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ss Chemical Engineering Dept., Loughborough Umkrsity
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Nassehi, Vahid. Practical aspects of finite elcmcnt modelling of polymer processing / Vahid Nassehi p. em. Includes bibliographical references. ISBN 0-471-49042-3 I. P o ~ y m e r ~ - ~ a t h e m ~ t imodels. cal 2. Chemical processes -Mathematical models. 3. Finite element inetliod. I. Title. TP1120 .N37 2001 668.9 dc21 2001045560 Lihrur-y ~
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1
1.1 Governiiig Equations of Non-Newtonian Fluid Mechanjcs 1.1.1 Continuity equation I , 1.2 Equation of inotion 1 .1.3 Thermal energy equatioii I .1.4 Constilutive equations
i T i ~ e - ~ n d e p e n ~Fluids e~lt L .2 ~ l a s s i ~ c ~ t iofo iInelastic 1.2.1 Newtonian fluids 1.22 Geiieralized Newtonian fluids
1.3 ~ ~ c~ ~~ r ~n cs ~ ~~ Fluids ~i ~~~ e n d c ~ ~ ~ 1.4 Viscoelastic Fluids 1.4.1 Model (material) paramcters used in viscoelastic constitutive equations 1.4.2 Diffcreiitial constitutive equations for viscoelaqtic fluids 1.4.3 Singlc-integral constitutive equatioizs for viscoelastlc fluids 1.4.4 Viscometric approach - the (CEF) model
eferences
2 2 2 3 3 4 4 5 8
9 9 11 13 14
15
7 10
2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.1.8 2.1.9
Intei-polation models Shape functions of commonly used Gnite elemeiits Non-standard elements Local coordinate systems Order of continuity of finite elements Coiivergence Irregular and curved elements - isoparanietric mapping Numcrical integration Mesh refinement - b- and p-versions of the finite element method
20 23 27 29 32 33 34 38 40
viii
CONTENTS
Numerical Solution of Differential Equations by the esidual Method 2.1 Weighted residual statements in the context of finite element discretizatioiis 1.2.2 The standard Galerkin method 2.2.2* Galerkin finite element procedure - a worked example 2.2.3 Streamline upwind Petrov-@derkin n~etliod 2.2.3* Application of upwiiiding - 2% worked example 2.2.4 Least-squares finite elemciit method 2.2.5 Solution oC time-dependent problems eferences
2.2
41
42 43 44 53 54 64 64 68
3 7
3.1
3.2
3.3 3.4
Solution o f the Equatioiis of Continuity and Motion 3.1.1 The U-V P scheiiie 3.1.2 The U-V-P scheme based on the slightly compressible continuity equation 3.1.3 Penalty schemes 3.1.4 Calculation of pressure in the penalty schemes - variational recovery method 3.1.5 Application of Green’s theorem - weak Corniulations 3.1.6 Least-squares scheme Modelling of Viscoelastic Flow 3.2.1 Outline of a decoupled scheme for the differential constitutive models Dcrivatiorz of the wwkipg equations 3.2.2 Finite element schemes for the integral coristitutive models 3.22 Non-isothernial vlscoelastic Row Solution of the Energy Equation Imposition of Boundary Conditions in Polymer
71 72
Models
93 93
3.4. I
3.5
3.4.2 3.4.3 Free 3.5.1
3.5.2 3.5.3 eferences
Velocity and surface force (stress) components i n k t eonditiom Lirw of symnletrj’ Solid walls Exit conditions Slip-wall boundary conditions Temperature and thermal stresses (teniperature gradients) SurCace and Moving Boundary Problems VOF method in ‘Eulerian’ frameworks VOF method in ‘Arbitrary Lagiangian--Ei~lerian’ frameworks VOF method in ‘Lagrangian’ frameworks
74 75 77 77 79 79
81 83 86 89 90
95
96 96 97 98 99 t01 101 102 1 04
I08
CONTENTS
ix
111 4.1
Modelling of Steady State Stokes Flow of a Generalized Newtonian Fluid 4.1.1 Goverriing equations in two-dimensional Cartesian coordinate systems 4.1.2 Governing equations in two-dimensional polar coordinate systems 4.1.3 Governing equations in axisyimnetric coordinate systeins 4.1.4 Working equations of the U -V- P scheme in Cartesian coordinate systems 4.1.5 Working equations of the U-V-P scheme in polar coordinate systems 4.1.6 Working equations of the U -V-P scheme in axisytnxnetric coordinate systeins 4.1.7 Working equations of the continuous penalty scheme in Cartesian coordinate sys tems 4 1.8 Working equations of the continuous penalty scheme in polar coordinate systems 4.1.9 Working equations of the continuous penalty scheme in axisyinmctric coordinate $ystGms 4.1.10 Working equations of the discrete penalty scheme in Cartesiaii coordinate systems 4.1.11 Working equations of the least-squares scheme in Cartesiaii coordinate systemv
