]-~
~,.,,~
where
1 e3 = {1 1 1 }
~
E3=
e3e~
=
1
1]
1 1 1 1
1 1
(I.12)
The membrane incompressibility condition is,
&l AII = 1
or from (I.3)
(1 + 2e I )(1 + 2ei] ) = 1 (I .13)
O1"
J1 + 2-/2
= 0
For small strains this condition reduces to,
J1-'-
F1 = E] + Eli = e~tE c = 0
The small cartesian (Cauchy) strains E
IE where
=
ctE
are related to the small total strains
(I ,14) ~t
through
f1.15)
t (I .15a)
47 and
rc23x Ct =
where
c23y
]/'~C23x C23y1
C31x
C31y
r C3,x c3, I
Cl2x
C12y
1/'2g2x clzy j
l
(i .16)
c23 x = cosine (23p 0 x ) e t c .
Note that
ctc
(i .]7)
= A
In large strain theory the parallel results are
fl .t8)
E t = ctE where
(I .18a) It w i l l be convenient in what foJlows fo use the notation x l , u 1 for
y,
v
for
X~U
and x 2, u 2
. Also we define,
(I .] 9) uI : I
c3U1 ax I
etc.
(I .20)
and
D
=
u ,1 I uI
,2
u ,1 2 u2
]
:
,2
[
au ax
it
(1.2])
With this notat}on the cartesian small strain tensor may be w r i t t e n ,
(I .22)
3 =
E21
E22
=
and the cartesian Green strain tensor is,
48
= [ El1 ~12]
3'
D + l)t+ DDt]
= ~1[
(I .23)
22J An alternative Formulation of the last r'.~sutt is obtained by writing
y
Thus
:
X
+
ILl
(I.24)
is the coordinate vector of the point x
¥
same axes
Ox I ~
Ox 2
=
H
• Then
c3y Ox
in its displaced position with respect to the
introducing the mapping matrix
= 12 + L]tt
f1.25)
the expression (I .23) becomes, .
If n
:
½[.,.
transformations
M i
- !" 2 ]
(1.26)
are applied the mapping of the final configuration with respect to
the original one is
J~v~ -x"
=
#~nlvln-1
which may replace lvl
f4(
.....
Ivl2/Nr t
in (I.26). Interestingly enough we may assign to each
distinct physical content. For example, .Iv!1 may describe a pure plastic and elastic deformation. By the cosine formula the angle c('
(1.27)
mapping a M 2 a pure
in the deformed state is cjiver, by,
Ap2 2lp + I.~,212~, ~212 2ApiBA~. i~ -
COS
~' =
~tp lp + When C(
is
f
and
2
~tz 1~
l# = 1~.
0.28)
2
-
Eto~
1~
÷
cos c,
lle along the cartesian axes,
4g
ell ÷ e22
-
COS (XI =
2eta
2 el2 ~.p~,
=
(I .29)
2. Three-Dimensional Strain In the same way as in the two-dlmensional case a point
tensionlnthedlrections
xl, x2, x 3
P ' ( ~'l xl" "Ell x 2 ' ~IIl x3 )
2~2
moves to
pl
under a uniform ex-
beco:"~les
.Thus P ( x I ~ x2, x 3 )
. Then
jl.~(xl)2 + A.~i(x2)2+/tm(x 2 3 )2
= [ OP' ) 2 \ OP =
r2
(I .30)
2 A2Ic2+ /1.i1c2 ÷ All2 I c2I
= t A2 -I) ~:T(
Defining as before
E =
P
C2
EI
+
E'|]C~]
+
•|1] C~]]
(l.31)
This result again shows that the principal Green strains transform by the same transformation rules as the small strains,
c I , c l ] , Cii ]
ore the direction cosines of
0P
with the principal
strain axes before deformation. Now consider a tetrahedron 1,2,3,4 . The total strains along the edges form the vector
Et
:: { ~t12 ~t34 ¢t/,t ¢t23 ~t31 Et24 }
(I .32)
ff the direction cosines of the edges with respect to the principal directions before deformation are c121
etc., tken,
t where
and,
(I .33)
•t
=
Cp Ep
Ep
::
{
~']
~']I
6]]|
}
(I.34)
SO
C~ [
C~3 I
G
cL cG 4~ G~ G
42m 4,m 4~ 43~ 4m
C24I]
2 C24m
0.35)
These equations are exactly as in the small strain case and we may again use the concept of component strains. Thus, the component strains are defined by
e t
= lie
Ec
= A-l~t
(1.36)
c
OF
(I .37)
where (1.38)
Ec = { ¢c12 ¢c34 ¢c41 ¢c23 ¢c31 ¢c24} and, ]
c2 12,34
2 c12,41
2 C12,23
C2 12,31
C2 12,24
1
c2 34,41
C2
C2
C2
c2
c2
c2
cz 23,31
cz 23,24
t A
=
34,23
1 sym.
34,31
1
3/~,24 (1.39)
c2
31,2&
1
All the transformations valid for small strain theory are valid also for the large strains .As before the component strains are no longer superposable. Then for the cartesian Green strains hasy
£
one
,L
Et
= C~
0.4o)
51 where
~t =
(1.4] )
C121 = cosine (12, OX')
etc.
and (I .42)
If ~p
isa principal strain with direction cosines
~p = ~_~6~c12 (C12' CoI ÷ c122 Cp2
Cpl , cp2, Cp3 + c123
then
Cp3 )2
6 (1.43)
-, 2%, %2 ~c,21 c,22 ~12 ÷ z%2 %3 Zc122c2'3~c125 2Cp3 Cpl ~,,c123 c121 £'c12 6 The principal strains will correspond to the values of
cp! , Cp2 and Cp3 making ~p
stationary subject toe
2
Cp l +
2 ÷
c23
=
1
(I .44)
This leads to the determinant
~, C121 C122 ~:c12 6
) sy m.
~,C121 C123 E'cl2 6 ~.~ C122C123 ~c12 6
=0
(I .45)
$2 This must be identical with the standard cubic equation for the principal strains in terms of the strain invar[ants
(I .46)
J3 = ~I ~rI ~111
~3p_ j1~2 + j2~p
_ J3
:
0
(I.47)
Thus, J1 : Z ( c 2 2 1
÷
6
c~22+ c~23)~c12
J2= Z sin2%,pq%q
15 1
The third invadant J3
(I .48)
cpq
0.49)
: T,1 t [E6
0.50)
is equal to the determinant of (I.45) with
J3 = 36 t/2~
ep = 0
and reduces to
Ec/j%pq%rs
0.5;)
where V = volume of tetrahedron ('1,2,3,4) and the terms in (I.5t) include all non-coplanar combinations of" the edges. The incompressibility is
2. I
Aii AII I = 1
or
J! + 2 J 2 + 4 J 3 = 0
(I.52~'
For small strains this reduces to J1 ~
F1 = C]
÷ E]]
+
E]]]
= e~ e C
=
0
(1.53)
$3 3.1 Cartesian Strain Increment-Relations We collect here some results in cartesian strains which are given for three dimensions but are immediately valid for two dimensions. The Green strain tensor following (1.26) is,
Correspondingly the Almansi tensor is,
f1.55) where
t4
ay a x
:
;
N
8 aH y
:
0.56)
Thus N since x
=
M-1
and y
0.57)
are referred to the same cartesian axes. The increments of ~11 and 3 A
due
to an increment ~U are, from (I .54) and (I.55),
For the purpose of calculating the virtual work of the stresses it is useful to have expressions for the increment of Cauchy strain hood of a point P
~i 6
basedon the final configuration ( y state). In the neighbour
which becomes P' in final state, one has
v
- y~.
]If
-
-
~
["
(1.60)
- "p]
But J~p, = ( M
" Mp)÷
( Li
Therefore, U
"-
-
-
Up)
$4 Then far an increment 6 u
cS[u
] :
- ~p
But Jll'
-
~[.
,1Nrp
:
H[y
up]
=
aHN[
- ,,p]
0.61)
- yp,]
So that
~[.Thus
~[.-..]
y
- yp, ]
(~.~2)
is or the form
o'[ ,r - yp,]
(see eq .(1.21 )
and hence
63t
=
Now from (I .58) with (I.63)
/~/t63 l~
=
(53
Nt6:~ N
=
63
or
0.64)
4. Strain in Bodies of Revolution In bodies of revolution with symmetrical loading and properties one may work in terms of triangular elements in the radial plane. For non-symmetrical loading there are two alternatives. The elements may be in the form of annular rings of tricngular radial section and the circumferential displacements developed as Fourier series. Or one may use segmental or even fetrahedron elements. Although for large displacements the Fourier components will no longer be orthogonal, it is thought that the total degrees af freedom for a given precision should be less by the Fourier method, at least for fairly smoothly distributed loading. 4 . 1 ~ i - S y m m e t r i c Strain Since the displacement normal to the radial plane is zero the strain in that direction,which is also the principal strain
~'1I
, is given by,
55
"C22
=
~]]
=
XI
+
T
k xl ]
(i .65)
In the radial plane the strains w i l l be defined as in (I .2) and are always reducible to the principal strains
eI
' ~']I[
4.2 Non-Axi-Symmetric Straln
xt
The cylindrical coordinates of a point of the body are
x2
J
x3
so that
x2
is now
an angle. To obtain in this system the strains
we require the matrix
114
in terms of the cylindrical coordinates. Thus M
reference to the orthogonaf triad of axes in the directions of x I s x 2
M
is expressed with
and x 3 at
P
: [ :3 ÷
where~
U1 ,1 *l
D
7(
U2 ,I
UI
_ U2 )
U1 ,3
1
7 (
2
U3 ,1
÷ u' )
U2
,3
1
U3
,,
(I .68)
U3
j3
The corresponding Green strain tensor is then given by eq. (I .54). We now define the Fourier components of the displacement in symmetric and anti-symmetric groups with reference to the generating triangle of each ring at x 2= 0 • Thus using
u
for the symmetric and
v
for the anti-symmetric terms the nth Fourier
compo-
nents are
U1~COS
n X2
.
V1
sin n x z
u 2 (~: sin n X 2
;
V 2 cc cos n x 2
U30CCOS n x 2
~
v 3 oC sin n x 2
(i .69)
$6 Deft ni ng,
% = F cosnx2
sin nx 2
cos n x Z J (I .70)
sn
=
f sin n x2
=
fCn
- cos nx 2
sin nx 2 J
or more briefly
Cn
Sn CnJ
;
= O¢n
;
Sn:
fSn-Cn
sn_]
1,7,1
then
ds n dx 2
d Cn dx 2
= -/~ $ n
(I .72)
In what follows we restrict ourselves to the symmetrical component U For the TRIAX3 Finite element presentaHon it is convenien~ to define two types of displacement vector. Thus the displacements at the ith node are
(I .73)
and the vector o{the displacements
"J
uj
at the nodes 1, 2, 3 is,
'4)
:
Three ways of expressing the nine nodal displacements are now defined.
u!
=
[ u~
u2
u 3}
ul
=
{u I
u2
u 3} 0.75)
The vector Thus
U
within the basic triangle is interpolated linearly between the nodal values.
$7
and also
where the ~ 's are the natural (areal) coordinates of the triangle.
uJ = ~ i . i j
Alternatively
For the displacement in the nth mode we introduce the inferior index n
un =
':,~%~,
IM t = n
([d nUi C n
and write,
(i .78)
or
With the above notation we find, for the Cauchy strain
~cn
in the nth
mode
/1 -
c,~;~I
I , , i
1
T~"[°',, f ~÷ 7 = '
I
(_n~1_.2;1t
n 'Jw
Cn ~, ~' [~'" n~ ~l
I I symmetric I
sin
~ o [ ~ ~ - 7n
J
=~1 n"
CnI~,3 U3n (i .79)
The individual components
ai ] u}' . In the a ILl
~i]
etc., of this tensor can easily be written in the matrix form
above matrix, 1
a x 1 = ~7
=
2.Q
a~, ax 3 =~,3
=
2~
x23 = x 3 - x 2 etc.
[ ;(33
3 Xl 3
3 Xl 2 ]
0.80)
'[4~ 4 4] and ~
To evaluate the Green strain
~
=origlnal area of triangle. we must expand the expression
DIll t We do not multiply out this expression but form instead the contribution of the symmetrical and anti-symmetrical Fourier components. Thus there will be terms of the form,
58 1
1
I ~n
1
When m = n
where
0.8])
t
the general expressions reduce to the R.H .S. members.
Then the total large strain tensor
.~ :
t
3
is
~ + ~, [ ~E}cmcn + ~ smcn + ~)crncn +
m,n
(3x3) 0.82)
31 is the total Cauchy strain given by
(I .82a)
~" (]cn ÷ ]sn) n Introducing the notation for a typical
11 ~crncn ~cmcn =
~I)
matrix,
12 ~cmcn 22 ~; Pi)cn
sym
/3
~cmcn 23 ~;tT'tcPi
(I .~3)
33
~cmcn
the individual entries of which are effectively contrlbu?ions to the higher order strain components, and also
jr~,11 = f~t,1 I~,1
,
J~,13 = ILIIt,1 1~,3
,
-~,33 = trot,3 ~ 3
(, .84)
co
II
E
o
Q
--2
0
~-~
I
iI
u,
÷
÷
~
~
iliJ
i|
÷
II
i ~---~
L
f'l
J
"
l
J--
4.
I
0
f
!
~I ~
~
4"
I
Ill
l~ | I
L~
I
II
I
01
60
For the case
m =/1
we have,
~i,,~, = T
12
~':/1c/1-
n
c;
1 uIt
2 /1
~,17J uz/1
,
-,s/1~I
-s/1~7
03
C/1 Jrz, l
ncnJrz, t
03
c,
03
~cncn = T
n
03
- nsnJr~,l -
J:
13
Cn
ILlI
22
22
~cncn =
1
u z
uIt
-2- n
03
~cncn =
,,
23 r~cncn
,:
derives from
u,,,F n
Cn
12 r'~cncn
]I"1,33
03
jr~.3 3
by substituting
jr~,~
Jr~,3
n2~ l ~
J-' n
in place
]r'~1 (I .86)
The terms corresponding to the anti-symmetrical components
~}crncn The terms
by changing
~}snsn
~cmsn ( g s m c n ) ( cm into s m)
c
into
s
,
s
are found by the same rules from are obtained From
, sn
into
~$msn
into - c
are obtained from and
~cncn ~cmcn
- Cn ( Sm into - cm)
U
into
v
. The coupled terms
by changing and
cn ~nto s n
u n into
vn
61
( u m into
vm )
~)cnsn
For
and
~)sncn
put
m =n
in the corresponding general case.
Eij
It is finally necessary to express each component linear and quadratic Forms in the nodal vectors
ISli n
and
of the Green tensor as the sum of l/J" n
. The(1 x 9)linear transforma-
tion matrix and the(9 x 9)matrix of the quadratic form are only dependent on the initial geometry and the Fourier modes involved.
5.
Virtual Work and Stress
5.1 Two-Dimensional Case - TRIM 3
(I .87)
are the component stresses on the final configuration the work done during an increment of the total strains may be obtained by considering only one strain at a time. We take Then if
6Eto:
is an increment in the Cauchy strain on the Final configuration
6 W = T1 a c ~ h ~ t ~ l ~ ~ But
Ctar
1
T i l ( ~~o ¢/ i c~ /
- ~.~E1
Sot
514/ = ~1
£t~
0.88)
, the final area.
tac~5 Etc,
Now Therefore,
6W = gl t~c~25¢tc~
0.89)
For all strains,
6 Hi = ~1 t crct~I.-2 ~ ~t
where
"
=
""J
In terms of the cartesian stresses and strains, since
(I .90)
62 % where and
tD' C0
and
5W
= ~E
and
: { o,, C1
=
417 = £ ~C
(1.9|)
]
refer to the original and final geometry,
.£21tutc(1
'
tA.-2 ~'0t (~ E
(1,92)
which may be shown to be equivalent to ~W
=
E21tcrt~IE
(1.93)
where 51E is the increment of cartesian Cauchy strain on the Final configuration. 5.2 Three-Dimensional Case - TET4 The parallel results for the three-dimensional case may be given Briefly.
Thus the component stress vector on the final configuration of volume Yl
°c :
{ ~c12~c34 ~c41
¢c23 ~c31
~c2Z,]
is
(1.94)
Then By a similar argument to that used For TRIM 3,
(I, 95) In terms of the cartesian stresses and strains
5W
: VI~'tsIE
:
Vl~'tc-l't"12C05~"
(,.96)
where
{0"110"22 0"33 ]/~12 V2~23]/r2~31} 5.3 Bodies of Revolution In cylindrical polar coordinates the internal virtual work equation for an increment of strain on a volume element of the final configuration is,
f1.97)
63
If
./1 , y2 , .y3
is the triad of orthogonal directions along the cylindrical coordinates in
the final position we require ~E we have
to be referred to these directions. But in terms of 6~"
~IE ~ referred to the original directions x I ~ x 2 ~ x 3
531' where ~
= #H/t'-163H
is given by eq. (1.25) with
Then transforming t o t h e a x e s
so that from (I.64)
-1
/Qt
y l , y2~ y 3
0.98) by (1.68). by the rotation operation
(;J
=
x2 )
-sin(y20
o
cos(y2 _ x 2) 0
0
0.99)
1
we obtain
6,3
=
J6
3'J
t
(I .]00)
or finally
= j J~t,-16:~ j ~ - I
fl .]0])
This equation may be rewritten in the form of the vector transformation
~E
-
~W
=
(I. ] 02)
T~E
and hence,
dr1 (I. 103)
6. Strain Energ[ Functions The choice of a strain energy function to represent the behaviour of a real material is a difficult one .This quesfi on is discussed in many Bookson fi ni te elastic ity ~ , 2 , 3 , 5 , 6, ~ . From the computational point of view it is o great advantage if the strain energy can be represented in the form of a polynomial in the Green strains, or in the case of isotropic materials, in the strain invariants. Of course, the advantage of this approach depends on the polynomials being of relatively low degree which may not always be possible for a true representation of the material for large strains.
64 6.1 Strain Energy Functions for !sptropic Membranes For isotropic material we may assume for the strain energy density thickness
f
based on a nominal
to be of the form, n
(I .lO4) m=2 where
(I. 105)
~m : ~ , A r s J l r J ; r,s r+ 2s = m
and
J1
and J2 are the strain invariants defined in eqs. (I.10) and (I.11). The first two
members are~
:
A20J12 + A01J2 (I .106)
:
4304 3 + 411 4
For incompressible isotropic material n (I.107)
: T, Am4 ~ m=l
with the incompressibility condition
4 More general forms for ~
÷ 2J 2 =
0
(1.13)
are
or in the incompressible case = Function
(21
)
or
Function
(22)
(I.109)
From (I .]08) the principal stresses are obtained by virtual work (see also three-dimensional case below) as
6S
(I.]lc) and similarly fo~
at[
. In the incompressible case /
\
6]
(I.]ll)
~h is the total hydrostatic stress equal to TI {~I
"~ ~IT }
• The part of this stress
depending on the strain energy function is
(I.112)
when
is given in terms of
J1
or
a# ~/~ when ~
(I.113)
J2 a j 2
is g i v e n in terms of
"]2
. The remainder of
o"h
is due to the compressibility
constraint.
6.1 .1 Some Special Cases The simplest strain energy expression is
~2 = A J21 -
B J2
(1.52)
which gives
AI
-
(I .I ]4)
Belt
The simplest incompressible case is, N
1 From which
Bj 1
Or
BJ 2
0.1 ]5)
66
fl.1]6) o1i Note that
.
- o-
°h
-
T
:
-
-
~I]
)
so that the shear stress and strain are in thls case proportional. Membranes characterised by (1.115) are analogous to the Mooney three-dimensional material (see below). 6.2 Three-Dimensional ,,Isotropic Ma,!,erlals Equation (I .104) is still valid but
-
and
n
s
fl.117)
t
~rn = ~,, ArstJ1 r J2 J3 m=2 r +2s÷ 3t= m
]'he first two members are,
and
£•2
= A200J12 ÷ A010J2
~3
= A300J13 + Al10 21 22
(I .i 18) +
A001 J3
In the hJcc,npresslble case /3 where
m=l
(J .1 ]9) and
P,q
p+2q=m with which is associated the incompressibility condition
J1
+
2J2
÷
4J3
= 0
We note that all the strain energy expressions proposed satisfy the condition that ~ = 0 and also if is a minimum for zero strain. A more general strain energy expression is
(I .52)
67
for which the principal stresses are typically
-
This equation is obtained from the virtual work due to
a#
6A 1
]
(I .121)
on a unit cube.
Thus
1
so thor
AI
a#
AIIXIIl O "1"I
a
(I .122)
~IRIII a ~I
from which (1.121) follows. In the incompressible case only two of the and one may express
=
~
J's
are independent
in terms of any two of them. We take
Function (J1 ' J2 )
(1.123)
from which, with the incompressibility condition (I .52),
aJl
which follows from
f1.124)
(] .121) by introducing the meat1 hydrostatic stress
1 and remembering that J3 When
~I ' uI]
and
does not appear in
(~
~III are specified any two of eqs. (1.124) together with the volumetric
constraint (1.52) suffice to determine the strains. However, in a practical problem the
0" 's
are unknown. ['he hydrostatic part of the stress will consist of two parts. The first part is associated with the strain energy and is given by
f1.125)
The second part may be considered as the hydrostatic constraint which maintains the volume constant
and is denoted by
~'
. This has the nature of a Lagrange multiplier although it is pre-
68 ferable to regard it as the pressure of an actual fluid which can be varied at w i l l . 6.2.1 Some Special Cases The simplest special case is the material characterised by ~2
of eq. (I.106). This gives,
(1.126) For incompressible materials (1.126) takes the form,
.~
1
:
~
÷
~
A2oJ~ ( "~ - ~ J~ )
(1.127)
Another example is
=
A Jt
÷
(1.128)
BJ 2
which is the so-called Mooney material. This does not apparently fit into the general class (I .119), but since for incompressib~llty
J1 = -2(Jz*
2J3 )
it may be construed as a
special case of the form
The stress iS,
6h + 2A (~'I
)
0.129)
~.J2,. 2~(J~
_
~-
and
5; - 5 Note that the equivalent form in terms of J2
(I .130)
and
"]3 yields
This may be compared with the membrane case (I .112) and (I .113).
0"hi = 0
at zero strain.
Bg
6.3 Strain Energy Density for Anisotropic Materials The simplicity of the isotropic case is due to the fact that the strain energy is a function onty of the principal strains without reference to their directions.However it will still be possible to approximate the strain energy By a polynomial in the six strain components and advantage may be taken of any axes of symmetry. In general the strain energy density function will vary with position. There seems little point in attempting to take account of this variation within a finite element. The assumption of uniform anisotropy within each element may Be regarded as part of the discretization process. Many technical materials, although not isotropic, may be well approximated by the assumption of orthotropy. In such cases there will exist three orthogonal directions (two in membranes) for which the principal stresses and strains co-incide. The idea developed by Varga in reference [1 ] of making the principal stresses and strains the basis for a discussion of large strains in isotropic materials has great conceptual simplicity o In anisotropic materials some of this simplicity is retained if the strain energy is expressed in terms of the principal strains but with coefficients depending on their directions. As an illustration the case of an anisotropic membrane is considered. Thus,
:
fl
fl ,
÷
l
os
..... (I .131)
+g21~] ' ~ll )sin2(P+g~(~I'~II)sin L,~+ The omission of the odd
~p
terms is justified By the fact that
~
.
.
.
.
.
.
wilt not be altered by a
change of 180° in the directions of the axes. tf there is an axis of symmetry from which measured the sine terms witl vanish. In the particular case of an orthotropic sheet the smatl strain energy density may be written
(I .132)
70 If the principal strains
El , Erl
are inclined to the axis
OX
by angle
~
, then
Eli = COS2@E[ + sin2e En E2z= sin2tpE I + cos2~ E n
(1.133)
E12 = sin~o cos~o (Ez- ED) Substituting these values in (I.132) we find,
1
+ 4 ( E,,-
(
+ (El, ÷ E22- 4 (3)(El Thus
~
) cos 2
(1.134)
- E.) 2 cos /.~}
i s o f t h e form (I.131). The same result is obtained if (I.132) is considered valid
for large strains expressed in terms of
7.
Stiffness Matrices
The stiffness matrices for the TRIM 3, TET4 and axisymmefric TRIAX 3 elements with large strains are now derived. These elements are also the basis for developing the properties of higher order simplex elements using the natural stresses and strains. 7.1 TRIM3 The virtual work eq. (I.90,) For the case where the stress arises purely from the elastic strain of the element is
5~ = ~ltl~tA'ZSl~t where
(!.135)
is the total strain energy of the element.
Thus
1 ,,112 g~ (I. 136)
71
In terms of the strain energy density
n0
(! .137)
From I1~c we may obtain
I~t
and the cartesian stresses •
by the usual transformations.
The strain energy density of an isotropic element will be supposed given in the form (~ .106). 21
and J2
in terms of
~t
are found from eqs. (l.10) and (I.11).
Thus
E1 : e~A-1Et
u2=
t
-1 (1. t38)
4-
~ A-1 E3 A'I ~t
In the small strain case
E
(I. 139)
2(I-~) This is of the form (I.106) with
A2o -
E ~ 2 (1 - ~ )
i
A°I
=
E 1÷v
(I.140)
If the use of expression (I.139) is extendedto large strains thequanfity w will lose its significance as Polsson's ratio. Then in terms of
IE"t
the strain energy density
~2 : T tE~ kN2 E t
~2
is,
(l.141)
where
E
/~N2 = "1 ÷ ~
[Aq÷
w A-1E3Aq]
i -
t~.~42)
is the natural stiffness matrix for large strains. The natural stiffness for the TRIM3 element ;s hence
72 m
kN2
=
.('20 t k N 2
(I .142a)
A physical interpretation of the generallsed forces based on the total strain energy
~2
of the
TRIM3,
63@2
'63E~
(1.143)
= kN2 Et
is as follows. Denoting the component stress resultant force along the side
a@2
10r by
Pa
we have,
0@2
63E:tc~
63Sta
Therefore,
P~ _
A~
a@z
_
a@2
0.144)
The vector of the P ' s is called the natural force vector
a@
= A I-1 kN 2 .E t
0.145)
We now see that the action of the TRIM 3 finite element is fully represented by three pairs of opposing forces
P0~ , P/3,/~
acting parallel to the sides at the vertices, For the particular
case of an equilateral triangle with
v = 1/3
pin-jointed bars each with area h t l 2 must obtain the total strains sian coordinatesx = {x I
Et
= ll~ltl4
the matrix
kN2
may be represented by three
. For application of the TR1M3 element one
in terms of the positions of the nodes expressed in the carte -
x 2 x3}. Since /aJ l/3s/~
are the lengths of the sides of the un-
strained triangle - including however any effect of temperature considered uniform over the element - then For the side
to~
,for example, the strain is,
73
,,o,:
: (I .146) t 2 [ J1¢23 ,1Nt23 - l(x ]
2tJ |
where etc. and
(1.147)
When using an i[ncremental procedure it is useful to have E t
E t
::
where Et0
in the form 0 .]48)
EtO ÷ ( ~ E t
is the value at the beginning of the step and
due fo displacements
GEt
is the increment of ~t
U = { u I u 2 U3}
Thus
1
t
t
Etc~
(I. :,49)
where
1~23
:
0.15o)
U3
Expanding (I .149), we obtain
(t .151)
eta =
The vector ~t
moy be written concisely as
74
1"/23]
~tp
Et
Eta,}
= Et0 +
(1.152) U12 J
where
rK23 '%
u~ =
"12J
FU23
U31
-ld
(I .153)
and
U31 ] = AU l
f1.154)
u12j where
0 3
o3
-13 f1.155)
13
o3
and remembering that,
u~ u~}
Then
'~,
=
+
'~,o ~;["J"
1
-s-uJ]4*'~
0.156)
Now the vector of nodal forces in frhe directions of the cartesian axes may be found by direct resolution, or by the virtual work on an increment in
u I as,
u~--{u, .2 .3}-{uf Ul~ u~ u~ u~ u~ u~ u~ u~} (I. 157) : .at[ ,.~ • .~,] x -~ t-,
75 Finally with
PN from (I.145) and
UZ
=
,~[-,, ~rt[j(z/
+ Thus U u2
+
÷
Et
from (I.152) eq. (I.157)yields,
,,,]
t -2 ~ N 2
Et0
uz/]
~-2
II-2
J'.~
(I.]58)
t ~" 1
["~
T"J]'~",
turns out to Be a cubic function of the nodal displacements. If is negligible
U
becomes linear in
~l~ t llzl I-2 kN2 + ~tMz~
!-2
In small strain theory [4]
I,I
I~tO + ~ t Uzl
kE
is small so that
I - 2kN2 EtO (I .159)
kN2 | - 2 i l g W U Z
the first term is the initial load vector
corresponds to the contribution of the geometrical stiffness tangent stiffness
U
and reduces to,
11"6
JZ
, the second term
and the third to that of the
at the beginning ofthestep. Theferm ~ t u z /
1=2kN2EtO
is inconvenient for direct step-by-step solution of a load increment since if is required in the form of some matrix
/"
I-2kN2 where the suffix
Et0 0
post-multiplied by
-
UI
. To find
T we have from (I.145),
/1.~ 1 4 ~ 0
(I.160)
refers to the beginning of the step. Then,
.~ A;~ t -~~o B
(~o/~o
I~) - ~
m
03
- #~ot~
4ot3
( ~o / ~o ~) ',~
'~o 13
03
-'% 13
1,12
( ~o/~'~o l) e12
-~oI~
~ol~
o~
U 3
(1.161)
76 where
(I. 162)
etc.
Hence
' ~ " , I~ i"N2~o - ~0 "I
f1.163)
where J
- ~o13 9
-~oX3
P2o =
-~o~
(I. 164)
- ~oo~
with this notation eq. (t.159) becomes
l-2klV 2 l-2~t4 ~ ]Ul
0.165)
which is equivalent to the Form,
(I. 166)
7.1.1 Higher Order Strain Energies The cubic strain energy for an isotroplc material as derived from
~3
in (I .105) may be
written t
(I .167)
77
/~N3
where
is a symmetrical (3 x 3) matrix. The quartic strain energy can be written in the
form
-1 k;',/6 Et
I
where J~N14 and
k~.
E~A -~ IE t E tt kN4 -2
mI
E' t
(1.168)
are also symmetrical. In general the energy expressions may be
written as polynomials of increasing degrees in
e't~ , et/'J , eta'
. Thus, in general,
one may express the vectorial derivative of the ruth degree energy ~m of the TRIM3 as
(I .169)
= kNm IE t
kNm
is not unique except in the case m = 2.
the symmetrical special secant modulus
However it may be conveniently defined as
kNsm
b~tween zero strain and
let
. This is
equal to
kNsm where
klqTm
1 m-1
=
0.]7o)
kNTm
is the tangent modulus at
polynomials of degree (m - 2) in the ct
le t
. Note that the elements of
kNsm
are
's. The analysis now proceeds as after eq. (I.143)
and one obtains
u~ = at[N~
+
u~]
l-2kNSmEt
(1.171)
This may be rewritten as,
Oi = , ~ t
NzI t-2kNSmO Et 0
+
4tuzl |-2klVS m I t (I .]72)
+
a tx~, I-2[ kNSr~ Et - kNsmo '~t0]
When the increment U I
is small eq. (I.172) is linearised as,
?8
u: :
,et~ t- 2%smo ~to÷ m',,,d m-2%s~o~to
which with (I .163) and (! .152) becomes
(I. 173)
zl The derivation of
was given,
for m = 2, in eq. (I.163) and is repeated here for
mo
convenience .
0.174) where
(I .175)
P
mo
and
(¢o)m : (P°O)m/~oo'o Note that eqs. ( [ . 1 7 l ) , ( l . 1 7 2 ) a n d (I.173) are valid for a material characterised by
k
(I .176)
® : 2]~m m=2 with
k
m
in place of
m:2 and
k
J•NTo= m : 2
kNTmo
efc.
P
/99o
(I .177)
79 The assembly of ~he global equations for the complete structure proceeds as in the standard linear case. 7.1 .2 Anisotropic TRIM3 The strain energy function for an anisotropic material may also be approxlmated by a polynomial in the ¢t
%. Other strain energy functions may be used but they will not result in a polynomial
form of the displacement functions and for this reason are computatlonally inconvenient. Once the strain energy function is fixed, which is of course the most difficult and highly technical question (see ref. [ 1 , 2 ] ) the analysis proceeds exactly as before. 7.1.3 Incompresslbl e Materials Although no really incompressible materials exist, in the case of certain rubbers the assumption of incompressibility is justified and circumvents the computational difficulties encountered with the nearly incompressible case. For the TRIM 3 element the incompressibility condition is most 11
conveniently considered by first supposing that each element has a constant internal pressure O"h . The area of one element is
: ~o(~ " 2~,)"~(~ + 2 ~ ) ''~ so that
.£2 = ]'2 0 ( 1 4- 2 J t , If the fluid constant pressure is
4 J2) 1/2
6"/~'
(1.178)
the potential of
0"/~' is
Then the corresponding natural loads are
PN0 = A I-1
8 ( - ..¢2o'/~' )
I -t. a f1.179)
= - o,;' ~.l-'
~-]
,a
Hence we find
: - G,;, ~.,c2 A , - ' A o ' [ ~ ÷
[ E ~ A ~ - Jr~] E,]
f1.18o)
80 The nodal loads, following eq. (I .157) are therefore,
u~. : ,a~[,,,,. ,,, ]~-~ I -~ eNo (J .181 )
• [E,~ ,, ].~,1 This expression must now be added to the R.H.S. oF the equilibrium eq. (1.171). The constraint equation for incompressibility is, J1
+
222
=
0
(I.13)
or
[~" -~ t";~
- "~ ]1 ~'-,
Thus when the constraint condition is applied the quantities and in this case .,r2Z/J2
reduces to
0"/~~ 320
,~,
in (I.182) must be regarded as unknown . The complete set oF equations for the
incompressible membrane element is therefore,
u~ (I .183)
with the constraint conditions (I.182) . When the increment U Z. is small eq. (1.183) is tlnearls~d in the same manner as eq. (I.173) in conjunction with (I .177), and becomes
u~ = [~o-~" ,~o~"~o"~'~ r~"o ~e~~o r~', "I] "i hz~
~'""'
81
where
(I .185)
-¢o~
-~,~
(~+ @ I3
a
with
~:
{~,, ~
4~}:S~oi~sliT[-~+lE~*l;7-1~]'<,o]
(I .186)
At the end of the step |1
#!
(i .187)
It
a/,o + ~4~
The Iinearised constraint condition is,
[,'~t A0-I -2.b,q? [A;; E~- i~]] I ~,,;au, : 0
f1.188)
The linearised equations may be assembled by the usual Boolean process - each ~t;b will appear in three sets of equilibrium equations. It should be noted that the hydrostatic stress
0'/;'
will not,
in general, include the total hydrostatic component of stress in the material. It is additional to the stress
0"/;
obtained from eq. (I.1!2) or (I.113) for isotropic materials.
7.1.4 Mass Matrix The consistent mass matrix for large strain TRIM3 elements will be exactly as for the small strain element. Thus with ~
rn where
as the displacement vector of the element the mass matrix is~
= btr~eb
f1.189)
82
2 1 1] = --~goQo
t
1 1
2 1
ft. 1~,o)
1 2
and
={ (I)(/,) (7) (2) (51 (81 (3) (6) (9) }
b
(1.19~)
where the numbers in the brackets represent the position of the Boolean instruction 'one' in the relevant rows. If the simpler lumped mass is considered adequate this is
= ~" 9 0 "~0 t Jl[g
(I.189a)
7.2 TET4 The TET4 element may be treated in an analogous way. It is important that the ordering of the Et
vector should follow that of eq. (I.32) and that all other relevant matrices such as A , u
,
efc., should also conform to this ordering. The volume is more simply expressed
in terms of the nodal coordinates than in terms of
Et
Thus
(I. 192)
The potential of the hydrostatic constraint pressure is,
- VG" h
(I.t e3)
and the corresponding nodal loads are
v22 ~3 v~ v32 v33 v~ v~2 v~3 } /,.,,,,~
83 where
V/j
is the co-factor of the element
UI~
integration of dynamic problems
j]
of the determinant
V
. For small step
is conveniently left in the above form. When llnearised
one may write
F uz a u~
I~ ]
The elements of
fl.] 95)
z
are either zero or of the form
aU i~U~r 0
The constraint equation may also be written
[° 1ou, "[
lu,. ......
so that the linearised form is,
"~u~ ° uz=O
(1.197)
7.3 TR I____AX3 ,Axlsy mrne,t ri c Referring to 4.1
the principal strains will be
¢1 '
¢tn
in the axial plane and
¢]I
in
the circumferential direction. For simplicity the equations will be written starting from zero initial strai n. Thus
~u:
"7
" -T
1
1
: ~'T [ 1÷
0.] 98)
u 7
2 x,] u,
Also X1 =
ILl ,I~[ 1
;
U1 =
tr~ U 1
(1.199)
84 where
: and
Xl
ul
]
(1.7~)
are respectively the vectors oF ~/ and
u.1 at the nodes
1 2 3. The strain
I
vector at any point is now,
E
= { eta
6tp
~]] }
-ct~
The strain energy density for the quadratic case may be written
where
# J
:
I~1 + 611
'
p J2 : el E:lI
(4xl)
(I.2,~)
in the form,
are strain invariants in the axial plane
which can be expressed (see Section 2 ) in terms of the natural strains
~:t~
etc. Thus (I.201)
may always be written as
£~ :---~ et~N2m where
kN2
(I.202)
is a symmetric square matrix independent of
densities can be expressed as polynomials in
~t~ ' E~# j ~f~
E
. Higher order strain energy and
el! .
uI
now takes
the form
(I .203)
and ~
=
etu
z
where
[ t],,]
(I .204)
(I .205)
-
JL-I, uxt
I ~,uT]~,b 2 ~xl J
8S and the Boolean matrix,
b
l
=
Then
I
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
]
(I .206)
(I .207)
To obtain the force vector Now since
But
kN2
[
UI
is symmetrical the increment of
1
,/Tt
i -2
z
[
,st
u~!
-2
~
is,
' I
I
I ~U t
a~
one must first calculate
2
(,,,.,)~
]
' I
II I
Therefore,
a~
-. = ~t~N2 a u I ÷ I[-{-atud.'[ "2
( 3 ~ ~ .j""~" " (1.208)-
.. ~t ~N2~ IuI
86
where
&
:
~ {~xl
~x
(!.2o9)
I J
Note the agreement of the first sub-matrix with the TRIM 3 result (eq. (L.158)). The element force vector
UI
is, q
UI = 2~T.p.o/wxl
cq~
a U~. d~ld~2
When the energy is of a higher order one may express ~
(1.210)
as a polynomial in the entries of E
Adapting an argument akin to that used for TRIM3 the energy is expressed in the form,
fl.211)
= 1-" EtkA/sm m e: where kNs m is the special secant modulus defined in (J .170). Then
1
4
t
m
:
= ~ouza
t
i'Nsm" kNsm""
But
6(l(Nsrnm} : kNTrn6m = {m - I )kNSm6~ -
( m - I ) kNsera
u z
87 Therefore,
t-t
Thus
8~
t
a %s
"z
(I .212)
which is of the same form as (t .208). The equation for I[II
, (I .210) is valid also for the
present case. The stresses are most conveniently obtained from the principal strains and their directions. ~I
and
E.II[
are found from
~'l~r J E:I,~ ~ ~t;r
Thus
j,,2]
(1.213)
with
VJ' = -~tan -1 2
Ecz sin 2i9 - ~cp sin 2~" + ~'cpcos2~" + ~ca.COS2/3
(I .214)
88 8. References 1 Varga, O.H., Stress-Strain Behaviour of Elastic Materials, tnterscience Publishers,Wiley, New York, (1966) 2 Green, A.E. and Adkins,J.E., Large Elastic Deformations, Clarendon Press, Oxford (1960) 3
Oden,J.T. and Key, J.E., Analysis of Finite Deformations, High Speed Computing of Elastic Structures, Tome 1, University of Liege (1971).This reference contains a usJul bibliography.
4 Argyrls,J.H., Recent Advances in Matrix Methods of Structural Analysis, Progress in Aeronautical Sciences, Pergamon Press, (1964) 5
Prager~W., Introduction to Mechanics of Continua, Ginn and Co.Boston, (1961)
6 MurnaghQn, F.D., Finite Deformation of an Elastic Solid, Wiley, New York, (1951) 7 Fung,Y.C., Foundations of Solid Mechanics, Prentice HalI,Englewood Cliffs (1965) 8 Argyris, J.H.~ Three-dimensional Anlsatropic and Inhomogeneous Elastic Media Matrix Analysis for Small and Large Displacements, Ingenieur-Archiv, 34, 33-55 (1965) 9 ArgyristJ.H., Continua and Discontlnua, Opening paper to the Air Force Conference on Matrix Methods in Structural Mechanics at Wright Patterson Air Force Base,Dayton, Ohio (1965).
89 1I = OPTOP
= rVr
I~ll, i'll I - -
--
~f
8
x
P
I 2'
Fig. I - 1 Two-Dimensional Strain
Fig. I - 2
'~
2
Large StrainTRIM3
4 x3
uS~
2 u
13
~ xI
½
= finat length ~'12 = 112/112 %12= T(~iz -t)
x 2 is angle tSz iS tangential 1
Fig. I - 3
Large Strain TET4
Fig. I - 4
Axisymmetric Strains Coordinate System
Fig. I - 5 Virtual Work of Natural Stresses ~c
go tl.
NON-LINEAR DYNAMICS OF STRUCTURES
1. Introduction These notes will attempt to give an account of recent experience [ 1 , 2 ] at 1SD in the field of structural dynamics. Although modal analysis by the DYNANpackage ~3] has been thestan dard method of treating large scale dynamics problems its application is limited to linear problems. The methods to be discussed here are based on step-by-step integration algorithms. These algorithms are all single step and work in terms of the inertia force vector and its time derivatives at the beginning and end of a step. Thus they are implicit methods and in general each step is solved iteratively. No excuse is made for concentrating on this type of algorithm. Our initial experience in applying these algorithms encouraged us to limit our efforts to improving and extending them, rather than making an attack on a broader front which would include explicit methods, extremum methods and weighted residual methods. In fact some of the algorithms developed here may be obtained by the latter method. Non-linear problems are very often treated by incremental linearisation. One of the byproducts of the present investigation has been the development of large time step integrating algorithms which are a powerful alternative to modal analysis of linear systems. By large time step we mean a step of any number, perhaps hundreds,
of times the least period of a dynamic system. Exten-
sion of large time step methods to non-Iinear systems is difficult for technical rather than theoretical reasons.
Broadly speaking one may say that to attempt large step solutions of non-llnear
systems is to swim against the tlde. The solution of a large time step is similar to the solution of a non-linear static problem by direct iteration - t h a t is~ not incrementally. Nevertheless if really large time steps can be achieved the effort may b~ worthwhile. Although the large step algorithms were originally developed as modifications of the small step algorithms their practical implementation is so different that the two types of algorithm will be considered separately. 2. SmaJJ Step or Conditionally StabJe Algorithms Starting from the equation of motion
i:
:
where Jr" =
(IJ .1)
M-117(r,i-,t) displacement vector
R
=
inertia force vector
M
=
mass matrix
we represent R
in terms of the values of R ,
interval of time
z" : t t - t o
. Thus R
/~
, R
etc. at the begi'nning and end of an
is represented by Hermltian interpolation polynomials
gl of order 2 , 4 , 6 - - - 2
n corresponding to polynomials which are linear, cubic, quintic - - -
(2 n -. 1) tlc. The first member of the family n = 1, is the well-known l i n e r acceleration algorithm. The second, n = 2 r is the algorithm first used in references [ ~ 4 ) , ,
where the term "finite ele-
ment in time" was coined by analogy with the cubic beam displacement function of standard finite element theory. Integrating eq. (1) twice one obtains tl
....
]dt
t t
(ll.3)
=
where the ~
are the Hermifian interpolation polynomials, The first four algorithms are sefout
in Tab!e I . If R
werea polynomial in
#" , r ,
r
t
of degree (2 n - 1) the nth algorithm would integrate both
exactly, Since R may includea forcing function [ ( f )
should be represented bya (2 n - 1)tic or less in t
it is clearfhat
. Impulse loads should be applied at the be-
ginning of a time interval and require that velocity discontinuities are included in the program. However real impulse loads are not meaningful in the context of finite step integration and in practice all loads should be applied over at least one integration step. This avoids special consideration of impulses. 2.1 Stability of the Algorithms When the algorithms are applied to a simple oscillator
F ÷ W2/" = 0
(11.4)
one has
~; = f? = - ~ j 2 r so that
#
:
_ wz/:
#
:
(/.)4/:
Thus all derivatives of R
,
#
:
etc. are expressible in terms of
r
or J" and the algorithms lead to
[ ][::] [:1-i::] A 0 ÷
A e
01.5)
g2 The matrix A 0 is the harmonic operator
AO = [_COS eJl"
u,,-lsin
(# sin u~1-
wr]
O).6)
cos ~1- j
and A e is composed of the First terms of the error matrix. The error matrix For the First and second algorithms are
(11.7)
2 and
1
F. W61-5
]
/ "~" l"
L For the nth algorithm A e
is of order T 2n +1
(l1.8)
. The error matrix gives a good idea of the
accuracy of the algorithms applied to linear systems with small time steps. It may ba shown that the eigenvalues of the matrix A
have modulus equal to
about half the period TO = 2 ~ / u ~ l" = 0.503 To
for integration step T less than
. For example, n = 1, z" = 0.55t TO ; For n = 2
For greater values of T
reason called conditionally stable. Quite
unity
,
the algorithms become divergent and are For this
independently of
the time step required for sufficient
accuracy of the response in the main modes of a multi-degree o£ freedom system, the maximum integration step is determined by the highest frequency mode of the system. 2.2 Implementation of the Algorithms Because of the limitation on the integration step discussed in the previous section these algorithms are most suited to a direct Jacobi type iterative solution of each time step. The time step for convergence of the algorithms with Jacobi iteration when applied to a linear oscillator is o£ the same order as that required For stability of the algorithm. Thus the allowable maximum integration step is again conditioned by the highest natural frequency of a linear system, or in the case of non-linear systems by the highest pseudo frequency corresponding to the current tangent stiffness. For the simple oscillator of eq. (11.4) one may obtain by inspection from Table 1 the equation
g3
=
+
r~
c
d
terms not involving i'1 ÷ t](l I
r~
Convergence of the Jacobi iteration requires that the eigenvalues ~. of the matrix have modulus less than 1 . The second algorithm, for example, gives, a:
and
~21:2
,
b :
= W2'r2//,'V~
-T ~r
,
c =-~2r3
,
d
:
~2r2
. Then " r / T O < 0.63 and for rapid convergence one requires "C/TO
less than half this - say " r / r o -: 0.3
. The criteria for the higher algorithms come OUt almost
the same. A number of variations of the actual implementation of the method are discussed in the references quoted. Experience has shown that the most economical and accurate procedure in non-linear systems is to carry out the iteration without incremental linearization. This avoids the necessity to store the global tangent stiffness matrix - only the element topology,geometry and material properties have to be stored. In most problems the inertia force R
R
where
=
- R s - Ci"
is of the form,
+
f(t)
Ot.lO)
RS = internal elastic force vector 1;:
= damping matrix, generally assumed constant
f(f) = applied loading. In linear systems #~j = K r
where /'(
be considered as Ksr where K s
is the stiffness matrix. In non-linear problems R S
is the secant stiffness matrix which is a function of r
eq. (11.10) may be written without using the global matrices C bution of each element
Rg
where ~g
=
g
and K
or K s
. Now
. The contri-
to the internal elastic and damping forces may be calculated from,
- k0sg0
-
o
:
-
(II .11)
-
and Pg are the element nodal displacements and forces and kgs
modulus of the element g
may
. The time derivatives of R
for application of all but the first (]inear) algorithm.
and therefore of Rg
is the secant are required
g4 Thus
-~
Rg: -
- ,:,,~,,--(k9,9~,)"
-,:g~g (11.12)
_ kg, C,g - ,:gOg
kg t is the tangent stiffness matrix of the element. For the large displacement small strain case kg t is ec,,uivalent to the sum of the elastic and geometric stiffness keg + kC~,, where
For ll~g one has,
R,,-Note that
('Ch. llt)
-~
- ,:gg,, : - ~,,,,9,, -
%~9~- *~
~,,.~>
kg t is best left in this form.
Continuing in this way the higher derivatives may be derived. Assembling the Rg etc. by premultiplication by the Boolean
location matrices a~
, and noting that,
~g = ~lg r we obtain by summation over
R :
Z agR~, fctl t
(ll .14)
g
= _Zat g
t
(11.15)
g
g
g
01.16)
g
g
+ f(t)
g
g
95 Equations( It. ] 4) to (11.16) are equivalent to the global expressions •
R
=
-K~r
-Ci'÷
R
=
- I,(ti-
- Ci-
= - KtE- - Kti"
f(t) ÷
(l1.17)
f(t)
- Ci,:,
i(t)
The global equilibrium equation is
R
= /v/~
(11.~8)
and this equation must be used to substitute for the derivatives of r-
higher than
IP in the
above expressions for R , R
etc.
fore greatly simplified when M
is a diagonal lumped mass matrix. I t w i l l in general be easier
The element wise programming af this procedure is there-
to carry a greater number of lumped mass freedoms than to use a smaller consistent mass matrix. A previously developed procedure for solving non-linear problems was the so-called /'('~ method discussed in ref. E 1..] . This method uses an effective global tangent stiffness which is the mean of the tangent stiffness at the beginning and of a time step. Accumulation of the increments of the internal elastic loading R S is used although it is also possible to incorporate a periodic check of R S calculated directly from the current displacements. This procedure may be the most economical for problems with less than about 600 degrees of freedom. Each step is again solved by Jacobi iteration. There are several other variations possible in the detailed implementation of the small stepa]lgorithm. In Fig. I1.1 areshawn three possible flow charts t w o o f w h i c h refer to non-linear and one to linear problems. For very large linear systems the nan-linear algorithm without linearizafion (Fig. 11.2) may often prove the most accurate and economical. Some examples of applications of the algorithms are shown in Figs. II- 3- 17. 2.3 Non-Structural Applications of the Small Step Algorithms Although the algorithms are especially suited to multi-degree of freedom structural systems their high accuracy and ease of programming should be useful in other applications. For example, the classical n-body problem is rather like the problem of masses connected by non-linear elastic cords. The accuracy achievable in thls type of problem is demonstrated in Figs. I I - 1 9 - 2 2 . Another problem is that of elastic or elasto-plastic wave propagation. Figures 11-23 -33 show some examples of applications in this field. Problems depending on systems of first order differential equations may be solved by use of the first lines only of the algorlthms in TaBle I, with Ik replaced by r
, R
by IP and soon.
g6 3. k,Qrgj,e,,lStep or Uncondlfiona,!ly
Stable Algorithms'
Our interest in large step algorithms was stimulated by a paper by Goudreau and Taylor ~6] They showed thaf the Newmarkaverageaccelerafionalgorithm, which may be considered as a modification of the linear acceleration method, was superior to other existing algorithms. The linear acceleration method is the case n = 1 of the previ~'us section. We therefore attempted to modify the higher members of the family in order that they should have unconditional stability. It was noticed that the Newmark algorithm when applied to a simple oscillator of circular frequency always gave the following relation between the vectors {q
[ cos.
/=1} and { r o
#O} "
[ro]
- w sin ~
(l1.19)
c o s ~p
where "-~ Wl" ~p =
2 t a n -1 I _
For small time steps
~ = ~'r
(l1.20)
(wt) 2 but when l"
is large the effective period of the oscillation
is increased. The total energy, howevert remains unaltered. It was found possible to modify the coefficients of the second lines in the algorithms of Table I so that all of them reproduce eq.(l1.19) when applied to a simple oscillator. Afterwards it was realised that the synthesis of the algorithms could be described very easily as follows. Calculate used to calculate
~:ro*
rl
~ 1-
~
by the same Hermitian quadrature as is
. For example, we give in detail the second member, v i z .
T [Jro ÷ J~l] ÷ ~
T2M
q [r O-
rl]
(11.22)
Substituting for #'1 from (11.21) in (II.22) one has
,i-
'
r0, ti'0 ÷ T
2M-'[z
R0+
+
I t3M4
[*0-
The complete algorithms are set out in Table II. Note that for some purposes it may be more convenient to use the second algorithm in the form of (1t .22) rather than (J1.23). In Table III a
summary of some important properties of the algorithms is given. Fig I I - 34 shows curves of the period elongations together with that for the method of Wilson.
g7 Naturally there is some reduction in the small time step accuracy compared with the conditionally stable algorithms. The error matrix for the second member, n = 2, is
[ :1 T
1
Ae=
W6T 5
~
(11.24)
1
which is of no greater order then the A e o f e q . (11.8). It was noted that the conditionally stable algorithms integrate exactly forcing functions up to the power
(2 n - t) in the time. The present
algorithms will integrate the velocity exactly but will give the displacement correctly only up to the (2n - 2)th power of
t
Compared with previously developed algorithms, known to the authors at least, the algorithms for n = 2~ 3 and 4 are remarkably accurate. 3.1 Implementation of the Algorithms The implementation of the large step algorithms is ~:ntirely different from that for the small step algorithms. The large time step precludes the use of the Jacobi ;terative solution and it is necessary always to farm the global matrices and to solve each step by a direct method such as elimination or matrix inversion. The algorithms are therefore ideal for linear systems. It may be shown that the algorithms retain their unconditional stability for multi-degree linear systems with or without damping C
. The damping need not be proportional to the stiffness or mass matrix but
it supposed positive. When
R'
is as in eq. (ll.10) all algoHthms lead to on equation of the form
O1
= O o
+ F
where OI 1 , 010 a~e square matrices of order(2 n x 2 n) and F D 1 , D O and
(11.25)
is a vector of the toad terms.
~F" are given in Figs. 11.-35,36 for the first three algorithms. Some examples
of applications to linear systems appear in Figs JI-37 - 42. 3.2 Algorithmic Damping Many authors have considered algorithmic damping to be a desirable property of algorithms designed for mulH-degree of freedom systems. By algorithmic damping we mean the eventual disappearance of the harmonic components of the higher frequencies of the system. This is considered desirable for two reasons. Firstly, because higher mod~s are often inaccurately represented by the inevitable discretlsation of the system. Secondly because the period elongations of the higher frequencies becon~e so large that~ although the maximum amplitudes of the high frequencies are present1 their occurrence in time is completely different from the exact solution .However, we
98 have not introduced algorithmic damping into the present algorithms. The reason is that the #pplication of the algorithms only Becomes meaningful in relation to the loading function. From .'he pseudo-indicial admittance curves of Fig. I1 r 43
, one sees that for large integration steps
compared with the period of the oscillator the high frequency modes will not be excited appre ciaBly if the changes in the loading function occur over at least one integration step. Real step functions or impulses will excite the higher modes and should not Be present in the load function. Such ]oadings obviously require very small time step if their response is to be accurately represented. If for some reason algorithmic damping is desired it can be introduced by weighting the terms i~1 , R 1 , Jl~1 ,etc. more than r o , R 0 , R0
in the algorithms of Tabl~ II or eqs .(l! .21)and
(11.22). The effect of this can Be shown By considering a general algorithm of the form (single degree system)
= ro ÷ A [A/=o÷ ( 2 - A ) ~ ] +
r1
= ro
B[/X?o -(2-/X)rl]
÷
c
.
.
=
.
.
.
+ A[X/'o+
(2-A)fl]
+
B[,,1.~o - ( 2 - ; L ) / ~ ]
+C[A? o+
(2-A)~]
+
.....
where .4,B,C, etc. are dependent on the tlme step );
.
(,,.26)
l"
(11.27)
. Applying this to the simple oscillator
w2r
(11.28)
we obtain
[1+ ( 2 - ~ l x ]
i-~ - (~ ÷ , t x l , ~ o
- - Jv[~,-
o . (2
(li .29) (2-~lY/-~.
~Yi- o -
[~.
(2-~lx]q
- (1 + ,1.X)r o
where
X
Y=
.,.
A
-JB
4- w z ' D
+
......
- w2C
4-. ( f l 4 E
+ ......
fl1.3o)
99
Solving for ";'1 and r 1 in terms of
~2
÷
/:O and r"0 and writing the total energy
wZrz )
(11.31)
one obtains
:'
Thus,when
3. =1
as in the algorithms of Table II we have
when the coefficients
A,B,C~ efc.
~1 = ~O . It may be shown that
have the values of Table II (or values of the same order of
magnitude)
l,,.33/
acccrd[ng as 3. -~ 1 It follows that our family of algorithms may be modified for algorithmic damping by simply taking 3.
a little less then unity. This will reduce a little the accuracy of the tow modes and that of
the free mass response to a forcing function. It may be noted finally that if we modify only the coefficients in the last term of each line of the algorithms the algorithm becomes convergen: for large "r but there will Be range of
"r" for which the algorithms become divergent. This may
not preclude the use of such algorithms because in the range of theoretical divergence for a lower mode the truncation error will be in any case very small. 3.3 ApplicaHon ~'o Non-Linear Systems The unconditional stability of the algorithms is defined only in relation to linear systems. It must not be expected, therefore, that these algorithms will perform better than the conditionally stable ones when applied to non-linear systens. In fact their small time step truncation error will be greater. An idea of their behaviour with non-llnear conservative systems may be seen by applying the first member to a non-linear oscillator (for a unit mass)
Y
=
(11.34)
- R S{F}
Thus -
*
1
•
(11.35)
100 and eliminating
T
one has,
-
,-o)(Rso
(11.36)
÷
Thus the change in kinetic energy does not equal the change in potential energy - see Fig.I1-44.
Although an individual time step will never become divergent it is possible by changing the integration step periodically to arrange for arbitrary in-put of energy to the system. The higher algorith'ns lead to similar but more complex energy balance conditions. The general conclusion must be that the present large step algorithms must not be applied to non-linear problems. 4. Large Step Algorithms for Non-Linear Problems There are many structural problems in which the non-linearity is elastic and in which the force displacement relation is well approximated by a polynomial. The non-linear problem of Fig .11-3 is approximately cubic for reasonably large deflections and materials with a polynomial strain energy function of the large strains (Green strains) lead to polynomial load deflection relations. 4.1 Single Degree of Freedom System The equation of motion of a non-linear undamped system may be written
Y = -R s+ f(t)
(..37)
If the displacement is given in terms of the velocity as in the linear algorithm
q = r0 ÷
(l1.38)
r(i'o + h)
Now suppose that exact energy balance is maintained. Then
2 where Rsa v of
r
and
~v
are the average values of
. Since from (11.38) i
one has, dividing the two sides of (l1.39) by ly,
considering them as functions
RS
ro ) I (~0 ~ h}
and
-~- I r o - r 1 )
respective-
101
Equations (11.381 and (11.40) constitute the algorithm, but in order to apply it a meaning has to
Rsa v
be glven to
RSav(rl
and
fay
- rO) :/'Rsdr ro
OI.41) ~r
r
=
[/?SO + r0
fR] dr]dr
Rso(r, - to)+
Jf R~dr dr roro
r0
where
R' S"
dR~
:
dr
Kt
(11.42)
is the tangent modulus. The double integral in (II .41) may be evaluated to various degrees of precision by Hermitian quadrature. Thus, if
Rs
is a quadratic function of
r
,
IF RS is up to quartic in r ,
R~av- Rs0 ~ ( ½ K0 ÷ ~ K~)(,'l - r0) 1
÷ (-~Ko-
-~1 KI')( r~ - r0) 2
(It .44)
Note that the coefficients in (11.43) and (11.44) are as in the second lines of the conditional algorithms of Table I, and higher order approximations may be obtained immediately from this Table. For
or
fay
we may use the approximations
102
1
1
~- ( ~o • ~) + qT , (~o
~-
which amounts to replacing the
average
f's
~)
~,,.~
over the displacement by those over the
time, In conjunction with
(11.43) and (11.45) the equation (11.40) becomes,
~ = I'0- T[RSO + ( + K o + If we substitute for
-
+
/:'1
- 61- K , ) ( r l
-
to)] ÷
+~(fo
+
f,)(,,.47)
from (11.47) in (11.38)
'-L
-
3Ko
÷
+K,)(~,
Equations (1t .47) and (11.48) may be extended to multi-degree of freedom systems but a more accurate algorithm will now be considered which represents an extension of the second algorithm of Table II to non-linear systems. We consider a free single degree of freedom system and assume that the equation for the displacement is obtained as before by Hermitian quadrature
"~ ; ,o,
f
~ (,~o + ~,) " @ ~~( i'o - ~'~)
(,..-)
with
f
-
-Oslr )
(tl.50)
For energy balance we have
'--2 (~
- ,:?) - - R~.,.(,-~ - ,o)
(H .5])
Equation (11.49) may be written,wlth (11.50)
r~where K c
ro
1
: TT(~o÷
r'1) + 1J2- ~-2Kc(r~ - ro)
is the chord stiffness modulus between
r0
and r 1 . From (11.5])
(11.52)
103
(II.53)
Multiplying each term of (11.52) by one side oF (11.53)
rl
::
Substituting for
1 .C2Kc
f'O-'rRSav /'1
12
(11.54)
in (11.52) one has,
1 l-2 rl = ro + "c£0 - --~ [Rsav ÷ + ( R s o _ 1..!_ ~ 3 K c ( f
0 _
24
Now
RSav
RS1)]
(l1.55)
fl)
and Kc must Be expressed wlfh consistent accuracy. Thus if we wish the energy
balance equation to remain accurate for up to a quartic toad dlsplacment law
RSav
wilt be
expressed by eq. (11.44) and
K~ :: 7 where
K'
=
I
( Ko + K, ) ,,
I
,
~2 ( Ko " K; ) ( ,~ - ro)
(,,.~
dK
dr If the loading terms are included as in the linear case the complete algorithm is
,[R 0 + fRc (11.57)
+"T
(It .58) where
Kc
appears in (ll.56) and
104
"/ R = WKo+ } & + ( ~ K ; ,-
An alternative, but equivalent, form for r 1 in (11.57). When the system is linear K c and
1 TgK()(firo)
(,,.sg)
may be obtained by substituting for J'l /~
are both equal to the stiffness
K
in (11.58) and the
algorithm reduces to the standard unconditionally stable algorithm with n = 2. A flee oscillator with a conservative load displacement relation up ta quartic will give a bounded response when integrated By the above algorithm. As in the linear case the period will be alongated. When the integration time step is very large eq. (ll.58) showsthat the static deflection provided
ro
already corresponds to
f0
ri
will tend to follow
. However, it cannot be can-
cluded from eq. (11.57) that /~1 also tends to follow the rate of change of the static deflection as it does in the linear case. Ideally if this condition could be satisfied one would be confident of the behaviour of the algorithm for really large steps. Nevertheless the algorithm should Behave well far the free system and also for the extren~e case of a Free mass. To extend the use of the non-linear algorithm to multi-degree of freedom systems it is necessary to define what is meant by /~
and
Kc
in this case.
4.1 Multi-Degree of Freedom Conservative Systems If the vector
Rs = Rs (r)
(11.6ol
is written in the form
R s = R s o + ~rc[ r~ - to]
~H.6,1
where K c is a square matrix there are in general many ways of writing K c . However we require K e
to be expressed in terms of the states r 0 and lr1
• Now
/r
Rs= RsO+
(Ii.62)
f l~tdr
which is independent of the path of integration. K t is the tangent stiffness matrix. The integral in (11.62) may be written with various degrees of approximation and by analogy with the single degree of freedom eq. (11.55) we take
/1
2I
÷ M,]÷
|
rg)[ m=l
0
K 'I ]
(II ,63)
105 where
:[ .,] and
r m
(11.64)
r=~
is themth element of the vector r
• Similarly
Rsav
may be defined by the relation,
Jr"
;d rt K dr - [,~ - ,'o i t R,o
~-[~ -,~]',~ [,~ -,-o]
Thus
~Sat/ = ~SO + where n
E ( ~ m- ~ K
°
m=l n
(ll.6,~;)
~ 15 7, ( "~ - "if') K ,m m=l The complete algorithm is as follows
M'I .r2 -1 • -:r,~- [lo÷f,] ÷ ~ ,,, [ t o - t , ]
(11.66)
(11.67)
106 This algorithm will maintain exact energy Balance for a free system in which the load displacement relation is up toquartlc in r
- o r w h e n the elastic energy is up t o o quintlc in
r
Higher order algorithms may be obtained butwill involve higher derivatives of the /4[t matrix. 4.2 Implementation of Algorithm The algorithm of eqs. (11.66) and (11.67) will be competitive with the small step algorithms of Section 2only if it is possible to use it with really large steps. Since K c r 1 , the solution of each time step is iteratlve in the sense that culated several times. However, for given
Kc
and K
Kc
and
and tl~
the vector {1~
1~1}
K
depend on
must be recalhas to be ob-
tained by a direct method. Thus the method is similar to the large step linear case except that the matrices 1]11 and
D O are no longer independent of
r0
and
/'1
As mentioned in the introduction the solution of a large flme step is similar to the solving of a static problem without using an incremental method. If it is necessary to solve incrementally the whole object of the exercise would be defeated. It appears from the examples calculated by the method <.hat convergence is obtained with no special precautions provided the time step does not exceed about one quarter of the period corresponding to the maximum amplitude. This time step is not large by the standard set by the linear algorithm. It would mean that the presence of a high frequency and unimportant non-linear "mode" would preclude the use of a large time step. Because the large step algorithm already requires sophisticated programming an 7 further complications to ensure convergence would be uneconomic. From eq. (11.67) it may be seen that when is large and the initial conditions are
rl : But
/'re : /'12
F'I : J'g = fO = 0
then
fl
(ll.6 )
in this case. Thus the algorithm will diverge. However, the simple device of
taking the average of successive iterates will ensure convergence. In general, averaging will increase convergence when the static load deflection curve is concave upwards for positive loads. In other cases it will slow but not destroy convergence. In a multi-degree freedom system there may be opposing criteria for the various "modes". The method has been tried out on the plane net problem which was solved previously by the small step algorithm. The results for a time step of "r = 0.01 sec are given in F;g. II - 45
where for
comparison the accurate curve for "r = 0.0005 sec by the conditional small step method is shown. Above "C = 0.01 the method does not converge. This is about five times the limiting time step for the condltlo:~al algorithm. Other examples are shown in Figs. II- 46 - 4 8 .
107 5. References Argyris~J.H., Dunne,P.C.and Angelopoulos, T., Non-linear Oscillations Using the Finite Element Technique, Computer Methods in Applied Mechanics and Engineering, 2, 203-250 (1973) 2
Argyris, J.H., Dunne, P.C.and Angelopoulos,T., Dynamic Response By Large Step Integration, Earthquake Engineering and Structural Dynamics, 2, 185-203 (1973) Braun,K.A., Br~nlund,O.E., B0hlmeier, J., Dietrich,G.,Frik,G.,Johnsen,T.L.,KiesBauer, H .T., Malejannakls,G .A., Straub,K . and Valllanos,G., DYNAN Lecture Notes with Computational Examples, ISD Report No .109, University of Sfuttyart (1 971) Argyris,J.H./ The Impact of the Digital Computer on Engineering Sciences1 Twelfth Lanchester Memorial Lecture, The Aeronautical Journal of the Roy .Aeron.Soc., 74, Nos.709,710 (1970)
5
Argyr]s,J.H. and Chan,A.S.L., Application of Finite Elements in Space and Time, Ingenieur Archly, 41, 235-257 (1972)
6
Goudreau,G.L. and Taylor,R.L., Evaluation of Numerical Integration Methods in Elastodynamics, Computer Methoths in Applied Mechanics and Engineering, 2, 65-68 (1 972)
7
Haase, M., Note on Stability of Large StepalgorithmsApplied to Multidegree Freedom Arbitrarily Damped Systems, ISD Internal Report, University of Stuttgart (1973)
8
Newmark, N.M., A Method of Computation for Structural Dynamics, Proceedings ASCE 85, EM 3 (1959)
9
Wilson, E.L., Farhoomand, I. and Bafhe,K.J., Non-linear Dynamic Analysis of Complex Structures, Earthquake Engineering and Structural Dynamics, 1, 241-252 (1973)
108
gwen initial
Conditions
I Assemb|e K ° ' C ' M ' tl : t0*T 1 Compute imtia| Forces Rs0 .
t J -o:--.o-C*o'"o~
,
.o:--o.o-C~o...o~
j
t J l st Appr0x l e t
" 1 : R0" R , : R 0 * l " " 0 ;
Boo[= FALSE. }
t =
,io. ~H'12,-o-~-o-9",-2,',~
~ : -x0r , - c , ~ . ~(t~) #, -~,o- Xo~" -c,~.tt!!)
NO
J
~ : -Xo, ~ - c ~ . wlt~l # , : - ' , o - Xo~d • - c , ~ , r l
|,,o:~-L,,o.*,] Bool = .TRUE,
i
N0
Rs° : ~,o" Ko*dr
.o: ,~, -o:~, r 0 = IF1 Assemble K 0
Fig .I1-1 a Flow Chart for Non-llnear Oscillations Using Modified Stiffness Matrix with Accumulation
109
Set t o ,
i,0 and r0 according to
given initial Conditions
f f
[-o
.o :-..o--o-,,,o;]
t 1st Approx : let J~l = Jl~O ; R1 = RO * l" R 0
dr
7
J
Tk0* 6~-'C2M-l(21R0 + 3rl~O* 9i~ 1- 2"Z'RI) M-'iR,~, # l : i ' o . d k ,
rl = t o * d r
f #l
= - Krl
- C #I * f(tl)
I RI:R1 t 0UTPUT
t,r 1 ,i~ ,#',' ,',,,,}
e0:#, a;O = r l r0 = r1 tI = tt * T
Fig. II-] b
Flow Chart for Linear Oscillations
110
Set tO, ir0 and m:O according to given initial Conditions
Assemble C, AIW ~ t 1 = t o + "t"
'~o=-"'[(%%~+
% = -,,'~..o~ - C,o • t,,o~ ,
%C-o~)/'%" ¢~o" i¢'o~
,st Aop.o.:,.t .,o ~,o ~ ",="o..'~o
l:
d,= ~.,,-1(6R o. .,,o. 6R, - .e,) d.= .%. ~ . 2 . - ' ( 2 , e o • 3,R o +ge,- 2.R,) ~, : . ' R 1 , i~:~0.dl ~ r 0+d. #~ = -,,'[c%~,. ) . hGC,~))]=~-~ - cfi •ict;) ~
= -atPi.~)
- c~
......
~YES
I OUTPUT
~
+ e(t 1)
t,,rl,'1*~]
YES
~NO eo=~ ,
i0=i, ro:r~ t1 : t l + T
I
Fig. II - 2
Flow Chart for Non-Linear Oscillations Using Total Displacements
111
~x-f
m 10-~
I...i..I
"j" .....
i.
200 cm
J
tO-6
m#+Rs(r)
= 0
TO = 0.2¢',~79~69 sec i0-8
z = ~-~
IO-~
-----
F,E in Time ~K~
I=I
Wilson 9 = 1.0
sec
EE, in Time D.E.
/ io-~2 S
Fig .[I - 3
~
iS
.....
Wilson e = 1.4
----
R u n g e - K u t t a D.E
zo ~s~¢.I
Graph Showing Relative Error in the Displacement During 20 Seconds of Free Oscillation
A
x
~0 ~m]
/
%,
T-EAB "R
%B
EA B = 2.1 x 107 kgf
EA~ = 2.1 x 106 kgf ~o8 = t 8x 10~ kgf
~= rA
Fig. II - 4
= 2o~ to3
kgf
/:'Jz = -10000 kgf= - 162,9545 crn
Plane Prestressed Network Loaded Vertically
112
....
.... ~ = 0.0005 sec ~ = 0.002 sec
rA
[cm] 200150 tO0
L
50
--
0
__
o. . . .
-50 -100 -150 -200
Fig. II - 5
Free Oscillations of Plane Net. Modified Stiffness Matrix with Accumulation (Members in Compression included)
~;=0.0005 Present Method| * 1 *K .... ~=0002 P. . . . . t M e t h , ~ ] ~ d - ~ - [ H . ~1 .... 1; = 0,0005 Witson - - - - - ~ = 0`002 Wi{son
y "
Zt
~
x\
X
[crn]
2oo1 t501
/~
IL\~~
_5ot/ ~ -IO0
!"Di~ ,
I i l
/,~
\ ~! lYl's ,ii
\\4V
Fig .11 - 6 Displacement-Time Diagram of Node A according to Various Methods (Members in Compression Eliminated)
113
y
0.55501 0 5
4
9
9
(
k
,
~
~
~5450
110
0',S
Fig. II - 7
115
~'l[sec]
Calculated Maximum Eigenvalue of the Iteration Matrix with Respect to Time
Y EA B = = EAi
[Metric7 t30 500 L tonsJ Metric
22000[ tons]
Fig. II - 8
- ~ n rMetric]
PNoB . . . . L tons.} = J.n [Metric] PNoI "VL,onsJ
=60m
Three-Dimensional Prestressed Network (Eastern Grandstand of the Olympic Stadium in Munich, 288 Unknowns)
114
~J
Mf*Ck+
[cm]
Rs(r) = 0
~
A"
~
30-
20I0-
I
~..
0.4
___
03
o
-10 - -
-20
.
.
[Cjj] = 0 1 .
.
-.-
Fig. 11 - 9
[ c j j ] : 0.25 [cjj] = o5
Damped O s c i l l a t i o n of the N o d e
A
F,
[cm]
Ni=+ Cf, Rs(r) = f(t) 132~ T
100
! =
2.0 t[sec]
80604020
00
0.5
....
F i g . II - 10
1.0
~--- I'.5
2'.0
rB , [cjj]
_- 0
....
re, [cjj]
= o 25
r B , [%]
= o.25
....
rA' [%]
: 0.25
2',5
~
l [s~cc]
Transient plus Constant Wind Loading on Net w i t h InFluence of Damping
115
152.5m ~m ~
!
A
2'5m
\
] cA;--~
\
Av = 108 160 000 kgf ~A D = 21 120 000 kgf E~b = 8 ~ o o o o kg~
i
L
V
\
E4~: ~
273 ooo kgf
L Fig .11 - 11
Transmitter Tower (549 - Unknowns)
~y
[cm]
o 3.0 t[sec]
150~
._
,oo
fll~nn
qw[kgf/m]
y
~qd[kgf/m]
Wind Load
Dead Weight
O1.0
-50-100-
2.0
........." ,/~
,
"'"
~
•
. . . _ rD
FIg. II - 12
Transient Wind Loading, Displacement of Sections C , D
,,..
t [sec]
Q
!
I
I
I
J
I
117
i i-
/ / /
Fig. II - 14 Deformation of the Tower at Various Times During the Earthquake
118
Fig. II - 15
OsttriloUne, Olympic Stadium Munich 6 m Net 1164 Unknowns
119
/
/ Fig. 11 - 16
Os~tribune, Olympic Stadium Munich 6 m Net 1164 Unknowns
120
I,
ri [cm~ 4O
/
l/
30
\
"\ "
i i
2O 10
/
/
/
/
I -II
"",
fFt
[~ ~2]Ii
~32i ~
\ \
oL-~-
\
\.L~',----<.~A~
30
Fig. I1 - 17
Dynamic Response of Various Points of Net
--
I21
Fig. I1-18 Three Planets and Moons - Small Step Algorithm - Hermltian Fifth Degree
122
Fig. 11 - 19
Earth - M o o n - M e r c u r y Problem - Small StepAIgorithm Hermitian Fifth Degree
123
Fig. II - 20
R e n d e z - v o u s - Hohmann's Transfer Small S~ep A l g o r i f h m - H e r m i t i a n Fifth Degree
124
Fig. II - 21
Eight - Body Problem - Small Step A l g o r i t h m H e r m i t i a n Fifth Degree
125
Fig. II - 22
Comet Passing Earth and Sun Small Step A l g o r i t h m H e r m i t i a n Fifth D e g r e e
126
:Z
Etement Force
b-l.s~-I
o 32 Elements, 'r : #sec/12 t x 64Elements, 'r =/ssec/24 j After 2/ssec Exact
-- *[.:..1
6/~'~\o
/j\ l~ro.p_p.¢~_o~_~_¢~_~_
~/o
-~o-~-~-~-~-~-~°-is-~-9../
i
~
1
Fig. II - 23
~'s
20
is
2o
30
&O
~0 32 Numbe~'"-
50
60 64
r° ~,-- T0~.t
o 32 Etements, v = #se¢112 ] After 2#see x 64 Elements, T =/ssec/24 - - Exact
t [#sec]
-~--D-~
2
i
5
10
1'5
20
~
3'0 :32
1
10
20
30
40
50
60
Fig. II - 2 4
of Elements
Triangular Pulse Through Elastlc Slab
r~, _r==½0,.-ooo~(~))
Etemer Force
0
lb
1O
Cosine Pulse through Elastic Slab
6~
Number of Elements
127
_~ _ Monoton=c Load,rig ~ 7 ~'-'-'-Aft. ... . . . . . . . Element I Force
Loading
Z: ~ * b Ramberg-Osgood Stress-Strata Curve and Kinemab¢ Hardening
I00-
Inttla[ Yield Point
075-
699Q--
1
5
1
10
Figd].25
r0 : 1,~sec
r° "
6 o x
6
x O O -Q-Q Ox Q --x--~-x--~
f = ½ (1 - cos ( 2~t )'~
'
J-- TO--.~ t [ / z ~ ]
o
050-
f
32 Elements,r=l/12p.sec~ . . . . 64 Elemenls,v =1/24p.sec ~ATier Zl~SeC
?
~ - O - b- 6 ~
~O-o-~-O-D-B-D~
l0 15
15 30
20 25 30 40 50 60 Number of Elements
32
6/-,
Propagahon of Cosine Pulse in Elastoplastic Free-Free Stab Comparison for Different Idealisations ~n Time and Space
7~ Element Force !
,={0 .... I~l)
TO = |/zsec
}-- r0-q
tc.i.~l
1.00: India[ Yield Poml
075-
~o-o-o. \ % - ~ \ - - ~o\-,o/ \\ /, o
O50
// o x o x
0..
~
~
"~o
7/ /_~ "x \
I.-, ---"°°°=°°i¢ Lo,,d,ng
//
\×'× \--'~ _x.,~? -/
7:~*% ~
///
// //
\\
Ramberg-Osgood Stress-Strain Curve and Kinemabc Hardening
i
x 4--x~---x
~
~
~1]" L...---~After inverse Loading
-100
E[ast,c 5o[ut,on
\% 0..
4 - 0.50
32 El. . . . t s . r :,/,2 p.sec --o--Afer 2 ILsec~ ... _ ~' Elastoplastic 5oluhon
lb
\
t~
2b
tnit,al Yield Point
/-
2s
36 3'2
Number of Elements
Fig. H.26
Propagation of Cosine Pulse
in
Elastoptastic
F r e e - F r e e Slab
---"
128
~ ~
_.E . . . . . . E = 100 9 =0.1 TO = 0.004
T= o.ool
- Z£(1-cos2-~t ~ ts½To To''
f - 2 S~'
pressure in hote
~l -1 ,,2To~
-t
/
/
40
Fig.II-2?
Underground Exptosion with Large Strain Simplified Material with Quadratic Green Strain Energy
.,
129
rj O.B- - 4 0 0,7- '35
I 0.6- 30
!
0,5- ,25
04-
2O
&3
15
0,2 0,t
j, /
/
0
/
-5
-0,2
-10
-0.3
-15
-0,4
-20
i
Yelocity t=
/
0.002
-0.1
; /
o,b~s
J
V
Fig.II-28 Displacement and Velocity of Point A up to 55 Time Steps "c
t = 0.029 . . . . t=0.4
/,.
'\
[ \
/
T
-1-
-2-
Fig.II-29 Underground Explosion with Green Strain Energy Displacement of Points along Axis of Symmetry at t = 0.029 and 0.4
3
3
0
0 3 e~
0
0
I
3
:j-
~
o
o
.<
s. c~
o
r-~
3
_,2"
3 _3
r~
l
r
t
I
I
~l~l
o..
~l~
o~.
i
t5
1
1,1
~I~
1
I
~1~
~I~
t
~'l~
I
i
t
~
i
~
i
r
n
~:
~
~ 7
f ~
n
n
f
z
n
,~17, ~t~ n
I
t
I
~1%
~1~ n
~i ~ n
~J~
z
~
i
i
r
r,i-
L
n
n
I'1
~1~
n
n
~1~
n
~1~
~1~
n
~i%
n
~1~
~ 7
I
I
~
B"
NQ
0
r-n
0 O_
-0 CO
O CO 0
o
I
o',-,11~
o
o.
.~.
~'\
oo
\
\ \
~
X o
"\.
.
~.°
•....~
.-.
ooo ...==....
~
ii
~.
~.
3
| z
°• ----.....=..
~s
°" ~ ' " ~
~
~
--~.~.~'
t31
l?m, + 20 !++. 01
+ ++ c m - ' K - 2~0 +~ \(K.+-'- c c,,+-' 2~
)+/h +<] i
[
+-. ~ - - ' - - ~ + . , - ' - , - - ' , 4
- ++++.++++..-.++c+,+-.-I+,+-,++++l
0+ =
I~
+,+ - ~0+',: - +++(--+'+-'-) + +---++ 1 ~+ T2 . . . . 4 +>++( ++ . . . +i [,+-++-+( . (,<,+-,_~+,+-,,+>+>]]
[
_+,,+ ~+0,:,+-'.++ (..-'. ~+,.+'>+).]
=
Jl ,I
LI
F
~ (K - ~,,I-'¢) - ~4~o ~l+-'c
~c-
=
II
T
1Ill ..... ++1HI+ +
12
_~
~-3
-I
II -
36
q 2
T
T2
-i
T
-~
4
+
~-2
tI+.-++,+-++-+., >l[+.. ++.+-°~-,-,+,+, >nJl'+lP
Fig.
T3
-i
~2
~a
-i
.
~3
L
++'++++'-'++'+]Jt"J l_+('0+'++J
Matrices for Third Algorithm of Family
132
ReLError Displa -
IR = 500 Newton --, Jcm
~ k ~
cement
~ ~ t
0 ~lcm
= 100cm~
tdeahsallon with 20 Elements In,tial Conditions %= 0 , r o = r~at,c 10-2
,.J
i
i
"t
10-~ 5th Degree, n : 3
lO"6 -----.--
10-e 10
8 16
32
3~ Degree, n = 2 Wilson
64
Famln
Fig.
II - 3 7
Free
Oscillation
o f Beam
lcm
! Re{,
Error
I
OispLar':d'? ~ t
= I00¢m~
Ideahsation with 20 Elements Jniha[ Condltlorl$ ~ = 0 , ~ = 0
r[.ewto~ ,o-21
~.'<.~"
."
.... I
~/~ ~ . . . . . .
/.. l /
I'/
~
f.t'"
I1S
(f ){ I I
. . . . . .
I"
I/I
.I -I
for,'.~T:
~= fro.., <÷>~oo-15++ 6<÷)b
I,/
11""
rl
-'::-''-<-" ..........
I
t z(2"~
.
.
_
r = 2s6" tom,n
t [sec]
- " - - -
t,x,~ : 0.02~677225
t [sec]
lO-6 .
lO-e lO-
D~,pl. . . . . . t't -I ~menl" I ---
/~'
2 8 ~6
32
s~
~e Tomm
Fig.
II - 38
Forced Oscillation
o f Beam
t -'I ....
I--t I ....
I
133
A : lcmz
~or~eI
•
.<:3::=
,(,)
[Newton] f [Newton] ,= sin (T°xt)
2000°
o
o
o
•
.
•
•
'r
150C
- - Exact x Wilson ~ : 1 . 4
~ •
~ •
•
"~
1000
x
x
x
x
x
x
x
x
•
•
x
x
~ ~
x
x
•
x
•
x •
7
F,i g .
~1 - 3 9
13
19
25
F o r c e A l o n g L e n g t h o f Bar a t
•x
31
40 Elements
t l : TO m a x
Force J
,4/ = 1 m2
Ot ' ~ t t L •
....
:41-
"~o
•
500
~
Present Method : o 5thDegree(n:3)/'c..J.. • 3 rd D e g r e e ( n = 2 ) } TOm,; 2
~
.
f [Newton]
= |--f~ !
t500 / /
x
x
~ x
,=,in(2-~-) =
m x
/ 1000~ !
Exact • x Wilson O = 1.4
•e " ~ o x
. ~ x -~ x x~
I "I T~ =
Present Method : o 5 th Degree(n=3)1 ~ . 1 • 3 r d D e g r e e ( n = 2 ) l T o , ~ . ~"
500
~\, ?
Fig.
II - 40
13
19
25
F o r c e A l o n g L e n g t h o f Bar a t
\%
31
fl
40 Elements
= TO m a x
134
Ret Error Oispla cement
IR = 500 Newton
o
T-I,,
o
Idealisation w i t h 20 EJements In,halConditions ~ = 0 . r o = rs~,c
/ rl
lO-S~
/ ~
te~:t=O.O43~,82Z~l
/
10"t'.
/
/
/
/ ----
3rd Degree, n = 2
----- Wilson
10"e.
J io-,O~e
t[sec]
z,.e32?so~z "
2;~=-
~6
32
Fig. II - 41
~,
~e
~Omln
Damped O s c i l l a t i o n of Beam
\\Z
77\\,
/
o=
//\\~ I
~o--200cm~ idealisiltioflwith 32 TUEIA6 Elements
Ret.Erm¢
Dispia cement
Load aphed aq A
[~°"1 / ,,~
r~..=-2s0 N,,to,
./
-2] tlO Modes / i /
/ ~ ~ ~ ~ - ....
---
5~ Degree , n : 3
lOSt
2ram ?hick
t~7 . . . . . I 2
Fig. II - 42
4
8
16
---
3~ Degree , ,~ = 2
----
Wilson
T0~,.
Forced O s c i l l a t i o n of Square Clamped Plate
135
Static Deflection E"xact
Solution
for 0 < t ~ l Load~
is
-----~-[ ,g
Load : t - s i n ( 2 / r n t ) 2~r~
for t >1
r 0.1 --'-- -~0 =
sin (nTr) cos(n~ (2t -I))
Load=l-
= 0.02
o
D/t"
100
T0
Where t = Time and n = lIT0
:2.0 ~~
i~
r\
/\
! \/
_-t i'~\
/'\
I \
i3
\\i \
I \
~ , I /i ! \\ k\ 1\ : ~. ~/: I i \\ I 1 o Io o !o o-o Vo o ~ o 4 0 o ~ o-o :o o ~..o oio 0.4 o
~°~T--~-2~-
1//
°'I//
~/i
_/\
!
o//J__~/" V1'0 0 Fig.
II - 43
i--,--,.; T--ix-T-
\
I
\ / V2'0
\ j
\/
:d
40
rg
Newmark
-
50 Time in Integration Steps "r
Stable
F(r)
Shaded aria is t~er in energy balamce
$~sD
II - 4 4
\~:
for Unconditionally
~,
Fig.
!
\ h. \
V 3'0
Pseudo I n d i c i a l A d m i t t a n c e Algorithm n = 2
!-
!",.I
Algorithm
Applied
r1
iF
to N o n - U n e a r
Oscillator
136
- - - - - - 1: = 0.0005 sec ..........."{ = 001 sec
Step
Sinai( Algorithm Large Step Algorithm
~
%0max = 0 25 sec Approx,
~
~
'
~
'
~
'
~
~
X
".~,~.,~__
1:0rain = 0.01 sec Approx.
C~ [cm]
SmallStep
A|gorithm Unstable for ~; > 0.00
~
200 ^
'ii ' . -ioo'
i~
A
//,~
\I'll
1//1 I
r
1
I
/~ ,/'t~'~'°3
!\ ]I
I '° i
iv,
-'~°1'
\~
-2004
Fig .11-45
w
Free Oscillations oF Plane Net - Small Step and Large Step Algorithms
•
ReLI
Error 151
r3 rn,~" + r 3 = 0
Break-down ol Cond
t =0
r0 = 1
T-
'=0 = 0
i
163.