4.2 4.3
4.4
Variations of Viscosity Modelling of Steady-State Viscometric Flow - Working Equations of the Continuous Penalty Scheme in Cartesian Coordinate Systems ~ o d e l l i n gof Thermal Energy 4.4.1 Working equations of thc e upwind @U) scheme for the steady-state energy equation iii Cartesian, polar and axisymmetric coordinate systems 4.4.2 Least-squares and streamline upwind Petrov-Galerkin (SUPG) schemes odelling o f Transient Stokes Flow of Generalized Newtonian
and Non-Newtonian Fluids cferences
5.1
111 111 112 113 114
116 117 118 120 121 123 125 126
127 128
129 131 132 139
Models Based on Simplified Domain Geomctry
141
5.1.1 ~ ~ d e l l ofi ~the g dispersion stage in partially filled batch internal mixers Flow simulation in a .single blade partially filled mixer Flow simulation iri a portrali-vjifrlled twin blade mixer
142 142 146
x
CONTENTS
5.2
Models Based oil Simplified Governing Equations 5.2.1 Sirnitlation of the Couette flow of silicon rubber - gencralized
150
Newtoiiian model Siinulation of the Gouette flow of silicon rubber - viscoelastic
151
inodd
152
5.2.2
epresenting Selected Segments of a Large 156 5.3. I
Prediction of stress overshoot in the contracting sections of a r)inunetric flow domain 5.3.2 Simulation o I slip in a rubber mixer
5.4
Models Based on oupled Flow quations the Flow Inside a Cone-and-Plate
-
156 158
Simulation of
5.4.1 Goveining equations 5.4.2 Finite element discretization of the governing equations
ased on ‘Thin Layer A ~ ~ r o x i m ~ t i o ~ ite element rnodelling of flow distribution in an extrusion die 5.5.2 Generalization of the Hele-Shaw approach to flow in thin curved layers Asymptotic expansion sclwtne
iiffiiess Analysis of Solid Polymeric 5.6.1 Stiffness analysis o f polymer composites filled with spherical particlcs
eferences
160 162 166 170 113 175 117 183 184 188
1
6.1
Gencral Considerations Generation
elated to Finite Element
rocessor Progmms
191 192 193 193 194 195 196
6.3.1 Direct solution methods Pivoting ~ ~ ~ ¶ ~elimimtion , ~ i a n with pnrtiul pivoting Nutnbc.~.OJ operictions in the Gimsiun eEiwzirzation method
199 200 200 20 1 202
6.1.1
Mesh types Block-ctructured g i k h Ovrrsrf grid5 Hybrid grids 6.1.2 Conmiou methods of mesh generation
6.2 6.3
6.4
ain Components of Finite ELem Lrmerical Solution o f the Global Equations
~ ~ l ~ ~a lt ~i oor ni t h based ~s on the Caussian e l i r n i ~ ~ ~ i o n method 6.4.1 LU dccomposition technique 6.4.2 Frontal solutioil tzchniquc
203 203 205
CONTENTS
6.5 Computational Errors 6.5.1 Round-off error 6.5.2 Iterative improvement o f the solution of systems of linear equations
References
xi 206 206
207 207
2 209 210 213 21 5 217 220 150
8.1 8.2
Vector Algebra Some Vector Calculus 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5
Divergence (Gauss’) theorcm Stokes theorem Reynolds transport thcorein Covariant and conlravanant vectors Second order tensors
8.3 Tensor Algebra 8.3.1
8.4
Invariants of a second-order tensor ( T )
Some Tensor Calculus 8.4.1 Covariant, contravariant and mixed tensora 8.4.2 The leiigth o f a linc and metric tensor
253 255 256 257 257 258 258 255) 26 I 262 262 263 7
9
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~ o ~ p u t a t i ~o u~ i~dday~n a ~ i c sis a major investigative tool in the design and analysis of complex flow processes eiicountered in moderii iiidustrial operations. At the core of every computational analysis is a iiurnerical method that determines its accuracy, reliabil~ty,speed and cost effectiveness. The finite element method, originally developed by structural engineers for the numerical modelling o f solid mechanical problems, l ~ i been s eslablislied as a powerful technique that provides these require~entsin the solution of fluid flow and beat transfer problems. The most significant characteristic of this technique is its geometrical flexibility. Therefore, it is regarded as the method of choice in the analysis of problems posed in g e o ~ ~ t r ~ ~complex d l l y domains. For this reason the ana~ysis olymer processing flow regimes is often based on the finite element
~ ~ d u s t r ipolymer al processing encompasses a wide range of operat~onssuch as extrusion, coating, mixing, moulding, etc. a multiplicity o f materials carried out under various operating conditions. design arid o r g ~ n i ~ a t i oofn each process should therefore be based on il detailed quantit~tiveanalysis of its specific features and conditions. The common - and probably the most portajority of these analyses is, however, the s i ~ ~ ~ l aoft ~a ononn wtonian fluid deformation and flow p tion o f non-isothermal, noii-Newtonia with the f o i ~ u ~ a t ~ofo na matheniat~calmodel consisting of the g o v e ~ i ~ l i g e q i i ~ t ~ oarising ~ s , from the laws of conservation of mass, energy, inomenturn icli describe the constitutive behaviour of the fluid, together ropriate boundary conditions. The f o r m ~ i ~ ~ it ~ e da t h e ~ ~ a t ~ c a l ed via a computer based nuiiierical lechnique. Therefore, the mputer models for n ~ n - ~ e w t oflow ~ i ~regi~ies a~ in polymer t i - d i ~ c ~ ~ l i itask i ~ r yin which iiumeri ~ a lana~y~is, ~omput~r programming, fluid mechanics and rlieology each form ail i ortaiit part. It is evident that these subjects cannot be covered in a single te and an in-depth description of each area requires separate volumes. It is not, however, realistic m n e that before embarking on a project in the area of computer m yrner ~rocess~ng one should acquire a thorough thesrctical kilo all of these subjects. ~ i i d e ethe ~ norrrial time period allowed for conip~et~oii of
xiv
PREFACE
postgraduate research studies or industrial projects precludes such an ambitious requirement. The utilization of commercially available finite element packages in the s~mulationo f routine operations in industrial polymer processing is well established. However, these packages camot be usually used as general research tools. Thus flexible ‘in-house’aeated progranis are needed to carry out the analysis required in the investigation, design and development of novel equipment and operations. This book is directed towards postgraduate students and practising ei~~ineers who wish to develop finite element codes for non-Newtonian flow problems arising in polymer processing operations. The main goal has been to enable the reader to come to speed in a relatively short time. ~i~evitably, in-depth discussions about the fundamental aspects of non-Newtonian fluid mechanics and i n ~ t h e ~ a t i cbackground a~ of the finite element method have been avoided. nstead, the focus of the text is to provide the ‘parts and tools’ required for a s s e ~ b ~~i n~ igt element e models which have applicabi~ityin s i t u ~ t ~ oexpected ns to arise under realistic conditions. The illustrative examples that are iiicluded in the book have been selected carefully to give a wide-ranging view of the application of the described finite element schemes to industrial problems. The finite element program listed in the last chapter can be used to model n o n ~ i s o t h e ~ ~steady-state al generalized Newtonian flow in two-dimensioiial planar domains. The code solves Iaminar incompressible Navier-Stokes equations for a power law fluid. The program is written using a clcar and simple sty1 and does not include any special features and hence can be readily compile most Fortran compilers. The basic code given in this program may be extended to solve more complex polymer processing problems using the finite e~eiiientschemes derived in the book (Chapter 4). An illustrative example that shows the extension of the code to axisyinmetric flow problems is discussed in the text and the required modifications are highlighte~on the program listing. i
Computer modelling provides powerf'ul and conveiiient tools for the quantitative analysis of Auid dynamics and heat transfer in n o n - ~ e w t o ~ ~polymer ian flow systems. Therefore these ~echniquesare routinely used in the modern polymer industry to design and develop better and more efficient process equ~pmentand operations. The main steps in the development of a c o ~ i ~ u t e r model for a physical process, such as the flow aiid defoiination of polymeric materials, can be sunimarized as: formulation of a sct o f governing equations which in conjunction with appropriate initial and boundary conditions provide a ~ i a t h e n ~ ~ t imodel cal for the process. and solution o€ the foriiiu~atedmodel by a computer based~nL~fne~ca1 scheme.