10_5.
\, ~
10.6
~
t0-7
~
lo'~
I,.
"
-ro
J
TO=7.4162987508 TO T = ~-
LargeStep Step
isplacernents
___Small
( 3rd Order )
0 4 8
Fig. 11-46
16
32
6~
Simple Cubic Oscillator Free Vibration, Non-linear Large and Small Step Algorithms Error in Maximum Displacement and Time after One Period
137
Coefficients
of ~lrlx
-TT I
~T
2
1
~T -~T
0T2
I
2
~2 ~-2
- ~2 T2
3 2
-~T
7tT
~'r 2
I 2 --~'r
ST2
~t2
2-Y° T,33
- ~ST 3
3
I
TfGT
~T
~T
~t
3
_ 31,2
~r
-~6r 2
2!52I-3
- &T
3
~-Et
~
I-~5 ~
" I-g~6
3-6T~~
- 3-~6~
Table tf - I - Family of Conditionally Stable Algorithms
138
Coefficients
of H-IX
l
.~T2
~ 2
_ • T 2_I~ ~0T2
~T2
2
~60T2
I
2
- ~IT 3
~3
i ~73 T
11
- i~T
1-~ T
-~1 3
- ~]
T~ T ~
- ~1
~
3
3-g-~T
Table II - 11
U n c o n d i t i o n a l l y Stable Algorithm
- 3-g-~T
n = 1 to 4
139
Order
Period elongahon
tan ~ / 2
n
@ for
1 4 ~2T2 12
I
1
Period elongation for small COT
(Newmark) 2 ( Re~ | )
720
I +
3
(d6 1"6
100800 1
f
W2T2 ~ / / /
Table II - III
~t,j2~2
~4T4 ~
Summary of Properties of Matrix
4
3~
ELEMENTS FINIS FINITE ELEMENTS
ONE-SIDED APPROXIMATION AND PLATE BENDING
Gilbert Strang Massachusetts Institute of Technology
We discuss a problem which arose in the theory of one-sided approximation, tion.
but has at the same time a natural physical interpreta-
Consider a triangular plate,
with all edges free, from a vertex. small
A.
supported at its vertices but
subject to a point load acting at a distance
The problem is to compute the deflection for
We shall describe how this computation promised to be
useful in estimating the distance from a nonnegative function to the set of nonnegative linear functions
v < u.
u
The latter es-
timate leads to an error bound for finite element approximation of continuous quadratic programming problems,
including the obstacle
problem and the deformation of elastic-perfectly plastic materials.
This paper was prepared for the International Symposium on Computing Methods in Applied Sciences and Engineering at IRIA, Eocquencourt, France, in December, 1973. It will be published in the Proceedings of the Symposium. I am grateful for the support of the National Science Foundation (GP 22928).
141 INTRODUCTION
This paper represents one step, and unfortunately only a small one, toward an understanding of the rate of conver~enge of Ritz methods for variational inequalities. a subclass of variational inequalities,
It applies most directly to which we might describe as
continuous quadratic programming problems--the problem is to minimize a quadratic functional of the usual "potential energy" type, but with the class of admissible trial functions subject to inequality constraints as well as equations.
The central problems of plasticity
theory fall naturally into this framework. The Ritz method chooses a family of trial functions depending only on a finite number of free parameters
E qj ~j,
qj, and
minimizes the potential energy under the given constraints. produces an ordinary
This
(discrete) quadratic programming problem.
Our
goal is not to solve this latter problem--a number of good algorithms already exist,
and this source of quadratic programming problems
ought to inspire new ones--but rather to estimate the distance between the true solution
u
and its Ritz approximation
u h.
First, we recall how such estimates are established in the classical case, when there are no inequality constraints and the problem is linear. l) solution
There are three steps to an error estimate:
To establish some smoothness, u.
or regularity,
for the true
Without this step we can still prove convergence
(this has already been done for many variational inequalities),
but
normally we cannot establish anything about its rate. 2)
To show that if such a smooth function
u
proximated by the given family of trial functions, trial function
uh
can be well apthen the particular
chosen by the Ritz method will be close to
u.
142
In the classical case,
this step is made simple by the fundamental
theorem of the Ritz method:
uh
is the p r o j e c t i o n
inner product natural to the problem, nonlinear problems, to
u;
uh
of
u,
onto the trial subspace.
In
will no longer be the closest trial function
we have somehow to find an a priori estimate for
given only the approximation 3)
in the
properties
of the trial subspa@e.
To study the approximation properties
and extract the information
required
u - u h,
of the trial subspace,
in Step 2.
At this stage the
given variational problem and the Ritz method are no longer involved; we are assured that the unknown function number of derivatives, element method,
is discussed
When the problem than an equation, Nevertheless,
of the data,
becomes
inactive.
u - u h,
h2
of
Lewy,
u--limited
inequalities
not
but rather by
Step 2 has been studied by have led to
in Step 3, and ! believe that But in this paper we
in [3], and prove the approximation TaKen together with
[3], this paper es-
estimate for the strain energy in the error
when piecewise
variational
Stampacchia,
The two approaches
may prove to be simpler.
theorem which it requires. an
rather
"free boundary" across which an active
different approximation problems
tablishes
inequality,
as in linear problems,
and by Mosco and the author.
adopt the approach described
/,
Brezls,
a limited regularity
of an internal
FalK's analysis
this ap-
each of these three steps becomes more difficult.
inequality constraint Falk,
For the finite
trial functions,
is governed by a variational
and others have established
the appearance
it.
a certain
in [4].
some results are Known.
by the smoothness
possesses
and asked to approximate
with piecewise polynomial
p r o x i m a t i o n problem
u
linear finite elements
are applied to
like the St. Venant torsion problem.
143 PIECEWISE LINEAR APPROXIMATION FROM BELOW
We shall describe two attempts at proving a theorem on onesided approximation. dependent
interest
The first attempt ended in a problem of inin the theory of plate bending--to
deflection of a plate under certain loads. originally
motivated
(The applications
the theorem were not to plate bending;
only that the approximation be restated
which
it is
theorem which was finally wanted could
in terms of plates.)
Unfortunately,
too great to prove the one-sided approximation fication of this unsuccessful note,
find the
argument,
the deflection was theorem.
But a modi-
described at the end of this
does yield a proof. The theorem
in question
THEOREM I:
Suppose
plane,
and let
P2 = (I, 0),
T
can be stated in the following way:
u(x,y)
be the triangle with vertices
P3 = (0,!).
(I)
v(x,y)
0SvSu
P! = (0,0), c,
there
such that
on
u(pi) - v(Pi) < clul2,T,
The last quantity
function on the
Then for some absolute constant
exists a linear function
(2)
is a nonnegative
is the standard
T i = 1,2,3.
~2
seminorm,
which must be
finite or the result is vacuous:
lu
2
12,T =
f~
2
2
(Uxx + 2U~y ÷ Uyy) dx dy.
This is not the only norm of interest.
One could prove the result
more simply,
variables,
priate
~
and admit more independent rather than
L2
by using appro-
norms of the second derivatives.
In
144 fact,
this weaker result would be sufficient
we have made, in quadratic
in a joint paper programming.
for those applications, First,
vj
like
satisfying
~2
norm is the optimal
the consequences Suppose
the plane
Within each triangle (1) and
(2).
Let
of Theorem
w
Tj
value
chosen from the triangles
w
0 ~ w ~ u
(4)
u(Pi)
The seminorm
of
u - w
is the interpolate determined
surrounding
that vertex.
elul2, i.
- w(Pi) J
over the triangles
u--the
ui(Pi) theory
continuous
= u(Pi)--then
< --
clul2. Tj ;
(6)
lu-uilo, Tj
clul2,T.;J
(7)
lu-Uzll, %
clul2,Tj.
(7) are
respectively.
at the vertices.
each triangle,
and therefore
u I - w--except
that the
luI2
re-
[4] that within each triangle
• a lu-uil
uI - w
If
linear
it is a central
Tj
The left sides of (6) and
Pi"
immediate.
piecewise
(5)
first derivatives,
surrounding
in other norms are almost
of
by
sult of approximation
of
is the smallest
over the whole plane
is also computed
Estimates
function
piecewise
must satisfy
(3)
uI
Pi
by tri-
is a linear function
be the continuous
whose value at every vertex
Then
one
1 for piece-
is covered
there
linear function vj(Pi),
to error estimates
so we shall stay with it here.
we describe
T.
[3] with Mosco,
But the
wise linear approximations. angles
for the applications
L2 From
norms of
u - w
(4), we already
But this is a linear the estimates
and of its know the size
function within
(5-7) apply also to
norms are taken over the union
~j
145
of all neighbors mates hold for
of
Tj.
u - w.
!ation of the plane,
By the triangle In fact,
maps each triangle Finally, longest
c
depending only on the
side--a simple affine t r a n s f o r m a t i o n
into the original triangle
T.
we rescale the independent variables
side is
h,
the same esti-
they remain valid for any triangu-
with a constant
smallest angle and the largest
inequality,
so that the
and loom to see how this scaling affects
each seminorm:
T H E O R E M 2. ous p iecewise
Given
u > 0
on the plane,
linear function
(8)
wh
there exists a continu-
such that
0 £ w h £ u;
(9)
max u-wh i c h]/21ul2,nj; Tj
(lO)
lU-~hlo,Tj i C h21ul2,~j, lU-~hlOi C h21~12;
(zl)
lU-Whlz, Tj ~ C ~lul2,nj, The second inequalities
lU-Whll ! c hlul2.
in (10-11) come from the first,
squaring and summing over all triangles; only on the smallest angle.
the constant
The underlying
C
by
depends
domain can be altered
from the whole plane to a polygon. The theorem leads to error bounds for the finite element approximation
of variational
first application ing
IZ
Igrad vl 2
constraint
v < X
inequalities.
of this Kind,
We describe
in [3] a
to the obstacle problem of minimiz-
under the constraint
v Z ~-
(An additional
could be handled by the same approach,
if the two obstacles
are separated:
~ < X.)
The argument
at least can be
146
extended
to St. Venant's
cyllnder--where hyperbolic
torsion problem for an elastic-plastic
there is an unknown boundary between elliptic
regions
(and,
if the cylinder
well as an upper obstacle).
which need not distinguish
between the elliptic and hyperbolic
it simply minimizes over those piecewlse satisfy the stress constraints. methods,
a lower as
This free boundary appears automatic-
ally in the discrete approximation, during the computation
is hollow,
and
domains:
linear functions which
For an exposition
we refer to Glowinski-Lions-Tr~moli~res
of these
Ill.
147
AN UNSUCCESSFUL
PROOF OF THEOREM
Suppose we choose any linear 0 < v < u
in the triangle
T,
function
and which
v
1
which
is tangent
satisfies to
u
at a
M
point
~
in
T.
(Everything
now takes place
this subscript
from the norms.)
(i.e.,
with
functions
differentiable, we mean that v(Pi) in
T.
about the word
without
violating
(2) of the theorem,
Since i_~s tangent
Ig12 = luI2" to zero at
we have on the function
then
class
is by way of the interpolates at the vertices; (12)
in
"tangent":
the constraint
is to show that i f
g(Pi ) A clgI2"
satisfies
g k 0
The best grasp
~,
in fact almost the only one,
gl"
We Know that
suppose we can evaluate
d(~) = max{f(~)
g = u - v
v ! u
by the argument which follows.
our problem
~,
u
and tha~ none of the nodal values
We hoped to prove that the difference
the estimate
and we drop
which are not pointwise
we must be more precise
can be increased
T,
Since there are functions
lu12 < ~)
v(~) = u(~),
in
g - gI
vanishes
the quantity
I f(Pi ) = O,
If12 = 1].
Then we will have
gz(~) : (gz-g)(~) <_ d(~) Igl-gl2 = d(~)Igl2. Since
gI
is nonnegative
the vertices suppose from (13)
~
and linear,
it cannot get very large a t
and remain small at the point
it were true that to the nearest
d(~)
~.
could be bounded
vertex:
d(~) < C A,
More precisely,
A = minl~-Pil.
by the distance
148
Then, from
gI({) ! C A IgI2
the slope of
gl
gl Z O,
and
could not exceed
it would follow that
CIgl2--in the extreme case
gI = 0
at the vertex nearest to
sible.
(A limiting argument would be required if
fore,
gi
cannot exceed
~
J-2 Clgl2
and climbs as fast as posA = 0.)
at any vertex, which (since
at the vertices) is what we want to prove. g = gl So the problem is to test the bound (13) on d(~). d(~)
has a natural interpretation
square of the deflection
(at
supported at the vertices point
~.
~)
Pi
Fortunately,
in engineering terms; it is the of a triangular plate which is
and acted on by a unit load at the
To verify this equivalence,
(14)
There-
renormalize
(12) to
l/d2 = rain[Ill2 2 I f(Pi ) = O, f(~) = I].
Introducing Lagrange multipliers and the Dirac functions 6 (x-Pi),
6~ = 6(x-~),
(15)
(fL
+
6i=
the problem is to minimize
+
-
f-
f)
dy.
T The Euler equation for the minimizing
F
is the biharmonic:
3 A2F = E ki6 i + k~6~. 1
(16)
This corresponds to the normal deflection of a plate with supports at the vertices and free edges
(and Poisson ratio equal to l, but
that could easily be changed) under a point load at tude
k~.
The four multipliers
k
T
IFI~ = k~ F(~) = k~,
of ampli-
are chosen, as usual, to
satisfy the four constraints in (14). integrating over
~
Multiplying
(16) by
(by parts, on the left side), we find so that the Lagrange multiplier
k~
F
and
149
coincides with tion
I/d 2.
Since a load
I/d 2
produced a unit deflec-
F(~) = I, a unit load would produce a deflection
F(~) = d 2.
There is still another equivalent statement of the problem: to find the fundamental frequency
I/d 2
of a free triangular plate,
supported at the vertices and carrying a concentrated mass at
~.
The difficulty is now to solve the boundary value problem (16). The lists of exact solutions to biharmonic equations yielded nothing for triangular plates with free edges--not surprisingly, solution must be very complicated. mate for
X~
(and thus for
since the
But we want only a good esti-
d), not necessarily its exact value.
Here some familiar and comparatively deep results from function theor~ can actually p!ay a useful role.
In fact, this is our main
justification for pursuing the problem; we want to show how an engineer's guess for the deflection
(he would surely have got the
estimate right) can be rigorously confirmed in spite of difficulties with the geometry. Our first step is to compare the minimum in (i~) with f2 b(~) = min{ ~-~ f2 xx + 2 f2xy + f2 yy + I f(Pi) = O, f(~) = 1].
Certainly,
the minimum has been increased by including the
and integrating over the whole plane instead of the triangle On the other hand, we claim that the new minimum a constant times the old one. f2,
T.
is bounded by
This is true of the inclusion of
because according to (6) ~
The change from
f2 = Ifl~,T = If-fl 120,T -< Clfl2 2,T" T
to the whole plane
Calderon's extension theorem: that
b
f2 term
llfll2,p ~ C, IIfll2, T.
(l+C)(c'/d)2.
P
is made legitimate by
there is an extension of Therefore,
b([)
f such
is no larger than
150
The boundary simpler than
value problem
(16):
associated
the boundary
with
disappears,
b
is very much
and the extremal
tion now satisfies,
for
(17)
AeF + F = Z ~i 6i + Xg ~g.
func-
-~ < x,y < ~,
3 I By superposition, loads, G
F
must be the sum of the responses
acting at the vertices
be the fundamental
Pi
solution,
and at
~.
or Green's
to four point
More precisely,
function,
let
over the whole
plane: (18)
A2G + G = 6(x,y).
Then
F
is a combination
of translates
of
G:
F = k I G(x,y) + k 2 G(x-l,y) k 3 G(x,y-l) The weights yields
k
(denoting
~
+ k~ G(x-~I,Y-~2).
are chosen so that
a set of four simultaneous by
P4
F(Pi)
= O,
equations,
F(~) = I.
This
and by Cramer's
rule
for convenience)
k~ = ~det , M'
(!9)
+
Mij =
G(Pi_Pj)" i,j = I ..... 4.
M'
is the submatrix
and column.
of order 3,
Therefore,
it is independent
in (19) is a fixed constant. when
a
is small,
The matrix cause the Fourier
M
say when
transform
definite
matrix
of
P4 = ~
of
is close to
definite G
is bounded
the denominator P3"
by Bochner's
is everywhere
Surprisingly,
the last row
~, and the numerator
We want to estimate
is positive
= ((el 2 + e2 2 )2 + I) -1. positive
formed by deleting
theorem,
positive:
the determinant
[2] by
of a
be-
ISI
z]9 M341 (20)
det M < MII M22 det < M 4 3
The factors
MII = M22 = G(O,O)
maining factor is
M44]
are independent of
G2(O,O) - G(P]-~) G(~-P]).
[, and the re-
Obviously,
G
in
(18) depends only on the radial distance from the origin, and for small
r
it is Known that a(r)
- o(0)
Therefore the 2 × 2 determinant c G2(0) g2 log A-I,
~
o r2
log
r -I
in (20) behaves like
and this provides an upper bound for
and therefore a lower bound for a similar upper bound for
k{.
X~.
det M,
It is not difficult to find
Therefore it must be that the d_ee-
flection of the plate includes a logarithmic term, and (21) The unlucky estimate
d = ( ~ ) - ~ ~ A(log A-I)½. (13) is not correct, and our proof fails.
152
A MORE SUCCESSFUL PROOF
Since the argument
came to grief only for small
A,
for a modification w h i c h will avoid this possibility. shall sketch is u n n e c e s s a r i l y
crude,
use the facts already proved,
that if
then C
g(Pi) ~ C(~)Igl2--as
~
we look
The idea we
but it seems to work. g ~ 0
approaches
at that vertex actually goes to zero
in
T
and
a vertex,
(because
could not establish that the coefficients
It does g(~) = 0,
the coefficient
d
does),
but we
at the other vertices
re-
main bounded. We shall denote by
K
the specific constant which occurs in
(5): (22)
max lu-ull ~ Klul2,T. T
Two cases of Theorem i are particularly at all three vertices uI
-
~lu12~
then
v ~ u,
at all three vertices and again inequality Suppose that but
u(P3)
P2'
c
v = 0
point
lowing device:
The point
~
is less than
exceeds
and u > 0
KluI2 v =
at the vertices.
KIul2,
then we take
If v = 0
(2) in the theorem holds. and
u(P2),
say, are less than
Then we construct
v ~ u
in
T.
was maximal,
KIuI2,
v = cy, choosing
At the vertices
is still less than
P3 = (0,1). and
v < 0
g(~) = 0.
KIuI2;
P1
and
and at some
We introduce the folare at
On this triangle we still have in the new part below the x-axis.
is certainly not near either of the two new vertices,
and even if it is near the old g(P3) ~ CIgI2.
- v = ~tu12
u
is not.
c
u
think of the larger triangle whose vertices
(2,-1),
g > 0, because
u
g = u - v
~, because
(0,-I),
and
for which
and
If
T, we may choose the approximation
u(P1)
= u(O, 1)
the largest
of
easy.
P3'
it would still follow that
So this case is settled.
153 Suppose finally that
u
is near zero at only one of the ver-
tices, for example the origin: origin by a vertex at T' = Q P2 P3" which
If we still choose
v < u
in
P2" because by hypothesis
for some absolute constant v
Q.
C.
g
[
and
P3
for
exceeds
is small at
P3:
If it happened that
be near
KIuI2. g(P3 ) ! CIul2 g(P2) ~ g(P3)
would establish the theorem; but it is more
g(P2) > g(P3), and we must look for another
P2
c
will again be on or
Nor will the point
If we carry out the same argument for roles of
~
g(P2) = u(P2)
Therefore, we can conclude that
likely that
v = cy, with the largest
T', a coincidence point
then this choice of
Then we replace the
Q = (-i,-I), and consider the larger triangle
above the x-axis, and not near to
u(P I) ! KluI2"
v
v.
of the form
cx, the
are reversed; either it happens that
g(P3) ! g(P2) ! clul2' or
else
g(P3) > g(P2)
and t h i s
v
is also
not satisfactory. Our last idea is this:
consider the family
V e = ce(x cos 0 + y sin e), For each v ~ u Q!). e.
e, c e
in
is chosen to be maximal under the constraint
T', so there is a coincidence point
The values Therefore
ge(Pi) = u(Pi) - ve(Pi)
corresponding Theorem i..
(not near to
depend continuously on
v
6 = a
for which
ga(P2) = g~(P3).
The
is the one-sided approximation we want in
It is zero at the origin, where by assumption
U(Pl) ~ KluI2. P2
~e
(unless the proof was settled by one of the previous
cases) there is a value
vertices
0 ~ e ~ ~/2.
Evidently, and
the point
P3" so that
other--and therefore at both.
~
ga ~ cIuI2
cannot be close to both at one vertex or the
154
This proof appears to extend to three dimensional Whether the theorem holds in all dimensions, order of one-sided approximation nomials of higher degree,
problems.
and whether a higher
is assured for piecewise poly-
is at this moment unknown.
155 REFERENCES
I.
R. GlowinsKi, Numerlque Dunod,
2.
.
des In4quations
Paris,
L. Mirsky, Press,
and R. Tr~mo!i~res, de la Mecanlque
Resolutlon" "
et de ! a Physique,
197~.
An Intrqductiom
to Linear Algebra,
Oxford University
1955.
U. Mosco and G. Strang, inequalities,
.
J. L. Lions,
0ne-sided approximation
Bull. Amer. Math.
Soc.,
G. Strang and G. Fix, An Analysis Prentice-Hall,
New York,
1973.
and variational
to appear.
of the Finite Element Method,
QUELqUES METHODES D'EL~IENTS FINIS POUR LE PROBLEME D'UNE PLAQUE ENCASTREE
P.G. CIARLET Analyse Num~rique, Tour 55 Universit~ de Paris VI II, Quai Saint-Bernard, 75230 PARIS CEDEX 05
Co-..unication pr~sent~e au Colloque International sur les M~thodes de Caleul Scientifique et Technique, I.R.I.A., Le Chesnay, 17-21 D~cembre 1973.
157
O. INTRODUCTION L'objet d e cet article est de pr@senter dlverses m~thodes d'~l~ments finis, effeetivement utilis~es par les Ing~nieurs, pour r~soudre num~riquement le probl~me d'une plaque eneastr~e. Nous n'avons pas consid~r~ toutes les m~thodes : C'est ainsi que nous ne dlsons rien des m~thodes "hybrides" ou "mixtes" (cf. Oden [ 31] , Pian [ 32]), qui sont associ~es ~ d'autres formulations variatlonnelles que eel!e que nous donnons ici. Nous renvoyons le lecteur int~ress~ par l'analyse num~rique de ces m~thodes Brezzi
[ lOI , Oden
[ 31]
et aux travaux de Johnson [ 23,2~] . De m~me, nous
renvoyons ~ Glowinski [ 21] et £ un article ~ para~tre de Ciarlet & Raviart [ 16]pour des m~thodes "par d~composition" qui, tout en ~tant des m~thodes non eonformes, correspondent ~ une fa$on de poser le probl~me discret diff~rente de eelle d~crite darts cet article. Le plan de l'artiele est le suivant : Au §I, on rappelle la formulation variationnelle du probl~me d'une plaque encastr~e. On donne ensuite au §2 des exemples varies d'&l~ments finis conformes utilis~s pour approcher la solution de ce probl~me. Enfin, nous examinons au §3 diverses m~thodes d'~l~ments finis non eonformes pour lesquelles un essai de Oustifieation a priori du "patch test" de B. Irons a ~t~ fait, suivant les travaux r~eents de M. Crouzeix, P. Lascaux, P. Lesaint, P.-A. Raviart, G. Strang, e t p a r l'interm~diaire d'un "lemme de Bramble-Hilbert sur les formes bilin~aires", introduit en [ 12] . On indique aussi une m~thode de p~nallsation, introduite par BabuShka & Zl~mal [ h] . Chaque fois qu'elle appara~t dans une in~galit~, la lettre C d~signe une constante ind~pendante des diverses fonetions intervenant dans l'in@galit~ en question, ainsi que du sous-espaee V h consider@. i. LE PROBL~ME CONTINU Dans tout c e qui suit, on d@signe par fl un ouvert born~ du plan. E t a n t
dorm@ un
entier m, les expressions
I.,o.o-(,
~
~
fo ;l" o,j
,~/2
• ,,o.o:
m
2
)i/2
repr6sentent les semi-normes et normes usuelles de Sobolev; on rappelle que sur l'espace de Sobolev ~o(~), la 5~i-nor~e I-Im,~ - ~
= e n o ~ e ~quiv~ente ~ la norme
11v Urn,~. Consid~rons le probl~me de l'~quillhre d'une plaque, tel qu'il est d~crit dans le livre de Landau et Lifchitz [ 26] , par exemple. En l'absenee de forces, la plaque est repr~sent~e par l'ensemble R du plan, suppos~ horizontal. On note respectivement
158
e, E, et u, l'~paisseur, le module d'Young, et le coefficient de Poisson de la plaque. La fonction inconnue u, qui repr~sente la cote de la plaque par rapport au plan horizontal lorsqu'une force verticale de densitg F est appliqu~e ~ la plaque, rend minimale l'~nergie de la plaque qui, pour une cote donn~e v, est donn~e par l'expression
E(v) =~.I'~-' {(~)2 ÷ ~(~-~~-k~.,~y ~2~ ~y~2~}d~dy-.[ F~xd~.
(L1)
n
n
12(i-u2) l' ~nergie de la plaque s'~crit (£ un facteur multisi l'on pose f = ~---~-T---F, plicatif constant pros)
(1.~)
j(v) =
1 a(~,v)
- (f,v)
oG (.,.) est le produit scalaire de l'espace L2(2), en supposant par consequent l'appartenance de f ~ L2(fi), et oG la forme bilin~aire a(.,.) est donn~e par (1.3)
a(u,v) = .r {Au Av + (i-~) f~ ~2u
B2v
B2u S2v
B2u B2V~}dxdy
n .
~r~2u ~2v
~2u B2v
_B2u
=I {~ ~u Av + (~-~JL~x--Z~--z+ ~y-~~-9z+ ~ y
B2v
~}a~dy.
n La forme bilin~aire ci-dessus est continue sum l'espace H2(fi) × H2(e) et, de plus, elle est H~(~)-elliptique; pour le volt, on remarque que
(~.~)
a(~,v) = ~I~I~,~ + (1-~)M~,~
et on utilise le fait que, physiquement le coefficient de Poisson ~ v~rifie les in~galit~s 0 < ~ < ~1 •Dans ces conditions, il existe une et une seule fonction u dans l'espace V = H2(~) telle que o
(1-5)
J(u) = min J(v), veV ou, de fa9on ~quivalente, qui v~rifie les ~quations (1.6)
a(u,v) = (f,v) pour tout v e V. Pour interpreter - au molns formellement - le probl~me variationnel (1.6), on
utilise les formules de Green suivantes : (1.7) (1.8)
.~ Au Av dxdy =.~ A2u v d x d y - ~ .r {2
S2U
B2V
_~n vdyBAu
S2U B2v
Au~dy,BV
B2U ~2vI~ ~
~xBy sx~y ~--z~-9~-~-97~ -zt~x~y = ~2u ~v ~
o G ~ n e t ~repr~sentent
+#
~2u ~v ~n~t
respectivement les d~riv~es normales et tangentielles le
long de la fronti~re ~fi de l'ouvert ~. Dans ces conditions, si la solution u du
159
probl~me variationnel (1.6) est suffisa~ment r~guli~re, elle est aussi la solution du probl~me
A2u = f dans n,
(1.9) (I.i0)
~u ~n = 0 S~r ~ ,
u =
qui est effectlvement le module "classique" le plus simple pour une plaque encastr~e, l'encastrement ~tant pris en compte par les conditions aux limites (i,i0). L'application de la seconde formule de Green (1.8) montre que la contribution ~2v ~2u ~2v ~2u ~2 du terme .r~(1-~){ ~ ~x~y ~x-~Z~-~2-~-~ ~x-~dxdy est nulle. En d'autres termes, on pourrait se contenter de la forme bilin@aire a' (u,v) --,~Au Av dxdy, et c'est d'ailleurs ce choix qui est fr~quemment fait, puisqu'il conduit ~ la formulation variationnelle la plus simple qu'on puisse attacher au probl~me (1.9)-(1.10). La forme bilin~aire a' est encore V-elliptique car la semi-norme IAVIo,~ est une norme sur l'espace v -- H2(~), ~quivalente ~ la norme Hv 112,~. En fait, la possibilit~ de remplacer la forme bilin~aire a par la forme bilin~aire a' tient ~ ce que l'on se place dans l'espace H2(G), et non dans un espace strictement plus grand, comme l'espace O
H2(~); autrement dit, cette possibilit~ serait supprim~e si l'on choisissait des conditions aux limites autres que celles d'une plaque eneastr~e. On volt aussi que la forme bilin@aire a' correspond ~ la valeur ~ = 1 dans l'expression (1.3) et qu'en fait, toute valeur W de l'intervalle [O,1] conduira ~galemerit au m~me probl~me (1.9)-(1.10), la V-ellipticit~ ~tant toujours assur~e; cf (l.h). De la m~me fagon, toute l'analyse falte au §2 pour les m~thodes conformes s'applique sans changement ~ toute valeur ~ de l'intervalle [ O,1]. Cependant, pour certaines des m~thodes non conformes du §3, certains r~sultats ne sont plus valables si W e s t ~gal ~ l, alors qu'ils restent vrais pour les valeurs physiquement admissibles de ~, i.e., dans l'intervalle ]0,~ [ . En ce qui concerne la r~gularit~ de la solution u du probl~me (1.5), on peut montrer (voir Kondrat'ev [ 25] ) que celle-ci appartient ~ l'espace H~(~) O H2(~) des que l'ensemble ~ est un polygone convexe, ce qui est effectivement souvent le cas des plaques. A cet ~gard, il est int~ressant de remarquer que (sauf pour l'~l~ment de Morley; cf §3), l'hypoth~se "u ~ H~(G) " est aussi l'hypoth~se minimale qu'on utilise pour obtenir la convergence des m&thodes d'~l~ments finis, aussi bien conformes que non conformes. 2. METHODE s CONFOHMES On suppose comme au §i clue f E L2(R). On notera ~. II l'expression 1.12,R, qui est une norme sur l'espace V = H2(R). Ii existe alors des constantes M e t ~ > 0 0
telles que (2.1) (2.2)
la(u,v) I ~ M flu " fly U pour tout u,v e V, ~ fly 112 ~ a(v,v) pour tout v e
V.
160
Etant donn~ un sous-espace de dimension flnie V h de V, le probl~me discret consiste ~ trouver une fonction u h E V h telle que
a(~,~- h) -- (f,vh) po~ vh e Vh,
(2.3)
et ee probl~me a une solution et une seule, d'apr~s (2.2). Utilisant les in~galit~s
Uu-~ , z ~ ~(u-u~, u-uh) = a(u-~h, u-v h) ~
M llu-uh B ,u-v h ,
v~rifi@es pour to=re fonction v h E Vh, on en d~dult l'in~gallt~
(2.~) ~u-uh~ ~ C
inf UU-Vh~,
o~ la constante C = M/ct est ind6pendante au sous-espace Vh, de sorte que le probl~me de l'@valuation de l'erreur
Hu-u h II est ramen~ ~ un probl~me de th~orie de l'appro-
ximation - l'~valuation de la quantit~
inf H u-v h H • Vh~-Vh Pour obtenir l'in~galit~ (2.4), on a utilis@ de fagon essentielle l'inclusion
V h C V. On dit alors que la m~thode d'approximation est interne et clue, dans le cas o~ l'espace V h correspond ~ une m~thode d'~l~ments finis (cf. les exemples d~crits plus loin), la m~thode, et les ~l~ments finis, sont conformes, ou encore compatibles. Pour construire un tel espace Vh, on se donne :
(a) Une triangulation
de l'ensemble ~ en ~l~ments finis, i.e., ~ = K E ~ K , o~ A
A
les ~16ments K sont des triangles, ou des ,~uadrilat~re,s, d'int~rieurs deux ~ deux disjoints, et tels que tout cSt~ d'un ~l~ment soit ou bien un cSt~ d'un autre ~l~ment, ou bien une pattie de ~ h =
on pose
max hK, avee h K = diam(K) pour tout K 6 ~ h ~
(b) Un espace P, de dimension finie, de fonctions r~elles tel que s i v h est une fonction quelconque de l'espaee Vh, alors restr'vh 1K ~ P pour tout K 6
~h •
(c) Un ensemble de de6r~s de iibert~ attaches £ un ~l~ment fini "co,rant" K qui, d'une part, d~finissent une base de l'espace P et qui, d'autre part, sont ehoisis de telle fa~on que !'inclusion V h C H2(~) ait lieu; comme les fonctlons de l'espace P sont le plus souvent tr~s r~gu!i~res dans la pratique, l'inelusion pr&c~dente sera une consequence de l'inclusion V h C CI(5), ce que l'on pourra v~rifier clans chacun des exemples dorm,s ci-apr~s. On v&rifiera aussi sum ees mSmes exemples que Its conditions aux limites (i.iO) peuvent ~tre satisfaites exactement dans Its sous-espaces V h eorrespondant s. Si u est une fonction suffisamment r&guli~re d~finie sum 5 (resp. sur un ~l~ment fini K), on notera HhU (resp. HKU) la fonetion de V h (resp. de P) qui interpole
161 l a f o n c t i o n u s u r ~ ( r e s p . sur K), i . e . ,
dont l e s d e ~ s
de l i b e r t ~ s u r ~ ( r e s p . sur
K) sont i d e n t i q u e s £ ceux de l a f o n c t i o n u . De l a s o r t e on v o i t que r e s t r , pour t o u t K e ~ h " On v a e / o r s majorer i a q u a n t i t ~ lit~
(2.4) pat" Jlu-IlhU tl, et comme
(2.5)
inf
nhU
Ilu-vh II de l ' i n ~ g a -
= ~Ku K
VhEVh
tu-nhul2, £ ~,
lu-,Ixul2,£/
,
le probl~me d'~valuation de l'erreur se trouve ramen~ ~ un probl~me de th~orie de l'interpolation "locale", i.e., Bur un ~l~ment fini "courant" K. Ce probl~me a ~t~ ~tudi~ par de nombreux auteurs ces derni~res ann~es; voir notsa~ent Babu~ka & Aziz [ 3] , Bramble & Hilbert [ 8] , Ciarlet [ ii] , Ciarlet & Raviart [ 14,15] , Raviart [ 33J , Strang [35] , Strang & Fix [37] , Zl~mal [40]. Le r~sultat fondamental est le suivant: on consid~re une "famille" d'~l~ments finis pour laquelle le param~tre h tend vers z~ro, et qui ne deviennent pas "plats" la limite (pour une d~finition precise de cette notion, voir Ciarlet & Raviart [ lh] ). L'hypoth~se fondamentale est l'inclusion
Pk C P,
(2.6)
o~ Pk d~signe l'espace vectoriel des polynSmes (ici : de deux variables) de degr~ ~ k. Alors pour tout entier m ~ k+l tel que P C Hm(K), on a
(2-7)
tu-IIKUlm,K g C
hk+l-m
K
u k+l,K
o~ la constance; C eat ind~pendante de la fonction u et de h K. Par application de (2.4), (2.5) et (2.7) avec m = 2, nous obtenons
(2.8)
Ilu-u h
, < c hk-llulk+l,£,
en supposant que l'inclusion (2.6) a lieu. En consequence, on obtient une conver6ence d'ordre O(h) d_~s que !~ solution u appartient ~ l'espace H3(£)
que l'incluslon
"minimale"
(2.9)
P2 C p
est satisfaite. Utiliaant les techniques de dualitY, d@velopp~es par Aubin [ 2] et Nitsche [ 30] , on peut montrer que si l'inclusion (2.6) eat v~rifi~e avec un entier k > 3~ on a Uu-uh
Ito,q
= O(hZ),
U-U h U 2,£ mais ce r~sultat est ~tabli en supposant que la fronti~re de £ est suffismment r~guli~re pour que la solution u soit dans l'espace H~(£) pour tout second membre f~
L2(~) et qu'il existe une in~alit~ du type ~u 114,.£ ~ C lJf ~0,£ pour tout
f ~ L2(£); voir l~e~as [ 29, Th~or~ma 2.2, page 216]. Examinons maintenant un certain nombre d'exemples. Dana les figures, on utilise les notations suivantes pour lea degr~s de libert~ :
162
•
u°
(9
I
" o,1.
d~riv~e normale au milieu du cSt~.
ax~y Exemple i ( c f . Figure 1). Cet ~l~ment, que nous appellerons "~l~ment ~ 21 de~r~s de l i b e r t Y " , est apparu en 1968, simultan~ment dans au molns s i x p u b l i c a t i o n s ; v o i r
Figure i ce sujet Zienkiewicz [ 38, page 209] , et Zl~mal [ hO] o~ la th~orie de l'interpolation pour cet ~l~ment a &t~ faite pour la premiere lois. L'espace P ~tant ici l'espace P5, de dimension 21, on obtient donc llu-uh II ~ C h~lui6,n , en supposant que u 6 H 6 (~). A partir de cet ~l~ment, on peut construire un "~l~ment ~ 18 de~r~s de libertY" (cf. Figure 2), qui est ~galement apparu en 1968 (voir £ ce sujet Zienkiewicz [38, page 209] ). L'espace P, de dimension 18, es% form~ par les polynSmes de degr~ 5 pour lesquels la d~riv~e normale le long de chaque cSt~ du triangle est un polynSme (d'une variable) de degr~ 3. Darts ce cas, on a donc les inclusions P~ c p c P5, de sorte que, si u E H5(~),
,u-h, ~ c h31uls,n
163
Fi6~r e 2 La thgorie de l'interpolation pour cet ~l~ment a ~t~ faite par Bramble & Zl~mal [ 9] = Exemple 2 (cf. Figure 3). Cet @l~ment a ~t@ introduit par Bogner, Fox &
Figure 3 Schmit [ 7] ; il ne s'applique qu'K des plaques dont les cSt~s sont parall~les aux axes de coordonn~es. L'espace P, de dimension 16, est form~ des polynSmes du type p(x,y) =
~ ~i~xiy j, Ogi ,j~3
c'est-~-dire qui sont de degr~ 3 par rapport ~ chacune des variables. De l'inclusion
164
P3 C P, on d~duit que, si u q H~(~), ~u-u h
les majorations locales pouvant ~tre d~duites par exemple des r~sultats de Ciarlet
& ~aviart [l~l. Exemple 3 ( o f . F i g u r e h ) . Cet ~ l ~ n e n t , a p p e l ~ ~l~mem~ de Clou6h e t T o c h e r , du nom de s e s i n v e n t e u r s [ 18] , e s t t r ~ s i n g ~ n i e u x . I 1 p r ~ s e n t e l ' a v a n t a g e
de r ~ d u i r e l a
Fi6ure d i m e n s i o n de l ' e s p a c e
P t o u t en r e s t a n t
eonforme ; c f . l a d i s c u s s i o n au d ~ b u t du §3.
L ' ~ l ~ m e n t f i n i K e s t e n c o r e un t r i a n g l e , triangles
Ki, et ltespace
Pest
sur K dont les restrictions
q u i e s t lui-m~me l a r ~ u n i o n de t r o i s
form~ des f o n c t i o n s une f o i s c o n t i n ~ m e n t d ~ r i v a b l e s
K. s o n t des polynSmes de 1 d e g r ~ 3. Une f o n c t i o n de P e s t done d ~ f l n i e p a r 30 p a r a m ~ t r e s q u i s o n t eux-m~mes obtenus ~ partlr
~ chacun des t r o i s
des 21 ~ q u a t i o n s r ~ s u l t a n t
e t des 9 ~ q u a t l o n s que l ' o n o b t i e n t ainsi
que s e s d ~ r i v ~ e s p a r t i e l l e s
m a t r i c e du syst~me l i n ~ a l r e du p o i n t a ~ l ' i n t ~ r i e u r t h ~ o r i e de t ' i n t e r p o l a t i o n si ue
triangles
de l a donn~e des 12 d e g r ~ s de l i b e r t ~
en ~ e r i v a n t que l a f o n c t i o n e s t c o n t i n u e s u r K,
p r e m i e r e s . On m o n t r e ( c f . C i a r l e t
correspondant est r~guli~re,
de l ' ~ l ~ m e n t , r~sulte
[ 13] ) que l a
q u e l l e que s o i t
la position
e t q u e , moyennant q u e l q u e s p r e c a u t i o n s ,
e n c o r e de l ' i n c l u s i o n
PS C p . On o b t i e n t
la
ainsi,
H~(n), ~u-u h L ' o r d r e a s y m p t o t i q u e de c o n v e r g e n c e e s t done i d e n t i q u e ~ e e l u i de l ' ~ l ~ m e n t de
i ' E x e m p l e 2, avec l ' a v a n t a g e
de p o u v o i r m a i n t e n a n t c o n s i d Q r e r des domaines p o l y g o n a u x
q u e l c onques. De m~me qu'on passe de l'~l~ment ~ 21 degr~s de libert~ ~ l'~l~ment ~ 18 degr~s
165
de libert~ (cf, Exemple i), de m~me peut-on r~duire de 3 le n ~ b r e de degr~s de libert~ de l'~l~ment de Clough et Tocher (ici encore, les d~riv~es normales aux milieux des cSt~.s) en assujetissant la d~riv~e normale le long de chaque cSt~ du triangle ~tre une fonctlon lin~aire (d'une variable), ce qui conduit ~ un espace P de dimension 9 v~rifiant l'inclusion P2 C p, et donc ~ une erreur en O(h) si u E HS(£).
•
Exemple 4 (cf. Figure 5). L'~l~me~t K, que nous appellercns ~l~ment de Fraei,~s de Veubeke et Sander [ 20,34] , est ici un quadrilat~re convexe, et sa conception
a l
a2
aB
~4
rel~ve de la re@me idle que celle de l'~l~ment de Clough et Tother. Suivant les notations de la Figure 5 on note K I le triangle de sommets ala 2 e t a 4 et K 2 le triangle de sommets ala 2 e t a 3. On pose
R~ -- {p ~ c1(K)~ p : 0 s = ~Kl, p ~ P3 sur K~}, R 2 = {p E CI(K); p = 0 sur CK2, p E PS sur K2} , les compl&nentaires
~tant pris par rapport ~ l'~l~ment K. A!ors l'espace P, de
dimension 16, est la s o ~ e directe de P3 et des deux espaces R I e t R 2. Ciavaldlni & N~d~lec [ 17] ont r~ce~aent fair l'analyse de cet ~l~ment : compte-tenu de l'inciusion PS C p, on obtient encore, si u e H~(9),
Liu-uh n ~ c h21u]~,a.
°
D'autres ~l~ments finis conformes sont ~galement employ~s: par exemple, on peut "ajouter" ~ des espaces de polynSmes des fonctions "singuli~res" judicieusement choisies. Ces fonctions sont singuli~res en ce sens que certaines de leurs d~riv@es, secondes par exemple, sont non born~es. Pour pouvoir appliquer les majorations du type (2.7) avec m = 2, ii faudra donc d'abord s'assurer que l'inclusion P C H2(K) est satisfalte. On trouvera des exemples de tels ~l~ments page 199 du livre de Zien-
166
klewicz [ 38] , ainsi que dans Birkhoff & Mansfield [ 6] o~ se trouve
de
sur-
croit faite une th~orie de 1'interpolation "locale". 3. METHODES NON CO~FORMES Les ~l~ments conformes d ~ e r i t s c a r l a d i m e n s i o n de l ' e s p a e e
Pest
dans l e §2 s o n t d i f f l c i l e s relativement
~ m e t t r e en o e u v r e
~lev~e et la structure
de l t e s p a e e P
est parfois compliqu~e (cf. Exemples 3 et 4). Naturellement, ces difficult~s r~sultent de la n~cessit~ d'avoir des d~riv~es partielles premieres continues lorsqu'on passe d'un ~l~ment ~ un ~l~ment adjacent. Par ailleurs, si les ~16~nents precedents conduisent ~ des ordres asymptotiques de convergence ~lev~, c'est en supposant que la solution est plus r~guli~re qu'elle ne iIest en pratique. Nous avons vu en effet ~ la fin du §I quton avait seulement "u fi H3(R) '' si ~ est un polygone convexe. Dans ces conditions, l'incluslon minimale P2 C p slgnal~e au §2 est aussi optimale et des espaces P plus grands que P2 ne condulront pas ~ une meilleure convergence. Or il
se
trouve
que si P = P2 (le cas
ideal l) la seule fonction de classe CI(~) v~rlfiant les conditions aux limltes y
(1.10) est la fonction nulle (cf. un article ~ paraltre de A. Zen~sek). Les considerations qui precedent conduisent donc naturellement ~ la conception de m~thodes pour lesquelles 1'inclusion Vh c CI(~) n'est pas satlsfaite : effectivement, dans les deux premiers exemples que nous donnerons, on a seulement I' inclusion V h C C°(£) et le dernier exemple est encore molns "conforme" puisque cette derni~re inclusion n'est m6me pas satisfaite. D'une fa~on g~n~rale, on dit que les ~l~ments finis sont non conform.es, ou incompatibles, d~s que l'inclusion V h C V = H2(~)o n'est pas v~rifi~e. On supposera que l'inclusion V h C L2(~) a lleu, ce qui permet d'~crlre les membres de droites des relations (3.1) ci-dessous. Puisque les fonctions de V h sont localement r~guli~res, la fagon la plus naturelle de d~finir le probl~me discret associ~ ~ un sous-espace Vh d'~l~ments finis non conformes conslste ~ chercher une fonctlon uh ~ Vh qui v~rlfie
(3.l)
%(u~,v h) -- (f,vh) pour
tout
Vh e
V h,
o~, par d~finition, --
(3.2)
ah(Uh 'Vh)
KE~h
K
l'expression { "''} ~tant la m~me que celle qui intervient dans la forme bilin~aire a(.,.) donn~e en (1.3). De la mSme fagon, il nous faut d~finir une norme sur l'espace Vh; I~ encore, il est naturel de poser
(3.3)
"Vh "h = ( E
2 hI/2 Ivht2,K/ ,
167
mais encore faut-il v~rifier qu'il s'agit effectivement d'une norme, la positivit~ n'~tant pas automatique; on d~montre que c'est effectivement une norme pour les trois exemples que nous d~crivons plus loin; cf. Lascaux & Lesaint [ 27]. Des expressions (3.2) et (3.3), on d~duit que (3.h)
a n v h II~ ~ ahCVh,Vh) pour tout vh 6 Vh,
o~ la constante e = (I-~) est ind~pendmute du sous-espace Vh, et c'est d'ailieurs cette "'uniformlt~" de la Vh-elllpticit~ qui permet d'obtenir la majoration fondamentale (3.6) de l'erreur. Slgnalons que si l'on avait choisi la valeur ~ = 1 darts la forme bilin~aire a(. ,.), valeur physiquement irr~aliste mais admissible pour les m~rhodes conformes, la Vh-ellipticit~ ne serait plus n~cessairement uniforme pour les exemples que nous considgrons plus loin. Dans ce qui suit, nous eonsid~rons que la forme bilin~aire ah(.,.) et la norme II. U h que nous venons de d~finir sur l'espace Vh sont ~galement d~finies sur l'espace V o~ elles sont respectivement ~gales ~ la forme a( .,. ) et £ la norme II. I]. Une simple application de l'in~galit~ de Cauchy-Schwarz montre pour commencer qu'il existe une constante M ind~pendante de l'espace Vh teile que (3.5)
[ah(Uh,Vh)I g
M Ilu h Ith ilvh IIh pour tout Uh, vh 6 V h.
Soit ensulte vh = uh-w h un glgment queleonque du sous-espace Vh. Utilisant les relations (1.6), (3.1), (3.4) et (3.5), on obtient
[twh
"~ < % ( W h , W h) =
ah(U-Vh,Uh-V h)
+
(f,w h)
-
ah(U,W h)
M Ilu-v h IIh llwh li h + I (f,wh)-ah(U,Wh)i , de
sorte
que Wh flh ~ M flU-Vh IIh +
i(f ,~h )-%( u ,Wh)l IIw h IIh
Cette derni~re in~galit~, jointe ~ I' in~galit~ triangulaire U u-uh ilh ~ ilu-v h flh + ilWh fib' conduit ~ l'in@galit~ I(f'wh)-ah(U'Wh) I ) (3.6)
Uu-u h U h ~ C ( inf flu-v IIh +
\Vh Vh
O~ la constante C = max{1 + M
h
sup
.h Vh
"'h "h
i} est ind~pendante du sous-espace V h. Cette in~ga-
lit~, fondamentale pour l'~tude des m~thodes non conformes, est due K Strang [ 36,37]. On remarque que l'in~galit~ (3.6) g~n~ralise l'in~galit& (2.4), puisque l'expression (3.7)
Eh(U,W h) = (f,wh) - ah(U'Wh)
est nulle pour tout w h E Vh, d~s lors que l'inclusion V h C V a lieu. Compte tenu de l'expression (3.1) de la forme bilin~aire ah( .,.), on~obtient en utilisant les formules de Green (1.7) et (1.8) sur chaque ~l~ment K {~n.. et 8tK
168
dgsignant respectivement les dgriv&es normales et tangentielles !e long de la fronti~re 8K de l'~l~ment K):
~(u,~) : ~(U,Wh ) + ~(U,Wh) ,
(3.8) a~e c
(3-9)
El~(U,Wh) =
~ ~.~K{~Au
wh + (l-u) 8nKS-----~K~tK }dy, ~Zu
~wa
~)2U ~)wh
o les formes bilin~aires Eh(U,W h) et ~ ( u , w h) ~tant respectivem~nt d~finies sur los espaces H4(~) x V h et H3(~) x V h. Observons clue si le sous-espace V h v~rifie l'inclusion V h C C°(~), ce qui est le (as du rectangle d'Adini et du triangle de Zienkiewicz, le terme ~(u,w h) est nul et l'hypoth~se "u q H3(~) '' est suffisante. Consid~rons le terme
inf llu-vh llh, qui intervient dans l'in~galit~ (3.6). VhEV h ~u supposant l'inclusion "minim&le" P2 C p v~rifi~e, et l'appsmtenance de la solution u ~ l'espace H3(~), on obtient, compte tenu de l'expression (3.3) de la norme I]. llh, (3.11)
inf
2vh
llu-vh H h < C hlul3,~.
Dans cos conditions, au vu de la majoration fondamentale (3.6), l'id~e naturelle est de d~montrer une in~galitg du type
(3.12)
IEh(U,Wh) ] < C h l u l 3 , n IIwh IIh pour t o u t u e ~ 3 ( n ) , Wh e Vh .
Or on a vu en (3.8), (3.9) et (3.10) que l'expression Eh(U,W h) est elle-m~me une somme du type (3.13)
~ ( u , w h) = ~
Eh,K(U,Wh),
chaque expression Eh,K(U,W h) ~tant associ~ h &u soul ~l~ment K; une telle d~composition n'est d'ailleurs pas unique et c'est pr~cis~ment du choix d'une d~composition Oudicieuse que r~sulteront les ma~orations d~sir~es; cf. l'expression (3.15). L'objectif est alors d'obtenir des majorations du type
(3.1h)
l~,K(U,Wh) I < C hluI3,K tWhl2,K pour tout u e H3(~), wh e Vh
l a majoration (3.12) en d~coulant imm~diatement. Nous allons pr~ciser sur un exemple la d~marche qui conduit ~ une majoration du type (3.14). On consid~re le terme ~ ( u , w h) correspondant au triangle de Morley (of. Exemple 7). Et~nt donn~ un triangle K et une fonction g d~finie sur la fronti~re du triangle, appelons HKg la fonction qui, sur chacun des trois cSt~s ~iK, i ~ i ~ 3, I du triangle K, est constante et ~gale ~ la valeur moyenne ~i '~SiKgaY de la fonction g sur ce cSt~, h i ~tant la longueur du cSt~ ~iK. On pout alors transformer l'expression ~ ( u , w h) de (3.10) en remarquant, suivant Lascaux et Lesaint [ 27] , que l'on
a aussi ~wh KE~h
~wh
~K
~wh
En effet, Bur un cBt~ BiK de milieu a i , la fonetion nK ~
~wh
vaut ~
(ai) puis-
~wh
que, le long de ce c~t~, la fonction ~
est un polynSme (d'une variable) de de@r~ I.
La contribution des deux int~grales curvilignes associ~s aux deux ~l~ments adjacents ce e~t~ est done nulle si ~.K est un cBt~ "int~rieur" ~ ~, et si ~.K est une pari i
~ h a i) = O, compte tenu de la deuxi~me condition aux limites, de tie de B~, alors ~--~" sorte que l'int~grale eorrespondante .F (...)dy est encore nulle. @.K 1
On est donc amen~ ~ ~valuer des expressions du type 2u
Bw.
@wh
Ah,i ,K(U,Wh) = .r~. K{-Au+(I-~) ~+--~)~t K (~nK - "K ~-~-}v.~d7 1
(3.16) On constate
alors que
(3.17)
Ah, i,K(u,wh ) = O pour tout u e H3(O), w h • PI,
(3.18)
Ah,i,K(U,Wh) = O pour tout u • e2, "h • Vh~w~
En effet, si w h • PI, la fonction ~
est constante le long d'un cSt~ ~iK du
triangle K et la relation (3.17) provient de l'invariance des fonctions constantes par l'application ~K" La v~riflcation de la relation (3.18) r~sulte de ce que ~.K(%-~K¢)d 7 = 0 s i ¢ • PI, d'apr~s la d~finition de l'application ~K' ce qui est I ~wh bien le cab pour la fonctlon ¢ = ~ , puisque W h • P2" On d~montre (cf. Ciarlet [ 12] ) le 1creme suivant, qui est la g~n~ralisatlon au cas des formes bilin~aires du Leone de Bramble-Hilbert | 8] , ~tabli pour des formes 1in,aires. l~mmg. Soit ~ un ouvert de R n de fronti~re suffisamsent r~6uli~re, soit k et deux cutlers, et soit V un espace de fonctions v~rifiant les inclusions _ p~ c V c H~+I(o). L'espace V ~tant norm~ par U • ~ + 1 , ~ ' _solt A :
H k+l (
~) x V -~ R
une forme bilin~aire continue, de norme ~IA ~ , et telle ~ue (3.19)
A(u,w) = 0 p.gur tout u • Hk+l(~), w ~ P£,
(3.20)
A(u,w) = 0 pour t0ut" u • Pk
, w • V.
Alors il existe une constante C, q u i n e d~pend que de l'ouvert ~, telle que (3.21)
IA(u,w)I ~ C HA ~ lUlk+l,R lwI£+l, ~, pour tout u • Hk+l(R), w • V.
Une lois ce lemme ~tabli, on l'applique aux expressions Ah,i, K de (3.16): on passe comme ~ l'accoutum~e de l'&l~ment "courant" K ~ un ~l&ment "de r~f~rence" sur lequel est appliqu~ le lemme, ce qui conduit ~ une majoration du type
170
(3.22)
I.'%,i,K(U,~h)l~ C h luI3,K J~hl2,K po= tout ~ ~ ZS(~), ~h ~ V h ,
le facteur h provenant des changements de variables dans les int~grales lorsqu'on passe de K K ~ , puis de ~ K int~grale sur
K. La dlfficult~ r~sultant de ce que l'on consid~re une
un cSt~ plutSt que sur tout le triangle
se traite ccmme dans
Crouzeix & Raviart [19, Lemme 3].
Les a u t r e s t e r m e s i n t e r v e n a n t darts l ' e x p r e s s i o n
~ ( u , w h) se t r a i t e n t
de faqon
a n a l o g u e , avec ~ v e n t u e l l e m e n t des m o d i f i c a t i o n s " t e c h n i q u e s " , e t c e l a a u s s i b i e n pour l e t r i a n g l e de Morley que pour l e t r i a n g l e
de Z i e n k i e w i c z ou l e r e c t a n g l e d ' A d i n i
(of. les exemples). Si nous raisons la somme de routes les relations (3.18) sur tousles ~l~ments finis K, nous end~duisons que (3.23)
Eh(u,wh) = 0 pour tout u e P2, w h E Vh,
or cette derni~re relation n'est autre que le c~l~bre "patch test" des Ing~nieurs, trouv~ empiriquement par Irons [ 22] comme une condition n~cessaire de "convergence" des m~thodes d'~l~ments finis non conformes. Nous nous sommes efforc~s ici pr~cis~merit de faire appara~tre cette n~cessit~ en consid~rant le patch test 'comme l'une des de ux invariances pol[nomi~es n~cessaires ~gur °btenir une majoration du t ~ e
(3.14),
une lois connu le lemme ~noncg plus haut. Examinons maintenant quelques exemples. Pour chacun des sous-espaces V h correspondant aux trois exemples qui suivent, les conditions aux limites (I.I0) sont prises en compte de la fagon la plus simple: tousles degr~s de libert~ sont nuxs lorsqu'ils correspondent ~ des noeuds situ~s sur la fronti~re. On constatera par ailleurs dans le cas du triangle de Zienkiewicz que le patch test se traduit par une restriction sur la g~om~trie des ~l~ments. Exemp1e 5 (cf. Figure 6). Cet ~l~ment, que 1'on appelle "trlangle de Zienkiewicz"
Figure 6
171
a ~t~ introduit dans Bazeley, Cheung, Irons & Zienkiewicz [ 5] • L'espace P e s t icl f o 1 ~ des polyn6mes p E PS pour lesquels
6p(a) - 2 I p(ai) ÷ I i=l O~ les points ai, 1 ~ i ~ 3, e t a
-- 0,
i=l sont respectivement les sommets et le barycentre du
triangle. L'espace P v~rifie les inclusions P2 C p C Ps,et sa dimension est 9. Lascaux et Lesaint { 27] ont montr~ que le patch test est v~rifi~ si et settlement si tousles c~t~s de tousles triangles d'une triangulation donn~e sont parall~les seulement trois direction, et qu'alors si u E HS(~),
,u-~ "h < C b lu13,~, ce qui est la solution du "probl~me de l'Union Jack" (cf, Zienkiewicz [ 38, pp. 188189] ). Par ailleurs, le fair que la valeur au centre de gravit~ ne soit plus un degr~ de libel~.' (cf. la relation (3.24)), condition ~galement trouv~e par les Ing~nleurs comme une condition empirique de convergence, volt ici sa justification d&ns le fait que la fonction de base correspondant au barycentre "ne passe pas" le
patch test.
•
E_xemple 6. (cf. Figure 7). Cet @l@ment, que l'on appelle "rectangle d'Adini",
()
x
Figure 7 a ~t~ introduit dans Adini & Clough [ i] • L'espace P consiste en tousles polynSmes du type "
p(x,y) = ~isxY 3 + ~slxSy +
~ijxlY J , O~i+j<3
et sa dimension est 12. Lascaux et Lesaint [ 27] ont montr~ que, si u e HS(~), llu-uh U h
C h .Jut3, , n,
172
le patch test ~tant ici toujours v~rifi~. L'absence de condition g~om~trique n'est qu'apparente puisque l'on ne peut consid~rer que des domaines ~ cSt~s parall~les aux
axes.
•
Exemple 7 ( c f . F i g u r e 8 ) . Cet ~t~ment e s t connu sous l e nom de t r i a n g l e
de Morle~
Figure 8 (cf. Morley { 28J ). L'espace P e s t ici P2 et sa dimension est 6. Cet ~l~ment est "hautement non conforme" puisque les fonctions de l'espace V h correspondant ne sont m~me pas continues. Cependant, cet @l~ment passe tou~0urs le patch test, quelle que soit la configuration de la triangulation. Le terme Eh(U,Wh) de (3.9) n'~tant pas nul, on arrive a une majoration du type (cf. Lascaux & Lesaint { 27] )
"h C(hlui3. ÷h lul)' , c'est-~-dire encore une majoration en O(h), mais en supposant que la solution u est dams l'espace H~(~). La th~orie de l'erreur lorsque la solution est "seulement" dans H3(~) reste ~ faire.
•
Pour les trois ~l~ments precedents, on peut ~galement ~tablir des majorations de l'erreur dans les normes ~. flO,~ ou
fl. Ul, ~ en g~n~ralisant au cas non conforme
les techniques de dualit~ d'Aubin-Nitsche d~j~ signal~es au §2; ~ cet ~gard nous renvoyons ~ Lascaux et Lesaint [ 27]. Terminons par quelques consld~rations sur une autre fa~on de mettre en oeuvre des ~l~ments non conformes. Pour fixer les idles, supposons qua l'on ait l'inclusion Vh C C°(~), mais no~ l'inclusion Vh C Cl(~). Les fonctions de l'espace Vh ~tant localement "r~guli~res", la conformit~ de la m~thode serait assur~e si le long de tout cSt~ "int~rieur", adjacent ~ deux ~l~ments K et K', on avait
(3.25)
h
h
an K + %TK,= 0 pour tout Vh E Vh,
173 et si l'on avait aussl sur ~Q (3.26)
~--~-- = 0 p o u r t o u t
v h e Vh -
Si les relations (3.25) et (3.26) ne peuvent ~tre satisfaites, du moins peut-on les eonsid~rer comme des eontralntes et les p~naliser, suivant en cela une technique eouramment utills~e en pratique; el. par exemple Zienkiewiez [ 39]. A v e c l a d~finition (3.2) pour la forme billn~aire ah( .,.), le probl~me discret consiste done ~ chereher le minimum de la fonctionnelle i Jh(Vh) = ~ ah(Vh,V h) - (f,vh),
(3.27)
les fonctions v h ~tant assujetties aux contraintes (3.25) et (3.26), que nous ~crirons sous la forme ¢(v h) = 0 avec
~vh
(3.28)
%(Vh) = K,K'E~_ ~ .~K.I...~K,C ~
3vh + --
anK,)
2
~vh 2 dy +
.rr [ ~ ) dy.
K~K' h La m@thode de p~nalisation eonsiste alors ~ chereher le minimum de la fonctionhelle
(3.29)
*
1
Jh(vh) Jh(Vh) ~--[£') ~(Vh3 =
+
lorsque v h d~crit le sous-espace Vh, ou ~(h) est une lone%ion de h convenablement cholsie qui tend vers z~ro lorsque h tend vers z~ro. G~n~ralement, la fonction c e s t de l& forme c(h) = C h a , o~ la constante ~ > 0 est ehoisie de faqon ~ obtenir le meilleur ordre de convergence. Exemple 8 (cf. Figure 9). Le point a es% ie barycentre du triangle, et l'espaee Pest
l'espaee P3 , dont la dimension est i0. Cet ~l~ment a ~t~ consid~r~ par
Fi6ure
9
174
Babu~ka & Zl~mal [ 4]. Avec le choix , = 0 dans la forme billn~aire, ils ont montr~ que au-u h ilh ~ C h 1/2 gu Ji3,n
(3.30)
pour le ehoix optimal £(h) -- C h 2. Du fair de la p~nalisation, il n'y a plus de patch test ~ passer, ce qui permet de conserver la valeur au baryeentre comme degr~ de libertY, K l'inverse de ce qui se passait pour le triangle de Zienkiewicz (cf. Exemple 5). REF~ENCES [ i] Adini, A.; Clough, R.W. : Analysis of plate bending by the finite element method, NSF R e p o ~ G. 7337, 1961. | 2] Aubln, J.P. : Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by @alerkin's and finite difference methods, Ann. Scuo! a Norm. Sup. Pisa 21 (1967), 599-637. [ 3] Babu~ka, I.; Aziz, A.K. : Survey Lectures on the Mathematical Foundations of the Finite Element Method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differentiai Equations (A.K. Aziz, Editor), pp. 3-359, Academic Press, New York, 1972. [ ~] Babu~ka, I.; Zl~mal, M.: Nonconforming elements in the finite element method Technical Note BN-729, University of Maryland, College Park, 1972. [5] Bazeley, G.P.; Cheung, Y.K.; Irons, B.M.; Zienkiewicz, O.C. : Triangular elements in bending-conformlng and nonconforming solutions, Co._nferenee on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. [ 6] Birkhoff, G.; Mansfield, L. : Compatible triangular finite elements, J. Math. Anal. Appl., ~ para~tre. [ 7] Bogner, F.K.; Fox, R.L.; Schmit, L.A. : The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas, Conference on Matrix Methods in Structura I Mechanics, Wright Patterson A.F.B., Ohio 1965. [8] Bramble, J.H.; Hilbert, S.H. : Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1971), 362-369. [ 9] Bramble, J.H.; Zl~mal, M. : Triangular Elements in the finite element method, Math. Comp.2~ (1970), 809-820. [ i0] Brezzi, F. : Sur la m~thode des ~16~ments finis hybrides pour le probl~me biharmonique, ~ paraltre. | ii] Ciarlet, P.G. ; Orders of convergence in finite element methods, The Mathematics of Finite Elements and Applications (J.R. Whiteman, Editor~,pp. 113-129, Academic Press, London, 1973. | 12] Ciarlet, P.G. : Conforming and nonconforming finite element methods for solving the plate problem, Conference on the Numerical Solution of Differential Equ ations, University o-f Dundee, July 03-06~ 1973. | 13] Ciarlet, P.G.; Sur l'~l~ment de Clough et Tother, K para~tre. lh| Ciarlet, P.G. ; Raviart, P.-A. : General Lagrange and Hermite interpolation in R n with applications to finite element methods, Arch. Rational Mech. Anal.
~6 (1972), 177-199. | 15] Ciarlet, P.G. ; Raviart, P.-A. : Interpolation theory over curved elements, with applications to finite element methods, Computer Meth. in Appl. Mech. and Engnrg i (1972), 217-249.
175
[ 16] Ciarlet, P.G.; Raviart, P.-A. : A nonconforming method for the plate problem, para~tre. [17] Ciavaldini, J.F.; Ngd~lec, J.C. : ~ para~tre. [18] Clough, R.W.; Tocher, J.L. : Finite element stiffness matrices for analysis of plate in bending, Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. [ 19] Crouzeix, M. ; Raviart, P.-A. : Conforming and nonconforming finite element methods for solving the stationary Stokes equations, I, ~ para~tre. [ 20] Fraeijs de Veubeke, B. : Bending and stretching of plates, Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. [21] Glowinski, R. : Approximations externes, par ~l~ments finis de Lagrange d'ordre un et deux, du probl~me de Dirichlet pour l'op~rateur biharmonique. M~thode it~rative de r@solution des probl~mes approch~s, Conference on Numerical Ana].ysis, Royal Irish Academy, 1972. [ 22] Irons, B.M.; Razzaque, Ao : Experience with the patch test for convergence of finite elements, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, Editor ,~---pp. 557-587, Academic Press, New York, 1972. [ 23] Johnson, C. : On the convergence of some mixed finite element methods in plate bending problems, K para~tre. [ 24] Johnson, C. : Convergence of another mixed finite-element method for plate bending problems, Report No. 27, Department of Mathematics, Chalmers Institute 0f Techn°l°$7 and the Universit[ of GStebor6, 1972. [ 25] Eondrat'ev, V.A. : Boundary value problems for elliptic equations in domains with conical or angular points, Trud[ Mosk. Mat. Ob~6". 16 (1967), 209-292. [ 261 Landau, L.; Lifchitz, E. : Th~orie de l'Elasticit~, Mir, Moscou, 1967. [ 27] Lascaux, P.; Lesaint, P. : Convergence de certains ~l~ments finis non conformes pour le probl~me de la flexion des plaques minces, ~ para~tre. [ 28] Morley, L.S.D. : The triangular equilibrium element in the solution of plate bending problems, Aero. Quart. 19 (1968), 149-169. [ 29] Ne~as, J. : Les M~thodes Directes en Th~orie des Equations Elliptiques, Masson, Paris, 1967. [ 30] Nitsche, J. : Ein Kriterium f~r die quasi-optimalit~t des Ritzchen Verfahrens, Numer. Math. ii (1968), 346-348. [ 31] Oden, J.T., Some contributions to the mathematical theory of mixed finite element approximations, Tok_T_~Seminar on Finite Elements, 1973. [ 32] Plan, T.H.H. : Finite element formulation by variational principles with relaxed continuity requirements, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equat~0ns (A.K. Aziz,Editor), pp. 671-687, Academic Press, New York, 1972. [ 33] Raviart, P.-A. : M~thode des El~ments Finis, Universit~ de Paris Vl, Paris,1972. [3hl Sander, C. : Bornes sup~rieures et inf~rieures dans l'analyse matrlcielle des plaques en flexion-torsion, Bull. Soc. Ro)-. Sci. L i ~ e 33 (i96h), 456-hgh. [ 35] Strang, G. : Approximation in the finite element method, Numer. Math. 19 (1972), 81-98. [ 36] S~rang, G. : Variational Crimes in the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differentlal Equations (A.K. Aziz, Editor), pp. 689-710, Academic Press, New York, 1972. [ 37] Strang, G.; Fix, G. : An Anal[sis of the Finite Element Method, Prentice-Ball, Englewood Cliffs, 1973.
176
[ 38] Zienkiewicz, 0.C. : _The Finite 'Element Meth0d in En~ineerin~ Science, McGravHill, London~ 1971. [ 39] Zienkiewicz, 0.C. : Constrained variational principles and penalty function methods in finite element analysis, Conference on the Numerical Solution of Differential Ecr0~tions, University of Dundee, July 03-06, 1973. [ ~0] Zl~nal, M. : On the finite element method, Numer. Math. 12 (1968), 39~-~09.
S
B'r,~ce M. I r c ~
Reader, University of Wales, Swansea.
The general purpose thin shell element described here is essentially a nonconforming quadratic Ahmad element, with ~ o r d e r nodal parameters which accept multiple junctions.
integraticm, and with A historical introductiom
clarifle~ the develDpmemt and leads smoothly into the fcrmalati~, and the diacus~iem of patch-test convergence.
The geametry and the shell theory are new
and are ccmsistent with rigid body responses of a general patch.
~plementatlon
will be via a foolproof shape function routine. Introduction et ~oti~ation "Semiloof" est prohablement la derni~re phase de la recherche de l'auteur pour un ~l~ment de coques~)une recherche motiv~e par un m~contentement profon~ envers presque tousles ~l~ments de toques familiers.
La liste suivante des conditions
requisea abr~ge cette m~fiance: (i)
II f~ut un ~l~ment ~e raideur direct, car heauccup de millions de dollars
sent d~j~ perdus sur les progranm~s d'ordinateur qui admettent tels ~l~mauts meulement. (ii)
L'~l~ment dolt ~tre utilisable ("mixable" en anglais) - au s e ~ exact de
l'~preuve de rapi~gage (q) ("patch teat" an anglais) - avec lea ~l~ments voisins tris~agulai~s ou quadrilatSraux, ~
avee lea membranes isoparam~triques, et avee u,
cerZain ~L~ment eonvenable de poutre~ (iii)
Lea ~l~ments de toutes formes ne violent Jamais lea mouvemeats rig%des, J
J
~insl que taas lee assemblages de tels elements.
Sewiloof est le premier quadrilat~re de !a famille Ahmad qui a eatisfait *
.
J
nln~erlcuement aux epreuvea (ii) et (iii).
.J
I
J
Lee raplecagea d' elements
triangulaires et quadrllateraux Imbr!ques ont r~ussi aux essais d'ordinateur. Um ~l~.ment de poutre tres special real~se" " r~cemment _oar F. Albuquerque, de Louren~o Marques.
178
(iv) La plupart des coques technologiques ont des angles vifs et des embranchement~ ~Lultiples. Dc~e, un ~l~ent dolt modeler ceux-ci, sans exceptions pathologiques, et sans complications telles que transformaticas locales etc. (v)
Quelque soit le module choisi, on doit le traduire par l'interm~dialre d'une
routine de fonctions de forme nolre"
("shape function routine" en anglais), une "boite
('black hcx" en anglai~) de laquelle l'utilisateur ne dolt l~aS se
acueier et ~
laquelle se trouvent toutes los complications telles que
transformations et conventions de signe. De cette fa~on le chercheur peut garantlr fermement ~ e l'utillsateur pourra incorporer facilement l'~l~ment dane /
.
sea progran~ne, et que toutes les caracterlstiques matricielles exotiques de l'~l~ent, dcnt c~ aura besoin plus tard - In~vltahlement ~ ccttrt d~lal pou~t
Wo~e:
~tre aJout~es sans peine.
Si les ~l~ments iso-P sc~t touJours en vogue, e'e~ en grande pattie que
la routine des fcnctions de forme est cr~e, v~rifi~e et bien documaut~e~ et que tout le monde peut lui donner des t~ehes multiples.
Nous euvisageons maintenant
une routine plu~ ccmpliqu~e, mais conservant ces avantages. (vi)
La routine de~ fonctions de forme doit rester abordable au Programmeur de
maintenance grace ~ une co~e lucide, une documentation complete, et des ordres d' i~pression intermediairs facultatifs. (vii) Toute routine dolt %ire convenable et ~ r ~ l a b l e .
Par exemple le
proe~d~ se change souvent ~ la premiere rencontre avec un nouvel ~l~ment: ll faut que l'utilisateur ne doive jamais l'en avertir. (viii) Encore c'est un bienfait de fournlr ~ l'utilisateur des dia~lostics abondants~ L'endroit naturel pour bien d'entre eux est la routine dee fonctions de forme. (i~)
Ii faut que l'~l~ment ne montre auoun caprice en fonctionnement@
Par
°~
exBmple i! dolt ~iter tousles pleges tels que rang defectueux. (X)
Use pr&oision extreme n' est jamais requise, mais un mailla~e ~oseier
dolt donner des r~su!tats acceptables.
Solon le consensus general, lee
contraintes qui varient lin~airement satisfont le mieux lee besoins teehnologique s~
179 Di_~position des Noeuds Semiloof
II es% di~ficile de critiquer la disposition de la ~g.1 du poimt ae rue de l'utilisateur - elle a semBl@ la plus attirante parmi celles examlm~es ~ laref. q.
Aux coins et ~ mi-c~tes^ " nous avons u, v, w, les fl~che~ an directions
globales.
Cecl suffirera pour un~ membrane, mais ne peut pas emp~cher le
pivote~ent pour time coqme avec flexion.
Afin d'assurer une conformit~
approximative pour les pentes, nous Introdui~ns les rotations aux points de J
G~ss~ le long de chaque cSte.
Les 32 degr~s de libert~ devraient suffire
d~terminer lem champs de contraintes lin&aires, ~ la lois pour les actians de flexion et de membrane.
Pour la logique du programme nc~s groupons les 5
vaz@ables le long d'un cSt~ ccmme si elles agissaient en son milieu.
La
convention de signe - ~ laquelle l'utilisateur s'interesse rarement - d~pend des num~ros de noeuds des coins voisins, disons N I , N 2.
Ceux-ci ne sont
jamals ~gaux, car il me rapportent ~ des cordonn~es nodales.
Disons N2> Nq.
Le sens des rotations aux points c et d est fix~ par la r~gle de la main droite pour le segment orient~ Nq-~ N2, c pr~c~dant d.
wia ~es~?a2:::t~eco~ ~i~,1 Pr~curseurs de Semiloof
.
On peut resoudre~ une probl~me 2D avec noeuds de Loof
~ng1~is)s~tues .
("Loof nodes" en
. . . atux ~eu~x .points ae Gauss. sur chaque cote d'un el~ment, comm~
indiqu~ ~ la fig.2.
I1 faut ajouter un 9me noeud au centre de l'~l~ment: la
recherche de~ fonctions de forme est donn~e a la ref. 7 f ~
A cause des noeuds
manquants a1~x coins, on ne peut mSme pas assurer C ~0) entre lee elements. z
180
•
•
.
J
P~rtant, le dlscontl~11te tend vers P2(~), la polynSme de Legendre qui s'annule aux deux points de Gauss (ici ~ arpente le cote, -I~ ~ ~ I ) dent l'int~grale et premier moment sent nuls.
Nous examinons l'~preuve de rapid,age, les contraintes
~tantmaintenues constantes.
Touteperturbatic~de
cet ~tat unlforme provoquera
des disconti~uit~s de d~placement entre les ~l~ments. comme P2(~) le travail r~sultant est nul. suffise, I e ~ u v e
Si les sauts varient
Par consequent, po~u~u que Is rang
de rapze~age est satisfaite.
Not__._~e. Nous raisons honneur a Leer.
A la ref. ll il mentionna des points de
Lobatto mais il parla de son dessein d'ess~yer lee points de Gauss ouand l'occasion se pr~senterait.
~. ,.,
2.
.
3
(l';,~
L element de Visse
(fig.3), parmi d'autres, qui postulent une variation A
i
lin~aire de moment de flexion sur chaque cote.
Nous pouvons f ~ e r
sans
peine de tels ~l~nents en termes de moments localis~s: M = (moment/cm.). d(c~t~)/d
(I)
aux deux points de Gauss, en sorte que tout travail est l'addition de M fois la rotation aux deux neeuds de Leer.
Ensuite une inversion partielle (~) ("part-
invermion" en anglais) donnerait preclsement " " ~ la version "eq~ivalente en raideur directe avec rotations de Leer.
3.
La membrane isoparametrique de fig.~ adapte lea /
fonctions de forme N.(~,~) qui etabllssent une 1
correslx~enee (~,~)-~(x,y,z) en 3 dimensions.
Y
(2)
(z;i
Ctest tr~s simple.
.
p
p
Lea complex~tes se presentent
seulement en calculant les d~formations etc., qu'au doit exprimer en cordonn~es locales X, T darts le plan tangentiel de la membrane.
Nous arena
181
ou ~ar exemple X est le veoteur unit~ en direction X, ~ est le vecteur base covariant ~(position)/~, ainsi ~Ni/'~X,
BNi/~Y
et
.
est le produit scalaire.
Nu~s calculons
au point do~n~ ~, ~ . Les @~formations et autres
quantit~s de ce genre suivent facilement.
Par example, la contri~uti~
~W/~ X ~ cause d'un d~placement ~i au noeud i e s t •
4-
x
,_I~. toque feuillet~e d ' A ~ d
("membrane-stack" an ~
~
originairement con~ue quand les raisonnements Pi~. 5
vectoriels ~taient manifestement absents dans
Q,) Son
les manuels d'~l&ments finis, ~tait r~alis~e comme ~tude introductive. utilisation g~n~rale @tait inattendu.
Eous discutons seulement le quadrilat~re ~ 8 noeuds et 40 degr~s de liberteo Chacune des 8 lignes rigides de la fig.5 se soumet aux fleches u,v,w a la misurface du feuillet, et lie les membranes. l'intervalle
(~ , ~+d~ ),
Chaque feuille correspond
comme indiqu~ ~ la fig.5. G~om~tri~Aement noms
avons une brique iso-P 20-noeuds, mais~ x,y,z varient llnealrement aveo ~, en sorte que chaque membrane peut avoir une ~paisseur nonuniforme. Elastiquement nous n'avons pas encore une coque r@aliste.
Le~ membranes ressemblent ~ un livre
dont les pages Elissent sans contrainte.
ments
~
.
Ncus
nous proposons de les coller afin de leur donner une raideur de flexion. (i)
Ii y a deu_x techniques:
Fi~.6
Abroad ajouta l'~nergie de d~for~tion de
glissemsnt ~ mesure que ohaque membrane glisse contre les voisines. imag~ grossi~re: le glissement aux ~
s
ermines est faible ou ~
O'est une en
r~alit@, hors du cas rare o~ des charges tangentielles sensibles s'appliquent la surface. Toujc~rs nous p r ~ o n s
que le glisse~ent varie comme (~ - ~2), e%
il ~Ait que les normales courbent comme indiqu~ ~ la fig.6. Ahmad effectivement divise YXZ
et ~YX
par ~
&fin de rectifier le module dont les normales
sont rigides. Notons que ~XY' une d~formation d~ns la membrane, n'est pas modifi~e.
182
(ii)
Alternativement, pour une coque mince nous voulons annuler
YXZ
et
YyZ o
En ca~ de oertains ~l~ments d~llnquants mais comp~titifs ("delinquent elements" en anglais, avec de~ "discrete Kirchhoff assttmptions")
nc~s annulcr~
~'XZ et
/
~'YZ aux points isole~, cholsis avec soir~
Les contraintes nous permettent
d'~liminer des variables nodales convenables avant d'assembler lea ~l~ments. Note.
Les deux techniques ne ~'excluent pas.
Par exemple, en principe on pout
annuler toutes variations de glissement, afin @e rendre eelui-ci constant. Autrement, on pout le contraindre ~ varier lir~air~ment avec ~ et ~, etc.(~) 5.
,,feuillet~e
La coque
d'Ahmad avec int~Tation 2x2.
Par suite d'un examen des
r~gles d' int~gratica, Too remarqua que Gauss 2x2 dcane des r~sultats remarquable~nt amellores, pouvu qu'il pre.sente seulement los contra~ntes aux ^
ll~.
memos points 2x2.
t
/
Cette observatlon
est toujcurs inexpllquee, cepen~nt
ells a inspir~ au moins deux innovations importantes. Discutons pourquoi les r~sultats sages nous ~tonnent ~cialsment,
surtout
/
lorsque la plaque c~ la coque est mince. En tel cas l'energle de d~formaticn de glissement applique de fair 8 contralntes rigi~es, car on s~it que le~
I
o~--~
3 noeuds/element
2 glissements aux points de Gauss 2x2 sent o
reli~s lin~airement et ind~pendamment aux variables nodales. Donc examinons un mailla e raffin~ c o m e indlqu~ & la fig.7 p~ar une plaque d'Ahmad.
Chaque element ajoute 3 noeuds, et
~g.7
c h a ~ e noeud a 3 variables - une fl~che et deux pentes - de sorte que chaque J
J
element ajcute 9 variables.
Soustrayons les 8 contraintes.
Alors ncus
n'aJoutons qu'un soul degr~ de libert~ par ~l~ment de plaque suppl~mentaire. N@anmoins los rSsultats nous plaisent. 6.
Le premier ~l~ment d61inquant iso-P se sugg~ra d'une part par la sagesse
s Lmprevae de la coque feuillet~e avec •
in
~ tegration 2x2, et d'autre part par son
Inefflcacit~ technologlque intrinsiq-e.
Car le glissement entre les membranes
voisines ne connote que pour quelques pour cent de l'~nergie de d~formation I
totals.
.
s
En reallte, il imports peu qu'on l'inolue cu pas.
Pcurtant la toque
feuillet~e fournlt une variatica de glissement quadratique en ~, ~ , maim •
p
.
llnealre en d~formation de flexion. •
I
Jusq'ici nous avers trop de degr~s de liberte; mais quels 8 devons-nous omettre? La fig.8 nous donne un indice. lin~airesa
Imaginon~ que AB soit une poutre aux flexions
On pourra la d~terminer entmerement'~ ave~ la fl~che et la pente aux
deux bouts, sans los deux variables.
Nous nous demanderons sl cn pout aussi
183
%ter les variables semhlables en cam de plaque ou de coque.
• Les glissements
" ~
¢ £d
J
soit oms Fig. 8
Note.
Par consequent, pour un grand probl~me chaque ncuvel %l~ment ajoute 5
variables au lieu d'une.
Si nous averts an trouv~ 4 en sus o'est que la fl~che et
ls pente ~ ms-cote ~ " ne doivent plus se conformer entre les 61&merits. Los nouveaux degr&s de libert~ a~liorent les r~sultats, un petit peu. Razzaqae rut le premier K coder les &l@ments d&linquants iso-P, et rapporta que la precislon rut parmi los meilleures publi~es pour los elements de forme g&n~rale.
II se peut que la plaque d~linquante soit l'aboutissement de la
formulation isc-P. (')~@ )
Mais la toque d~linquante fsit d~faut en pratique. .
A
Deux variables, une fl~che
I
et une pente, disparaiss~nt &e chaque ram-cote. On dolt choisir avec soin leurs directions, et par consequent on doit choisir aussi les 3 variables restantes. ,,
+,
Razzaque a d~termin~ cos directions in~pendan~nent dans chaque element, sans ~e soucier des l~g~res inconsistanee~. On soup~onne qu'il s reussi ~ cause de la smmpllclte geometrlque de ses probl~mes.
Sans doute il n'aurai% pas obtenu de r~sultats utilisables pour des
toques ~ angles vifs et ~ embranohements ~altiples.
Note.
Los % l ~ e n t s d~iinouants, sent justement denornmes." I Dlrmges" " par !'intuition
physique plus que par route theorle de ooques admise, nous avons trouv~ un element competztlf.
Bien qu'il ne satisfasse a l'epreuve de r~aple~ge que pour
I i un maillage de parall~logr~mes (~ventuellement inegaux), los s elements difformes r~ussissent presque. (et il es%
D'ailleurs, quand nous commettons ce d~lit particulier .
J
d
diffieile de preclser en termes generaux ce o~e cola signifie
quand la glissement s'sr~uule aux points choisis), quel statut conserve-t-elle, I' ~preuve de rapid,age?
184
D&placements de Semiloof Libres de Contraintes Avant d'aVpliguer le~ contra~ntes, nous donnons le d~placement de tout point comme l'addition de trois classes: I.
• D~placements aux coins st ~
u,v,w oausent une translation des 8
" ^ •
m l - c o t e s
Nous comptons 2& degr~s de libert~
lignes rigides comme indiqu~ h la fig.5. de cette sorte. 2.
Rotations aux noeuds de loof et au centre, qui introduisent une difference de •
4
.
.
s
.
depiaoement entre les surfaces superleure et inferleure.
Nous con~ptons 18
degr~s de libert~ de plus. 3.
Une seule fonction bulle.(1-~2)(1-~ 2)
qui donue un d&placement dans la direction perpencliculaire a l' element au centre .%
(~ = V = 0), et qui s'annule aux front~eres. de satisfaire ~ l'~preuve de rapi&oage.
(fig.9)
•
Ce supplement permit
Nous devons fournir tout ~tat de
flexion constants dans un element, en general quadrilateral et plat, disons
ou Ni( ~ , ~ ) sont les fonctions de forme de coins et de m~-cotes.
~is
is
quadrilat~re n'exige pas touts la base, et par exemple x
=
A + B~
+ C?
+ D~ •
I
Si nous ajoutons le 43me degr~ de liberte, la fonction bulls, c'est cue 2 2 . A la ref.~ nous avons manqu~ cette w = x renferme le terms ~2 condition requise. Semiloof st les Contraintes de Glissement Nous pcursuivons en r~duisant les 43 degree de libert~ aux 32 de la fig.1. t " (~ la suite de plusieurs qui echouerent) se Les 11 contraintes qui ont reuss~ groupent naturellement en trois classes: I.
~
Les 8 pentes~ Indeslrees~ ~ i mlx noeuds de L o of. La rotation
de la normele octane indiqu~ ~ la fig.t0 engendre un glissement ~YZ"
~
~
~
Done il est naturel de contraindre la pente a v e c l a
condition ~YZ -- 0.
~ig.10
185
Note.
Lea 8 glissements aux noeuds de Loof sont reli6s h peu pros via la
matrice identlt6 aux 8 rotations ela~namees. dire¢:te, leg condition~
~XZ = ~YZ = 0
En outre, l'alternative la plus
aux points de Gauss 2x2 ne sont m~eme
paa ~md6pendant as. .p
I
L'~preuve cle rap~eoa~e. Consid~ons encore l'inflmence de l'epreuve sur la feu&lle au niveau ~
~
~+ d ~.
Perturhons les noeuds "int~rieurs"
(c-a-d cerne~ ~ completement • d' "elements ~ - se reporter ~ la ref.~po559,
583)
et demandons-nous si le travail virtual g'annule dang un champ de contraintes planes uniformes.
.J
Aux noeudg @e Loof A e t B ~ la rue en plan du raplecage, 16
fig.t1, lea fl&ches~ normalea a RS sont continues
~
.
.
]
A
.
.
.
entre lea ~l~mm2~ts voisins ~ cause des pentes nodales costumes.
Si leg fl~ohes dens la
directicm de AB se~t ~galement continues, c'est qua WR, Wset wll, lea fl~chea aux coitus et ~ mi-cSt6, moat communes., e~ qme le glissemen%
Fifo11
~(YZ s'annule ~ cause des 8 contraiutes
~anposees. Par ccms~quent nous devoms satisfaire l'&preuve de rapxegage quelques soiemt lea 3 derni~res contrai~teg qu'il nc~s plalse d'imposer. 2.
Lea 2 pentes au centre.
a lea glissemen%s lat~reux
Observation pr61imlnalre. ~(XZ' ~YZ"
Une plaque, en plan XY,
Or, poar Z fixe, e% X,Y tournant, la
quart%it~ bidimensionnelle
me %ransforme cOnm~ un vecteur. Lea 2 contraintes. et n o u a
Noua z~s~rvons les vecteurs unit~s au centre, X 9 et Yg'
fixtures x9 .
e(gupe
±cie)
employant lea points de Gauss 2x2.
2 points de Gauss. Exact. Note.
Pr~lable~ent une version plus simple
l'~preuve de rapid,age. con%raintes ncuuniformes~
S~
4
mi-cgte. Trop souple.
a ~chou~, malgr6 sa reussite
Ella se d6formait trop souplemen% d~ns leg champs de Au lieu des deux in%~Erale8 de (7), nous avons
contrain% lea deux glissemen%s au centre.
D'une mani~re semblable l'~l~ment
quadratique de poatre indiqu~ ~ la fig.12 est %rop souple avec une seule ccmtralnte.
{
186
3.
La font%ion bulle
es~ ~videmment fiche d'une cer%aine c~nbinaison ~e
glissement=:
est ~me e o n t r a ~ t e cr~dibleo
•
/
•
Nou~ ~ceferons transformer l ' ~ t ~ g ~ a l e a
(gllssemen% normal) d(arc)
=
0
(9)
frontiere e% oelle-oi es% la forme employ~o G~ometrzqaes~
S em~loqf
nSemiloof= est un module de coque mince, et nous faiscms les supposi%ions I
/
geome%ri~ues oz~dihles c o ~ e suit. (i)
Pour commemmer, nous ~enclons approximativement normales ~ la surface
moyenne du feuillet les lignes nodales rigides de la fig.5. J
Par conqueror, lea
I
el~ments ne m'imbriquent en general pas aux embranchemen%s. (ii)
I
J
La coque es± si mince que l'~lement de surface d(superficie) pour route
membrane peut ~tre confondu avec am valeur sur la mi-surface. (ill)
Le~ membranes sont presque parall~les.
L'&preuve de raple~age a aeja admis %ous les quadrilat~res pla%es.
Un autre
o
prinoipe aomine ~os main%enan%: desormals nous exigerc~s qu'tm rapie~age d'elements de tou%es formes ne viole j ~ i s
les mouvements rigides,
%
nous avons fair les hypotheses sans assez de soins,
A Is ref.1
No%ons:
(a)
Les &l~men%s strictement iso-P pourvoien% les mouvements rigides,
(b)
T~/te ligne rigide - comme on la ooque feuillet~e d'Ahmad - es% tu~e
contr~mte d'accord avec un mouvement rigide. (C)
Un couple d'~lements fournlrait un mouvement rigide seulement sl les
variables nodales qui les llent ensemble (en particulier les pentes)
se
d&finissent ex~ctemen% pour %c~% mouvement rigide donn~. Il faut que route lizne rigide ~ ,~?= constante soit normale, ~ chaque noeud de Loof, ~ la tangente frontlere.
Autrement un mouvement de rotation ~
normal
la surface moyenne, con~ne lux~iqu~ ~ la fig.~3, provoquerait tune "rota%ion"
187 A
f a u s ~ autour de Sj.
Si nous devons
@,
/% K
~ ) ~ -" ccastante
compenser les ~paisseurs dorm's, c'est que eette ccr~/tion n' eat pas assuree automatiquement.
Quand nc~s rencontroms
um nou~el ~l~nent, nous devons remplir
FIR.i
plusieurs t~ches: Stade !.
Nous intez%oolcns les ~paisseur~ Ti, i = ~ a 8, donn~es aux coins e%
mi-c~te~, afin d'emgendrer los Tj, j -- ~ ~ 9, aux noeuds de Loof et au centre. Nous ~levons ~j (la premiere ~ apprccch~ation) perpendiculaire ~ la mi-surface. St ado 2"
A f ~ d'assurer que le~ Tj, j = I a 9, d~terminent une base
geometr~que darts l'espace de fonctions de coins et de m~-cotes, " " " ^ "
N~(
~, ~ ),
i -- ~ ~ 8, il faut imposer une condltion aux ~pai~seurs Tj, c'est ~ dire
• j : ~j
multiplions par (-I) j
(-~
(~-p.sjsj)
(9)
et l ' a d d i t i ~ o ~ :
i
I
- p.sj sj) = 0
(io)
Trmduisc~ en f o z ~ ~tricielle:
~e~ucc~p de techniques sont possibles: celle-ci emt imdereglable, c ~ 3x~ e~.~ d~inie positive,
la matrice
N o t o ~ que les ajusteme~ts f o r ~ s sont de m~eme ordre
~e grandeur que les variations ~' ~paisseur normale T d ~ s un ~l~ment iso-P dont lea T i sont co~.~t~ts aux c o i ~ et mi-cotes~ Puis nous construisons et conservons les d~placements relatifs
~. = ~. x ^
qui engendrent la pente ur~ite conm~_ue avec les el~nents voislns: aussi m.O : qul domnerait une pente approximativement unitaira le long du cote, J . . . une variable destin~e pour l' elLm2matlon. Stade ~.
Nous constrtLisons et r~luisons la matrice des contraintes.
Sta~e ~.
Nous nous sermons de cette matrice-ci pour creer les fonctions de
#
¢
I
force a l'endroit donne ~, 7. ~auf dans le cas d'un nouvel element, nous sauto~e directement au 4me ~tade.
0
188
Le~ ~esures @e D~formation pour une Coqae Peuillet~e Ninoe Ccmme dans~ les theories clasmiques de coques, notre but est simplifier le calcul ~anm introduire des inexaetit~des excesmives+
Nc~s employons les symboles
suivants: A
Z
=
normale unit~ au plan tangent ~ la surface moyenne.
/%
=
/ / S nermale dirig~e vers i t emtezaeur ~ana le plan tangent ~ I' element ~ la
X
.
fronti~re. =
d~placament ~ la surface moyemae. i
•
.
,p
•
deplacement relatif entre les surfaces superieure et inferzeure, tel que ~
rj Ni, Lj Ainsl T
=
= &
au meme point ~ i +V•
S j,
s
8 fonctio~s ae forme de coins et @e mi-cb~es, e% 9 foneti~ms ~e Loaf. = INid i
et
~
=ILj
j,
i= I ~ 8
et
j =I ~9.
vecteur d'~pais~eur
: ~ +~+__ :
~
+ s~ + ~ ,
<~i~<~.~.
Par cc~s~quent T es~ la composante normale, et R, S introduisent une pente Initiale.
T
n' ~ale pas ~ LjTj.
car X et Y ~e trouve~t ~
Nous in~e~ons:
le plan ~ - ~ ,
quand $ varie de -~ a ¢, tenant ~ et ~ ~l~z
=
(2~/~!~
:
(~.x
- ~'+I~ x
e@ car le changement ~e positic~ constants,
esPY.
- s~l~)z)/~
Or, (~5)
Par exemple, uz
-
Les~emx ~llssements,
~XZ
~u x =
termes de correction entra~nant
-
U Z + WX R
(+~+)
s,z)/~
et
et ~YZ : V Z ÷ Wy S,
~ cause de
exigent les
la pente initiale
de T. Les ~riv~es qui d~termiment les d~fcrma%icr~ de membrane e% le mouvement de rotation sont simples. Par example, :
ux
:
(+5)
189
au lieu Re oourbares no.as utillsons directement UXZ, Uyz, VXZ et Vyz: oUxz
au lieu de W XX
-~Z-VXZ
-V~z ~ n ~
au lieu~e
2Wxy
de W n
(I6)
Afin ~'~riter la complication de (73), nous rempla~nS UXZ par:
Car en pratique nca,s multiplierons UXZ par T,
/
~
afln d~ calcttler I' ecart ~e d~f°rmati°n entre lee / surfaces s~p~rieure et inf~rieure~ De eette fa~on n ~ co~arons la ~formation en A, dmns la fig.?4, / avec eelle en B, pas orthogonalement au-~essus de A. To~s les aeux s'annulent ,i le mouvement e~t rigi~e.
I
J
7
II y a enI~ore deux %ermes supplementalre~ de flexion, facilement "l~noTes9 " causes i "
"
Par d.
(a) Si ~ R / } ~
et B N i ~ V
sont les ~mem aux surfaces sup~rieure e%
inf~rleure, c'e~t que N i es% une function de ~, ~ seulement, ms.ls un point X devient X + R, Ydevien% Y + S. Or,
[" "]i:t Sx
e
!%
tSx
e
(b) Si X,Y scat dmas la surface inf~rieure, la directi~a normale ~ la surface superzeure devient (-Tx, -Ty, !). A cause &'une fl~che W n~as avons les J
.
compesante~ ~ U = TXW,
J
.
~ V = TyW dan~ la surface superle~re. Par exemple,
190 R&sumons so=malrement:
(2.~)
sy (22) i
p
f
(Voir (23) aussl, la version preferee.) aux m~v~ments ri~i~es, Mettons V -- -Z a@ W = Y, de sorte que ~ = Maintenant mettoms ~ Note,
= Z;
D~mcntrons que les forn~les ob~issent
= X;
sur la surface infer~e~re + SZ.
U = O,
Les d~formations s'annulent.
etc.
(22) donne une ma%rice an%isymetr~que au lieu de nulle, qui peu% %
p
aggrav~c les prohlemes ~e m~uvais oonditicmnement,
#
Nous preferons modifier (22):
Semiloof En Pratique Cette explication eourte era% developpee ~ la ref.g qui eomprend une versio~ anclaune et d~fectueuse du programme.
L'ensemble actuel des fonctions de forme
e~t aussl disponible sans contrainte.
I1 oc~prend 450 instructions FORTRAN et
il f~ut ~600 memo~res " " (nombre reels) pour les tableaux. et les ~
Avecla documentation
essais diagnostiques, il devrait Stre facile ~ incorporer~
On peut
am&li~er la vitesse, pourtant, car le progr~mme a chang~ continuellement pendant les~ ~4 mois de raise au point, la vi%esse etant negllgee au profit de le clart~ et la faeilite de modification. •
o
Le ran~ et les mecanlsmes/0arasltes. Id~alement la matrice de raideur pour un I
I
element qusdrilat~ral a l e rang de 26 = 32 (le hombre des variables) - 6(le
nombre ~es mouvements rigides disponibles); 18 = 2 @ - 6.
.
r
s pour un /element triangulaire,
.
Mals chaque point d'integratlou peut contribuer au m~ximum i 4
•
•
6 (le rang de la matrice d elastlelte), dormant 24 pour le quadrilatere Integra%lon 2x2, et 18 pour le triangle int~gr~ par !a regle de mi-c~t~s, ar c~%~eq/ent le quadrilatere a aux molns deux
.~
mecanismes parasites, com~ne oeux indiqu~s ~ la
fig.15 p = .
~l~ment
car~. (L'ex~riencen'a p a s ~ l ~
encore montr~ ~'autre m~canisme ni pour le triangle ni pour le qu~arilat~re.)
I1 es% bien possible qu'un tel
~
191 m~aa.ui:~e puls~e ae transmettre ~ travers les elements voisins et de i~ contamine t~te
la solution.
Nous pouvon~:
(a)
savoir ~
qu'il faut faire apropos du probleme et l ' ~ t e r
(b)
employer toujours les triangles.
ordir~irement.
Quoique ce~i entraTne davs~tage de
variables, les inexactitudes sont neanmo~ns cinq lois plus grandes que lorsqu' o~ utilise les quadrllateres. (c)
employer une r~gle d'int~gration ~ 5 points: s ¢ -1
-~
4 1
t
ou b : I - ¼a
et
B = (3b) "~, disons a = .O~, b : °9975, et B = ,5780733131
au lle~ de .577350~69. /
#
La performanc.ede l! ~l,ement, ?°
Les elements de toques iso~P ~ 8 noeuds aveo mntegrat~on 2x2 et avec le~
mesures de d~foxm~tion ~ai ne violent jamais les mouvements rigides, ont donn~ les r~ultats excellents pour les coques regul~ere~.
Surtoat la performance
avec un maillage grossier est remarquableo 2.
Pui~que la convergence est assur~e, l'auteur s'est concentre a l'~tude des
maillages grossiers. •
s
La plaque eirculaire charge .
~
i
ur~formement, c o ~ e indlque ~ la rigor6 ( ~ = 0,3) a donn~ des ~carts de contrainte de ~ 3% m a x ~ /
La plaque carree de la fig.1 7 a donn~ une fl~che centrale +O@13%.
,J
192
3.
La reponse de la plaque carree/ a une charge ponctuelle est moins satisfaisante
oomme i n a _ i q ~ (a) s
la fig.t8, et bien pire que laref.
Les coins peuvent pivoter: voir la fig.19. I
I' element de Visser (b)
Nous l'admettons car:
Probablement
co~ettrait cette faute.
Ordinairement une charge ponctuelle est
reprise par une poutreo Reccmnaissance. L'auteur veut exprimer sa gratitude a son coll~gme M. Ivan Cormeau de Bruxelles pour l'aide patiente propos de la granm~ire et du choix de sots.
Fig. i 9
References I.
Ahmad, S., "Curved finite elements in the analysis of solid, shell and plate structures" Ph.D. thesis, University of Wales, 1969.
2.
Al~querque, F., "A beam element for use with the Semiloof shell element", M.Se. thesis, University of Wales 1973.
3o
Asplund, S.0., "S~ructural Mechanics: Classical and Matrix methods" PrenticeHall, Englewood Cliffs, N.J., 1966.
4.
Baldwin, J.T., Razzaque, A, and Irons, B., "Shape function subroutine for an isoparametrie thin plate element", Int. J. Num. Meth., to be published.
5o
Irons, B., Lecture notes , International Research Seminar on the Theory and Application fo Finite Elements (A NAT0 Advanced Study Institute), JulyAugust 1973, Calcary University, Canada.
6.
Irons, Bo, "Engineering Applications of Numerical Integration in Stiffness Methods", JAIAA vol.4 (1966) po2035-2037.
7.
Irons, B., Conmment on "A Higher Order Conforming Recan~alar Plate Element" by S. Gopalacharyulu, I. J. Num. Methods in Eng., vol.6 no.2, p. 305-309.
8.
Irons, B° and Razzaque, A., "Shape function formulations for elements other than displacement models", Conference on Variational ~ethods in Engineering, Southampton University, September 1972.
9.
Irons, B. and Razzaque, A., "Experience with the Patch Test", P.557-587, from "The Mathematical Foundations of the Finite Elemeut Method with Applications to Partial Differential Equations" Academic Press 1972.
I O.
Irons, B. and Rasza?me, A., "A fagther modification to Ahmad's shell element", I. J. Num. Methods in Eng., Vol.5, no.& (1973) p.588-589.
11.
Loof, H. W., "The economical computation of stiffness of large structural elements", Int. Syrup. on use of Comp. in Struct. Eng., University of Newcastle-upon-Tyne, 1966.
12.
Pawsey, S. F. and Clough, R. W., "Improved numerical inte~ation of thick shell finite elements", I. J. Num. Methods in Eng., volo3, p. 576-586 (1971)
13o
Razzaque, A., "Finite element analysis of plates and shells", Ph.D. thesis, University of Wales, 1972.
1&.
Zienkiewioz,0.C., Too, J.J-M., Taylor, r.1., "Reduced Integration technique in general analysis of plates of shells" I.J.Num.~ethods in Eng., vol.3 ~1971) 15. Visser, W., "The Application of a curved mlxed-type shell element", S~,p.~Ant.~ • Union of Theor., Applied Mechanics, Liege, August ~ 970.
NUMERICAL
SOLUTION
EQUATIONS P. J A M E T
I.
OF THE
STATIONARY
BY F I N I T E
ELEMENT
(~) and P.A.
NAVIER-STOKES METHODS
RAVIART
(~)
INTRODUCTION Let ~ be a b o u n d e d d o m a i n of ~ N (N=2 or 3) w i t h b o u n d a r y F. We c o n s i d e r the s t a t i o n a r y N a v l e r - S t o k e s p r o b l e m for an i n c o m p r e s s i b l e v i s c o u s fluid c o n f i n e d in ~. Find f u n c t i o n s ~ = ( U l , . . . , u N) and P d e f i n e d over ~ such that
N ~ Z u -- ÷ i= 1 i ~x.1
rAp
=?
in
(i.i) div ~ = O
in ~,
u = O on F, + w h e r e u is the fluid v e l o c i t y , p is the p r e s s u r e , ~ are the body forces and 9>O is the v i s c o s i t y c o e f f i c i e n t (the d e n s i t y of the fluid has been set equal to i). In a p r e v i o u s p a p e r [ 4]! C r o u z e i x and the second author have d e v e l o p p e d a g e n e r a l th-eory for the finite e l e m e n t a p p r o x i m a t i o n of the Stokes p r o b l e m and have c o n s t r u c t e d c o n f o r m i n g and nonconforming t r i a n g u l a r e l e m e n t s (N=2) or t e t r a e d r a l e l e m e n t s (N=31 well suited for the n u m e r i c a l t r e a t m e n t of the c o n s t r a i n t dlv u=O. In the p r e s e n t paper, we shall extend the a n a l y s i s of 4!to the finite e l e m e n t a p p r o x i m a t i o n of the n o n l i n e a r o lem (].I). On the other hand, it has been found w o r t h w i l e to use n u m e r i c a l q u a d r a t u r e for e v a l u a t i n g the v a r i o u s i n t e g r a l s w h i c h appear in the finite e l e m e n t m e t h o d and p a r t i c u l a r l y those a s s o c i a t e d w i t h N ~ the n o n l i n e a r term ~=I u i - . Thus, we shall a n a l y z e in this ~x i paper the effect of n u m e r i c a l q u a d r a t u r e . •
(~)
Service de M a t h ~ m a t i q u e s A p p l i q u ~ e s , B.P. 27, 94190 V I L L E N E U V E - S T - G E O R G E S ,
(~)
A n a l y s e N u m ~ r i q u e , T.55, U n l v e r s l t ~ 7 5 2 3 0 PARIS CEDEX 05, FRANCE.
Centre d ' E t u d e s FRANCE. de Paris
Vl,
de Limeil,
4 Place
Jussleu,
t94
An outline of the p a p e r is as follows. In §2, we shall recall some s t a n d a r d r e s u l t s on the c o n t i n u o u s p r o b l e m (i.i). We shall give in §3 a general f o r m u l a t i o n of c o n f o r m i n g finite e l e m e n t a p p r o x i m a tions of the N a v l e r - S t o k e s p r o b l e m (L;I). S e c t i o n 4 will be d e v o t e d to the d e r i v a t i o n of general error b o u n d s for the v e l o c i t y and for the p r e s s u r e . We shall i n t r o d u c e in §5 the n u m e r i c a l i n t e g r a t i o n m e t h o d and we shall analyze in §6 the effect of n u m e r i c a l q u a d r a t u r e on e r r o r e s t i m a t e s . Finally, we shall b r i e f l y d i s c u s s in §7 an e x a m p l e of a n o n c o n f o r m i n g method. For the sake of b r e v i t y , we have only c o n s i d e r e d a g e n e r a l theory for c o n f o r m i n ~ finite e l e m e n t m e t h o d s but, u s i n g the ideas o f [ 4 3 , we could have also d e v e l o p p e d a general a p p r o a c h i n c l u d i n g n o n c o n f o r m i n g m e t h o d s . S i m i l a r l y , we have c o n f i n e d o u r s e l v e s to p o l y h e d r a l d o m a i n s £ and to the use of s t r a i g h t e l e m e n t s . The case of general curved d o m a i n s can be h a n d l e d by using isop a r a m e t r i c finite e l e m e n t s as a n a l % z e d in C i a r l e t & R a v i a r t [ 2 ], U J [3 ](see also Sco tt i l l ] a n d Z l a m a l [ 1 4 ] ) Let us m e n t i o n that some of our results are r e l a t e d to those of Fort in [5 ] who f i r s t gave a m a t h e m a t i c a l a n a l y s i s of the finite e l e m e n t a p p r o x i m a t i o n of the N a v i e r - S t o k e s e q u a t i o n s . For r e l a t e d work by the E n g i n e e r s , we refer to T a y l o r & Hood [13] and the r e f e r e n c e s therein. 2. N O T A T I O N S
AND P R E L I M I N A R I E S
We shall c o n s i d e r denote by
real-valued
(2.1)
u(x)v(x)dx
(u,v)
the scalar
(2.2)
=~
product
UVlo,~
=
defined
on ~. Let us
in L2(fl) and by
(v,v) 1/2
the c o r r e s p o n d i n g norm. w i t h the q u o t i e n t n o r m
The q u o t i e n t
space L 2 ( ~ ) / ~
is p r o v i d e d
inf Iv+clio '~ UVlIL2(~)/~ = c~
(2.3)
where, for s i m p l i c i t y , class v & L 2 ( ~ ) / ~ . For any i n t e g e r m~O, (2.4)
Hm(R)
In (2.4), and
functions
we also
we c o n s i d e r
= { v l v ~ L2(~)
e is a m u l t i i n d e x
Ba=(~---Xl)
denote
...(~.--/--)aN
the usual
, 8ev K L 2 ( ~ )
Ilvll,~,a
(2.6)
Ivlm,e
Sobolev
. We p r o v i d e
1/2
= (i~,<m ~ 11~%112 ~)l/2 I,~l,~m o,
in the
space
, l=I~ m }
: ==(al,.,.,=N),
seminorm
(2.5)
by v any f u n c t i o n
ai~O,
Hm(~) w i t h
laI==l+-.-+=N
the norm and
195 Let ( L 2 ( ~ ) ) N ( r e s p . ( H m ( ~ ) ) N) be the s p a c e of v e c t o r - v a l u e d f u n c t i o n s ~'v--(vl,...,v~) w i t h c o m p o n e n t s v. in L 2 ( ~ ) ( r e s p . i n The s c a l a r p r o d u c t i n ~ ( L 2 ( ~ ) ) N is g i v e n b~ (2.7) We
(UiV)
shall
use
=
u(x).v(x)dx
the
N
(2.8)
IIV~m,~ =
(2.9)
IVlm,n=
Let
us
llvilI2m,~)
!vit2m,q )1f2
( i
i=l the
spaces
= (H~(n)) N = {vlv ~ ~ ~(HI(Q)) N, ÷vIF=0} +
(2.10)
X
(2.11)
V -- {v
Note
:
1/2
( E i=l
introduce
(uiiv i)
i=l notations
following
÷
=
Hm(Q)).
I;6 X
, div
v=O}
+
that
i~ a n o r m over the s p a c e s X and V w h i c h is e q u i v a l e n t to the n o r m [vii I ~. We e x t e n d the s c a l a r p r o d u c t in (L2(~)) N to r e p r e s e n t the d u a l ~ t y b e t w e e n V and its dual space V'. We p r o v i d e V' w i t h the dual n o r m
[ (fiv)l (2.13)
ll~" = sup v ~.v
Let
us d e f i n e
(2.14)
, ~v'
II;!
:
a(u,v)
=
-. ~ dx ax i ax i
i=l N
(2.15)
bl(UiViW)
(2.16)
b ( u+, v÷, w÷)
=
--
, u,v K(HI(a)) N
av
~ ~ i=l
ui ~ . w dx ~x I
= ~I (b I (~,v,w)
- bl(U,W,V))
By the S o b o l e v ' s imbedding t h e o r e m , we have that the t r l l i n e a r form b(u,v,w) is d e f i n e d XxXxX. Moreover, we h a v e : (2.17)
b ( ~ , v-~ i w+)
= bl(U,ViW)
(2.18)
b(u,v,v)
-- 0 for
Then, (i)
two w e a k
Given such
(2.19)
~V',
all
u ~V
and
all
v,w~
X
all ~ , ~ K X .
formulations
a function
for
X ~ (L4(~)) N (N~<3) and c o n t i n u o u s on
of p r o b l e m find
(I.i)
functions
are
~ ~V
as and
follows
peL2(p~)/~
that
va(u,~)+b(ulu,v)-(p,div
v)
=
(f,v)
for
all
:
v ~X.
so
196 ->
(ii)
Given
a function +
(2.20)
~6V', +
~ a(u,~)
+
find
a function
u ~.V such
->
that
+
+ b(u,u,v)
= (f,v)
for
all v ~ V.
In fact, these two f o r m u l a t i o n s are e q u i v a l e n t and we have f o l l o w i n ~ r e s u l t (cf. L a d y z h e n s k a y a [ 9 ], Lions [ I O ] ) . Theorem
I : Define
(2.21)
8 =_~ su~
the
I ib (u,v,w) + +÷ u,v,w~v il]Itl;gII~ and
assume
(2.22)
that
8___ ~ [ , ~2
the f u n c t i o n
; satisfies
.< 1-5
for some c o n s t a n t O<5<1. Then, there exists a unique pair of . . f u n c t i o n s (u,p) £ V x L 2 (~)/fR s o. l u .t i o n of e q u a t i o n (2.19). M o r e o v e r , the f u n c t i o n u ~ V can be c h a r a c t e r l z e d as the unlque s o l u t i o n of e q u a t i o n (2.20). We
shall
need
the
following
estimate
for ~
:
V
+
w h i c h follows at once from (2.18) and (2.20) c o m b i n i n g (2.22) and (2.23), we obtain :
with
+
v=u.
By
(2.24)
II~II < ~ B Note that the e x i s t e n c e result r e m a i n s valid w i t h o u t any r e s t r i c tion on the f u n c t i o n f. But we shall always assume in this paper that c o n d i t i o n (2.22) holds in order to e n s u r e + t h e u n i q u e n e s s of the s o l u t i o n (~.p). We shall also assume that f b e l o n s s at least to the space (L~(9)) N and that the s o l u t i o n (~,p) is as sm°°th as we need for d e r i v i n g the error e s t i m a t e s . 3. A C .O.....N..~F O R M I N G
FINITE
E L E.... MENT
METHOD
For the sake of s i m p l i c i t y , we shall that ~ is a p o l y h e d r a l domain of ~ N ;
assume
in all
the
sequel
Let us recall some d e f i n i t i o n s and n o t a t i o n s given in [ 4 ] . Let h>O be a p a r a m e t e r ; we c o n s t r u c t a t r i a n g u l a t i o n ~ h of the set with n o n d e g e n e r a t e N - s i m p l i c e s K ( t r i a n g l e s if N=2, t e t r a h e d r o n s if N=3) w i t h d i a m e t e r s ~ h. For any K E ~ h , we let : (3.1)
[ h(K)
= diameter
of K,
p(K)
= diameter
of
o(K)
= h ( K ) / p (K)
We shall assume a>O i n d e p e n d e n t (3.2)
o(K)
Z c
Let k and k' be
the
inscribed
an the f o l l o w i n g of h such that for fixed
all
K ~
integers
that
sphere
there
of K
exists
a constant
h. such
that
l~k~k'.
With
any K & ~ h ,
197
we a s s o c i a t e (3.3)
a finite-dimensional
space of p o l y n o m i a l s
PK such
that
Pk ~ PK ~ P k '
where, for any i n t e g e r m%0, Pm d e n o t e s the space of all p o l y n o m i a l s of degree ~ m in the N v a r i a b l e s X l , . . . , x N. Let us c o n s i d e r
the f i n i t e - d i m e n s i o n a l
(3.4)
W h = {VhlVh~ C0(~),VNIKeP
(3.5)
X h = {~hI~h K (Wh)N,~hlr
Then, we may (3.6)
introduce
Vh =
K
spaces
:
for all K 6 ~ h} ~ HI(~)
= ~} C X
;
(I)
the space
~
q div ~h dx = 0
for all q £ Pk-I
and
all K E ~ h } w h i c h a p p r o x i m a t e s the space V. Notice V h is not a s u b s p a e e of V. Remark
1 : At first +
glance,
it w o u l d
however
that,
in Keneral,
seem more n a t u r a l
to set
:
,
O}
Vh = {VhlVh~ Xh,div vh
Unfortunately, as simple e x a m p l e s
show, this d e f i n i t i o n may lead to the rather u n d e s i r a b l e s i t u a t i o n V h = { ~ }. Thus, in order to get a no~ trivial s u b s p a c e V h of Xh, we have to w e a k e n this c o n d i t i o n div Vh=O as has been done in d e f i n i t i o n (3.6). E x a m p l e 1 : Just for s i m p l i c i t y , we shall r e s t r i c t o u r s e l v e s to the case N=2. Let K be a triangle of the t r i a n g u l a t i o n ~h with v e r t i c e s ai,K, I$i~3. D e n o t e by aij,K the m i d p o i n t of the side [ai,K,aj,K] , l~i<J~3,, and by a123, K the c e n t r o i d of the triangle K. a3
FiE.
aI
Let us denote
by PK
a12
I
a2
the space of p o l y n o m i a l s
spanned
by
12'~22'~32' ~I~2'~2 ~ 3 ' ~ 3 ~ I ' ~ I ~ 2 ~ 3 where the li's are the b a r y c e n t r i c c o o r d i n a t e s with r e s p e c t to the v e r t i c e s of the triangle K. Then, we get (3.3) with k=2, k'=3.
(I) This
space X h is d e n o t e d by
(Wo,h)N
in [ 4 ]-
198
Moreover, vh(ai,K) , For other N=2 or 3)
a f u n c t i o n V h ~ W h is u n i q u e l y d e t e r m i n e d by its values i.
Going back to the general case, we i n t r o d u c e the space ~h of all f u n c t i o n s ~n d e f i n e d on ~ such that ~ h i K e P k _ l for all K E ~ h. (Note that these f u n c t i o n s are g e n e r a l l y d i s c o n t i n u o u s ) . Thus, we may w r i t e : -~
(3.7)
->
->
V h = { V h l V h K X h , (~h,div v h) = O for all
Consider
now
the d i s c r e t e
(i) Find f u n c t i o n s -~
(3.8)
analogues
of p r o b l e m s
~ h ~ Vh and ph K ~ h / ~ S u c h
-~
->
-+
->
(ii) Find a f u n c t i o n U h E V h such -~
(3.9)
->
-~
->
(2.19)
and
(2.20):
that
.~
~ a(uh,Vh)+b(Uh,Uh,Vh)-(Ph,div
~he~h}.
->
->
Vh)=(f,vh)
for all ~ h ~ X h
;
that
->
->
~ a ( u h , V h ) + b ( U h , U h , V h ) = ( f , v h) for all ~h E Vh.
Before e s t a b l i s h i n g an e x i s t e n c e the f o l l o w i n g d e f i n i t i o n s :
and u n i q u e n e s s
Ib (Uh,Vh,Wh) (3.10)
Bh =
we need
I
sup
]h,~h,;h ~ Vh (3.11)
theorem,
II~ ~ = +sup
I(f'vh)!
vh e vh Theorem
2 : Assume
(3.12)
8h
~f~
II~hna~hH ~hl
II~h~
that the f u n c t i o n
f satisfies
< I.
Then, there exists a unique pair of f u n c t i o n s (u~,p~) ~ V L x ~h/~ s o l u t i o n of p r o b l e m (3.8). M o r e o v e r ~ the f u n c t i o n u h K V h can be c h a r a c t e r i z e d " a s the unique s o i u t i o n of p r o b l e m (3.9). Proof : We only sketch the proof. Let ÷u h b e given in V~ ; by the ~ilgram theorem, there exists a unique f u n c t z o n v h ~ V h such that a ( +v h m W~ h ) + b ( u~ h ,4 V h , W+ h ) = (~,w÷ h) for all ~h ~ v h " We have thus d e f i n e d a m a p p i n ~ T : u ~ V h + V h = T ( U h ) ~ V h. Using (2.18), it is easy to check that, under the a s s u m p t i o n (3.12), T is a strict c o n t r a c t i o n m a p p i n g with resnect to the norm (2.12). Hence, p r o b l e m (3.9) has a unique s o l u t i o n ~ h ~ V h. Then,
consider vh +
the linear
functional
defined
on X h :
~ a ( ~ h , V h ) + b ( U h , U h , V h ) - ( ~ , v h)
w h i c h v a n i s h e s on V h, By a p p l y i n g [ 4 unique f u n c t i o n Ph ~ ~h/~ such that
,Lemma 7],
there
exists
a
199
a(uh,Vh)+b(Uh,Uh,Vh)-(f,vh)=(Ph,div
~h ) for all ~h 6-Xh"
Thus, (uh?Ph) is a solution of problem ~3.8). Conversely, let (uh,Ph) ~ V h x ~h/~be a solution of (3.8 . Then, u h is the solution of problem (3.9). By taking ~h=~h estimate : (3.13)
in (3.9) and using condition
IIUhll ~< --~)llf~h
(3.12), we get the
8h
Here a g a i n , one can easily show that the existence result valid without the restriction (3.12) on the function ~.
remains
4. ERROR ESTIMATES Note : In all the sequel, we shall denote by C or c.z various constants which do not depend on the parameter h. 4.1. An estimate
for Uh-U in X
In order to derive bounds for the error u -u, we shall need a hypothesis concerning the approximation o~ an arbitrary smooth function of V by functions of V h and which is essentially hypothesis H.I o f [ 4 ]. Hypothesis~_ _ ~H'I : There exists an operator (Wh)N){]~((Hof(~)Og°(~))N;x h) such that : (i)
(4.1)
(div(rhv-v),~h)
(li)
(4.2)
-~ -~ ~< Chmlvlm+l,~ '÷ IIrhv-vil
Note (4.3)
that condition /q K
= 0 for all ~ h £ ¢ h and all ~ ( H I ( ~ ) ~
Co(~))N;
for all ~ ( H m + l ( ~ ) ) N, l.<m-
(4.1) can be equivalently
div(rhv-V)dx=O
for all qEPk_l, N E(HI(~)(] C°(~)) .
Now, assuming that Hypothesis H.I holds, rh e ~ ( V o (C°(~))N;Vh) and by (4.2) (4.4)
inf ~heVh
IVh_V~ ~ ÷
rh £~((HI(~)(]C°(~))N ;
~ ChmI~Im+l,
Lemma I : Assume
we have by
(4.5)
llm 8h= B , h-*O
(4.1)
for all ~ V q ( H m + l ( ~ ) ) N, l~m~k
the convergence
that Hypothesis
in the form
all K 6 ~ h and all
As it has been shown in r 4 , § ~ , Hypothesis in the case of example 1% First, we consider zero.
stated
of 8h and
H.I holds.
H.I holds with k=2
Ilf~ as h tends Then, we have
:
to
200
lim lllh
(4.6)
: I~11"
h+O
Proof : We shall only prove (4.5) since (4.6) can be o b t a i n e d in a similar way. For each value of the p a r a m e t e r h, let U h , V h , W h be f u n c t i o n s in V h such that : ~ ~Uh[
I+ = ~Vh
= i ,
~Wh~ +
=
]b(UhiVhiWh) I = Sh Since
Bh is b o u n d e d
{hn}n> I such that I
as h tends
to zero,
we can find a s u b s e q u e n c e
:
~hn ÷ um , ~hn ÷ v~ , W h n ~ W m
weakly
in X,
Bhn ÷ 8~ . Let us show that uK, v~, w ± ~ V. Denote by O h the o r t h o g o n a l p r o j e c t i o n o p e r a t o r in L~(~) upon ~h" Then, one can prove
(of.
[
(4.7)
2]) : U ~ - p h ~ o , fl ~ c I h m I ~ I m , D
Since Hk(~)
is dense
llm 0 h ~ = ~ h÷O
in L2(Q),
in L2(~)
for all we get
for all
~EHm(~),
O~m~k.
:
~L2(fl).
Therefore (div ~ , ~ ) which
implies
= lim
div ~
(div ~ h n , 0 h n ~) = 0 for all ~ 0 and
Now, since the i m b e d d i n g we may assume that : Uhn ÷ ÷u~ Thus, we o b t a i n
of X into
+ ~ v~
, ~hn
div ~
= div w~ = O.
(L4(~)) N is c o m p a c t
, ~Whn ÷ +w~
strongly
in
for N~3,
(L4(~ ) ) N .
:
B~ = lim Bh = lim n~ n n~ and
similarly
L2(~)
Ib + ÷ + I = Ib(uliv~iw~ ) [ (Uhn'VhniWhn)
therefore
(4.8) since
~m ~ B
ut,v±,~ ~ ~ V with ~u,~,~v~l,Jlw,ll ~ I
continuity
of the norm
in the weak
(by the
lower-semi-
topology).
Let us prove the r e v e r s e i n e q u a l i t y . Let u, v, w be a r b l t r a r y f u n c t i o n s in ~ = V 0 (H2(fl)) • By H y p o t h e s i s H.1, we have : rhu ~ as h tends
, rh
to zero
+ v , rhw ~ w in X so that
201
Ib (U~V ~ +,w) + I
÷ , r h v+, r h w+) l Ib(rhu
Hence
: for all
But,
since ~
13 =
is dense
u,v,wE
.
in V, we may write
:
sup
~,;,~
I~a I;I ll;I
and we c o n c l u d e that 8~ ~ 8. C o m p a r l n K with (4.8), we get B~ =8 • Also, it f o l l o w s that the whole s e q u e n c e {8 h} c o n v e r K e s to B as h tends to zero. As a c o n s e q u e n c e of L e m m a (3.12) for h small e n o u g h
I, c o n d i t i o n (2.22) implies c o n d i t i o n so that T h e o r e m 2 can be applied.
T h e o r e m 3 : A s s u m e that H ~ p o t h e s i s H.I holds and that the f u n c t i o n ~ s a t l s f [ e s c o n d i t i o n (2.22). A s s u m e ~ in the s o l u t i o n !uep) of p r o b l e m (2.19) v e r i f i e s the s m o o t h n e s s .properties :
addition~that
(4.9)
~ G V O (Hk+l(~)) N , p ~ H k ( ~ ) .
Then, for h small enough, p r o b ! e m Uh-~Vh--and we have the e s t i m a t e (4.10)
~Uh-U ~ $ c h k ( l ~ I k + l , ~
where
the c o n s t a n t
Proof
: By Lemma
(4.11)
(3.9) has a u n i q u e
solution ......
+ Iplk,~ )
C is i n d e p e n d e n t
i, we may choose
of h~ ~ a n d
h small
p.
enough
so that
8_h ]~n* v2
h 6 I-~
Then, e x i s t e n c e and u n i q u e n e s s of the s o l u t i o n ~ h ~ Vh follow from T h e o r e m 2. Now, let v h he an a r b i t r a r y f u n c t i o n of V h and W h = U h - V h. We c o n s i d e r the e x p r e s s i o n : (4.12) Using
D =
v a ( w h , W h ) + b ( U h , U h , W h ) - b ( V h , V h , W h)
(2.18),
= using
~ ~Wh~ ~ (3.11) +
D
and
:
~ a ( w h , W h ) + b ( W h , U h ,Wh)-b (Uh,Wh,W h) =
D =
Thus,
we may w r i t e
~
therefore
v
b(Wh,Uh,Wh).
and 2
(3.13),
IWhll-,hl~hllll~hll
by
(4.11)
we get
2
8h >( ~--
: ÷ *
+
Ifllh)lWhll
~
202
~ ~ D >s 2
(4.13)
On
the
other D = =
we
may b(~
hand,
using
(3.9)
and
, +U , W+ h ) - b ( V h+ , V h+, W h +)
since
Wh~Vh,
(pjdiv and
we
have
÷ Wh).
write = b ( U+ - V+h , U ÷, W h+ ) + b ( V h ,+ U - V÷h+ , W
.< Cl Next,
(2.19),
( f , w h ) - ~ a ( v h , W h ) - b ( V h , V h , W h) = ÷ _~ _~ _~ ÷ ÷ -~ ÷ + v a(u-vh,Wh)+b(u,u,wh)-b(Vh,Vh,Wh)-(p,div
(4.14)
But,
2 h~
Wh)
we =
have
+
h) ~<
(IIII1+ IlVh~) RU-Vhll IlWhi
:
(p-phP,div
w h)
therefore
I(p,dlv Thus,
we
(4.15) Now,
Wh)I .< ~2~P-OhPlo,aTIWhH
obtain D ~< { ( ~ +Cl(llulI+llvhlT))ll~-Vhll+c2~D-phPllo,~}IIWhl I
comparing
(4.13)
and
(4.15),
we
Eet
:
llwhi .< ~-~ { ( ~ +e I ( llul+ IlVhl ) ) 11U-Vhl +c 2 llp OhPlI o, ~ ] so
that
.<
lUh-Ul + + + + + i .< llwhll+nuvhll
~< C 3 { ( l + l ~ + ~ h ~ ) ~ U - V h l l + I l P - P h P [ I o , C h o o s i n g v h to be the o r t h o g o n a l and u s i n g "(2.24), we o b t a i n : (4.16)
Then, (4.7)
~Uh-Ull
the and
~< c
desired (4.16).
projection
{ inf llU_VhlI+ 4 ~h e VN
inequality
(4.10)
~ } in X of
u upon
inf ljp_~h I } ~hE¢ h O,~ follows
at once
from
R e m a r k 2 : If we a s s u m e o n l y i n t e g e r m w i t h l~<m.
-~ m+l N u E V O (H (~)) , p E H m ( Q ) the w e a k e r e s t i m a t e
(4.17)
a + [ p l m , fl).
l]Uh-Ull ~< C h m ( l U l m + l
M o r e o v e r , +if we do not a s s u m e functions u and p, it f o l l o w s a r g u m e n t that : (4.18)
lira u h = u h+O
in X
.
Vh
any r e g u l a r i t y f r o m (4.16) by
p r o p e r t y on a density
(4.4),
for
the
some
203
.+
4.2. An e s t i m a t e
->
for Uh-U
in
(L2(fl)) N
In order to derive a L 2 - e s t i m a t e for the error Uh-U , we shall use the c l a s s i c a l A u b i n - N i t s c h e ' s duality argument. We write :
(4.19)
]]Uh-U]I0,~
I (%-~,I)) )]~I[
snp
~i(L 2 (~))N
O,a ~et u con ider the f o l l o w i n g linear p r o b l e m ~ g i v e n a f u n c t i o n g ~ ( L ~ ( ~ ) ) ~, find f u n c t i o n s ~ e V and X e e 2 ( ~ ) . , such that .... .+
(4.20) Lemma
.+
2 : Problem
Proof
_~
_~
.+
->
-~
-w
->
~ a(v,~)+b(u)v,?)+b(v,u)T)-(x,div (4.20) has
a uniRue
solution
: P r o b l e m ( 4 . 2 0 ) is e q u i v a l e n t a function ~V such that
(4,21)
-+
->
.+
v)=(v,g)
for all v ¢ X
(~,X)EV x L2(O.)~
to the followinF~ one
~ a(v)?)+b(u)v)~)+b(v)u)~)=(v)g)
:
for all v ~ V .
Since a(v,v)+b(u,v,v)+b(v,u)v)= >~
~ a(v,v)+b(v)u)v)
( ~ -~]l~)II~] 2 >s v ~I]~ 2
e x i s t e n c e and u n i q u e n e s s L a x - M i l g r a m theorem.
of the
>p
for a l l - @ E V)
solution
~
V follow
from
the
We shall also need the f o l l o w i n g r e g u l a r i t y p r o p e r t y for the solution (~,X) of p r o b l e m (4.20) : there exists a c o n s t a n t A>O w h i c h does not depend on ~ such that :
ll~)r2)+ IXll, ~ ^ll~Ilo,~
(4,.22)
In fact) by u s i n g t e c h n i q u e s of K o n d r a t i e v [ 8 ] and G r i s v a r d [ 7], one can prove that this p r o p e r t y (4.22) holds p r o v i d e d t h e ~ p o l y h e d r a l d o m a i n Q is c o n E e x and the s o l u t i o n ~ is smooth enough (for example ~ K (CI(~))s). T h e o r e m 4 : Assume that H y p o t h e s i s H.I! h"~Id. Then| for h small enough, we h a v e (4~23)
II~h-~l[O,~ ~ Ch k+l
for
constant
some
Proof (4.24)
: Taking
V=Uh-U
in
which
is i n d e p e n d e n t
(4.20), we get
and
(4~=22)
of h.
:
(Uh-U,g)= v a ( u h - u ) ~ ) + b ( U , U h - U , ~ ) + b ( U h - U , U , @ ) - ( X , d i V ( U h - U ) )
On the other (4.,25)
C=C(u~p)
(2.22)) (4.9) the e s t i m a t e
hand,
using
(2.19)
and
(3.9), we may write
v a(uh-U,?h)+b(Uh,Uh)~h)-b(u,u)~h)+(p,div
for all ~ ~ V . Now, c o m b i n i n g +h n that d i v ~ =O, we obtain :
(4.24)
and
(4.25)
:
~h )=0 and n o t i c i n g
204
(Uh-U,~)= v a(uh-u,~-~h)+b(Uh-U,Uh,~-~h )+ + b(~,'+Uh-U"+ ,~ - - ~ h ) - b ( U h - U , ~ h - U , ~ ) - (×,diV(~h-~))-(p ,dlv(~O-? • -~ ÷ h)) Thus, we get for all ~h E V h l(Uh-U,g)I
(4.26) Next,
we
have
by
+Cl(Tlul+[lUhll)}llUh-URl~-~hll c I llUh-Ul2 l]~l+l ( × , d i V ( U h - U ) ) I + l
.< { ~
+
+
(p,div(~-~h))
(4.7)
1
:
lXll,n
(4.27)
] (×' dlV(Uh-U))I=I " "+ "+ "~ "+ "+ "+ (X-PhX'diV(Uh-U))l-
(4.28)
I (p,div(~-~h))I=I (p-0hP,div(~-~h))l.
II~-~hlIlplk,~
Now, we choose ~ h = r h ~ so that we get by Hypothesis
(4.29)
H.I :
II~-~hll.< e4hI~12sa
By combining (4.26),..,, (4.29) and by applying Theorem 3 and inequality (4.17), we obtain for h small enough : (4.30)
I(Uh-U,~) I ~< e5hk+
(lUlk+l, +Iplk,fl)(l+lu[2,
+IpIl,~)
+I×I Using
(4.19), (4.22) and (4.30), we get the estimate
)
(4.23) with
C = c5A(I~Ik+l,~+Iplk,~)(l+I~I2,~+Ipll,~) 4.3. An estimate
for ph- p in L2(~)/~
In this section, we shall need a new hypothesis "H.3" as in [ 4 ] .
that we call
H~pothesis H.3 . . . .:. .Given . . . . . .any function ~h E ~h such t h a t ~ h d X = O there exists a function Sh ~ X h
(i) (ii)
(4.31)
(div ~h-~h,~h)=O
(4.32)
IlSh~ ~ C~?h~O, ~
,
such that :
for all ~ h e #h'
As it has been shown in [4 , §6], Hypothesis H.3 holds provided the polyhedral domain is convex and the triangulation ~ h satisfies the uniformity condition (4.33)
0(K) ~ Ch
for all K ~ h
(2)
(2) This condition (4.33) is easily seen to be equivalent to the inverse assumption : IVhll, ~ ~ Ch-IIlvhlI0, for all V h ~ W h.
205 Theorem 5 : With the same assumptions as in The.orem 3, we assume ~ a ~ o n , that hypothesis H.3 holds. Then : (4.34) where
J]ph-P~e2(~)/~ the constant
,< chk(lulk+l,
C is independent
+Iplk,~) of h! ~ and p.
Proof : The proof is similar to t h a t of [ 4 , Theorem will be only sketched. We choose p and Ph such t h a t j From
pdx = ~
(2.19)
and
63
and
PhdX = O (3.8), we get for all Vh~ X h :
(Ph-PhP,div
~h ) = (p-0hP,div
~h)+~ a(uh-u,v h) +
+ b(Uh-U,Uh,Vh)+b (U,Uh-U,V h) Now / ( p h - P h P ) d x = 0 . u +v h 6 X h so that :
Hence,
(div ~h,~h)
by Hypothesis
= (ph-PhP,~h)
H.3, we may choose
for all ~h 6 ~h'
I]Vh] ~< ClIIPh-OhP~o,Q. Therefore -> -> -~ -+ ~Ph-Ph pjO,~ "< IIP-OhP~IO,~+ ~ Cl~Uh-U~+C2 ( ~ h ~ +~II) llUh-U~ and the desired 5. THE EFFECT
estimate
OF NUMERICAL
(4.34)
follows
INTEGRATION
at once.
: GENERALITIES
The practical application of the finite element method (3.8) requires the computation of various multiple integrals. Although most of these integrals involve only polynomials and can be computed exactly, it is easier and faster to use numerical integration techniques ; we shall show that it can be performed with no loss in the order of accuracy of the method. Note that these numerical integration techniques are essential when using curved isoparametric finite element methods. Let us describe the numerical quadrature method that we shall use. We proceed as in Ciarlet ~ Raviart [ 3 ] - L e t K be a fixed nondegenerate N-simplex of I~m. We choose- ~ certain quadrature formula over the reference set ~ : L (5.1) ~(~) = / ~(~)d~ is approximated ~X ~a(~)=£=~ I ~ ~(~£) K for some specified points ~ £ ~ K and weights ~£ which will be assumed to be positive. For each K 6 ~ h , let F K : ~ FK(~)=BK~+b K, ~K ~ ( s N ) ' hK~N, be an affine invertible mapping which maps K onto K. We may assume t h a t the Jacohian determinant JK=det(BK ) of the mapping F K is positive. Then, there corresponds to (5.].) the quadrature formula over K : L (5.2) IK(~) = K / ? ( x ) d x is approximated by IK,a(~)= ~-£=i ~£'K~(b£'K)
206
where (5.3) Then,
~,K
= ~£JK
any integral
(5.4)
I(~)
la(~)
over
' I~£~L.
the polyhedral
domain
Q
= /~p(x)dx
is a p p r o x i m a t e d (5.5)
' b£,K = FK(b£)
by
=K~
IK'a(~)" h
In all the sequel, we shall assume that the function ~ belongs to the space (C (~)) . Let a~(u~,v~), b h ( U ~ , V ~ , W h ) and (~,vh) ~ be the a p p r o x i m a t i o n s of a ( ~ , ~ b ) , - b ( ~ h ) and (~,~h) resulting from numerical integratioE': . . . . N ~h + ~ u~ h ah(Uh,V h) = la(i__~I ~ , ~ ), ..... = ~x i ~x i Note
that
(5.6)
the analogue
bh(Uh,Vh,Vh)
of property
= 0
(5.7)
functions
holds
:
for all ÷Uh,V+ h e X h
Now, we replace the discrete following ones : (i) Find
(2.18)
problems
(3.8)
~U h E V h and ~h e # h ~ such
and
(3.9) by the
that
~ % ÷ (~h,div ~h ) = (~,Vh) ~ ah(Uh,V+ h )+b h (Uh,Uh,Vh)÷ h for all Vh~ X h
(ii) Find (5.8)
a function
u h ~ V h such
that
v ah(Uh,Vh)+bh(Uh,Uh,Vh)~f,v
h) for all V h ~ V h.
Remark 3 : We shall ignore the p o s s i b i l i t y i n t e g r a t i o n in the d e f i n i t i o n of the space V h = {~hI~ h ~ X h , la(~hdiV
of using Vh :
~h ) = O
for all ~h e ~h }
and in the c o m p u t a t l o n of (ph,div vh). The interest p o s s i b i l i t y is limited because we shall prove in §6 error ~Uh-Ull is optimal when the q u a d r a t u r e formula exact for all p o l y n o m i a l s of degree 4 k+k'-2, which to an exact c o m p u t a t i o n of (~h,dlv ~h ). For
studying
problems
(5.7)
and
(5.8),
H y p o t h e s i s H.4 : The q u a d r a t u r e ' formula f o l l o w i n ~ p Koperties :
we
shall
(5.1)
(i) The....w...e..i...g..h..t...s. g.• are p o s i t i v e and the set P k , _ l - u n i s o l v e n t subset, i~e~ ....... (5.9)
P ~ Pk'-i
' P(bz)=O
' I~Z6L
===~ p=O
numerical
need
of this that the (5.I) is corresponds :
satisfles
the
{~£~L ~=I contains
;
a
207
(li) There exists an i n t e g e r s ~ k'-I such that the ~ u a d r a t u r e formula (5.1 ~) is exact for . . .all . . . p o l y n o m i a l s of desree ~< s0 By H y p o t h e s i s H.4 (i) and [3 , Theorem 3] for instance, exists a constant a>O i n d e p e n d e n t of h s~ch that (5.10)
ah(Vh,V h) >~ a~Vh] I
As a first
consequence
is a norm over us now
(5.10)
X h.
introduce
the following
quantities
:
sup Ibh (~h'Vh'Wh) I Bh = + + ÷ V + + + U h ' V h ' W h ~ h IlUh~hlIVhlIh~Wh~h
(5.12)
+
~2 (5.13)
_>
I (f ,Vh) h I
If h =
sup +
v
Vh~- h
Then,
for all V h ~ X h.
+ + + I/2 ~VhN h = ah (Vh,V h )
(5.11)
Let
of
there
we have
~I +
~ h h
the analogue
of Theorem
T h e o r e m 6+: Assume that H y p o t h e s i s function f satisfies
2.
H.4
(i) holds
and
that
the
(5.14) ~ ll~l[h < I Then,
there
exists
a unique
pair
of functions
solution of p r o b l e m (5.7). Moreover, ~'~aractgrized as the unique solution Proof
: The proof
By taking
Vh=U h in (5.8), 1
"+
(5.15)
follows
tl"Uh Ith
the same we get
(~h,~h) ~-V h x ~h/~
the function Uh~_V h can be o-T Drob--"-~e-~~5.8)
line
as that of Theorem
2.
the inequality
~
"< --
! ~11h
"
Before e s t a b l i s h i n g the convergence of B h and l]f|h as h tends to zero and before deriving error estimates, we need introduce some notations. ~qe set : (5.16)
E(~)=i(~)-ia(~),
EK(?)=IK(~)-IK,a(?),
E(?)=l(?)-la(?)-
Given a fixed K~=~,,, the m a p p i n g F K deteKmines o n e - t o - o n e c o r r e s p o n d e n c e s between the points ~ of K and the points x of K by X=FK (~) and between
and
the functions
1 We shall (5..17)
~=FKI (x) ,
make
and
constant
IK(?)=JK.~(~)
~ : K--~
and+ ~ : K - - ~
by
?° use of the above , IK,a(?)=JK.~a(~),
correspondences. EK(?)=JK.E(~).
Note
that
208
Let us introduce now m~O and any q~l : (5.18)
wm'q(~)
We provide
the Sobolev
= { v l v e Lq(~)
wm'q(~)
with
,~
~vllq
i~i~ m "
that
we need
Hypothesis
integer
, lal6m }
seminorm
)l/q
eq(~)
=
Era(~)
'
a slightly
H.~I : There
llv~m,Z, n = Ivl~,n ,
,
refined
exists
version
such
Ivlm,2,~
of H y p o t h e s i s
an operator
(Wh)N)/~ ~ ( ( H I ( ~ ) O C ° ( ~ ) ) N ; X h) o (i)
for any
:
wm'2(~)
Then,
wm'q(~)
v ~ Lq(~)
the norm and
(5.19) Nvlm,q,a = ( ~
Note
spaces
rh£~((Hl(~)
=
IVlm,
~
•
H.I. ~C°(~))N
;
that
(4.1) holds
+ + qs,q,K ) i/q ~ C h m + l - S l v÷l m + l , q , ~ ( ~-F Irhv-vl , O~ szm+l Ke~ h N for all ~ ~ (wm+l'q(~)) N and all m with O~mSk and m+l--- > O (3) q
(ii)
(5.21)
In practice,
Hypothesis
H.I holds
as soon as H y p o t h e s i s
H.I does.
Now, we want to study the convergence of B h and ~ f ~ as h tends zero. To this purpose, we need some technical lemmas.
to
Lemma 3 : Assume that H y p o t h e s i s H.4 (ii) holds for some i n t e g e r s=r+kr'/-I with r~O. Then, there exists a constant C>0 i n d e p e n d e n t of h and K such that
(5.22) IEK(2vi ~xj for all K ~
w ,
vI
l
I~xjlr+l,K
and all vi,w i
O,K
Wh•
Proof : This result is similar to L 3 , T h e o r e m 6J but it will be e s t a b l i s h e d in a slightly different way. Since H y p o t h e s i s H.4 (ii) holds with s=r+k'-l, we may write : ~(~vf __
~wi)
~xj ~xj
^ =
~v i
E((---~
~xj
~ v i ~w i r
)
3xj ~xj
)
,
where, for any integer m>O, ~m denotes the o r t h o g o n a l p r o j e c t i o n o p e r a t o r in L2(K) upon Pm" Since ~vi -~r ~vi and ~w. belong to i
~xj (3) so that wm+l'q(~) ~ C ° ( ~ )
by the Sobolev's
~xj
~xj
imbeddin~
theorem.
209
the finite-dimenslonal space Pk,_l ~ there exists a constant Cl=Cl(K~k'pr) such that
l[(('x......j _ .~r ~xj By the Bramble-Hilbert we get :
I .< c I
il~xI
-
~'
o,[
r ~xjllO,R
lemma [ i ] in the form given in [2 ,Lemma 7],
r ~xj~O,K
~ ~xj ~r+l,K
and therefore
t~(~xj ~xq I -< c3 l~xj r+l,K l~x111O,K Since (cf. [ 2 , f o r m u l a <5.23)
(4.15)])
j-I/q
£
l~Iz,q,R "< K
IIBKII l~l£,q,K
.< c
j-I/qh£ 4 K
I~I
£,q,K
for all ~ O ~ W g ' q ( K ) , £ ~ O , we obtain
~ ~. (@vi ~wi) c5hr+l ~ = jKl~ .... ] ~< ~xj ~xj ~xj r+l,K
?v. ~w. (___.l ._ i) IEK ~xj ~xj
i 0,K
N Since a(vh,Wh)-ah(VhJWh)-
EK(____I i), where v , wi, ~X. ~x. i 3 3 l.
I
(5,24)
la(~h,Wh)-ah
h
~< Chr+l(~K&~hlVh 12r+2,K)I/21[~'hII• rU
Lemma 4 : A s s u m e f o r any s e q u e n c e
t h a t H y p o t h e s i s . H.1 ( i i ) and H.4 ( i i ) h.ol..d. Then., {Vh } o f f u n c t i o n s ~ht~ Xh s u c h t h a t ~h ~" ~ w e a k l y
in X as h -> 0, We hive (5.25)
I~B .< lira infll~hllh h+O -w
Proof : Define l=limh-~oinf~VhlIh" First, we extract a subsequence yet denoted by {~h } such that lira I[~h~ h" h÷O
= ~.
Now~ sinceOC=X(~(H2(~)) N is dense in X, we can find a sequence {~n}n>.l of functions of O~ such that lim ~ n = ~ n÷~
strongly in X.
210
Then~
consider
(5.26)
the expression
:
ah (vh-rhVn, vh-rhVn ) = ah (Vh, Vh)-2 ah (rhVn, Vh) +ah (rhVn, rhVn).
By using
(5.24) with r=O and Hypothesis "~ + + + la(rhVn,Vh)-ah(rhVn,Vh)
H~.I (ii), we ~et
I ~< Clh( K ~
:
+ 2 K-) i / 2 ~ h l I IrhVnl2,
n
4
2hlVnI2,a I h
and therefore lim h-~O Similarly,
ah(rh~n,~ h) -- lim a(rh~n,~ h) = a(~n,~). h÷O
we obtain
lira lira a (rh~ n $ (~ ,v ) ÷ ) = h÷O h÷O a h (rh~ n ~rhVn ,r h n ) = a n ÷ n "
Hence h+olim a h ( v h - r h V n , v h - r h v n) = 12-2a(~n,V)+a(Vn,Vn) and lim lim ah($h-rhSn,~h-rh~n) n+= h÷O Since
the expression
(5.26)
= ~2-a($,~).
is >4 O, the conclusion
follows
at once.
Lemma 5 : Assume that Hypothesis H.4 (ii) holds and that the triangulatfon ~ h verifies the uniformity qondition (4.33). Then, we have : N (5.27)
Ib(Uh,Vh,Wh)-bh(Uh,Vh,Wh)
I .< Ch
tluh~ IlVhll IlWh/I
for all U h , V h , W h ~ X h. Proof
: First we note
that condition
(4.33)
implies
-N/q
(5.28)
~VhIIL~(~) ~< Clh
IlVhlILq(f~) for all V h ~ W h and all q>~l.
Let us prove this classical inequality for the reader's convenience. Let v, 6 W and let K ~ , ~ ; since ~ h E Pk,, there n h ^ exists a constant c2=c2(K,k') such that ll~h~L~(~ ) .< c2~hllLq(~ ) and therefore llVhllL~(K) ~< C2JKl/q~vhIILq(K ) But j-i = det(BK I ) ~< K
IIBKlll ~ t
where denotes the s p e c t r a l Lemma 2J and (4.33), we get
°atrix-noro.
hus, usin
r2,
211
j-I h~/ N K ~< ( ~ ) o(K)
-N "<
c3h
so that IIVh[IL~(K) ~< c2c~/q Hence
h-N/q
IlVhI[Lq(K)
: ~VhlIL=(~) = sup IlVh~e~ (K) ~< c2c~/q K~e h
Nextj
we have
(5.29)
:
h -N/q
llVhllLq (~)
N I
~v. 3w . . . . . . I {E(u ']w ) - E ( u i ~v )}. b(Uh'Vh'Wh)-bh(Uh'Vh'Wh)=2 i,j=l ijx J ~x. J i l
where ui,vi~wi~ l.
the components
~vj EK(Ui
of Uh,VhtW h 6 X h. For
~v.
... 3 ~.) ~x. wj) = JK ~ (~i .3x. J I
•
l
~vj Now,
....6. P k ' - I and, by Hypothesis ~x i I"-.
H.4
(ii), we have /"-.
- - ,~j) = E((uiwj-~o(uiw j) ........ ) , ~x i ~x i where, again ~o denotes L 2(K) upon Po ° Since ~i.~j-~o(~i.Oj) spaces P2k' c4=c4(~,k')
and Pk'-I such that
the orthogonal
in
~vj
belong to the f i n i t e - d i m e n s i o n a l ~x.l respectively, there exists a constant
~xi
Thus, by the Bramble-Hilbert c5=c5(K,k') : 3v.
IEK
operator
and
IE((~i~j-~'o(~i~j))--~J)I
and by using
projection
(5.23) ~v]
.< c 4 I~i~j-~o(~i~j)lo,~ • - -
l~xi~o, ~
lemma~ we get for some constant
h
n
(ui ~xi wj) I ~< c6hluiwjll, K ~ x i ~ O , K
Hence [E(ui . 3 w )] ~< c6hIuiwj Ii, a ~xi O,f~ ~x i J
.< o
I
ll, +luiI , lw
~xAIO,n 1
212
Now, applying
(5.28) and the Sobolev's imbedding theorem, we obtain l-N/4 w j ) l ~< c7h (iiUi~L4(~)lWjil,~ +
avj ~
+luill,alwjIL4(a))
~< J~xiliO,q
.< cshl-N/4~hlIll~hllll~hl Clearly, a similar estimate is valid for each term in the righthand side of (5.29) so that inequality (5.27) is proved. We are now able to prove Lemma 6 : Assume that Hypotheses H.I and H.4 hold. Assume. in addition,that the trian~ulat~gn ~ h verifies condition (4.33~'T Then, we "have
:
. . . . . . . . . . . . . . . . . . . . . . .
(5.30)
lim ~ = 8 , h-+O h
(5.31)
lira II~llh =
o,,
h-+O
Proof : We shall only prove al'~'f-~h,~h,~h e X h : + + [bh(~h,Vh,Wh)I
(5.30). First, by Lemma 5, we get for
÷ ÷ +clhl-N/4 [[Uh~ ÷ ÷ ÷ ~ [b(Uh,Vh,~h)[ I[Vh[[ I]Wh[I
IluhlI IlvhlI IlwhlI ÷
.< c
Using
(5.10),
2
(l+h l'N/4)
(5.11), we obtain
~h ~< c2 a-3/2
~
÷
(N~3)
(l+hl-N/4) ~< c3 "
Next, we proceed as in the proof of Lemma I : for each value of the parameter h, let ~h' ~ ' ~ be functions in V h such that h h I
h l[h = []~h[lh = I[~hlh = I , Ibh(Uh,Vh,W h) I = Bh •
Then, we can find a subsequence ~hn + u/~ , v hn ÷ v I
in (L 4_~ (fl))N I
{hn}n>~l such that
, Whn ÷ w
÷
weakly
in X and strongly
÷
%
-~
as n -~ . Like in the proof of Lemma I, we have Morevoer, by Lemma 4, we have Ilu,II
-< 1 ,
[Iv,II
~ 1
.
Ilw,II
.< 1.
On the other hand, by Lemma 5, we ~et :
÷
÷
: u~,v ,w ~V.
213
and
l i m b h (Uhn,Vhn,Whn) n+m n therefore
~,
= lim b(u h ,Vhn,W h ) n~m n n -~
,+w h )[
lira Ibhn(~hn,~h
=
n ~
which Now,
implies using
n
Hypothesis
H.I,
formula
lim
Ibh(rh]'rh~'rh~)l~
h+O
Irh~n h ~rhvl h ~rhwl h
Furthermore,
it follows
8 as
to
tends
6. THE EFFECT 6.1.
Error In
and Lemma
5, we have
= Ib(~';'$)l ~II ~
Hence,
that
OF N U M E R I C A L
to
(5.24)
~
8x ~ 8 so that
the whole
sequence
Bz
{~h}
converges
to
zero.
estimate
order
Ib(],,v,,w~)I~
n
8~ ~ 8.
for all u,v,~ ~ V ~ ( H 2(~))N
h
=
->.
INTEGRATION
: ERROR
ESTIMATES
in X
derive
an
estimate
for
the
error
Uh-U
proceed as in § 4.1. We assume the h y p o t h e s e s we m a y c h o o s e h small enough so that 8h ~ ~ (6.1) --~ llfllh # I ---2
in
X,
of Lemma
we
6. Thus,
%
Then, existence from Theorem 6.
Wh=~h-Vh. (6.2)
We consider
Dh =
Using
(5.6),
(5.12)
we
get
and
(5.10)
~6
+
(6.3)
D h ~ --IIWh~
On t h e
other
2 hand,
the solution u of arbitrary function
(5.15),
-
bh(Vh,Vh,Wh)
we m a y w r i t e
+ bh(Wh,Uh,W h)
and
:
~
v (1-'-~1~
(2.1q),
we h a v e
I )limb ~
(6.1)
2 using
Dh = ( ~ ' W h ) h - V
(5.7)
and
ah(Vh'Wh)-bh(Vh'Vh'Wh)
~ a(u-vh,Wh)+b(u,u,wh)-b(Vh,Vh,Wh)-(p,div
+
~ {a (~h ,+W h ) - a h ( V+h , W+ ~
with
(4.14)).
Hence,
:
=
=
w h) +
+ b ( V+ h , V+h , W+h ) - b h ( V ÷h , V h+ , W h+) } -
- { (f,wh)-(f,wh) h} (Compare
(5.8) follow o f Vh a n d
:
+ DhtUh,Uh,Wh)
v ah(Wh,W h) by
of an
the e x p r e s s i o n
~ ah(Wh,Wh)
Dh = and
and uniqueness Now, l e t v h be
we obtain
:
214
I D h ~< {(~ +Cl(I]ulI+IIVh~)) ~U-Vh[J+c21P-phPIo,~}~Wh~ (6.4)
+ ÷
Comparing
(6.5)
rt.
+
~la(vh,Wh)-ah(Vh,Wh) 1 +Ib(Vh,Vh,Wh)-bh(Vh,Vh,Wh)l
+
•I(f,wh)- ( f->, W÷ h ) h l
(6.3)
and
..~
lUh-U I .~ c
3
(6.4),
we g e t
for
all
~h~..Vh
:
(AI*A2+A3+A4)
where (6.6)
A I = (l+~+IVh|)~U-Vh~
(6.7)
A
= sup 2 ~h~. Vh
+ IIp-phP~o, ~
la(v h,wh)-a h (Vh,Wh)l
ll~hll {b (VhPVh,Wh)-bh(Vh,Vh,W + ÷ -~ ÷ -~ ÷ h) I
(6.8)
A 3 = sup
WhO h (6.9)
A 4 =+sup
l;WhU I (f,wh)- (~,Wh)h I
Wh~- V h
'
fl~hiT
-~ -> We choose v. =r. u. Then, we must estlmate each of the terms A.. n Note that A 1 i~ the same as in § 4.1 and it will be evaluate~ in a similar way. The three terms AIJ A 2 and A 3 come from numerical quadrature. The term A 2 has been evaluated in (5.24). Thus, it remains to evaluate the terms A 3 and A 4. N First, ÷ i ,3~. ~~vj ~w~ ~ E_K( v ' -?x- iW '3- V i V j - -?x) 1i b ( V->h ' V÷h ' W->h ) - b h ( V h-~' V h-~ p W h ) = -2 i =1 KE~h Hence~
for estimating A3, we need the following
lemma.
Lemma 7 : Assume that Hypothesis H.4 (ii) holds for some inteser s=r+kr'/-I with r~O. Then, there ex£sts a constant C>O independent of h and K such that :
(6.10)
IEK(ViVj
~x.J)I ~
Bv. I EK(Vi ~x. (6.11) l
wJ )I ,<
Chr+ 1
{Rvillr,4,m
~vj
~r+l,
4'K~@xi[IO'K
l~vjl
~wJ I' 4,K
I,K
+
II~Xi~ r'
+ Ivi[Ir+l,4,K ~~xlnr+1,K ilw j II0,4,K } for all K 6~h Proof
: First,
and all vi,vj,w j e W hlet us consider
~wj .... ). As a the term EK(ViVj~x. i
215 consequence of...Hypothesis H.4 (ii), we may write : ~w ~w ~ ^(vlv ^ j ~x.~ ) = ~( (vivj ^- -gr(~i~j)) ~x.J I i and by using the Bramble-Hilbert lemma IE(~i~J~xi I "< e I l~i~j Ir+l,~|~xil~O, ~ Hence, using (5.23), we get (6.12)
IEK(ViV j axi)l 4
2
Iviv j Ir+l, K U~Xino,K
But, using Leibnitz's rule and HSlder's inequality, we obtain r+l (6.13) Ivi v J Ir+l,K ~< c 3 ~ Ivil s,4,K Ivj Ir+l_s,4,K -< s=O ~< c4 ~Vi]Ir+l, 4, K ~vj]lr+l, 4,K The proof of (6.10) is achieved by combining inequalities and (6.13).
(6.12)
Consider next the term EK(V i ~vj wj ) and assume that r>.l (the case ~x.l r=O is left to the reader). We write : ~v
f~v
°,,"
•
l I By Hypothesis H.4 (ii), we get
i
1 and by the Bramble-Hilbert lemma (6.14)
i
i
~xi~V"(~j-#'o~j)) I ~< c5 1~i---l~xilr,KlWj^ ~v. I l~(~i---~J II,K
Similarly, we_get by H.4(ii) E(~ i ~x
9o~j) = E((~ i
~x
-f 1
r
(~iJ))~oWj) ~x i
and therefore~.. (6.15)
I~(¢~i~vj ~'o~J)I ~< c6 ~x.~
~Wj~o,4,K ~Xi r+I,4/3,~
Thu~, it follows from (5.23), (6.14) and (6.15) (6.16)
IEK(Vi~--xwj) I ~< c6hr+l{ v i lwj Ii, K + i l exi|r,K
~v_!l +
IVi xi[r+l,4/3,K
lwjlIO,4,K
216
Then,
(6.11)
is a consequence
of (6.16) and
v.-~J ~ c8¿Vi~r,4 l~x i r,K
iv
ax i r+l,4/3,K
K
laxi~r,4,K
,
.< e9 Ivillr+l, 4, K
Consider next
[
•
r+l~K
N
(f'wh)(~'~h)h = ~.=I ÷~
K~I
EK (f iwi )
For estimating A4, we shall use the following been proved in [ 3, Theorem 4].
lemma which has
Lemma 8 : Assume that Hypothesis H.4 (ii) holds for some integer ~ I with r~0. Assume, i'n addition, that the'fu'nction f.~.wr+l'q(~) .constant C>O (6.17)
for some q>~l with r+l-N>0. Then, q independent of h and K such that
IEK(fiwi)I
for all K ~ h
there exists
a
~< Chr+l~fi~r+l,q, K llWi~l,q, K
and all w i ~ Wh, ! + --1 =1. q q'
We are now able to prove Theorem 7 : We make. the followin$ (i) H.~p..o..thesis H.I holds
assumptions
;
(ii) Hx.~.qthesis .}!......4 holds for some inte$.er s=r+k'-I with O~.r~k.71 ; (iii) ~ ~wr+l'q(~)) N with q~2, r+l-~>O and satisfies (2.22) ; q
condition
(iv) The solutio.n. (u~p) of (2.19) verif.i.es the smqothness properties (.4.9) ; (v) T.h..e.triansulat, ion ~ h (4.33). Then,
for h small enough,
UhG Vh a n d we h a v e
(6.18)
satisfies
the
the. uniformit%, condition
pro.blem (5.8..)...has a unl.q.ue .solution
estimate
l~Uh-]~ ~< Clhk(lulk+l,a+Iplk,a) + C2 hr+l(lul
where C 1 .and C 2 a r e
positive
^ r+z,a
+
+lull2r +2, a+ll~Ilr+l,
constants
independent
2.-
Proof
: Taking vh=rh u in (6.6), we obtain A 1 ~< ( 1 + 2 ~ )
~u-rhUll+l[u-rhulm+llP-OhPllo,f~
q,~) of
h I u and
217
By applying (6.19) Next,
(2.24),
(4.7)
and (5.21),
we get
A I ~ c3hk(I~Ik+l,Q+Iplk,fl)+C4h2r+2 using
(5.24) with vh=rhu
and
:
I~2 r+2,~
(5.21),
we obtain
(6.20) Now,
A 2 ~ Cshr+ll~ I r+2,~ using Lemma 7, we get Ib(rh~,rh,u,wh)-bh(rhu,rhu,w
h) I -<
.< C6hr+l { (
+ ( E and by using into L 4 (fl)
|rh~
(5.21)
A3 ~< C7 hr+l
(6.21) Finally,
using Lemma
I]rhu~ 4r+l, 4, K )I/2
)1/4( I
+14
and by continuity
llrh~ll2
r+2
K )I/2
I~Wh~
llWhll
of the imbedding
+
}
of HI(~)
iI~IIr+2,~ 2 8, we obtain
_~ -~ ÷ + I (f,wh)-(f,Wh)hl
+ ~< cshr+l Ilf~r+l,q,~
-~ ~WhIIl,q,~
and since q'~<2 (6.22) Thus,
A 4 4 C9hr+l inequality
~llr+l,q, ~
(6.18)
follows
from
Corollary : Assume that the hypotheses with r=k-l. Then, we have : (6.23)
lUh-U÷ ~ ~< chk(I~Ik+l,
(6.5),
(6.19),...,
of Theorem
(6.22).
7 are satisfied
+Iplk,fl + llu~k+l ,~ +~fllk ,q,~ ).
Therefore, the order of convergence in X of the finite method is not lowered when we use a ~uadrature formula which is exact for all polynomials ' of degree .< k+k'-2.
element (5.1)
Remark : We want to mention that we need the uniformity ~ i o n (4.33) only for proving the convergence of "~h to ~ as h tends to zero and not fo r deriving the error estimate (6.1g). Wether this condition (4.33) is necessary for obtaining (5.30) or not is an open question. Example 1 (continued). We go back to Example 1 which corresponds to the case k=2, k'=3. Thus, in order to get an optimal error estimate in X, it is sufficient to use a quadrature formula (5.1) which is exact for all polynomials of degree < 3 and such I that {~£}£s I contains a P2-unisolvent subset. This is an important simplification since the exact computation of b(Uh,Uh,V h) would require the integration of polynomials of degree 8. In particular, we may choose the quadrature formula
218
2 (6.24)
K/
-- meas( )
Y
iEI "=
60
~ ( a i ' K ) + ~ O' I i~<j--~3 r
+
27 + 60 ~(a123, K)} since ~ i s f o r m u l a is exact for all p o l y n o m i a l s of degree ~ 3 and the set {ai,K}l$i~ 3 ~ { a i j , K } l $ i s j $ 3 is a P 2 - u n i s o l v e n t set. F u r t h e r m o r e • note that in this case, for each finite the i n t e r p o l a t i o n n o d e s c o i n c i d e w i t h the q u a d r a t u r e 6.2. E r r o r
estimate
Let us state
in
the
element nodes.
K•
(L2(~))N theorem
first.
T h e o r e m 8 : The a s s u m p t i o n s are the same as for t h e o r e m 7 with the s t r o n g e r r e q u i r e m e n t that h y p o t h e s i s (ii) of t h e o r e m 7 holds for s = m a x { r + k ' - l , k } . A s s u m e also k ' ~ k + l . Then, for h small enough, we have the e s t i m a t e : + I]Uh-UJIo, ~ Z chk+l
(6.25)
where C=C(u,p) of h.
+ C' hr+2
and C ' = C ' ( ~ , p )
are certain
Proof : We p r o c e e d like in section d e f i n i n g ~ by (4.21), we get : (6.26)
4.2.
constants Starting
[ (~h-u,g)]+ Z ]BII+ ~ IB21+IB~]+IB41
independent
from
(4.19)
and
, where~ :
BI= ~ a ( u h - u+ , ~+- ~+h ) + b ( _~ h -~ ,~h ,~-?h + ~ ) +b(u, + ~h -~ ,?-~h ~ ÷ ) + -b ( U h - U , U h - U , ? J - ( x • d i V ( U h - U ) ) + ( p • d i v ( ~ - ~ h )
)
2=a Vuh •~h ) - a h ( U h ' ~ h ) ~ ~ ~ ~ ,~) B3=b(Uh,Uh,~h)-bh(~h,U h -
L;
B4= ( ~ , ~ h ) - ( f , ~ h ) h The term B I is identical to the term o b t a i n e d in section 4.2 and the three terms B2, B 3 and B 4 stem f r o m the use of n u m e r i c a l q u a d r a t u r e . We must e s t i m a t e these terms for ~ h = r h ~ . For the first one we get like in (4.30) : (6.27) where For
IBII ~ c h k + l ( u ~ 2 , C=C(u,p)
the
+]×Ii,~ ) ,
is i n d e p e n d e n t
of h.
(6.28)
term B • we can e s t a b l i s h 3 & Chr+2~2,.
We will
only
an e s t i m a t e
of the form
IB31
sketch
the proof
of
B3 = B3•I+B3,2+B3,3+B3•4
(6.28). • where
B3, 1 = (b-bh) (rh~•rh~,~h) B3, 2 = ( b - b h ) ( U h - r h u , r h U , ~ h )
We write :
:
:
219
B3,3 = (b-bh) (rhU,Uh-rhu,~ n) B3,4 = (b-bh) (uh-rhU,Uh-rhU,~h) We have : (6.29)
B3, I
o
~vj
E 2 i,J=l Ke~h
~x i
~xi
where vi, ~i denote the components of vh=rhu and
h'
We can write : ~vj ~v. ~v. EK(V i -~X i ~j) = EK(Vi ---! ~ (vi~x3 ~X. i ~j )+E K ." ( ~j -~i ~V I I Let z=vi---~ H I ~..j ~ Since r+l~
))
e
IE(~)I = l~(~-~r+lE)I -< Cll]E-~r+t~II0,K -< C21~Ir+2, ~ It follows that :
K~
~vj
•
(Vl~xi
?vj l?j )I 4 C3 hr+2 K6eh
I
C3hr+2{(Z v i K
r+2,l,K
2
~xi
) I/2 IKI~j I O,=%K
~
~vj '4 1/4 4 1/4} .~ ) (Z I~l?j II,4,K ) + (K-iVi~x r+l,4,K K But :
IV _~v.3 I i~x i r+2,l,K
~vjl
since vi=(rhu) i 6 P k , ~ P k + l . v K
Hence, by H I :
J )1/2 i-~x i r+2,l,K
$
C4 IIullk+l, ~
In the same way : I ( v i ~vj I4 1/4 -) K ~x i ]r+l,4,K On the other hand :
2
4 C6 I]]lk+l
IKI~j I0 ~,K < CTI~hlO,~,K & C71~hlo . ~ < +
%II~hll,4,~ ~ c9~I11,4,~ < cioI17112,~ by H~I and the imbedding
+ 2
~ ~vilIr+2,K4 ~rhn~k+l, K ll~xiU r+2,K
: H 2 ( ~ ) ~ W I ' 4 ( ~ ) ~C°(~).
In the same way, we have :
1/2 .<
220 ([[~l~j 14 )1/4 K 1,4,K Taking
the foregoing
(6.32)
Now,
.<
CIIII~2,~
estimates
into
KI~ h IEK(Vi ~vj ~x~ ~l~J)l'<
we must
consider ~N
^
(6.31),
term of ~
~vj
~v.
Hence,
we
Cl 3 *i
(6.30).
l
12,
conclude
as before,
:
s_k, (~i~vJ))(~j-~l~j))I ~x i
l
~Xils-k'+l,K
We have JN
^
i j)l - l?((,i--J ~x.
l~v " -<
:
-~~k+l 2 '~ II?I + 2 '~ Cl 2hr+2 flu
the second
•
we get
•
since
s-k'+l=r~
:
~v°
KE~ h~ IEK(Vi~x'~'(~J-NI~J)Ii "< C14hr+2~2'4'Q~2'fl ~< Cl5hr+2 which,
together K~h
with
(6.32)
~v.i IEK(Vi~x
~)I
yields "< Chr+2
lU~k+l,~
~< 2,a
: IIu~2
k+l,~
II~II
2,~2
A similar estimate for the second term in the right handside member of (6.29) is obtained by w r i t i n g :
=
E(viv j 3x i
(vivj o(
3x i
(viVj
~x i
~x i
The e s t i m a t e s for B , B 3 and B 3 4 are o b t a i n e d by identical 3 2 techniques and we leave t ~ m to the'reader. Finally, estimates for B and B 4 in terms of o have been derived in CiarletRavia~t [ 3 ] , w h i c h ends the p~oof of the theorem.
,,.,,~2
Example
I (continued)
In the case of example i, if we use the numerical quadrature formula given at the end of section 6.1, the h y p o t h e s e s of theorem 8 are satisfied with k=2, s=3j r=l. Hence we get the estimate : ]l~h-~II o,e 6.3.
Error
estimate
The following
for
-
O(h 3 )
a
the pressure
result
holds
:
T h e o r e m 9 : Let us make the same a s s u m p t i o n s as in theorem and assume in addition that h y p o t h e s i s H.3 holds. Then : (6.34)
~ph-PllL2(fl)/~
.< CN k + C,N r+l
,
7
~<
22~
where
C and
C' are
constants
independent
of h.
Proof : The proof is identical to the proof of t h e o r e m 5. There a p p e a r s three a d d i t i o n a l terms w h i c h are i d e n t i c a l to the terms A2, A3, A 4 of section 6.1 and w h i c h have a l r e a d y been e s t i m a t e d . 7. AN
EXAMPLE
OF A N O N C O N F O R M I N G
METHOD
For s i m p l i c i t y , we have r e s t r i c t e d our p r e s e n t a t i o n to the case of c o n f o r m i n g finite elements, i.e. to the case X h ~ X. However, all the f o r e g o i n g r e s u l t s can be e x t e n d e d to n o n c o n f o r m i n g e l e m e n t s p r o v i d e d these e l e m e n t s satisfy the c o m p a t i b i l i t y c o n d i t i o n H.2 of C r o u z e i x - R a v i a r t [4] . The d e r i v a t i o n of the e s t i m a t e s is i d e n t i c a l e x c e p t that we must c o n s i d e r a d d i t i o n a l terms i n v o l v i n g i n t e g r a l s on the faces of each e l e m e n t K ; these terms can be e s t i m a t e d in the same way as in . We will not develop these c o m p u t a t i o n s here ; we will only give a simple e x a m p l e for w h i c h we refer to C r o u z e i x - R a v i a r t [4 , example 4].
[43
Example 2 : Let K be a N - s i m p l e x of ~ h w i t h v e r t i c e s Denote by ~i K the c e n t r o Y d of the ( N - l ) - d i m e n s i o n a l w h i c h does n~t contain ai,KO
a i K, I$izN+I. fate of K
a3
~
~
a
2
aI We choose PK =PI÷' i.e. k = k ' = l and we define V h to be the space of all f u n c t i o n s v h d e f i n e d on ~ w h i c h are c o n t i n u o u s at the p o i n t s ~ ~ and whose r e s t r i c t i o n to each K £ ~ h s a t i s f i e s : ~hir ~ (PK)N aE~div ~.. = O. The h y p o t h e s e s H.I, H.2 and H.3 are ~ E t i s f l e d
hIK
(see ~4] ). On the other hand, the c o r r e s p o n d i n ~ a n a l o g u e of t h e o r e m 8 shows that h y p o t h e s i s H.4 must be s a t i s f i e d w i t h r=O s=l. Let I~h~=( I I~hI~,K) I/2~ ; then, the f o l l o w i n g e s t i m a t e s hold
:
K£~h +
(7 • i)
+
~Uh-Ull ~ + =0 (h)
,
]~h-~ll O ,a = 0
Note that we can use for e x a m p l e quadrature formulae : (7.2) center (7.3)
and
IK(~) of g r a v i t y IK(~)
= J~(x)dx
(h 2 )
one of
~ Meas
,
the
IPh-P ie2 (~)/~=O (h) two
following
K. ~(a123) , where
a123
is the
of K, or : i N+I ~ -3 Meas K i~=l "= ~ ( ~ i , K )
Indeed, if we use (7.3) all the i n t e g r a l s are c o m p u t e d e x a c t l y e x c e p t those c o r r e s p o n d i n g to the right h a n d s i d e m e m b e r ~. Remark : We have not d i s c u s s e d in this paper the p r a c t i c a l solution of the finite e l e m e n t e q u a t i o n s . In this respect, we refer to C r o u z e i x and R a v i a r t [4 , II], where basis f u n c t i o n s for the
222 various spaces V h are exhibited and to Fortin [5] and Fortin Peyret - Temam [6] , where various iterative methods are discussed. REFERENCES [I]
Bramble,
J.H. and S.R. Hilbert.
Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SlAM J. Numer. Anal. ~ (1970), 112-124. [2]
Ciarlet,
P.G. and P-A. RAVIART.
General Lagrange and Hermite interpolation in ~ n applications to finite element methods. Arch. Rat. Mech. Anal. 46, (1972), 177-199. [3]
Ciarlet,
with
P.G. and P-A. Raviart.
The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. "The mathematical fundations of the finite element method with applications to partial differential equations" A.K. Aziz ed. Academic press, New-York (1972), 409-474.
[4]
Crouzeix,
M. and P-A. Raviart.
Conforming and non-conforming finite element methods for solving the stationary Stokes equations (I, II). To appear in RAIRO, SErie MathEmatiques. [5]
Fortin,
M.
Calcul numErique des ~coulements des fluides de Bingham et des fluides newtoniens incompressibles par la mEthode d e s ~l~ments finis. Th~se, Paris, 1972.
[6]
Fortin,
M., R. Peyret et R. Temam.
REsolution numErique des Equations de Navier-Stokes fluide incompressible. Journal de mEcanique, IO, 3 (1971), 357-390. [7]
Grisvard P. Alternative de Fredholm relative au probl~me un polygone ou un polyh~dre. Boll. Un. Mat. Ital. ~ (1972), 132-164.
[8]
de Dirichlet dans
Kondratiev V.A. Boundary problems for elliptic equations with angular points. Trans. Moscow Math. Soc. 16 (1967), 227-313.
[9]
pour un
Ladyzhenskaya,
conical or
O.A.
The mathematical theory of viscous Gordon and Breach, 1963.
incompressible flow.
223
[i0] Lions,
J.L.
Quel~ues m~thodes lin~aires, Dunod, ~iI]
Scott,
de r@solution des probl~mes 1969.
aux limites non
~.