liidustrial scale polymer forming operations are usually based on the combination of various types of i~ dividua~ processes. Therefore in the c o ~ ~ L ~ t e r - ~ i d e d design of these operations a section-by-section approach can be adopted, in which each section of a larger process is modelled separately. An i ~ e q ~ i ~ ~ nini ethis n t approach is the imposition of realistic boundary c at the limits of the sub-sections of a complicated process. The division of a complex operation into simpler sections should therefore be based on a systematic procedure that can provide the necessary boundary conditions at the ~imits of its sub-processes. A rational method for the identification of the subs of polymer forming operations is ~ e s c r ~ b eby d Non-Newtcmian flow processes play a key role in many types of polymer en~~iieeriiig operations. Hence, f ~ ~ ~ l a to if ~o i~a it h e i ~ dmodels ~ ~ c ~ for l these processes can be based on the equatio~isof n o n - ~ e w t o n i ~fluid ~ n mechanics. The general equations of ~ o n - ~ ~ w fluid ~ o mechanics n i ~ ~ provide expressions in terms of velocity, pressure, stress, rate of strain and t ~ ~ ~ c r a tin~ 1a -flow e d o n ~ ~ These ~ n . equations are ~erivedon the basis of physical laws and
2
THE BASIC EQLJATIONS OF NON-NEWTONIAN FI,UTD MECHANICS
rheological experiments. ecause of the predominant role of non-Newtonian flow equations in the modelliiig of polymer processes it is important to understand the theoretical foundations of these equations. However, detailed explanation of the theoretical foundations of non-Newtonian fluid meclianics is outside the scope of the present book. The subject is covered in many textbooks devoted to the topic. For example, the reader can find detailed derivations of the basic cquatiovis of non-Newtonian fluid mechanics in Bird et a/. (1960) and Aris (1989) and more specifically for polymeric fluids in ~ i d d l e m ~ n ird et ul. (1977). In this chapter the general equations of laminar, non-Newtonian, noni s o t h e ~ ~inco~pressib~e al~ flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity o f presentation, vector notations are used and all of the equations are given in a h e d ( s t a t i o ~ ~ or ~ r yEulerian) coordinate system.
The conti~iuityequation is the expression of the law of con~erv~tion of mass. This equation is written as
where '7 is the operator nabla (gradient opcrator) and v is the velocity vector. quation (1.1) is also callcd the in~ompressibilityconstraint. The absence of a pressure temi in the above equation is a source of diffkulty in tbe nttnierical simulation of incompressible flows.