Finite element techniques for curved boundaries. Ph. D. Dissertation~ Mass. Inst. Tech. (1973). [12]
Strang,
G. and GoJ. Fix
An analysis of the finite element method,
Prentice llall (1973).
[13] Taylor C. and P. Hood A numerical solution of the Navier-Stokes equations using the finite element technique, Computers and Fluids, i (1973), 73-1OO. [14] Zlamal M. Curved elements in the finite element method. SlAM J. Numer. Anal. I0 (1973), 229-240.
FINITE
ELEMENTS
METHOD
IN
AEROSPACE
ENGINEERING
PROBLEMS
B. FRAEIJS de VEUBEKE Head~ Aerospace Engineering Laboratory, Univ. of Ll~gep Belgium
SUMMARY Some key areas of finite element development are briefly reviewed on the basis of IO years experience in the analysis of aerospace structures. The emphasis is on the practicability of the dual analysis concept for linear elastostatics, elastodynemlcs and steady state heat flow. Typical displacement and equilibrium models and test problems are described, that were found to be of practical value and belong to the library of elements currently used in the ASEF and DYNAM programs developped by the Aerospace Laboratory. The oral presentation will include samples of large scale computations involving static strain energy bounds, eigenvalues (frequencies and critical loads) and panel flutter under thermal constraints.
INTRODUCTION Past the era of Matrix Structural Analysisp concerned essentially with the topological problems of element intereonnexions, the developments of the finite element method have been largely centered on the nature and qualities of finite element models with respect to computational efficiency 2 ease of stress output interpretation and convergence characteristics. Improvements in mathematical models will not be without interaction with the methods of matrix structural analysis because of the growing need for overall economy as larger and more complicated problems are investigated. The volume of computations required to handle non-llnearities, geometrical and material, and structural optimization is such thatj in some cases~ advanced versions of the Force Method or combined methods may well be preferred to the presentj almost universally favored, Displacement or Direct Stiffness method.
225
MODELLING
OF
FINITE
ELEMENTS
From the basic work of PRAGER and SYNGE it is well known that upper and lower bounds to the strain energy can be obtained from discretizations based on the dual principles of minimum total energy and minimum complementary energy. The application of this idea to finite elements E I, 4 ~ requires properties of interelement connexions known respectively as conformity and diffusitivity. The possibility of a quantitative assessment of energy convergence by displacement conforming, or (C, C)p elements and equilibrium diffusive, or (E, E), elements has certainly weighed heavily in favor of their development and inclusion in the library of ASEF. For triangular membrane elements and tetraedral 3-dimensional (C, C) elements, complete polynomial expansions of arbitrary degree in the displacement vector u ( x ) = Q(x) q + B ( x ) b
(1)
are easily expressed in terms of shaping functions Q(x) related to a set of independent boundary displacements q and a set of "bubble functions" B(x). Conformity requires no more than CO continuity of displacements and this coincides with nodal identification of the q coordinates, The Kirchhoff-Love assumption in plate bending and shell theory is responsible for difficulties associated with C I continuity, hut those can now be revolved, for any polynomial degree, by a subdomain formulation (see T y p e 1 5
hereafter)
or by explicit formulation of dependency constraints between the q coordinates. (E, E) elements are obtained from complete polynomial expansions of a stress vector O = S(x) S + H(x) h
(2)
in terms of a set of internal stress redundancies s and body loading modes h, or by a stress function discretization E 2 , from either C
5, 8, 1 6 ~
. Diffusitivity follows
continuity of the stress functions, as in Kirchhoff plate ben-
o ding, or C I continuity, as in the case of the Airy~ function of membrane elements. In the latter case, the subdomain formulation again resolves the difficulty by virtue of static-geometric analogies E 2, 4 3
•
The extension of the use of (C, C) models to elastodynamics is straightforward through the use of the HAMILTON principle and the production of a consistent mass matrix in addition to the stiffness matrix. As shown by GERADIN and TABARROK L 9, IO, 19 ~
, (E, E) models are also applicable through
the discretization of the TOUPIN principle, the dual of Hamilton's principle E4,
7 ~ , in which case a consistent inverse mass matrix is produced and the
226
self-stressing states can be statically condensed, leaving a reduced eigenvalue problem. Eigenvalues are, as a rule, underestimated by this process, but an economical algorithm to predict the exact nature of the bound is still lacking. Similarly steady state heat conduct ion can be approached by dual variational principles on a "dissipation", or linearized entropy production functional E 6 J here by C
. The properties of conformity and diffisitivity are replaced
continuity of piecewise differentiable temperature fields and C
o continuity of equilibrated heat fluxes.
o
A bewildering variety of other models can be constructed, either by relaxing conformity or d iffusitivity (the hybrid elements), or by discretizlng the more general two-field variational principles E 1 ~
(the mixed elements).
Such combination of elements do no more generate energy bounds but may well present superior convergence characteristics E 13 ~
.
In the hybrid (E, C) the internal assumed stress field is no more connected at the boundary ~E of the element as it is in (E, E) by the weak displacements
q " t
TT(x) u(x) dS
(3)
8E which are averages of the unknown displacements, weighted by the surface traction modes. It is connected through a strong assumed displacement field that needs only to be defined at the interfaces, advantages over the (E, E) element may comprize : easier elimination of kinematical modes and increase in nodal valency. The loss of the guarantee in energy bounding seems however a high price to pay; the more so since experiments E 5 ~ have shown that an injection of bubble modes of higher degree in (C, C) elements raises, as would be expected, their flexibility to a level comparable to (E, C) elements without losing the bounding property. A similar remark applies to the dual hybrids (C, E), where the weak surface traction coordinates
g = ( QT(x) t ( x ) dx $ 3E
(4)
are replaced, for connexion purposes, by assumed strong surface traction modes. Injection of bubble stress modes of higher degree in (E, E) models increases their rigidity to a level comparable to the hybrid without loss of the bounding property. The mixed elements have not been explored to such an extent that conclusions may be derived regarding their merits. We would venture to say that before embarking on such an exploration program, better guidelines are needed from
227
mathematical studies of convergence. Nor is the field of model development exhausted by the previous classification, Convergence studies based on the patch test El2, 18 3
will give birth to new finite element models of the non-
conforming variety, some of which may prove to be simple and efficient. Quasidiffusive elements E 8 3
, in which we sacrifice the transmission of some
statically zero portion of the surface tractions at an interface, have duals in the static-geometric sense which are non conforming elements passing the patch test (an unpublished result). Another recent development leading to a new family of elements is the discretization of rotational equilibirum L 8 ] . It enables C O continuity to prevail in all cases for stress functions implementing the translational equilibrium equations. The static-geometric duals of such elements are yet to be investigated, ASSEMBLING
FINITE
ELEMENTS
(C, C) and (E, C) models are most easily assembled by the direct stiffhess method E 20~
, by passing the need for a topological analysis of the
self-stresslng states, (C, E) and (E, E) models can be assembled by a dual type of direqt f iexib[lity method E3, 4 3
, using the local stress function values to generate automa-
tically the self-stressings and applying the complementary energy principle to the whole structure. However the direct stiffness method is also applicable to (E, E) elements by considering the weak boundary displacements as the connectors and conversely for the (C, C) elements, where weak stress function values can be introduced at the interfaces. In general any assembling software type is applicable to any type of model, the choice is governed by the efficiency in the process and reduction in the size of the final system of equations.
STRESS
OUTPUT
EVALUATION
This remain a problem, especially with (C, C) and (C, E) elements, where local stress values are known to be unreliable. The strong stress output, obtained through the constitutive equations with Hookean matrix H : O = H [ D Q(x) ~ q + H E D B(X) ~ b
(5)
is preferably replaced by the weak stress information
I
~E ~ D Q(x) 3 T O dV " Kqq q + Kqb b
(6)
228
f
ED B(x)3
T a dV ffi~ q
q + Kbb b
E
which states the weighted averages obtained for the equilibrium equations. The strong information (5) satisfies (6). Often the analyst will not be satisfied by the mere knowledge of the average stresses (6) and will want a smoothing interpolation procedure between the elements without the need for reshuffling the whole stress information. This poses the interesting problem of invertlng~ so to speak~ the Gaussian quadrature procedure. How can one derive from the weak information (6) a set of reliable local estimates and in which local points of the element ?
ELEMENT LIBRARY AND TEST PROBLEMS
The following pages are devoted to a short survey of the finite element models which are operational in the ASEF and DYNAM programs up to 1969. Whenever new models are introduced,they are normally tested by comparison with the others in at least one of the test problems described on pages 30,31 and 32.
229
.TYPE I -
FRAME
References z
AFFDL-TR-66-199
ELEMENT
(1966)
- A_ssu~p~io_nE : u - a t + a 2 x v
=
t +
aI
' a2
x +
, x2
a3
w - a~ + u~ x + ~"~ 2
3 +
a~
x
+~ ; x ~
¢ x = 81 + ~32 x
- The frame element is a straight prismatic member for which deformation modes in extension, bending !n two perpendicular planes and in torsion are assumed. It follows the simplest engineering beam theory. -
The local axes of the element are the inertia axes wlth the Ox axis oriented from node I to node 2 while a third node defines the local X Oz p l a n e of c o o r d i n a t e s . ~ x /
---~v ==~ ,y
j,
~
U
v W
~Z
y..-
!JJJ.."'" ssSo
.......
230
TYPE 2
BAR
- References :
ELEMENT
OF
VARIABLE
DEGREE
AFFDL-TR-66-199 (1966)
-_AsEtn~p~ion : u = a I + e 2 x + a 3 x 2 + u 4 x 3 , . .
9 ÷ ulO x
(complete polynomial truncated a t a specified degree) ~e~c~iEt_io2 Thle element has only the axial deformation mode. It is in~ented to reinforce the conforming membrane elements of types 3, 4, 6 or 20 and the hybrid element of type 14. The degrees of the polynomia] approxlm~tlov for u and that used to represent the axial distribution of the cross section can be fixed Independently to any value between O and 9. Analytical integration is always used for obtaining the stiffness matrix.
ffS~X f"
.
I
"
~
- Local axis
3
is directed
ND = 3
f r o m node I t o n o d e 2.
- De~re.es_ o f _ f r e e d o m
The length
of the bar is subdivided in ND sections, ND being the degree
of the polynomial used
to represent the axial displacement u. This
defines ND-I points between the ends i and 2 which are numbered 3, 4 ...
(}~ + 1) f r o m I t o 2 .
The d i s p l a c e m e n t
sequence in global axes is
q~ " ( d l d2 d3 d4 " ' " riND + 1 ) where
d i - (u i v i v l )
:
231 TYPE 3
CONFORMING
- Reference :
TRIANGULAR
MEmbRANE
ELEMENT
4.9 (internal)
- As!u~!io__n z u = a I + a 2 x + a 3 y ÷ a4 x2 ÷ a5 xy + a 6 y2
+ a7 x3 + a8 X2y + a9 xy2 + alO y3 +
y5 • ,. + el5 y4 + . . . . . .
v = aI
+ a21
~4 x2
a2
,
2
+ a~ x 3 + a' x2y + , xy2 + , y3 8 a9 alO
÷ "'" =~5 y4 ÷ . . .
÷ =~i
y5
(complete polynomial in x. y t r u n c a t e d a t a s p e c i f i e d degree) - ~e~c£1~o~ This element generalizes the classical triangular membrane elemen~in this sense that it can be of any degree between i and 5 along any of the 3 interfaces and inside r independently. The thickness is assumed to vary linearly between the 3 vertices. The Hooke's law modification necessary to use the element by the Southwell analogies as a plate bending
element is a l s o p r o v i d e d ( i n t h i s case t h e t h i c k n e s s has however to be constant), -Local
axes an d . d e ~ r e e s of freedom
3
ND - 3 for all edges.
)
I
6
4
5
2
The local axes are directed as indicated above with the axis Ox coinciding wlth the edge I-2, They influence only the stress ouput.
232
Each edge is subdivided in ND equal segments, ND being the degree assigned by the phase I to each edge. These segments define a certain number of points along the interfaces which =re numbered following the same se~ise of rotation as for the 3 geometrlcal nodes 1.2.3. The example given SbOvo #ppl~es ~or ND = 3 on a~L adgos. The d~splacaman~s sequence i s
qt = (dI d2 d3 d4 d5 . . .
I
dn )
where d i = (u I v i wl) in global axes. Eventually~ if along an Interfacel only 2 membrane elements are connected (plus eventually a bar of type 2) which are coplanar and not in a plane of coordlnatesp the phase ~ decides to express the dlsp!acemen~of
the interface in special local axes.
These axes are defined by the tangent and normal to the interface. Their orientation is given by a reference node printed in the output. If this node is positive the normal is directed in the half plane containing this node. If the reference node is negative the normal is directed in the opposite half plane. The advantage of these speclal axes is that the out
of plane stiffness therefore
is automatically
is automatlcally
zero at the interface
p o i n t s and
f i x e d by t h e p r o g r a m . I f t h e 2 membrane e l e -
ments are "almost" coplanar,
the speclal
local axes are selected
if the
a n g l e i s s m a l l e r t h a n .01 r a d i a n t s . - Bubble functions of displacement If the degree specified for the element is higher than the degree of the Interfaces~ bubble functions of displacement are automatically included. Thls is also the case if the degree is higher than 2 in which case internal degrees of freedom have to be defined. They can be interpreted also as bubble functions. The presence of bubble modes can be forced in various rays : for instance by specifying the variable NDB or by specifying a degree higher than that of the surrounding adjacent elements. Eventually bar elements of zero cross section can be used to force the reduction
of d e g r e e a l o n g c e r t a i n
interfaces.
233
TYPE 4
-Reference
CONFORMING' ,, qUADRILATERAL
|
MEMBRANE
ELEMENT
A.9 (internal)
-Auu~ptlo=~n s u = a I + a 2 x + a 3 y + a 4 x 2 + a 5 xy + a 6 y2
+ a7 x 3 + a 8 x2y + a9 xy2 + al0 y3
' + ' x÷ V = aI a2
' y÷ a3
, x3 a7
' X2 ' , 2 a4 + a 5 xy + a 6 y
, x2y + a8
, xy2 * a9
,
3
+ alO Y
(complete p o l y n o m i a l in x . y truncated at a s p e c i f i e d degree in each triangle - ~sc£i~ti.on
independantly)
z
6
ND=2 3
5
l
4
This quadrilateral
i s a s u p e r - e l e m e n t composed o f ~ t r i a n g u l a r
membrane
elements of type 3. The internal interfaces are defined by the diagonals of the quadrilateral, The internal degrees of freedom cGrresponding to the internal interfaces and eventuall 7 to internal modes of the triangles a r e o l l m l n a t e d by c o n d e n s a t i o n , The d e g r e e s o f t h e 4 t r i a n g l e s a r e a s s u m e d t o be i d e n t i c a l degree along the external and f o l l o w s t h e same r u l e s
and n o t h i g h e r t h a n 3, The r e d u c t i o n interfaces
of the
is possible
fo~ t h e t y p e 3 e l e m e n t . The p o s s i b i l i t y
using the element as a plate is also provided.
of the q u a d r i l a t e r a l
b e n d i n g e l e m e n t by t h e S o u t h w e l l
of
analogies
234
The thickness can vary linearly along each external interface and i~ defined by the local values at the vertices. The thickness at the diagonal intersection point is interpolated as follows : a linear variation is assumed along the two diagonals. The thickness at the central node is t h e average of 2 thicknesses so defined at this point. If the 4 vertices of the quadrilateral are not coplanart a correction of the warping is achieved by defining a mean plane and projecting the 4 nodes in this plane, The importance of t h e warping in NOT checked. - ~ e ~ r e e s of freedom The sequence of the generalized displacements is
q' = (d I d 2 d 3 d 4 d 5 d 6 d 7 d 8 ..- ) f o l l o w i n g the same conventions a s f o r the type 3 e l e m e n t . The s e l e c t i o n
of the special
l o c a l a x e s on t h e i n t e r f a c e s
follows
a l s o the same rules, - Bubble functions of displ w_acem.en~t Such functions can be introduced following the rules defined for the element of type 3.
i 8
I
I i
7 •
6
..._>. x
i
0
I 0 ~ _ _ _ _ _ _ _ _ _ _ / ~ 4
11
I
12
Local axes for stress output and numbering
of the reference points for NDT - 3
235
TYPE 6 - Reference :
SPECIAL
BULKHEAD
ELE~NT
A.7 (internal)
-~s~ump~ion~ ~ u - =I + a2 x + a 3 x 2 + =4 x3 + y (~5+~6 x+ a7 x2+ =8 x3)
V " B1 + B2 x + S3 x 2 + B4 x 3 + y (BS+86x+ 87x2+ £8X 3)
(eventually reduced to 2 d degree along the edge I-2)
- ~ c _ r le~o~ The special bulkhead element is a kind of frame or spar element designed to reinforce conforming membrane elements for out of plane bending and for which the neutral axis is not in the plane of the membrane, This situation is often met in the idealization of bulkheads in fuselage analysis, The element is a quadrilateral membrane which can be connected with the membrane elements of the fuselage skin only by the edge 1-2 along which it is conforming. The opposite edge is always supposed to be free. The 2 lateral edges 2-3 and 4-1 can be connected with similar elements or with general membrane elements. This connection is strlcly conforming only if the lateral edges are perpendicular to the edge 1-2. Otherwise it introduces a slight lack of conformity which, for bulkhead analysis, is not important, It has the advantage over standard membrane elements (llke TYPE 4) of a better representation of the bending modes which include the effects of shear deformation and shortening of the cross section, but with less degrees of freedom. Note that connection is not allowed along the edge 3-4 due to the condensation of the 4 interface degrees of freedom along t h a t edge.
The high of the element can vary linearly and two different reinforcing flanges of linearly varying cross section are provided. The thickness of the web is constant but can be different in extension and in shear to allow a correct representation of~effect of holes in the web. The loading can be achieved~ in addition to tip concentrated forces, by a linearly distributed llne load applied on the edge 1-2 in the plane of the web. This allows to input the pres~urlsatlon of the fuselage with the consistent loads. -
Local axes are oriented with Ox along 1-2
236
y
i i
i
t v4
~
4
3
h.._~
>
angles
v21
a and B s h o u l d b e s m a l l .
TYPE~,,,,,7 -Reference
~ .....
~
v12 -The
---'~ u3
FRAHEELED~NT INCLUD,ING :
- A s y . u ~ _ t i o _ nE "
A.8 u-
DEFORMATION
(internal)
a I + a 2 x + a 3 X2 + a 4
¥ m" e + aI
SHEAR
t X + a2
x3
t X2 + a~ X3 a3
i!
2 t x - eI ÷ e2 x ÷ e3 x
%y m B1* ~z
"
+
~2t
X+
" + B~ el x +
(eventually 2rid d e g r e e )
e3!
X2
" X2
~3
the dlaplacements
up v t w a r e r e d u c e d
to
X
237
- D escr ip_~on
ThSs element is a straight prismatic member which deforms in extension. bending and torsion. The local axes are the inertia axes of the cross section. As the shear deformation is included in the theorY used for bendlng, the displacements u~ v~ w are represented by modes independent ~w of those used to describe the rotations. (~y is not equal to ~ , etc...). The difference between the elements of type i and type 7 is the same as between the plate bending elements of type 15 (Kirchhoff)
and type 8
(Hencky). This element should be used to represent beams reinforcing membrane elements and which have a bending stiffness in the plane of the membranes. The connection can be conforming with membrane elements of
2nd and 3rd degree (Type 3, 4 or 20). A11 t h e c h a r a c t e r i s t i c s
o f t h e c r o s s s e c t i o n can v a r y l i n e a r l y
along
the element neutral axis.
/x
,7/ J"
,,< ,/~_G~" .! +-,
/,
./
'~/"21
i
*~,.,=/J.~M[,._!y,~
e"
~ v " / " .,.-', ~ y
,,, 'Yl "i
e.:.
u--
"/~//
"/
.I jf
wl2
~
-21 -
"12
"
, /
,, "
!
n
'. -~
238 TYPE
8
CONFORMING }DDERATE
TRIANGULAR
TtilCK~ESS
PI~TF, BENDING
(TI!EORY OF
ELEMENT
OF
HKNCKY)
- Reference • G.SANDER,Dc.Thesis,Collection des Pub.Fac.Sc.Appl.L~ge,N°i5(1969) - _ A s i u ~ i o . . n . _ , u = s $x ( x , y)
v.,
sly
(x, y)
#x " al + a2 x + a3 y + a4 x~ + a5 xy + a6 72 ty
' ÷ ' X + ' y + ' X 2 + ' xy + , y2 " uI a2 a3 a4 a5 a6
w
= B I + B 2 x + B 3 y + B 4 x 2 + B 5 xy + S 6 y2
B7 -
x3 + B8 x2y + 89 xy2 +
810 Y
3
Descri£tlon Thls plate bending element is derived according to the theory of llencky for
plates
of moderate thickness. It differs from the Kirchhoff plate
theory by the independence of the rotations ~x' ~y of the cross section and the slopes ~w/~x, ~w/~y of the mean plane of the plate, It allows to take into account the shear deformation and the edge effect due to the transverse rotation around the axis normal to the mean i>lane. The element Is derived with fixed degree and constant thickness. The Young's modulus used to compute the bending energy and the shear modulus used to compute the transverse shear energy can be independan~. This allows ~o use the element to model non homogeneous materials llke sandwich, multi-layer 7
trusses etc.
/ '~
I t~
t2.
I / i
"-L
!t
/
~ !
A+
1[ Y23
"0
i
/ *
,
/
ft'l
•
U
I-
.~o ....
~
- +h
,
.,_
w
/
1
-'--~.~ u ~2-~
ii h=/ o/ •yA
i. i"~-.~ ~" L23
o.. "ix_-
L
239
- D e ~ r e e s of freedom The element can only be used in the X-Y p l a n e . The sequence of the gener a l i z e d displacements Is. in the global XY axes.
q'
= (wl gxl Syl
w2 @x2 @y2
w3 ~x3 ~y3
w12 4x12 ~y12 w21 w23 ~x23 gy23 w32 w31 ~x31 ~y31 w13)
The i n t e r n a l degree of freedom w necessary to d e r i v e the element i s e l i m i nated by condensation. To allow a correct representation of the boundary conditions, special axes can be defined at the 6 points where rotations are expressed. Only one set of special axes can be defined per element, by giving their direction cosines in the LOCAL,,axes. This transformation of coordinates does not necessarily affect all the rotations.
TYPE. 9
~UILIBRIUM OF
qU~)RILA%RAL
PLATE
BENDING .ELE,~f~NT
MODERATE ..THICKNESS (REISSNER ~{EORY.)
- Reference : G.SANDER,Dc.Thesis,Coll.Pub.Fac. Sc.Appl.Li~ge,NOl5 (1969) -Assumptions ; In e=ch triangular region, the bending moment field in tha oblique axes defined by the 2 internal interfaces of length a and b is : Mx
- B1.
S 2 a + 83 ~b
x =s4+ss~+~6 "
l b
÷ s9 Zb
÷ s8
A p a r t i c u l a r s o l u t i o n f o r c o n s t a n t d i s t r i b u t e d load p i s superimposed pslnn
l
" ~6
~lO
(n is the angle of she oblique axes)
240
This plate bending element is derived by the equllibrium theory of Relssner for plates of moderate thickness. It differs from the equilibrium formulatlou of the Kirchhoff theory by the form of ~he complementary stress energy which includes the contribution of the transversc shcar and by the continuity requirements for the surface tractions. In this element the £endlng and twls=ing moments are continuous across an inter-
face as well as the shear forces. There are no corner loads. Although the form of the bending ~oment field is the same as in the equil~rlum Kirchhoff plate bending element of ~pe 13, it is impossible to derive a triangular element free of additional constraints (or klnematie deformation modes)due to the increased number of interface generalized ~rces, The solution of building a super-element composed of 4 triangles allows to reduce the additional constra~ts ins~e the assemblage. The element is derived with fixed degree and constant thickness. The Youngts modulus can be ~dependent of the transverse shear modulus. -De~rees of freedom
The element can only be used In ~ e X-Y plane..The sequence of generalized tortes
is :
;' " ~ n l 2 M6n!2 V12 Mn21 Man21Hn23 Man23 V n23 M n32 M sn32 I~34M=34
V
n34 Mn43
M
an43
M
M
n41
V M M ) an41 n14 n14 snl4
/"
M
M
){In
Mn
3~34.~
In32
Pn
Jb II
/
y
./" /
./M
n.-.~..,,~.
~ ~
~
M
sn23
sn2i
~'21 %
M Mn 4
~
snl2
V n14
"~
M
nl 4
~n41
Msnl~
"~
X
241 TYPE
11
-Reference
E_~UILIBRIUM BAR ELE~h'T,,,,,,,ylTH ,,,CONSTANT, SHEAR
:
AFFDL-TR-66-199
(I966)
- As_,~_p~ion : ~ - constant
or N(x) " ~I + ~2 x -
~ezc_riZ~o~ This e q u i l i b r i u m b a r e l e m e n t i s s t r e s s e d by a c o n s t a n t l o n g i t u d i n a l i h e a r and two t i p f o r ~ e s . I t i s i n t e n d e d to r e i n f o r c e t h e e q u i l i b r i u m membrane e l e m e n t o f type 12. The s e c t i o n i s supposed to be c o n s t a n t .
- De~rees of freedom
2
1
~ .
l j,-
N1
~ x
Local a x i s
N2
13,e l o c a l a x i s i s o r i e n t e d from node 1 to node 2 and t h e g e n e r a l i z e d f o r c e s a r e Ni, N2 and T. £12 = N12* These 3 l o c a l f o r c e s a r e e x p r e s s e d i n g l o b a l axes i n t h e sequence
s'-C zrzy zz=
x2
z
Y2
F
z2
z
x12
,
YI2
,
z12
)
E v e n t u a l l y , i f o n l y 2 membrane e l e m e n t s meet along the i n t e r f a c e which a r e c o p l a n a r and not i n a p l a n e o f c o o r d i n a t e s
1-2
the phase I d e c i d e s
to keep the l o c a l a x i s f o r t h e i n t e r f a c e f o r c e . The t o t a l number of g e n e ralized
f o r c e s remains the same but F i s r e p l a c e d by N12 w h i l e a z e r o YI2 mtiffness is given to 2 other components. TYPE 12
E~UILIB,RIUM QUADRI,LATEP~L ~mtmRA~ CONSTAN~ 51"?ZSSES
WITH
-Reference
ELF~E,NT,
: AFFDL-T~:,-~G-199
(I966)
-_Amsumvt_ion~ Z In each triangular regions, the stress field i n the oblique axes defined by the 2 diagonals is : ox = B 1
Oy
Lie
B2
Tzy " B3
242
This quadrilateral is the simplest equilibrium membrane element. It is eubdlvlded by the diagonals in 4 triangular regions in each of which a constant stress field Is assumed. The 12 parameters are reduced to 5 by the constraints of continuity of the nermal and tangential surface tractlons along the internal interfaces. The element is free of remaining o0nstralnts
(or kinematical deformation modes). However the connection
with other elements is achieved by identifying the simple averages of the displacements along each Interface which is equivalent to a pin Joint. Therefore special care has to be taken in expressing the boundary condilions to avoid possible mechanism in the structure. Such mechanisms are
always avoided if the element is bordered by bar elements of type ii, If the 4 nodes are not coplanarj a correction of twist is achieved by defining a mean plane in which the 4 nodes are projected. The importance
Of the warping is NOT checked.
I
/
i'2~,
y2~
2 /
i
F
3
~ t12
~i . 6 ,
. . . . . . .
"34' .C.~ ~ , "
4
X
~
(In local
nl~ o.t a~es~
. . . . . . . . . . .
\(legal ~es
..,-
global axes
F
\\ ~
f or s t r e s s
Z
(oblique axis)
243
-VeKre_es of freedom Along each interface of length Lij the generalized forces are in local
c a r t e s l a n axes normal and t a n g e n t l a l to the i n t e r f a c e Fnl j = On ' LiJ
and
F t l j = Tsn " l i J
They are expressed in global axes in the sequence
s' = (~12~ z z F F z z F F YI2 z12 x23 Y23 z23 x34 Y34 z34 r
x41
F
F
Y41 z41
)
If ~long an interface there are only 2 adjacent membrane elements which are coplanar and not in a plane of coordinatesp the local cartesian axes are kept for the interface and thethird component of force is given a zero
TYPE
values 13
K IRCHHOFF
-- References :
EQUILIBRIUM
PLATE
de Veubeke and Sander,lnt.JI.Sol.
- A.ssu.mpt..io.~n : Mx -
S1 + 8 2 x + B3 y + ~ I 0
(1 - x )
BENDING
TRIANGLE
Str.,4 (1968) x
My = 84 + B5 x + B6 y + BIO (I - y) y Mxy"
8 + 88 x + B9 y - BIO xy
The parameter BIO controls the particular solution under a unlformdistributed
load.
-VescrIEt_ion 1~alS classical equilibrium plate bending element is derived from a linear bending moments field. Such a field satisfies the homogeneous equilibrium equations. A particular solution is superimposedwhich is in equilibrium vlth a constant distributed transverse load. The thickness is assumed to be constant. - D eKre_es of freedom The element can only be used in the X Y plane. The generalized forces which insure the equilibrium along the interfaces (in the sense of Kir=hhoff) are : 2 local values of the normal bending momen~ Mn, the constRnt value of the normal Kirchhoff shear force K n - Vn + ~Msn and the 3 ~s corner loads equal to the jump of twlsclng moment Z i = M
- M sni+ o
sui_ o
244
They a r e c o n ~ u t e d i n t h e s e q u e n c e :
- (z I z 2 z 3 ~
g'
K
M
H
K
M
M
K
n12 n12 n21 n23 n23 n52 n31 n31
The ~ n C e r n a l d e g r e e o f f r e e d o m c o r r e s p o n d i n g to t h e d i s t r i b u t e d el/~nated
by c o n d e n s a t i o n .
z3/
/
H I 3)
load is
/,
/
/ X
Zl
M hi2
X
TYPE 14
(global axes)
HYBRID , RECTANGULAR
MEMBRAN E . ELEMENT
- R e f e r e n c e = T.H.PIAN,US-JAPAN Seminar,~OKYO (1969) -Ajsu~i_on~
s I ) u " a I ÷ a 2 s + a 3 s 2 + ~4 s3 + a s s4 + a6 s$ " ' " ' ÷ ' S + o h 12 ' S3 + ' s4 + ' s5 v - aI a2 + a4 a5 a6 ,..
•S b e i n g a c u r r e n t
c o o r d i n a t e t a n g e n t ~o t h e e d g e s .
~y2
y
;ix 2
xy
~x~y
( x . y ) b e i n g a c o m p l e t e p l y n o m i a l i n x and y . The p o l y n o m l a l s f o r u . v . ~ a r e t r u n c a t e d r e q u e s t e d by t h e u s e r .
a t ~he d e g r e e s
245
This rectangular membrane element is derived according to the theory of hybrid elements dcduclble of the Relssner principle and presented In the chapter 2 of the present report, It covers the family of such elements up t o the 5th degree for the displacement field assumed along each inter-
face independently and with practically no limitation for the degree of
the stress field assumed inside. Numerical difficulties arise however if the degree of the stress field exceeds 8. The nature and the sequence of the generalized displacements defined alon E the interfaces are such that
the element can be Joined indifferently to any conformln E =embrane element of types 3~ 4, 6~ 20 and to the bar or beam elements of types 2 or 7. It should be noted that no check is incorporated that the element is effectively a plane rectangle. The thickness is supposed constant. - D e~rees of freedom and local axes
Y
3
8
I , I
__
~
-
9
2 6
90e °
I I
10
4
II
'5
.---->x
for ND - 3 on all interfaces
-i
I 12
The s e q u e n c e o f t h e g e n e r a l i z e d d i s p l a c e m e n t s f o l l o w s t h e same l o g i c as for the conforming membrane elements of types 3~4 or 20 : the displacements of the vertices are followed by those of points defined on the interfaces
taken sequentially
turnin E anticlockwlse.
q' = (d I d 2 d 3 d 4 d 5 d 6 d 7 , . . d12) w l t h d i - (u i v i wi ) The d i s p l a c e m e n t s o f t h e v e r t i c e s
a r e always e x p r e s s e d i n g l o b a l axes
while on each individual interface the special local axes defined for
the elements of tvves 3 or 4 can be selected by the phase I,
246
TYPE
15
,CONFp?~IING QUADRILATERAL
PLATE
BENDING,,,,ELLMENT
(KIRCHI{OFF THEORY) -Reference : de V~UB~K~.Int.JI.Sol.Str..4 (1968~ -~s~p!i~n
£ : In each triangular region 2 w m al + o2 x + a 3 y + a 4 x
÷
-
2 + n 5 xy + a 6 y
"7 x3 + °8 x2 Y + n9 xY 2 + ~i0
y3
£ezcai£t!o~ This plate bending super-element is obtained by assembling the 4 trlan8ular elements defined by the diagonals of the quadrilateral. In each of these triangles the deflection is represented by a complete cubic. The 40 corresponding parameters are reduced to 16 by expressing the continuity requirements of th~ deflection and normal slope along ~he internal interfaces of the quadrilateral. These 16 lhdependent parameters are finally expressed in terms of the 16 generalized displacements (w, ~x ' ~y ' at the 3 vertices and ~n along each interface) necessary to insure a strict continuity of the deflection and normal slope along the external interface of the quadrilateral. The element is programmed with the possibility of using a variable thickhess D anlsotroplc stress-strain relations and various special support options. The element can be used as an equilibrium membrane element by the stress function method. It is the Southwell analog of the element t y p e 16. In this case the thickness has t o be constant.
- ~e~rzes of freedom The element can only be used in the global X-Y plane. The sequence of t h e 16 generalized displacements is :
q'
~w
~w_~_ w2 ~w
(wl °~! ~Y2
~w
~x2 ~Y2
w3 ~w
~w
~x3 ~Y3
w4 ~w
~w
~w
~w
~w
~w
~ ~Y4 %n12~nz3 ~n34 ~n41
247
~,~'Y
Ij
w2//
/
%'
x3
n34 V
x4 w1
~.
n41
Y$
~X
g
0
Oxy
oblique l o c a l axes
u ~
Ox y
c a r t e s i a n l o c a l axes
Oxg y8
global axes
The local axes O~ is directed from 0 toward the middle of the edge 12. The normal slopes w
are expressed in the middle of each edge. nlJ
- S~ec!a! ~ i o n ~ I*/ Variable thickness : the thickness is constant unless the variable ICHOI - 1
In this case the thickness at point O is interpolated from the values defined alon~ the 2 diagonals at this point by a linear variation. The average of the 2 values is assumed.When the thickness is variable all the other special options are ineffective.
2"/ A n i s o t r ~
: it is controlled by the variable IANISO. IANISO = 0 corresponda to the Isotropic case, Fou~ anisotropic layers can be superimposed to the parent plate. Each of the~e layer can have an independent thickness tan i an4 I E tani 3 • the bending rigidity of each layer is ~-~
248 TYPE
16
E_qUILiBR!UM OUADRILATERAL (LINEAR
- References :
MEMBRANE
ELEMENT
STRESS FIELD)
deVVEUBEKE,AFFDL-TR-66-88 (1966)
- A ss_umpt_io_n~ : In each triangular region :
°x
" ~1 + ~2 x + B3 y
Oy
= B4 + 8 5 x + ~6 y
x
= B7 - B6 x - B2 y
- ~eEc~iEt!o ~
This e q u i l i b r i u m membrane is a super-element o~tained by assembling the 4 triangular elements defined by the diagonals of the quadrilateral. In each of these triangles the stress field is linear as indicated in the
assumptions. The 21 corresponding parameters are reduced to 13 by expressing the continuity of the normal and tangential surface tractions along the internal interfaces of the quadrilateral. The 16 generalized forces necessary to determine uniquely the linear variation of the surface tractlons along the external interfaces are expressed ~n terms of the 13 parameters. As the 16 forces satisfy 3 global equilibrium equations, the element is free of spurious kinematic modes. The generalized forces are the local values of the surface tractions Tsn . O n (times the thickness)
at each vertext along an interface. Note that although expressed at a
vertex,
they are interface
variables,
The e l e m e n t i s programmed w i t h t h e p o s s i b i l i t y
varLable t h l c k n e s 3 , -De~es
of freedom
of using a linearly
249
\ T
T43
3
,t, ~
4
"='=W T41
4 (local oolique
• ~N4I
axes)
Y8 (local axes for stresses) :X
(global axes)
0 The s e q u e n c e o f t h e g e n e r a l l z e d
f o r c e s i n t h e l o c a l axes Oxy i s
8' = (H12 TI2 N21 T21 N23 T23 N32 T32 N34 T34 N43 T43 N41 T41 NI4 TI4 ) where
face
Nij P Tij are cartesian normal and tangentlal components of the surtractions
multiplied
by t h e l e n g t h o f t h e i n t e r f a c e .
These forces
ere transformed in global forces S" -
F_
(x12 Eventually
F
F
F
F
F
is given a zero stiffness. along an interface
dinates,
.,. F
)
z14
t h e p h a s e i can d e c i d e t o keep t h e l o c a l n , t axes on an i n t e r -
face in which case the 3rd co~onent
wet
F
YI2 z12 x21 Y2I z21 x23
of t h e g l o b a l f o r c e s a t e a c h p o i n t
T h i s i s ~he c a s e when o n l y 2 membrane e l e m e n t s
and a~e c o p l a n a r w i t h o u t b e l n ~ i n a p l a n e o f c o o r -
250
TYPE
,EQUILIBRIUM BAR
17
VARYING
- Reference : - Ajs_u~p~io.n~
SH~
ELE!m~ AND
M.KIEFFER, SF-6 (internal) :
~(x)
CROSS
WITH.,,,,,,,,,,LTNE,ARLY SECTION
ARF~
(1969)
= S I + ~2 x
2 X
or S(x)
-
_Des_~ZZ~on
= SO + SI x + S 2 ~ - -
-
This b a r element i s i n t e n d e d to r e i n f o r c e the membrane element o f type 16. It is stresses by a linear axial shear and 2 tip loads. The cross
section can vary linearly
between t h e 2 e n d s .
(local)
- D e g r e e s o f freedom
r
X
/.
/x
7"
(global axes)
0 The sequence of the generalized forces in the local axes is
g' = {N 1 N 2 TI2 T21 ) whexe Ni i c a t i p 1cad and T i j t h e l o c a l v a l u e o f t h e s h e a r a t an end tfmes
the length of the bar.
Theae f o r c e s a r e e x p r e s s e d i n the g l o b a l axes i n t h e s e q u e n c e :
25t g' - (FXl F F F F F T T T T T T ) Yl Zl x2 Y2 z2 x12 YI2 z12 x21 Y21 z21 Eventually the forces TI2 and T21 are kept in local axes if this special
choice of axes has been decided by the phase I. In this case, zero stiffmess is given to the components T T and replaces T in the sequence given above, x12 z12 Tij YlJ TYPE
20
CONFORMING VARIABLE
- Reference : -_As~um_p~ion :
PARALLELOCRA}! DEGREE
A~D
ELE~N~
BUBBLE
WITH
FUNCTIONS
A.II (internal) u and v are of the form
2
2
PI (x, y) + (l - a ~ ) (1 - bye2) P2 (x, y) where both functions PI and P2 are P£ = (u 1 + a 2 x * ~3 x2 + "" + an+l n ) ( 8 1
+ B2 y + ~3 y2 + . . . ÷ g n + l
yn)
t~uncated at the degree requested by the user. - D es_cEiEtIo~ This element is a generalization of the classical rectangular membrane elements in this sense that it is extended to variable degree polynomials (from i to 9) and that it includes bubble displacement modes of degree independently variable (from O to 9). In this purpose the polynomials used to represent the displacements are split in two parts controlled by the functions PI and P2" PI describes the basic field and the interface modes Whi~e
P2
represents only bubble
displacement modes.
To allow easy comparison with other elements (type 4 or 14) the number of terms retained in the functions P1 (x, y) corresponds exactly to the number of generalized displacements necessary along the interfaces. This r
reduction of parameters is obtained by dropping in PI the terms of power greater or equal 2 in x ~ y
and the terms of power greater than n in
x O~R y, n being the degree of the displacements in the direction of the
edges. The complete form given above is always used for P2 (x, y). The thickness is constant and the material isotropic. The element can be used as a plate bending equilibrium element by the Southwell analogies. - ~e~rees of freedom
252
3
9/
s
/
//
/
10
4
/'
/
/
2
/ ~
x
/ II
,
! ! !
12
/
I
°
r
1
/
.
I
1
~ L14
~ ~14
n o d a l p o i n t s e q u e n c e f o r ND = 3 The g e n e r a l i z e d d i s p l a c e m e n t s a r e e x p r e s s e d a t e q u i d i s t a n t
points along
the i n t e r £ a c e , d e p e n d i n g o£ t h e d e g r e e s e l e c t e d . The s e q u e n c e o f t h e d i s p l a c e m e n t s , t h e c o n v e n t i o n s f o r r e d u c i n g t h e d e g r e e a l o n g an i n t e r f a c e , t h e o p t i o n o f l o c a l n o r m a l - t a n g e n t axes a r e e x a c t l y t h e same a s f o r e l e -
-
ments t y p e 4 o r 14. S t r e s s out~u~ The s t r e s s e s
a r e computed a t r e f e r e n c e p o i n t s which form a r e g u l a r mesh
d e f i n e d by ND-1 o r ITENS and o r d e r e d i n t h e s e q u e n c e i 1 1 u s t r a t e d The l o c a l c a r t e s i a n
a x e s £or s t r e s s e s
have t h e Ox a x i s p a r a l l e l
below. to the
!,
raise 4 - 1 .
! 10
1
.........
~o
Ii
12
t/
/
........ /I ............
1
~£erence
points for stress
2
1
3
! o u t p u t : 2nd d e g r e e i n x and 3rd i n y
253
Z
i
L h!2_. i"
,
-1
m
E,, 2.!0l°
~
v:o. t
=
~
.
0,01
TEST
CASE
NBK
1
:
CANTILEVER
U
BEAM
/,.
~"
/
\
'/
\
,
,, l ~ s
"
,
"/
.... "~"
~/,
'¢
,
1
2
',/
-
TEST
/'2
"
I/
"
: I I
i
/3
CASE
3
' /
•
/
2
l:6
CLAMPED
NBR
~
:
',~.',/
',
5
/s
CANTILEVER
,
SKEW
6
",/
"--,/6
PLATE
)
7
/
- 10 3
p
- 0.1 ,- 10 4
t p
E 2 ", 10 8
- 0.3
1.092
;
v
E1 -
.... V _ . . . . . . K
", /
i /,',,/,-,./,./,o,,/,,,,/,~,,/ O'A V" :/ x , '/ "," '4. i / \ ,o, /',, ,o~/', ,o~/", ,o~/', ,o~/". ,os/" /
%/
101°
"A.
,,./",o, / "',,/" ";./" .',.,/," "/", , , / ' ,-/'" ',.X / / , , ,oo / ,, , o , / , . , , o / . . ,,, /
I
\
"/
\ I~6
~I
,
. . . . . .
.y
',, n / ' , , 2 ~ /
"
, , / ; /
,
,, I ~
\ ,~ /
I~
,,
Ii
',u, /
IJZ
~
/~.., ,,,/',, ,,~/.. ,,~/,, ,,~/,, ,,~/,, , , , /
" uo/
\
'/
,,
,
!
",,,,~/
-
I
I
,s/
!
131
42
•
!
0 P
i
f
¥
/
,,
"/~
,, I
,
x
/,9
/
0 P --
255
Y,I, i
,i•Y ~ •
15"
15'"
i-3
!
i,".-'"
3G
23
16
J1
24
17
xz
]~---.---~-.~-.-.-:
z I"('-... // g
-1 ti 2~
t2 ........
~.'OX
I3
6~L ~
"N.
! ~
26
IS
27
20
2e
s4
21
7?"
.!
! !
J
9
--
\I
......
I
e
29~
!_,;.; ...¥ i~;: ..-'" \\ o~:
i
lSo,,
! I
.P: 1000 k 9.
v.t, 29
22 ~. rll
I5
@
2~
30
9
I6
®
~,
®
17
2,c
31
~
I
"
(2)
,,,
@
2
3
Io t,,
25 ".
I8
@
® 26
.~ ® @ 35
V = 0,33 ~9
12
i'.
®
@ ....13
~.
2e
®
E = 1,06
I!
27
~4
(~
~.
21
@
e,i
14
®
!07
256
REFERENCES I. FRAEIJS de VEUBEKE, B. "Displacement and equilibrium models in the finite element method" Stress Analysis, ed. O.C. Zienklewlcz and G. Hollster, Wiley, 145-197, (1965) 2. FRAEIJS de VEUBEKE, B. and ZIENKIEWICZ, O.C. "Strain energy bounds in finite element analysis by slab analogies" Jnl of StralnAnalysls, 2-4, 265-271, (1967) 3. FRAEIJS de VEUBEKE, B. "Basis of a well-condltloned force program via the Southwell slab analogies" USAF Report AFFDL-TR-67-80, (1967) 4. FRAEIJS de VEUBEKE, B. "Duality in structural analysis by finite elements, static-geometric analogies, the dual principles of elastodynamics" NATO Advanced Study Institute lectures on finite elements, Univ. of Alabmma Press in Huntsville, 299-377, (1971) 5. FRAEIJS de VEUBEKE, B., SANDER r G. and BECKERS I P. "Dual analysis by finite elements. Linear and non linear applications" Air Force Flight Dynamics Laboratory, Wrlght-Patterson AFB, Ohio, Technical Report AFFDL-TR-72-93, (1972) 6. FRAEIJS de VEUBEKEp B. and HOGGE, M. "Dual analysis for heat conduction problems by finite elements" Int. Jnl Num. Meth. Eng., 5, 65-82, (1972) 7. FRAEIJS de VEUBEKE, B., GERADIN, M. and }lUCK, A. "Structural Dynamics" CISM, Udine, (1973) 8. FRAEIJS de VEUBEKE, B. "Diffusive equilibrium models" University of Calgary lecture notes, (1973)
257
9. GERADIN. M. "Computational e f f i c i e n c y of e q u i l i b r i u m models in e i g e n v a l u e a n a l y s i s " P r o c e e d i n g s o f t h e IUTAM Symposium on High Speed Computing of E l a s t i c S t r u c t u r e s . Congr~s et Colloques de l ' U n i v e r s i t ~ de L i e g e . Place du XX AoQt. 16. 4000
LIEGE. 589-623. (1971)
10. GERADIN. M. "Analyse dynemique duale des s t r u c t u r e s p a r l a m~thode des ~l~ments f i n i s " C o l l e c t i o n des P u b l i c a t i o n s de l a Facult~ des Sciences Appliqu~es de l ' U n i v e r s i t ~ de Liege. 36. 1-173, (1973) II. IMBERT, J.F., GIRARD, A. and GERADIN, M. "Modal analysis of a satellite primary structure using a finite element procedure" Sympositnn on s t r u c t u r e s of space v e h i c l e s and s p a c e c r a f t . U n i v e r s i t y College London. (1973) 12. IRONS, B. and RAZZAQUE, A. " E x p e r i e n c e s w i t h t h e p a t c h t e s t f o r t h e convergence of f i n i t e e l e m e n t s " Conference on t h e m a t h e m a t i c a l f o u n d a t i o n s of t h e f i n i t e element t h e o r y " Univ. of Washington ( B a l t i m o r e ) , Academic P r e s s ,
(1972)
13. PLAN, T.M. and PIN TONG " B a s i s of f i n i t e element methods f o r s o l i d c o n t i n u g ' I n t . J n l Num. Meth. Eng., 1, 3-28, (1969) 14. SANDER, G., BECKERS, P. and NGUYEN, H.D. "Digital computation of stresses and deflexions in a box beam" C o l l e c t i o n des P u b l i c a t i o n s de l a Facult~ des Sciences Appliqu~es de l ' U n i v e r s i t ~ de Li~ge. 4, 87-137B (1967) 15. SANDER, G. "Dual a n a l y s i s of a multlweb swept back wing model" A i r c r a f t E n g i n e e r i n g . 6-16. (1968) 16. SANDER, G. " A p p l i c a t i o n of t h e dual a n a l y s i s p r i n c i p l e " P r o c e e d i n g s o f t h e IUTAH Symposium on High Speed Computing of E l a s t i c S t r u c t u r e s I Congr~s e t Colloques de l ' U n i v e r s i t ~ de Liege. Place du X.X AoGt, 16. 4000
Liege, 167-207. (1971)
258
17. SANDER, G., BON, C. and GERADIN, M. "Finite element analysis of supersonic panel flutter" In,, Jnl Numo Meth, eng., 7-2, (1973)
18. STRANG, G. "Variational crimes in the finite element method" Math, Foundations of the finite element method~ ed. A.K. AZIZ, Academic Press, (1972)
19. TABARROK, B, and SODHI, D.S. "The generalization of stress function procedure for dynamic analysis of plates" Int. Jnl Num. Meth, Eng., 5, 523-542, (1973)
20. TURNER, M.J., MARTIN, H.C. and WEIKEL, R.C. "Further development and applications of the direct stiffness method" Matrix methods of structural analysis. AGARDograph 72, 203-266, Pergamon Press, (1964)
VISCO-PLASTICITY AND PLASTICITY AN ALTERNATIVE FOR FINITE ELEMENT SOLUTION OF MATERIAL NONLINEARITIES by O. C. ZIENKIEWICZ Professor of Civil Engineering, University of Wales, Swansea and I. C. CORMEAU Aspirant F.N.R.S., Unlverslte Libre de Bruxelles (now at Swansea) •
.
J
Summary In this paper, authors present a formulation and some computational details dealing with a general elastic/visco-plastie material where nonlinear elasticity is admissible and the flow rule and yield condition need not be associated. If, in a visco-plastic solution method, stationary conditions are reached for the displacements, the solution to an equivalent plasticity problem is obtained. The visco-plastic approach thus provides an alternative technique to solve elastoplastic problems, and which is found to possess considerable merits vis ~ vis other iterative processes. In particular, non-associated flow rules and strain softening can be dealt with inageneral purpose program without requiring specific numerical artifices.
Further,
by providing always an equilibrating solution (within the approximations of the finite element discretisation) and, at displacementss~a~i~rmri~, ensuring a plastically admissible stress distribution, results always give a lower bound to collapse. The paper includes several examples to illustrate the application of the method to some problems of practical interest.
i.
INTRODUCTION
It is now customary to apply finite element techniques to obtain the solution of non-linear material problems (ZIENK!EWICZ27).
A great variety of specialised
numerical models were developed to deal with particular situations such as viscoelasticity (WHITE 19, ZIENKIEWICZ, WATSON and KING24), creep (GREENBAUM and RUBINSTEIN 9, SUTHERLAND 17, TREHARNE 18) or classical elasto-plastieity (ARGYRIS and SHARPF 2, MARCAL and KING II, NAYAK and ZIENKIEWICZ 12, ZIENKIEWICZ, VALLIAPPAN and KING25'26).
Not only metals but also polymers, cracking materials, rocks and soils
have been idealised by means of several ad hoc models (ZIENKIEWICZ27). The idea of a visco-plastic medium, involving both time and plastic effects is not new (BINGHAM4, FREUDENTHAL and GEIRINGER 8, REINERI6).
Surprisingly, however, it
280
arose only limited interest among structural analysts and engineers, despite its conceptual simplicity, generality and relative ease of implementation on digital computers. Early applications of visco-plasticity theory dealt with rigid/visco-perfectlyplastic plates and axisymmetric shells, under linearization assumptions, for static (APPLEBY and PRAGER I) and mainly dynamic loadings (WIERZBICK121, WIERZBICKI and FLORENCE22). The viscoplastic model does equally well in quasi static situations where the inelastic strains are of the same order as the elastic ones.
Closed form solutions
exist for simple geometries where the whole material is assumed to be above the static yield limit (WIERZBICK120) and numerical results are reported by CHABOCHE 5 for an elasto-visco-plastic structure subject to time varying thermal gradients. The formulation and experimental determination of viscoplastic constitutive relations were discussed by ZARKA 23 (microscopic approach - metals), LEMAITRE IO and PERZYNA 15 (macroscopic approach) among others;
these references contain
extensive bibliographies on the subject. Numerical methods for the solution of quasi static elastic/visco-plastic problems of arbitrary geometry were described by NGUYEN and ZARKA 14, ZIENKIEWICZ and CORMEAU 28, but so far only little numerical work has been published.
2.
FINITE ELEMENT FORMULATION OF QUASI STATIC SMALL STRAIN ELASTO/VISCO-PLASTICITY
Let a body ./~ bounded by a regular surface S under body forces ~
and surface tractions ~
be in quasi static equilibrium
, and subject to the boundary con-
ditions for displacements, velocities and surface forces
The material is supposed to be capable of transforming a mechanical energy input into both stored and dissipative forms, with possible interaction of externally prescribed thermal effects. It is well known (ZIENKIEWICZ 27) that a finite element approximation of a displacement (and velocity) field in _ ~ , by interpolation in terms of 'nodal' values .~ such as Z=o
c,~ 5=.
(2)
leads to a velocity strain distribution in /L (3)
261 where the matrices N imal s t r a i n s
and ~
depend on spatial coordinates
only if infinites-
are assumed.
With the introduction of stresses conjugate (in the sense of virtual work) to the velocity strains
T
and of consistent nodal forces .~
equivalent to ( ~
,~
), the principle of
virtual velocities yields the discretised equilibrium equations
(4a) (4b) as long as the changes of geometry are disregarded and assuming that rotations remain small to allow ~
to be simply taken as
(5) With regards to the reversible part of the total behaviour in view of isotropic cases, a specific stored-energy function, depending on the stress invariants, is supposed to represent the mechanical energy that can be released on instantaneous isothermal stress removal
where
The rate at which mechanical energy is stored can always be written as = c~-T M~ ~ where ~
(7)
is a symmetric, generally stress dependent, matrix given in Appendix I.
Further, it is assumed that the total stress power
~J = ~c~ ~-T~
can
be written as a sum of three independant terms
where
~
= (~_r~ 8
denotes the contribution of prescribed thermal effects and
~o : 0 "T ~~vF>.O ~-
designates the irrecoverable power of dissipation due to
viscous and/or creep phenomena.
Here,
~e
and
~vp
have dimensions of strains and
can be considered as internal state variables describing thermal and dissipative effects.
The balance equation for the stress power gives
T ( ~ _ M ~ _~o_ ~ ) : o
(9)
262 from which the stress velocities are obtained
Since dissipative and thermal effects are present, some heat supply per unit volume and unit time and heat flux will exist in ~ h principle of conservation of energy;
and on 5
to satisfy the
the quasi static hypotheses are therefore
extended to the consideration of processes where the retro-actlon of is negligible, such that
$o
onto ~ B
remains externally controlled everywhere in -f]- at
any time. Combining the local equation (I0), the general equilibrium equation (4b) and the strain displacement relation (3) one obtains
K T =J.o~.~r ~'J'~B
where matrix.
~.fl-
is a stress dependent tangential stiffness
Assuming instantaneous elastic stability, it is possible to solve (II)
for the velocities
A constitutive equation remains to be chosen for
~
; herein a generalized
form of the elastic/visco-plastic model proposed by PEI~IYNA15 is adopted since it covers several other models as particular cases.
PERZYNA's constitutive equations
state that the irrecoverable dissipation occurs only if the stresses exceed the static yield condition
)> where
a) b)
~
is a positive, possibly time dependent, fluidity coefficient
~(~-)
is a positive scalar-valued monotonic increasing function
in the range
~>~
such that
~-'f(~)
exists and possess similar
properties in the same range c)
the notation 4 >
stands for
d)
F= ~(o-~ ~vF')-7~< )
o
if
represents the yield function, being zero
when the static yield condition is satisfied. e)
y(~)
f)
q=Q(~,~.v~4-~
g)
~
is a static yield stress is the visco-plastic potential
is a hardening parameter, either state or more generally history
dependent and whose value is then given by the integration along the path of past states of an evolution equation
h)
~o
is a positive quantity introduced to make F / ~ o
dimensionless to
263
allow arbitrary forms of the function Specific forms for Fj ~
and ~
were derived by NAYAK and ZIENKIEWICZ 12 for
some popular isotropic yield criteria and plastic potentials, in a form convenient to computations and can be found in Appendix II. In isotropie situations F
and (~
depend on the stress invariants;
as shown
in Appendix I, it is also possible to write
where f
i s ' another s t a t e dependent symmetric m a t r i x which should be at l e a s t
positive semidefinite since
and that vanishes for any stresses below the current static yield surface ~--O The mathematical nature of &he final problem to be solved can be better appreciated after ~ •
-4
and ~ -4
*
have been eliminated between (3, IO, 12, 14) T
-I-9
T
-1. v F
(16)
~t thus g i v i n g r i s e to a n o n l i n e a r f i r s t
order system of p a r t i a l d i f f e r e n t i a l
f o r the stresses and hidden v a r i a b l e s ~ v p J k~,
equations
In the f o l l o w i n g sections,
simplifications will be achieved by considering linear elasticity and numerical integration. 3.
LINEAR ELASTIC/NONLINEAR VISCO-PLASTIC BEHAVlOUR
Isotropic linear elastic properties result from a particular form of the specific stored energy function:
B(4-=~) =
where E
0.,~2 i.
&E
3Z
(17)
E
and ~} are Young's modulus and Poisson's ratio respectively.
Equation (I0) becomes
-_ _D where D
is t:he usual elasticity matrix;
(18) integration in time gives O- I
~ / d e n o t i n g an arbitrary co-ordinate dependent stress distribution. The tangential stiffness matrix reduces to the ordinary stiffness matrix
264
which can now be assembled, part inverted and kept in tkis form once for all times ready for further operations involving
~-~ --
Initial conditions determining a solution to the initial value problem (16) may consist of an elastic set as follows: for
~
)~
~~ ~~
being given arbitrary values one has
- ~. ~,
(21)
i
,/0=¥C
o
eo
vPo
o)
4.
PARAMETRIC DISPLACEMENT FIELD AND NUMERICAL INTEGRATION
The diseretization of the velocity field (2) is generally not sufficient to eliminate fully the space variables ~ (~c.,,, ~ v p
and ~
which are still implicitly presence since
are allowed to vary within every finite element unless
simplex constant stress elements are used (ZIENKIEWICZ and CORMEAU28). Even if, at a given stage ~
of the calculations, the analytical form~[.~,~'~)
were known, the exact integration of
J-O.~T.~ ~vp ~-/I-
would remain
difficult since the instantaneous visco-plastic boundary crosses some elements; moreover
considering the possibly complex form of
~vP
, it is concluded that
numerical integration will be required, a context in which (iso)parametric elements find their natural justification.
These elements, fully described by ZIENKIEWICZ 27
have proved their capabilities in many non-linear problems. Displacements and velocities are interpolated over an element by means of parametric equations where the parameters
~
and ~
(2-D problems) are curvilinear
co-ordinates ranging from -i to +I: =~
~ (~,~ ~
I = ~ N ~ (~,~) ~
displacement interpolation
(22a)
co-ordinates transformation
(22b)
Introducing the Jacobian matrix
one has for planar cases,
and for axisymmetrie cases, where
$~.=o
is the axis of revolution
265
The matrix ~
depends on Cartesian derivatives of the displacement interpol-
ation functions NL
uonvergence,
and on the radial co-ordinate in axisymmetric cases
with
uniform mesh refinement, of the finite element approximations,
to the solution of the continuum problem, will occur if the following (sufficient) conditions are met: a)
the approximate displacement field can represent the exact one in the limit;
if the constant strain and interelement compatibility criteria are satisfied, this condition is fulfilled provided the exact strain field is bounded in-f~ . b)
nodal equilibrium equations are exactly satisfied when constant stresses
and body forces prevail in all elements, provided the exact stress field is bounded in YL . c)
Constitutive relationships are satisfied for all stresses and strains
appearing in the discretized equations (i6). Conditions (a) are always satisfied by isoparametric ( ~ by subparametric elements if linear relations
W~g= ~ N ~
~
elements and exist
(ZIENKIEWICZ27). Condition (b) requires an exact integration of
J~_ N ~
~-/k
for constant vectors o- ,
if body forces are present, ~v~
, ~ @
, -- .
Relevant 'square' Gaussian rules can
be found in Appendix III for various 2-D cases. Condition (c) is related to the convergence properties of the numerical method used to integrate (16) in time. In order to examine the effect of numerical integration on the discretization, integrals are approximated by the symbolic formula
266
where T
denotes the total number of integrating points in .O- and
This leads to a further diseretization since the approximate fields O~ , --~ vt , t~. depending on ~
and ~
are replaced by a finite number of time dependent
values at the integrating points only. Introducing 'structural' vectors grouping all integrating point quantities such
(27) it is possible to rewrite the diseretised equations in a compact form:
(28b)
where: B
~"
~
(28e)
(28d)
.
.
.
.
.
.
.
.
.
E
(28e)
is a constant structural strain displacement matrix made of elemental matrices computed at the integrating points is a constant diagonal matrix of -GA
D
coefficients defined in (26)
is a constant syrmetric matrix made of ~
matrices following each other
along the diagonal of ~) is a state dependent symmetric positive semidefinite matrix with structure analogous to that of and such that
C
and C
D = D~
~
=~-n C
In the case of linear elasticity, Eqns. (16) are replaced by (29a)
(29b)
(29c)
where
5=D S K
-4
T
D -D£
-4 (BOa)
267
is a constant, symmetric, negative semidefinite matrix and where
--
=
D B K- B c
~
'
(3Oh)
is a known, time varying, vector. For ideal visco-plasticity, Eq. (29c) does not exist and ~ '
depends solely on
so that finally the problem reduces to a nonlinear ordinary differential vector
iAl
equation
= where ~
=(~)(~
± ~-=
~)
(3l)
, though itself non-symmetric, is the product of two
symmetric matrices. Alternatively it is possible to derive a similar expression for weighted viscoplastic strains
I,,,
~
~
~-" ~..~
(32b) - T
but where the stresses are still implicitly present in
A'
and l '
It is seen that the size of the numerical problem is proportional to the number of integrating points in./')- ;
it is therefore of prime importance to use the lowest
possible order of numerical integration compatible with convergence conditions ( b ) mentioned earlier in this section if economy requirements are to be met. 5. 5.1
COMPUTATIONAL STRATEGY
Basic alg£rith~ The solution to (29) starting from initial conditions (21) can be obtained by a
time marching procedure. Let an equilibrium situation be known at a time t~a~
A vector of current pseudo loads is kept permanently up-to-date (33) Increments
interval
of hidden
variables
and pseudo
loads
are
calculated
over
a time
~ ~;~ = ~z,t~ - gzn ~_e . e (34a)
(34b)
268
(34c)
(34d)
v,,, +
(34e)
and the total displacements, strains and stresses are fully recalculated at time m~4
~
,,,v , ' 1 ' 1 4 4
(35c)
so that nodal equilibrium is maintained exactly at all stages of the computations. This procedure is slightly different from that presented earlier by ZIENKIEWICZ and CORMEAU 28 since it avoids the possible accumulation of errors due to the summation of incremental displacements;
the proposed process, however, would not apply in
the case of nonlinear elastic properties, where K T
is to be used with incremental
displacements only. 5.2
Time intervals selection Finite time intervals add new errors to the usual finite element space dis-
cretization errors.
Though a constant time step is the simplest procedure, it is
uneconomic near steady state if it is chosen as to give accurate results in early transient stages.
A variable interval is desirable, increasing when stresses
approach stationary and decreasing when the visco-plastic flow accelerates (when strain softening occurs in highly stressed regions or upon instantaneous load application). Two empirical criteria were used with success: Criterion I:
limitation of the incremental visco-plastic strains I
where
~
~
Z
~
Z
~
X
is the minimum taken over all the integrating points in.(]- and
269
where ~
is a time increment parameter specified by the user.
indicates the ranges
~4~O.~
Practical experience
for simple problems of contained plastic flow
o.O~
for problems of stress concentrations on structures loaded slightly below their static bearing capacity.
This criterion is well behaved in transient analysis but fails if steady state conditions are reached for stresses and viscoplastic strains;
hence the introduction
of Criterion 2:
limitation of the growth of successive steps
~L~4< -~ ~
~=4,~''
In all results presented herein, a value reported the range
@.Z~-~
~=4.5
(37) was assumed.
SUTHERLAND 17
in a transient creep analysis.
These two criteria do not offer an absolute safety regarding the propagation of truncation and roundoff errors;
for most problems of contained visco-plastic flows,
however, a plastically admissible stress vector ~
is generally achieved before
significant instabilities manifest themselves. For visco-plastic laws where the second order derivatives
~
can be
computed easily, CHABOCHE 5 reported a more elaborate method based on the second order TAYLOR expansion of the visco-plastic strains;
where ~ is a precision parameter; here the stress velocities ~ ated at each step. 5.3 S o m $
must be calcul-
schemes for integration in the time domain
So far, the mean rates of Eqns. (34) have been left undefined. values can be computed for known quantities [ ~
~
While exact
only approximate values can he
{~g~}
used for the unknown hidden variables ~=
(39)
The simplest, and crudest, approximation is EULER's rule ~ = ~ m giving a per step truncation error on ~/ steps exceed a critical value and difficult to estimate. estimate
~V[ --
(40) O [ ~ 2 -~ -"
~ > A ~
and being unstable when the
which is generally stress dependent
It was also found that EULER's rule tends to over-
and can cause premature collapse of elastic/visco-perfectly-plastic
structures loaded below their bearing capacity.
Despite these undesirable features,
270
EULER's rule remains attractive because of its simplicity. More accurate results can be obtained from the trapezoidal rule _
-q
-- ~ but since quantities at quantities at
~
predictor
I-a+Y,)
~*~
(41)
cannot be expressed explicitly as functions of
, an iterative scheme must be used: ~~/(~) +4
=~/4~~c~ ~
corrector
~
(42a)
+ t~f/~,~l/
i= ~ ~/'"
(42b)
The predictor and each repetition of the corrector involve a complete resolution (34b-c-e, 35 a-b-c).
If iterations converge, the trapezoidal rule (41) is satisfied,
providing a per step error om ~/ O ( a ~ ) . This is true even when only one cycle of the corrector is performed (HEUN's rule);
sincere there is no guarantee that con-
tinued iterations would yield more accurate results in the end, HEUN's rule is preferred for reasons of economy. Convergence of iterations require the time intervals to obey precisely the same condition as necessary for stability of EULER's and HEUN's rules;
this could
possibly offer a method of error control during the computations. In a variant of the midpoint rule, averaged values of
~
s ~WPj ~
are
considered : predictor corrector
where
X//~4} ~ + ~ = ~_/~ + ~
~
x//(~*~ y/
-" ~
~_~+__~I=
2.
(43a)
~ ~+4 =.~-~ ÷ & ~ ~*~_.~-+~
7'
(43b)
2.
~
2_
Converged iterations do not give a standard numerical integration formula. converged and one-cycle midpoint rules have both a per step error (~.~/
(~(~)
appear to have a behaviour very similar to their trapezoidal equivalents. 5.4
Storage requirements The amount of computer resources needed increases when predictor corrector
methods are used;
particular attention must be paid to integrating points data
which are frequently accessed during the computations, whose transfers between central memory and backing store should be optimised requirements are given in TABLE I.
(if any) and whose storage
The and
271 TABLE I ?roblem Algorith~ Time type s tapping
PP
E
C
IH
E
C
PP
E
V
Integrating Example points S forage (K) *
Nxl xds
I. 6
e
Nxl x(d +i)
2
*
2xNxl xd e c
3.2
IH
E
V
*
2xNxlex(d +1~ 4
PP
T
V
*
3xNxl xd~
IH
T
V
*
3xNxleX(de+l ~ 6
PP
M
V
*
NXleX(2d +dol 4.8
IH
M
V
*
NXleX(2d ~ +d
?P: ~:
perfectly plastic Midpoint
4.8
e
C:
IH:
constant
isotropic hardening V:
variable
N:
E:
EULER
T:
number of elements
5.6
+2)
Trapezoidal I : nr. integ. e
points per element ~xample:
d :
dimension of ~
d :
dimension of ~gr"
i00 quadratic elements, axisymmetric, 2x2 Gaussian integration
6.
LIMIT ANALYSIS BY THE VISCO-PLASTIC APPROACH
This section deals with two discretised bodies J~J~ and _~Av~ proportional loading ~ ~
(where
~
, subject to a
is the load parameter), and which are
identical in all respects except for the constitutive relations of their respective materials. For -fl4y an elastic/perfectly plastic associated behaviour is assumed
~ = i ~ " ~ #~ ~ while for
-W~LP
t~ ~'> O=l~ "gil~F~/O F=I'~{~°'')-'=° ~ =lt~ °F=o= ~al~'O F <.o
a corresponding ,elastic/yisco (perfectly) plastic law of the
associated type is postulated
r>o
Underthe additional assun~Qtionsthat f~(O~_]
is a homogeneous function of degree ~
~(O')-~=O
o
represents a non-concave surface in the space of .~°"
it is possible to prove two bound theorems:
272
lower bound theorem If, under the constant load~ ~ L ~ reached
in
s u c h that
, total stationarity conditions are
=o
o
then
where
the latter denotes the perfectly plastic collapse load parameter associated with-i~.~ and
~ This is an i~mediate application of the well known lower bound theorem of
DRUCKER, PRAGER and GREENBERG 7, since at tc~l stationarity, ~
is plastically
admissible and in equilibrium with ~ L ~ upper bound theorem If, under the constant loa~s ~v ~ reached in
such that ~ = O
,.partial stationarity conditions are and
= constant ~ O
The proof relies on the fact that, if ~
then ~ f ~ V
is an arbitrary plastically
admissible structural stress vector,
~ >O
, and follows lines
very similar to those of the lower bound theorem. These theorems bear out a very useful feature of the visco-plastic approach in limit analysis:
whatever constant loads act on a structure, the final viscoplastic
results will indicate their position with respect to the corresponding collapse loads.
Therefore, the safety of a structure designed to bear
specific maximum
service loads can be tested by a unique visco-plastic analysis under full loads; this contrasts with the incremental elasto-plastic approach where load increments must be specified ab initio according to some predicted value of
~ -
, an
estimate that is often not available with great accuracy. 7. As a first example,
APPLICATION
a rectangular plate of aspect ratio 2/1, unit thickness and
initially free of stresses, is clamped to a rigid support and instantaneously cooled; the restraint put on shrinkage strains near the fixed edge induces stresses which are assumed to violate the static yield condition of the plate material considered to be in a state of plane stress.
Symmetry permits to analyze only a half-plate, as
shown on Fig. i where the element subdivision is refined in the region of the expected stress concentration. Elastic and asymptotic elastic/visco-plastic displacement fields are almost identical, except for the stress peak region where a slight increase of displacements, parallel to the support, is observed.
The stress reduction is therefore
mainly due to relaxation at quasi constant total deformation. Elastic and asymptotic stress distributions (Fig. 2) show a significant stress variation along the entire bonded edge while the plastic straining -
~p=
~
~
remains localised at the corners.
(~
: strain hardening form, Appendix II)
273
RIGID
Plastic region
SUPPORT
~F///////~
T,.- . . . . .
+. . . . .
+ - - - ~ - - - ~ -
t-l--r**~,t
E=100 bars
~=0.52 Y=I bar
~T=-O.01
\
.
.
.
.
.
.
.
.
.
.
=0.03 time param. Perfect yon Mises pl ; Fo = Y ¢Ix)=x T =I seeG2 Gaussian integrat ~ l e r ' s rule
i.
.
'
,o
X
Deformed mesh(exaggeration ~ )
Mesh subdivision(96 elts.)
RECTANGULAR VISCO-PLASTIC PLATE. UNIFORM TE~Ph~ATURE DROP FIG. I Another typical application, concerned with limit analysis, elastic/visco-perfectly-plastic
is that of the
cantilever viewed on Fig. 3, and subject to either
an incremental prescribed displacement or an incremented concentrated load.
For
comparison with bending elasto-plastic beam theory it is useful to introduce the dimensionless variables
for which the theoretical load-deflection formula for the mid depth node at the free end is
5
~ 5
Fig. 3 also illustrates the time propagation of the viseo-plastic boundary when the beam is subject to a single step load increment
k = 1.5, for which no collapse
was encountered. Fig. 4 shows how various finite element results compare with beam theory. an incremented load, again, no collapse occurred for
k = 1.5;
With
EULER's rule failed
to converge under ~ = 1.51 after 50 additional time steps but HEUN's rule did reach a fully stationary state for ~
= 1.51 and ~
an even better lower bound to collapse, time steps were used. last results:
= 1.52.
~ L = 1.55;
Incremental displacements give in this case, however, smaller
There is no doubt about the lower bound character of these
stresses are in equilibrium with the concentrated reaction and no
longer violate the yield condition when steady state is reached.
274
IB
js
f
"
/"
c~ vp
~
O.Sm
o: ¢last |¢ ( ~[ ] BAUER & REISS
%
Ibar
]A RECTANGULAR PLATE. UNIFORM TEMP.DROP.STRESSES(section A-B) e:instantaneous elastic response vp. asymptotic state
B
is
o I o:eZasllc [~ l -,bar
0
"C'X~
A
I0
O.OS
~p
IA FIG. 2
The negligible difference between the deflections given by a unique algorithm (EULER here) for a load
~ = 1.5, applied either in a single step or incrementally,
is of prime interest and renews the suggestion that proportional incremental loading is superfluous in visco-plastic limit analysis (unless load-deflection characteristics must be fully determined). The comparatively poor deflections predicted by the fully integrated (G3) parabolic elements confirm previous tests in favour of the reduced integration technique in isoparametric elements, while the use of more complex cubic elements had no significant effect on the results, To finish with this problem, Fig. 5 represents the time variable stress distribution, under one step
~ = 1.5, in section AA of Fig. 3.
275
I^ I
time
Stcp~ I ° y ~__°-l~Ct~o o ~ ~ o
; / ~ /
Dimensions:l=36 cm ha6 cm
~
/
o
E=50000 bars 9=0.2 Perfect von Mises pl. F =Y=50 bars
~ ~
o.
~ ~-6"--o/
:il:o,
^
Linear flow rule4~(x)=x°
~ '\
o
(
kkQuadratic or cubic elements or G3 Gaussian integration
o
~h |
,
ELASTIC/VISCO-PERFECTLY PLASTIC CANTILEFER BEAM LOAD OR DISPLACEMENT BOUNDARY CONDITIONS FIG. 3
~.5
/
/
°/. (D
)
E
~
i< - -
NCREktENIAL OISPLACEMEt~*PARABOLIC **
O
• /F/
///
.
.
X
,/~/,--~ ~de~t~ca!
~/
//
not4,
"I
BEAM |HEORY
IINCREHENTAL
LOAD
;dem
idem
"dem
"dem
ONE STEP
LOAD
|or ~z~or 3x~
G2 ELTS*
EUN ~/~0~03
+PARABOLIC G2 £LTS+EULI[R
G3
.pARAgOLIC
2
HEUH
idem
I~em
~dem
G2ELTS.EUIER
GAu SSIAN INTEGRATION in "
T-0.~0
idem
space
3
ELASTIC/VISCO-PERFECTLY PLASTIC CANTILEVER BEAM NON-DIMENSIONAL LOAD-DEFLECTION CURVE AT FREE END. FIG. 4 The more realistic application which follows consists of an axi-symmetric pressurized thin shell for which experimental (DINNO and GILL 6) as well as numerical 15 elasto-plastie results (NAYAK and ZIENKIEWICZ) are available. Fig. 6 gives full details about problem specification, together with the localized deformation, at the sphere/branch junction.
Quadratic isoparametrie elements,
and EULER's rule were used once more. incremented.
2x2 Gaussian integration
The pressure was applied in a single step or
276
??
??
[! -
0.5-
-I i
O
i
L.ol .,4I ~, 1
(a) (h)
ELASTIC/VISCO-Ph~WFECTLY PLASTIC ~ CANTILEVER BEAM. STRESS REDISTRIBUTION IN SECTION A-A. (a)tShearing stress (b):Normal bending stress. FIG. 5 Two load/displacement curves (Fig. 7) at the branch end T and at junction A
show:
a)
a good agreement with the elasto-plastic analysis based on the same mesh pattern
b)
that an incremented or a one-step load give identical results
e)
an appreciable sensitivity to time interval size: overestimate the inelastic strains.
larger time steps
The number of time intervals needed in various analyses can be found in Table 2. When the incremental approach took 386 steps to reach stationary conditions under 1140 p.s.i, only 120 intervals were sufficient under full pressure. TABLE 2 Incremental
p (psi) 900 950 IOOO 1020 1040 1060
"6= O.15 time steps Incremental
= 0.03
14
31
5
16
20
p (psi) 900 920 940 960 980 iOOO 1020 1040 1060 1080 IiOO 1120 1140 1160
= 0.03 time steps One step
4
IO
6
p (psi)
1140
time steps
120
8
7
6
8
8
36
34
76
43
82
62 > IOO
1180 No convergence after 400 steps
As a last example, the relaxation of a rock mass around a lined tunnel is analyzed.
In this problem, described in Figs. 8 and 9, the rock behaves like an
277
SCALES
l
!
!
f'~
"°g'"'
/ l
, ,
\ •
/- //
~
displac,mcnt s [magnification
:20]
, / I~i//
L
I
N i
2s13
~ o 127
L, All materials
:E=2.912x107 pal ¢=0.3 Branch material:Y=38750 psi Weld material :Y=40540 psi Sphere material:Y=42340 psi Perfect von Mises plasticity Linear flow rule ~ (x)=x Mesh: 54 quadratic isoparametric elements. G2 Gaussian integration Loading: internal pressure applied incrementally or in one step.
AXISYMMETRIC ELASTIC/VISCO-P~ECTLY PLASTIC PRESSURE VESSEL.
FIG. 6
278 pressurcCpsl)
4000
~
...........
o- . . . . ~C
X incremental ¢iastoplastic F.E.~I.%| rl v.p.relaxotion under tt40pSi(~'=.03)
5oo
/
o ,nc~..,.,o, ..p.~.,o.o~io.(r..o3) A
. Id,~
__
steady state
....
continuing v.p. flow
(r=, s)
branah/sphere junotion disp.
VA
pr¢ ~surc(psi) expcrlm. [£]
-
~
........
13 v.p.rcloxotion under 144Opsi(T=.03)
0 Incrcmentol v.p.rmloxotlon (C=-03)
/
a /
--
, 0
. . . . . ~-----e-
.04"
id=m
(I:,4s)
Steady star"
branch top disPlacement .02x .OY
PRESSURE VESSEL: PRESSURE-DISPLACEmENT CURVES
FIG. 7 associated COULOMB visco-plastic material with a linear flow rule ~(~)=~: the sake of simplicity,
For
no displacements or relaxation effects are supposed to take
place until total completion of the tunnel. The instantaneous elastic response is depicted in Figs. 9, iO and 12(a).
The
yield condition is violated in separate zones (Figs. 8 and i0) and some tensions, less than
C c ~
, appear in areas distinct from the plastic zones.
The state
of stress in the lining consists of high compressions in the wall and small stresses (some of which are tensile) at the crown and in the floor.
MOHR's diagram (Fig. II)
suggests that a considerable proportion of the loads are transmitted to the lining by shearing stresses distributed along the inclined parts of the rock/lining interface.
279
Mesh subdivision(63 quad.elsi Deformed mesh(exag.loo) Lining:E--4.32xS0|psf $ =0.15 Rock: E=0.72xS01psf $=0.20 C~14400 psf ~=~'-30 ° Mohr-Coulomb perfect plasticity(associated)7 Fo=C c o s ~ ( x ) = x Euler's rule; time increment parameter ~=0.05 ROCK RELAXATION AROUND A LINED TUNNEL
FIG. 8 Asymptotic displacements
~'~
Fig. 8 and their inelastic components
obtained after 27 time intervals are given in ~
-Go
show (Fig. i0) a shortening
of the upper and lower vaults and an elongation of the lining wall, thus causing a transfer of compressions from wall to vaults.
The mechanism of this transfer
appears from the orientation of the slip facets families
S~ and S~
in Fig. IO:
MOHR's diagram shows a reduction of the shearing stress and an increase of the normal stress acting upon these faeets3a family of which remains parallel to the rock /lining interface. Finally Fig. 12(b) depicts the asymptotic state of stress in the lining (suppression of lining tensions) as well as shrunken rock tensile zones.
Although the
present material properties did not cause pronounced plastic flow, the general conclusions drawn here remain~valid, with amplification, when the cohesion
was
reduced, resulting in the expansion of viseo-plastic zones and greater stress variations in the lining.
280
gO
80
70
60
50 70
80 90 LI~1ING Dt~"O~-ATI ON
~~0 ~
FIG. 9 8.
CONCLUDING REMARKS
Due to space limitation, only four examples have been presented and discussed. Problems involving non-linear flow rules, isotropic hardening or softening and nonassociated viscoplastic laws have been solved without creating any particular difficulty and bear out the versatility of the method in materially non-linear analyses. Though the present formulation deals with isotropic situations, it should be possible to extend it to anisotropic cases such as laminated and joint materials. The approximations related to the visco-plastic approach (finite element space diseretization and numerical integration in the time domain) stand out to be well defined;
this contrasts with the elasto-plastic approach, where several
computational
artifices such as corrections in plastic stress estimates must be used to restore the stresses to the correct yield surface and where equilibrium must be permanently checked. However the visco-plastic method was shown to lead ultimately to a non-linear system of first order differential equations, the numerical solution of which can, in principle, be obtained by a great variety of classical procedures.
281
LINED TUNNEL/PLASTIC COMPONENTS OF DISPLACEMENTS FIG. I0 In practical engineering applications, extreme accuracy is not the goal.
An
ideal scheme of integration should: - minimize the number of time intervals for a prescribed accuracy compatible with engineering practice -
prevent the appearance of numerical instabilities or, at least, detect them and take appropriate measures after their detection
- give an estimate of the truncation errors cumulated as the integration proceeds in the time domain -
-
remain reasonably simple to implement on computers (coding and especially storage requirements) be conceived to deal with systems of realistic size (at least 500 simultaneous differential equations)
To the authors' knowledge, such an ideal scheme has not yet been produced in relation to visco-plasticity applications and opens a wide field to future research.
282
z"I
'
~
~-
~
J
x
/
-
'
-
~
%
~'
'
l
:
- - . . . . . . i.i~io, .t~t.
.....
¢tosUc sta~,e msymptot,cs t a t y
~x1@thsltt 2
:
\\
~/-__~-
-2
/
STRESS C:KA.NGEAT I N T U I T I N G POINTS G1 ,G.2(de~±nec]. on P i g . 8 ) FIG. 1].
J
.
~o s
,b~I,~,,
,,
i
~I00bars
,o5 ,b,l,q,,
~
~OO bars
%erosion ELASTIC PRINCIPAL STRESSES
(a)
ASYMPT0r'2IO PRINCIPAL SrfR.I~SES (b)
\ FIG. 12(a)
FIG. 12(b)
283
APPENDIX I:
The derivatives
of
~
where
FUNCTIONS OF STRESS INVARIANTS
the
stress
invariants
~-~--
~
~
~ ~ o 3 ~j~
may be written
: -.
as
~-~
are symmetric matrices 0
0
symm.
0
0
9_ 0
o
o
o
X
-J¢~+9 5
symm.
one has
if
~'= ~4&
crTI~e T
~J
where
Similarly,
where
is a symmetric, stress dependent, matrix.
if
+~73
:
angle
"vl- ~t " v 2.
cohesion
friction angle
cohesion
~:
~:
friction
.
Z
uniaxial yield stress
uniaxial yield stress
Yield function
L~:
~(~'('):
~(~<,]:
Isotropic ha rdenin $
MohrCoulomb
DruckerPrager
Tresca
von Mises
Criterion
Yield functions and viscopl,,a,sticpotentials
O=T'~'~ ~
APPENDIX II: Together with the notation of Appendix I, let
7"'~
%VpZ- . vF~,,~q
2
~Z
~
for work hardening
for strain hardening
flow function angle
Q: 3-~a~?/ cry+ V ~
q
Viscoplastic potential
- ~ (04-~
SPECIFIC VISCOPLATICITY FORMS
Z
E-
a f fine NI~ [ ~ ~ ]
subparametric ~ [g~]
isoparametric
affine N ~ [~s ~ i
' ~
~
~g ~.__¢
~
"-
~
GI
GI
GI
GI
G2
subparametric ~ [g~]
G4
-
G3
G2
+
G2
G2
G2
G2
G2
GI
GI
G2
G3
G2
G2
G3
GI
G2
GI
G2
G5
GI
G2
GI
GI
G3
GI
GI
G2
GI
GI G3
GI
G2
-
-
-
-
-
G4
-
-
G3
-
-
G2
-
G4
G4
G4
G3
G3
G3
G2
G2
G3
G3
G5
G2
G2
G3
GI
G2
G2
G3
G5
G2
G2
G3
GI
G2
+the function is rational and cannot be exactly integrated by Gausslrule; ++assuming that IJI * constant matrix when mesh is refined; +++assuming that ~ ~ v p ~0 * constant values in every element; ++++assuming that ~ + constant value in every element; ~ neglecting the variation of the radius x inside element ~ : (++++ and ~ ) simultaneously;
** N'. ~ N. i i **~*constant IJI throughout the element;
is bracketed;
G4
G4
G3
G3
G3
G2
G2
***mid-side nodes linearly interpolated between corner nodes;
*the higher order terms of the polynomial in ~ ,~
'~ubic
quadratic
GI
GI
~~ Jl ~ ~ ~ S S ~ v ~ ~~~ exact approx exact approx limit ~ = L % exact exact lim 5~B D ~ % ++ +++ ++++
Iq~E~,~]
~,
sketch
Axial synmletry
GAUSSIAN INTEGRATION RULES
Plane Stress & Plane Strain
isoparametric N~[~ ~s ~.],
displacement geometrical and velocity interpolation fields N' . i N.l isoparamet ric ~41[~] linear affine****
Element characteristics
APPENDIX III :
286
REFERENCES i.
APPLEBY, E. J. and PRAGER, W. 'A problem in visco-plasticity' J. Appl. Mech., 29, 381-384 (1962).
2.
ARGYRIS, J. H. and SHARPF, D. W. 'Methods of elasto-plastie analysis'. Symp. on Finite Element Techniques, Stuttgart (1969).
3.
BAUER, F. and REISS, E. L. 'On the numerical determination of shrinkage stresses'. A.S.M.E. Trans., J. Appl. Mech., March 1970, 123-27 (1970).
4.
BINGHAM, E. C. 'Fluidity and Plasticity' McGraw-Hill, New York (1922).
5.
CHABOCHE, J. L. 'Calcul des deformatlons vlsco-plastlques d'une structure soumise ~ des gradlents " . ~ . , FaculteI des Sciences d'Orsay. thermlques evolutlfs Paris. Th~se (1972).
6.
DINNO, K. S. and GILL, S. S. 'An experimental investigation into the plastic behaviour of flush nozzles in spherical pressure vessels' Int. J. Mech. Sci., ~, 817 (1965).
7.
DRUCKER, D. C., PRAGER, W. and GREENBERG, H. J. theorems for continuous media' Quart. Appl. Math., ~, 381-389 (1952).
8.
FREUDENTHAL, A. M. and GEIRINGER, H. 'The mathematical theories of the inelastic continuum' Encyclopedia of Physics, Vol. VI, 229-433 (1958).
9.
GREENBAUM, G. A., and RUBINSTEIN, M . F . 'Creep analysis of axisymmetric bodies using finite elements' Nucl. Eng. and Design, ~, 4, 378-397 (1968).
IO.
LEMAITRE, J. 'Elasto-visco-plastic constitutive relations for quasi-static structures calculations' ONERA publication TP 1089 - Chatillon (France) (1972).
II.
MARCAL, P. V., and KING, I. P. 'Elastic plastic analysis of two dimensional stress systems by the finite element method'. Int. J. Mech. Sci., ~, 143-155 (1967).
12.
NAYAK, G. C. and ZIENKIEWICZ, O. C. 'Convenient form of stress invariants for plasticity' Proc. A.S.C,E., 98, ST4, 949-954 (1972).
13.
NAYAK, G. C. and ZIENKIEWICZ, O. C. 'Elasto-plastic stress analysis. Generalization for various constitutive relations including strain softening'. Int. J. Num. Meths. in Eng., ~, 113-135 (1972).
14.
NGUYEN, Q. 8. and ZARKA, J. Quelques methodes de resolutlon numerlque en elastoplastlclte classlque et en alastovlscoplastlclte S~minaire Plasticit~ et Viscoplasticit~, Ecole Polytechnique, Paris (1972). •
•
/
•
'Extended limit design
/
!
f
Chapter VIII, 215-218,
/
.
/
.
.
•
/
•
P,
15.
PERZYNA, P. 'Fundamental problems in viscoplasticity' Advances in Applied Mechanics, ~, 243-377 (1966).
16.
REINER, M.
17.
SUTHERLAND, W. H. 'AXICRP. Finite element computer code for creep analysis of plane stress, plane strain and axisymmetric bodies' Nucl. Eng. and Design, ii, 269-285 (1970)o
'Rh~ologie theorlque'~ "
Dunod, Paris (1955)
287
18.
TREHARNE, G. 'Applications of the finite element method to the stress analysis of materials subject to creep' Ph.D. thesis, University of Wales, Swansea (1971).
19.
WHITE, J . L . 'Finite elements in linear viscoelasticity' Proc. 2d. Conf. Matrix Methods Struct. Mech. AFFDL-TR-68-150, pp. 489-516, Wright-Patterson A.F.B., Ohio (1968).
20.
WIERZBICKI, T. 'A thick-walled elasto-visco-plastic spherical container under stress and displacement boundary value conditions'. Archiwum Mech. Stos. 2, 15, 297-308 (1963).
21.
WIERZBICKI, T. 'Impulsive loading of rigid viscoplastic plates' Int. J. Solids and Struct., ~, 635-647 (1967).
22.
WIERZBICKI, T. and FLORENCE, A. L. 'A theoretical and experimental investigation of impulsively loaded clamped circular viscoplastic plates'. Int. J. Solids and Struct., ~, 553-568 (1970).
23.
ZARKA, J . 'Generallsatlon de la theorle du potentiel multiple en viscoplasticitY'. J. Mech. Phys. Solids, 20, 179-195 (1972) o
24.
ZIENKIEWICZ, O. C., WATSON, M. and KING, I. P. 'A numerical method of visco-elastic stress analysis' Int. J. Mech. Sci., IO, 807-827 (1968).
25.
ZIENKIEWICZ, O. C. and VALLIAPPAN, S. 'Analysis of real structures for creep, plasticity and other complex constitutive laws'. Int. Conf. on Struct., Sol. Mech., Eng. Design and Civ. Eng. Mat., Southampton (1969).
26.
ZIENKIEWICZ, O. C., VALLIAPPAN, S. and KING, I. P. 'Elasto-plastic solutions of engineering problems. Initial stress, finite element approach'. Int. J. Num. Meths. Eng., !, 75-1OO (1969).
27.
ZIENKIEWICZ, O. C. 'The finite element method in Engineering science' McGraw-Hill, London (1971).
28.
ZIENKIEWICZ, O. & and CORMEAU, I. C. 'Viscoplasticity solution by finite element process'. Archives of Mechanics, 24, 5-6, 873-888 (1972).
/ Y
°
,
I ,
SOME SUPERCONVERGENCE RESULTS FOR AN H 1-GALERKIN PROCEDURE
FOR THE HEAT E Q U A T I O N
Jim Douglas, Jr., Todd Dupont , and M a r y Fanett Wheeler
1.
Introduction.
We s h a l l c o n s i d e r a G a l e r k i n method b a s e d on the u s e
of the i n n e r product in the Sobolev s p a c e
HI(I)
i n t r o d u c e d by Thomee
and W a h l b i n [ 4] i n a more g e n e r a l s e t t i n g for approximating the s o l u t i o n of the simple p a r a b o l i c b o u n d a r y problem 2 Ou_ 0 u = - f ( x , t ) , x ~ I = [ % 1 1 , t ~ I = (O,T] , 8t 2 8x (1. I)
u(O,t) = u(1,t)
-- O, t ~ I ,
u ( x , O ) = Uo(X) , x ~ I .
Let
d e n o t e a f i n i t e - d i m e n s i o n a l s u b s p a c e of
define a m a p
U :[O,T] ~,
with
I-I2(I) fl HO(I) ,
and
U(O) to be determined l a t e r in a
rather p a r t i c u l a r f a s h i o n , such that (the i n n e r product b e i n g on real L2(I)) (i 2)
(82U
•
~'~
dv ' dx ) +('
~2U
dZv
2'
8x
dx 2
dZv ) = (f'
dx 2
)
V ~~
'
'
This map provides the HI-Galerkin process to be analyzed. and uniqueness of
U follows whenever
by choosing the test function
f ~ L2(I × J) ~
t ~ I •
Existence
as can be seen
v = U.
Sponsored by the United States Army under Contract No. DA-31-12zI-ARO-D-462 and the National Science Foundation• University of Wisconsin - Madison ~vlathematics Research Center Technical Summary Report No. 1382
289 W e s h a l l a l w a y s a s s u m e t h a t w e h a v e a f a m i l y of s u b s p a c e s , 0 < h < 1 ,
~=~h'
s a t i s f y i n g t h e a p p r o x i m a b i l i t y c o n d i t i o n for a n y
function z ~ HP(I) R HI(I) (1.3)
given by
(I[z-xII +hI[z-xIll +hZ[Iz-xIlz)
inf
2
x E~ h
for some integer r >_ 3. The norms indicated above are the Sobolev norm s
IIzII = IIz[I0 = I{ZHL2(1), {Izllk = [IZIIHk(l) • W i t h no further r e s t r i c t i o n on t h e s u b s p a c e
(1.4)
llu - u{t
it c a n b e s h o w n [ 4 ] t h a t
= ess sup H(u - u)
LO°(H j)
for s u f f i c i e n t l y s m o o t h to
~
u ,
provided that
U(O)
t < j _< 2 ,
is chosen subject
rather mild constraints.
some
The m a i n p u r p o s e of t h i s p a p e r is to d e r i v e s u p e r c o n v e r g e n c e estimates 6:0
at t h e k n o t s for t w o p a r t i c u l a r c l a s s e s
= Xo < Xl < ' ' -
Denote by the s e t
Pr(E)
E c I
of subspaces.
< x M = 1, O < h i = x i - Xi_l, m a x h i = h , i
t h e c l a s s of f u n c t i o n s on
w h i c h a g r e e on
I
Let I i = [Xi_l, Xi] .
h a v i n g r e s t r i c t i o n s to
E w i t h a p o l y n o m i a l of d e g r e e n o t
greater than r . Let ~k(r, 6) = {v ~ c k ( 1 ) I v
~ Pr(li),
i = I,...,
M}
and s e t (I. 5)
~ k = {v ~ ~k(r, 6) Iv(O) = v ( i )
= O} .
#1382
290 The two f a m i l i e s t h a t w i l l i n t e r e s t us a r e in s e c t i o n 3 t h a t , p r o v i d e d
(1.6)
U(0)
N1
and
N2 " W e s h a l l s h o w
i s c h o s e n in a p a r t i c u l a r w a y ,
I(U - U)(xi,t) I <_ C(u)h 2r-2, H = Z~1 or H2 ,
and that
(l.7)
[--~-88(u x - U)(xi, t) I _< C(u )h 2r-2, H = H 1
The proofs of these results will be based on a quasi-projection approximation procedure introduced by the authors [ 1 ] in order to demonstrate knot superconvergence when the space ~ 0
is employed in the standard
Galerkln method for parabolic equations based on an L2-inner product rather than the Hl-inner product. The results of this paper can be extended in several ways.
It
is clear from the development of section 3 that, if ~ ~ (0, I) is a fixed point such that ~ = xi(6) as holds at the point x
h -~ 0 , the estimate (i. 6) or (1.7)
whenever the space
Hk , k > I , is employed
in (I. 2) after modifying the smoothness constraint at the single knot to be
C2(Ii(8) U ii(8)+l)
or
cl(li(8) U li(8)+l) ,
respectively.
particular, it follows that the heat flux at the end points x = I
always
of the space.
satisfies (1.7) for any choice of This is in marked
standard Galerkin method
using
k
x = 0
In and
without modification
contrast with the situation for the ~k
[2, 5].
It is of interest to dlscretize
(i. 2) in time in such a fashion as to preserve the superconvergence associated
#138Z
with the
HI-Galerkin
process in the space variable; see
291
[ 1 ] for a c o l l o c a t i o n - i n - t i m e method t h a t c a n be a d a p t e d to the p r e s e n t c a s e in s u c h a w a y t h a t the e s t i m a t e s at the k n o t the form s
O(h z r - 2 + (At) zs) ,
are e m p l o y e d in time.
(xi, t k)
w o u l d be of
where p i e c e w f s e - p o l y n o m i a t s of d e g r e e
F i n a l l y , more g e n e r a l d i f f e r e n t i a l o p e r a t o r s
s h o u l d be c o n s i d e r e d , and the authors w i l l return to t h i s q u e s t i o n in another paper.
#1382
292
2. • G l o b a l E s t i m a t e s .
Let us q u i c k l y p r o d u c e an
H z estimate s i m i l a r
s
to t h a t of T h o m e e - W a h l b i n [ 4 ] .
Let
~ = u - U.
Then,
7
(z.l)
r-9-7-~ v'~+( K~z' v") =o, v ~ ~, t~J . ~BxBt ' " 8x
Take
v=~t
+ O-~t(X-u), 88~t ~ .
Then,
1 d 0Z~ ~
(8°xZa~,ax~at)+ ~7{ ( z, 8x
8x
Z
z)
= (,~_7.!
o
Z
"SxSt , @xBt (X-u))
+(0z~ '~3 8x
2'
(x-u)).
2
8x 8t
It f o l l o w s e a s i l y t h a t
(z.2) II~llTz(~} + II~IIT?o(HZ)_< C {11~(0)Itz+ I1~(×-u)IIT.z(~I)+ II~(x- u) IILI( H 2) }" Select the i n i t i a l condition U(O) so that
I1¢(o)11+hll¢(o)ll~ +hZlI¢(o)IIz<_cli%
(z.3)
this can be done in many ways.
r+l h
r+l
;
In particular, we shall note below
that the "biharmonic" projection
(Z.4)
((U(O) - u0)" ,v'') = 0, v ~ ~ ,
satisfies (2.3); it will also be the case that the more complicated initial condition (3.5) needed to obtain superconvergence at the knots satisfies an inequality similar to (2.3). It then follows from (Z. 3) and (I. 3) that
(z. 5)
)h~-I flu- U IlT.(~z) _< C( II~ HLI(ff+I) + flailLz(~f)+ llu011r+l I~ow, let W : [O,T] --~
11= u - W ,
#1382
be the biharmonic projection of u . If
293
Z
2
(-%( u -
(z. s)
w),v,,) =
v,,)
8x
=
O, v ¢ ~ ,
t~ J.
8x
It is obvious that 2
2
ll-e--~zll <_
X c Z4
8x
Since
rl(O) = D(1) = 0 ,
(z.7) Let
119-~ - x"ll
inf
then
llnll z <_ cllullr+/-I, z
•
ax
t~ j.
be the solution of the t w o point b o u n d a r y problem
z" = - q ,
X e I, z (0) = z(1) = 0 . Then, b y (2.6)
2
2
= (~ ax and t r i v i a l e l l i p t i c r e g u l a r i t y for
z
implies that
II~II I--< cllnll 2 ~ -< cllull
(z.8) Let
z" - ×")
ax
~ c Hs(1)
and l e t
~ HS+4(1)
(iv)
=4,
r+l h r , t ~ 5.
be the solution of
xcI,
(2.9) : ~" = 0 , x c aI • Then, for
0 < s < r- 3 ,
2
2
ax
ax
(~, ¢) = (-t-~z, *") = ( - ~ z , *" - ×")
-- o( 11~11z IIq, lls hs÷z) . Thus, if w e define
H-s(1) = HS(1) ' using the n o r m
#1382
294
Ilg ll_s --
sup
~
"~"s
0 ¢ q~ e H s ( I ) we have shown that
Itll
< Cllu
-
•" h " - s
r+~
hr+s+l
'
0 < s < r-3.
An o b v i o u s m o d i f i c a t i o n o f the argument above g i v e s the s l i g h t g e n e r a l i z a tion
(z.
10)
Itht]_s_< CttUtlqh q+s , 2_< q _ < r + 1 , - 2 < s < r L2
We can derive an
estimate for the error using the first step
of the development of the next section and the bounds on = U - W ,
3 .
~.
If
then
--~ - ~ v,), v, ~, t , j . (oxat'v') + (- ~ z ' v,,) = (~xat'
(z. 11)
ax
Set
YO
=
h
and define (mostly for future use) a sequence
of p r o j e c t i o n s
by the r e l a t i o n s
,2 (z. lz)
-
-
~ V
8x
II
_--
--
)
8 Y~-I v' , V~, axat '
t¢
If w e s e t
(2.13)
@~ = (~ + YI + "'" + Yf) : [O'T] - ~ ,
then an easy calculation shows that
(2.14) "
#1382
t~2~-,v,
+ \~x ---2,v,
-- t a- x-a t , v, ] , v ~ .
J, 1 > _ 1
.
295
For t h e g l o b a l e s t i m a t e it is s u f f i c i e n t to look a t the test function
@1
Use
v = @1 i n (2. I4) to s h o w t h a t
[[L°°(H1)
IILa(HZ)
-
La(H_I)
-
(z.15)
<_c~lh(o) II1
10,ri) LZ(L z)
It i s c l e a r t h a t we n e e d to e s t i m a t e bounds on Lemma I°
8kYi/St k If
2 JqJr
S i n c e we s h a l l n e e d
l a t e r , we s h a l l t r e a t a l l of them now. + 1
and
- 2 < s < r-
2f - 3 ,
then
8 k y l ][ / 8k+~u I[ -< C ~ h q+s+2~ t~ J . [I ~t k - s q '
(2.16)
• Proof.
8Yl/St .
The inequalities (Z. 16) hold for ~ = 0 ,
by (2. i0). Now,
ax
8x and
sY~_~ IIYi II2 <
Cll~l
i •
By differentiating (2. IZ) with respect to t , it follows in like manner that
8k+i Y~- I
#1382
296
Next, let
t~ ~ HS(I)
and d e f i n e
~
by (2.9).
ak+Zy~_l) _--
,X I
_
( ak+zY~ +
0xat k + l
Then, for any × ~ ~,
- -
Ox2 8tk '
-
By ( 1 . 3 ) ,
there exist
X E~
(z.17) 11~"-×"If-< ctl~lf
8tk+ I
-
~" - X"
\+
= [ak+2y, _ 8k+iY~_l
~Sx2atk
)
, ~,,_ X" }
/ak+Iy~_ 1
~
atk+l
such that
< cll~l[ Shs+z , s+4 hs+z --
s + 4 < _ r + i.
For the global L 2 bound we shall need (2.17) only in the case however,
'
s = 0 ;
s i n c e we r e a l l y are a f t e r t h e s u p e r c o n v e r g e n c e r e s u l t s in
t h i s p a p e r , w e s h a l l n o t c o m p l i c a t e t h e s t a t e m e n t s of o u r r e s u l t s b y treating this s p e c i a l c a s e .
Now, i t follows t h a t
,,
s
Jl_s_2+ 0,k
h
t h a t c a n be a c h i e v e d is
2r-
Note that the maximum exponent 2 .
I'.et us a p p l y t h i s l e m m a to (2.1 5).
First,
IIIFL21, 02__ -- ,c HLZ(Hr- 1)hr+, " 1fat2
#1382
0tk+,
I ~tk+1-III-s-2 + IIak+1~-I IIhs+ Otk+l
The l e m m a t h e n f o l l o w s by i n d u c t i o n . on
cIIJ-,
h s+z
297 Also,
IIYl(O) II l _< c IIa~(o)II r hr+: so that the
choice u(0) = w(0)
(Z. 18) is adequate to imply that
Ilol(o)li 1 < c llau/at(o) II r hr+ 1 .
As sume
_
more generally that the initial condition is selected so that
lle1(0)II, <_Co(Ulh r+1
(z. 19) Then, since f[~tl
<_ cJluil
L°O(Hj )
h r+l-j
L°°(Hr+ 1)
'
0<__j<__z,
we have shown that 2u ~hr+l-j llu - ull LOO(Hj) < C0(u)hr+l + C{HUHLCO(Hr+I) + [[ a0t-'-~ L2(Hr-I-J)" '
(z. z0)
0 < j < l
Theorem 1. Let ~ = ~h that (1.3) holds.
.
be selected from a family of subspaces such
Then, if U(0)
is chosen so that (2.19) is valid, the
error in the solution of the Hl-Galerkin process (1.2) satisfies the inequality (Z. 20). U0 ,
In particular, if
then
U(0) is defined as the biharmonic projection of
Co(U) has the form
o(IIau/at(0)lIr). Moreover, if
is given by (3.4), then
Co(U) has the same form as above for
vanishes for
5,
r = 4
or
and is given by (3.16) for
U(O) r = 3 ,
r> 6 .
The l a s t sentence of the theorem is proved at the end of section 3. The
L2 estimate above was derived by use of the argument to be
#1382
298
u s e d i n t h e n e x t s e c t i o n for o b t a i n i n g s u p e r c o n v e r g e n c e ; t h i s d e r i v a t i o n d o e s n o t l e a d to m i n i m a l s m o o t h n e s s r e q u i r e m e n t s on t h e s o l u t i o n
u . We
s h a l l u t i l i z e the f o l l o w i n g l e m m a to r e d u c e the s m o o t h n e s s r e q u i r e d of u
to e x a c t l y t h e s a m e a s n e e d e d by the s t a n d a r d G a l e r k i n p r o c e d u r e [ 3].
Lemma 2.
Let
~ : [O,T] -~
satisfy the equation
2 ~ v') + (~2'v") (axSt' = (p~v"), v ~ ~ ,
(z. Zl)
ax
t~ I.
Then, (2.2Z)
11~11L~(Lz) -< c
Proof. Let ~ :[0, T] ~
(z.z3)
satisfy
( z'v")=(ox'V')'
v~.
8x
Then, , Ox t h u s , the c h o i c e
(z. z4)
) = II
II
and
(8__~.6__~
1 d
8x
8x
v = ~
i n (2.21) s h o w s t h a t
2 ! d ll2ll + ll Jl 2 =
2 dt
8x
) ax
Next, let × : [ 0, T] -~~ be the projection given by 2
(z.zs)
(p_a~ 8x
Then, for any v ~ ~ ,
#I38 2
7
v")= 0
,
v~
, "
299
(p, o2~) = (oaxz, o_~) = 8x
2
8x
8x
(ox' ~x )
2
= - ( ~ - v", a x 8x
2
a_i
2
=
_ (¢, a ~) ax
p) - (~, p)
and
l(p,~ ) I _< C {h 11p It + 1tp tl_i }tt ~ II 1 .
(2.26)
8x
It then follows from (Z. 24) and (2.26) that
11¢ IIL~(H2) <_c{ I1~(o)II z + Jlo IILZ(H_I) + h II p IILZ(LZ)} • If the right-hand side of (2.23) is integrated by parts, the choice
v =
implies t h a t (2. Z7)
l]~il ~C{ll~(0) L°°(H 2)
ll + llp[IL2(H_l) + h l J p IILZ(L2)} •
If v = ~ in (2.21),
ll~ IJL®(H1) <_ C{ II,~(o)II i + llo tlLZ(LZ) .
(z. z8)
The proof w i l l b e a c c o m p l i s h e d if we s h o w t h a t
1t,~11<_ c{ lip fi z + h It~ 11i} .
(2.29t ;2
Since
(o_~ + ~,v") = 0 8x
II~ll2=c~+ ~ 2' ~)-( 8x
for v e 9,
8x
then
' ~)=(~+
8x
2'
~
_v,,,+ II ]I2 '
v~.
8x
Hence,
#1382
300
11¢II2<_(tt~1t
z~
+
tl,~ zll) 8x
inf
It~- v,,ll
+
and (2.29) follows from (1.3) applied to the function ~"=
~
11/~21tz 8x
v ~~
¢
given by
X ( I , 9,(0) = ~(1) = 0 .
for
Now, a p p l y Lemma 2 to (2.11), a f t e r i n t e g r a t i n g t h e r i g h t - h a n d s i d e
by parts.
Then,
I1~11
C°(T.2) <_ c{ll~(o)11
+ hll¢(0)ll l
+
II~IILz(H_I )
h lie(o)I11
+
IiaU ~t IIT.2(~ ) hr+i } '
+h
II~t IILz(L2)}
(2.30)
< c (1I¢(o)II by(2.10).
If U(0)
+
is chosenby(2.4),
then
~(0)= 0 . If U(O)
is
chosen by (3.4), then (3.17)indicates that minimal smoothness is retained w h e n s u p e r c o n v e r g e n c e i s s o u g h t for odd s p a c e d e r i v a t i v e o c c u r s i n the e v e n c a s e .
r,
b u t a l o s s e q u i v a l e n t to o n e
The r e s u l t s of the a b o v e a r g u -
m e n t c a n be s t a t e d a s f o l l o w s .
Theorem 2. If U(0) = W(0)
(2.31)
Ilu- ull
If U(0)
L°°(L2)
_<
or, more generally, if (2.3) is valid, then
c{llull
L°°(Hr+l)
+ II.~au II L2(Hr) }hr+l .
is given by (3.4), then (2.31) holds for r = 3 and for r_> 4
llu - ull
< o(llull L°°(L2)
L~(Hr+l)
+ II~II 2
r
L (H)
(2.32) m
~, IIg-~(o)IImax(4, r_2~+l) }hr+l , L=I 8t where
m
is defined by (3.3).
The inequalities (2.31) and (2.32) are best possible for odd
#1382
r in the
301 sense tions
that, at
(Z. 33)
when
x = 0
f = 0 and
l,
and
u0
satisfies
sufficient
compatibility
condi-
then
flU - U IILOO(LZ) < c IIuo IIr+lhr+l ;
see [ 3].
#1382
302
3.
Knot E s t i m a t e s .
I, w h e n
r
We are d r i v i n g at
O(h 2 r - 2 )
estimates.
From Lemma
is even,
_r
8Y
Ot 2 For
r
odd,
r+l
Ot Set r m =-~-I,
(3.3)
m -
r-3 2 '
r
~ = ~,
r-I ~ = T'
O = Om = U - W ,
r
even,
r odd ,
W=
Let us r e q u i r e t h a t t h e i n i t i a l c o n d i t i o n
(3.4)
for
W(0)
U(O)
Ym
be c h o s e n s o t h a t
8(0) = O:
can be computed from the data of the problem alone;
t h i s a s s i g n m e n t is i m p l e m e n t a b l e , ~ =~1
or
and a c t u a l l y r a t h e r e a s i l y s o
~2.
C h o o s e the t e s t f u n c t i o n follows directly that
#1382
Y1 . . . . .
u(o) = w ( o ) .
Note that
thus,
W-
v = @e/St
in (2.14).
For
r
even, it
303 (3.5) For
I1~11 r
LZ(H I)
odd ,
integrate
t,
JTII
< cll-~tl 2 • < c £s L°°(H2)L (Hi)the term on the right-hand
to
x
and
then, a f t e r
with respect
to
t .
Thus,
first with respect to
+ tloll
(H)
h2r-2 , r even.
side of (2.14) by parts
integrating
(Z. 14) w i t h r e s p e c t
r~ < c~jI-z-ll
11~11 2 : + Iletl I,
LZ(Hr+l)
LO°(XZ) -
L°°(L 2)
(3.6)
i C
8~+lu
a~u
+ ~
~ LI(Hr+I/h
2r-2
r odd.
'
If
(3.v)
Ilgit oo = Ilgll W1 L°°(IXJ)
+
~11 , 8x L°°(I×j)
then
Icll at°-~HL2'Hr+I'~IhZr-2 , (3.8)
r even ,
IIe II ® <_ a+l W1
C
O---U-U[
We need to determine function first.
and its x-derivative Let us define
Hr+l)
bounds for
two Green's
[I
IIL1(Hr+lll
at the knots ~] '
for the values
YI' " " " ' Yrn "
functions
, r
odd .
of the
We shall treat
~1
as follows:
#1382
304
o4G0
ag4(xi'~)= 8x'i'
~,I,
(3.9)
82G0
%%
T(x ,
~eaI,
and 84Gl(xl,~) =
-
6'
,
~I,
(3. lo) aZGt Gl(x i, ~) = - - ~ z ( x i ,
=
0 ,
~caI.
f u n c t i o n s and that
Go(Xr "), cZ(i)
G l ( x i , - ) c c l ( I ) . M o r e o v e r , we c a n require that
a3Gl(xi , ~)/a~ 3
Note that both are p i e c e w i s e - o u b i c and
6)
be continuous. Then, for any X e g~, 4 n(xi, t) = (n, '~ a~ 4. %. ( x i. , - t .)
=
(~2~,
2 a-~G0(xi,')
- X")
2
= o(llnlt z Ifa---;Q(x., ") - x"ll) a~ u 1 2
= o( Itu II+l h~-I 1ta-~T%(xi, .) - ×,, li). It is at this point that we need to s e l e c t can approximate
7~ = N1 or ~2 '
8 2 G 0 / a ~ 2 well in e i t h e r c a s e .
Indeed,
2
(3.11)
#1382
inf X e ~k
II-%G.(x.,.) - x"ll < Oh ~-I 8~ ~
1 1
,
r >__3 ,
s i n c e we
305 for
k = 1 or
2
when
(3.12) when
j = 0
a n d for
k = 1 when
j = 1 .
Thus,
ln(x i, t) l <_ c flu II LOO(Hr+l) h zr-2 ~ = ~1
or 9 2 . The function
G 1 can be used in like manner
to show that
lax( o~ ~i,t)l
(3.13)
C
_<
llullFo(ff+1) h 2r-2
for ~ = ~'I ' r>_3 . Note that for the very simple case of the heat operator~
GO ~ ~ 2
and
G1 ~ ~I ; hence, the infima above are zero.
Consider the functions
Y~ . If a bound on
Y~(xi,t) is to be
obtained:, it necessitates redoing the duality argument considering the space H s = [HS(0,xi) equipped
®Hs(xi,1)]
®JR,
s >0
,
with the norm
z + ii~i12 III (~, q) Ill s -- ll~l12s H (o, xi) s
+ q2
H (xi, 1)
If bounds are desired on both ~s equipped
=
Y~(xi,t) and
[HS(0, xi) ® HS(xi, I)] ®
aY~/ax(xi, t) , ]R 2
the space
, s>_O
with the norm
lll(¢'q)lllsz should be studied.
ll¢llZs =
H
(o,x i)
+ ll¢llZ +q20 + q ~ HS(xi,1)
In e i t h e r c a s e w e c o n s i d e r
[g'(%q)] = (g'•)+
i
the duality
g(xi)q, (¢, q) ~ ~s ,
[g(xi)q 0 + g'(xi)ql,
~s (qJ, q) ~
,
#138 2
306
for functions
g ¢ H2(I) ,
lllglll_s= It is clear that
lllg I11- s
and
~s
g ' ( x i)
for
Hs ,
[ g. (~. q)]
iH(%q)ltls,
sup
(% q) # 0
H2(I)
g~
.
A
dominates
The definition of For the s p a c e
and define
g(xt)
for
Hs
and both
g(x i)
given by (2.9) has to be changed as follows.
let ~ ¢ CZ(1)
satisfy
(iv)
¢ = ~" = 0 , X E ,,,
For the s p a c e
it s
(xi +
let
81,
o)-~,,'(x
~ ~ CI(I)
9"(x i + 0 ) -
i
-
0)=
q
=
qo
.
s a t i s f y the above plus the relation ~"(x i - 0) = - ql "
It then follows that aZY1 [Yl,(,,q)]
= (----~, ~':)
8x in either case. (q~,q) ~
When
. When
7fl = ~2 ' ¢'' can be approximated as before w h e n T~ = ~I ' ¢'' can be handled w h e n
(4, q) c
,
Thus, w e can carry out the argument leading to L e m m a 1 to see that 8kY~
[l[7
k +l l[[- s - < C [[ 8 8 t ~ [[qh q+s+2~ , Z < q < _ r + l ,
0<s
Also, it follows that 0Y itgu It1-~IiI < c 0 atq
#1382
r+l h
2~-2
r
even
•
Zl-
3,
tc J.
307
Thus, we s e e t h a t
k I19-~ II oo
[Y~(x i, t) l <_ c
st
k=l
+I h z r - z ' I < t < m ,
~ = ~I
or ~Z '
(:.:r)
n
(3.14)
k=l
l[sk-'-P-utl h 2 r - z 1 <_ f < m, ~ = Y~l " Stk L°°(Hr+l) ' -
Let =+1
k
(3.15) Q(u)= 2
II - II
k=0
+l + :l -(-llr)ll
L (H r
8t
)
If l
+l "
L(H r
at
)
The a b o v e b o u n d s c a n b e c o m b i n e d to c o m p l e t e the proof of t h e ~ollowing t h e o r e m .
Theorem 3,
Let U(0) = W(0) and let x = xi(5)
I(u u)(:,t)l
as
~ = ~1
lax-9-(u - v ) ( : , t) t <_ C0(u)h zr-z
~ -- ~l
-
In t h e c a s e the d e r i v a t i v e for
h - * 0 . Then, or
7~z
~ = 7~1 t h i s r e p r e s e n t s a s u p e r c o n v e r g e n c e r e s u l t for r > 3 ;
the knot values for r > 4
i n e i t h e r c a s e s u p e r c o n v e r g e n c e o c c u r s for .
W e have left to m a k e s o m e small remarks to complete the proof of the last sentences of Theorems i and 2.
Ilel(o) II1 = II(Y z
+
...
+
W e need a bound of the form
Zm)(O)111 < C0(u)h r+l ,
r >__S .
It f o l l o w s e a s i l y from Lemma 1, (3.1), a n d ( 3 . 2 ) t h a t m
II(Yz +
• . . +
~)(0)IIt < c 2 ~=2
ll:-~(0)llr_z~+zhr+1 8t
#1382
308 ~[.e.
I,
m
(3.16)
Co(U)
-- c
~
~=2
ll~-~(o) ll_z~+z
•
at
It a l s o f o l l o w s t h a t , for t h e c h o i c e
U(O) = W(O) ,
m
(3.17)
#1382
lie(o) II + hlle(o)II l <_ c
[19--~(0) [Ima×(4, r-2~+l) =I at-
hr+l
309
REFERENCES 1.
J. Douglas, Jr., T. Dupont, and M. F. Wheeler, A quasi-projection approximation method applied to Galerkin procedures for parabolic and hyperbolic equations, to appear.
Z.
J. Douglas, Jr., T. Dupont, and M. F. Wheeler, A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems, M R C report # 1 381 and to appear.
3.
T. Dupont, Some
L 2 error estimates for parabolic Galerkin methods,
The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential ~
,
A. K. Aziz ( e d . ) ,
Academic Press, New York, 1972. 4.
V. Thomee and L. Wahtbin, to appear.
5.
M. F. Wheeler,
A G a l e r k i n procedure for e s t i m a t i n g the flux for two
p o i n t b o u n d a r y v a l u e problems, to appear in SIAM J. Numer. Anat.
#1382
310
A R 70-31 .......
Unclassified
~,,.,~
D O C U M E N CONTROL T DATA .' R & O
c,.~.,¢,, .,,~.
(.¢e~'u~H¥ ~f~lff~¢~f~on o f ~H~, t ~ d y o f ~bsft~¢t ~ d t
OMI~INAYING
ACTIVITY
(CO~O~le mu~
fnde~t~ ~n~t~t~
mul~ ~ entered w h ~ th~ o~er~l! ~ep~et l~ t / ~ l l J ( J ~ c t ~
~ l . flEPOfl? S E C U R I T Y ¢ ~ A I S l F I C A T I O N
3
Mathematics Research Center I Unclassified U n i v e r s i t y of W i s c o n s i n , . M a d i s o n ~ W i s . 53706 ,]~"""°""None SOME SUPERCONVERGENCE RESULTS FOR AN H1-GALERKIN PROCEDURE FOR THE HEAT EQUATION 4, OESCRIP~rIVI[ N O T E | (TYpe o t re~,cRt ~ f l Ine|umlve 1~411feJll)
Summary Report: no specific reporting period. S. * ' u T . o * q ~ s / (Flpet creme, m~dWe InllJal, lalelr rotate)
Jim D o u g l a s , I t . , Todd Dupont, and M a r y F a n e t t W h e e l e r • ~ . . . . . . . . 22 ....... co&',rn',,.c T or
~A,'r
~o.
Contract No.
b, P m O ~ K C T N 0 ,
[. . . . . .
' "~5""
N. ONIG~NATOR*I Iq(PORT NUMISKM{IJ
DA-31-124-ARO-D-46Z 1382
None
I ~ . O T H E R R E P O R Y NOIS) {Jw~. e ~ * l ~ i i ~ l l i M t l l IIPNIINF b ~t~ mpo~)
111|41ltal4
None , Q o, S T . , s u . , o .
~*~*E~E.T
.....
Distribution of this document is unlimited. t-;~ S~P~LEMENTARV
NOTZl
IZ. SPONSOmNG M~t.IT*ItV
None |l,
ACTIVITY
Army R e s e a r c h O f f i c e - D u r h a m , N . C .
A O ~il'TR'A'¢'T
,
,
,
L
,L
Thomee and W a h l b t n h a v e i n t r o d u c e d a G a l e r k i n method for the h e a t e q u a t i o n in a s i n g l e s p a c e v a r i a b l e b a s e d on t h e H l - i n n e r product a n d have o b t a i n e d
Hz
and
estimate is given here.
H 1 e s t i m a t e s for the error.
An L z
The main o b j e c t is to show knot s u p e r c o n v e r g e n c e
p h e n o m e n a when the s u b s p a c e i s a p i e c e w i s e - p o l y n o m i a l s p a c e . Z C - p i e c e w i s e - p o l y n o m i a l s of d e g r e e is
O(h z r - z ) ;
for the
r,
.
the error in the knot v a l u e s
C 1 c a s e , both knot v a l u e s and knot f i r s t x -
d e r i v a t i v e s are a p p r o x i m a t e d to within
D D , ,0. . o , . , 1473
For
.
.
.
.
O(h z r - z ) .
.
.
.
U n. c l a s. s i f i e d Security C|l, E s i f t © a t i ~
AGO 6SD~A
i I I
i
error in the knot values is O(h zr-2) ; for the C 1 case, both knot values and knot first x-derivatives are
u e oo e ence
approxi_m_ated t o wi_thi_n _O_(h2_rf2) "_ . . . . . . . ,: _ ........... a p p r o x i m a t e d to w i t h i n O(h2r - 2 ) . I 7 UNCLASSIFIED UNCLASSIFIED ~Mathematics R e s e a r c h C e n t e r Mathematics Research Center SOME SUPERCONVERGENCE RESULTS S O M E SU~ERCONVERGENCE RESULTS l Galerkin methods FOR AN H1-GALERKIN PROCEDURE FOR AN H -GALERKIN PROCEDURE sGalerkin methods FOR THE HEAT EQUATION FOR THE HEAT EQUATION uperconvergence Iim D o u g l a s , J r . , Todd D u p o n t , and ;ira D o u g l a s , I r . , Todd D u p o n t , a n d Mary Fanett Wheeler Mary Fanett Wheeler 22 pp. 22 pp. vIRC Report No. 1382 AD VIRC Report No. 1 382 AD : o n t r a c t No. D A 3 1 1 2 4 A R O D 4 6 2 ~ o n t r a c t No. DA-31-124-ARO-D-462 The m a i n o b j e c t i s to s h o w k n o t s u p e r c o n v e r g e n c e The m a i n o b j e c t i s to s h o w k n o t s u p e r c o n v e r g e n c e phenomena when the subspace is a piecewise-polynomial phenomena when the subspace is a piecewise-polynomial 2 2 For C - p i e c e w i s e - p o l y n o m i a l s of degree r, the space. For C - p i e c e w i s e - p o l y n o m i a l s o f d e g r e e r , th~ s p a c e . e r r o r in t h e k n o t v a l u e s is O(h 2 r - z ) ; for t h e C 1 c a s e , e r r o r in t h e k n o t v a l u e s i s O(h 2 r - z ) ; for t h e C 1 c a s e , b oth knot values and knot first x-derivatives are b o t h k n o t v a l u e s and k n o t f i r s t x - d e r i v a t i v e s a r e 2r-2 2r-2 a pmr Qx_im a t e_d_t9 _wi t hi n _ Qlh_ _ _ i) • . . . . . . . . . . . . . . . . . . . r xim t e d io w i t h i n O ~ _ ) .
error in the knot values is O(h 2r-z) ; for the C 1 case,. both knot values and knot first x-derivatives are
Mathematics Research Center UNCLASSIFIED ~lathematlcs Research Center UNCLASSIFIED SOME SUPERCONVERGENCE RESULTS r ............ 1~,S O M E SUP E R C O N V E R G E N C E RESULTS Galerkln m e t h o d ' s F O R A N HI-GALERKIN P R O C E D U R E IGalerkin methods If F O R A N HI-GALERKIN P R O C E D U R E F O R T H E H E A T E Q U A T I O N [ FOR THE HEAT E Q U A T I O N s uperconvergence LS uperconvergence lim Douglas, It., Todd Dupont, and Iilira Douglas, Ira, Todd Dupont, and M a r y Fanett Wheeler i M a r y Fanett Wheeler 22 pp. 22 pp. M R C Report No. 1382 A D i,MRC Report No. 1 38 2 AID ~ontract No. DA-31-124-ARO-D-462 bontract No. DA-31-124-ARO-D-462 The main object is to show knot superconvergence The main object is to show knot superconvergence phenomena w h e n the subspace is a piecewise-polynomial phenomena w h e n the subspace is a piecewise-polynomial 2 space. For C -piecewise-polynomials of degree r , the space. For C 2 -piecewise-polynomia Is of degree r , the
1
APPLICATION DE LA METHODE DES ELEMENTS FINIS - UN PROCEDE DE SOUS-ASSEMBLAGE -
J.M. BOISSERIE Ing~nieur Chercheur ~ Electriclt~ de France
Une des idles les plus utiles au calcut automatique des solutions d'~quations aux d~riv~es partielles est celle de Ia d~composition du domaine ~ sous-domaines
@l~mentaires. La reconstitution de ,~
de Fronti~re F e n
aboutit ~ poser un syst~me d'@qua-
tlons ?~ grand nombre de variables que beaucoup de praticiens traitent comme un tout apr~s avoir choisi une m~thode it~ratlve ou une m~thode directe.
II existe cependant une autre fa~on de fake. Le domaine ~
est reconstitu~
par ~tapes, c'est ce qu'on appelle le sous-assemblage [R@f. 1 , 2 , 3 ] . A la place de I'unique et grand syst~me d'~quations pos~es, se trouve une suite de syst~mes satellites compl~tement d@coup!~s et coordonn@s par un syst~me principal de faille et de largeur de bande r~duite. G~n~ralement, ce proc~d~ est un cas particulier des m~thodes directes. Pourtant le syst~me principal est susceptible d'@tre r~solu it~rativement.
313 Ce proc~d@ est rendu possible parce que la signification de I'op@ration de reconstitution du domaine ~
est tout ~ fait ind@pendante des fronti~res 7" (e)
des sous-
domaines ou des domaines ~l~mentaires. Ce proc~d~ est naturet. It facilite t'utilisation de la machine 6 calculer. Les m@moires lentes de I'ordinateur conservent les r~sultats des calculs relatiFs au sous-domaines d~i~ trait6s. La m~moire rapide ne contient que ce qui est n~cessaire au traitement d'un seul sous-domaine.
Le d@couplage des diff~rents
syst~mes satellites ralentit la propagation du bruit num@rique. EnRn, les procedures de d~coupages, pourvu qu'elles soient adapt~es 6 I'algorithme de r~solution~ trouvent leur efficaclt~.
Nous allons presenter un sous-assemblage tr~s simple qui est a la base d'un des codes de calcul utills~ ~ la Direction des Etudes et Recherches d'Electricit~ de France pour les probl~mes d'~quations aux d~riv~es partielles du second ordre elllptiques et auto~adjointes. Les sous-domaines sont des files simples d'~l@ments discr~tis@s de fagon analogue et dont la f'orme est quadrangulaire curviligne pour les probl~mes ~ deux dimensions et hexa~drale curviligne dans le cas de trois. Le sous-domaine est appel6 r6glette. Le nora symbrollque du code est RULE.
I.'aspect modulaire du sous-assemblage a pour cons6quences de simplifier I'extr@me la structure du syst~me principal et des syst~messatellites qui est tri-diagonale par blocs identiques. L'aJgorithme de Gauss par blocs est adapt@e ~ leur r~solution.
314 1 - N O T I O N DE REGLETTE Soit ,0.
le domaine consid~r~ et t"
= 3..0, sa fronti~re. A la suite de la
d4composltion apparoissent des domaines ~l~mentaires ou ~l~ments finis cu(e) y(e)
de fronti~re
et nous avons:
1,1)
a)
U
~(e)
= ~,
b)
N
~ (e)
=
#
c)
U
T (e)
=
£ + £ oux
£
est donc I'ensemble des fronti~res nouvelles qui apparaissent apres la
au×
d~composltlon de .0, . Dans tout ce qui suit nous n'examinerons que les situations bi-dimensionnelles. La g4n4ralisation 6 trois dimensions est en effet imm4diate. D'autre part nous imposerons toujours sur
F
la condition de Dirichlet. Cela est effectivement rendu possible par
aux
la nature des problemes trait~s : celui du Potentiel et de l'Elasticit~ en formulation d@lacement. Enfin les 41gments finis utilis~s ant une discr~tisation particuli~re [R~f. 4 ] . Sur la figure 1) est repr4sent~ un des ces ~l~ments et ses quatre c6t~s 7"(~) ,
YC) '
7"(~) ,
¥(~) . Les points de dlscr~tisation qul le d~crlvent peuvent ~tre r@artis en
cinq ensembles que nous d~signerons par
S. I
avec
1 -.< i
~ 5 .
Nous avons: Si
2,1)
~<4
]..
=
si c z
5
S.
C
0 ~
Nous disons que cet 4t4menf est "6corn~" parce que
si
n si
= #
v~
l i
l <~ b i
Aucun point de discr4tisation n'appartient simulton4ment 6 deux @l@ments par
(e)
les notations
oa (e)
et
Si(e)
,.< 5 Si. Si nous rep@rons les
ant une signification @vidente.
315
%
FIGURE 1
FIGURE 2
°
316 Nous appelons r4glette la file d'~l@ments hum@rotes - en chiffres romalns- : I, II . t £ , . . .
NR
La figure 2 repr4sente une r~glette a 4 416merits. Les conditions de voisinage entre ~16ments et fronti~re sont restrictives. Nous les groupons en trois rubriques : 1) Conditions de recouvrement de la fronti~re donn@e N E r~glettes suffisent a couvrir La premiere r4glet~ touche if' composent. La derni~re r@glette touche r'
~
.
par toutes les faces (~) par toutes les faces
des 414ments qui la
(~)
des ~l@ments qul
O
de leur premier
la composent. Enfin toutes les r6glettes recouvrent if' al@ment et par la face
(~)
du
par la face
NRi~me 61~ment.
Toutes ces conditions font que le domaine ~ est topologlquement ~quivalent un carrY.
2) D4finition de la fronti~re auxilia|re int~rleure de r~glette La fronti~re auxiliaire int~rieure de r~glette est d~finie par I'ensemble des points de discr~tisation
S2(e)
et
$1(e)
avec :
2
~< e
~< N R
I
~
~ NR-1
avec e
Le but du sous-assemblage est d'~liminer les inconnues relatives ~ cette fronti~re. 3) D@finitlon de la fronti~re auxitiaire ext@rieure de r~glette Les ensembles de points de dlscr4tisatlon I ~
e ~ NR
une
et
$4(e)
avec
discr4tisent la fronti~re auxiliaire ext~rieure de r~glette. Rappelons
que pour la premiere et la derni~re des avec
$3(e)
N E r@glettes, la fronti~re
partie de leur fronti~re auxiliaire ext4rleure.
r'
est confondue
317 2-
ASSEMBLAGE D'UNE REGLETTE ET SYSTEMES SATELLITES Le code RULE r~sout des 6quafions de type eiliptique et auto-adioint :
1,2 a)
.~ (u) - f =
0
dans
1,2 b)
B(u) - g =
0
sur F
ob
u
est fonction inconnue. ll existe donc une autre faqon de poser le probl~me :
2,2)
Min V { J ( V ) = f , ~ F(V,Vx, Vy ---) d'~ }
les conditions
1,2 b)
6tant satlsfaltes.
Pour r~soudre (2,2), on exprime que ta variation solution
de
u est incr~ment~e de 8 u , est nulle . L'expression de ~J
3,2)
8J = f f ~
~u
[F]u
J, Iorsque ta
est
:
[F]u
# + ~/1F/ ~u (Fux dy - Fuy dx) avec
8J
= ~J1 + 8J 2
='~(u) - f et Fux dx - Fuy dy = Lorsque
h(u) ds F
F est donn,~, il est possible de calculer
h(u) . La condition
h(u) = 0 est appel&e la condition naturelle du probl~me (2,2) . Cette relation est satisfaite sur tout le pourtour si on minlmlse J(V)
sans oucune confrainfe.
,~ est d~compos6 en une suite de domalnes
~u(e) et on a :
o ,/ut oux GuN
Comme F
est ffonti~re entre deux ~i6ments - et deux seu!ement si les aux 61~menfs sont "~corn6s" - , les contributions ~ 8J 2 de deux 61~ments conqg,Js par l'interm6dlaire de leur fronfi~re commune doivent ~tre de somme nulle. En natant F *a u x
318 cet ~l~ment de fronti6re commune aux ~l~ments (e) 5,2)
S ~:
~u (h(e)(U) + h(e,)(u) ) dsF~
aux
6,21
REMARQUE :
=
0
"raisonnables" on a :
h(e)(U) + h(e,)(u)
sur tousles points
(e') on a :
OUX
la relation 5,2) volant pour tous les ~u
discr~tisation
et
=
0
de la fronti~re commune ?:k
aux
et en particulier sur les points de
Si(e) . La relation 6,2) est la condition d'assemblage. Le fait que les ~l~ments soient "~corn~s" exclut de la frontigre discr~tl-
s~e les coins oO la fonction inconnue perd de la "r~gularit~". La discr~tlsation par ~l~ments finis que nous employons dons RULE joue un double rSle. Le premier est celui de IqnterpolaHon. Lorsque routes les conditions aux limites g(x) ~ue
ou
sont connues sur la fronti~re 7'(e )
u est calcul~ par interpolation sur
s5(e). Le second est de produire la correspondance discr~tis~e entre
h(u)
e,
u en tous les points : 7,2)
7"(e)
I"t
[' OUX
Nous notons cette correspondonce : 8,2)
h.(e) =
R. Ie)
,
(e)
II
L'ensemble des Rij (e)
5
gj
e
fix~, avec
1 ~< i , j ~< 5 est souvent appel~
matrlce de rigidit~ de I'~l~ment parce que dons le probt~me de l'~lasticlt~ une contrainte superficielle et
gj(e)
hl (e)
est
un d@lacement.
Trois cos sont ~ distinguer : o) r~glette, gj
Si
si(e)
est sur la t:ronti~re auxiliaire ext~rieure ou int~rieure de
est la valeur de la f'onctlon inconnue puisque la condition de Dirichtet est
impos~e s u r r b)
aux" Si
si(e)
est sur la fronti~re
r de ~
• les conditions aux J~mites
sont donn~es et tr~s g~n~,ralement elles sont du type m61~. Les gi donn~es de fronti~re du probl~me.
sont les v~ritables
319
c)
Si
S.(e) I
est t'int~rleur d'un ~16ment. C'est le cas
i = 5
Les g5
sont les seconds membres f les ~5
des ~quations ~ r6soudre. Lorsque l'6quation est homog~ne (e) sont nuls et les termes Ri5 ne sont pas calcul6s. Dans le premier de ces trois cas~ les valeurs des
gi = ui
ne sont pas
initialement connues. Dans le cos de la m~fhode de sous-assemblage de RULE elles sont fournies au fur et ~ mesure de la r6solution, Nous sommes maintenant en mesure de poser le syst~me d'6quafion satellite qui exprime la continuit6 de la r6glette (6quation 612) sur les faces appartenant ~ la fronti~re auxiliaire int6rieure de r6glette, Ces ~quations figurent dans le tableau I. Les inconnues sont t'ensemble complet des valeurs de discr~fisation de la fronti~re auxillalre int~rieure
avec
1 < e < NR_ 1
ui
aux points de
Uin t
et
2 < e'
< NR
IJop6rateur lin~aire du premier membre dans l ~ q u a t i o n du tableau 1 est
6gal au nombre
de nature t r i - d i a g o n a l e par blocs. C'est une matrice dont le rang est d'~l~ments de
Uint, ~ savolr : e
(N R - 1) x dimension de Associ~e ~ un probl~me de
e J
S2
ou
S1
m~me nature num~rique que le probl~me g6n~ral ~ cette
matrice est positive et inver sible, kes seconds membres ne sont pas consfitu~s par un seul vecteur mais par une suite de rrmtdces
Bk
qui op~rent sur :
a) des valeurs initia!ement inconnues sur les faces
@
et
@
qui
constituent la fronti~re auxiliaire ext~rieure~ en g6n6ral. b) le vecteur des valeurs gi
sur la portion de .1" recouverte par la
c) le vecteur des valeurs
donn6 ~ t'int~rleur de ~
r~g tette.
Si nous d~signons par le tableau
Gk
fi
les valeurs de
gi
,
ordonn6es comme le montre
I ,, la forme des ~quations d'assemblage de la r~glette est :
320 k=4 9,2)
A
Uin t
=
~ k=l
Bk
Gk
En multiptiant les deux membres de cette relation par une correspondance entre les valeurs de
Uin t
et ceJles des
A -1
nous @tablissons
G k ,relation qui est :
k=4 10,2)
Uin t
=
~ k=l
Ok
Gk
Cette relation va @tre celle qui permet la construction de ta matrice de rigidit@ de ta r@glette.
321 3-
MATRICE DE RIGIDtTE D'UNE REGLETTE ET SYSTEME PRINCIPAL Nous venons de volr comment assurer ta recompositlon d'une r~gtette. Mals le but de I'algorithme est de reconstituer le domalne entier ~
en posant te syst~me
d'~quations principal. Cela est possible parce que la relation de voislnage des r~gtettes entre elles est analogue ~ celle des ~t~ments entre eux dans cette r~glette. Done le syst~me d'~quations principal va avolr la m~me structure que les syst~mes satellites d~j5 pr~sent6s et attaches ~ chaque r~glette. Si nous examinons le tableau 1, nous voyons que les premiers membres du (e) syst~me d~quation satellite ne fait intervenir que des Rii avec : 1~
i
et
S. et S. I I liaire int~rleure de r~glette.
c'est-5-dire que les
j (2
correspondants appartlennent tous 5 la fronti~re auxi-
Le syst~me d'~quations principal fait jouer aux fronti~res auxilialres ext~rieures de r~glette le r61e jou,~ dans les systames satellites par les trronti~res auxiliaires int~rieures. Nous allons construire les op~rateurs T.. (r) II
3 ~ i ~< 4
et
] ~< r ~< N E
I <~ j ~< 4
nombre de r6glettes
attaches ~ chaque r~glette~ qui vont permettre de poser le systame d'6quations principal. En utilisant la d~finitlon des
R.. ~ej II
nous pouvons construlre la relation
discrete entre : a) les valeurs des b) I'ensemble
h3
Uin t
et U
h4
sur 1~ fronti~re auxiliaire ext~rieure r
G k.
Cette relation figure au tableau II et attire les commentairessulvants : a) le tableau r~sulte de la seule d~finitlon des tous.
b) la m~me d~finition est celle de
T4j(r)
R3i(e) . Ceux-ci y figurent
et fait intervenir tous]es R4i
c) les op~randes des dlff~.rentes matrices sont d,~finis et ranges de £o~on identlque a c e qul figure au tableau I.
(e)
322 La matrice de rigidit& de la r@g!ette sera d@Finle comme la relation discr~tis@e entre les valeurs de et
h(u)
prises sur la fl-onti~re auxiliaire ext~rieure de r~gtette
:
a) les valeurs de
u.!
sur les faces
(~
F
et
(~)
qui constituent pour chaque
rgglette sa Fronti~re ext~rleure. b) les valeurs de
gi
sur la partie
recouverte par la fronti~re de la
c) les valeurs de
f.
donn~es ~ I'int~rieur de la r~glette - la fronti~re
r@glette. I
auxillaire int~rleure ~tant exclue On aura donc une relation tout ~ fait analogue ~ 9t2) k=4 ~ TBk Gk T3j (r) = TA Uin t + k=l
1,3)
Mais nous pouvons ~liminer
Uin t
en tenant compte de 10,2).
II vient : 2,3)
T3i(r)
= { T A x O k + TBk }
Gk
11 est alors possible de construire le syst~me d'@quations principal qui se d@dult du syst~mes satellites : - en rempla~ant
N R par
- en rempla~ant les
R.. (e)"
NE avec
1 ..< i , i
..< 2
avec
3 -.< i , j
..< 4
par les
II
T.. (r) II
Les seconds membres du syst~me d'@quations principal ne font plus intervenir que les valeurs connues des g
sur F
et des
f
sur ..0..
La fin de I*algorithme - souvent appel~ la redescente - consiste e d~terminer les
Uin t
par utillsation de 10,2). Apres que ces dernlers alent ~t6 calcul6s t la pro-
print@ d'interpolation des @lements permet d'avolr le champ inconnu en tous les points de dlscr~tlsation. Dans le cas du programme tri-dlmenslonnel le sous-assemblage se Fait lul-m@me par ~tape. Les 616ments sont d'abord assembl6s en reglette du premier type. Les r@glettes sont elles-m@mes associ@es en r~igtettes du second type qui par un troisieme assemblage reconstituent le domaine ~
.
323
4-
APPLICATIONS Les applications du code RULE ant @t@ assez nombreuses. La comparaison
avec I'exp~rience et avec les r~suttats d'autres codes de calcul a ~t~ faite. A) Calculs glastic].ues Le code RULE a ~t@ ~crit en vue de calculs de b~ton pr@contraint dans la fili~re fran~aise (MAGNOX),
Le b@ton est un mat~riau non lin~aire. I1 y avait a pr~-
voir une re@rhode incr~mentale de calcul. De cette sorte d'~tude il reste un example qui mantra bien I'int~r@t d'@i@merits ~ fronti~re curviligne. Sur la figure 3) sont repr~sent~es les lignes'lso-valeur" de la contrainte circonf~rencielle dans le cas de deux caissons en b~ton pr~contraint soumis a la pression de service. Le premier est I'ouvrage r~el. II a un goussei" de raccordement entre le fond et le cylindre. II figure ~ gauche de la figure 3). Le hombre d'~l~ments utilis~s est de 608. Chaque @l~ment a 6 4 degr~s de libertY. Le second r@sultat figure b droite de la Figure 3 ) . I I s'agit d'un ouvrage fictif dont le raccordement iupe-datte est r@alis~ par un arc de conique. Le nombre d'~l~ments utilis~ est beaucoup mains grand. 11 y en a 100. Les r~suhats sont aussi r~guliers qua darts te cas pr@c~dent. l..es caract~ristiques de ces deux ~alculs figurent dans le tableau ttt. B) Calculs de potential Un autre prob!~me dont nous montrons le r~sultat est du type potential. Le r@sultat cherch~ est celui de l'int@grate du flux du gradient autour d'une cavitY. La moiti~ du domaine est discr~tis~, ke maillage est represent@ sur la figure 4 . C'est un probl~me m@t@. Conditions aux limites : BC
Neumann
CD
Dirichlet
DE
Dirichlet
EF
Neumann
FA
Dirichlet
324 Sur le segment
AB , 2 conditions sont possibles Neumann ou Dirichlet.
L'~tude a consist~ ~ comparer les r~sultats de plusieurs d~coupages. Une erreur de calcul infgrieure ~ 10-3 @tait le but ~ atteindre.
TABLEAU IV
Nombre de degr~s de llbert~ par l~ment 21
Cas 1
NR
NE
7
35
0.37_792
Cas 2 0. 38087
21
9
35
0. 37788
0. 38067
21
11
35
0. 37785
0. 38029
32
6
35
0. 37797
0. 38042
32
6
25
0. 37789
0. 38048
4S
5
25
0. 37793
0. 38036
Les tigngs iso-valeurs du potentiel sont repr~sent~es sur les figures 5 et 6. Les difficult~s de cette ~tude
provlennent du
volsinage de AB oO les lignes iso-poten-
tielles ont une courbure tr~s variable. C) La version tri-dimenslonnelle de RULE Cette version permet dans 1'6tat de la programmation actuel de sous-assembier en deux ~tapes 16 ~16ments hexa6driques ~ 96 degr~s de tTbertg. La largeur de bande maxlmale du syst~me princlpa~ est de 576. C'est actuellement le seull critique. Les autres seuils critiques sont la taille des syst~mes satellites et la capaclt~ des m~moires tentes. Le sous-domalne de type r~glette a, au plus r 1272 degr~s de libertY. Le syst~me satellite de premier niveau a un rang au plus ~gal ~ 168. Le temps de traitement d'un tel sous-assemblage est de l'ordre de 80 secondes en CDC 6600. II n'y a pas ~ I'heure actuelle de traceurs de r~sultats associ~s ~ cette version.
La description de ce calcuF est la seule qui figure au tableau I11.
32S CONCLUSION ke code RULE a permis de poser et de r~soudre la plupart des probl~mes pes@s par la programmation de l'id@e du sous-assemblage.
La slmplicit~ de ta notion
de r~glette @tait n~cessaire pour r@dulre I'effort de programmation et surtout de maintenance. Des situations ~ grand nombre de variables ont permis de s'assurer de la stabilit@ des algorithmes mis en [eu.
326 BIBLIOGRAPHIE [1 ]
ARGYRIS
Recent advances in matrix methods of structural analysis Perg:mon Press - 1964.
[2 ]
SCHREM ROY
An automatic system for kinematic analysis ASKA High Speed compreting oF elastic structures Proc of Symposium tUTAM- August t970.
3]
Wl LLIAMS
Comparison between sparse stiffness matrix and substructures methods - Int Journal for Numerical Methods in Engineering 5-3 p.383.
4]
BOISSERIE
Generation of Two- and Three-dimensional finite elemnrs International Journal For Numerical Methods in Engineering - Vol. 3 327-347.
A
0
(3) (3)(4) 21 22+11
Premiers membres
0
(2)(3) 22+11
(2) 21
0
0
12
22+11
(1) 23
(3) 12
(2)
(1) (2)
Uinter
I
B1
(4) 13
0
(3) 23
0
0
0
(2) (3) 23 13
(2) 13
faces @
Seconds
0
0
(]) 24
membres
B2
(4) 14
0
(3) 24
0
0
0
(2) (3) 24 14
14
(2)
Faces @
(2)!
0
B3
(2) 25
15
0
R(e) i J (e) iJ
(3) (4) 25 15
(3) 15 0
0
Les op6rateurs sont repr6senf6s
0
0
25
(1)
ensembles @
SOUS-ASSEM~LAGE D'UNE REGLETT.E_A 4 ELEMENTS DAN SR 2
TABLEAU
0
0
21
34
(1)
(4) 12
0
0
feces F
328
o
o
0
0
~
r, u 0
0--
+ 0
0
0
0
o
®
:tZ l
~
~0 ~ 0
~
o I--
0
E
v
CO
0
0
0
0
0
0
0
0
e0
Z
+
®
°t
u
0
0
0
0
0
0
vC'O
<
+ 'N"
®
0
0
0
0
~
r~
N
o
~
0
0
0
0
0
m
N
u 0
~m
v
im
+ II 0
4-. °-
.< 0
~
0
0
~
Probl~me potentiel
616ments curvilTgnes
Calcut 61astique axi-sym6trique
EI6ments quadrangles
Calcut ~lastique ax]-sym~trique
TITRE DU CALCUL
25
20
76
Nombre de r~glettes
d'ordre 20
25 syst~mes
4525
25x 5 = 125 616ments 6 45 degr6s de llbert6
20 syst~mes
d'ordre 54
76 syst~mes
60
96
144
600
760
4800
75
120
192
largeur bande
SYSTEME PRINCIPAL
premiers membres nombre de seconds i ordre membres
SYSTEMES SATELLITES
d'ordre 32
2500
14928
Nombre de points
III
20x 5 = 100 61~ments 64 degr~s de l ibert~
76x 8 = 603 616ments 64 degr6s de ITbert6
Nombre d' 616ments
TAB LEAU
CO
330
Figure 3
o
•
,
./
. . . . . .
.........
. ~'~....
~
. . . . .
~
. . . . . .
~
~
.~."
~...,... : , ~ . - - . . _ ~
~
....
.~--.
: ;-,-,-,-.-..-.
~. . . . . . . . . . . . . . , , . , , , , , . :
~
- -
,
,
~
~
m
~
~
7
~
1
-
7
~
~
"11
-rl
332
Figure
5
\ \
\,\ 1
i
/
/
. . . .
÷
\
\
_
~
r
j
~
j
~
J
J
C~
CD
w~
ITI C~ ~w
PROBLEMESNON-LINEAIRES] [NON-LINEAR PROBLEMS
,[
F O R M U L A T I O N AND A P P L I C A T I O N OF CERTAIN PRIMAL AND M I X E D FINITE E L E M E N T MODELS OF FINITE D E F O R M A T I O N S OF ELASTIC BODIES J.
T. Oden
Texas Institute for C o m p u t a t i o n a l M e c h a n i c s The U n i v e r s i t y of Texas i.
INTRODUCTION
The evolution of automatic computing m a c h i n e r y and the advent of m o d e r n numerical techniques
such as the f i n i t e - e l e m e n t m e t h o d have had
a p r o f o u n d effect on finite elasticity. ject was largely an a c a d e m i c one,
Not m a n y years ago the sub-
studied only by purists at a few
institutions and, owing to its complexity, anything of m u c h practical importance.
scarcely ever applied to
Today finite e l a s t i c i t y theory
has become an important tool in the analysis of a variety of complex systems,
including air cushions and bags,
ing pads,
shock absorbers, balloons,
inflatable structures, organs,
flexible storage tanks bear-
d e c e l e r a t i o n systems, membranes,
as well as the study of veins, arteries,
and other b i o l o g i c a l tissue.
human
To be sure, the subject is still
in its infancy, but new d e v e l o p m e n t s in c o m p u t a t i o n a l methods give strong promise that this infant will grow to maturity. The numerical analysis of problems of finite e l a s t i c i t y by finite element methods began around seven years ago with a series of studies of elastic m e m b r a n e s
[1-5].
Since then, a number of papers have
appeared on the subject dealing largely with specific details of the formulation and with applications to bodies of revolution, strain problems,
plane
stretching and inflation of thin sheets, and with
certain c o m p u t a t i o n a l details such as methods for solving the large systems of n o n l i n e a r e q u a t i o n s inherent in such analyses.
Summary
accounts of these and related investigations can be found in the monograph
[6].
For more recent applications,
see
[7-9].
The mission of the p r e s e n t paper is four fold:
First, we sum-
marize certain features of several formulations of primal and m i x e d f i n i t e - e l e m e n t models of both q u a s i - s t a t i c and dynamic b e h a v i o r of highly elastic bodies.
Secondly, we present; when possible,
error estimates and c o n v e r g e n c e results,
certain
and, thirdly, we discuss a
number of c o m p u t a t i o n a l m e t h o d s that have proved to be effective in recent calculations.
Finally, we cite numerical results obtained by
applying these methods to r e p r e s e n t a t i v e problems in finite e l a s t i c i t ~ F o l l o w i n g this introduction, we describe basic properties of G a i e r k i n models of the equations of finite elasticity, while in Section 3 we
335
describe
in some
detail
properties
appropriate
assumptions,
and a m i x e d
approximation.
sistency tion
and
stability
4 we c o m m e n t
incremental
on
loading
duce
here
wave
calculations.
which
include
We
of the
several methods
of a o n e - d i m e n s i o n a l
able
2.
also
introduce
finite
element
computational
e.g.
,
xE
= g(x,t),
~
is the m a t e r i a l
type
we cite
of conIn Sec-
emphasis
We a l s o scheme
for s h o c k
numerical
examples
stability
motions,
on
intro-
and p o s t -
and
shock waves.
ELASTICITY
~ is g o v e r n e d
by the
system
of
[i0,ii])
+ pF = pa
- n = T
with
problems.
IN F I N I T E
body
notations
methods,
problems,
APPROXIMATIONS
(see,
these
Under
of p r i m a l
approximation.
transient
of a h y p e r e l a s t i c
q~S(u(x,t))
u(x,t)
section
large-amplitude
equations
S(u(x,t))~ ~ ~
fifth
model.
convergence
Lax-Wendroff
elastostatics
GALERKIN
The m o t i o n nonlinear
to p r o v e
for q u a s i s t a t i c
In the
nonlinear
behavior,
q
are
a finite-element-based,
buckling
Here
we
in
~i
'
t L 0
~2'
t ~ 0
gradient,
x =
(2.1)
(xl,x2,x 3) d e s c r i b e s
labels
of a
X
material
point
xsn,
S(u(x,t)) ~
merit v e c t o r
u(x,t) ~
tensor,
is a n o n l i n e a r
operator
on the d i s p l a c e -
~
which
represents
the
first
Piola-Kirchhoff
stress
~
F is the b o d y
force,
p the
a = ? 2 u ( x , t ) / ~ t 2 ~ u ~ D u the
initial
mass
acceleration.
density,
and
It is u n d e r s t o o d
• S ~ d i v S. The b o u n d a r y ~ is the sum of two parts, ?x ~ on w h i c h the s t r e s s v e c t o r S . n, n b e i n g a u n i t n o r m a l
~i to
that
and ~,
~2' is p r e -
~
scribed For
as T a n d the d i s p l a c e m e n t
such h y p e r e l a s t i c
tial
function
volume,
referred
in the
sense
~ W y
;
S =
where
W,
¥ -
• F
bodies,
is p r e s c r i b e d
the
stress
to as the
as g, r e s p e c t i v e l y .
S is d e r i v a b l e
strain
energy
from
for u n i t
a poten-
initial
that F = I + H
(H + H T + H T H ) / 2 ~
;
H = ?
is the
x
6~ u
strain
(2.2)
tensor
and H is the dis-
~
~
placement
gradient.
Let A(x),
B(x) ~
defined
Here
be s e c o n d - o r d e r
over
~.
We
B> .
=. I tr. A . . Bd~
tr d e n o t e s
tensors
and a(x),
b(x) ~
~
be v e c t o r s
~
the
define
trace
inner
of the
products
, (a,b)
according
= I a
traction
A ~
to
• bd~
• B and L e b e s q u e
(2.3)
integra-
336
tion is implied. The completions in the norm induced by <',-> and associated with (-,.) are Hilbert respectively. The completion in vector-valued
of the space of second-order tensors the space of vectors in the norm spaces, denoted. J(~)~ and L2(~) the norm (.,.)1/2 of the space of
functions
which vanish on ~ , is denoted L~I(~).^ 01 Let Z(x) be an arbitrary vector in ~2 (~)" Then, upon taking the inner product of (2.1) with v(x) and using the Green-Gauss Theorem, we obtain from (2.1) the nonlinear variational problem: find ueL~l(~)x^ (0,t*] such that (p~,v) + <S(y),Vx®Y>
V ysL~l(~
= ~(v)
(2.4)
and t s [O,t*], wherein
(2.5)
~Y) = (P~'Y) + I T'Y ds Moreover,
3n 2 u is also subject to the initial conditions, (z(.,o)-~u0(.),v)
:
0
;
-- 0
v
(2.6)
Let g~x) be a basis of L2(~) ; then each v in L2(~) is of the form v(x~=~vl(x)~x) vi(x) sL2(~) Let W01(~) be the Sobolev space of ~
~"
,
~
°
m
functions
whose derivatives of order m are in L2(~). We now construct of W01(~) a subspace S k(~) h m ' of finite dimension G, by identifying a collection of G linearly independent functions ¢l(X),~2(x), .... CG(X). The identification of these basis functions defines an L2(~)-orthogonal projection K h of W01(~)m into S~(~) such that the image of any ~(x) s W 0 1m( ~ ) ~ under Hhk is of the form ~h~(X)
where
=
(VllV2) O ~
The subspace S~(~)
G ~ ~ Z (~,~ ~ ) o~ (x) ~ V(X) e=l aVlV2da and ~ ( x ) ~ GZ ( ~ , ~ B ) ~i,6 (x). ~ ~ is assumed to have the following
(i) There exists a constant C I l~hV I Im <_ Col Ivl Im wherein, index:
Iv,, 2 = , Iv, ,2
~ I
o
independent
(2.7)
properties:
of h and v such that
V h
(2.8)
~m~ID vl lod~, ~ being a multi-
~ = (~i,~2,~3) ; ~i = integer >_ 0, I~I - ~i,~2,a3
;
337
+
(ii) If p(x)
is a polynomial Hhp(~)
(iii)
of degree
< k,
= p(x)
(2.9)
For any r such that 0 < r < k, there exists
a constant
C > 0 such that If (I-Kh)Vl Is _< chk+l-s Iv Ik+l wherein
tv ik+12 =
~ 2da. f ~(I_~I--z Gk+iDav)
h is a real number,
generally
(2.10)
In all of these relations,
selected
so that 0 < h < 1 (i.e.,
h is a mesh parameter). Let E be the real numbers interval.
Let B[O,T]
be a linear
(generally we take B[O,T] S (~, [O,T]) Galerkin elements
= Sh(a)xB[O,T]
[O,T]-~E be a finite
space of functions
= cr[O,T],
r=0,1,2,or3).
which satisfy
3 G ~ ~ ~i i=l ~=I
=
(2.4)
defined
Here D~U = a and gIz,t)
(x,t) Aia(t)~
to seeking
among
(x)
V, > = £(V)~
the
(2.11)
¥ V ~ S kh (~)
= 0 ; (DtU(.,0)-Vo('), ~) = 0
= ZFmi(x,t)gm(X,0).. ~
Upon introducing
(2.9), we obtain a large system of nonlinear
tions in the coefficients
differential
+ <S( E giAi~(t)% i,~
)o,~6> j
(2.8) equa-
= FiB(t)
(Uo,gJ%8)o;
Z (gi+a,gJ+B)Al+(0) i,a
= (Vo,g3,B) o
Here M~
zat3 and F3~ denote the mass matrix and the generalized
M3+'le6= (pgi+~,gJ+6)O ; ~
(2.12)
Aid(t) :
Z (gi*a,g3}B)i l~ (0) = i,~
i,j = 1,2,3;
[O,T]
h (~) :
(Q(.,0)-Uo(.),V)
with
on
The space
for v ~ S k
(pD~U,V)°~ . . .+ . <S(U),VX ~_
Z M3 8Ale(t) i,a
time
is termed the space of semidiscrete
functions. Galerkin's method amounts k in Sh(~, [O,T]) the function(s) U(x,t)
into
and let
F!8(t) = £(gJ+8)O
a,B = 1,2,...,G.
(2.13) forces, (2.14)
338
Finite
Element
for a d i r e c t arbitrary systems ments)
Formulations.
and s y s t e m a t i c
domains
~ which,
of equations.
The f i n i t e - e l e m e n t application in general,
We p a r t i t i o n
concept
of the G a l e r k i n leads
provides method
to
to w e l l - c o n d i t i o n e d
~ into E subdomains
(finite
ele-
so that E e=l
On each e l e m e n t tions
X~(~)N
e
(2.15)
; ~e(~ ~f = ~, e @ f
e we i n t r o d u c e
a system of local
interpolat ion func-
such that
D X N~~~ e~ "xM") = 6MN 6~~ ; X e~ N- (x) = 0 for x ~ L here x M d e n o t e s Global
one of N e p r e s c r i b e d
"coordinate"
functions
N E e = U Z e=l N
%A(X)
(e) N ~ A is a B o o l e a n
where
coincides
functions
w i t h h = die ~e for the b a n d e d
[see
(2.10)
I~I = 0 in % (X) - U
for each
Z
defined
element
in
(2.16)
(e) N . ~ A=± if node N of L
element.
problems,
approximation
For example,
are p r e s c r i b e d
let gi(x,t)
are in sk([ e)
(2.16) 2 is r e s p o n s i b l e
in linear
a local G a l e r k i n
only values Also
generally
Property
of <_SCA la ) , 3j#B>
independent
(i.e.,
~ N XN(X).
(e)(t) NZ m N(e) M ~iN
and of the
suppose
at nodes)
: ii(i[ ij : 6ij).
and
Then,
for
we have the local equations, + <S(Z (e) (t)ijXN(.)),~iXM(.)> e = feiN(t) ~ i NZ ajN
<.,->
of the
(2.17)
transformation:
to f o r m u l a t e
(~}2)
each finite
where
(e~ e~ ~A XN ~ (x)
(2.6)-(2.8)]..
character
it p o s s i b l e
type
in ~ e ( M , N = I , 2 , . . . , N e ) .
}~ (x) are then given by
w i t h xAe~ and is zero if otherwise.
The local
makes
nodes
(2.16)
is the inner p r o d u c t e a r g u m e n t s t o ~ and e
in
(2.3)
obtained
using
(2.18)
restructions
(e) (e)_ mNM
(0 XN' e X~)
(e) = E a Na A l~ ; feiN = ~ ( ! i X ~ ) ; aiN
The g l o b a l
equations
are o b t a i n e d
-
by " c o n n e c t i n g
(2.19)
the e l e m e n t s
together" using (2.15) and the fact that ~ it), the global generalized force, is p r e c i s e l y E (e)N fe See [6,12] for a d d i t i o n a l E Z ~ e iN (t). details, e N
339
It is w e l l - k n o w n constructed h = ma~ [13],
so t h a t
{hl,h 2,
a . .
that
finite-element
(2.6) a n d
'he}' h e = d i a m e t e r
in e x t e n d i n g w o r k of F r i e d
class of e l e m e n t s
(2.8)
if Pe is the d i a m e t e r
scribed
in an e l e m e n t % ,
so t h a t
(2.ZO)
reduces
3.
[e
Indeed,
[14], h a v e
incompressible, the s t r a i n
p=m~n
to
{pe }.
sphere
Often
that can be in-
p=vh for v = c o n s t a n t > O ,
(2.10).
ACCURACY
AND CONVERGENCE
(2.9).
and prove
STUDIES
several
special
forms of
In some c a s e s we c o n s t r u c t
convergence.
homogeneous,
energy
and R a v i a r t
(2.20)
of the l a r g e s t
approximation
m a t e s of a c c u r a c y
Ciarlet
as
shown that for a w i d e
IVlk+l
In this a r t i c l e we s h a l l i n v e s t i g a t e the G a l k e r i n
can be e a s i l y
with h defined
can be r e p l a c e d by
hk+l < C -p- s I I (I-Kk)Vl Is __ where
interpolants
(2.8) a r e s a t i s f i e d ,
isotropic,
In p a r t i c u l a r , hyperelastic
esti-
we c o n s i d e r
bodies
for w h i c h
f u n c t i o n W is of the f o r m
W = W ( I I , I 2)
I3=I
(3.1)
where
I. are the p r i n c i p a l i n v a r i a n t s of the d e f o r m a t i o n t e n s o r 1 G = I + 2~ [see (2.2)]; i.e., u s i n g the s u m m a t i o n c o n v e n t i o n r r + 4, r s r s. I 1 = 3 + 27 r ; 12 = 3 + 47 r Z tYrYs-Tsy r)
; 13 =
16r+27rls s (3.2)
where
ysr = g r m y m s ,
are the c o v a r i a n t x i at t=0. Among
¥ms are the c o v a r i a n t components
components
of the m e t r i c
of ¥, and g r m
tensor associated
forms of W ( l l , I 2) in use in the c h a r a c t e r i z a t i o n
and s y n t h e t i c
rubbers,
we m e n t i o n
W = M l(I1-3) the H a r t - S m i t h
as e x a m p l e s
the M o o n e y
with
of n a t u r a l
form,
(3.3) 1
+ M 2 (I2-3)
form
W = C{ fe x p
[k!(Ii-3)2]dIl
+
k 2 1 n ~I2 }
(3-3) 2
and the B iderman form W = BI(II-3)
+ B2(II-3) 2 + B3(II-3) 3 + B4(I2-3)
(3-3) 3
340
H e r e M1,
M 2, C, kl,
examples
are c i t e d
Owing
(2.2)
= ~ • F
in 013
called
; Z = ~ W ~ ~
a static length
force
S(1)
Denote
this
D ~ d/dx.
We
even
problems
(S(Du),
(S(1) by s e t t i n g
v=V
Assuming S(1)
in
(3.6)
= S(A)
simple rod of
fixed
at xl=0,
h is
case,
subjected
normal
stresses
can be e l i m i n a t e d stress
we
to
per
unit
are
ab i n i t i o
is f o u n d
to be
i.e.,
if u(x)
ratio; then
S(1)
= S(I+Du).
type,
=
find
S(I)
a n d S(Du)
Likewise, (A,B) o.
u such
for one-
The w e a k
problem
Z(V),
f o r m of
that
V v ~ HI(o,L)
Galerkin
is the
i = 1 + u'.
use the n o t a t i o n
is then:
(3.6)
is f i n d
k U s Sh
(O,L)
such
¥ V s S~(O,L)
arrive
DV) ° = 0 , and
multiplier
end of p, and to a force
extension
Dv) O = £(v),
S(I)
(3.4)
~W
shall
we e a s i l y
~I + h 3 ~Yij
As a f i r s t
and u' = du/dx,
problem
- S(A),
the s t r e s s
cylindrical
transverse
~W
though
(S(DU),DV) o =
A=I+DU,
BVP.
pressure
of this
the a s s o c i a t e d
Denoting
free
the
longitudinal
boundary-value
whereas
its
~I ~ ~Yij
thin
Piola-Kirchhoff
displacement
interchangeably, dimensional
other
[6,pp.236-242]).
material,
that
2(1_i-3)
i is the
longitudinal
(3.1)2,
the L a g r a n g e
(see
Two-Point
the h y d r o s t a t i c
longitudinal
Z and
of a long,
at
Assuming
negligible,
of
hyperelastic
tensile
constraint,
2 ~ ~W ~=l~I~
=
pressure
Quasistatic
of f.
and the
; ij
the s t r e t c h i n g
incompressible,
Numerous
by,
are c o m p o n e n t s
3.1 N o n l i n e a r
where
is g i v e n
the h y d r o s t a t i c
consider
constants.
[6].
to the i n c o m p r e s s i b i l i t y
t e n s o r S of
where
k 2, and B i are m a t e r i a l in
3.7)
at the o r t h o g o n a l i t y
condition
V V e S~(O,L)
subtracting
that
(3.8)
(3.7).
~ C I, we h a v e
+ sl(~)el
;
~ =
I +
8(~-i)
;
eX =
i-
A
(3.9)
341 with 0 < ¢ < i, S' (~) =- dS([)/dl.
Hence
(3.10)
(S(1) - S(A),I-A) ° = (S' (~)el,e l) For most hyperelastic materials,
there are numbers Po' ~l > 0
such that ~o ! S ' ( ~ ) ! Ul for every I > Ic, where I c is some critical extension ratio > 0. Assuming this is the case,
~o I [et tl 2 !
(s' (~-)e t , t - A + ~ h t - t I h t ) = (S' (1)el,El) < pll leiIl O I IEll 10
where E l = I- Hhl
= I+u'=KhI-HhU'=(I-~h)DU=Eu,
and el=l-A=D(u-U)=De u.
Consequently
~_!
( 3. ii)
i lDeul I° <- ~o lIEu' I IO We summarize these results in the following theorem: Theorem 3.1 homogeneous
Let the first Piola-Kirchhoff
stress S(1) in a
rod of isotropic hyperelastic material be such that
constants ~o' ~i > o exist so that 0 < Po -< dS(1) dl V I s [Ic,~].
< Pl --
(3.12)
Let U denote the Galerkin approximation
of the solution
u of
(3.6) and e = u-U be the error. Then (3.11) holds. U k if U s Sh(O,L)_ for which (2.6)-(2.8) hold, then I IDeul Io ~ khklulk+l
;
Moreover,
(3.13)
I [eull O ~ khk+iIulk+l
where k is a constant independent of h. The results of
(3.13) follow from
with the observation that
(2.8) and
(3.11) together
I leul ll ~ el IDeulIo and
IIeulIo ~ hell IDeul Io
for the simple case under consideration. Thus, if polynomials of k Sh(O,L), the approximation is con-
order k are used in constructing vergent and I IDeul Io = o(hk), Remark:
I Ieul 10 = o(hk+l).
Observe that u - ~1 DS(u x) = u
C(Du)
/
1 dS dX
1p dl dS Uxx"
Hence
(3.14)
342
is the intrinsic wave speed in the material. S' (Du) exists V Du and
(3.12) holds,
with real and finite wave speeds.
is such that
in the body travel
9~en this occurs, we shall say that
S(Du)
satisfies
3.2.
Stretchin 9 of a Rod of Mooney Material.
the wave condition.
can be obtained illustrate,
When S(Du)
disturbances
if specific forms of W(II,I2)
suppose the material
More precise results are identified.
is of the Mooney type
[see
To (3.3),].
Then it can be shown that S(1) = 2(1
-
I-3) (IMI + M2);
h
=
-
1 -!
[M 1
+
I-IM2
(i
+
13 )
(3.15)
so that dS
dl = 2MI(I + 21-3) + 6M21-4 For stretching of the rod,
I > 1 --
condition and,
c
(3.16)
= i; hence,
S satisfies
the wave
in fact, dS 0 < 2M 1 <_ ~-~
<_ 6(M 1 + M2)
(3.17)
Thus llDeull o ! 3(I+MI/M2) IIEu, l 1o 9onver~ence
of the Hydrostatic
the hydrostatic
l lh-Hll
pressure
h(x)
eI < I IM 1 + {12(I---~) o --
Pressure.
(3.18)
The approximation
also converges, 21el-e 1
H(x) of
and, in fact from
(3.15)
2
+
} M2110 x I le11-2(l-el/1)-i I Io 12(i_ei/i) (3.19)
Here e I = I - i = Du-DU = Deu, which is bounded above by the interpolation error DE u in (3.11). 3.3
Mixed Finite Element Models.
that
[15] improvements
In the linear theory it is known
in the accuracy of approximations
tives can be obtained by using mixed models dependent variables are approximated
of deriva-
in which two or more
simultaneously.
We shall explore
certain properties of a mixed nonlinear
formulation
cular,
form of the basic nonlinear
consider the following canonical
quasistatic
two-point boundary-value
here.
In parti-
problem
V(X)
-~ S(Du)
;
u(o) = 0
Dr(x)
= - f
;
v(L)
(3.20) These equations
corresponds
= P/A °
to the longitudinal
rod of initial cross-sectional
deformations
of a thin
area A O and length L fixed at x = 0 and
subjected to a force P at x = L.
Independent approximations
are now
343
made
of the d i s p l a c e m e n t
S(D u) is c o n s i d e r e d u(x)
a U and v(x)
u and the stress v.
N o w the K i r c h h o f f
s V.
L e t S<(O.L)
be a f i n i t e - d i m e n s i o n a l
subspace
of U and let TT(O,L) be a f i n i t e - d i m e n s i o n a l s u b s p a c e of V. k m H h : U ÷ Sh(O,L) and P z : V ÷ T%(O,L) are o r t h o g o n a l p r o j e c t i o n s these subspaces, are
the p r o j e c t i o n s
of a r b i t r a r y e l e m e n t s
G (u,¢~)o¢
(x)
each generated
L e t u* and v* be the a c t u a l appropriate
; ~v
=
H {WA}A= 1 are b a s e s
{¢ } =i and
respectively,
Then,
if
into
u a U and v e V
H
Zh u = ~ ~=i where
stress
to be a mappingu f r o m a S p a c e U into V, w i t h
solutions
using
[ (v,~A)oeA(X) A=I
k of Sh(O,L)
m and Tz(O,L)
appropriate
solution
of
are the e l e m e n t s
(3.21)
finite-element
models.
(3.20). The f i n i t e - e l e m e n t U*
e Sh(O,L)k
and V*
¢ TZm(O,L)
such that P%(V* w i t h V*(L)
- S(U*))
= 0 , and ~h(DV*
= P / A O ; U* (o) = 0
Lemma
3.1.
Let
+ f) = 0
(3.22)
(say) .
(3.ZZ) h o l d and d e n o t e
eu= u* - U*
; e
= v* ~ V* ; E = u* - KhU* v u E v = v* - P~v* ; au = U* - HhU* ; Cv = V* - PZV
(3.23)
T h e n the f o l l o w i n g hold: P~(S(u~)
- S(U~))
+ ev = Ev
(3.24)
Hh D e v = 0 Proof:
(3.24)
v* + P£S(u*)
in
upon replacing errors
(3.25)
is o b t a i n e d
(3.22), w h e r e a s H h f by -KhDV*.
by a d d i n g (3.23) In
and s u b s t r a c t i n g
follows
(3.23)
from
(3.20)
a v are t e r m e d p r o j e c t i o n
L e t u~U andiEV. approximation
of
(3.20)
lim L£(v) 4+0 where
Lh(V)
L~(v)
- (Dv,v)
(3.22)
errors,
respectively.
(3.22)
a consistent
Galerkin
if
= 0 and
and M h z ( U , V )
-
errors,
We s h a l l t e r m
and
eu, e v are a ~ p r o x i m a t i o n
a s s o c i a t e d w i t h u and v; E u, E v are i n t e r p o l a t i o n
and Eu,
Pzv* +
(DPzv,v)
lim Mhz(U,V) h,~÷0
= 0
are the l a c k - o f - c o n s i s t e n c y
; Mhz(U,V)
-- (S(Du) ,v) -
(3.26) functions
(S(DHhU,Pzv)
(3.27)
344
A simple calculation
reveals
that
L£(v) ! l lv! Io [ IDEvl [o and Mh£(U,V)
<_ l lVIIo II s(Du) - s(D~hU) IIo + lls(D~hU) IloII~vllo
(328)
While several different defintions of stability suggest themselves at this juncture, we shall choose to refer to the Galerkin approximation that
(3.22) as stable whenever
IIKhDP~S'(DU)~hUI[ o ~ aII~hUlIo for every uEu and v~V. stability.
and
constants
a,8 > 0 exist such
[IPzS'(Du)P~vll o i~l[P£VIlo
This may be a rather strong requirement
(3.29) for
Theorem 3.2. Let the Piola-Kirchhoff stress S(Du) satisfy the wave condition and let the subspaces S kh (O,L) and T~m(O,L) be such that (2.6) - (2.8) are satisfied and ]IDEvIlo and I IDEul ]o vanish as h, + 0. Then the Galerkin approximation is consistent. Proof:
This is obvious.
in (3.28) and use
(3.14).
Simply set S(Du)
- S(DHhU ) = S, (Du)DEu
It then follows that Lh, M£h ÷ 0 as h,Z+0
Theorem 3.3. Let the conditions of Theorem 3.2 hold and let the Galerkin scheme be stable in the sense of (3.29). Then it is convergent in the sense that IIDeu[Io vanish as h,£+0. Proof:
According
to the triangle inequality,
I IDeul IO ! IIDEul IO + I IDSul 10 Since S(Dul)obeys
the wave condition,
(3.30)
(3.24) yields
P£S' (~)Deu = Ev - ev = Sv = P£S' (~)D(Su - Eu)
(3.31)
Hence, nhDP~S' (~)De u = ~hDP~S' (~)DEu + ~hD~u However, ~hD~u = HhDEv, in accordance thesis, (3.29) holds, we have 1
JlD~uJlo ~ ~ lJHhDP~S'(~)Jlo Combining
this with
(3.30) and
with
(3.32)
(3.25).
Since, by hypo-
1
JJDE~IIo + ~ II~hDEvJI o
(3.31) gives
345
llDeulIo i (i + !~ IIHhDPzS, (~) Ilo ) I IDEu[ Io + !~ IIDEvl Io
(3.33)
and IIevIIo ! IIEvIlo + cllDeuIIo Since s([) satisfies
(3.34)
the wave condition,
DP~S'
is bounded.
Conse-
quently, both De n and e v are bounded above by IIDEul Io, I IDEvl Io and IIEvIlo, which vanish as h,z ÷ 0. 3.4. T i m e - D e p e n d e n t
Problems.
we shall use the combined
In the case of time dependent
finite-difference/finite-element
problems
approxima-
tion p (~
ui,V)o + (S(DUi), DV) O = Z(V~
(3.35)
VeS (O,L), wherein ~t2 is a second-order central difference operator and U l = U(x,iAt); 0 = t o < t <...< t R = T; ti+ 1 - t i = At. If the exact solution ucC3[o,T], then p (~ui,V)o
+ (S(Dui),DV)o
. a bounded function where m i is
of t.
= ~(v i) + p (~iAt,v) The following
(3.36)
lemmas are proved
in [15]: Lemma 3.2. wave condition.
Let (3.35) and Then
(3.36) hold and let S(u) satisfy the
P(62t ~i'si) + ~o I IDei112 <- Po (62Ei' i ) +
~lh2 + e2(At) 2 (3.37)
and P(6~ei,E i) ~ 80 I IDEil I~ + 81 I IDeil I~
+ B2(At) 2
(3.38)
wherein ~i' ~i are constants > 0 and independent of At and h and e i = u(x,i~t) - U(x,iAt), E i = u(x,iAt) - KhU(X,iAt), E i = U(x,iAt) - ~hU (x,iAt) . Theorem 3.4. Let (3.35) and (3.36) hold and let S(u) satisfy the wave condition. Then the finite-element/difference approximation error e(x,t) is such that for sufficiently small h and At, (3.39)
IIDeil Io = O(At + h) Proof: by observing
This result is obtained
immediately
that p (6t~,e) - p (~tE,s) =-p (~te,s)
from
(3.37) and (3.38) in (3.37) and then
346
using
(3.38).
such that
Since the q i , B
are arbitrary,
find constants
ilDeili~ ~ kl h2 + k~At 2, from which
The question
of numerical
stability,
(3.39)
Mi and ~
follows.
of course,
arises here.
[15] it is shown that the approximation
(3.35)
of the consistent
in the calculations,
In
is stable in energy i 1/2 whenever (h/At) > ~i C ax//2, where C max i = max[S'(Ux)/p o] ' i = 1,2 ' with ~i = 2//~ and ~2 = 2. The case involving ~i pertains to the use ~2 corresponds some specific
numerical
approximations 3.5
mass matrix of
(2.14)
to the case in which masses results
in Section
of the type described
Shock Waves
in Elastic
are lumped.
Rods.
whereas
We describe
5 that were obtained
using
here.
Consider
once again the one-dimen-
sional hyperelastic rod problem and again denote bv X(x,t) the ,,. longitudinal extension ratio, I = 1 + Du. Since C (i) ~ S / S X | is the intrinsic
material
(.acoustical)
wave speed{
and since Du = i,
we have the equation -- D[C2(X)DX] This alternate waves
form of the wave equation
in nonlinear
materials.
+ AtD2s(ln(x)) P
xn+l(x)
+ ---qAt n(x) p
= xn(x)
integration
scheme.
in the next section it in Section 5.
+ ~At2D 2 [S' (ln(x)qn(x)]
(3.41), q(x,t)
SYSTEMS
that have proved
systems
of nonlinear
We describe
We begin with the description
of nonlinear
algebraic
Incremental
system of ordinary
differential
algebraic
of methods
described
for systems
in static problems.
Method. equations
equations
of the large
with the models
encountered
Loading/Newton-Raphson
forming a system of nonlinear
using
FOR NONLINEAR
in the solution
previously.
4.1
obtained
a number of computational
associated
equations
results
~ X(x,nAt)
such a scheme
OF EQUATIONS
to be effective
equations
type
= X (x,t) and ln(x)
METHODS
In this section we shall describe methods
(3.41)
based-Lax-Wendroff
and At=ti+l-t i.
COMPUTATIONAL
shock
+ ~ ( A t 3)
+ At2~=---u2S(Xn(x)) + O(At 3) 2p
and cite some numerical
4.
in studying
~ s C 3, the expansions
a finite-element
In
0~tl~t2~'''~tR=T
(3.40) is useful
Whenever
qn+l(x ) = qn(x)
can be used to develop where
= 0
The idea of transinto an equivalent
and then solving
them by
347
by n u m e r i c a l integration, was introduced i n d e p e n d e n t l y by Lahaye Davidenko
[17], and G o l d b e r g and Richard
of the method can be found in of the m e t h o d d e v e l o p e d in
[19] and
[6] and
§3 leads,
[18], and a brief history
[6].
We shall outline one form
[20] which has been used to solve
very large systems having m u l t i p l e roots For nonlinear problems,
[16],
[21].
the f i n i t e - e l e m e n t m e t h o d d e s c r i b e d in
for q u a s i - s t a t i c problems,
to a system of n nonlinear
equations of the form f(x,p)
= 0
(4.1)
w h e r e i n f is an n - v e c t o r of n o n l i n e a r equations in the n unknown T g e n e r a l i z e d d i s p l a c e m e n t ~ = (Xl,''',x n) and a p a r a m e t e r p representing the applied load.
By assuming that x and p are functions of a
parameter s, we can compute dE 8f (x,p) ~f (x,p) . = d-~ = ~ = -- ~X ~ + Zp p ~
(4.2)
~
Thus
(4.1)
is t r a n s f o r m e d into the system of differential equations
j(x,p)@ + g(~,p)~ : 0 where {(x,p)
(4.3)
~ ~f(x,p)/~x is the J a c o b i a n m a t r i x and g(x,p)
~ ~f(~,p)
is a "load-correction" vector.
~P
We now p r o c e e d to integrate
(4.3) numerically.
While a number
of s o p h i s t i c a t e d numerical schemes could be used, numerical experiments have repeatedly indicated that, for large systems, none have any definite advantages over the standard Euler technique,
provided
a corrector of some type is i n t r o d u c e d to reduce errors at the end of each i n c r e m e n t As.
Hence,
tion 0 = So<Sl<...<Sn=l,
suppose s s [0,i] and introduce the partir Si+l-Si=As. Denoting xr=x(Sr), p =P(Sr),
and £pr=pr+l_pr, we arrive at the recurrance relation ,xr ,p r) (xr+l_xr) J[~ ~ ~
+ g(xr,pr) Ap r = 0
(4.4)
~
O r d i n a r i l y we set Apr=As and p r e s c r i b e the load increments hence the term "incremental loading." convenient append to
(in fact, necessary) (4.4)
However,
Ap r
it is sometimes
to treat Apr as an unknown.
Then we
the a p p r o x i m a t i o n
(£xr-l)TAxr + Apr-l~p r = As 2
(4.5)
~
of the arc length ds 2 = dxTdx + dp 2 in E n+l.
Here Axr=xr+l-AN r. ~
348
We now propose to reduce the a c c u m u l a t e d round-off error at the end of each load i n c r e m e n t using N e w t o n - R a p s h o m iterations such that x r'm+l = x r'm - J-l(xr'm,pr)f(xr,pr)
(4.6)
where the s t a r t i n g value is xr'°=xr and m=0,1,2,...,k.
The number k
is d e t e r m i n e d by p r e s c r i b i n g i n i t i a l l y an exceptable error E such that [(xr'm) T(xr'm)]I/2 = boldt
IIxr'mlIEn ~ e for m > k.
Ortega and Rhein-
[19] prove a version of the following theorem T h e o r e m 4.1
(Cf[19]).
where f: D~___En x
C o n s i d e r the system of equations
[0,i] ÷ E n.
to x and let its derivative J(x,p) singular on Dr[0,1] [or x(p(s))]
= ~f/3x be continuous and non-
for all s ~ [0,I] and assume a solution x(s)
exists.
integers kl,k2,..,
(2.1)
Let f be d i f f e r e n t i a b l e with respect
Then there exists a p a r t i t i o n of such that the sequence
remains in D and, after N load increments,
[0,i] and
{ r,m}, m = 0 , . . . , k r _ l lim x N'm = x(1). m÷~
In applications,
the major p r o b l e m w i t h this m e t h o d is its
inability to handle, w i t h o u t m a j o r modifications, J(x,p)
is d i s c o n t i n u o u s or
singular.
cases in w h i c h
Such cases are e n c o u n t e r e d
frequently in n o n l i n e a r e l a s t i c i t y in the form of bifurcations and limit points on the e q u i l i b r i u m path F: 4.2.
Stability,
Bifurcations,
x = x(p).
and Limit Points.
We consider a
m o d i f i c a t i o n of the procedure d e s c r i b e d above w h i c h can be used to determine limit points and points of b i f u r c a t i o n and to carry the solution beyond these along stable e q u i l i b r i u m paths. Ideally,
at critical points ~c such that det J(Xc,Pc ) = 0, we
introcuce a change of v a r i a b l e s x = Ay such that the m a t r i x H = ATj(xc,P)A
(4.7)
is diagonal and of rank r < n, w i t h zeros in the last n-r entries. Let y =
(YI'[2)T' [I being the first r rows of ~,
arbitrary r - v e c t o r of constants. constant.
and let ~o be an
We set ~ = A(Zo,~2 ) , holding p
This moves the solution off an e q u i l i b r i u m path F but
in a d i r e c t i o n tangent to F at the critical point ~c" constant, we iterate on p until f(x,p*)
Holding
= 0 (approximately).
P* > Pc' the p o s t c r i t i c a l e q u i l i b r i u m path is stable.
If
If p* < Pc'
it is unstable, whereas if p*-p, the test fails, a new ~i is selected (IZll
> IZol) and the process is repeated.
e q u i l i b r i u m path is reached,
Once a postcritical
the incremental loading process is
c o n t i n u e d w i t h p r e s c r i b e d load increments Ap such that Ap > o if
349
P* < Pc and Ap < 0 if p* < Pc" To determine x with sufficient accuracy, we employ a procedure ~c described by Gallagher I22], and evaluate the sign of the determinant of J(x,p)
at each load increment.
undertaking.
This is a numerically
Generally J must be appropriately
is only estimated by linear or quadratic the test is necessary by-passed
in the incremental
loading process.
point
for post-critical
too slow and expensive
scale nonlinear problems. effectively
for the practical
An alternative
is described in
[9].
analysis
just described
study of large
that has been used
In this process,
each bifurcation
(i.e., each critical point through which two or more stable
equilibrium paths cross)
is interpreted
critical point involving
only one equilibrium path)
system obtained by introducing system of equations. of perturbations of loads, ~.
and x ~c However,
interpolation.
since bifurcation points can be inadvertently
The elaborate procedure is generally
scaled,
sensitive
as a limit point
imperfections
These imperfections
in either the stiffness
(i.e., a
of a perturbed
into the original
are generally coefficients,
in the form the location
or both, and are represented by an imperfection
The incremental
scheme is based on the observation
, the post-critical approaches
equilibrium path of the imperfect
asymptotically
tion points
that of the "perfect"
parameter
that for small system
system and bifurca-
in the perfect system are reduced to limit points in the
imperfect system.
Thus, we proceed with the usual incremental
loading
Newton-Raphson
technique, checking det J as described previously, r r-i . until a critical point x =x c is reached. The system at x is
perturbed,
and the e q u i l i b r i u m path of the perturbed
beyond x c.
Newton-Raphson
iterations
system is traced
(with ~=0) then return the
system to the correct e q u i l i b r i u m path and the incremental process
is re-initiated.
Examples
of postbuckling
problems
loading solved
in this way are given in the next section° 4.3
Explicit
grating
Integration
Procedures.
(4.3), w h i c h r e q u i r e s
construct
the system of differential % + Cf(x,p)
where C is a damping matrix, x* of
(4.1)
lected so that
system.
(4.8)
integration
inte-
scheme, we can
equations
= 0
and ~(~)
is the steady-state
stable dynamical
Instead of numerically
an implicit
(4.8)
is usually 2-
solution of
Then the solution
(4.8), provided
(4.8)
is a
The damping matrix C can generally be se-
is stable.
Taking C to be the diagonal matrix
350
CI, one choice of an explicit integration scheme for
(4.8)
is
x r+l = x r + As c f ( r , p )
(4.9)
~
AS = Sr+l-Sr,
s ~ [0,i].
Since p is new held constant, x(s) may not
intersect with an e q u i l i b r i u m path at any points other than s=0 and s=l.
We cite an example p r o b l e m solved using this m e t h o d in the
next section. 4.4
Simple E x p l i c i t Scheme for T r a n s i e n t Response.
to the e l a s t o d y n a m i c s p r o b l e m
(2.9), which,
We now turn
for the present,
can be
thought of as a system of s e c o n d - o r d e r n o n l i n e a r differential equations of the form
~
+
x(0)
f(x,p) =
=
0
(4.10)
; ~(o)
x °
-
v°
Here M is the mass m a t r i x d e s c r i b e d in
(2.14).
~
We describe some
results in the next secion o b t a i n e d using the following scheme: The m a t r i x M is replaced by a "lumped" mass matrix mI.
It can be
~
shown that this does not deteriorate the accuracy of the approxim a t i o n for sufficiently smooth x(t) tion of wave fronts. solve
and it leads to a better defini-
This step also makes it p a r t i c u l a r l y easy to
(4.10) using explicit schemes.
Next, we replace
(4.10) by the
equivalent system = -!fm~(x,p)~ ,
v~(0) = Yo
(4 .ii) =
v
,
x(0)
=
x
~O
w h i c h we approximate using the divided central d i f f e r e n c e scheme v r + } = v r - 21 - mA_~t f~ (xr,p) ~ r+l x
r = x
1 + At v r + 2
(4.12)
This scheme is easily p r o g r a m m e d and has yielded surprisingly good results for some large problems. 4.4. ....A... F i n i t e - E l e m e n t Based L a x - W e n d r 0 f f - T y p e Scheme for Shock Waves in Elastic Materials. R~(g~) w i t h bases Suppose
C o n s i d e r two f i n i t e - e l e m e n t subspaces S~n(~) and
{ ~ ( x ) } G = 1 and {~h(x) }A=I' H respectively,
(~ =
[O,L]).
351
G ~ = (¢~, ¢~) A (x,t)
;
HA~ = (~A' ~r )
[Ae(t)~(x)
;
If A and Q are finite-element finite-element/Lax-Wendroff into
Q(x,t)
= ~BA(t)~A(X) -
approximations
of X and q of
(3.41), a
scheme is obtained by introducing
(3.41) and equating the projections
R~(~)
(4.13)
(4.13) k(~) and in S h
of the residuals
to zero:
y~HAFBF (n+l) =
~HArBr (n+l) = ~{HAF r - 2pAt-~2 (DS' (An) @ F'D@A) } By (n) At ~(S' (An ) - 7D~,
_ G 8A$(n+l ) = ~ { G B _ _At2(s , (in)D%~ 6
P
D~ A
)A~(n)
n + ~2
+ __ At ! (~A' ~ )BA (n)
D%B)}Ae(n) '
P
n + z2 Here A e(n)
~ A~(nAt),
generalized
forces.
"lumped" matrices
etc.
We conclude
5.1.
we replace G 8 and HAF in
so as to produce an explicit SOME NUMERICAL
this investigation
tained by applying representative
n n and £i' £2 are terms contributed
In general,
5.
integration
by
(known)
(4.14) by scheme.
RESULTS
by citing numerical
the methods described previously
results
ob-
to a number of
problems.
Large Deformation
comment on the numerical
of an Elastic Frame.
jected to a vertical
load P.
sionalized horizontal
As a first example,
analysis of large deformations
buckling behaviour of a Hookean two-bar
and P
(4 14)
and post-
frame shown in Fig.
i, sub-
If u = u/b and v = ~/b are non-dimen-
and vertical displacements
pd3/aoEb 3 is a non-dimensional
area and E the modulus,
we
of the center node
load, a ° being the initial bar
then the system is described by the equations
[2O ] (v-~) (V2-2~V + u 2) = P ; where ~ = c/b.
For ~ < /~, limit-point
since u = O V P. exist,
u[u 2 - (2~v - 2(2-v2))] behaviour
For ~ > /2, bifurcations
as indicated
in Fig.
i.
Figure
= 0
(5.1)
is encountered
in the equilibrium path
2 shows numerical
obtained using the u n c r e m e n t a l - l o a d i n g / N e w t o n
solutions
Raphson procedure
scribed in Section 4.1, together with a postbuckling
de-
analysis of the
type m e n t i o n e d
in Section 4.2, for the cases ~ = 1.0 and ~ = 1.5=
Good agreement
is obtained.
352
5.2.
Biaxial
has b e c o m e nonlinear using
Strip-Problem,
a standard elasticity.
the e x p l i c i t
on the finite
in.,
Another
0.05
5.3.
version
bodies in.,
solution
behavior
considered
degree
of AP = 0.4 ibs. from a positive (P = 2.8 ibs.)
Hence, 2.60
The
value
technique
failed
load pC was
ibs.
ibs.,
ated in the d e f o r m e d
placements
center
were
state
linear.
0.51 in. or 6.4 p e r c e n t
equations,
&P = 0.05
isolated
if the modulus of inertia
vertical,
the column
was applied at the top initial
deformation.
the v e r t i c a l
load was
(x2=L)
for which
of e l a s t i c i t y
within
case
the
ibs.
load w h i c h is approxi-
load.
Before
evalu-
buckling,
and the v e r t i c a l
dis-
at b u c k l i n g was
with a maximum
of the original width. the d e f o r m e d
shape.
After load
of the column to give the h o r i z o n t a l
increased
of load
at a load of
the critical
a small h o r i z o n t a l
While h o l d i n g again
restarted
at state of zero load,
and the p o s t b u c k l e d
determined,
changed
p o i n t occurred
height,
4
load was
matrix
displacement
Figure
critical
loading
in increments
I and length L were
near the critical
of the original
are sixth-
at a load of 2.65
well with
The v e r t i c a l
x2 = 0
to the range
of 0.07 in. or 3.5 p e r c e n t
load,
which
increment
ibs,
to converge
in w i d t h shows
The
is sub-
of incremental
at the seventh
change
at the critical
and
element m o d e l
load was applied
solution was
line r e m a i n e d
strain.
nodal displacements.
of the stability
This agrees
is pC = ~2EI/L 2 = 2.35
[24].
we cite
the surfaces
The finite
that a b i f u r c a t i o n
incremental
m a t e d by E = 6C 1 and the m o m e n t
the v e r t i c a l
The
load increments
the critical
with
and 72 unknown
equilibrium
corrections.
in
[9] on s t a b i l i t y
at finite
solved by the m e t h o d
indicating
< pC < 2.65
in
bodies
rotation.
The d e t e r m i n a n t
P = 2.4 ibs, with Newton-Raphson
were
.000[.
4, a 2.0 x 8.0 in. body
elements
to n e g a t i v e
the increment.
initially
(C 2 = 0) with C 1 = 24 ibs. per sq.
In Figure
72 n o n l i n e a r
with N e w t o n - R a p h s o n
obtained
in §4.2
For completeness,
load in the x 2 - d i r e c t i o n
polynomials,
strip in
obtained.
o b t a in e d
are n e o - H o o k e a n
of 64 t r i a n g u l a r
The resulting
c = .001 to
Bghavior.
results
fixed and x 2 = L fixed against consists
results
described
the same p r o b l e m was d e s c r i b e d
times were
of h y p e r e l a s t i c
thick.
to an axial
technique
of damping:
and P o s t b u c k l i n H
all 1/3 in.
jected
3 we i l l u s t r a t e d
methods
thick, M 1 = 24.0 and M 2 = 1.5 ibs. per
values
4-6 some recent
postbuckling
of a r e c t a n g u l a r
for c o m p u t a t i o n a l
of a strip of M o o n e y material,
of b a s i c a l l y
Stability
in Figures
In Figure
in.
for various
Agai n very rapid
stretching
time-integration
e l e m e n t model
8.0 in. square, sq.
The
test p r o b l e m
shape
the first
(p=0.15
ibs.)
the system an load constant,
from zero to the critical
by
353
the i n c r e m e n t a l technique.
For this case the e q u i l i b r i u m path of the
imperfect s y s t e m was stable to a load of 3.60 Ibs. where l i m i t - p o i n t type b u c k l i n g was indicated.
Holding the v e r t i c a l load c o n s t a n t at
3.0 ibs., the h o r i z o n t a l load was incremented to zero to p r o j e c t to the p o s t b u c k l e d path of the perfect system.
The m o t i o n followed by
the structure in removal of the horizontalload is shown by line segment I-J in Figure 5.
With the system now p r e s u m a b l y on the p o s t b u c k l e d
path of the perfect system,
the v e r t i c a l load was i n c r e a s e d from
3.0 lbs. to the second critical load of 3.75 ibs. < pc < 3.80 Ibs. w h e r e l i m i t - p o i n t type b u c k l i n g was experienced. creasing load increments from p o i n t J
The first two in-
(Figure 5) indicates that the
system has not r e t u r n e d to a point of complete relative m i n i m u m of the total p o t e n t i a l energy. c o n d i t i o n of the system.
This action is a t t r i b u t e d to the weak
The p o s t b u c k l e d path of the perfect system
from p o i n t K to the second c r i t i c a l point at point L does appear to be of c o r r e c t form in that it follows nearly p a r a l l e l to the imperfect system. A second example is indicated in Figure 6.
Here we see an arch-
type structure m o d e l l e d w i t h 72 triangular elements and ii0 degreesof-freedom.
A load was applied along the axis of symmetry in incre-
ments of 0.2 ibs., and the b i f u r c a t i o n point was passed in the increment from 1.4 to 1.6 ibs.
This is shown g r a p h i c a l l y in Figure 6.
The i n c r e m e n t a l s o l u t i o n was r e s t a r t e d at a load of P = 1.4 ibs. with the load i n c r e m e n t ~P = 0.02 ibso
In this p a r t i c u l a r case,
the
incremental solution did actually pick up the p o s t b u c k l e d path at a load of P = 1.50 lbs. Geometric deformationswere measuredrap~dlyfromthis point
(Figure 6),
and the critical load pC was isolated to the inter-
val 1.515 ibs < pc < 1.520 ibs. structure w i t h increasing load
S u c c e s s i v e plots of the deformed (Figure 7) are i n t e r e s t i n g in that they
show the step-wise t r a n s i t i o n to a b u c k l e d mode.
It is noted that one
of the members reverses curvature, which is typical of this type structure.
A small couple was applied at the vertex of the structure
and the load was increased to the critical.
The critical load of the
imperfect system was found to be 1.00 < p C <
1.05 lb.
The critical
load of the p e r t u r b e d system occurs well b e l o w that of the p e r f e c t system, which indicates that the perfect system exhibits unstable symmetric bifurcation.
This,
of course,
is typical of this type
structure. 5.4.
N o n ! i n e a r E l a s t o d ~ n a m i c s - S h o c k E v a l u a t i o ~.
rod of M o o n e y m a t e r i a l
We c o n s i d e r a thin
(M 1 = 24.0 psi, M 2 = 1.5 psi) with the follow-
ing u n d e f o r m e d characteristics:
length = 3.0 in., c r o s s - s e c t i o n a l
354
area = 0.0314 in 2, mass density = 10 -4 ib.sec2/in. %. element model, we take 60 evenly spaced elements, in. and N O = 61, and we consider a concentrated,
For the finite
so that h = 0.05 t i m e - d e p e n d e n t load
which varies sinusoidally is applied at the free end; a complete loading cycle occurs in 0.002 seconds.
It is clear from the computed
response shown in Figure 8 that shocks develop quickly for this kind of loading.
Unlike the response for the tensile step load where the
unloading wave is p r o d u c e d by simply removing the load, the sinusoidal load actually "pushes"
the end of the rod:
The instant the load
starts to decrease is the m o m e n t w h e n the first w a v e l e t is g e n e r a t e d which propagates
faster than the preceding one.
Thus, at some time
subsequent to when the compression cycle starts, a compression shock forms in the rod. A comparison between two integration,
v e l o c i t y - c e n t e r e d central
differences and the f i n i t e - e i e m e n t / L a x - W e n d r o f f scheme, is also shown in Figure 9 for the sinusoidal loading.
In this case,
it is clear
that the internal energy behind the c o m p r e s s i o n shock renders the central d i f f e r e n c e scheme unacceptable. however,
It is interesting to note,
that the tension cycle evidently
"absorbs" the large
oscillations p r e c e d i n g it and again produces a smooth wave front. The d e t a i l e d response to this loading is shown in Figure 10.
From
the response shown, we notive several interesting features of nonlinear wave motion°
First,
the compressive shock wave is r e f l e c t e d
from the wall as a compressive shock wave by almost doubling the compressive stress, but the tension part of the stress wave is reflected with only a small increase in stress. milliseconds,
Secondly,
at t = 4.7
two compressive shocks are to collide, with relatively
little deterioration,
p o s s i b l y owing to the fact that m e c h a n i c a l work
of the external forces is c o n t i n u o u s l y supplied to the system.
In
addition, w h e r e we compare the response at t = 3 m i l l i s e c o n d s to that at t = 5 milliseconds, we find that it a p p r o x i m a t e l y repeats itself,
again indicating relatively little deterioration.
Finally,
we note that, as in the d e v e l o p m e n t of shocks from Lipschitz continuous data, the shock forms subsequent to initiation of the compressive cycle. characteristics type of loading.
Thus we are led to examine the positive slope
in the X-t plane to see if they preduct tCR for this Figure
ii shows that if we assume straight
compression characteristics of positive
slope, the cusp of the
c o r r e s p o n d i n g envelope in the X-t plane does, in fact, give a good estimate of the tCR o b s e r v e d in the stress-time plots.
355
Acknowledgement.
Portions of this work were sponsored by the U.S.
Air Force Office of Scientific Research under Contract F44620-69-C0124 to the University of Alabama in Huntsville.
The work reported
on nonlinear elastodynamics was completed with the assistance of Mr. R.B. Fost under the support of a grant, GK-39071,
from the U.S.
National Science Foundation to the University of Texas at Austin. 6. i.
Oden J.T.
2.
Oden, J.T. and Sato, T.
REFERENCES
Int'l Congress on Large-Span Shells, Leningrad,
1966. I, 471-488, 3.
Becker, E.B.
Stuttgart,
Berkeley,
Proceedings,
Int'l. Coll. on
82-107, 1967.
"A Numerical Solution of a Class of Problems
of Finite Elastic Deformation", California,
J. of Solids and Structures,
1967.
Oden, J.T. and Kubitza, W.K.
Pneumatic Structures, 4.
Int'l.
PhD Dissertation,
The University of
1961.
5.
Oden, J.T.
J. St. Div., ASCE,
6.
Oden, J.T.
Finite Elements of Nonlinear continua, McGraw-
93, No. ST3, 235-255,
Hill, New York, 1972. 7. Oden, J.T. J. Comp. Structures, 8.
Oden, J.T.
Truesdell,
by S. Fluggee, ii.
(To appear).
Sandidge, D. and Oden, J.T.
Appl'd. Mech., Pittsburg, i0.
2, No. 7, 175-].94, 1973.
Nonlinear Elasticity, Edited by W. Dickey,
Academic Press, New York, 9.
Proceedings, Midwestern Conf: ....
1973.
C. and Noll, W.
Encyclopedia of Physics, Edited
III/3, Springer-Verlag,
Green, A.E. and Adkins, J.E.
New York,
Oden, J.T.
1972.
Mechanics T0day-1973, Edited by S. Nemmet-NasseD
Pergamoon Press, Oxford, 13.
1965.
L_ar~e Elastic Deformations,
Second Edition, Clarendon Press, Oxford Press, 12.
1967.
(To appear).
Ciarlet, P.G. and Ciarlet, P.A.
Arch. Rat. Mech. and Anal,
46, 3, 177-199, 1972. 14. Fried, I. "Discretization and Round-Off Error in the Finite Element Method Analysis of Elliptic Boundary-Value Problems and Eigenvalue Problems,"
PhD Dissertation, Massachusetts
Technology, Cambridge, 1971. 15. Oden, J.T. and Fost, R.B. 357-365, 1973. 16. Lahaye, E. 1948.
Institute of
Int'l. J. Num. Meth. Engr'g u, 6,
Acad. Roya! ' Belgian Bull. CI. Sc~., 5, 805-822,
356
17.
Davidenko,
18.
Goldberg,
19.
Ortega,
D.
Dokl. Akad~ Nauk0 USSR r 88 r 601-604,
J. and Richard,
J. Struct.
Div. ASCE,
1953.
89, 333-351,
1963. J.M.
Nonlinear Equations 235,
and Rheinboldt,
W.D.
in Several , Variables,
Iterative
Solution of
Academic Press, New York,
1970. 20.
Oden, J.T.
Continuum Mechanics, The UAH Press,
Huntsville,
21.
Oden, J.T.
22.
Gallagher,
Washington, 23.
(NATO)
D.C.,
Lectures
in
1972.
J. Comp. R.H.
1972,
Structures,
Nat'l s~rm. Comp.
(See also J= Comp.
Oden, J.T. and Key, J.E.
(To appear).
on Finite Element Methods
Edited by J.T. Oden and E. Arantes e Oliveira,
Struct'! Anal.
and D e s i ~ n ~
Struct.
Int'l J. Num. Meth.
in En~r'g.
357
3.0
C
b 2
"
0
~=l
.7
~=0
I,/= 1.5
~/= 1.7 1,0
i 0
O.
4.0
--i.0
"~
--2 .0
~U
~|~=
V
Figure
1.
Nonlinear
Response
of
a Two-Bar Frame.
1.8
f
1,8
/
1.4
Exact Path
QO00
1.2
Calculated Path
1.0
.8
.6 =
°
.2
Figure
2.
Analysis
of
a Two-Bar Frame.
358
iiiiii~i
I Static Solution I~
_~-~-~
~ -
_
.....
. . . . .
~x
|1 r Figure 3.
• O 0 o ~
~.
. . . .
c:~oo 2 . 0 o o l
llt (milliseconds) Stretching of a Thin Strip
P
'
=
'~"
0
p ~ pC p > pc
N,d/~ N,d/IA a
Figure 4.
b
c
Deformed and Undeformed Geometry of Buckled Structure.
359 4
x(72) ~ / O ~
L x(72)
O= 0_........__.L
i
x(Tl)
Pl
[~x
o
i
0
i
21
~.0
.0
DISPLACEMENT Figure 5.
,
3.0
Load-dlsplacement
4.0
(in.)
Curves.
2.0 s, • / s • •
•
s s
s
sj
t f
1.0
~q o~q
0
0 . 1 ........ 0'.2
Figure 6.
....
I
O. 3
,,
0 . 4'
0 .'5
0
16
!
0.7
DISPLACEMENT (in.) Frame Instability at Finite Strain.
360
Figure 7.
Computed Buckled and Post-Buckled
Frame.
361
II 4~ L.~
°1 li
o r-t
r-I
,~, 1,,
,
~----_
?
T
1~ ~t ~
-"4 I~
° ~o
\
'
i
.rt
° r0 (i)
~
H < @
•rl
i
®
!
0 0
0 4~
-.7 @
E E~
CO
CO
\ I
o
~
o
o
o
I
T~d
r
i
,
o
I
~
9
I
~s~ea%E
B,
362
VELOCITY CENTRAL DIFF. t= 2.0ms
1.0ms
LAX-WENDROFF t= 2.0ms
1.0ms
l+O
,2
• 2
--40
40i,3.0ms 0
© CO
--40
--80
I20
--16C
--20C
12
13
II
I2
Distance Along the Rod ~ Inches Figure 9.
Comparison of Response Computed Using Two Integration Schemes.
363
3
~:.
t=l.6~,
~
1,'1'
~--1.Sms
m,
.__...___~x
__.~_
k
D I S T A N C E ALONG ROD ~ Inches Figure i0.
Change in Stress During Shock Formation.
Time
0
~v
sec.
.001
.002
I tc ~
Figure II.
Computed Characteristic Field
X
D
364
ERRATA For "FORMULATION AND APPLICATION OF CERTAIN PRIMAL AND MIXED FINITE ELEMENT MODELS OF FINITE DEFORMATION OF ELASTIC BODIES" By J . T . Oden I.
The tilda (-) should be inserted under letters in the following places : x+x ~ u÷u v÷v o+o Y+X ~÷i a+~
; Equation (2.1), Eq. (2.16), 3 rd line below Eq. (4.6), a line below Eq. (4.8), 2nd line below Eq. (4. ly ; Eq. (2.6) ; a line above Eq. (2.7), 3 places in Eq. (2.7), Eq. (2.12) ; Eq. (4.2), 2nd line above Eq. (4.7), a line below Eq. (4.8) ; 2nd and 3rd line below Eq. (4.7) ; 2 placed in Eq. (2.7) ; 2nd line below Eq. (2.8).
2.
Place subscript "o" after parenthesis and Eq. (2.14), Eq. (3.7).
")" 8 places in Eq.
3.
Place a dot ..... after F in Eq.
(2.13), --FJg(t) + FJ-g(t).
4.
Place a bar on first fie in Eq.
(2.15), ae ÷ Ke.
5.
A line below Eq. (2.15): interpoltaion + interpolation Line 9 above Eq. (4.7): nonsingular ÷ singular Line 5 below Eq. (4.3): repeated ÷ repeatedly Line 19 below § 5.3: agress + agrees A line below Eq. (3.29): For ÷ for Line 29 above § 5.4: were ÷ where Line 13 above ~ 5.4: measured + were measured.
6.
Line 3 Line 3 Line 2 A line Line 9 Line 3 A line A line
and 4 below below above above above below above
7.
In Eq.
(2.18): place a parenthesis
below Eq. (2.17): (2.12) + (2.16) Eq. (2.20): (2.16) ÷ (2.20) and (2.8) + (2.10) Eq. (3.19): (3.13) ÷ (3.11) (2.19): (2.2) + (2.3) Eq. (3.40): (2.11) + (2.14) Eq. (3.22): (e.20) + (3.20) Eq. (4.10): (2.10) ÷ (2.14) Eq. (3.23): (3.2) ÷ (3.22) ")" before
8. Aline below Eq. (2.8): /allj~i<_m z DvII_~ +/a 9. i0 "
II.
In Eq.
(2.19):
In Eq.
(3 8): "
(2.13)
i (liXN) • e + £(iixe). S h ÷ Shk k
Above Eq. (3.11): Next line: delete
"
(I'=Hh) Du ÷ (I'-~h)Du (Q[12]).
,~iX~(-).
z IIDjII. I~I<_m
365
Errata
12.
In Eq. In Eq.
(3.21) and Eq. (3.24): P + PZ (3.23) : e + e u and ¢ v = V~-P£ V~ + ¢ v = V~-P£ v*
13.
Line
14.
In Eq.
(3.27):
M h Z ( U , V ) = ÷ N h z ( U , V )-
15.
In Eq.
(3.28):
take off a dot "'" in front
16.
In Eq.
(3.29):
place
17.
Take off "o" from p i e. p + p in Eq. (3.35) and Eq. (3.36) Take off p a r e n t h e s z s.o ,,( ,, 1.e. . o , 6~(÷ ~ . Take off comma from t o < t, < ...t 2ndLline b e l o w Eq. (3.35).
18.
Place
19.
Line Line
20.
In Eq.
(3.40),
X = D + X - D.
21.
In Eq.
(3.41),
2 places:
22.
Change
p to p in Eq.
23.
Line b e l o w
24.
Line Line
25.
Line 8 b e l o w § 5.2: c = .001 to .0010 + c = .001 to On the same page at Line 6 from the bottom: 4a + 4.
26.
Line 6 from the top of § 5.4; load shown + load is shown. On the same page at Line ii from the top: (Figure 5 ÷ (Figure
2 above
Eq.
a minus
(3.26):
u U ÷ ucU and v V ÷ v¢V.
~ after
"-" b e f o r e
of S(Du)
">"
p(~2e,s)
at 2nd line b e l o w Eq.
(3.39).
I0 above § 3.5, di,S i ÷ ~i,Bi and KI,K 2 ÷ ki,k 2. 2 b e l o w § 3.5, C2(~) = Id~/3~p ÷ C(X) = ¢ i ~ c ~ / ~ .
Eq.
Figure
27.
In the
28.
Line
29.
Place
30.
0÷ o in E q u a t i o n
2 above
(4.3);
(4.9):
3 above Eq. 6 b e l o w Eq.
i., take
superscript
and a line b e l o w
As + ST+l-ST,
(5.I); (5.I);
(2.18):
0(At 3) ÷ O(At3). (4.3)
in 4 places.
s c [0,I] ÷ As = ST+l-ST,
s ~ [0,i]
P = pd3/aoEb3 + P = pd3/aoEb3 p = 1.2 + ~ = 1.5.
off Q and place gi (x't)
= ii'ij
"i" on V in Eq. (4.3).
Eq.
a bar on P.
.0001.
i.e.
= ~ij + ~gi (x,t)
(3.30):
~(v)
P÷P.
= ii,
÷ Z(v i)
5).
ii'ij
= 6ij
METHODES NUMERIQUES POUR LE PROJET D'APPAREILLAGES INDUSTRIELS AVANCES
par S, ALBERTON,,,,,,!,, ' Universit&
de P a v i a e t ARS SpA, M i l a n o
Introduction 11 e s t b i e n connu que,
grace soit
au d 6 v e l o p p e m e n t r a p l d e
n i q u e s n u m & r i q u e s a p t e s & r 6 s o u d r e des probl&mes t e c h n o l o g i q u e s qu&s, g r a c e s o i t
au d ~ v e l o p p e m e n t de la t e c h n o l o g i e
w a r e ) des c a l c u l a t e u r s
modernes,
donner des r&ponses v a l a b l e s , la c o n d u i t e
et
le c o n t r ~ l e
aupr&s de d i f f 6 r e n t s
ces de I ' i n d u s t r i e pression
d'une s&rle
crest
de ta c o m p ~ t i t l o n ,
D'ici
la n & c e s s i t ~
rats
technologlques
qui
o5 b i e n c ' e s t
d6cld&ment
sophistiqu~e
tr&s
importants
se r a t t a c h e
aux e x i g e ~
t y p e de p r o c e s s u s avanc~ p u i s q u e , bien & partir
des s o l u t i o n s
de la r ~ a l i t ~
d'iei,
au t e c h n i
classiques.
habltuets
physique bien d'ava~
de r o u t i n e .
Cela e n t r a ~ n e
des mSthodes de c a l c u l
que de la f i a b i l l t & . les d e r n i e r s
conv~
de m e n t i o n n e r , En p a r t l c u l i e r ,
temps a u p r ~ s du s e c t e u r
e t de I ' O c h a n 9 e t h e r m i q u e .
de p l u s en p l u s
la
le n & c e s s l t & d ' u n e r & a l i s a t l o n
des modules q u ' o n v i e n t
¢es e x i g e n c e s se s e n t r ~ v $ 1 ~ e s dans
souvent c'est
le p r o j e t ,
de mod&les m a t h ~ m a t i q u e s d ' a p p a -
~ la d i s p o s i t i o n
de r u e de la p r & c l s l o n
I~ s o n t en e f f e t
qui
Iointaines
s'approchent
des mod&les f l u l d o - d y n a m i q u e s
de
marquent de p l u s en p l u s c l a i r e m e n t
aux r a p p r o c h e m e n t s
n a b l e s & la s t r u c t u r e du p o i n t
la s t i m u l a t i o n
de la mlse au p o i n t
la n 6 c e s s i t O aussJ d S a v o i r
soit
pour
avanc~e pour de b u t s s p 6 c i a u x , q u l s ' i m p o s e n t
c i e n s p o u r des s o l u t i o n s
ta9e par rapport
et hard
~ present,
de vue n u m S r i q u e , d'appareillages
compt i
de l ' i n d u s t r i e .
moderne, qui
teehnologiquement
e s t devenu p o s s i b l e ,
du p o i n t
secteurs
Plus pr~cisement,
il
(soft-ware
des t e c ~
importantes pour p a r exemple dens
Ces q u e s t i o n s
l'anatyse
de cheque
le g~nJe ¢ h i m l q u e ,
e t non pas t e l l e m e n t
de I ' a s p e c t
p~
367 rement c h i m i q u e du p r o b l ~ m e , lyse d~taill~e
ob I ' o n
doit
combattre,
des ph~nom~nes de t r a n s p o r t ,
~ G r a v e r s une an~
la b a t a i l l e
du rendement e t
de la f i a b i l i t ~ . Malheureusement
Iorsqu'on
les modules c o n c r e t s , fiabilit6, bien
le c h o i x
I& la n a t u r e
parle
on ne c o n n a i t
du probl&me-m~me,
des c o n s i d 6 r a t i o n s
toujours,
pas a s s e = ,
des m6thodes g 6 n 6 r a l e s
de la m a t h 6 m a t i q u e e x p ~ r l m e n t a l e dologie
de m6thodes de c a l c u l
ainsi
qui,
techniques
bien naturellement,
tr~s
du p o i n t
optimales;
concernant
de vue de la
e'est
que I # e s p 6 r l e n c e
~ la f i n r
toutefois du g 6 n i e e t
en impose s u r
g6n6rales,
importantes
ees d e r n i S r e s
la m6th~ 6rant
p o u r une r u e d ' e n s e m b l e
des choses. II sufflt de mentionner concret,
qui naissent
des conditions
que des non-lin~airit~s, ~quations,
et encore,du
pos du comportement
les difficult~s, aux
llmites,
du type d'accouplement degr~ d'information
~ prevoir en avance
lequel d~pend,
naturelIement,
l~on a pr~fer~
dans cet~e communication,
ees difficult~s
aupr~s de I'ARS dans d'appareiIIages
1.
MOldllele
puisque
n'~tait pas teIlement
du r~seau d'apr~s II suit de I~ que
I~analyse th~orique signifieative,
d~velopp~s
de se
num~riquement
la suite.
de p o l y m ~ r i s a t i o n
du pr,9,,b,l~me le comportement
depolym~risation
lange partieIJement
thermo-fluido-dynamique
inf~rieure
est exothermique
d'un r~acteur
~ flux continu en r~gime stationnaire.
polym~rls~
entre dans
teur, et apr~s avoir ~t~ compI~tement
r~action
et
des solutions pour des projets
qu'ont ~t~ r~alis~s par
On a ~ t u d i ~
section
m~me si approximatif,
Gout solution precise.
le but d'obtenir
les diff~rentes
l'on peut obtenir ~ pr~
poulr,,,,le c o n t l r ~ l e tlhermiqu,e ' d ' u n r ~ a c t e u r
a) D e s c r i p t i o n
tubulaire
de [a nature sp~cif~
I'~pesseur
r~ferer ~ un certain nombre de cas, concrets,
~ I'~gard du cas
entre
que
"a priori" de la solution,
enfin de la difficult~
concernant
toujours
polym~ris~,
en sortie du r~acteur.
et d~pend fortement
a lieu. ke polds moI~culaire
la partie sup~rieure
Un m~
du r~ae-
on le pr~I~ve de la
La r6action
de polym6risation
de la temperature
~ laquelle
la
moyen du produit
obtenu est
inve~
368 sement p r o p o r t i o n n e l
& la t e m p e r a t u r e d e r & a c t i o n .
Le but de c e t t e nir
~tude a 6t~ de v 6 r i f i e r
un c o n t r 6 1 e t h e r m i q u e du r 6 a c t e u r
bles distributions L'output
b)
la p a r o i
& Iralde
d'obte-
de convena
e x t e r n e du r & a c t e u r .
donc dans la d & t e r m i n a & i o n
( ~ , p) (champ de mouvement), C i ( i ces p o l y m ~ r i q u e s ) ,
est possible
satisfaisant
de t e m p & r a t u r e s sur
du mod&le c o n s i s t e
s'il
= 1, 2; c o n c e n t r a t i o n
des f o n c t i o n s des deux esp&
T (champ t h e r m i q u e ) .
Mod&le math6matique ta 9 & o m ~ t r i e du syst&me e s t de t y p e a x i a l - s y m ~ t r i q u e
1.1).
Les ~ q u a t i o n s qui d & c r i v e n t I r ~ t a t
des 6 q u a t l o n s h a b i t u e l l e s
~/ ( f o n c t i o n
de c o u r a n t )
et
z) ( F i g .
du syst&me sont c o n s t i ± u 6 e s par
de c o n s e r v a t i o n
I r & n e r g i e e t de Inesp&ce c h i m i q u e ,
(r,
de la masse, du momenL, de
les q u e l l e s ,
rJ ( t o u r b i l l o n )
ont
en t e r m e s des v a r i a b l e s la forme s u i v a n t e :
(:.:)
t e r m e s de c o n v e c t i o n
t e r m e s de
diffusion
source
r me s d e
diffusion
source
t e r m e s de
diffusion
source
diffusion
source
"~-e
t e r m e s de
convection
3~
-3"7
5g
termes
3c
* de
convection
o~ :
C~=
r"
_~Z 3~
DrY D~
termes
de
369 composants s e l o n r ,
z de la v i t e s s e
v
densit6 viscosit6
dynamique
Cp
chaleur
sp6cifique
T K
temp6rature constante cln6tique conductibillt6 thepmlque chaleup de p6action
AH
fraction
C I3
de la masse du monom&re
coefficient
de d i f f u s i o n
A' ces 6 q u a t i o n s dependence ( 1 . 6 ) rh6ologique) piable
et
il
faut
joindpe
de la v i s c o s i t ~
IS6quation d~6tat
par r a p p o r t
la dependence ( 1 . 7 )
(1.5)
~ la v a r i a b l e
et
la
T, c ( I o i
de la c o n s t a n t e c i n 6 t i q u e
de l a v a
T.
(1.s)
(1.6)
(1,7)
,
soo
Les c o n d i t i o n s
aux l i m i t e s ,
se P 6 f 6 r a n t & le F i g . V=O
Z:O
Z--H
~=0
/
a s s o c i & e s au syst&me
(1,1)
1.1 s o n t -
~___ -_
0.,
des
_ rat
"~4; '
C= C% " T = T ~
-
----
(assign&es)
-
(1,7)
et
370
OC .0
T=T~(~)
(distrlbution
O~
c)
ture
assignee de la tempe~a-
~ la p a r o i )
M~thode de s o l u t i o n
Le syst&me ( i . 1 ) s~ en u t i l i s a n t tion,
tandis
a~e~ le~ c o n d i t i o n s en c o n t r e v e n t
que l e s d i f £ ~ r e n c e s c e n t r a l e s
t e r m e s de d i f f u s i o n , non-lin~aires
(1.9)
- (i.7),
les d i f f e r e n c e s
ayant
On a a i n s i
(1.8)
a ~t~ d i ~ t i
dans l e s t e r m e s de convec
o n t St~ u t i l i s S e s
pour
les
o b t e n u un syst~me d ' ~ c l u a t i o n s a l g ~ b r i q u e s
la forme s u i v a n t e
(Fig,
I ):
~P;, = C,%~ ~÷,,~+ C=;, (tOg~÷~÷C~4k~_,,z+C4~,2L~;,~_i+ ~
05 f
peut reprgsenter
ture
et
la f o n e t i o n
de c o u r a n t ,
le t o u r b i l l o n ~
a ~t~ r ~ s o l u
it~rativement
la t e m p ~ r ~
la c o n c e n t r a t i o n .
Le systSme a l 9 ~ b r i q u e
(1.9)
~ l'aide
de
la m~thode de G a u s s - S e i d e l , E t a n t donnSe la t r & s qui
concePne la v i s c o s i t $
forte et
d~pendence de la t e m p e r a t u r e p o u r ce
la v i t e s s e
de r ~ a c t i o n
de ¢ o n v e n a b l e s t e c h n i q u e s de s o u s - r e l a x a t i o n ee de la s o l u t i o n
d) R ~ s u l t a t s
la p a r o i
la d i s t r i b u t i o n
la c o n v e r 9 e n -
obtenus
La p r e m i & r e a p p l i c a t i o n
sur
pour o b t e n i r
num~rique.
un r ~ a c t e u P & d i m e n s i o n s ture
on a d~ i n t r o d u i r e
de la m~thode de c a l c u l
industrielles
comme d ' a p r S s
de ~ ,T , c,
On p e u t r e m a r q u e r
la F i g .
1.2.
sont report6s
I'existence
I ' a x e dans la zone de s o r t i e
ayant
a 6tg r6alis6e
la d i s t r i b u t i o n Les r 6 s u l t a t s
en F i g .
1.2,
sur
de temp6r E obtenus pour
1.3.
d ' u n e zone de r e c i r c u l a t i o n
p r o s de
du r 6 a c t e u r .
En c o r r e s p o n d a n c e de c e t t e
zone i l
y e un maximum dans la d i s t r i -
371 bution
de t e m p 6 r a t u r e d o n t
I'on obtiendrait
dens un r ~ e c t e u r
La c o n c e n t r a t i o n la p a r o i , par
la v a l e u r
tandis
retie
qu'elle
r~lative
est ~gele & celle
adiabetique,
r~guili~rement
dens la couche q u i
est remarquablement diff&rente
l'effet de l'augmentatlon
que
de temp&rature
adore
pr&s de I ' a x e
qui se produit
dens cet en
droit. Dens le b u t de d i m i n u e r du r ~ a c t e u r ,
on e p r i s
de la p a r o i
les r & s u l t a t s
une d i s t r i b u t i o n
o b t e n u s on p e u t r e m a r q u e r que
p r o s de la p e r o i ,
& l e v & e , une v i t e s s e polym&risation rature
de t e m p e r a t u r e
a y a n t e une d & r i v 6 e a x i a l e
e s t p a r t a g & e en deux p a r t i . e s e y e n t
premi&re pattie,
mineure
est caractAris&e
du monom~re, e l l e - e u s s i
de la p a r o i
on a caus~ donc
oblig& d'6couler
la r ~ g i o n de
un c o m p o r t e m e n t d i f f e r e n t .
dt&coulement plus petite
pPesque s t a g n a n t e q u i est
croissante,
de t e m p & r a t u r e & l ' i n t & r i e u r
ligne b).
D'apr&s flux
en c o n s i d & r a t i o n
lin~eirement
1.4 -
(Fig,
la v a r i a t i o n
et
mineure.
La
p a r une v i s c o s i t $
une e f f i c i e n c e
en ta
En d i m i n u a n t
la t e m p e -
la c r o l s s a n c e d ' u n e couche de f l u i d e
edh&re & la p a r o i
externe.
Presque t o u t
pr&s de I ' a x e evec un temps de s & j o u r
le f l u i d e
insuffisant
p o u r une c o m p l & t e p o l y m ~ P i s a t i o n du monom&re, On a p r i s de la p a r o i
en c o n s i d & P a t i o n d ' a u t r e s
sans o b t e n i r
c e u x que l ' o n
vlent
des r ~ s u l t a t s
de d & c r i r e .
tefois
o b t e n u e en c o n s i d & r a n t
rieur.
D'apr&s
t.4,
1.5,
ration
la d i s t r i b u t i o n
les r ~ s u l t a t s
PegulieP d&sir&,
qui
dill&rent
Une a m & l i o r a t i o n
de t e m p e r a t u r e s e n s i b l e m e n ~ de
r e m e r q u e b l e a &t& t o u
un r & a c t e u r avec un d i a m & t r e 6 f o l s de t e m p & r a t u r e de la p a r o i
la zone c e n t r a l e
la c o n c e n t r a t i o n
du r & a c t e u r e s t t r ~ s
du monom&re p r e n d e
inf&-
r e p o r t ~ e en F i g .
o b t e n u s m o n t r e n t comment, dens ce c a s ,
de t e m p & r a t u r e dens
eL, p a r c o n s e q u e n t ,
distributions
I'augme~ limit~e
le c o m p o r t e m e n t
372
2. U I t r , a , g e n t r i f u g e (x)
a) D e s c r i p t i o n
du p r o b l ~ m e
On a ~ t u d i ~ tlons
le comportement
stationnaires,
nulaire
?luidodynamique et thermique,
d'une ultracentri?uge
axial-sym6trique
?erm6 ~ I ' e x t r e m i t 6
mouvement de c o n v e c t i o n
~ I'int6rieur
(Fig,
ment du m~lange d o n t veulent
d~terminer:
appliqu~e
entre
Le
est provoqu~
l e s deux c o u v e r -
la p r 6 s e n c e de I ' a l i m e n ~ a t i o n
les composants,
les [ o n c t l o n s
des c o u c h e s
iimites
T,
dans
de I ' u l t r a c e n t r i f u g a t i o n .
p (champ de mouvement),
de I ' ~ t u d e
a ~t6 celui
la r ~ g i o n
que l e s e a r a c ~ 6 r i s t i q u e s
e t du p r 6 1 g v e -
a y a n t un p o i d s m o l ~ c u l a l r e
se s 6 p a r e r p a r e f f e t
Le b u t p r i n c i p a l e
ainsi
p a r deux c o u v e r c l e s .
2,1).
On n ' a pas c o n s i d ~ r ~
rent,
par un c o r p s a n -
de I ' u l t r a c e n t r i ? u g e
p a r une d i ? ? 6 r e n c e ~ T de t e m p e r a t u r e cles
constitute
en cond~
la p l u s
di?f6On v e u t
T (champ t h e r m i q u e ) .
de d ~ t e r m i n e r
int~ress~e
I'extension
& la r e c i r c u l a t i o n ,
des composantes de la v l t e s s e .
b) ModUle m a t h 6 m a t i q u e
La g ~ o m 6 t r i e Contrairement a consider6 qul
au cas p r e c e d e n t ,
aussi
d~erivent
du syst~me e s t
de nouveau a x i a l - s y m 6 t r i q u e
(r,
z).
e t ~ cause du mouvement de r o t a t i o n ,
la composante s e l o n
le syst~me s o n t c e l l e s
0
de la v i t e s s e .
de c o n s e r v a t i o n
on
Les 6 q u a t l o n s
de la masse, du
moment e t de 1 " 6 n e r g l e . Elles
ont 6t6 d~crites
se a n g u l a i r e par r a p p o r t pos6 e n t r e
p a r un syst~me de r o t a t i o n ,
de I ' u l t r a c e n t r i ? u g e , ~ la s o l u t i o n
e t en o u t r e
o b t e n u e en
elles
avec
o n t ~t~
absence de g r a d i e n t
la v i t e s lin6airls~s
thermique
les plats.
Les ~ q u a t i o n s c o n s l d ~ r 6 e s
s o n t done:
(x) - Travail execut6 sous contract avec le C N E M l'Energia Nucleate), R o m a .
(Comitato Nazionale p e r
im
373
(2.2) termes de diffusion
source
t e r , rnes de d i f f u s l o n
source
termes
SOUPCe
(2.3)
(2.4)
(2.5) C,,~o
de d i f f u s i o n
3E" ÷ ~ o~ 4
termes
v
de c o n v e c t i on
~r
or
'
c o m p o s a n t e s de pressiont
~"Z"
t e p m e s de d i f f u s i on
I
?o ,90To
l
la vitesse
denslt~,
selon
temperature
T= Tort
rl de
source
I
~ , ~. It~et
statique
(~ T = 0)
viscosit~ C~
chaleur
sp~cifique
conductibilit~ oJ
thermique
vitesse
an9ulaire
vitesse
de r S f 6 r e n c e
dW~tatn p o u r
Comme ~ q u a t i o n que:
(2.6)
I
T'
~ = - ~o-~o
I t a c h S v e m e n t du s y s t ~ m e r
on a s u p p o s ~
374 les c o n d i t i o n s
aux l i m i t e s
a s s o c i & e s sonG:
9T
~_--O
JY
r=R E=O
(2.7)
x i = ~Y-_ ~ .
T'%
0
+ AT 2
T': To. AT 2
c)
M6thode de s 0 1 u t i o n
Le s y s t ~ m e d ' & q u a t i o n s tes
(2.7)
a &t& r & s o l u
(2,1)
- (2,6)
en c o n s e r v a n t
ave¢
les conditions
les v a r i a b l e s
aux
naturelles
limi-
u, v, w,
p', T'. Comme r&seau d'int&gretion 2.i, ob
les composantes
on e aonsid&r&
le r&seeu montr& en Fig.
radiales et a×iales sont d&plec6es
pas par rapport eu point central
ob
l'on calaule
d'un demi-
[a pression
et
le te~
p&rature. Le proc&d& adopt& pour etteindre
la solution pr~sente
les phases
suivantes: i)
Le & q u a t l o n s song d l s c r 6 t i s & e s di{{~rences Lion
2)
en c o n ~ r e v e n L ,
avec
pour
les dlf£&rences
les termes
centrales
de c o n v e c t i o n
et
les
de IS6qua -
(2.5).
L'&quaLion
(2.3)
pour
vent par rapport
la v l t e s s e
azimu±ale est elimin&e
& la composanLe r a d i a l e
en la r&so&
e t en la s u b s t l t u e n t
dens
la ( 2 . 2 ) . 3) Pour cheque c e l l u l e pour
les p o i n t s
le p o i n t 4)
de c a l c u l
(i-½,
g6n&rique
(i,
j),
(i+½,j),
pros
I ' u n e de l r a u t r e
A q u a t i o n s dans Lion
tin&aire
I'&quation qui
les 4 ~quations
(i,j-½),
(i,j+½)
du moment
qui
entourent
j).
En s u p p o s a n t que l e s v a l e u r s lules
on d & c r i t
donne p"
des d l f ~ r e n t e s solent
connuesl
de c o n t i n u i t & ,
i,j.
variables
dens
e~ en s u b s t l t u a n t
on o b t i e n t
les ce~ ces 4
une s e u l e &qua-
375 . dans les 4 & q u a t i o n s du moment, emplo irj l e s v a l e u r s des 4 v i t e s s e s r e l a t i v e s .
5) En s u b s t i t u a n t
la v a l e u r
y6es on o b t i e n t 6)
Lr6quatlon Iteide
7)
p'.
de la t e m p 6 r a t u r e
a 6t6 r6solues
cellule
par c e l l u l e u
de la m6thode de G a u s s - S e l d e l .
Le p r o c 6 d 6 e s t r e p 6 t 6 recouvrent
it6rativement
pour r o u t e s
le domalne d ~ i n t 6 9 r a t i o n ,
ce de la s o l u t i o n
jusqu'&
les c e l l u l e s
obtenir
qui
la c o n v e r g e D
num6rique.
d) R 6 s u l t a t s
Le mod$1e m e t h 6 m a t i q u e a &t& e p p l i q u 6 anguleirer
nus, concernant
la v i ~ e s s e axiale e t azJmutalef ainsi que la temp6ra-
(en forme a d i m e n s i o n e l l e ) ,
Dans la p r e m i & r e f l g u r e axiale
en f o n c t i o n
f6rentes
valeurs
de la d i f f 6 r e n c e duit
le r a p p o r t
~ = 10.
tournante
& haute vitesse
Lure
avec
~ une c e n t r i f u g e
s o n t m o n t r 6 s en F i 9 .
on r e p o r t e
de la c o o r d o n n 6 e a x l e l e .
Par e f f e t
qui
de le v i t e s s e de 3 d i f
de la r o t a t i o n
les c o u v e r c l e s ,
interf&re
obt~
2 . 3 eL 2 . 4 .
en c o r r e s p o n d a n c e
impos6e e n t r e
un mouvement de r e c i r c u l a t i o n
2.2,
le comportement
de le c o o r d o n n 6 e r a d i a l e
de t e m p 6 r a t u r e
Les r 6 s u l t e t s
sur route
il
et se p r ~
la r 6 g i o n
dtint6gratlon. On p e u t c o n s t a t e r la p a r o i
externe
air
aussi
comment
une e x t e n s i o n
la couche tr&s
petite
limite
dynamique pr&s de
par r a p p o r t
au r a y o n de
la c e n t r i f u g e . Dans la deuxi~me f i g u r e la v i t e s s e
azlmutale
sont report6s
eL de la t e m p & r a t u r e ,
les c o m p o r t e m e n t s r61atlfs
radiaux
de
aux m~mes v a l e u r s
de la c o o r d o n n 6 e a x i a l e . Dans la zone c e n t r a l e , avec r dans re.
la v i t e s s e
la , r 6 g i o n oO la v i t e s s e
En s ' a p p r o c h a n t
un maximum a s s o c i 6 s
de la p a r o i
azimutale axiale
lat6rele
au c o r r e s p o n d a n t
augmente
reste lav
pratiquement
pr6sente
comportement
lineairement
de w.
eonstan-
un minimum eL
376
Lrexamen du comportement de la temperature montre t e r m e s de c o n v e c t i o n r
qui
d~trulsent
la s y m ~ t r i e
Irimportance des
par rapport
au p l a n
moyen de la c e n t r i f u g e .
3. D,i s ~ r i b u t i o n
de la ch,ar~e d,ans un 6 c h a n g e u r de c h a l e u r
a) ,D,es,c,ription ,d,u' pro,blame
On a ~ t u d i 6
le champ de mouvement dans le c i r c u i t
6changeur cylindrique
~ falseau×
pr61~vement du f l u i d e
ont
de
lieu
de t u y a u × ( F i g .
sur route
la p a r o l
principal
3.1).
drun
Lrentr6e
la±6rale
et
le
limltrophe
Ir~changeur. A Itlnt~rleur
yau×, met
il
de I ' ~ c h a n g e u r ,
y a une s S r l e
la d i s t r i b u t i o n
distrlbutions
de
de g r i l l e s
de la c h a r g e .
outre
la p r e s e n c e
radiales
ayant
du f a i s e a u
la f o n e t l o n
de t u -
d'unifor-
Le module m a t h ~ m a t i q u e d o i t
donner
les
V, P (champ de mouvement).
b) ,M,lO,dele math,elm,a,tlque Lteffet geur,
q u i e s t dG ~ la p r e s e n c e du f a i s e a u
a ~ t 6 s i m u l ~ en s u p p o s a n t que le f l u i d e
de la v i s c o s l t g
e t que l e s p e r t e s
axiale w soient
charge
principal
soit
de la c h a r g e en d i r e c t i o n
donn6es par des c o e £ ¢ i c i e n t s
La p r e s e n c e des g r i l l e s
de t u y a u x dans
d6pourvu radiale
diff~rentsentrteux
a ~ t ~ s l m u l 6 en i n t r o d u i s a n t
It6cha~
et
et connus.
des p e r t e s
de
Iocalis~es,
Pour ces h y p o t h e s e s , ment song
les ~ q u a t l o n s
les suivantes:
V
3~
a~
qui
d~crivent
le champ de mouv~
377
K,. ~
(3.2)
R
termes
3"0"
(3,3)
~
de c o n v e c t i o n
30" +
"u'- ,~'0-
i&i"
SOLIPCeS
=
4
DP
R
2 J
,de c o n v e c t i o n
termes
sources
o6
.,tz,i ~"
?
composantes
de
la v i t e s s e
pcession
Krqz } K~ ~i~
co&fficients co~fficient densit&
pour
te c a l c u l
de la p e r t e
appamante =
Les c o n d i t i o n s
au×
de
la p e r t e
local is6e par
de c h a r 9 e Ies grilles
~(3#l.i~e. V#~.;~e
l imites
z=O
0
~
z =H
0
~ r~
r=O
0
~
P =R
0
Zzgh
soni:
r~R
les s u i v a n t e s
v=O R
v=O
z~H
u=O 1
h2 g z ~ h3
u =0
h4~z~
h1 ~ z ~ h3 ~
h2
z ~ h4
U = U.(z) !
O._.~u = 0 Or
assign&e
378
c)
M6thode de s o l u t i o n Les 6 q u a t i o n s
dens
ont 6t6 discr6tis6es
ont 6t6 r6solues
en u t i l i s a n t
le methode Marker e t
(MAC). Les t e r m e s de c o n v e c t i o n
f6rences
ont 6t6 discr6tis6s
directement diagonal
en u t i l i s a n t
des d i f
en c o n t r e v e n t u
LW6quetlon de P o i s s o n p o u r la m a t r l c e
la p r e s s i o n
des c o 6 f f l c l e n t s ,
a 6t6 r6solue
~ Itaide
en i n v e r s e n t
de I r a l g o r i t h m e
tri-
~ blocs.
La s o l u t i o n de la s o l u t i o n
d) R 6 s u l t a t s
stetionnalre
a 6 t 6 o b t e n u e comme l l m i t e
obtenus
On p e u t r e m a r q u e r du f l u x
asymptotique
transitoire.
Un example de d i s t r i b u t i o n
tion
le r 6 s e e u d 6 c r i t
le p a r a g r a p h e p r 6 c 6 d e n t . Les 6 q u e t i o n s
Cell
en e m p l o y a n t
que dens
coincide
les dues ~ la s o r t i e de ces d e r n i ~ r e s
avec
de le c h a r g e e s t r e p o r t 6
le r 6 g i o n
la d i r e c t i o n
et A Itentr6e
entr6esl
centrele axiale,
en F i g ,
de I t 6 c h a n g e u r ,
3.2.
la d i r e ~
les composantes r a d i a -
6rant presentesseulement
tout
prSs
379 BIBLIOGRAPHIE
ProblSme 1 G. A s t a r i t a ,
R. Sala, A. Tozzi,
"StyrSne polymerization G. A s t a r i t a ,
R. S a l a ,
"A p a r a m e t r i c drical
reactor",
A. T o z z i ,
F. Valz Gris ARS
RT70/35 (1970)
F. Valz Gris
study f o r s t y r e n e p o l y m e r i z a t i o n
r e a c t o r s having c y l i n -
and annular geometry", ARS RT71/5 (1971)
A. D. Gosman, W. M. Pun, A. K. Runchal, "Heat and Mass T r a n s f e r
in R e c l r c u l a t l n g
D. B. S p a l d i n g ,
M. W o l f s h t e i n
F l o w s " , Academic Press,
London
(1969)
Probl~me 2 L. S. C a r e , t o ,
A. D. Gosman, D, B. S p a l d i n g
"Removal of an i n s t a b i l i t y
in a f r e e c o n v e c t i o n p r o b l e m " ,
EF/TN/A/35
(1971) R, S a l a ,
A, T o z z i ,
F, Valz Gris
" S o l u z i o n e numerlca del eampo di moto d e l l e u l t r a c e n t r i f u g h e
a control
Io termico", ARS RT 72/40 (1972) R. S a l a ,
A. Tozzi
"Comportamento f l u i d o d i n a m i c o ARS RT 72/38
delle ultracentrifughe
a controllo
termico"
380 Probl~me,,3
F. H. Harlow, J. E. Welch "Numerical calculation of time-dependent of fluid
with free surface",
viscous
incompressible flow
Phys. F l u i d s 8, 2182 (1965)
L, Biasi~ A. Colombon A. Tozzi "Codice per
la s o l u z i o n e di problemi di f l u i d o d i n a m l c a
re di c a l o r e " ARS RT
73/11
(1973)
in uno scamblat~
N
+l
C,. t..
"O
r..
IL
C..
p~,,,
.J
r.... i
382
i-
0 ,,
383
U
e~
~
o
I"-I
N
t
I
T
110
..........F i g .
.
.
.
6
.
.
.
Tw
385
,,
,
,,
,
,
O
b3 !
t ~
b..
N
o.
t~
"fl
I
~
Z
¢m
~
+
_c~
~
~ ¢m l
om 0 ¢N +
¢w.
!
tq
,ll
387
0
/ i I
iE
l,i
~J
O'
/
"7
¢0
V/vo 6
y
2
/ r - Rt
/ J
R - RI
I
Fi9. 2.3
389
z/,
/
T'=.2
T~==I
I
T'=.O
T""
- .1
J
T'= - . 2
0
J
J R-R1
0 Fig, 2 , 4
H
0__~ ~
h3
f grids
\ h2
l
i ~-
E
r
R
Figo 3,1 t
~
.
.
.
.
.
.
.
.
.
.
39t
I I ! ].0~.02
\.2
i/~////////i//////i//// F I g , 3.3
~--
ETUDE NUMERIQUE DU CHAMP MAGNETIQUE DANS UN ALTERNATEUR TETRAPOLAIRE P A R L A METHODE DES ELEMENTS FINIS
R. Glowinski Universit6 Paris VI A. Marroceo IRIA - LABORIA
I - INTRODUCTION Nous pr6sentons ici, l'analyse num6rique du probl+me de la r6partition de l'induction magn6tique dans une machine tournante (alternateur t6tmpolaire). L'6tude est faite pour une coupe transversale m6diane de ta machine, ce qui ram~ne le probl~me, qui en toute ggn6ralit~ est tridimensionnel, ~t un probl~me bidimensionnel. Le ph6nom~ne 6tudi~ est stationnaire. L'approximation des 6quations pour le calcul effectif par ordinateur a ~t6 faite par la technique des 616ments finis. L'61gment fini de r6f6rence retenu est l'61gment triangutaire de degr~ 1 ; ce choix correspond 6galement g l'approximation suivante :dans le fer, la perm6abitit6 mag~tique sera constante sur chaque ~16ment. La r~solution num6rique du syst~me alg6brique non tin~aire r6sultant a 6t6 faite par des m6thodes it6ratives de surr61axation ponctuelle. Les N °s 2, 3, 4, 5 , 6 reprennent succintement (avec une approche un peu diff~rente au n ° 2) tes consid6rations d6vetopp6es dans R. G L O W l N S K I - A. MARROCCO [8] avec au n ° 7 des d6tails suppl6mentaires sur la raise en oeuvre pratique; des r6sultats num6riques partiels sont donn6s au n ° 8. II - PRESENTATION DU PROBLEME Rappelons les 6quations de Maxwell de la magn6tostatique (2-1)
rot H = j
(2-2)
"B
= . -->
(2-3)
divB
= O
I~ est le vecteur champ magn6tique et de Maxwell-Ampere. B
j le vecteur densit6 de courant; l'6quation (2-1) est la relation
est l'induction magn6tique et ta la perm6abilit~ magn6tique; l'6quation (2-2) donne la relation entre ..~ .+
te champ magn~tique et l'induction; ta relation entre B e t /s = t%
H est lingaire dans l'air car
= 4 n 10"7MKSA, mais elle devient non lingaire dans le fer car la perm~abilit6 ta est elle -m6me
une fonction de
IH 1. La relation (2-3) exprime la conservation du flux d'induction.
393
La relation (2-3) permet d'introduire le potentiel vecteur (2.4)
~
=
A = (A 1 , A 2 , A 3 ) li6 ~ ff par
rot
Si nous utilisons (2-4), les relations (2-1), (2-2), (2-3) deviennent (2-5)
rot(vrot
~ ) = ~"
avec
v = 1 /1
Le probl6me ~tudi~ est bidimensionnel; certaines consid6rations physiques permettent de choisir dans ce cas particulier, le potentiel vecteur A comme_~un vecteur n'ayant qu'une composante non nutle A 3 (le courant j est aussi donn~ sous la forme j (2-6)
- ~ l: ~
~- (v ~ A 3 ) = ~Xi ~ Xi
= (0, 0, J3 )' si bien que l'6quation (2-5) s'6crit dans ce cas j~
REMARQUE 2-1
La relation (2-4) ne d6termine pas uniquement A ; pour avoir cette unicit6 il suffit par exemple [ cf.0] d'imposer une valeur ~t div A (par ex. div A = 0 se trouve implicitement v6rifi6 parnotre choix du potentiel vecteur; L'6quation (2-5) ou (2-6) est valable th6oriquement dans tout le plan. D'un point de vue pratique, on se limite ~ u n domaine born& La figure 1 indique le domaine retenu; nous avons consid6r~ une <> ~t l'ext6rieur du stator, constitu6e par de Fair 6videmment. Sur le bord ext6rieur de cette bande nous imposons la condition A 3 = 0. Cette condition entraine en particulier que toutes les lignes de champ (ou d'induction) sont contenues dans le domaine consid6r6; et donc que le ph6nom~ne magn6tique se trouve confin6 au domaine choisi. Les r6sultats num6riques obtenus montrent en fait que l'on aurait pu imposer cette condition A = 0 sur le bord du stator. Cette condition aux limites entraine l'unicit6 du potentiel vecteur comme nous le verrons en III et IV. REMARQUE 2-2
La machine effectivement 6tudi6e est un alternateur t6trapolaire; si l'on tient compte des conditions d'antip6riodicit6, il suffit de n'6tudier qu'un quart de la machine. L'6tude th6orique va 6tre fare sur le domaine entier, il faudra 6videmment modifier 16g6rement le cadre fonctionnel si on veut prendre en compte ces conditions d'antip6riodicit6. R E M A R Q U E 2-3
Pour le probl6me bi-dimensionnel 6tudi6, les lignes d'induction dans la machine sont donn6es par les lignes 6quipotentielles (ou lignes de niveau) de la fonction A 3 (x I , x 2 ) III - LE MODELE MATHEMATIQUE 111.1. Equations aux d~riv6es partielles Le domaine [2 consid6r6 est un cercle dont le bord est not6 F. La d6termination du champ magn6tique (ou de l'induction magn6tique) dans ~2, se ram~ne dans la cas pr6sent,/t la r6solution de l'&tuation aux d6riv6es partieUes
10x, ou x = ( x 1, x 2) avec les conditions aux lirnites
(3-2)
A = 0 sur F
~x 2
]
= j dans ~2
m
av1
!!~,k~z~ . r ~ ~Q
~Q
~0
~o
~k.ILr~ii ~i!
dJ
r
~ r~
c~
395
Nous avons indiqu6 la d6pendance de v par rapport ~ la variable d'espace x et A v ( x , A ) = v0 = t s i x estclansl'air mais v d6pend non lin6airement de l'induction, et donc de A par (2-4) lorsque x est dans le fer. t11-2. Fonctionnelle d'6nergie I1 y a 6quivalence entre la r6solution de (3-1), (3-2) et la minimisation (sur un espace h pr6ciser) de la fonctionnelle d'6nergie (cf, [1 ] [2 ])
ou rappelonsqe (3-4)
B = rot A
avec A = (0,0, A)
L'6quation d'Euler du probl~me de calcul des variations redonne (3-1) Pour le fer nous tirons la fonction v des caract6ristiques magn6tiques du stator du rotor et nous l'exprimons comme fonction de IBI- Ceci va simplifier l'6criture de la fonctionnelle d'6nergie (3-3), qui devient, si on note ff la primitive de v consid6r6e comme fonction de IBI- qui s'annute pour ta vateur 0, (3-5)
~ (A)= 21f~
~(x, lBl2)dx-~
~ (A) =
~ (x, trot AI ) dx -
jAdx
soit aussi (3-6)
j Adx
ou encore (3-7)
~(A)=
1 f~
ff(x, l g r a d A l 2 ) d x - f ~
jAdx
puisque dans le cas bidimensionnel on a iB'~ = Irot ~1 = lgrad AI 111.3. Approximation de v L'inverse de la perm6abilit6 magn6tique peut s'6crire v=v
0
vr
OU v0 -
1 7 MKSA 4rclO_,
et vr e s t la valeur relative par rapport ~ Fair; nous avons 0
(3-8)
4
vz(lBI ) = a 4 " ( 1 - a ) ( I B I ) (IBI)+ 3
Voir sur les figures [ 2, 3] l'approximation r6alis6e. Cette approximation a 6t6 r6alis6e par une m6thode des moindres carr6s et a donn6 pour valeur des param6tres
10-4
of
10-"
10-1
0
j
i 1
I 2
I 3
Fig. Z
i 4
t ...... 5
f 6
I 7
(MKSA)
11312
APPROXIMATION,de Io CARACTERISTIQUE 5TATOR
10.4
10-3
10.2
10 J
,I
2
I
1
I
3
Fig. 3
4
, I
I
5
6
f
7
........ I,
IB I~ (MKSA)
APPROXIMATION de ,,!,q., CARACTERISTIQUE ROTOR
W 0')
397
(3-9)
~ ~ = 4,5 13 = 2,2
(3-10) t ~ = 3 ~3 = 1 , 6
10 -4 10 4
dans l e r o t o r
10 -4 10 4
danslestator
On n'a pas tenu compte de l'effet d'hyst6r6sis qui conduirait ~ des repr6sentations multivoquespour vr Nous avons dans tout le domaine ~2 (3-11)
0<e
<~5~<1
IV - ETUDE THEORIQUE DU PROBLEME IV. 1. Formulation pr6cise du probl~me On cherche A e H~o(~2) (4-1)
N(A)
solution de
~< ~ (v)
VveH~(a)
ou ~ est donn6 par (4-2)
~ (A)
= 1 Z
q~(x, lgradAI ~ ) d x - Z
jAdx
avec qJ d6fini en 111.2 (4-3) [
l
H~(~2)= ( v , Iv,~iav e L: (~2), i = 1,2 , v{F = 0 } IlVllH~(ft) = ( f ~
Igradvl 2 dx) ~/:
1V.2. R~sultat d'existence et d unieit6 THEOREME 4-t - Le problbme de minimisation (4-t) admet une solution et une seule. D6monstration On fait un changement d'unit6 tel que v = vr * la fonction ~ -, ~ (x, ~) est contractante, V x e f t d'apr~s (3-11), cela entraine que ~ est fortement continue sur Hlo (~2) * Ia fonction ~ -~ ~ (x, g) est stricternent m o n o t o n e croissante, la fonction ~ est doric strictement a~ convexe
* d'apr~s 3-11 on a f ~ (x,lgrad012 ) d x /> e/l~rad012 d x = e o' d~2
I10112 H~(a)
donc lira : (0) = +
De ces propri6t6s on d~duit par un raisonnement classique &analyse convexe (cf. [3]) l'existence et l'unicit6 d'une solution pour (4-1) 1V.3. Relation entre le probl6me aux limites (3-1 ), (3-2) et le probl6me (4-1 )
THEOREME 4-2 A solution de (4-1) est solution unique de (3-1) (3-2) dans H ~ ( ~ )
398
Ddmonstration est diffdrentiable au sens de OATEAU X sur Hi(fZ) c'est4t-dire (cf.[4]) Vu e Hi(~2) il existe ~'(u) • H'I(S2) (dual topologique de Hi (fz)) tel que V0 e Hi (S2) on a (4-4)
lim
~ (u+tO)-~
t-~o
t*o
(u)_(~,(u),O)
t
Donc si A est solution de (4-t) on a (4-5)
~(A+tO)
-~(A) ~>
Vt>o v o e Hi ( n )
o
D'oO ~ la limite (4-6) (~' (A), O) / > o
VO e Hi(I2)
En prenant t < o on trouve (4-7) (~' (A), 0) < o
VO e Hi(s2)
Ce qui donne (4-8) (~ '(A), O)
VO e H i (f~)
d'ofi (4-9)
~'(A)
= o
= o
Les formes explicites de (4-8) et (4-9) sont donn6es respectivemem par (4-10) et (4-11) (4-t0)
~v(x,A)gradA.
(4-11)
t -~i=1a'x i a (v(x,A) aA)~x i -j = o
(
ALF=
gradOdx-~
jOdx = o
VOeH:(~)
o t
A est donc bien solution de (3-1), (3-2). ~ 6rant strictement convexe sur Ho(~2), la relation (4-9) caractdrise compl~tement A. V - APPROXIMATION PAR ELEMENTS FINIS V.1. Triangulation de f~ et notations On se donne une triangulation ~'la de I2 (h dgal au plus grand cStd de tousles triangles par exemple) (5-1)
u T c Te~ h
Avec les propri6tds habitueltes si T 1 et T2 e I~ h l (5-2)
ou
T1 n
T2 =
ou
Tl e t
T2
ou
T~ et T a
ont un cSt6 commun ont unsommet commun
Les interfaces air-fer coincident au mieux avec des cStds de triangles
399
Notations
0
(5-3)
ah
=
~ Te h
,
Ph =
0ah
(5-4)
col~ = { P I P e a h , P sommet de T e~ h
(5-5)
"/h
=
(5-6)
~h
= cob + 7h
{P t P e r h , P sommet de T e~ h
)
V.2. Approximation de Ho (s2) Pour chaque nceud M e cob ( 5 4 ) on ddfinit la fonction WM de la faqon suivante W~
(V) =
i si P =
M
o siP~M (5-7)
W hM
affindsur chaque triangle
WM
= o
sur ~2 "f~h
Les fonctions W~i(x) ainsi ddfinies sont continues sur ~ ,
1
elles sont dans Ho(S2) et lindairement
inddpendantes (5-8)
Voh est l'espace engendr6 par les WM pour M e co!a
Tout 61dment Oh de Voh est compl~tement d6termin6 par les valeurs prises sur coh' En supposant les 616ments de Wh ordonn6s et index6s par i pour i = 1, 2 . . .
N h = Card (coh) = dim Voh on notera
(5-9)
Nh
O.1 =
0 h ( M i)
i = 1, 2 . . .
REMARQUE 5-1 La restriction de 0~ e Voh ~t un triangle est affin6, cela entraine que le gradient de Oh e s t constant sur chaque triangle de meme v (2-5) qui est une fonction de igradOh 12 sera constant sur chaque triangle. Un 616ment Oh e Voh s'exprime expticitement de la faqon suivante
(5-1 O)
Oh(x'Y) =Te~fh 21-'--~(T)\ (~j=I
(PT+ qTx + r 7
y) ~jT)®T (x,y)
ou a(T) reprdsente la surface du triangle T OhT e s t la fonction caractdristique du triangle T pT, q.T, rT, sont des fonctions des coordonndes des sommets du triangle T
J
J ]
0 T est la valeur de la fonction au sommet j du triangle T J V.3. Formulation du probldme approch6 L'analogue discret du probl6me (4-I) est (5-11)
trouver A h e Voh tel que (Ah) ~< ~ (Oh)
VO h e Voh
ou ~ est toujours donnd par (4-2)
400
THEOREME 5-1 Le probl6me d'optimisation (5-11 ) admet une solution et une seule caract6ris6e par (5-12)
~ (An) = aOi
o
i = 1,2...
N~
On peut montrer le r6sultat suivant
THEOREME 5-2 Si
i)
~2-~2 h-+ o
lorsque h ~ O
ii)
les angles de tous les triangles T e g h sont born4s inf6rieurement, uniform6ment en h par Oo > o o n a
(5-12)
lira Ah = h-+o
A
1
dans H~ ( ~ 2 ) f o r t A solution de (4-1)
Le probl~me de l'estimation de l'erreur d'approximation en fonction de h est Ii6 ~t celui de la r6gularit6 de la solution A de (4-1) qui est un probl~me ouvert ~t notre connaissance'. VI - RESOLUTION NUMERIQUE Le probl~me approch6 (5-1 I) peut se mettre sous la forme variationnelle 6quivalente (6-1)
t
J~
v ( x ' A h ) g r a d A~ grad~9 h d x =
j~
j 0 h dx
VOhe Voh
A. ,Vo~ V1.1. Lin6arisation Dans ces conditions, il est tr6s naturel pour r6soudre (5-t t) de songer h utiliser l'algorithme suivant qui ram6ne la r6solution du probl6me approch6 ~ celle d'une suite de probl~mes lin6aires (h coefficients variables avec x et n les inconnues 4tant bien entendu les valeurs nodales P~ir~l) (6-2)
A ° donn6 dans Voh h A n connu, on calcule A n÷l par h
(6-3,
tf ~
h
v(x,A~)grad
An÷l h
grad O dx = h
f~
jOhdx
VO e V h
oh
A n÷l e V h oh Une variante de cet algorithme est la suivante : au lieu de r4soudre compl6tement le probl6me lin6aire 6-3, on se contente d'effectuer un balayage de type surr61axation ponctuelle limit6 ~t un seul cycle. Pour ces deux algorithmes, il n'y a convergence que pour des valeurs de j (densit6 de courant d'excitation) assez petites (j ~< 0.5 A/ram: ). La divergence pour j plus grand 6tant visiblement li6e h la variation trop rapide d'une it6ration ~t l'autre du coefficient x "-~ ~ (x, A~), il a donc fallu mettre en oeuvre des algorithmes tenant plus implicitement compte de cette variation. V1.2. R4solution par surrelaxation ponctuelle non lin6aire On utilise le formalisme du paragraphe V, on notera N = N h = dim Voh et 0h = (0 t , 0 2 . . . 6quivalente
ON), d ' o 6 p o u r l e s y s t 6 m e n o n l i n 6 a i r e l a f o r m u l a t i o n
401
(6-4)
OOi (A 1 , A~ . . . .
AN )
=
i=
O
1,2...
N
V1.2-1 Description de l ' a l g o d t h m e SNL1 (6-5)
A~
donn6 dans Vola
(A~ = o
A~
connu, on calcule A~~
coordonn6e par coordonn6e par
(6-6)
(A~+, , "'" Ain.~, "- , a ~ ÷~ =
An +
par exernple )
Atn+, , Ai+l n "" . A ~ ) =
¢o (A~n÷~ - A n) 1
o o<¢o<
¢oM ~< 2
i = 1,2 .... N VL2-2 Convergence de l'algoritlune SNL1 Proposition
6-1. Pour ¢0~ suffisarnment petit, la suite A~ d~finie par (6-5), (6-6) converge vers la
solution Ah du syst~rne (6-4) I~monstration L'application 0 h ~ I~(0 h) est c ~ de fftN dans 11 et strictement convexe; par ailleurs (3-11) entraine 6galernent que la matrice hessienne ~h *
11
~
00lad j
(O h) II
est d6finie positive
11 r6sulte alors de S. SCHECHTER ([5],[6],[7]), l'existence de ~o~ , o < ~oM ~< 2 pour lequel ii y a convergence de SNLI. V1.2-3
Description
(6-7)
d e l ' a l g o r i t h m e SNL2
Ah° donn6 dans Voh A~
(6-8)
(A~ =
o par exemple)
connu, on calcule A~÷1 coordonn6e par coordonn6e par
l " " • A in÷~ t ~-0i 0~ a(-n+l _ , , A n+, i , A ~ , . .. . . o < ~ < ~ c o M ~<2
pour
A ~n)
=(l'co)
i = 1,2...
0~ ~ i (A~+~ " " An*l i - , ' A ni ' A ni+, . . . .
AN n)
N
Le probl~me de la convergence de l'algorithme SNL2 sernble ouvert sauf bien entendu, pour ¢o = 1 car SNL1
-
SNL2. Pour notre probl~rne pr6cis les deux algorithmes SNL1 et SNL2 o n t donn6 des r6sultats
6quivalents du point de rapidit6. VI.2-4 Cornpl6rnents et r e m a r q u e s sur S N L 1 e t SNL2 - Algorithme EGSN L'utilisation de SNL1 et SNL2 conduit ~, r6soudre u n e suite d'~quafions non lin6aires ~t une variable pour d6terrniner A n*l
et A n+l par (6-6) et (6-8) respectivement. Dans ces conditions, il est tr6s naturel
d'utiliser la m 6 t h o d e de N e w t o n ~ une variable et ceci conduit de faqon 6vidente g consid6rer les variantes des algorithmes SNL1 et SNL2 darts lesquelles on prend respectivernent pour A-~-a
et A T M
le premier
it6r6 de Newton avec initialisation par An. On v6rifiera que ceci conduit, en fait au m 6 m e algorithme dont la forrne explicite est donn6e par ~6-9)
A~ donn6 dans Voi~ A~
(A~ = o par exemple)
connu, on calcule A n÷l coordonn6e par coordonn6e par h
m
4~
Ctrj
C~ Z
C~
4~
o
403 a_~ A~+I ,An+1 An÷l= A~-co 00i ( " • i-I '
(6-10)
An' An1+1 . . . .
A Na
)
~( AT*' , . . . A'in~l, Ai.n , Aln1. . . . A~) aO i
o
~
2
pour i = 1,2... N O n ddmontre dans S. S C H E C H T E R (loc.cit.)qui compte tenu des propridtds de ~ ddjfidnoncdes, ily a convergence pour w ~ suffisamment petit de la suite A~ ddfiniepar ralgorithme (6-9) (6-i0), algorithme que nous noterons E G S N (Extrapolated - Gauss - Scidel - Newton) Nil - QUELQUES REMARQUES SUR LA MISE EN OEUVRE Nous devons effectuer une triangulation d'un cercle (domaine gZ); il n'y aurait vraiment aucune difficultd, s'il n'y avait/t l'intdrieur, des frontidres naturelles gt respecter, constitudes par les interfaces air-fer (voir fig.1 ). Pour la gdndration de la triangulation nous avons tenu compte de la rdpdtition de la gdomdtrie (24 dans la pattie stator et 4 dans la r6gion rotor) et il a suffit de d6couper en triangles un secteur de couronne de 15 ° pour la partie stator et un quart de cercle pour la partie rotor, il a ensuite fallu faire des r6pdtitions. I1 a donc dr6 ndcessaire d'entrer les donndes topologiques (numdro du n~eud de la triangulation correspondant au ki~m e sommet, k = 1, 2, 3, du triangle consid6r6 ; les sommets sont toujours num6rot6s dans un sens ddtermind, celui adoptd est le sens trigonomdtrique) pour 305 triangles ce qui fait 915 hombres entiers. Le calcul des coordonndes de certains noeuds de la triangulation a dfi ~tre fait avec prdcision cela pour approcher au mieux la gdomdtrie de la machine. La discr6tisation du domaine adoptde a conduit au ddcoupage de celui-ci en 3.240 triangles. Le nombre total de nceuds est 1.645 avec 1.597 noeuds int6rieurs. La num6rotation des triangles n'a pas dtd quelconque; on doit connaitre les triangles se trouvant darts le fer (stator ou rotor), ainsi que ceux qui sont supports du courant d'excitation (voir rigA. triangulation 1/4 du domaine). On remarque au milieu de l'entrefer une ligne sur laqueUe les nmuds sont espac6s rdguli6rement; nous appeIons cette ligne >. La disposition rdguli~re des neeuds sur cette ligne permet une rotation discrete de pas 1° 30' du rotor (ou plus pr~cisdment de la r6gion rotor) par rapport au stator ; il suffit pour ceta de recalculer les coordonndes des n0euds qui ont dtd tranformds par cette rotation et de modifier en consdquence la topologie au voisinage de la ligne de rotation. Nous obtenons ainsi 10 positions relatives rotor/stator diffdrentes. Le calcul des surfaces des triangles par produit vectoriel permet de ddceler des erreurs grossi~res de triangulation, mais ce n'est pas un moyen efficace de vdrification; un moyen plus stir et plus direct est le tracd automatique (par l'interm6diaire d'un traceur BENSON par exemple) de cette triangulation. Ce procddd n'est cependant pas infaillibte. Par exempte (Fig. 5) les triangles ABC et BAD d'une part, et ABC et BCD d'autre part, donneront le mdme tracd ; les premiers font partie d'une triangulation rdguli~re, les seconds non. Ainsi lorsque la triangulation se fait en partie (ou enti6rement) manuellement, il est pratiquement ndcessaire de recourir e Fig. 5
au procddd suivant pour ~tre certain du non chevauchement de triangles. Un triangle ABC dtant donn6, on trace effectivement
par le BENSON un triangle A'B'C' homothdtique de ABC et intdrieur g celui-ci. (Fig. 6). Chaque triangle est ainsi individualisd, et au premier coup d'ceil ~
;
A
sur l'ensemble du tracd, on se trouve renseignd sur la validitd de la triangulation.
B ":
;:'" c Fig. 6
Pour l'utilisation des algorithmes SNLt, SNL2 et EGSN iI est utile d'avoir calculd une fois pour toute, pour chaque nmud I
404
* le nombre M. de trianglesayant le nceud I pour sommet, * le num6ro du I~i~me triangle associ6 au n0eud I 1 < £ < M, * d a n s c e ~ m e triangle associ6 au nceud I, ~ quel sommet 1, 2 ou 3 correspond le nceud I. I1 faut aus~i ¢onnaitre pour chaque triangle de la triangulation, la surface de celui-ci, ainsi que les termes (combinaiso~_s de_s pj, qj~ rj intervenant dans 5-10) utilis6s pour le calcul de Igrad AI~ et pour -~lgradAl ~ Prise en consid6ration des eonditions d'anti-p6dodieit6 Le d6coupage en triangles est le m6me que pour le domaine entier. Pour une certaine position du rotor par rapport au stator le domaine
~2' est un quart de cercle (Fig. 7)
°"
F~g. 7 .
,.~
Fig. 8
j
O
O
Pour retrouver les I 0 positions diff6rentes du rotor par rapport au stator, on fait glisser la r6gion stator sur la ligne de rotation (Fig. 8), mais alors ici le nombre de nceuds int6rieurs, le nombre total de noeuds, ainsi que le hombre de ri~euds pour lesquels il faut tenir compte de la condition d'anti-p6riodicit6 varient avec chaque position. Nous nous trouvons ici devant un probl6me de topologie assez difficite. Nous pensons avoir r6duit consid6rablement les difficult6s en proc6dant de la faqon suivante. La num6rotation des triangles et des noeuds de la r6gion stator est faite ind6pendamment de cetle des triangles et n~euds de la r6gion rotor ; la topologie de chaque r6gion reste donc fixe quelle que soit la position relative rotor/stator. Nous sommes ainsi pratiquement en pr6sence de deux domaines ind6pendants; la liaison est faite tors de la r6solution en <>les 6quations aux n~euds communs de la ligne de rotation, et en tenant compte de la condition d'anti-p6riodicit6 pour les autres n0~uds de cette ligne. Pour la r6gion rotor, la discr6tisation est faite par 296 triangles et 187 n~euds. I1 y a 516 triangles et 304 neeuds dans la r6gion stator. On aurait pu, bien entendu, conserver le domaine ~2' ~ sa forme initiale (un quart de cercle), et changer la g6om6trie int6rieure, cela ne simplifie paste probl6me, car il faut ~ chaque position rotor/stator g6n6rer une nouvelle triangulation. VIII - RESULTATS NUMERIQUES Les algorithmes SNL1 ((6-5), (6-6)), SNL2 ((6-7), (6-8)) et EGSN ((6-9), (6-10)) ont tous trois ~t6 test6s num6riquement. Des essais ont 6t6 fairs aussi bien sur la structure compl6te que sur un quart de machine, tenant alors compte des conditions d'anti-p6riodicit~. M6me dans le cas d'6tude de la structure complete les calculs ont 6t6 faits sur ordinateur CII 10.070 sans utilisation de m6moires anxiliaires ; le temps de caicul est environ quatre fois cetui utilis6 pour la r6solution sur un quart de machine avec les conditions d'antip6riodicit~. Les algorithmes SNL1 et SNL2 ont donn6 des r6sultats 6quivalents du point de vue temps de calcul, EGSN s'est montr6 beaucoup plus efficace. VIII.1
Valeurs de diff$rents paramStres
Les valeurs des densit6s de courant retenues sont 0,5 ; 2 ; 5 ; 7,5 ; 10 (en amperes par mm ~ ), I0 6tant la valeur limite acceptable physiquement en raison de ph6nom~nes thermiques destructeurs que peut entrainer un trop fort eourant. Les m6thodes pr6sent6es ne sont pas limitees par cette contrainte therrnique. Le probl6me a 6t6 aussi r6solu pour J = 20 et J =
100.
405
1,95
X
1.9 CHOIX O P T I M A L de
RELAXATION
du PARAMETRE ~
dans EGSN
J (A/ram2) 2
5
10 Fig 9
400 ITERATION.5
300
200 _
100 ,
=1o
J
zu=S test lO_S
EGSN 0
3 d =2 4
,J
=0.5
I
r
CO I
,
,
I,
I
1. g
1.95
Fig
10
~
i
i
406
%
~ITE
DE ~UR^~
ROTOR ST^TOR
2,0 O.
O,
VALEUR k8S-11^I OU P O ~ I E L UECTE~ 0°13090E-01 Vk~UR DE LA FOI~ICTIONSIJFIL^ LIG~ I 0o30000E-03 OIFFERE~E ENTRE 2 LIGNES COI~JSECUTIUES OoGOOOOE-03 POSITION ROTOR-STkT~
A
P~
A
rAnmn~
~SA ~IKSA MKSA
0
POSITION COURANTSTATOR 0
0
I,t,~l^
Fig I I
/~,e,~lOCCO
407
DE'NSITEDE COURANT
ROTOP, STATOP,
10,0 O, O,
A A
tARn~ r~
VALEURABS-HAXDU POTENTIELUECTEUR 0,2~.332[-0"L HKS^ U^LEURDE LA FONCTIONSUBLA LIGNE "I 0.30000E-03 HKSA DIFFEBENCEENTBE2 LIGNESCONSECUTIUESOoGOOOOE-03 rIKs^
POSITIONBOTOB-STATOB 0 POSITIONCOUBANTSTATOB 0
LABORIA A.P,AF~CO
0 Fi 9 12
408
On utilise pour SNL1, SNL2 et EGSN le test d'arr6t d6fini par N .E{A~ +1 - Ain } (8-1) RES (n) = 1=~ N
RES (n)
~< e
On a choisi e = 10 -5 . I1 convient de noter que ce test correspond ~ la limite de precision de l'ordinateur utitis6, pour obtenir une valeur de RES(n) inf6rieure ou 6gale ~ 10 -4 ou 10 -7 , il faut n6cessairement faire les calcuts en double pr6cision. D'un point de vue pratique, on peut se timiter ~t e = 10-4 et m~me 5.10 -4 , la solution obtenue coincide 3 chiffres significatifs, pris avec celle obtenue avec e = 10-5 . Vlll.2
R6sultats obtenus
Les r6sultats donn6s ici concernent la r6solution num6rique du probl6me tenant compte des conditions d'anti-p6riodicit6. Le temps de passage de At ~t A~÷1 est de t'ordre de 1,2 s
sur CII 10.070 pour
l'algorithme EGSN ; bien entendu pour SNLt et SNL2 ce temps de passage est variable et d6pend du hombre d'it6rations internes effectu6es clans l'algorithme de Newton. Les temps globaux de r6solution varient entre 3 et 7ram suivant la valeur de e choisie dans (8-1) et la valeur du courant d'excitation. Ces temps peuvent augmenter si le param6tre de relaxation ~ est mal choisi. La figure 9 donne la d6termination exp6rimentale du param6tre optimal de relaxation pour l'algorithme EGSN. On peut voir sur la figure 10 que ce choix est d'autant plus critique que le courant d'excitation est petit. Par exemple, pour J = 10, o~ = 1.92 et e = 10 "s , SNL2 n6cessite 162 it6rations et un temps de calcul de 41Ms, EGSN demande 151 it&ations pour un temps de 230s (les temps d'impression et d'interpr6tation des r6sultats sont compris dans les temps indiqu6s). Pour J = 2, 380s.
co = 1.97 et e = 10 "s SNL2 n~cessite 229 it6rations pour 520s, EGSN en demande 290 pour
409
BIBLIOGRAPHIE
0 - G. Bruhat - Electficit6
1
-
P. Silvester - M.V.K. Chari - Analysis of t u r b o a l t e r n a t o r magnetic field by finite elements IEEE. PAS 90 N°2 4 5 4 4 6 4
1971
2 - P. Silvester - M.V.K. C h a d - Finite element analysis of magnetically saturated D.C. machines IEEE. PAS
2.362-2.372
Oct. 1971
3 - J.L. Lions - Contrdle optimal de systdmes gouvernds par des 6quations aux dddvdes partielles DUNOD. Paris 1968 4 - J. Cea - Op timisation - DUNOD. Paris 1971 5 - S. Scheehter - Iteration m e t h o d for non linear problems TRANS. A.M.S. 104 p. 179-189 1962 6 - S. Schechter - Relaxation m e t h o d s for convex problems SIAM J. on numerical analysis 5 p. 601-612 7 - S. Scheehter - Minimisation of convex function by relaxation INTEGER AND NON LIN EA R PROGRAMMING ABADIE-EDITOR North Holland 1970 p. 177-189 8 - R. Glo win sk i - A. Marrocco - Analyse numdrique du champ magndtique d ' u n alternateur par 616ments finis et surrelaxation ponctuelle non lindafi'e C o m p u t e r m e t h o d s in applied - mechanics and engineering 1973
UNE NOUVELLE METHODE D'ANALYSE NUM,ERIQUE,DES PROBLEMES DE
FILTRATION DANSLES MATERIAUXPOREUX par C. BAIOCCHI (Pavia, Italie)
I - DESCRIPTION DU PROBLEME PHYSIQUE On consid~re une classe de probl@mes de #iltration dont un cas typique peut ~tre sch@matis@ sous la forme suivante d'eau, de n~veaux dill@rents, riau poreux;
: sur une base imperm@able deux bassins
sont en communication ~ travers une digue en mat@-
l'eau filtre du niveau le plus @lev@ eu niveau le moins @lev@; et
on veut d@terminer la "partie mouill@e" de la digue ainsi que les grandeurs physiques (telles qua vitesse, d@bit, etc..] associ@es au mouvement.
II - TRADUCTIONEN PROBLEMEMATHEMATIQUE.DE FRONTIERE' L.IB.RE (Cf [ 13], [ 15] ). Supposons la digue inflnlment @tendue et & section constante @tudier un probl@me bidlmensionnel];
[de fagon &
en absence de cepillarit@ pour un fluxe
stationnaire,irrotatlonael,ineompressible,
la ioi de Oercy assure qua le mouve-
ment de l'eau dens la digue est li6 ~ un potentlel de vitesse; plus pr@cis@ment, si ~ est la partie mouill@e de la digue D, et si p[×,y] est la pression au point Ix,y) de ~ Ix axe horizontal, y axe vertical) nelle au gradient de perm@abilit@~
Y + p[x,y)
le coefficient K[x,y] @tant un coefficient de
[qui et la function Y u[x,y] = y + P(~,,',,Y,] Y
satisfait, grace ~ l'incomprimibilit@
[1]
div
on a qua la vitesse est proportion-
k zrad u = 0
e,s,t une h a u t e u r
pi@zom@~rique)
:
dens
A cette @quation aux d@riv@es partielles on dolt ajouter des conditions aux limites qui sont de deux types; du type Neumann
[2]
3o - 0 ~n
le long des ligues de courants
[~-- d@signant la d@riv@e normale; en particulier [2) dolt @tre impas@e le long ~n de la base imperm@able et le long de la "ligne llbre", DO - 2D] at de type Oirichlet
(3]
p[x,y]
= 0 ~
[4]
u(x,y] = Yl
u{x,y)
= y
sur la partle ~
pour
les points
B~ 8 c o n t a c t
avec l'air
~ contact avec le premier bassin,
411
(5)
u{x,y)
(Yl' Y2
= Y2 sur la partie de ~
d~si~nant
1as hauteurs des deux bassins).
I1 s'a~it d'un incannu ~
~ contact avec 16 deuxi~me bassin.
classique probl~me ~ fronti6re
on dolt r~soudre un probl~me aux limites
{lJ, avec comd~tlon surhabondan~es
fibre : sur le domaine {2) ,.. (5) pour l'~quation
((2) ~t (~)) ~ur 3m pattie Inconnue ~
- ~D
de la fronti~re de ~,
III
- ESQUISSE DES METHODES NUMERIQUES TRADITIONNELLES Une idle elassique pour la r6solution
libra est la suivante bibliographie
pour le problbme consid~r~ icl)
comma premiere approximation ximation ~
num6rique de prebl@mes ~ fronti~re
[Cf [10 ] pour une vue d'ensemble;
de la franti6re
Cf aussi [14 ] et sa
: on fixe une ceurbe
y = ~o(x)
libra: dams la correspondente
appro-
de la pattie moui116e de D on r6seud un prob16me m616 pour l'6quation
o [I) en imposant
16 long de y = ~o[×) une seulement des deux conditions
puis l'on modifie ~
de fagon qua l'autre condition
o ume nouvelle courbe ~lCX);
on it~re le proc6d6
soit remplie,
~o ÷ ~ I
{2) et (3);
en obtenant ainsi
pour passer ~ une nauvelle
courbe ~2 et ainsi de suite. Par exemple si le probl~me m~l~ que 1'on r6soud correspond ~ imposer le lon~ de y = Me(X) gl(X)
par la formula
~l(X)
la condltion de Neumann
= Uo(X, Me(X))
et la solutiom du p~obl~me ~ frontibre
(2), on obtiendra
{u e 6tent la solution du probl6me m616]~
libra est un point ~ixa pour
le transfor-
~o ÷ ~1"
mation
Toutefois on dolt remarquer qua :
a) du point de vue th~orique on ne sait
pas justifier
ces precedes
(on ne connaSt ni existence et un±cit~ de la solution,
ni stabilitY,
convergence,
majoration de l'erreur etc,., pour 1as solutions appre-
ch6es)~
bJ du point de vue num~rique
le prae~d~
est tr6s lourd, d~s qua 1'on dolt r~soudre une famille de prob16mes m~l~s,
sur des
domaines qui variant ~ cheque 6tape en fonction de la solution de 1'iteration
pr6-
c~dente,
IV - REDUCTION A INEQUATIONS VARIATIONNELLES Dens une ~tude t h 6 o r i q u e du prob16me J ' a i 16 p l u s s i m p l e o~ k ( x , y )
montre {Cf [ I ] )
= 1 e t D e s t un r e c t a n g l e ,
r e s p o n d e n t ~ une d i g u e homog~ne, ~ base h o r i z o n t a l e ,
0 = ]0,~
qua, dens l e cas x ]O,Yl[
Ices c o r -
~ parois verticales,
seur a) on peut ramener le probl~me & une In~quation variationnelle,
d'~pais-
quitte b
412
remplacer
l'inconnue
blement prolong~e Pr~cis~ment
(6)
w[x.y)
u(x,y) d@~inle dans ~ par une primitive
de u[x,y)
convena-
hors de ~. si l'on pose,
= fYl [O[x,t)-t
pour
[x,y) e O :
)dr
Y oOO
est la #onstion
les relations
:
[7)
j
w ~ 0
qul prolonge u ~ y hors de ~, on peut montrer que w satis~ait
Aw ~ 1
~
w(I-Aw]
at on peut @valuer une #onction
[8)
= 0
g[x,y]
dons
O
d@#inie sur 30
telle que
Wl~ o = g
[g d@pendant
des donn@es ~ , Yl ' Y2 )'
Si l'on pose alors
(9)
K = {ve
an peut traduire
I w e K
]
HI(O)
;
(HI(D) d@signant
v ~o
~
l'usuel espace de Sobolev)
:
vl~ o = g}
(7) et (8) sous la #orme d'in~quation
variationnelle
:
~ v ~ K :
(10)
(fD
• vcv-.)
d× dy
[0
dx dy
o5 @quivalemment,
dane l e probl@me de minimum :
[I1)
sur K la #onctionnelle
Vice
w minimise versa,
connaissant
[12]
0 = {Ix,y)
[13)
u[x,y)
:
> O}
dans 0
st, & partir d'ici et des r@sultats obtient
dx dy + ~O v dx dy
w, on remonte & ~,u par les #ormules
e O I w[x,y)
= y-Wy[X,y]
1
~ f~O
un Th@or~me d'existence
connus
sur les in@quations
st unicit@ pour le probl@me
du type
& fronti@re
[10) on fibre
413
(pour les d@tails Cf toujours [ I] ).
Successivement, [16]]
dans une s@rie de travaux
(Of [2], [3], [4], [5], [6], [8],
oe r@sultat a 6t@ adapt@ ~ des prebl@mes de filtration plus compliqu6s
[perm6abilit6 immiscibles
variable,
g6om@trie plus g@n@rale,
,..) et ~ des probl@mes & frontiers
lement subsonique autour d'un profil,
problems de Stefan
Tout en restant dans les probl~mes g@n@ral,
pr@sence de plusieurs
de filtration
aussi un int~r@t physique ligne d'@mergencs,
ii faut remarquer
: par exemple le d@bit et 3'abscisse simple pr@o@dent
qua, en
(tells qua (10)]
d@pendante d'un ou plusieurs param@tres
qui, dens l e c a s
(@eou-
...; Cf [4],[6],[11]],
on dolt @tudier non plus une in@quation variationnelle
mais une #amille d'in6quetions
liquides
libra de nature dill@rents
(ayant eux
initials de la
sent connus a priori].
V - RESOLUTION NUMERIQUE
La r@solution
num@rique d'in~quat/on
on dolt ajouter un algorithms de dlfficult@s
: on connait
tisation parfaitement de vue pratique programmation
pour la d@termination
des param~tres)
(en g@n~ral,
quadratique),
apr~s discr6tisation~on Par example
(Of [ 9],[2]
finies on remplacera
de disor@tisation]
n'offre pas
(Of par example [12]) de nombreux proc6d@s de discr~-
justifi@s du point de vue th@orique
tisant en diff@rences (h pas
telles qua [10) [dens le oas g@n@ral
et efflc£ent
du point
pervient ~ des probl@mes de pour 3es d@tails]
en discr@-
Ze conve×e K par une famille {Kh} h
de aonvexes constitu6s par fonctions constantes
par mor-
ceaux, qui sent non n@gatives et qui, sur le "bard discret" de 0 prennent des valeurs gh discr@tisantes tisation de (7],(8]
wh > 0
la fonction g; on tombe sur le s~st@me
~ Ahest
j
la discr6tisation
Ah Wh < 1
;
[qui est le discr@-
~ 5 points de A} :
Wh(1-A h whI = 0
dens ~' "int6rieur discret" de O Wh = gh
sur le "bard discret" de O ,
syst@me qul peut, par exemple, On remarquera finalement
@tre r@solu en adaptant que les r6sultats
essurent qua w h m w dans H I discret, non pas w mais l'ensemble ~
Oh = { [ x , y )
l'algorithme de S.O.R.
olassiques
or si l'on pose (analoguement
e O I Wh(X,Y)
sur les in@quations
mais qua la "vraie" inconnue Ou probl~me est
> O}
Q (12]]
:
414
la c o n v e r g e n c e de w h., k w dens H 1 dlscret n'assure p a s l e En e f f e t
on p e u t m o n t r e r
[ C f [ 4] ) q u e
= int~rieur
C. BAIOCCHI
,
Su un problema di frontiera
di idraulica. 107-127;
[ 2]
C. BAIOCCHI,
de ~h ~ ~"
[lim' ~h ] . h+O
B I BL I O G R A P H
[1]
"convergence"
~h c o n v e r g e v e r s ~ au sens s u i v a n t
I E
libera connesso a questioni
Ann. di Mat. pure e eppl.
(IV] vol. XCII,[1972),
note aux C,R. Acad. Sc. Paris, t.273
V.COMINCIOLI,
L.GUERRI,
(1971), 1215-1217.
G.VOLPI.
Free boundary problems in the theory o4 fluid flow through perous
[ 3]
C. BAIOCCHI,
media
: a numerical approach.
V.COMINCIOLI,
E,MAGENES,
Calcolo X[1973)
1-86.
G.A,POZZI;
Free boundary problems in the theory of fluid flow through porous media (1973)
[4]
C. BAIOCCHI,
: existence and uniqueness
E.MAGENES, Problemi di frontiera
libera in idraulica.
"Atti del Convegno Internazionale magematica"
[ 5]
theorems,Al1~, di Nat. XCVII
1.82.
V. BENCI,
: Metodi valutativi
A paraitre aux helle fisica
15-20 dicembre 1972. Academia Nazionale dei Lincei, Roma
Su un problema di filtrezlone
attraverso un mezzo poroso.
A praraitre aux Annli dl Mat. pure e appl.
[ s]
H. BREZIS,
G.DUVAUT, Ecoulement
incidence.
[ 71
H. BREZIS,
avec sillage autour d'un profil sym~trique
sans
C.R. Ac. Sc. Paris 276 [1973)
G.STAMPACCHIA, Une nouvelle m@thode pour 1'@tude d'@coulements
res. C.R. Acad. Sc. Paris 276(1973).
stationnai-
415
[8]
V. COMINCIOLI,
A theoretical
problems.
[ 9]
V.CDMINCIOLI
and numerical approach to some free boundary
A paraitre aux Annell di Mat. Pure e appl.
, L,GUERRI,
G.VOLPI,
Analisi numerica di un prohlema di frontiera
libera connes-
so col moto di un fluido attreverso un mezzo poroso. Publication du Laboratorio
[10] C,W. CRYER,
di Analisi Numerica C.N.R. Pavia
On the approximate
finite differences.
[11]
6, DUVAUT,
[12]
R, GLOWINSKI, J . L .
[1971].
solution of free boundary problems using
J. Assoc. Comput. Mech.,
R@solution
z@ro degree),
I?
d'un probZ@me de St@fan
17, N. 311970),397-411.
(fusion d'un bloc
C.R. Ac, Sc. Paris 276 [1973).
LIONS, R. TREMOLIERES, R@solution num@rique des in@quations de l a M@canique et
de l a Physique,
[13]
M,E, HARR,
[14] E. MAGENES,
A paraitre,
6rounwater
Ounod P a r i s ,
and seepage. New York ; Mc Graw-Hi11
Su alcuni problemi ellittlci di frontiera
con il comportamento
1962
libera connessi
dei fluidi nei mezzi porosi. Symposia Math,
Rome, 1972,
[15]
M. MUSKAT,
The flow of homogenous fluids through porous media.
York : Mo Graw-Hi11
[16] A. TORELLI,
New
1937
Su un problema di flltramione
da un canale. A pa~aitre,
CIRCUITS ET TRANSISTORS NETWORKS AND SEMI-CONDUCTORS NUMERICAL METHODS FOR EQUATIONS RELATED WITH
S T I F F S Y S T E M S OF D I F F E R E N T I A L TRANSISTORS, TUNNEL DIODES, ETC.
by
W i l l a r d L. M i r a n k e r IBM Research Center and Frank Hoppensteadt New York University
1.
INTRODUCTION
T h e m o d e l s of c i r c u i t s w h i c h c o n t a i n e l e m e n t s s u c h as t r a n s i s t o r s or t u n n e l d i o d e s a r e d i f f e r e n t i a l equations as is w e l l k n o w n . Because of the h i g h s p e e d of p e r f o r m a n c e of t h e s e c i r c u i t e l e m e n t s , the range of t h e p a r a m e t e r v a l u e s a n d of t h e d e v i c e c h a r a c t e r i s t i c s of c u r r e n t interest in t h e c o r r e s p o n d i n g differential equations r e s u l t in s o l u t i o n s w i t h an e x t r e m e r a n g e of b e h a v i o r . These solutions m a y be c o m p o s e d of s l o w l y v a r y i n g components, highly damped components, highly oscillatory components, and combinations of s o m e or all of t h e s e . This variation in b e h a v i o r of the s o l u t i o n s is c h a r a c t e r i z e d b y the term stiffness. In S e c t i o n 2 we w i l l d e s c r i b e a simple circuit model for a t u n n e l d i o d e a n d we s h o w h o w t h i s r a n g e of s o l u t i o n s does arise. The numerical s o l u t i o n of s t i f f d i f f e r e n t i a l equations meets with difficulties because of the e x t r e m e s in t h e r a n g e of b e h a v i o r of the solutions. In r e c e n t y e a r s a n u m b e r of n u m e r i c a l methods for stiff differential equations have been devised. (cf. G. G. B j u r e l (1970).) S o m e of t h e s e m e t h o d s have been applied successfully to c e r t a i n c l a s s e s of s t i f f d i f f e r e n t i a l equations. Nevertheless it is s t i l l n o t k n o w n h o w to d e a l e f f e c t i v e l y with this computational problem in g e n e r a l . Especially difficult is t h e c l a s s of d i f f e r e n t i a l equations which contain highly oscillatory components in its s o l u t i o n s . In p r e v i o u s studies (cf. W. L. M i r a n k e r (1973) and W. L. Miranker and J. P. ~rree~,~ (1973))one of us h a s p o i n t e d o u t t h e r e l a t i o n s h i p between stiff differential equations and differential equations subj e c t to s i n g u l a r p e r t u r b a t i o n s , and has exploited this relationship to d e v e l o p e numerical methods f o r t h e s o l u t i o n of s t i f f e q u a t i o n s . In t h i s s t u d y we w i l l e n l a r g e on t h i s p o i n t of v i e w to d e v e l o p e numerical m e t h o d s w h i c h c a n d e a l w i t h t h e f u l l r a n g e of b e h a v i o r in s o l u t i o n s just described. To d o t h i s r e q u i r e s t h a t a u n i f o r m m e t h o d be d e v e l o p e d for c h a r a c t e r i z i n g the s o l u t i o n s of s i n g u l a r p e r t u r b a t i o n problems throughout the f u l l r a n g e of i n d i c a t e d solution behavior. As is w e l l known (cf. J. C o l e (1968) a n d A. If. N a y ~ e h ( 1 9 7 3 ) ) t h e c h o i c e of asymptotic techniques u s e d to o b t a i n d e s c r i p t i o n s of s o l u t i o n s of d i f ferential equations d e p e n d s on t h e n a t u r e of t h e s o l u t i o n (e.g. s t r o n g ly d a m p e d or h i g h l y o s c i l l a t o r y ) . In S e c t i o n 3 we s h o w h o w the m u l t i t i m e technique of a s y m p t o t i c expansions m a y be c o m b i n e d w i t h t h e m e t h o d of a v e r a g i n g of B o g o l i u b o v
417
to p r o d u c e a procedure for deriving the asymptotic f o r m of s o l u t i o n s of s i n g u l a r l y perturbed differential equations o v e r the f u l l i n d i c a t e d r a n g e of s o l u t i o n b e h a v i o r . We do t h i s for the s i m p l e m o d e l p r o b l e m du e d-~
=
(A0
+ cA1)u
(l.l)
a n d we i n c l u d e t h e p r o o f of v a l i d i t y of the e x p a n s i o n to w i t h i n O ( e 2 ) . We m a k e t h e s e l i m i t a t i o n s f o r t h e s a k e of c l a r i t y of p r e s e n t a t i o n and because t h i s m u c h of t h e a s y m p t o t i c development is a d e q u a t e for the s p e c i f i c a t i o n of t h e n u m e r i c a l method. We e m p h a s i z e that this asymptotic theory (and t h e n u m e r i c a l m e t h o d ) m a y be c a r r i e d o v e r to non-linear problems a n d to a l l o r d e r s , b u t we d e f e r t h i s t r e a t m e n t to a n o t h e r study. In t h e s i n g u l a r limit, highly oscillatory solutions may conv e r g e to i n v a r i a n t manifolds of d i m e n s i o n greater than one. Thus the meaningfulness of d e s c r i b i n g such a trajectory b y a s e t of its v a l u e s on the p o i n t s of a t i m e m e s h is l o s t . Indeed this suggests the reasons f o r t h e l a c k of e f f e c t i v e n e s s of e x i s t i n g numerical methods for such problems. To r e m e d y t h i s d i f f i c u l t y we i n t r o d u c e in S e c t i o n 4 a n e w n u m e r i c a l solution concept for d i f f e r e n t i a l equations. We accept a quantity as an a p p r o x i m a t i o n to the s o l u t i o n at a p o i n t in t i m e if the q u a n t i t y approximates a n y v a l u e w h i c h the solution a s s u m e s on a n e i g h b o r h o o d of t h a t p o i n t in t i m e , t h e s i z e of t h e n e i g h b o r h o o d being arbitrary but positive. With this numerical solution c o n c e p t we t h e n c o n s t r u c t a numerical m e t h o d b a s e d on the asymptotic theory introduced in S e c t i o n 3. We i l l u s t r a t e the numerical m e t h o d b y m e a n s of c a l c u l a t i o n s b a s e d on s e v e r a l s a m p l e problems. 2.
and
In t h i s show how
A CIRCUIT
section various
MODEL
AND
STIFF
BEHAVIOR
we w i l l d i s c u s s a model for a tunnel diode circuit solution classes with extreme behavior arise.
A simple circuit representing tically in f i g u r e 2 . 1 a
L
a tunnel
diode
is
given
schema-
R
(a) E
C
v V
Figure
2.1
The current through the non-linear element is g i v e n by ! = f ( v ) , where the tunnel diode characteristic, f ( v ) , is the S shaped graph as indicated in f i g u r e 2 . 1 b . The differential equations describing this circuit are dv C ~-~ =
i-f (v) (2.1)
di L ~
= E-Ri-v
418 For certain ranges of values of the parameters, the simple model (2.1) is a s t i f f s y s t e m w h o s e s o l u t i o n s exhibit a variety of extreme behavior. We will now give an indication of this behavior. Introduce
the
new
R x = ~ t, in
(2.1).
We
variables CR 2 L
s =
'
I =
Ri,
F(v)
(2.2)
= Rf(v)
get
dv e -dx
=
I - F (v) (2.3)
dI -dx
= E - v - I
W h e n £ is s m a l l t h i s s y s t e m is s t i f f a n d s o l u t i o n s m o v e a l t e r nately through regions of slow change and rapid change. A typical family of solutions in t h i s c a s e is s c h e m a t i z e d in f i g u r e 2 . 2 a .
(o)
/if\
/\V
,:oh
Alternatively
, ~ o ^ \ \\ i ,-eo^\ \ \
~y
we
may
Figure
2.2
introduce
the
t z = -RC in
terms
of
which dv dz
--
X //8=0
(b)
DECREASING
6 =
(2.3) =
i/£
/
/ i
variables
,
(2.4)
becomes
I-F
(v)
(2.5) dI ~z W h e n ~ is schematized in
= E-I-v.
small the solutions behave figure 2.2b.
in
A different form of extreme behavior of certain other ranges of parameter values may the variable y : and
writing
the
an
extreme
as
solutions of (2.3) f o r be seen by introducing
t//~
system
manner
(2.6) (2.3)
as
a single
equation
419
d2v
dy
2'
2 +
~
dv v
=
:<(v,
~-~y
(2.7)
)
where K =
of
(~2-1)v
- Rf(v)
In t h i s c a s e t h e r e (1.9) as i l l u s t r a t e d
(R
are one and sometimes in f i g u r e 2.3.
LIMIT
Figure The
frequency
of
dv f' ( v ) ) ~ y
+
these
limit
two
+ E.
periodic
(2.8)
solutions
/
2.3
cycles
is
approximately
2 = C ( L - R 2)
(2.9)
in t h e t - t i m e s c a l e . Thus for are oscillatory solutions with
c e r t a i n v a l u e s of C, L a n d possibly high frequency.
R there
Detailed s t u d i e s of t h e s o l u t i o n s of t h e t u n n e l d i o d e e q u a t i o n s w e r e m a d e b y o n e of us g i v i n g p r e c i s e characterizations of t h e solution classes just displayed. The methods used were typical asymptotic expansion techniques involving time stretching, matching, u s e of t h e so c a l l e d s e c u l a r i t y condition etc. We r e f e r to W. L. Miranker (1962 a,b) f o r d e t a i l s . 3.
ASYMPTOTIC
ANALYSIS
From consideration of t h e c i r c u i t e x a m p l e in S e c t i o n 2, we a r e l e d to s t u d y s t i f f s y s t e m s of o r d i n a r y differential equations whose solutions exhibit abrupt excursions b o t h in the s e n s e of r a p i d damping a n d in t h e s e n s e of r a p i d o s c i l l a t i o n s a n d of c o u r s e m i x t u r e s of t h e t w o . In t h i s s e c t i o n we w i l l c o n s i d e r a simple prototype of s u c h systems and develope a technique for approximating its solution. We h a v e r e s t r i c t e d our development to t h i s s a m p l e p r o b l e m to i l l u strate the method in a c l e a r m a n n e r . We e m p h a s i z e that the developm e n t m a y be c a r r i e d o v e r to n o n - l i n e a r problems b u t we d e f e r t h i s treatment to a n o t h e r study.
420
3.1
The
Two
Consider
Time the
Approach following du E ~
=
system
(A 0 +
of
differential
equations
CAl)U
(3.1)
w h e r e u is an n - v e c t o r , A 0 a n d A. a r e n x n m a t r i c e s a n d ~ is a s m a l l parameter. We s e e k to a p p r o x i m a t e the s o l u t i o n of t h e i n i t i a l value problem f o r (3.1) on a f i x e d t i m e i n t e r v a l , 0 < t < T. The fundamental m a t r i x of t h i s s y s t e m m a y be r e p r e s e n t e d - - i n T h e f o r m
~(t,£) However, the 0 ~ t ~ T is
= e x p [ (A 0 +
EAI) (t/E) ] .
(3.2)
numerical evaluation of t h i s m a t r i x o v e r the usually difficult w h e n e is v e r y n e a r z e r o .
interval
T h u s m e t h o d s w h i c h g i v e e a s y to o b t a i n a p p r o x i m a t i o n s to the fundamental matrix are useful. We will derive such a method which we w i l l u s e to o b t a i n t h e d e s i r e d a p p r o x i m a t i o n s . In a d d i t i o n o u r method illustrates the so c a l l e d m u l t i - t i m e perturbation method. Moreover o u r m e t h o d produces t h e e x p a n s i o n s developed b y t w o other w i d e l y u s e d perturbation techniques : the m e t h o d o f m a t c h e d a s y m p t o t i c e x p a n s i c n s a n d t h e m e t h o d of a v e r a g i n g . When
a new
time
scale
du/d~ and
the
fundamental
=
is
introduced,
(3.1)
becomes (3.3)
(A 0 + e A l ) U
matrix
~(T,£)
T = t/e
becomes
= exp[AoT
+ A 1 ~T].
(3.4)
W h i l e t h i s c h a n g e of v a r i a b l e s apparently m a k e s the p r o b l e m into one whose solution depends s m o o t h l y on the p a r a m e t e r e n e a r e = 0, the standard Taylor series approach for c o n s t r u c t i n g approximations to the s o l u t i o n g i v e s a r e s u l t v a l i d o n l y on b o u n d e d T intervals. However, the problem (3.3) is to be c o n s i d e r e d on t h e l a r g e i n t e r v a l 0 < T < T/E. The matrix scales,tand T:
~ may
= exp
be
[AoT
considered
to
be
This
will
be
~ r=0
u
r
a useful
(t,T)
e
series
U r ( t , t / E ) £r
of
the
two
time
(3 .5)
+ Alt].
This motivates us to s e e k a p p r o x i m a t i o n s t h e f o r m of a g e n e r a l t w o - t i m e expansion u =
a function
to
the
solution
of
(3.3)
r
in
(3.6) for
purposes
= o(r-l),
r =
of
approximation,
1,2, ....
as ~ + 0, u n i f o r m l y f o r 0 < t ~ T. W i t h (3.7) v a l i d we p a y (3.6) is an a s y m p t o t i c expansion with asymptotic scale £
if
we
have
(3.7) that A sufficient
42t
condition
for
(3.7) u
as
T +
m
for
is
(t,T)
r
r =
that
= o(T)
(3.8)
1,2, ....
The expansion resulting from this prescription of f o r m ( 3 . 6 ) (3.8) o f t h e s o l u t i o n w i l l b e d e r i v e d b e l o w . It is possible to o b t a i n m o r e i n f o r m a t i o n from the expansion by placing a stronger condition on the coefficients than (3.8). In p a r t i c u l a r we will determine conditions o n A 0 a n d A 1 so t h a t t h e r e q u i r e m e n t A0~ u as
T ÷
~
for
r
(t,T)
r =
= o(~e
1,2, ....
)
can
(3.9)
be
used
to
obtain
a valid
expansion.
N o w , if A_ is a n o s c i l l a t o r y matrix (all e i g e n v a l u e s have zero O real part), then conditions (3.8) a n d (3.9) a r e e q u i v a l e n t . If A 0 is a s t a b l e m a t r i x (all e i g e n v a l u e s have negative real parts), then condition (3.9) is m o r e r e s t r i c t i v e than (3.8). In t h e s t a b l e c a s e it m a y n o t b e p o s s i b l e to obtain an expansion o f t h e s o l u t i o n of (3.3) in t h e f o r m (3.6) w h o s e c o e f f i c i e n t s satisfy (3.9). However, we will describe another restriction on the problem which when used w i t h (3.9) g u a r a n t e e s t h a t t h e s o l u t i o n o f (3.3) c a n b e a p p r o x i m a t e l y s o l v e d in t h e f o r m ( 3 . 6 ) . This approximation technique proceeds via the two-time approach. T h i s r e s u l t is v a l i d w h e n t h e e i g e n v a l u e s of A ~ l i e in t h e s t a b l e h a l f p l a n e ; t h e r e f o r e , it contains both the stable and oscillatory cases. In t h e s t a b l e c a s e , t h e e x p a n s i o n found by this m e t h o d r e d u c e s to t h e o n e w h i c h w o u l d b e o b t a i n e d b y t h e m e t h o d o f matched asymptotic expansions. In t h e o s c i l l a t o r y case, this result r e d u c e s to a n e x p a n s i o n equivalent to t h e o n e o b t a i n e d by the Bogoliubov method of averaging. 3.2
we
Formal
Expansion
We consider the write the initial u(0)
To
simplify
=
Procedure initial value problem for conditions in t h e f o r m Z r=0
a
e
r
computation
the
system
(3.3)
r
and
(3.10)
let
-A0~ v(t,T) Since
v
is
considered
dv
Then
(3.3)
= e
u(t,T).
as
(£T,T) d~
a function ~v(t,T) - ~t
becomes
~v ~v e ~ + ~
the =
(3.11)
e B(Y)v,
the
two
variables
~v(t,T) ~
+
following
of
equation
v(0)
=
Z r=0
T and
t=
ET
(3.12) for ar £
v: r
(3.13)
where -AoT B(T) We
seek
a
= e
solution v =
Z r=0
A0T A1 e
in
the
(3.14)
form
Vr(t,T)e
r
(3.6)
which
becomes (3.15)
422
subject to t h e latter becomes
condition
(3.9)
~v r ~T ........ = B(T) v r _ 1 Here
r
= O(T)
as
Substituting (3.15) i n t o l i k e p o w e r s of e g i v e s
the
v
the
u
.
In
terms
of
the
r Vr(t,T)
v
on
~Vr_ 1 ~t
'
r = 0,i, . . . .
and
Vr(0,0)
, the r
T ÷ =,
(3.13)
v
equating
= a r,
(3.16)
coefficients
of
(3.17)
r = 0,i,...
E 0.
-i
The problem (3.17) c a n be i n t e g r a t e d to
is
The
underdetermined.
equation
(3.17)
for
give T
Vr(t,T)
= ~r(t)
+ f
~Vr_l(t,~) ~t
[B(a)Vr_l(t,~)
]d~,
r=0,1,...
o
(3.18)
where v Except
for
respect
to
r
(0)
= a
(3.19),
t
(3.19)
r
~r(t)
is
Differentiating
arbitrary.
(3.18)
with
gives
~Vr ~t
~r ~t
~V
/ +
22
r-i ~t
[B(0)
Vr-I ~t 2
]do
.
(3.20)
o Combining
this
with
(3.18)
gives
T
T v
r (t,T)
= ~ r (t)
+
B(o)d~
~ Vr_l(t) _
dVr- 1 T -
dt -
+fRr_l(t,~)da o (3.21)
where / Rr (t,c)
=
[B(~)
~Vr-l(t'a') ~t
~2Vr_l(t,o') ~t 2
Ida'
o + B(O)
f
[B(~')Vr-l(t'~')
-
~Vr_l(t,o, ~t
)
]do'.
(3.22)
o (3.21) a n d (3.22) h o l d f o r r = 0 , 1 , . . . , where v ~ R _ I E R 0 E 0. Let impose the growth condition (3.16) in ( 3 . 2 1 ) . - i T o do t h i s , d i v i d e (3.21) b y T a n d t a k e t h e l i m i t as T + ~ This results in t h e - +-following condition for v . r T T r = dT
(lim T+~
T
B(0lda)~
+ lim r
T~ ~
o
(~
Rr(t,0)d~)
, r = 0 , 1 ....
o (3.23)
When
these
limits
exist,
(3.23)
along
with
(3.19)
determine
Vr,
r=O,1, ....
in
This approach depends (3.23). The development
critically on t h e e x i s t e n c e of t h e l i m i t s w i l l be s i m p l i f i e d u s i n g the n o t a t i o n
us
423
T = l i m ~1
/
f(x)dx
o If f e x i s t s we w i l l c a l l it t h e notation (3.23) becomes
(3.24) average
of
f.
In t e r m s
of
this
dv dt provided
r
the
+ R
r
averages
We t u r n appear
which
= B" ~
r
(t),
~
r
(0)
= a
r
,
r=0,1,...,
(3.25)
exist.
n o w to the in (3.25).
question
of
the
existence
of
the
averages
Remark: We c o n t e n t o u r s e l v e s h e r e w i t h a s t u d y of B a n d R_ s i n c e the i ~ n c e of t h e s e t w o a v e r a g e s p r o v i d e s us w i t h the e x i s t e n c e of the approximation Vo + e v 1 t o v . This approximation is ad~equate for our computational purposes. Note that Vo(t,T) - Vo(t) = Vo(t)3.3
Existence The
of
example
Average,
the
B
-i 0 (0 -2 ) ' A1
A0 =
0 0 (i 0 ) s h o w s
=
that
B may
not
exist.
Restriction: We a s s u m e in t h i s a n a l y s i s t h a t the m a t r i x A has simple elementary divisors. I n w h i c h c a s e we m a y a s s u m e w ~ t h o u t l o s s of g e n e r a l i t y t h a t A~ is a d i a g o n a l m a t r i x . We d e n o t e the e l e m e n t s of A 0 a n d A 1 r e s p e c t i v e l y by A0 = Then
(i i ~ij ) a n d
1
A1 =
(aij) .
(3.26)
T IT
f
e
_A0~
A1 e
A0~
=
(a~j
fij)
(3.27)
o
I )3
where
e (lj-li -1 3(I -I ) j i
fij
computation
Theorem Remark holds.
3.1: 3.1:
The Cq[ol!ary
] (3.28)
,
demonstrates
exists
computation
X. the
if a n d
We h e n c e f o r t h
following
theorem.
1 if aij
= 0 whenever
only
assume
= X,, ]
that
also
has
the
then
Bij
= a~j
the
following
hypothesis
Re(lj-li)>0. of T h e o r e m
3.1
corollary.
3.1:
i) If B e x i s t s Kronecker delta. ii)
li
=
! This
'
If
the
eigenvalues
iii) If A^ a n d conjugate to ~i"
A 1 are
~ (li'
of A 0 a r e normal
kj) ' w h e r e
distinct
matrices
then
then
6 is
B =
B exists
the
1 (aij and
.). ~i3 is
424
iv)
If A 0 a n d
v)
If A 1 is
vi)
A 1 commute diagonal
If A 0 = II
then
then
then
B exists
In t e r m s represented
decomposition 1
II All
=
where
is
Proof: (iii) ~nalizable
3.4
Existence The
the
following
Theorem
3.2:
If
Proof: tation:
The proof Let
from
(3.22)
Rl(t,a) where
from
1
to
exist.
then
based
may
J
AA a n d A. a r e s i m u l t a n e o u s l y 1 o~her statements are immediate.
characterizes
B exists
0
B =
result
RI~
on
R1
the
existence
of
RI"
exists.
theorem
3.1
and
the
following
compu-
c = /
we
have
=
(B(~)
(B(~')-B)d~'
(3.29)
o p(~)
have
used
% dv 0 _ -~ = B v0
We f i r s t (3.28)
show
the
we
B this
-I A22
assumed
Average
0(~) Then
but
theorem
is
A 1 and
A ~
(iv) f o l l o w s i n c e these cases. All
of
of
1 A22
unspecified
and in
to A I.
= I^ = ... = I = 0 and t h e z m x m p ~ i n c i p ml e £ubmatrlx of B is A ~ I w h i l e B.. = 0 f o r iJ
AI2
1 21
A-i 22
conjugate
B = A I.
of a b l o c k by:_ 1
A1
is
B = A I.
vii) Let B exist and suppose that ~ a n d I m *.l. ~ 0, i = l . . . . . n-m. L e t A ~I b~ . of A I. Then the mxm principle subma%rlx • > m and i < m and for j < m and i > m
be
and
- p ( a ) B ) e Bt
which
existence
of
v0(0)
follows p.
The
(3.30)
from
(3.25).
ij-th
element
of
p is
(lj-li)a
Ill e
-i
3
(P) ij
aij •
(;
1
e
(lj-li)s 3
i
= I aij 0
I i = lj -1 I i ~ lJ 1 I i = lj
sall3""
'
I i = lj
425
Then 1 aij I.-I.
T
T
(~)da
p
T [if
e ( l J - l i )a
do-l],
I i ~ Xj
o
=
o
0
,
X.
= l
X.
. j
(3.31) The limit of Re(lj-li)>0.
(3.31), as T + ~ exists In this case we have 1 -a..
ij
l.-l. =
3
'
xi #
,
I i = lj
the e x i s t e n c e
n
Pk
k=l Bik
)
(3.32)
of the a v e r a g e
n
~ 1 J = k=l aik e
n
1 1 k=l aik akj
e
,
of p.
We now show the e x i s t e n c e
. . = (BP)I3
Xj
l
0 demonstrating
if an only
(Ik-I
i
)s
(l,-l.)cl (Ik -li)c 3 -e lj-lk
Ij~X k n k=l
1 if a.. = 0 w h e n e v e r 13
(~
Bp.
We have
aI kj fk 9 - ~ Bkj)
n
. 1 1 1 + k=iZ ( c a i k a k j - C a i k
Bkj)
Xj=X k 1 1 aik akj
e (k J -li)a -e ( l k - k ) i~ lj-i k
lj~X k
n = k=lZ
1 (ik-li)c aik e
1 akj
e
(ij-lk)~
-i
Xj_X k
(3.33)
lj~k k If the real part of any e x p o n e n t a p p e a r i n g in this e x p r e s s i o n is p o s i t i v e , then by h y p o t h e s i s the c o r r e s p o n d i n g e l e m e n t of A 1 a p p e a r i n g in front of that e x p o n e n t i a l term v a n i s h e s . Thus the sum a p p e a r i n g here c o n t a i n s only e x p o n e n t i a l s with e x p o n e n t s with n o n - p o s i t i v e real part. Thus Bp exists. This c o m p l e t e s Remark
3.2:
We may
the p r o o f of the theorem. show that
13
n
1 aik Bkj
k=l
Ik-li
XiMlk
(3.34)
426
3.5
Estimate
of
the
Remainder
As we remarked above, we will deviation of the approximation
the
restrict ourselves v ° + ev I f r o m v.
to
estimate
Let w We will estimate
= v
- v°
Ev I
(3.35)
derive a differential its solution.
equation
dv Using ~ = eB(T)V (cf. ( 3 . 1 3 ) ) (3.25)), we ~ifferentiate (3.35) to
for
w,
dVo and ~ obtaln
solve
=
Bv
this
(cf.
equation
(3.21)
and
and
o
dVl(C~,T) dw(ex,T)
£Bv(£T,T)
-
£Bv
d~
(e~)
-
e
o
d~ (3.36)
=
£B(W+Vo+CVl)
-
EBVo(~T)
-
dVI(ET,T) £ dT
Let c(~) Then
From
from
=
B(~)o(o)
(3.25) and (3.30) we obtain nj d v I (t) -~ dt = BVl + CVo(t)
(3.18)
and
(3.29)
Vl(t,T) Differentiating
we
this
Combining this with which when inserted equation f o r w. =
EB(W+Vo+EV
given
d d-T w(e~,T)
To
estimate
w d? d-~ =
respect
(3.39) to
T gives
=
we
I)
~(B-B)v
by
eBw
+
first
£BT,
(B(T)
- B)v
o
(eT)
+
ep ( T ) B v
(3.38) gives us an expression for into (3.36) gives us the following
-
is
with
+
dt
C
v o (t).
dv l(eT) £
Note that becomes
p(~)
+
relation
dT
W(ET,T)
(3.38)
have
= Vl(t)
dv I (£T,T)
d dT
(3.37)
- ~(~)~.
-
o
(3.34).
e2
+
o
Using
=
I.
the
2-~ _ ~ Bv I
-
E2pBv
[ ( B - ~ ) v% I +
introduce
T(O)
cBv
¢2Cv
o
(£T) .
(3.40)
dV](£T,T)/dT differential
o (3.41)
o (3.37)
and
(3.39),
(C-C)~ ° ]
fundamental
(3.41)
(3.42)
matrix
? defined
by
(3.43)
427
Note
that d (~-I) d--~
Let
B
denote
the
= _e~-i B
complex
d 2 d d-~ I I~1 I ~ dT
"
(3.44)
conjugate
of
B and
* (~T ~ ) = £ ~ T ( B T
B T its
* * + B )~
T * Then since B + B is H e r m i ~ i a n , , t h e r e exists the l a r g e s t e i g e n v a l u e of B + B ) such that d il~ll2 d-7
< -- ~ K l l ~ l l
transpose.
K
Then (3.45)
(e.g.
the magnitude
2
of
(3.46)
Then
[f~l{
2 ~ e ~K~
(3.47)
By o u r h y p o t h e s i s (cf. R e m a r k 3 . 1 ) , B(W) is a b o u n d e d f u n c t i o n of for T ~ O. T h u s K is i n d e p e n d e n t of T a n d (3.47) s h o w s t h a t I I~II is b o u n d e d u n i f o r m l y for w r e s t r i c t e d to an i n t e r v a l of the f o r m , 0 < T < T/E. Now
we
w(E%,%)
solve
(3.42)
and
= ~(T) [ a 2 c 2 + . . . ]
write
+
T + E2
~(~)~-l(o)
[(B-B)Vl(eO)+(C-C)v
o
(~o)]do.
(3.48)
o °
In
(3.48)
write
(B-B)
as
d~c
(B-BldG'
and
(C-C)
as ~
(C-C)dg'
T h e n u s e the f o r m u l a (FGI{) ' = FGH' + F G ' t I + F ' G H t o i n t e g r a t e by the i n t e g r a l w i t h r e s p e c t to g. U s i n g (3.29) a n d (3.44) in w h a t results, we obtain finally
parts
T W(ET,~)
= C 2 {~(~)
O(1)
+ P (T)~l(~T)
+ 6 ~o T ~ ( ~ ) ~ - i ( ~ )
+~o
(C-C)dg'
~o(~r)
[B(o) ~o O ( C - C ) d O ' ~ o ( e S ) - p
(g) ~d
"~ Vl(eO)
o + Bp ( C ) V I ( £ O ) aj
-
(C-C)dc' ~o
Bv ( e c ) ] d c } o -~
.
(3.49)
We h a v e r e m a r k e d t h a t I I~I I is b o u n d e d u n i f o r m l y on the i n t e r v a l 0 ~ T ! T / ~ a n d t h a t ~ is b o u n d e d for T _> 0 (cf (3.47) f . f . ) . The f u n c t i o n s v (~o) and v. (cT) a p p e a r i n g in (3.49) are d e f i n e d as continuous ~unctions and their arguments r a n g e o v e r the b o u n d e d i n t e r v a l 0 ~ e% ~ T. T h u s t h e s e f u n c t i o n s m a y be u n i f o r m l y b o u n d e d . ,o The quantities p (o) a n d ~ (c-C)do' appearing in (3.49) e x i s t as bounded
functions
for
0 < o < ~ since
they
are
continuous
functions
428
W h e n the e i g• e n v a l u e s of A^u h a v e n e g a t i v e r e a l p a r t s , so t h a t method reproduces the r e s u l t s of the m a t c h e d a s y m p t o t i c e x p a n s i o n technique, ~ m a y a l s o be t a k e n to be zero. 4.2
The
our
Algorithm
F o r the sake of s i m p l i c i t y we take the l e a d i n g term, u (t,T) of the e x p a n s i o n (3.6) as an a p p r o x i m a t i o n to the s o l u t i o n ~f the initial value problem (3.1) a n d (3.10). F r o m (3.11) and (3.18) Uo(t,T) ~(T)
is
the
fundamental
%T while
from
= A0~'
matrix
~(0)
(4.1) given
by
= I,
(4.2)
(3.25) dv
o =
B
Vo,
= lim
~1
dt
From
= ~(T)Vo(t)
Vo(0)
=
a O.
(4.3)
(3.14) /oT 9-1 (o)A 1 ~ ( 0 ) d 0
(4.4)
.
We d e s c r i b e the a l g o r i t h m for r e p l a c i n g a the a p p r o x i m a t i o n to u(0) by U(h) the a p p r o x i m a t i o n to u(h) (in t h e ° s e n s e of the s o l u t i o n c o n c e p t in s e c t i o n 4.1 a b o v e ) . T h e a l g o r i t h m is to be r e p e a t e d approximating u(t) at u(2h) , ..., u(nh) s u c c e s s i v e l y . Algorithm
self j=0,
i) S o l v e (4.2) on a m e s h of i n c r e m e n t k in the T s c a l e by s t a r t i n g n u m e r i c a l m e t h o d , o b t a i n i n g the s e q u e n c e ~(jk), .... N.
some
ii) U s i n g the v a l u e s ~(jk) o b t a i n e d in (i), a p p r o x i m a t e B by t r u n c a t i n g the l i m i t of i n t e g r a t i o n T a n d r e p l a c i n g the i n t e g r a l in (4.4) by a q u a d r a t u r e f o r m u l a , say N
1
C k #-l(jk)
A 1 #(jk).
j=0 The that
integer the
N is d e t e r m i n e d
elements
of
the
by
matrix
a numerical B are
criterion
calculated
to
which some
assures
desired
accuracy.
v
o
iii) (h) by
With some
B (approximately) determined in self s t a r t i n g n u m e r i c a l m e t h o d .
iv) Compute m a t i o n to u(h) .
Uo(h,Nk)
= ~(Nk)Vo(h)
and
(ii) , s o l v e
take
this
as
(3.3)
the
for
approxi-
Refinement: T~e m e t h o d m a y be r e f i n e d by a d d i n g an a p p r o x i m a t i o n of £ v _ ( h h/£) to v (h) p r i o r to m u l t i p l i c a t i o n by #(Nk) (step (iv)). Thls approxlmatlon in t u r n Is d e t e r m l n e d f r o m a n u m e r l c a l s o l u t l o n of the e q u a t i o n s d e f i n i n g v l ( t , T ) ; viz.
429
which
are bounded at infinity max 0
our
desired
I W ( ¢ T , T ) I <__ c o n s t
e
2
(3.49) yields : (3.50)
,
estimate.
4. 4.1
(cf. (3.29)). Thus
Solution
THE
NUMERICAL
APPROXIMATION
Concept
From our consideration of the c i r c u i t equations in s e c t i o n 2 a n d / o r t h e f o r m (3.5) of t h e f u n d a m e n t a l matrix, we see t h a t t h e solution of t h e i n i t i a l v a l u e p r o b l e m may describe a trajectory with components w h i c h are, s l o w l y v a r y i n g , highly damped, highly oscill a t o r y or a c o m b i n a t i o n of a l l of t h e s e . T h e u s u a l n o t i o n of a numerical approximation to a t r a j e c t o r y is u n t e n a b l e for problems w i t h t h i s r a n g e of b e h a v i o r in its s o l u t i o n . In t h e h i g h l y o s c i l latory case the solution m a y c o m e as c l o s e as we m a y m e a s u r e to a n y specified precision to a m a n i f o l d of d i m e n s i o n greater t h a n one. (Consider
the
example
A0
=
[~-~]
and
A1
=
[-~
_~]
As
e +
0 the
solution converges (in an a p p r o x i m a t e s e n s e ) to t h e " c o n e " o b t a i n e d by rotating the curve I { u ( 0 ) | l e -L a b o u t t h e t a x i s . ) T h u s the meaningfulness of d e s c r i b i n g a trajectory by a s e t of i t s v a l u e s on the p o i n t s of a m e s h { j h l j = O , l .... , [T/hi } is lost. A variety of a l t e r n a t e numerical lated. Consider the f o l l o w i n g one.
Solution concept: (¢,6) (numerical) such that
Iu(t) Of c o u r s e approximation. is g i v e n
Given ¢ > 0 and approximation to
- u(t+~)l<_
solution
concepts
may
be
formu-
~ > 0 w e s a y t h a t U(t) is an u(t) if t h e r e e x i s t s • w i t h ITI
¢.
6 = 0 for t h e c l a s s i c a l concept of ( n u m e r i c a l ) In f i g u r e 4 . 1 an e x a m p l e of t h i s a p p r o x i m a t i o n
!
~
430
T vl(t,T)
= Vl(t)
dv I _% dt = BVl
-~B
-
;o
B(q)da
Vo(t )
(4.5)
+ R1 (t)
~(t) = (~--~- p--E)Vo(t). 1
om)\
I ~
f uo(h, h/~'Eo)(t) h ~--~(t/el
Figure In f i g u r e 4.2 practice E w i l l be
t
4.2
we s c h e m a t i z e the computation. extremely s m a l l so t h a t u n l i k e
Of the
c o u r s e in schematic
an e n o r m o u s n u m b e r of o s c i l l a t i o n s of # w i l l o c c u r in the t i n t e r v a l [0,h]. Notice how far the computed answer #(Nk)v (h) m a y be f r o m the c l a s s i c a l approximation to the s o l u t i o n , u (h~h/~) . The fundao m e n t a l m a t r i x #(T) is c o m p o s e d of m o d e s c o r r e s p o n d l n g to the e i g e n v a l u e s of A . Since A is n o t an u n s t a b l e matrix, the profile for (a o component) of # w i l l a ~ t e r s o m e m o d e r a t e n u m b e r of c y c l e s s e t t l e d o w n to an ( a l m o s t ) p e r i o d i c function. T h u s the set of m e s h p o i n t s {jklk=0,...,N} m a y be e x p e c t e d to e x t e n d o v e r j u s t t h e s e c y c l e s (approximately). 4.3
Numerical
Results
In t h i s s e c t i o n we t a b u l a t e the r e s u l t s of c a l c u l a t i o n s with three sample problems, P., i = i , 2 , 3 . P~ c o r r e s p o n d s to a d a m p e d c a s e (A^u h a s r e a l e z g e n v a l u e s ~ , P2 to a p u r e l y o s c z l l a t o r y A~ a n d P3 to a mixed case. The numerical m e t h o d s u s e d w e r e c h o s e n t~ be the m o s t elementary (e.g. E u l e r ' s m e t h o d f o r d i f f e r e n t i a l equations and Simpson's r u l e for i n t e g r a l s ) so t h a t the r e s u l t s are a c c u r a t e only to a f e w p e r c e n t . Moreover £ / h = .i or .2 so t h a t the e x a m p l e s are not particularly stiff.
°
•
0
~
~ . . ° . °
° ~ . ° ° °
0 0 0 0 0 0
0 0 0 0 0 ~
~ N)o o
°
•
.
o o ~o~
~ p ~ ~ 0 ~
~ o o
~ o o
. , ~ ° i ° ~ o o
~ 0 o ~ 0 ~ o ~
°
o
0
0
v
~P
,
h~
0
II
oIs1o(.;11-J o
II
~ooo
p~ 0
~
II
I1~00
OP~O
0
II
I-~ 0
I.~ 0 0 0
0 0
'oooo
II
..I..
11~
~-
0 I!
0
~
~ ~
0
o ~
j , . , ~ .
I
~
. . . . . .
~ 0 ~ o ~
0 0 0 0 0 0
0
0
o
o
0
0
~
Z
~P
o
o
~
!
o
II
~
i
,
0 ~
0 o o ~ o
i
,
o
0
c~
(-t
~
I--.
cl
0
o"
o
o
~
o
0
0
~
0
~
~
0 0 0 0 0 0
0
~ 0
~
o
0
~
0
0 0
0
0
ff Z
~
CP
~
tl
~
0
0
II
I-~
I-,
I1
o ~
II
rQ~
I
~o
0
o
O0
432
REFERENCES G. G. B j u r e l et al, S u r v e y of s t i f f o r d i n a r y d i f f e r e n t i a l equations, R e p o r t N A 7 0 . 1 1 (1970), Dept. of Inf. P r o c . , T h e R o y a l Inst. T e c h . , Stockholm. J.
Cole,
Perturbation
Methods
in A p p l i e d
Mathematics,
Blaisdeil
(1968)o
W. L. M i r a n k e r , S i n g u l a r p e r t u r b a t i o n a n a l y s i s of the d i f f e r e n t i a l e q u a t i o n s of a t u n n e l d i o d e c i r c u i t , Quart. of A p p l . M a t h XX (1962) 279-299. W. L. M i r a n k e r , The o c c u r r e n c e of l i m i t c y c l e s in the e q u a t i o n s of t u n n e l d i o d e c i r c u i t , IRE T r a n s . on C i r c u i t T h e o r y (1962) 3 1 6 - 3 2 0 . W. L. M i r a n k e r , N u m e r i c a l m e t h o d s of b o u n d a r y l a y e r s y s t e m s of o r d i n a r y d i f f e r e n t i a l equations, c~tin~,
a
t y p e for s t i f f ii, (1973)
W. L. M i r a n k e r a n d J. P. M o r r e e u w , S e m i - a n a l y t i c n u m e r i c a l s t u d i e s of t u r n i n g p o i n t s a r i s i n g in s t i f f b o u n d a r y v a l u e p r o b l e m s , IBM Research C e n t e r R e p o r t RC 4458, (1973). A.
H.
Nayfeh,
Perturbation
Methods,
SUPPLEMENTARY
Wiley
(1973).
BIBLIOGRAPHY
F. H o p p e n s t e a d t , P r o p e r t i e s of s o l u t i o n s of equations with small parameters, Comm. P u r e (1971) 8 0 7 - 8 4 0 . J. A. M o r r i s o n , C o m p a r i s o n of the m o d i f i e d two v a r i a b l e e x p a n s i o n procedure; S I A M Rev. V. M. V o l o s o v , A v e r a g i n g in s y s t e m s of R u s s i a n M a t h S u r v e y s 17 (1962) 1-126.
ordinary differential a n d A p p I . Math. X X I V
m e t h o d of a v e r a g i n g ~ (1966) 66-85.
ordinary
differential
and
the
equations,
CONCEPTION,
SIMULATION,
OPTIMISATION D'UN
D'UN
FILTRE
A L'AIDE
ORDINATEUR
Agn~s GUERARD, Centre National d'Etudes T@l@communications
des
INTRODUCTION Dans le cadre tr~s g@n@ral de l'analyse et de la synth~se de r@seaux @lectroniques, nous proposons ici un algorithrne qui a pour but de lin@ariser le d@phasage de la r@ponse d'un filtre en ajustant les valeurs des @l~ments de fagon minirniser l'amplitude de la variation du temps de propagation de groupe dans une certaine bande de fr@quenees, en g@n@ral la bande passante du filtre en question. Rappelons bri~vement qu'un filtre analogique est un quadrip~le compos@ de cellules @l@mentaires. Certaines de ces cellules sont des r@sonateurs qui rejettent chacun une fr@quence. L'ensemble constitue un Nitre qui ne laisse passer qu'une bande de fr@quence. II existe des programmes de synth@se directe qui permettent de calculer les valeurs des @l@ments et des programrnes d'analyse qui permettent de simuler le cornportement de la r~ponse en amplitude. Le programme r@alis@ ici perme% d'optimiser le comportement de la r@ponse en phase. II
-
METHODE A
PROPOSEE
- ELEMENTS
s =
f2
DU
PROBLEME
s ~_ K = [ a l , b l I
x Ia2, biIx ... X[an, bnl
s est le vecteur des param~tres de base, composants du r@seau, sur lesquels se fair la recherche d'un optimum. Ces pararn~tres sont au nornbre de n et varient dans un domaine K qui est, en g@n@ral, un produit d'intervalles. 2 o
Pour
route fr@quence
f =
~o , C k (s, u~) est la matriee
de chafne
in
ki~me
quadrip~le complexes.
@l@mentaire
constituant
le r@seau,
compos@e
de 4 nombres
.I
ck(s,~O)
=.
(s,~O) + j ~ , 2 ( s
C k est une fonetion de ~ n + l
......>..~
~.~ )
du
434
~ la fonction
d~signant le produit de matrices complexes carr4es d'ordre 2, C est de ~n+l_~
~t8 d~finie par:
C(s,~ ) = ~J. k=l
ck(s, ~0)
m d4signe le nombre de quadripOles 414mentaires sont suppos4s @tre associ~s en cascade.
3°
qui
Pour une fr4quence donn4e, posons A = C(s, ~ ) =
Alors l'amplitude a e t le d~phasage ~ / ta relation ci-dessous : ea+j~ =
R0
constituent
et R
n
1 2 V RoRn
4rant les r4sistances On en d~duit
C11
F'z2
!:~21
12r"1
de la r~ponse sont donn4s par
(CIIRn + r12 + [~21
d'entr4e
le filtre : ils
RoRn + ['~22 R0)
et de sortie appliqu~es
au quadrip61e.
: ......
4°
Remarque :
Une application lin~aire de ~ 2.__)(~2 , repr~sent4e par un syst~me lin~aire complexe d'ordre 2 :
~1 + J ~ 2
I ~I +j ~ 2
~i
+J
6"1 +J
~2
~2
=
71 + j
~2
peut @tre consid4r4e comme une application lin~aire de t4e par le s "st&me ci-dessous :
L =
ol I
_4 2
~2
°~I
~I
•
4~
-~2 g2
71
72
"C2
et repr4sen-
435
B
- PROBLEME
On eherche dans le domaine K de variation des pararn&tres s, domaine qui a @t@ d~termin@ par exemple par un programme de simulation de fagon clue le gabarit en amplitude y soit toujours respectS, le point pour lequel la phase est "la plus lin@aire possible" dans la bande passante. Sur un ensemble dans la bande passante, cient de corr41ation.
de fr~quenees la lin4arit4
(U,~l) I = 1 & L convenablement
du d4phasage
est earaet4ris4e
r4parties
par le coeffi-
L
L (~IJ ~ 5--_ i=i Ce coeffieient
contraintes
Ce probl~me :
x
L (~o i_ ~)2 i=I
doit etre le plus proehe consiste
done
& minirniser
C
I.
pour s 6 K, sous les
L
C (s, ~ i )
- RESOLUTION
On introduit les multiplicateurs
R t , R 2 ...... et
I- p
de la valeur 2
!
Vl ~ ~" 1,2 A1 =
possible
le Hamiltonnien
:
Rt
:
~Is ;fl,r2 ..... ~L' L
~I'R2 ..... RLI = L
RIX (AI-C(s,W
I)
436
Remarque
: On note
(AI-C(s,~I))
R 1 ± (A 1 - C
eonsiddr~s
Minirniser pour s ~ K. On dolt ~
Mais
(s~ cO i )) le produit des
vecteurs
de
i- ~ 2 sous les contraintes
donc
chercher
(s ~) = 0
= _
~
cornme
~
point
s~
et
~2
+
~2
/I/1
scalaire
indiqu~es
point s ± on doit donc
D'autre s , on peut 4erire
part,
avoir
si on se trouve
revient
& minimiser
K tel clue : (s ~) = 0,
~I
et l
Au
de R 1 et de
~8.
=
I
:
en un point s relativement
proche
de
: n
i=l
~i 1
Al'optimum,
~
= O, et :
~) i
1 D
- ALGORITHME
1° - Choisir s (0), point de d4part. 2 ° - A l'@tape k,
s (k) @rant suppos~ connu, L analyses du syst~me
pour les diff~rentes valeurs (~ 1)1=1 & L fournissent 11 et A I, pour 1 = 1 & L.
437
O n en d~duit
:
et
3 ° - L e p o i n t s ( k + l ) e s t l a p r o j e c t i o n s u r l e d o m a i n e a d m i s s i b l e K du point s '(k+l) obtenu en modifiant une seule des composantes
~.'i(k+l) _ffi(k) _
.
d e s {k) :
~ (k)
~/~¢i l ' i n d i c e i d e l a c o m p o s a n t e modifi4}e e s t a u g m e n t 6 , it4ration.
m o d u l o n, de t ~ c h a q u e
4 ° - L e p r o c e s s u s e s t i t 4 r 6 j u s q u ' ~ c e clue s o i t a t t e i n t e u n e p r 4 c i s i o n r e l a t i v e r a i s o n n a b l e p o u r t e s v a l e u r s d e s c o m p o s a n t s , s u i v a n t l a n a t u r e du p r o b l a m e t r a i t G ou b i e n l o r s q u e l e p o i n t c o u r a n t s (k) s e t r o u v e ~ u n s o m m e t du d o m a i n e K. Note
:
Au cours des premieres it4rations, et si le domaine K est vaste, c o n v i e n t d ' a p p l i q u e r un c o e f f i c i e n t de s o u s - r e l a x a t i o n : 6"' ( k + l ) i
ff (k) i
_ ~
~(k)
:
p o u r 6 v i t e r un 6 1 o i g n e m e n t du p o i n t o p t i m u m .
, ,~<1
il
438 111
-
APPLICATION
Consid~rons par exemple le quadrip61e dont le s c h e m a ~quivalent est ci-dessous :
75~-
i
t~lasp f
alq'~qp F
~: ~z~,ootF
'~ =;t~'ofF
I|
j.
~,lzo p F
I~ : 1o8oor F
i
~-5~ ~sopf
w
C ' e s t un f i l t r e p a s s e - b a s dont l a bande p a s s a n t e e s t 0-200 KHz. On c h e r c h e d o n c & m a x i m i s e r l e c o e f f i c i e n t de c o r r ~ l a t i o n ~ _ dans la bande 0-200 KHz, en f a i s a n t v a r i e r u~Aquement l e s c a p a c i t ~ s C 1, C 2, C 3 et C 4, RESULTATS
A / AC max
s2
0. 964 0. 965 0.968 0. 972 0. 974 0. 975
= 2000
pF
C 1 7170 9170
C 2 12400
C 3
C4
10800
3550
10400 8988
5550 8800
439
B / ~ . C m a x = 5000 pF
2 s
0. 964 0. 967 0. 970 0. 974 0. 9768 0.97 69 0. 979 0. 981 0. 983
C 1
7170 12023
C 2
12400
C 3
10800
C 4
355O
10660 8927 52 84 12170 9750
8116 6635
440
Bibliographie
J. CEA Optirnisation,
th~orie et algorithmes,
Dunod
1971.
P. ALLEMANDOU Calcul des filtres en tenant compte des pertes des composants C~bles et transmissions, 25, n ° 2, avrfl 1971.
A. GUERARD et
A. J O L I V E T
C o n c e p t i o n a s s i s t ~ e des f i l t r e s p a r o r d i n a t e u r . A n n a l e s des T ~ l ~ c o m m u n i c a t i o n s tome 27, n ° 12, j a n v i e r - f ~ v r i e r 1972.
Y. B E N I G U E L - A. G L O W I N S K I - A. GUERARD Conception, s i m u l a t i o n , o p t i m i s a t i o n de f i l t r e s p a r o r d i n a t e u r . L ' E c h o des R e c h e r c h e s , n ° 71, j a n v i e r 1973.
dissipatifs.
COMPUTING METHODS IN SEMICONDUCTOR PROBLEMS by Martin Relser IBM Research Laboratory San Jose, California
1.
INTRODUCTION
The fast rate of progress in semiconductor and dimensions, adequate.
technology has led to structures
where the traditional analytical approximation models are no longer
What is needed is a two- or even three-dimensional
set of nonlinear partial differential
solution of the full
equations, which is known as diffusion model.
The success of treating two-dimenslonal
problems analytically was rather limited
[Fulkerson 68, Lewis 70] and one has to revert to numerical methods to obtain solutions
for the complicated
geometries of modern devices.
suitable computer programs for one- and two-dimensional
In recent years,
analyses have begun to
close the gap between the simple models and the needs of the device designer. rapidly growing literature documents
A
the increasing influence of computing in this
field. The object of this paper is to survey numerical semiconductor
device problems.
solution methods for
For an area of research which is still in a
relatively early stage as far as rapid progress is concerned,
such a survey cannot
be fully objective and personal judgement plays an important role. therefore,
are all subject to discussion and revision.
to their relevance
to numerical methods.
of the device problem. in one-dimensional
The findings,
Results are quoted according
Emphasis is on two-dimensional
The references are not fully comprehensive,
solutions
especially not
solutions.
The history of numerical methods in semiconductor pioneering work of H. K. Gummel about a decade ago.
problems started with the His sequential
iteration method
442
is still widely used today.
First transient solutions were attempted in the area
of Gunn diodes as early as 1966 [McCumber 66].
For p-n Junctions and Read-Diode
oscillators such solutions followed in late 1968 [DeMari] and early 1969 [Scharfetter].
In the latter publication D. L. Scharfetter and H. K. Gummel
introduced the best suited finite-difference approximation.
Among the later
publications of one-dimensional results [DeMari 68, Caughey 69, Gokhale 70, Arandjelovic 70, Hachtel 72, Seidman 72, Petersen 73] the method of G. D. Hachtel, R. C. Roy and J. W. Cooley deserves special attention because of its efficiency and generality.
One-dimensional numerical analysis plays an important role in many
areas but its most significant contribution was for the bipolar transistor. Two-dimensional solutions appeared in the literature in late 1969 [Kennedey, Slotboom].
The first two-dimensional version of Gummel's method was described by
J. W. Slotboom.
Around the same time, the simpler zero-current or capacitance
problem was successfully solved by numerical methods [Dubock 69, Wasserstrom 70]. The most important incentive, however, came from the development of the modern short channel field-effect transistor in the mid-sixties.
Unlike the bipolar
transistor, this device is based on two-dimensional flow patterns.
The scope of
the traditional regional modes for this device is quite limited and failed to answer many vitally important questions.
The urgent need for better models Justified the
large-scale use of digital computing and the field-effect transistor has become the first device extensively investigated by two-dimensional numerical analysis [Loeb 69, Kennedey 69,-71, Kim 70, Reiser 70,-71,-72, Vandorpe 71,-72, Heydemann 71, Jutzi 72, Ruch 72, Himsworth 72, Mock 73].
Some attempts were made to reduce
the amount of computing by substituting a two-dimensional potential solution in the regional model [Loeb 69, Amelio 72] or by simplifying the geometry [Kim 70]. These attempts to avoid a full solution, however, were not too successful. Understandably, the d.c. problem was tackled first.
It was, however, shown that
getting a complete transient solution is not necessarily more demanding than the d.c. problem [Reiser 71,-72] and a complete d.c.-, a.c.- and transient analysis of Schottky-gate fleld-effect transistors was achieved [Reiser 73].
443
Two-dimensional technology.
analysis played a less significant role in other areas of device
Only few results, for example, were published for the bipolar
transistor although the need for understanding parasitic effects (such as current spreading)
seems to have led to a revival of two-dimensional
[Heimeier 73, Slotboom 73].
analysis in this field
Other two-dimensional results were obtained for Gunn
diodes [Katakoa 70, Yanai 70, Suzuki 72, Reiser 71] to handle questions about domain nucleation and stabilization.
2.
THE PHYSICAL MODEL
The basic physical model for the charge transport in semiconductors has remained essentially unchanged since the pioneering work of Shockley and Schottky in the fifties.
Its success in understanding and quantitatively describing semiconductor
devices was so great that it is now generally accepted without further discussion. This is dangerous in physical sciences and technology is, indeed, rapidly progressing into dimensions where the model can no longer be taken for granted. In this section, we present the equations which are known as diffusion model. A short discussion of the basic assumptions underlying the diffusion model is also given.
2.1
The Diffusion Model In suitably normalized form, the equations are:
(i) Continuity Equation ~nn = ?.J + R ~t n n
~--P = - v J
3t
'
+R
p
p
(1)
(2)
(2) Transport Equation ~n = n~n~ + Dn'Vn
(3)
~p = p~pE - Dp-Vp
(4)
444
(3) Poisson's Equation -V2~ = N + p - n
(5)
(6)
The meaning of the symbols is: n: electron density, p: hole density, ~"• current density, R: recombination rate, ~: mobility, D: diffusivity, ~: electrical field, ~: potential field, and N: the ionized impurity density. The subscript n refers to electrons, p to holes. Additional state equations describe properties of different materials.
The
following assumptions are typically made: (I) R
n
= R
p
= R(n,p) according to the Shockley-Read-Hal! model.
(2) Global validity of Einstein's relation, i.e., D = ~ and empirical formulas for the dependence of the mobility on the electrical field-strength E, i.e.,
~n = ]/n(E)[~ + Vn]
(7)
(3) All impurities ionized, i.e., N = N(~) with x the position vector. The mathematical description is completed by specification of a domain, three boundary conditions for n, p and $ and an initial condition in the case of the time dependent problem.
The boundary usually consists of several segments, i.e.,
contacts and insulated surfaces where (I) ~, n and p have fixed prescribed values in the case of ohmic contacts. (2) ~ given and normal components of ~
n
and ~
p
zero in the case of insulated
surfaces. The following transformation of variables is often found:
V>+(hn n
=
e
(s)
-~+¢p p = e
(9)
445
with # called pseudo Fermi-potentials.
Equation
(7) becomes
~n = PnnV~ n = ~n e
and similarly for ~p.
(lO)
V~ n
It is also customary to introduce the logarithmic variables
~n = exp(~n) and ~p = exp(~p) yielding the form
~n = ~ne~?~n
In the zero-current or equilibrium case it is ¢ = 0 and equations
(Ii)
(8) and (9) revert
to the well known relations
n = e~
,
P = e-~
(12)
(13)
Combining Poisson's equation (5) with (12) and (13) gives the so-called Shockley-Poisson
equation
-V2~ = N + e -~ - e ~
(14)
which describes the zero-current case.
2.2
Limits of the Diffusion Model The diffusion model can be derived from the Boltzmann's equation by a set of
assumptions,
namely:
(i) The distribution can be fully characterized by the mean velocity v (or equivalently the second and all higher moments have negligible relaxation time). The problem can then be described by the first two moment equations. (2) Neglect of
~$/~t and
momentum balance) to get
of the Bernoulli term in the first moment equation (or
446 ->
nv : -J
= -n~
- DVn
(15)
n
(3) Neglect of magnetic forces and static treatment of the electric fields. Most serious is assumption (i) and its consequences, i.e., no thermal forces and no explicit energy balance equation. The electrical field-strength is not a good description of the physical state and equation (7) may lead to serious contradictions.
This becomes apparent in
regions where large density gradients are balanced by correspondingly high fields so that no current flow results. unphysical "hot-electron" effects.
In such areas, equation (7) would produce This problem has led to serious difficulties
in the analysis of the insulated-gate field-effect transistor.
A better form based
on the assumption of addltivity of the forces is [Reiser 71]
:
n
nV(F)~
(16)
with ~ = ~ + V log (n) and ~(F) the same form as ~(E) in equation (7).
2.3
Special Aspects of Various Devices Not all terms and equations of the diffusion model (i) to (6) are always
relevant and simpler models which are computationally less demanding may be adequate.
The following is a short discussion of the properties of various
important devices: (i) The bipolar transistor is based on injection of minority carriers and thus the full set of equations is relevant.
The solution depends critically on the
recombination and diffusion term and a density-dependent diffusivity may even be required to account for degeneracy in high injection levels [Hachtel 72]. A one-dimensional analysis is sufficient in most cases. (2) The field-effect transistor is a majority-carrier device and therefore neglect or approxi=mtive treatment of the minority carriers are usually justified. Recombination effects can be neglected, even if minority carriers are taken into account.
The solution is insensitive to changes in the diffusion term.
447
The proper nonlinear drift term, however, is important.
The shortcomings of
drift equation (7) are especially apparent in the current channel of the insulated gate field-effect transistor and use of the more appropriate form (16) is strongly indicated.
The fleld-effect transistor requires a
two-dimensional solution. (3) Gunn-effect devices.
In the case of "two-valley" semiconductors such as GaAs,
the solution is totally dominated by the nonlinear drift law which has a range of negative differential conductance. neglected.
Minority carriers (i.e., holes) can be
The proper form of the state dependent diffusion term is unknown
and many authors use a constant diffusivity.
Often no steady-state solution
exists and therefore a transient analysis is always required.
3.
}~R~ERICAL SOLUTION
The set of equations known as diffusion model are nonlinear partial differential equations of the parabolic type or of the elliptic type in the case of the d.c. problem.
All numerical solutions of these equations described so far in the
literature fall into the class of finite-difference methods on rectangular grids with low order centered difference approximations (so-called five-point formulas). The problem then is one of: (i) choosing optimal finite difference expressions, and (2) solving the resultant system of algebraic equations efficiently. The need for efficiency is absolutely predominant since especially in the two-dimensional case the demand on computing and the resulting costs may well be prohibitive for any extensive use of numerical analysis. The following well known properties of the solution are typical for an ill-conditioned system and make a numerical solution accordingly difficult: (i) Over short distances, the density variables may vary over several orders of magnitude. (2) The solution, especially the location of space charge areas is very sensitive to changes in the boundary conditions (i.e., the applied voltages).
448
(3) Important macroscopic quantities such as the device current are defined over differentials of the solution.
3.1
Mesh and Finite-Difference Approximation Regular rectangular grids are a preferred choice for the solution of
two-dimensional plane problems as they lead to simple finite-difference formulas and to sparse matrices with only few non-zero diagonals (a regular mesh is one where no grid line ends in the interior of the domain).
To save mesh points in
the uninteresting neutral areas nonuniform mesh spacings are often used.
Regular
nonuniform grids, however, also have drawbacks which in certain cases may even offset their advantages, i.e., (i) reduced order of accuracy, (2) numerically degenerate long-shaped rectangles in certain areas, and (3) computational overhead in the equation setup. There is a great deal of freedom in choosing finite-difference approximations to the original equations.
Two requirements are important, viz.,
(i) Consistency, i.e., the finite difference equation converges to the differential equation if the mesh is refined. (2) The conservation property of the continuity equation has its discrete analogue. The second condition turns out to be necessary in order to obtain a uniquely defined device current [Reiser 71]. class satisfying both conditions. in the literature.
We subsequently give several examples as found
For notational convenience a one-dlmensional uniform grid with
spacings h is assumed. straightforward.
Centered difference formulas are the simplest
The generalization to two space-dimensions is usually
Moreover, we give the examples for the electron current only and
omit the subscript n.
The basic centered difference approximation of the divergence
term is
V']Ix=ih = [~i+I/2 - ~i-I/2 ]/h + O(h2)
(17)
The various methods published differ in the approximations ~ of the current density, i.e., in basic variables n [Katakoa 70, Reiser 71, Suzuki 72]
449
~i+I/2
ni+l + ni ~i - ~i+l ni+l - ni 2 Vi+i/2 h + Di+i/2 h
(18)
or in logarithmic variables ~ [Slotboom 69]
$I+I12 $i+I - ~i ~i+i12 = ~i+i/2 e h
(19)
where exp($i+i/2) may he approximated by grid values by ll2[exp(~i) + exp(~i+l)] or exp(½ ~i + $i+I ) or better by [Mock 73] [exp($i+l) - exp($i)]/h.
Some
position-dependent scaling is required as ~ may become very large and, in fact, even exceed the range of floating point numbers of many present-day computers.
A
particularly simple form is obtained if the density function itself is used for scaling, in which case equation (19) becomes [Vandorpe 71]:
[P -~i+l- n i e -*i]
^ (~i + ~i+l)/21ni+l e Ji+i/2 = ~i+i/2 e -
h
(20)
"
As first observed by D. L. Scharfetter and H. K. Gummel all the above formulas may lead to gross errors in case of large voltage drops IV~I = I$i+1 - $i I over a single mesh cell.
This can easily be seen in equation (18) which in case of zero-current
condition, cannot be satisfied by both n i and ni+ I positive whenever IV~I > which is obviously an unphysical situation.
2,
As reported recently, the same type
of error is associated with equations (19) and (20) [Mock 73].
A serious
restriction of the mesh size may result in case of high bias voltages.
A scheme
which avoids this problem may be obtained by assuming J and E constant over the mesh cell and integrating the transport equation j -i = nE + dn/dx yielding
A~ ~i+i/2 = -~i+I/2 h
For
IA~!
ni+l ni l i - exp(A~) + i - exp(-A~)"
<< I, equation (21) reverts to the standard centered difference formula
(18), whereas for iA~I >> i a pure drift current of the form
(21)
450
Vi+i/2 hA--~ni+l
if
A~ > 0 (22)
~i+I/2 = ~i+i/2 hA--~ni
is obtained,
if
A~ < 0
The superiority of this Seharfetter-Gu~el scheme was recently
demonstrated theoretically [Mock, to appear]. The discretization of Poisson's equation offers no problems. five-point formulas [Verge 62] are most widely used.
Standard
Care has to be taken that
the discretization of the boundary conditions does not violate the current conservation property.
Techniques for treating boundary conditions are well known
(viz., the so-called phantom-point method) and need not be discussed in detail.
3.2
D.C. Problem In case of the d.c. problem, the time derivatives in equations (I) and (2) are
put to zero.
In discretized form a set of nonlinear algebraic equations results
which may be symbolically written as
A(~)n
= -R
(23)
~(~)m
=
(24)
L~ = N + ~ -
where ~, ~, ~, R and N are the vectors of grid values, and A(~), B(~) and L are band matrices (three-diagonal in the one-dimensional case, five-diagonal in the two-dimensional case).
The proper form of A(~) and B(~) is defined by the
finite-difference formulas used, i.e., in the case of the standard centered difference formula (18) A(~) becomes
(25)
451
i Di+i/2 + ~i+i/2 ~(~i - ~i+l )
h2A(~) = h2[aij(~)] =
if j=i+l
1 -(Di+i/2 + Di_i/2) + ~- ~i+i/2(~)i - ~)i+l)
i - 2 ~i-1/2(~i-1 - ~i )
Di_i/2 - ~i_i/2(~i_l - ~i )
0
(26)
if j=i
if j=i-i
otherwise
where in case of field-dependent ~ and D it is ~i+i/2 = ~(~i - ~i+l ) etc. Two methods have been in use to solve this system, namely: (i) sequential iteration or Gummel's method, (2) Newton's method. Methods of the first kind are simple but have only a first order rate of convergence whereas Newton's method requires a more complicated equation setup but promise second-order convergence. Sequential iteration is the most widely used method.
It is based on the fact
that the discretized continuity equation is linear in the density variables n and p or ~n and ~p (but not in ~n and ~p).
This suggests the following algorithm:
Step i: Compute mobility and recombination rate for current values of ~k, k
k = V(~k)
and k
, R k = R(nk,p k)
Step 2: For ~k fixed compute n#+I and pk+l such that the continuity equations are satisfied, i.e.
452
k)nk+l .
o
_Rk _
,
k+l =
Step 3: Compute updated potential values ~k+l as a function of the newly computed k + l and k + l Steps 1 to 3 are repeated until the desired convergence is achieved (or divergence becomes apparent).
Note that both Step 2 and Step 3 lead to systems of
linear equations which in the two-dimensional case are of five-diagonal form and may be solved conveniently by standard iterative methods [Varga 62]. density variables ~
n
Logarithmic
and # may substitute for n and p. p
Many variations of this algorithm have been used.
They differ mainly in the
following: (I) The density state variables, i.e., basic variables n and p [Dubock 70, Vandorpe 71], logarithmic varlables ~
n
and ~
p
[Slotboom 69, Heydeman 71, Heimeier 73]
or other [Mock 73]. (2) The finite-difference formulas (see section 31). (3) The methods used in Step 3 (see discussion below). (4) The methods for solving the systems of linear equations in Steps 2 and 3, i.e., point relaxation (SOR) [Slotboom 69, Dubock 70], line relaxation (LSOR) [Vandorpe 71, Heydemann 71, Mock 73], Douglas Racheford methods (ADI) [Vandorpe 72, Mock 73], or Stone's method (SIP) [Heimeier 73, Mock 73].
LSOR is
particularly well suited for the insulated gate field-effect transistor whereas SIP is best if different device geometries are to be analyzed. The simplest way to treat Step 3 would be simply to solve the Poisson's equation _?2~k+l = N + p - n.
This formula, however, was never actually used as it would
lead to slow convergence.
Most widely employed is the method of the original
publication [Gummel 64], i.e.,
V2~ k+l + (n + p)[~k+l _ ~k] + N + p - n = 0
(27)
453
This formula is derived by linearization of the nonlinear potential equation -V2~ + ~p exp(-~) - #n exp(~) = 0 around the current estimate ~ k
A different
approach to accelerating is the following two-stage iterative process:
~k+l = ~k + k [ ~ + l
where -~"'i is defined by -V2~ -+I--~
=
_ ~k] + 8k[~k _ ~k-l]
N + p
-
n and k
(28)
and ~k are acceleration
parameters for which appropriate Chebychev sequences can be constructed [Mock 73]. A mathematical analysis of the asymptotic rate of convergence was recently achieved under some simplifying assumptions [Mock 72].
Convergence problems are
predicted for forward bias condition and high recombination rates.
Some variations
of the sequential iteration procedure avoiding these convergence problems were subsequently described [Seidmann 72].
The idea is to treat a linearized part of
the recombination rate computation in Step 2 rather than in Step i.
Note, however,
that the analysis which is based on a perturbation argument says nothing about the global convergence behavior.
Unfortunately, the informmtion about this most
important question is only fragmentary. are found in the literature.
Figures from 20 to 200 overall iterations
Generally, convergence is slower:
(i) the more charge is stored in the device (i.e., the higher N); (2) the higher the applied voltages; (3) the higher the recombination rate. Arriving at a suitable initial guess of the solution is a nontrivial problem.
In
case of large voltages~ the full voltage drop may have to be applied in small steps at a time.
It may even be necessary to build up the doping level slowly in order
to avoid convergence problems. Newton's method has so far been used in one-dimensional problems only [Scharfetter 69, Gaughey 69, Hachtel 72].
The difficulty with two-dimensional
problems is the fact that now the Poisson and continuity equation have to be solved simultaneously, thus leading to a larger system of equations with more than five non-zero diagonals. form
If for simplicity holes are neglected, the equations take the
454
(29)
where x is the combined vector of mesh values, i.e., x identity matrix.
T
= (n,~) and I is the
Then, Newton's method is:
x
k+l
=x
k
-
j-i (x k) F ~k) ~
(30)
or more conveniently for numerical computations:
j k+l = j( k)xk _ F ( k )
where J(x) is the Jacobian (i.e., Jij = ~Fi/~xj)'
(31)
Since A and L are five-diagonal
matrices, J has the structure
(32)
where A, A' = ~A(~)/~ and L are all five-diagonal submatrices.
Classical methods,
such as SOR, are not suitable for solving this system of linear equations since J has more than five non-zero diagonals.
Stone's method [Stone 68], however, is
exactly tailored to such "coupled systems" and allows for a efficient solution. Promising results with this Newton-Stone's method have already been achieved [Hachtel, private con.].
3.3
The Transient Problem Finite difference schemas of the Crank-Nicholson type [Richtmeyer 57] are the
simplest method for solving the transient problem.
In case of electrons in the
absence of holes, the discretlzed continuity equation takes the form: k+l At
k .,,k+l, k+l mt~ )~ + (i - ~)A(~k)n k
(33)
455
with an error term of the order 0(~t2, h2) in case of a = 1/2, O(At,h 2) otherwise. In case of e = 0 one speaks of an explicit scheme, e > 0 are implicit schemes.
As
is well known, the explicit scheme, which is computationally very simple (no systems of equations) has a strong tendency towards instability and may require prohibitively small time steps, limited by the following stability condition [Reiser 7Z]:
h2D '
At ~ min
where v
max
= max(~)
2D 1 Vma x
(34)
is the maximum drift velocity which in the case of
field-dependent mobility is always finite and well known.
Implicit schemes, which
avoid this stability problem, lead to nonlinear systems of equations similar to those of the d.c. problem and which have to be solved for every time step.
This
is feasible in the one-dimensional case but may lead to an excessive amount of computing in the two-dimensional case.
Therefore, despite the small At, explicit
schemes were used for Gunn diode analysis.
A compromise between the speed of the
explicit method and the stability of the implicit method is the half-implicit method of M. Reiser.
This method is based on the same principle as the sequential
iteration, i.e., the fact that the continuity equation is linear in n, provided is held fixed.
This leads to the following algorithm for carrying out one time
step :
Compute ~k for given n k according to e~ k = N - n k. Step 2: Compute new density from the system of linear equations k+l n
k - n
At
i ~. k. k+l i k-nk) ~n + y A~ 2 ~ ~
-- - A ~
Both the above steps require the solution of five-diagonal systems of linear equations.
Very efficient direct methods exist for the Poisson problem of Step i
(35)
456
[Hockney 70] which should be utilized whenever possible.
Standard iterative methods
are appropriate for Step 2. The instability giving rise to condition (34) is totally avoided by the half-implicit method.
It was, however, found that decoupling of continuity and
Poisson's equation introduced stability problems of a different kind, leading to the following restriction on At [Reiser 73]:
At < -
i
(36)
~N
Note that unlike equation (34), the above stability condition is independent of the space increment h.
Also the behavior of the unstable solution is quite
different from the exponential growth pattern of the diffusion (or linear) instability, i.e., it exhibits stationary standing waves in the density function. These oscillations preferably in neutral areas do not grow as time goes on but prevent the solution from reaching a steady-state.
The stability condition (36)
restricts the applicability of the half-implicit method to small doping levels N < !ol7cm-3, as they are typical in Schottky-gate field-effect transistors and Gunn-effect devices.
Where applicable, however, it is an efficient method and,
owing to the increased stability, faster than the explicit method. The transient solution is the most general one and a computer program for a transient analysis provides answers to the following problem areas: (i) large signal responses; (2) small signal analysis by means of Fourier analysis of step responses; (3) steady-state or d.c. analysis. To compute d.c. solutions one starts from a suitable initial condition and proceeds until a steady-state is reached.
Unlike in the case of sequential iteration, this
initial solution is completely uncritical and o
= N has proven to be a good choice.
Furthermore, the method is very insensitive to large voltage steps and viewed as an iterative d.c. method, has a linear rate of convergence.
It is therefore
comparable in efficiency to the sequential iteration method but has the important
457
advantage of giving additional information about important time-constants of the device as a by-product of each d.c. computation.
3.4
On the Problem of Accuracy In many publications the adjective "accurate" appears in the title but they
generally fall short of convincingly demonstrating its justification.
The
mathematical theory is still far from producing error bounds for a given mesh and even the somewhat weaker question of convergence of a given finite-difference scheme (i.e., whether the solution of the difference converges to the solution of the partial differential equation) generally remains unsolved.
Some progress towards
a mathematical treatment of the convergence problem has been made recently [Mock, to appear].
Bounds on the error in the total device current, introduced by an
imperfect potential solution ~ were found.
The assumptions are (i) D = ~ =
constant, (2) ~ and p such that the continuity equations are fulfilled exactly [i.e., V-(V~ - ~V~) = 0], and (3) the current i is computed as a suitable average of the currents through the ohmic contacts.
Then the error II - ~i is found to be
bounded by
iI - ~i ~ C l ~ -
where C I is a suitable constant, the potential residuals.
(37)
li~resll
l i" I I is a norm and ~res = V2~ + N + p - ~ are
Note that this result is obtained by purely analytical
means for the general two-dimensional case with mixed boundary conditions.
In one
space dimension, equation (37) may be sharpened and applied to particular finite difference schemes.
It is found that standard centered difference formulas for
the Poisson problem together with Scharfetter-Gummel type formulas for the continuity equation yield an error O(h 2) in the computed device current. schemes are shown to have larger error, i.e., of the form O(h).
Other
This important
theoretical result supports the earlier finding that th Scharfetter-Gummel formula is better than other common discretization schemes.
458
In the absence of a rigorous mathematical theory, numerical experiments are the only means of investigating the properties of numerical methods.
The following
should always be performed and the results disclosed in the publications: (I) observation of the solution as the mesh is refined; (2) convergence properties of the iterative equation solvers; (3) check on the independence of the device current of the integration paths; (4) single and double precision computations. With regard to point 3, for example, it is generally found that a large portion of the required iterations is spent to slow and minute changes in the field variables in order to get a path independent current. Round-off errors are an additional source of inaccuracy.
Although there is
generally no error build-up in the Iterative equation solvers for the systems of linear equations arising, the finite precision arithmetic may lead to serious accuracy problems, especially in the case of fine meshes. computations are advisable when economically feasible.
Double precision
In the case of
Scharfetter-Gummel formulas, special attention should be paid to the evaluation of exponential terms in the denominators.
It is an advantage to test the value of
IA~I and revert to a standard flnite-difference formula in case of IA~I << i.
Such
a procedure is not only more efficient but reduces serious cancellation problems in computation of the terms [I - exp(A~)] and [I - exp(-A~)].
3.5 The Systems Aspect of Computer Programs for Device Analysis Computer programs which embody the numerical methods described above usually represent a long programming effort in the neighborhood between one to five man-years.
The ideal program has:
(i) a user oriented input language; (2) a numerically stable equation solver which requires no extra information; (3) a graphical result display and storage of already computed results in libraries; (4) programs for handling such result libraries. Experience shows that the auxiliary components (i), (3) and (4) are a significant part of the whole programming effort, especially if graphical output is desired.
459
It is most desirable The simplest,
to have at least some primitive form of a graphical output.
fastest and least expensive means are chain-printer plots.
The
resolution of such plots is adequate for debugging and gross characterization the solution. Perspective
More sophisticated
is the use of plotters or CRT displays.
drawings or contour maps have both been used successfully.
purposes, movie pictures were made by several authors, for steady-state
of
solutions and [Reiser]
[Slotboom, Nachtel]
for transient solutions.
system described so far makes use of an interactive evaluation and history file handling
i.e.,
For tutorial
The most elaborate
CRT terminal for result
[Reiser 72].
None of the computer programs known so far is general enough to handle several devices.
They are all more or less ad-hoc
(i.e., for one case only) solutions and
running the programs usually requires supervision by the authors.
There is still
a long way to general purpose simualtion packages for a two-dimensional like in common use for one-dimenslonal
4.
analysis
problems.
CONCLUSION
In about one decade, numerical device models have become an important tool for research and development a conclusive
of semiconductor
state is reached.
devices.
Scharfetter-Gu~m~el
with Newton's method have proven to be accurate,
In the one-dimensional finite-difference
formulas paired
stable and efficient.
program packages with automatic control of the dlscretization
case,
Elaborate
prarameters
such as
mesh size have been implemented and extensively used for design of the bipolar transistor.
Besides these d.c. programs,
obtained for wlrious devices. one-dlmensional
a.c. and transient solutions have been
The computing capacity necessary
for any desired
solutio~ is now freely available and procuring such solutions may
be considered a routine task. The situation in the two-dimenslonal
case is far from such a definitive state.
Computing costs are still a limiting factor to the proliferation the search for the best methods is in full progress. where a two-dimensional field-effect
of programs and
The only device,
so far,
analysis has played an important role is the short channel
transistors.
Finite difference methods using centered difference
460
formulas on rectangular meshes are common to all the published results. Understandably, the d.c. problem is being most intensively investigated and the relatively simple and slow sequential iteration procedure is the main method for solving the systems of nonlinear equations.
The Newton-Stone method is an
interesting alternative and may very well prove similarly superior as in the one-dimensional case.
Transient solutions have been obtained for low-doped devices
but the simple schemes used are not capable of solving the general case.
There Is
no way in sight to avoid this difficulty without exceeding reasonable bounds on the demand on computing. The finite-difference methods used in semiconductor device analysis are rather primitive if measured by the standards of other fields in computational physics such as structural mechanics or fluid dynamics.
The finite-element method, for
example, has led to significant progress in the solution of more-dimensional boundary and initial value problems, the main advantage being (i) the use of a triangular grid which allows arbitrary local refinement; (2) variable order of accuracy of the discretized equations. Application of the finite element method to the semiconductor problem is an open area for future research. The trend in semiconductor technology is towards integration of full circuits on a single chip of semiconductor material. devices only.
Analysis, so far, is limited to single
It is clear that the numerical analysis and optimization of a full
circuit would be the ultimate goal. three-dimensional transient solution.
In general, this would require a Such a solution is not yet economically
feasible and would approach the limits of the fastest computers.
Some limited
progress in applying numerical analysis to more complicated structures can be expected for charge coupled devices [Amelio 72] and CMOS memory cells which allow for a two-dimensional analysis. As the limitations of the traditional diffusion model have now become apparent, a revival of the search for improved models will have to take place. be expected along two lines, viz.,
Progress may
461
(i) improved "hydrodynamic" models based on moment equations (i.e., energy-balance model instead of diffusion model [Cheung 72]); (2) spatially non-homogeneous solutions of the Boltzmann equation by Monte-Carlo or iterative methods. The first approach, i.e., supplementing the diffusion model by the energy balance model goes hand in hand with a theoretical determination of the new energy transfer coefficients which enter into the model but at the time being are not accessible by experimental methods.
The computational complexity of the
energy-balance model is not seriously greater than that of the diffusion model. It might even be conjectured that the energy-balance model may lead to better conditioned equations. The second approach, i.e., a microscopic model for the charge transport is computationally very demanding and reaches the limit of today's fastest computer. Such models are a combination of particle models, as is common in plasma physics and of Monte-Carlo methods.
The idea of the method is to observe the time evolution
of an ensemble of particles.
At discrete points in time, the self-consistent field
is computed and then the particles are moved and scattered for another time increment according to conventional Monte-Carlo procedures.
Some first results in
this direction are already available, i.e., a full microscopic simulation of a GaAs sample in one space dimension [Lebwohl 71] and results for two-dimensional problems under the simplifying assumptions of (I) a fixed (i.e., non-self-consistent)
field
[Rueh 72] and (2) a simplified law of motion in conjunction with a self-consistent field [Hockney 72].
Owing to the tremendous computing power required it is not
likely that such models will supplant the "hydrodynamic" models in the near future. They are, however, a great scientific advance and will furnish important reference solutions against which simpler models can be validated.
ACKNOWLEDGMENT The views expressed in this paper are the essence of many discussions with eminent scientists in the field of semiconductor technology and computing, most notably Prof. E. Baumann, Prof. Dejon, Prof. R. W. Hockney, Prof. Thomas, Dr. P.
462
Gueret, Dr. G. D. Hachtel, Dr. W. Jutzi, Dr. P. Wolf and J, W. Slotboom. is grateful for the assistance of all these people.
The author
He would also llke to thank
Mrs. Bruellmann for careful manuscipt reading.
REFERENCES Amelio, G. F.: Computer modelling of charge-coupled device characteristics, The Bell Sys. Tech. J., 51, 705-730 (1972). ArandJelovic, V.: Accurate numerical steady-state solutions for a diffused one-dimensional Junction diode, Solid State Electron., 13, 865-871 (1970). Caughey, M.: Simulation of UHF transistor small-sisnal behavior to 10 GMz for circuit modelling, Proc. Second Biennial Cornell E. E. Conf., IEEE Catalog No. 69 C 65-CORN (1969). Cheung, P. S. and Hearn, C. J.: Energy balance model for Gunn-domains, Electron. Left., 8, 79-81 (January 1972). De Marl, A.: An accurate numerical steady-state one-dimensional solution of the p-n Junction, Solid State Electron., ii, 33-58 (January 1968). De Marl, A.: An accurate numerical one-dimensional solution of the p-n Junction under arbitrary transient conditions, Solid State Electron., II, 1021-1053 (November 1968). Dubock, P.: Numerical analysis of forward and reverse bias potential distribution in a two-dimensional p-n Junction with applications to capacitance calculations, Electron. Left., 5, 236-238 (May 1969). Dubock, P.: D.C. numerical model for arbitrarily biased bipolar transistors in two dimensions, Electron. Left., 6, 53-55 (February 1970). Fawcett, W., Boardman, A. D., and Swain, S.: Monte Carlo determination of electron transport properties in gallium arsenide, J. Phys. Chem. Solids, 31, 1963-1990 (July 1970). Fulkerson, D. E.: A two-dimensional model for the calculation of common-emitter currant gains of lateral p-n-p transistors, Solid State Electron., ii, 821-826 (August 1968).
463 Gummel, H. K.: A self-conslstent iterative scheme for one-dlmensional steady-state transistor calculations, IEEE Trans. Electron Devices, ED-II, 455-465 (October 1964). Gokhale, B. V.: Numerical solutions for a one-dlmensional silicon n-p-n transistor, IEEE Trans. Electron Devices, ED-17, 594-602 (August 1970). Hachtel, G. D., Joy, R. C., and Cooley, J. W.: A new efficient one-dimensional analysis program for junction device modelling, Proc. of the IEEE, 60 (January
1972). Heimeier, H. H.: A two-dimensional numerical analysis of a silicon N-P-N transistor, IEEE Trans. Electron Devices, ED-20 (August 1973). Heydemann, M.: Methode num~rlque d'~tude des structures M.O.S.T., Electron. Lett., 6, 735-737 (May 1971). Himsworth, B.: A two-dimenslonal analysis of gallium-arsenide junction fleld-effect transistors with long and short channels, Solid-State Electron., 15, 1353-1361 (1972). Hockney, R. W.: A fast solution of Poisson's equation using Fourier analysis, J. ACM, 12, 95-113 (January 1965). Hockney, R. W~: The potential calculation and some applications, Meth. in Comp. Phys., 9, 135-211 (1970). Hockney, R. W. and Reiser, M.: Two-dimensional particle models in semiconductors device analysis, IBM Res. Report, RZ482 (February 1972). Jutzi, W. and Reiser, M.: Threshold voltage of normally off MESFET's, IEEE Trans. Electron Devices, ED-19, 514-522 (March 1972). Katakoa, S., Tanteno, H., and Hawashlma, M.: Two-dimensional computer analysis of dielectric-surface-loaded GaAs bulk element, Electron. Lett., 6, 169-171 (March 1970). Kennedey, D. P. and O'Brien, R.: Electric current saturation in a junction field-effect transistor, Solid State Electron, 12, 829-830 (August 1969). Kennedey, D. P. and O'Brien, R. R.: Computer-aided two-dimenslonal analysis of the junction field-effect transistor, IBM J. Res. Develop., 14, 95-116 (March 1971).
464
Kennedey, D. P. and O'Brien, R. R.: Two-dimensional analysis of J.F.E.T. structure containing a low-conductivity substrate, Electron. Lett., 7, 714-716 (December 1971). Kilpatrick, J. A. and Ryan, W. D.: Two-dimensional analysis of lateral-base transistors, zeleetron. Lett., 7, 226-227 (May 1971). Kim, C. and Yang, E. S.: An analysis of current saturation mechanism of junction field-effect transistors, IEEE Trans. Electron Devices, E-17, 120-127 (February 1970). Lebwohl, P. A. and Price, P. J.: Direct microscopic simulation of Gunn-domain phenomena, Applied Phys. Lett., 19, 530-532 (December 1971). Lewis, J. A.: The flat plate problem for a semiconductor, Bell Syst. Tech. J., 49, 1484-1490 (September 1970). Loeb, H. W., Andrew, R., and Love, W.: Application of two-dimensional solution of the Shockley-Poisson equation to inversion-layer MOST devices, Electron. Lett., 4, 352-354 (August 1969). McCumber~ D.E. and Chynoweth, A. G.: Theory of negative-conductance amplification and of Gunn instabilities in two-valley semiconductors, IEEE Trans. Electron Devices, ED-13, 4-21 (January 1966). Mock, M.S.: On the convergence of Gummel~s numerical algorithm, Solid-State Electron., 15, 1-4 (January 1972). Mock, M. S.: A two-dimensional mathematical model of the insulated field-effect transistor, Solid State Electron., 16, 601-609 (1973). Mock, M.S.: On the computation of semiconductor device current characteristics by finite difference methods, to appear. Mock, M. S., On equations describing steady-state carrier distributions in a semiconductor device, to appear. Petersen, O. G., Rikoski, R. A., and Cowles, W. W.: Numerical method for the solution of the transient behavior of bipolar semiconductor devices, Solid State Electron., 16, 239-251 (1973). Reiser, M.: Two-dimensional anslysis of suhstrate effects in junction F.E.T.s, Electron. Lett., 6, 493-494 (August 1970).
465
Reiser, M.: Difference methods for the solution of the time-dependent semiconductor flow equations, Electron. Lett., 7, 353-355 (June 1971). Reiser, M.: Zweidimensiona!e Losung der instationaren Halbleitertransportgleichungen fur Feldeffekt-Transistoren, Ph.D. Thesis, Swiss Federal Institute of Technology (ETH) Zurich (July 1971). Reiser, M. and Wolf, P.: Computer study of submersion F.E.T.s, Electron. Lett., 8, 254-256 (April 1972). Reiser, M.: Large-scale numerical simulation in semiconductor device modelling, Computer Methods in Appl. Mech. and Eng., I, 17-39 (April 1973). Reiser, M.: A two-dimensional numerical FET model for dc, ac, and large signal analysis~ IEEE Trans. Electron Devices, 20, 35-45 (January 1973). Reiser, M.: On the stability of finite difference schemes in transient semiconductor problemsj. Computer Meth. in Appl. Mech. and Eng., 2, 65-68 (1973). Richtmeyer, R. D. and Morton, K. W.: Difference methods for initial value problems, New York,, Wiley-Interscience (1957). Ruch, J.G.: Electron dynamics in short channel field-effect transistors, IEEE Trans. Electron Devices, 19, 652-654 (May 1972). Scharfetter, D. L. and Gummel, H. K.: Large-signal analysis of a silicon read diode oscillator, IEEE Trans. Electron Devices, 16, 64-77 (January 1969). Seidman, T. I. and Choo, S. C.: Iterative scheme for computer simulation of semiconductor devices, Solid-State Electrons., 15, 1229-1235 (1972). Slotboom, J. W.: Iterative scheme for one- and two-dimensional dc- transistor simulation, Electron. Lett., 5, 677-678 (December 1969). Slotboom, J. W.: Computer aided two-dimensional analysis of bipolar transistor, IEEE Trans. Electron Devices, ED-20 (August 1973). Stone, H. L.: Iterative solution of implicit approximations of multi-dimensional partial differential equations, SIAM J. Numer. Anal., 5, 530-558 (September 1968). Stratton, R.: Semiconductor flow equations (diffusion and degeneracy), IEEE Trans. Electron Devices, ED-19, 1288-1292 (December 1972).
466
Suzuki, N., Yanai, H., and Ikoma, T.: Simple analysis and computer simulation of lateral spreading of space charge in bulk GaAs, IEEE Trans. Electron Devices, 19, 364-375 (March 1972). Vandorpe, D. and Xuong, N. H.: Mathematical two-dimensional model of semiconductor devices, Electron. Lett., 7, 47-50 (January 1971). Vandorpe, D., Barel, J., Merckel, G., and Saintot, P.: An accurate two-dimensional numerical analysis of the MOS transistor, Solid State Electron., 15, 547-557 (1972). Varga, R. S.: Matrix iterative analysis, New Jersey, Prentice Hall, Inc., (1962). Wasserstrom, E. and McKanna, J.: The potential due to a charged metallic strip on a semiconductor surface, Bell Syst. Tech. J., 49, 853-877 (May 1970). Yanai, H., Suzuki, N., Sugeta, T., and Tainlmoto, M.: Effect of electrode structure on dipole-domain formation, Proc. of the Third Intern. Symp. on GaAs and Rel. Compounds, the Inst. of Phys., London and Bristol, 153-162 (1970).
SIMULATION
NUMERIQUE DES
DE
LA
DISPOSITIFS
D. Institut de Physique
FABRICATION
Nucl4aire 69
ET
DU
COMPORTEMENT
SEMICONDUCTEURS
VANDORPE de L y o n
-43,
B d du II n o v e m b r e
1918
VILLEURBANNE
L'4volution e x t r ~ m e m e n t
rapide de la technologie des semi-conduc-
teurs a entrafn4 une modification radicale de la m a n i ~ r e de concevoir et de r4aliset les circuits 41ectroniques.
V u la complexit4 et le cofit 41ev4 des r4alisations
les concepteurs sont maintenant a m e n 4 s ~ utiliser de fa~on courante les outils de la conception assist4e par ordinateurs pour la simulation de leurs circuits avant r 4alisation. Ii est bien 4vident que la qualit4 d'une simulation sur ordinateur d4pend essentiellement de la qualit4 des m o d u l e s utilis4s, c'est ce qui a a m e n 4 les 41ectroniciens ~ d4velopper une
importante recherche dans le d o m a i n e des " M o d U -
les M a t h 4 m a t i q u e s". P a r ailleurs la r4duction importante de la faille des dispositifs dans les circuits int4gr4s a accru fortement l'influence d'effets p r 4 c 4 d e m m e n t
n4glig4s
(surface, bords des diverses zones des dispositifs ..... ) aussi il est indispensable de teniz compte de ces p h 4 n o m ~ n e s
qui deviennent pr4pond4rants pour certai-
nes structures ce qui a m i n e ~ affiner les m o d u l e s et ~ les rendre beaucoup plus complexes.
L e t r a v a i l p r 4 s e n t 4 i c i a 4t4 c o m m e n c 4 ~ l ' I n s t i t u t d e M a t h 4 m a t i q u e s A p p l i q u 4 e s de G r e n o b l e et f i n a n c 4 en p a r t i e p a r la D414gation G 4 n ~ r a l e h la R e c h e r c h e S c i e n t i f i q u e e t T e c h n i q u e ( c o n t r a t D. G. R. S. T. n ° 67 00 864) e t l e C o m m i s s a r i a t l'Energie Atomique (contrat C.E.A./ GR - 751. 066).
468
Enfin on a toujours cherchE k relier sirnplernent la technologie et les processus de fabrication avec les m o d u l e s de cornposants. dans le cadre de cette Etude apporte de n o m b r e u x me.
41Ernents de r4ponse ~t ce prob1~-
Elle perrnet en outre :
- de rnieux cornprendre certains p h E n o m ~ n e s de relier les p e r f o r m a n c e s
-
L a simulation rEalisEe
physiques fondamentaux,
du dispositif 5 ses caractEristiques technologiques,
- d'aborder l'optimisation des dispositifs eux-m~rnes. O n a vu apparaftre ces derni~res annEes de n o m b r e u s e s
Etudes sur la
simulation du comporternent des dispositifs, unidirnensionnelles tout d'abord, gtudiant diodes et transistors bipolaires en fonctionnement statique (5, 9, 24) et dynam i q u e (6, 30) puis bidimensionnelles statiques (Z, I0, Z2, 25) et rn~rne bidirnenssionnelles dynarniques pour des dispositifs particuliers (18). Toutes ces dtudes reposent en g4n@ral sur les m ~ m e s
hypotheses physiques fondamentales nEcessaires
la raise en Equation du probl~me.
Elles ndcessitent par ailleurs la connaissance
prEalable du profil de dopage du serni-conducteur qui intervient c o m m e mdrique.
Malheureusement
la m e s u r e
donn@e nu-
expdrimentale de ces profils est difficile et
relativement imprecise si l'on cherche ~t connaftre le dopage en fonction de la profondeur exclusivement, bidimensionnel
elle devient quasiment impossible si l'on desire un profil
r4el.
P o u r rEsoudre ce probl~rne on a donc ErE a m e n @ ~ rEaliser 4galement une simulation nurnErique du processus technologique de fabrication des dispositifs (1Z,
15,
16, 27). O n prEsente ci-apr~s les principaux p r o b l ~ m e s rencontres dans cette
Etude et la m a n i ~ r e dont ils ont ErE rEsolus. I
-
ETUDE
NUMERIQUE
DE LA DIFFUSION
D'IMPURETES
DANS UN SEMI-CON-
DUCTEUR. I.i.
S y s t ~ m e d'4quations
C o m p t e - t e n u des divers m 4 c a n i s m e s
physiques de diffusion des impu-
retds dans le silicium et de leurs interactions possibles (15) on peut Ecrire (apr~s normalisation des diverses fonctions) pour chaque type d'irnpuretEs les Equations : (1)
J
(2)
~ t
Oc
=
-
-
D
grad
div
-~
(J)
(C)
-
~-C.
E.
Z
469
o~
:
J
e s t le f l u x d ' i m p u r e t @ s ,
C
la c o n c e n t r a t i o n ,
E
le c h a m p ~ l e c t r i q u e , la m o b i l i t ~ de l ' i m p u r e t ~ ,
Z
la c h a r g e ~ l e c t r i q u e de l ' i m p u r e t ~ ,
D
le c o e f f i c i e n t de d i f f u s i o n .
La r~alisation d'un transistor m u l t a n ~ e de d e u x t y p e s d ' i m p u r e t ~ s
p a r e x e m p l e n ~ c e s s i t e la d i f f u s i o n s i -
( d o n n e u r s et a c c e p t e u r s
Une l o i s i n t r o d u i t e s d a n s l e r ~ s e a u c r i s t a l l i n l e s des charges libres.
impuret~s
P o u r ~ v a l u e r le c h a m p ~ l e c t r i q u e
E
notes
C A et
CD).
s ' i o n i s e n t et c r ~ e n t qui i n t e r v i e n t d a n s le
r n ~ c a n i s m e de d i f f u s i o n , il c o n v i e n d r a i t d o n c de r ~ s o u d r e l ' ~ q u a t i o n de P o i s s o n et d'~valuer les courants des porteurs
l i b r e s ce qui n o u s a m ~ n e r a i t N a n a l y s e r un
m o d u l e de c o n d u c t i o n en t r a n s i t o i r e
~ d o p a g e v a r i a b l e en f o n c t i o n du t e m p s .
Cette fa~on d'aborder
le probl~me serait sans doute envisageable sur
un m o d u l e ~ une v a r i a b l e g ~ o m ~ t r i q u e r n a i s s e m b l e i r r ~ a l i s t e variables
g~om~triques,
essentiellement
s u r un m o d u l e ~ d e u x
e n r a i s o n du t e m p s de c a l c u l n ~ c e s s a i r e
p o u r une t e l l e ~tude. On fair donc une h y p o t h ~ s e p h y s i q u e c o m p l ~ m e n t a i r e
en a d r n e t t a n t q u e
l a c h a r g e d ' e s p a c e e s t n u l l e et q u e l e s y s t ~ m e e s f ~ l ' ~ q u i l i b r e t h e r m o d y n a m i q u e ce qui p e r m e t d ' o b t e n i r le s y s t ~ m e s u i v a n t :
(3)
~ CD t -
CA (4)
¢) ~) x
[ DD
e)
C) t
(hD
r
- ~
LDA
~)C D ~ x
(h D - l)
c) CA t) x
~ CA (hA
t) x
)]
~ CD (hA - I) ~
)
S u r un m o d u l e u n i d i m e n s i o n n e l et s u r un m o d u l e b i d i m e n s i o n n e l
(5)
~CD "~ t
-
@ ~ x
[ DD
(hD
@C D ~ x
.~C - (h D - l)
] )
~ [DD (hD -~y 'lCD - ( h D - l ) @CA)] ~ Y
+ ~-~
:
470
CA (6)
~ -
e) t
(h A
~I o A
(h A
-(h A - i)
x
~x
)
cA
+
oh x et y sont les variables t
cA
r
[ DA
~ x
(h A - i )
~y
g4om4triques,
le temps,
les quantit4s h A
et h D
sont des fonctions de
CD (7)
hD
= i +
2n.
C A et C D.
1 ~
1
G D -GA)Z (
2n.
+
1
1 CA (8)
1
hA = 1 + ~ 2 n.
~',',ii,. C D _ CA Z (' 2 n . ) + 1
1
1
n.! 4rant la concentration d'~lectrons libres clans le semi-conducteur intrins~que
(ni > 0) De m~me de
CA
et C D
les coefficients de diffusion D A
et D D
varient en fonction
d'apr~s les lois : Z +~I
(9)
DD
= Do D
n. cn 1 Z +
(i0)
DA
= D°A
2 +~' Z +
÷ 13
CA '
~- ni
DOA, DOD sont les v a l e u r s des coefficients pour le s e m i - c o n d u c t e u r intrins~que,
@ et ~ '
sont des coefficients qui c a r a c t 4 r i s e n t le
p r o c e s s u s de
diffusion de l'impuret4 consid~r4e. Toutes ces qualit4s sont strictement positives de m ~ m e tions inconnues
CA
et C D
.
que les fonc-
471
I. 2.
- Conditions aux limites
L a fabrication des dispositifs se fait en plusieurs phases successives, les unes dites de pr4d4pSt ou k faible temp4rature,
on fair circuler un gaz porteur
d'impuret4s qui restent en surface du dispositif, les autres dites de redistribution qui se font h temp4rature 41ev4e pour permettre aux impuret~s de diffuser dans le volume. Les phases de redistribution sont a c c o m p a g n 4 e s
d'une oxydation en surfa-
ce ce qui va poser un probl~me particulier dfl ~ la pr4sence de deux milieux diff4rents (oxyde et silicium) les ~quations dans l'oxyde 4rant 14g~rement diff4rentes. Par ailleurs,, la position de la fronti~re g4om4trique ainsi que l'4paisseur totale varient au cours du temps. Les conditions aux limites ~ prendre en compte sont alors les suivantes : I. 2. i.
- ModUle unidirnensionnel
FLUX entrant
FIGURE
~
~
FLUX sortant (nul)
Si
j,,
;
t
0
A
B
..~
x
1 -. Diffusion unidimensionnelle (pr4d4p6t)
Dans le cas d'un pr4d4pSt, on suppose que le flux sortant est nul, soit
:
co (11)
x
CA (B)
=
~x
(B)
=
0
O n connaft les flux entrant en surface (IZ)
JD
= ~(Co
C D (A))
:
472
ou
sa
= o¢(Co
-c A(A))
On ne p r d d d p o s e en effet q u ' u n e i m p u r e t g ~ la fois. ~'
e t Co
sont des constantes positives.
On p o u r r a i t b i e n stir p r e n d r e
en consid@ration dans le m o d M e route
a u t r e c o n d i t i o n a u x l i m l t e s q u ' e l l e s o i t de t y p e D i r i c h l e t ou N e u m a n n s a n s a v o i r modifier sensiblement les programmes
num@riques.
La solution i n i t i a l e u t i l i s d e s e r a soit le r g s u l t a t d ' u n e p h a s e a n t 6 r i e u re,
s o i t u n e r g p a r t i t i o n u n i f o r m e ( c a s du p r e m i e r
p r g - d @ p S t ) p o u r la r e d i s t r i b u t i o n ,
o n n d g l i g e l ' i n f l u e n c e du c h a m p g l e c t r i q u e d a n s l ' o x y d e e t l e s v a l e u r s d e s c o e f f i c i e n t s de d i f f u s i o n sont d i f f 6 r e n t e s .
1
entrant
FIGURE
Z -
Si
sortant (nut)
,L
I
I
I
0
A
C
B
X
Diffusion unidimensionnelle (redistribution)
L e s c o n d i t i o n s a u x l i m i t e s en A (t)
et
B
sont les m ~ m e s que pour
l e p r g d @ p g t (aux v a l e u r s n u m g r i q u e s p r o s ) . Les gquations ~ l'interface
Si
SiO 2
sont
:
CSi =
m
m
CSi O2
(13)
: ~tant une constante positive (coefficient de s ~ g r g g a t i o n ) .
Csioz (~'- ±) v (t) -- Dsioz m
~ CSiOz
~
x
DSi
~ CSi x
~tant le rapport entre l'@paisseur de silicium oxyd@ et celle de silice cr@de V (t)
la vitesse d'oxydation. On tient compte par ailleurs du d~placement gdom@trique des points
473
et
A
C,
B
4rant s u p p o s 4 fixe. L a solution initiale est le rdsultat d'une p h a s e ant4rieure.
- Miod~le b i d i m e n s i o n n e l
I. 2. 2.
G
C
D
G
C
D
SiO Si
Si 0 2
YI
F
Si
B
l
E
A
0
F
E
A
x
FIGURE
3
- M o d h l e bidimensionnel diffusion (pr4d4pSt)
de
FIGURE
4
- M o d U l e bidimensionnel de diffusion (redistribution)
Nous raisons les m~mes hypotheses que pour le mod~te unidirnensionnel.
C e p e n d a n t d a n s c e t t e 4 t u d e , on d e v r a i t t e n i r c o m p t e de l a p r 4 s e n c e
d'oxyde
m ~ m e p o u r l e p r 4 d 4 p S t ( e n d e h o r s de l a " f e n ~ t r e " p a r l a q u e l l e s ' e f f e c t u e l a d i f f u sion). Cette pr4sence n'ayant qu'une influence relativement pas tenu cornpte rigoureusement e n s u p p o s a n t q u e , l e l o n g de On a donc
B C
:
B C
et D E
@c ~x
sur
C D
et A E
~y
sur
AB
J
-
~c
J
n
0
- 0
=
f(C)
e s t la c o m p o s a n t e n o r m a l e du f l u x d ' i m p u r e t 4 s . n
On a p r i s p o u r
J
la m ~ n
me fonction f que pr4c4demment
(~4)
A C D E
, il n ' y a p a s d ' ~ c h a n g e s a v e c l ' e x t 4 r i e u r .
sur
oh
faible nous n'en avons
et 4tudions u n i q u e m e n t le r e c t a n g l e
f(c)
=o~(Co
soit
:
- c)
P o u r la redistribution les conditions a u x limites sont identiques celles du p r 4 d 4 p S t (aux valeurs des constantes pros) et on ne tient plus c o m p t e
de
474
l'existence de la fen~tre de diffusion en supposant que la vitesse de croissance d'oxyde est inddpendante de se de redistribution.
Y
et que l'4paisseur d'oxyde est nulle au d4but de la pha-
L'~quation d'interface est la m ~ m e
que celle 4crite p r d c d d e m -
m e n t sur le m o d u l e unidimensionnel. I, 3.
- Position du p r o b l ~ m e m a t h g m a t i q u e
Nous sommes
donc a m e n d s
~ rechercher la solution, si elle existe,
d'un syst~me de deux 4quations paraboliques non lindaires. Ii ne nous a pas 4t4 possible de d d m o n t r e r l'existence et l'unicit4 de la solution de ce p r o b l ~ m e en raison des non lindarit4s figurant dans les 4quations. O n adrnet l'existence et l'unicit4 de la solution,
par analogie avec le probl~.~._e de la chaleur auquel on se r a m h n e
exactement s i l'on ndglige : les interactions entre les deux types d'impuretds,
-
-
la variation des "coefficients de diffusion",
- les effets du c h a m p 41ectrique. Ce syst~me dtant impossible ~ intdgrer formellement,
on remplace
le p r o b l ~ m e continu par un p r o b l ~ m e discret en utilisant pour ce faire une m 4 t h o d e de diffdrences finies. II conviendrait alors de d d m o n t r e r que la limite, quand les grandeurs des pas de discrdtisation tendent vers zdro, de la solution du p r o b l ~ m e discret tend vers la solution du p r o b l ~ m e continu. faite. N o u s s o m m e s
Cette ddmonstration n'a pu ~tre
conduits ~ admettre ce rdsultat, apr~s avoir vdrifid en faisant
plusieurs essais n u m 4 r i q u e s
sur le m ~ m e
cas, avec des valeurs de pas diff4rentes,
que la solution trouvde ne ddpend pas du maillage choisi. I. 4.
M d t h o d e s n u m d r i q u e s de traitement
P o u r tenir c o m p t e correctement des variations des p h d n o m ~ n e s
de
diffusion dans l'espace et le temps, nous somrnes a m e n d s ~ utiliser des pas de discrdtisation variables en X , Y , T. O n dcrit chacune des deux dquations A traiter sous la f o r m e (en bidimensionnel)
:
~u (15)
oh
L (u) =~)t
a
et b
C) x
(a
sont des fonctions de x
~u ~
)
~9
(a ~-~U~) Y
b
et t dans lesquelles on fair entrer tousles
t e r m e s non lindaires et ceux ddpendant de la seconde impuretd, syst~me complet en deux sous-syst~mes tion inconnue
CA
ou
CD .
= 0
sdparant ainsi le
oh l'on consid~re qu'il n'y a qu'une fonc-
475
O n approxirne les d~riv~es de
u
par une m ~ t h o d e
classique de diffe-
r e n c e s finies (7, 29). P o u r des raisons de stabilit~ (l i) n o u s a v o n s choisi une rn~thode purernent irnplicite. Cette fa~on de faire, n o u s arn~ne alors ~ utiliser un p r o c e s s u s de calcul it~ratif bas~ sur la m ~ t h o d e
du point fixe. Ceci n'est pas un inconvenient car, en
raison de la n o n lin~arit~ du problhrne, ment.
cette solution se serait i m p o s ~ e
naturelle-
P a r contre, un avantage i m p o r t a n t de ces choix r~side dans le fair que les
m a t r i c e s des syst%rnes d'~quations lin~aires rencontres dans le p r o c e s s u s de calcul sont
:
- tridiagonales pour le rnod~le unidimensionnel, - tridiagonales par bloc p o u r le m o d u l e bidirnensionnel, ce qui sirnplifie grandernent les calculs n ~ c e s s a i r e s h la r~solution des syst~rnes. O n pr~cisera ult~rieurernent les rn~thodes de calcul utilisdes. I. 5.
Algorithrne et prograrnrnes
L a rn~thode e x p o s ~ e ci-dessus a ~t~ prograrnrn~e pour les d e u x m o d U les uni et bidimensionnels. I)
L'algorithrne g~n~ral est identique dans les deux cas
:
E n t r e e des d o n n ~ e s et initialisation.
2) - Calcul du pas en temps.
Calcul ~ventuel de l'~paisseur d'oxyde c r ~ e .
3) - Calcul des coefficients de diffusion. 4) - R~solution des d e u x syst~rnes d'~quations. 5) - Test c o n v e r g e n c e et retour ~ventuel en 6) - Test final, retour ~ventuel en Les prograrnmes IBM
3.
2.
sont ~crits en F o r t r a n
IV
et utilis~s sur materiel
360 50 et 67 . Ils sont relativernent courts (1000 cartes environ pour le m o d u l e
bidirnensionnel sous syst~rne t e m p s n~cessite
CP/CMS)
et rapides. L e calcul pour un intervalle de
90 ~ 30 iterations p o u r le m o d u l e bidirnensionnel et de 15 ~ 3 it~-
rations p o u r le m o d u l e unidirnensionnel. Un module
de tracd autornatique fournit ~ l'utilisateur une representa-
tion graphique des r~sultats. I. 6.
- R~sultats obtenus
O n d o n n e ~ titre d'exernple sur la figure 5 le rgsultat de la sirnulation bidimensionnelle de la fabrication d'un transistor bipolaire sur du silicium. L ' a n a lyse physique de tels r~sultats a dgj& gig entreprise (I l) et les conclusions que l'on en tire sont trhs int~ressantes pour les ~lectroniciens, rant p o u r la c o m p r e h e n s i o n physique des p h g n o r n ~ n e s que p o u r l'arnglioration de la technologie et l'optirnisation
3 -I
J
J
Z
J
I D
J
i
i
~n
/
to
DoZ~
.....
-_.-
->
: l igne d '~quiconcentration
: jonction
~metteur
base
[ 40 zo
~(X,y)
404s
4 0 40
477
des dispositifs. Signalons s i m p l e m e n t
N ce propos que les p r o g r a m m e s
DIFFUSI
sont
utilis4s de fa~on courante dans plusieurs laboratoires. II - E T U D E
NUMERIQUE
DU
FONCTION-NEMENT
DE
CERTAINS
DISPOSITIFS
SEMI CONDU CTEURS L'4tude num4rique semiconducteurs solide.
pr6sent4e
On peut montrer
v4rifi4es
des ph4nom~nes
i c i r 4 s u l t e de c o n s i d 4 r a t i o n s
que sous certaines
on peut 4crire
classiques
hypothhses
raisonnables,
d e p h y s i q u e du g4n4ralement
le syst~me d'4quations.
(16)
J n
= q ~n
n
e + q dn
grad (n)
(iv)
j p
= q •p
p
e
g r a d (p)
-
q dp
(18)
-~t-~P- =
- R
- ~ q
div
(19)
¢)n ~-~-- =
- R
1 +-q
-'~ div On )
(20)
de c o n d u c t i o n d a n s l e s d i s p o s i t i f s
div (e) = ~
(n - p
(jp)
- Hop)
L e s s y m b o l e s utilis4s 4rant d4finis dens la table ci-dessous On transforme me (21)
ce s y s t ~ m e d'4quations en d4finisant un n o u v e a u syst~-
d'unit4s pour obtenir : J
-
-
(grad (P) + P
grad
(~)
grad
(~))
)
p
(22)
"~ J
1
-
~ (grad
(N)
- div
-') ( Jp )
N
n
(23)
-
p T N T
(24)
(25)
-
V
=
- U
U +
=
z~¢
= N
div
- P
--) (Jn)
hop
L e s inconnues principales seront les expressions analytiques de riv4es. L a fonction
DOP
variables g 4 o m 4 t r i q u e s.
~n' ~'p , U
est connue
N, P ,
~
, on supposera c o n n a ~ r e
en fonction de
N, P, ~2et de leurs dE-
(an g4n6ral n u m @ r i q u e m e n t )
en fonction des
478
TABLE
1
Syrnbole non normalisE
x, y
LISTE DES SYMBOLES
G r a n d e u r reprEsentEe normalisE
X, Y
Variables g4om4triques
T
temps
N
concentration d'41ectrons libres
P
P
concentration de trous
Jt
J
Jn
J
Jp
J
densit4 de courant externe
t
densit4 de courant d'Electrons n
densit~ de courant de trous P
--)
champ
E
e
41ectrique
potentiel 41e ctrostatique -1 -1
~p D
d
mobilitE des trous coefficient de diffusion des Electrons
n
n
coefficient de diffusion des trous
D
d
mobilit4 des Electrons
P
P
dop
DOP
dopage du semiconducteur
R
U
bilan g4nEration - recombinaison
TAUP
dur4e de vie des trous
TAUN
durEe de vie des Electrons
~P
q
charge de l'41ectron
479
N et
~p
et
P
sont des fonctions strictement
positives
de m ~ m e
que
~'n
. Z. 1.
Conditions aux limites
O n a 4tudi4 ce p r o b l ~ m e sur des m o d u l e s g4om4triques ~ une puis deux dimensions. r 4 g i m e dynamique,
L'analyse unidimensionnelle a @t4 faite ell r 4 g i m e statique et en en tenant c o m p t e 6ventuellement de p h 4 n o m ~ n e s
particuliers tels
la g4n4ration par effet d'avalanche (13) ou par effets des r a y o n n e m e n t s (18) l'analyse bidimensionnelle a dr6 r4alisde uniquement en r 4 g i m e statique. Z. i.I.
Conditions aux limites sur le rn0d~l e unidi!nensionnel statique Sur ce m o d u l e on ne peut 4tudier que des dispositifs ou les effets de
surface sont n4gligeables tels la jonction
PN
et le transistor bipolaire. base
emetteur
L
I I
Aux deux limites
m4talliques
sur lesquels
A et B
bipolaire
des potentiels
Aet
ModUle unidimensionnel de t r a n s i s t o r NPN
B
repr4sentent
les contacts
l e s c o n d i t i o n s de n e u t r a l i t 4
41ectrique.
il t a u t t e n i r c o m p t e du c o n t a c t de b a s e c e On peut r4soudre
partiellement
cette diffi-
(qui s o n t d 4 f i n i s h u n e c o n s t a n t e a d d i t i v e
pros) par une hypoth~se physique c o m p l 4 m e n t a i r e 2. i. 2.
-
B
on a a l o r s d e s c o n d i t i o n s de type D i r i c h l e t
on suppose v4rifier
de f a ~ o n r i g o u r e u s e .
cult4 en fixant l'origine
FIGURE 7
N, P , ~2 c a r l e s p o i n t s
Pour le transistor qui est impossible
de
i L
I 0
I
A
ModUle unidimensionnel jonction PN
sur les trois inconnues
i
I
x
B
A
-
!
®
® ®
®
FIGURE 6
eollecteur
donnant la valeur de
(o)
- C__qonditionsaux limites sur le m o d u l e unidimensionnel d y n a m i Q u e P o u r l'4tude d y n a m i q u e de ce p r o b l A m e il s'av~re que le choix des
variables principales dolt ~tre modifi4 et que l'on a int4r~t ~ 4tudier les fonctions N, P, E
les conditions aux limites sur
type Dirichlet,
N
et P
les conditions aux limites sur
E
4tant c o m m e
pr4c4demment
de
4tant par contre de type N e u m a n n .
480
(Z6)
~ E (A, T)
Ox
:
0E
(B, T)
e) x
V
=0
T
L e s conditions initiales 4tant alors soit le r4sultat d'un calcul en statique, soit le r4sultat d'un p r ~ c 4 d e n t calcul en d y n a m i q u e q u a n d on d4sire poursuivre une simulation. O n doit p a r ailleurs i m p o s e r rieur sous la f o r m e
(zT) 2 I. 3.
en outre la contrainte due au circuit ext@-
:
f ((~B(T)
~ A (T))'
Jr(T)) = 0
- Conditions aux limites sur le m o d u l e bidimensionnel statique N o u s ne d o n n o n s ici que les conditions a u x limites sur le m o d h l e
jonction
PN
par le n o m b r e le dispositif.
de
represent4 en figure 8. E n effet les divers dispositifs ne diffhrent que de contacts m4talliques et la fonction
DOP
(X, Y) qui caract4rise
P a r contre, on 4tudiera toujours les dispositifs en supposant l'exis-
fence d'une couche d'oxyde en surface, cette c o u c h e 4tant s u r m o n t 4 e d'une Electrode m4tallique (anneau de garde).
D
D'
C
~2
o~ ...... ! I !
A
FIGURE
8
- ModUle
A'
B
g 6 o m 4 t r i q u e de jonction
Sur le m o d u l e
PN
les parties h a c h u r 6 e s c o r r e s p o n d e n t k l ' e m p l a c e m e n t
des contacts m4talliques.
O n 4tudie les d e u x fonctions
A' B C D' et la fonction
sur le rectangle
Net
P
sur le rectangle
A B C D. C'est-~-dire que pour c h a q u e
481
dispositif les fonctions semi-conducteur
aans l'oxyde
Net
P
ne sont dgfinies que sur le rectangle repr~sentant le
(~'~z) alors que le potentiel est 4tudi4 dans le s e m i - c o n d u c t e u r
([l 1 Ufl2). ?Pour l e s c o n t a c t s
et sur
et
mgtalliques
Aa'o¢0/ on a
tels
7 ( X , Y) = 7 1
A'~'
D D'~'
et sur
D
et
on a
P (XI' Y)
= P1
N (X,
= N I
~I)(X,
'
Y) Y)
P (Xl'
Y)
= ~!12 = P2
N ( x 1, Y) : N z
sur
A' B
P o u r l'41ectrode de garde (X ~
on a
Pour
on a
et
Ies fonctions
n'
C
du d i s p o s i t i f
,)~'
ON --
0y sur
~
B C
0
et sur une interface 1
'
~'n
(zg)
c
3¥0 x
N
0N elx
0¥+ O x g
~x
--
,)y
0
~P -
~'p
0
:
1
) =
--
clx
relations
01~
( t) x
f et
=
:
,) P
-
ely
on ales
0N
(zS)
oh les fonctions
tellx'~
~
-
(Xo, Y) = %u3
z)- -P
( -~x
+ p
O x
) = f (N, P )
= g (~)
repr4sentent
les phgnom~nes
d'interface
et
C
est une cons-
tante. Dans les essais
num4riques
on a pris
:
A
(30)
f (N, P )
(31)
~(~) = B ( ~ -
oh A, B, ~ o
effectu4s
= p +--------~ ( P N - 1)
%0 o)
sont des constantes positives.
O n a donc u n p r o b l ~ m e
m i x t e ~ r4sou-
dre. Z.Z.
Existence et unicit4 de la solution
II ne nous a pas 4t4 possible de d 4 m o n t r e r la solution de ce p r o b l ~ m e
l'existence et l'unicit4 de
n o n lin4aire.
O n a pu alors obtenir quelques r4sultats tr~s parcellaires
d'une telle
d 4 m o n s t r a t i o n (Z5) m a i s les h y p o t h e s e s qu'il est alors n 4 c e s s a i r e de faire sont si
482
restrictives qae la port4e de ces d4monstrations reste tr~s limit4e. N o u s supposons donc que la solution du p r o b l ~ m e existe et est unique avant de mettre en oeuvre une m ~ t h o d e n u m 4 r i q u e pour en trouver une solution approch4e discrete. Z. 3.
M4thodes n u m 6 r i q a e s de traitement
P o u r chacun des m o d u l e s r4alis4s on a employ4 une m 4 t h o d e de point fixe de type it4ratif apr~s avoir discr4tis4 le p r o b l ~ m e par une m 4 t h o d e de diff4rences finies appropri4e.
L e choix de la m 4 t h o d e de point fixe it4ratif s'imposait en ef-
let assez naturellement en raison des non lin4arit4s du p r o b l ~ m e k traiter. P a r ailleurs on peut remarquer, ~
sur le syst~me k int4grer, que si l'on suppose connaftre
, alors on peut 4tudier s @ p a r 4 m e n t les p r o b l ~ m e s en
Net P
d4finis par les 4qua-
tions (22) et (24) d'une part, (Zl) et (Z3) d'autre part. Par contre si l'on suppose connaftre
Net P
on peut alors r4soudre le p r o b l ~ m e en ~7 d4fini par l'4quation (25).
Cette s4paration en trois p r o b l ~ m e s plus simples ~ traiter simplifiera notablement le traitement et s'ins6rera tr~s facilement dans la m 4 t h o d e de point fixe choi~ie 4galement en raison de ce d4coupage. Z. 3. i.
- M o d U l e unidimensi.onne ~ statique O n a repris pour ce p r o b l ~ m e l'algorithme propos4 par G u m m e l
(9) et
am41ior4 par D e M a r l (5) en lui apportant des simplifications importantes t24).'" L e syst~me d'4quations s'4crit alors :
(32)
J
(X)
:
1 ~[p
p
(33)
$
(X)-
P (X)
d ~) dX
N (X)
d dX
(X) n
:
U
=
-U
d X d J
(X)
P d X
(35)
+
dX
~'n
d J
(34)
,d,,N (X)
i
n
d P (X) dX
az (36)
- N (x)
P (x)
DOP
d X2 oh l'on c o n n a ~
~n (X)
,
~p (X) et D O P
(X).
U n e des principales difficult6s n u m 4 r i q u e s de ce p r o b l ~ m e provient du fair que les expressions (3Z) et (33) font intervenir les diff6rences de quantit4s tr~s grandes devant
J
et J c o m m e une 4valuation grossi~rement approch4e le montre. p n O n a alors int4r~t, pour faire apparaftre un t e r m e calculable de fagon pr6cise h in-
483 troduire
les variables
interm4diaires
(37)
N
=
h
e
(38)
P
--
f# e
A et
de variables
(39)
(X)
=
-
et de l'4quation
(40)
g
n
Y n (t)
XB
Jn (t)
e-
4tant calculable
(34), o n a
/Ax U
(X) =
les 4quations
(32) e t (33) t r a n s f o r m d e s
et l'on a pour une fonction
Cette expression limites
:
~' _~'
On peut ensuite int4grer changement
~l.d4finies par
PN , par exemple
~'(t)
car,
par ce
:
(B)
dt + ~
compte-tenu
des conditions
aux
:
A
(t) dt +
B
- A (A)
(8)
~,n(t)
e- ~l~(t)
dt
A
/.°
~n(t)
e-
U (y) dy
L"A
~B
~'n (t) e-
]
at
~(t)dt
O n voit appara~re dans ces expressions une autre difficult4 due h la prdsence des termes tels e
ou
e
. E n effet pour des polarisations appliqudes
de l'ordre de i0 volts ~Jvarie de 0 h 4000 environ. Par ailleurs les ordres de grandeur des quantit4s N e t
P
que les quantit4s Ai@et
varient de l'ordre de e~ou
e -~
I0 -I0 h 106 ce qui m o n t r e bien
sont impossibles ~ calculer sur ordlnateur.
O n r4soud ici ce probl~me en exprimant les 4quations de type (39) l'aide des quantit4s N e t P
(40
N (x) = e -~(x)
et on 4valuera fonctions
pour obtenir
telles
les expressions
:
N (8) e-
~'n :n (t) e-Y(t)
de type
~
~n
e
~(t)
dt
en introduisant
:
B (4z)
des
~
(x) : n
/B ~'n
X
e tf(X)
t~(t) dt : e (X)
e-q'(t)dt X
n
484 N (X)
ce qui permettra ensuite de faire a p p a r a ~ r e dans les expressions donnant des p r o d u i t s t e l s
e ~(X)
e- ~ ( X )
:
i.
O n aura done cette lois des quantitgs calculables sur ordinateur de fa9on simple, ceci quelque soit la valeur de la polarisation appliqu4e. E n outre, en reportant les valeurs obtenues dans les expressions type 41
on obtient i m m 4 d i a t e m e n t
la valeur de N e t
P
ou plus pr6cis6ment Aet ~ q u i
sont les inconnues que l'on cherche 5 d~ter1~niner . L a d4termination de ensuite ~ partir de l'4quation
d-----T - =
7
se fera
:
- ~£
- DOP
x
que l'on discr6tisera par une m 4 t h o d e de diff4rences finies apr~s avoir utilis~ un processus de quasi lin4arisation en posant, ~ la k + li~me it4ration :
(44)
"~k + 1 :
pour obtenir
I~k + O ( k
: d
z~ Z
(45)
_ O~k (Ae ~ k + /~e
Vk)
=
A e~k
-
d z ~k p-e "~k - DOP -----T
dx
dx
O n utilise alors l'algorithme g4n4ral suivant i) - D4termination d'une fonction
q2 ° (x) de d4part
:
(~ partir de consid4rations phy-
siques).
z) - Cal~ul ae A ot [~ ~ partir de ~y on en tire N et P. 3) - C a l e u l de
~
~ partir
de
i
et ~-
.
4) - Test convergence et retour ~ventuel en Z. 5) - Edition des r~sultats. Des programmes partir de cette m4thode,
correspondant aux divers dispositifs ont 4t4 4crits
lls sont 4crits en F O R T R A N
IV et comportent 5 ~ 600
cartes. Ils sont tr~s rapides et d'un emploi ais4. Ils sont d'ailleurs utilis4s de fa9on courante dans plusieurs laboratoires de consfructeurs et d'universit4s. Cependant, peuvent appara~re.
sous certaines conditions,
des difficult4s de convergence
U n e condition de convergence a pu ~tre m l s e en 4vidence par
M o c k (14) et des solutions k ce p r o b l ~ m e qui avait 4t4 constat4 exp~rimentalement ont 4t4 propos4s ce qui p e r m e t d'obtenir la solution dans tous les cas (21, Z7),
485
Z. 3. Z,
- ModUle unidimensionnel dynamique D a n s le cas d'une 4tude d y n a m i q u e
on se r e n d c o m p t e i m m 4 d i a t e m e n t
q u e le choix de la variable ~2 se r4v~le tr~s maladroit. de tenir c o m p L e de l'4quation d'4volution du c h a m p
(46)
-~)E ~
(X, T) T
:
J
(T)
_ (J n (X, T) + Jp (X, T) )
E comme fonction inconnue,
crivant en ce cas
:
(4s)
:
f) T
-Jx
:
4lectrique (4quation de Maxwell).
t
et donc de c h o i s i r
(47)
E n effet il est indispensable
les dquations donnant
s'4-
u
- ~ x
-
U Jp
Jn E x p r e s s i o n s clans lesquelles on r e m p l a c e U
Net P
par leurs valeurs en fonction de
N, P e t
~x
•
,)X
, J
n
, J
p
et
E.
D a n s ce m o d u l e unidimensionnel d y n a m i q u e
on a pu tenir c o m p t e de ph4-
nom~ne"s particuliers et fort importants de par leurs applications pratiques tels les effets de g4n4ration p a r avalanche (13) et effets de r a y o n n e m e n t s nom~nes
(8) qui sont des ph4-
essentiellement transitoires qul modifient e n t i ~ r e m e n t le c o m p o r t e m e n t
du
dispositlf en des intervalles de t e m p s tr~s brefs (quelques nanosecondes). C e m o d u l e utilise les m ~ m e s @voqu4es pr4c4demment par rapport a u t e m p s ,
techniques de discr4tisation que celles
et nous e m p l o y o n s toujours une m 4 t h o d e purement
de discr@tisation,
implicite ainsi qu'une quasi-lin4arisation syst~mati-
q u e p o u r 41iminer les non-lin4arit4s qui apparaissent dans les diverses expressions. C e qui nous a:m~ne de n o u v e a u ~ devoir r 4 s o u d r e des s y s t ~ m e s d'4quations lin4aires dont la m a t r i c e est tridiagonale en utilisant e n c o r e la s4paration du p r o b l ~ m e
sous-probl~mes
li4s, l'un donnant
E , l'autre N et P.
L ' a l g o r i t h m e g4n4ral est alors le suivant i) - Initialisation ~ partir du m o d u l e 2) - Evaluation du pas en t e m p s
en d e u x
:
statique.
et incr4mentation du temps.
3) - Evaluation 4ventuelle du courant ext4rieur. 4) - Calcul de
U
en tenant c o m p t e 4ventuellement de la g6n4ration par avalanche et
p a r effets ionisants. 5) - Calcul des corrections sur
E, N , P.
6) - Test c o n v e r g e n c e et retour 4ventuel en 4 ou 3 selon le signal appliqu@.
486
7) - Test fin et retour 6ventuel en 2. 8) - Edition des rdsultats. U n certain n o m b r e
de p r o b l ~ m e s restant ~ pr4ciser et en particulier
la gdn4ration automatique des pas de discr4tishtion en
X
et en t e m p s qui se fait
en fonction des r4sultats obtenus dans le calcul (13) pour r4aliser le meilleur cornp r o m i s possible entre l'accroissement de l'erreur, due ~ des pas trop grands et la dur4e du t e m p s de calcul ndcessaire, Des programmes
due ~ des pas trop petits.
de simulation ont 4t4 4crits h partir de cette m 6 t h o -
de. O n m o n t r e ~ titre d'exemple (figure 9) les r4sultats obtenus dans la simulation d'une diode hyperfr4quence en fonctionnement L'emploi de ces p r o g r a m m e s d~pend par contre 4 n o r m 4 m e n t
TRAPATT
.
est tr~s simple. L e cofit de la simulation
des p h 4 n o m ~ n e s
simul4s variant de quelques secon-
des C. P. U. pour un fonctionnement n o r m a l ~ plus de 20 minutes pour simuler une p4riode de fonctionnement hyperfr4quence en raison de l'extr~me rapidit4 des ph4nom~nes
(le t e m p s correspondant ~ une exploitation sur mat4riel C D C
6600).
Cependant la richesse des informations fournies p e r m e t d'expliquer des p h 4 n o m ~ n e s jusqu'alors peu clairs et justifie le coGt de telles simulations (8). 2.3.3.
- M o d U l e bidimensionnel statique .... C o m p t e - t e n u du c h a n g e m e n t de variables donn~ par les 4quations (37)
et (38) le syst~me d'4quations ~ int4grer d'4crit
(4:9)
(--"-~ e - ~ x ) + ~
~)x
4"
(51)
~)x
+
z
"~ ~'-~x
~
= ~e ~
~e-
-
~) z y
:
( T
e
1
1
"~y ) = U -~/ ~ " ) = U
- DOP
O n discr&ise chacune des 4quations par une m 4 t h o d e de diff4rences
f i n i e s s p 4 c l a l e m e n t adapt4e h c e type d ' 4 q u a t i o n e l l i p t i q u e a u t o - a d j o i n t e (29), soit si l ' o n appelIe r .1. j le d o m a i n e r e c t a n g u l a i r e e n t o u r a n t le point i, j d4fini p a r h. h. ij
l-i
x.i
Yj
2 m. j-i +
z
<x
/.. xi +
z
i
m.
<
y < Yj
+
z J
et
487
(m~/Uo^) ~nb!,~2,~p duI,:~lD
I ,
I
I~
ii
I
....
I
1
I
I
O
/ / v
!
'\ Z
O
.5
÷
Z
(em~/s~u~o~)
FIGURE
9
I
uo0~u~uo
D
- Simulation d'une diode h y p e r f r 4 q u e n c e
488
on pose par exemple
:
(52)
O X
~[n
e
~) x) + ~
( T
--
)
dx dy
ij U dx dy 1j et l ' o n a p p l i q u e l a f o r m u l e de G r e e n h c e t t e i n t g g r a l e d o u b l e . O n 4 v a l u e e n s u i t e l ' i n t 4 g r a l e c u r v i l i g n e s u r le c o n t o u r
C . . du d o m a i n e 1j des f o n c t i o n s ~ et ~ , h l ' a i d e de d i f f 4 r e n -
r . . en e x p r i m a n t l e s d 4 r i v 4 e s p r e m i e r e s 1j c e s f i n i e s . On o b t i e n t a i n s i u n e 4 q u a t i o n d i s c r g t i s 4 e en
~ et ~ - a v e c u n f o r m a l i s m e
q u i e s t u t i l i s a b l e 4 g a l e m e n t p o u r t e n i r c o m p t e d e s c o n d i t i o n s a u x l i m i t e s de t y p e
Neumann
et des conditions d'interface. On o b t i e n t a i n s i d e s 4 q u a t i o n s l i n 4 a i r e s en
duterme
(53)
U.. telle t.1
•
+ ~
(h i + hzi . 1) h i
i, j
-
+ k i, j+ 1
+ k
~i j
h l'exception
:
i + 1, j
I
~ i j" et
i, j - 1
~/in -1 + t/2 j
2
(i
(hi+hi-l)hi-1
2 (m. + m . 1) m . J JJ
- 1
2
(hi + hi - 1) hi - t
1/2, j
n
~ i ÷ 1"iZ + ~ ' i e
~'i,j+l/2
e
~ i - 1,,i +71, j] 2
~i, j + t
2
÷71, J7
~ in,
~ni+l/Z,
~" i - 1 / z j n
Yin '
.1
t~i, j_l + 9' i j'/ j-I/Z
j
e
e
e
Z
j +
I/z
e
J
Wi + 1, ,i ÷ ~"i, j 2 ~2i-
1, j
Z
+~i
1t'i, j+l + ~ i , j ( m j + zm j _ 1 ) m j ....
l
n
2 (mj + m .j - l) m j - 1
z ( h i + hi_1 ) hi
i. j
e
Z
j
489
Z (=] + re.j_ i) =j -1 Expressions
grandeur
sur ordinateur
e ~i, J-!Z + ....~/i, ,i3= U. ~, j
directement
et que l'on transforme
de faTon ~ faire appara~re
calculable
j-1/Z
hi'
qui sont utilisables
tr~s faibles polarisations mensionnel
~
comme
les quantit4s
uniquement
dans le cas des
indiqu4 pour le module unidi-
N et P
qui restent
et e n f a i s a n t a p p a r a ~ r e
uniquement
d'un ordre
de
des exponen-
tielles
~ / i , j - ~/i - 1, j par exemple. Cette discr4tisation pose cependant un e certain nombre de probl~mes li4s essentiellement N la pr4cision des r4sultats obte-
nus et ~ l'erreur t4grale
de discr4tisation
double le th6or~me
j
(54)
En effet on a utilis4,
de Green et l'on fair donc appara~re
1 ( ~
r,
commise.
I,~
~) ~ #) x ) dy
e
-
( i ~n
e
pour 4valuer l'in-
:
~2 , ~ ) ~ dx
1J
que l'on @value sur chacun des quatre segments constituant le contour en posant pour l'un d'entre
eux par exemple
x
= x.1 +
h(i)Z
2
Yj -
m'i'12 ,~ Y ~ Yj + mj+12
que l'int4grale curviligne sur ce segment est 4gale ~ :
(55)
_m(j) + m(,i -i) Z
a l o r s q u ' o n a u r a i t dO 4 c r i r e
I ~'ni+I/Z , j que cette int4grale
e @i+i/Z, j
I i+~, j - A,,i,,,i h (i)
e n a p p l i c a t i o n du t h 4 o r ~ m e
de la mo-
y e n n e e s t ~ga][e ~ :
(56)
_re(j)
m(j_l )
+
z
(
i
~
~
e
~
)
)x P
~1' n
la fonction appara~re
que de
e~
~~ x
4rant calcul4e en un point
ici une des principales
e
tV
causes d'erreur
P o du s e g m e n t .
On voit
qui est li4e h l'4valuation num4ri-
en un point d'un segment alors que l'on ne conna~ les valeurs de
qu'aux extr4mit4s de ce segment.
Cette constatation impose des contraintes assez
strictes sur le maillage (Z5). U n m o y e n e
o
d'am41iorer ce processus consiste k calculer
sur le segment et d'en @valuer la v a l e u r m o y e n n e
t i o n lin4aire de
~
en supposant une v a r i a -
le long du segment (i0). Cette fagon de faire permet d'obtenir des
r4sultats plus acceptables. Ii serait pr4f4rable d'envisager un lissage tr~s pr4cis de la fonction ~
; malheureusement
cette solution a coot prohibitif en raison du
grand n o m b r e d'it4rations n4cessaires. Ii a par ailleurs ~t4 expos~ (i) une modifica-
490
tion des 4quations discr4tis4es nentiets.
Malheureusement
pour 4viter le calcul syst4matique
cette am41ioration ne peut s'appliquer
de t e r m e s
expo-
qu'aux faibles
polarisations. L a discr4tisation de l'6quation en ~/ se fera en utilisant la m ~ m e rhode de discr~tisation que p r 4 c 4 d e m m e n t
m4-
ce qui p e r m e t de traiter ais4ment la con-
dition d'interface. O n utilise pour cette 4quation le processus de quasi-lin4arisation 4voqu4 p r 4 c 4 d e m m e n t . L'algorithme g4n4ral est alors I) - D4termination de vecteurs de d4par£ Z) - Calcul de la correction sur
3)
~,
A
N
et
: , P , q/ . o o o ~t fix4s.
Calcul de la correction sur I et
4) - Test convergence et retour 4ventuel en 2. 5) - Edition des r4sultats, Contrairement aux m o d u l e s unidimensionnels on devra cette lois utiliser des m 4 t h o d e s indirectes pour r4soudre les s y s t ~ m e s d'4quations lin4aires rencontr4s ~ chaque it4ration, en raison essentiellemen£ du n o m b r e tisation (on p r e n d c o u r a m m e n t
de points de discr4-
pour ce p r o b l ~ m e de 3 k 5000 points). E n conservant
l'algorithme ei-dessus on a utilis4 principalemen£ trois m 4 t h o d e s : - relaxation par points, - relaxation par blocs (lignes ou colonnes), - m 4 t h o d e des directions altern4es ( P e a c e m a n
- Rachford).
L a m 4 t h o d e de relaxation par points a 4t4 i m m 4 d i a t e m e n t abandonn4e en raison de sa lenteur de convergence (3 ~ 4 lois m o i n s vite que les deux autres). L e s m 4 t h o d e s de relaxation par blocs et de directions altern4es se sont r4v414es quasiment 4quivalente, m a i s s'il a 4t4 possible d'4vaiuer des coefficients de sur-relaxation efficaces et d'apporter une r4duction du t e m p s de moiti4 environ par diverses am41iorations de la m 4 t h o d e L.S.O.I~. m 4 t h o d e de P e a c e m a n - R a c h f o r d
(ZS) il n'a gu~re 4t4 possible d'acc414rer la
de fagon sensible sur ce probl~me.
O n peut aussi penser, acc414rer la convergence du processus, modifier l'algorithme propos4 en se contentant par e x e m p l e de p balayages complets des lignes du maillage pour l'~quation en ~2 suivis de et ~
q
balayages sur les 4quations en
at it4ration de ce processus plutSt que d'attendre la convergence de la re@-
rhode indireete de r4solution sur chacun des s y s t ~ m e s lin6aires. s'est r4v41~e tr~s efficace et il est m ~ m e
Cette faqon de faire
tr~s int4ressant de faire croftre le n o m b r e
de balayages au cours du calcul en se contentant d'un seul balayage au d4but du processus.
491
U n e autre m4thode,
bas4e sur une id4e expos4e par Stone (23), a 4t4
utilis4e (lO) , elle semble 4galement tr~s efficace m a i s sa sensibilit4 aux p a r a m ~ t r e s d'accgl4ration la rend d'un emploi relativement d41icat. L e gain de t e m p s de calcul reste n 4 a n m o i n s appreciable. O n peut enfin penser que l'application de m 4 t h o d e s d'414ments finis ce p r o b l ~ m e devrait permettre une am41ioration sensible des p e r f o r m a n c e s grammes
des pro-
et permettre d'aborder clans des conditions r4alistes l'4tude d'un m o d u l e
bidimensionnel dynamique, ce qui est inenvisageable actuellement v u l e co(it de l'analyse statique. D e s 4tudes sur l'application de ces m 4 t h o d e s k ce p r o b l ~ m e sont en cours k l'heure actuelle. U n autre avantage que devraient apporter ces m 4 t h o d e s r4siderait dans le fair que l'on pourra s'affranchir de l'obligation des maillages rectangulaires dus aux m 4 t h o d e s de diff4rences finies qui se r4v~lent trop coQteux pour l'analyse de certains types de dispositifs. II reste enfin sur ce m o d u l e un dernier p r o b l ~ m e 5 r4soudre : celui du calcul des courants. E n effet ceux-ei n'apparaissent pas explicitement dans les calculs et il convient de les 4valuer 5 partir des trois inconnues principales ~, O n a vu que l'on a par exemple
"-~ ~p
(sv)
1 - ~ ~'p
=
e
_~
-4
grad
~,
~.
:
([,..)
expression qui ne serait calculable que pour les tr~s faibles valeurs de polarisation (pour pouvoir exprimer que des d4riv4es de ~
e- ~ et g~'-~d ( ~ ) ) et qui n4eessiterait l'4valuation n u m 4 r i et ~.~.
Cette fagon de faire entragnerait une tr~s forte impr4cision dans l'4valuation du eourant. O n pr4f4rera, pour chacune des composantes des courants utiliser une int4gration de telle qu'elle a exemple,
l'4quation (57) entre deux bornes quelconques
de
J
soit Y P
Y
(Y + h) P
Yet
Y + h
et J
p
,
A et B
, l'4quation dormant la c o m -
. P P (Y)
(58)
n
4t4 r4alis4e pour le m o d u l e unidimensionnel statique ce qui donne par
si l'on int~gre entre deux points
posante en Y
J
e ~2(y)
= h
~p
fY Y
. ~2(y + h) +h
e
P (Y + h) dt
492
O n retrouve une lois encore la difficult4 due h l'4valuation n u m 4 r i q u e de t e r m e s tels aux extr4mit4s.
je ~ --
sur un s e g m e n t quand on ne connaft que les valeurs de
Cette lois cependant on pourra se permettre un lissage de la fonc-
tion %~car celui-ci ne devra se faire qu'h l'4dition des r~sultats (Z0). Cette fagon de proc4der donne des r4sultats relativement satisfaisants alors que la solution pr4c4demment
propos4e (25) et reprise par ailleurs (i0) a m h n e ~ des variations sensi-
bles de la valeur calcul4e du courant en fonction du maillage choisi. Ce point est cependant e x t r 6 m e m e n t important car c'est uniquement ~ partir du calcul des densit4s de courants que l'on peut r4aliser une c o m p a r a i s o n entre les r~sultats de la simulation n u m 4 r i q u e et l'exp4rlence physique. de simulation ont 4t4 4crits ~ partir des m 4 t h o d e s ex-
Des programmes
pos4es ci-dessus. II sont sensiblement plus importants que ceux des m o d h l e s unidimensionnels (2500 cartes F O R T R A N
IV environ) et n4cessitent une faille m 4 m o i r e
de l'ordre de 300 K octets. L e t e m p s n4cessaire ~ la simulation d'un dispositif varie de quelques secondes ~ quelques minutes (2 ~ 3) sur mat4riel
IBM
360/91,
selon le dispositif
4tudi4 et la polarisation appliqu4e. O n m o n t r e sur la figure 10 ~ titre d'exemple les r4sultats obtenus dans la simulation d'un transistor
M.O.S.
en r4gime de pergage.
L e s p r e m i h r e s analyses physiques qui ont 4t4 faites des r4sultats obtenus (3, Z0, 26) montrent une bonne concordance avec l'exp4rience et les r4sultats que l'on pouvait pr4voir qualitativement. Ii semble que ces p r o g r a m m e s se r4v41er dans un proche avenir un m o y e n
devraient
nouveau et tr~s puissant d'analyse physi-
que.
III
-
CONCLUSION B i e n q u e de n o m b r e u x
en suspens
d a n s c e t t e @rude, t e l s de l ' e x i s t e n c e
- d4monstration
de l a c o n v e r g e n c e
et d e l ' u n i c i t 4
a i e n t dfi ~ t r e l a i s s 4 s
de l a s o l u t i o n d u p r o b l ~ m e
de l a s o l u t i o n du p r o b l ~ m e
discret
pos4,
vers
la solu-
pos4,
- d4monstration
de la convergence
- d4termination
des coefficients
On a pu montrer constituer
math4matiques
:
- d4monstration
tion du problhme
probl~mes
l'application
lier des dispositifs
du processus
optimaux
it4ratif,
d'acc~14ration.
par ce travaill'apport
de m 4 t h o d e s
semi-conducteurs.
num4riques
extr~mement
riche que peut
de c a l c u l d a n s le d o m a i n e
particu-
493
-
o
g
g
g
-
~
v "1"
~
c
C C
5
O)
0
03
iO
CO
¢23 _1
FIGURE
I0 t
-
Etude
de per~age
sur un transistor
M.O.S.
494
O n arrive m ~ r n e ~ obtenir par ce m o y e n
des r4sultats qui sernblent ira-
possibles ~ obtenir par d'autres voles (il sernble en effet 4vident qu'il sera toujours impossible,
quelque soient les progr~s techniques de rnesurer directernent les con-
centrations de porteurs libres ou les potentiels dans les dispositifs) ce qui se r4v~le ~tre un m o y e n
d'4tude physique extr~rnernent int4ressant (4). P a r ailleurs, les rn4thodes et p r o g r a m r n e s nurn4riques utilis4s pour
cette 4rude et l'exp4rience a c c u m u l 4 e ~ ce propos pourront certainernent ~tre ernploy6s efficacernent pour des applications nouvelles dans bien d'autres d o m a i n e s que celui des dispositifs semi-conducteurs
(19).
495
BIBLIO GRAPHIE
(1)
ANDREW R. I m p r o v e d f o r m u l a t i o n of G u m m e l ' s a l g o r i t h m f o r s o l v i n g the Z - d i m e n sional c u r r e n t - f l o w e q u a t i o n s in s e m i c o n d u c t o r d e v i c e s . Electronics letters
(z)
vol. 8
n° Z 2
536-538
BARRON M.B. C o m p u t e r aided analysis of I. G. F. E. T. Techn. Report n ° 5501-i Stanford L a b s
(1972)
(1969)
(3)
BOREL J. , H A J - S A I D F., M E R C K E L G. , S A I N T O T A two-dimenslonnal analysis of a lateral transistor E. S. S. D. E. R.C. Munich (1973)
P. , V A N D O R P E
(4)
BOREL
J. , MAFFEI
J. , SAINTOT
STERN
M.,
VANDORPE
A.
, MERCKEL
G. , MONNIER
D.
P. ,
D.
Connection between technology and m o d e l s using computer analysis. I.E.E.E. int. Solid state circuits conference. Philadelphie (1973)
(5)
DE MARIA. A n accurate numerical steady-state one dimensional solution of the P N junction. SolidState Electronics vol. ii p. 33-58 (1968)
(6)
DE MARI A. A n accurate one-dimensional solution of the P N junction under arbitrary transient conditions. Solid state Electronics vol. ii pp IOZI-I053 (1968)
(7)
DOUGLAS J. Survey of numerical m e t h o d s for parabolic differential equations. A d v a n c e s in c o m p u t e r s vol. Z p. 1 A c a d e m i c press N e w - Y o r k
(8)
GAILLARD R. Photocourants induits par une impulsion de r a y o n n e m e n t s sitifs seml- conducteur s. T h ~ s e d'4tat Lyon (1973)
(1961)
dans les dispo-
(9)
GUMMEL H.K. Ik self consistent iterative s c h e m e for one-dimensional steady-state transistor calculations. I.E.E.E. Trans. on E.D. E.D. ii 455 (1964)
(lO)
H E Y D E M A N N M. R 4 s o l u t i o n n u m 4 r i q u e des 4quations b i d i m e n s i o n n e l l e s de t r a n s p o r t dans l e s s e m i - c o n d u c t e u r s. Th~se Ing. Doct. Paris-Sud (1972)
496
(ii)
KELLER J.B. R4solution d'4quations paraboliques aux d4riv4es partielles ("M4thodes m a t h 4 m a t i q u e s pour calculateurs arithm4tiques") Dunod Paris (1965)
(lZ)
KENNEDY D. P., M U R L E Y P. C., Calculations of impurity a t o m diffusion through a n a r r o w diffusion m a s k opening. I.B.M. Jnal I0 pp. 6-12 (1966)
(13)
L I O N S J. L., M A G E N E S E. P r o b l ~ m e s aux limites non h o m o g h n e s Dunod Paris (1968)
et applications
(vol. 1)
(14)
MAFFEI A. , M o d h l e m a t h 4 m a t i q u e unidimensionnel transitoire du fonctionnement des diodes ~ avalanche. Thhse de 3 h m e cycle Lyon (197Z)
(15)
MO CK M. S. On the c o n v e r g e n c e of G u m m e l ' s n u m e r i c a l a l g o r i t h m Solid State Electronics vol. 15 pp. 1-4 (1972)
(16)
MONNIER J. Simulation n u m 4 r i q u e de la diffusion d'impuret4s dans un semi-conducteur. Application au cas du bore dans le silicium.
T h ~ s e Ing. Doct.
Grenoble
(1971)
(17)
MONNIER J., M A F F E I A. , S T E R N M., V A N D O R P E D. N u m e r i c a l analysis of simultaneous diffusion of several impurities in silicon and silicon dioxide. Electronics letters 8 pp. 517-518 (197Z)
(18)
R E I S E R M. A two-dimensional numerical F E T m o d e l for DC, A C and large-signal analysis. I.E.E.E. Trans. on E.D. E.D. Z0 pp. 35-45 (1973)
(19)
R O U X J. P. , T O U R N E U R J. L., V A N D O R P E D. G E N E P I , G4n4rateur de p r o g r a m m e s d'int4gration. Rapport D . G . R . S . T . 72.7. 0434 (1973)
(zo)
SAINTOT P. Analyse bidlmensionnelle du transistor M. O. S. Th~se 3 ~ m e cycle Grenoble (1973)
(Zl)
SEIDMANN T. I. , C H O O S. C. , Iterative s c h e m e for computer simulation of semiconductor devices. Solid State Electronics vol. 15 pp. 1229-1235 (1972)
(22)
SLOT]3OOM J.W. Iterative s c h e m e for 1 and Z dimensional D. C. transistor simulation. Electronics Letters vol. 5 n ° 26 pp, 677-678 (1969)
497
(23)
(24)
S T O N E H. L. I t e r a t i v e s o l u t i o n of i m p l i c i t a p p r o x i m a t i o n o f m u l t i d i m e n s i o n a l p a r t i a l differential equations. S . I . A . M . Jnal N u m e r . Anal. vol. 5 n ° 3 pp. 5 3 0 - 5 5 8 (1968) V A N D O R P E D. Etude d'un module mathgmatique Ing4nieur D o c t e u r
(25)
(26)
Grenoble
V A N D O R P E D. Etude mathdmatique de la fabrication t i f s s e m i c o n d u c t e u r s. Thhse d'4tat Lyon (1971)
semiconducteurs
5
des disposi-
G, S A I N T O T P. analysis of the M . O . S .
pp. 5 4 7 - 5 5 7
tran-
(1972)
V A N D O R P E D. , M O N N I E R J. T w o - d i m e n s i o n a l a n a l y s i s of p l a n a r d i f f u s i o n . ]Electronics Letters
(28)
vol.
dispositifs
e t du f o n c t i o n n e m e n t
VANDORPE D. , B O R E L J. , M E R C K E L A n accurate t w o - d i m e n s i o n a l n u m e r i c a l sistor
Solid State Electronics (27)
de certains
(1969)
vol. 8
n ° 23
pp. 5 5 4 - 5 5 5
V A N D O R P E D. , X U O N G N, H. , ModUle bidimensionnel de semiconducteurs, A n n , du c o l l o q u e Int. d e m i c r o 4 1 e c t r o n i q u e
(1972)
effets d'avalanche avanc4e CHIRON
Paris
111970) (29)
(3o)
V A R G A R. Matrix iterative :Prentice Hall
analysis ( 1962)
X U O N G N. H. E t u d e du r d g i m e t r a n s i t o i r e d ' u n m o d u l e m a t h d m a t i q u e Thhse 3~me cycle Grenoble (1969)
de jonction PN