The equation of motion is based on the law of' conservation of inomenturn ~ ~ e w ~ osecond n ' s law o f motion). This equation i s written as
where p is fluid density, v is selocity, is the Cauchy stress tensor and g i s the r i cregimes the convection body force per unit volume of fluid. ~ o ~ y ~ eflow (1.2) is usually small and can e neglected. Tfiis is a eynolds number (creepiiig or okes) flow of highly
GOVERNING EQUATIONS OF NON-NEWTONIAN FLUID MECHANICS
3
viscous fluids. In the majority of polymer flow systems the body force in comparison to stress is very small and can also be omitted from the equation of motion. The Cauchy stress tensor is given as = -p6
+z
(1.3)
where p is hydrostatic pressure, 6 is unit second-order tensor (Kronecker delta) and z is the extra stress tensor. The equation of motion is hence written as
(1.4)
This equation is the expression of the conservation of theiinal energy (first law of thermodynamics) and i s written as T = kV2T+ Dt
(1.5)
pc --
where I' is specific heat, k is thermal conductivity, T is temperature aiid V2 is the scalar ~aplaciaii.The terms on the right-hand side of Equation (1.5) represent heat flux due to conduction, viscous heat dissipation and a heat source (e.g, heat generated by chemical reactions etc.), respectively. Thermal energy changes related to the variations of fluid density are neglected in this equation. processes iiivolving solidi~cationor melting of polymers, the specific heat varies substantially with temperature and it should be retained inside the material derivative in Equation (1.5). Thermal conductivity of polymers is also likely to be t e ~ p e r a t ~ dependent re and anisotropic and, ideally, should be treated as a variable in the derivation of the energy equation. In practice, however, the lack of ~xperimentaldata usually prevents the use of a variable k in polymer flow models.
the e
s
A c o n s t i t ~ ~ ~equation ve i s a relation between the extra stress (7)and the rate of deformation that a fluid experiences as it flows. Therefore, theoretically, the constitutive equation of a fluid characterises its macroscopic d e f o r ~ a t ~ o ~ behaviour under different flow conditions. It is reasonable to assume that the macroscopic bcliaviour of a fluid mainly depends on its microscopic structure. owever, it is e x t r e ~ e ~difficult, y if not i ~ p o s s i b ~to e .establish exact q u a ~ i t i ~ ~ t i v e
relationships between the microscopic structure of non-Newtonian fluids and their macroscopic properties. Therefore the derivation of universally applicable const~tL~t ive models for non-Wewtonian fluids is, in general, not attempted. Instead, semi-empirical relationships which give a reasonable prediction for the behaviour of specified classes of non-Newtonian fluids under given flow con~itionsare uscd. Depending on their constitutive behaviour, polymeric liquids are classified as: ti~e-in~ependent inelastic, tii~e-~ependent inelastic, or vi~coe~astic
In an i~~elastic, time-independent (Stokesian) fluid the extra stress is considered to be a function of the instantaneous rate of defornnatioii (rate o f strain). ~ ~ l i e ~ e fin o rthis e case the fluid does not retain any memory of tlic history of tlie defor~ationwhich it has expericnced at previous stages of the flow.
In the s i ~ p l e scase t of Newtoniaii fluids (linear Stokesian uids) the extra stress tensor is expressed, using a constant Auid viscosity p, as
is the rate of deformation (rate of strain) tensor re~~esenting the s y r n ~ ~ ~part r i c of the velocity gradient tensor. C o ~ ~ o n e of ~ tthe s rate of ion tensor are hence given in terms of the velocity ~ r a d i ~ as n~s
) and (1.7) are used to formulate explicit relationships between extra stress c o ~ p o n e n t and s the velocity gradients. Using these r e l a t i o ~ ~ h ~ ~ s the extra stress, T,can be eliininated from the governi~i~ e q u a ~ i ~ ~This n s . is the basis for the derivation of the well-known Navbvier- Stokes equations which re wtonian flow (Aris, 1989).
CLASSIFICA~IONOF INELASTIC T ~ M E - I N ~ E ~ ’ E NFLUIDS ~~Nr
5
Common expcrime~italevidence shows that the viscosity of polymers varies as they flow. Under certain conditions however, elastic effects in a polymeric flow can be neglected. In these situations the extra stress is again expressed, e~pli~itly’ in terms of the rate of deformation as
where q is the apparent viscosity, which is a function of the ~agiiitudeof the rate of deforination tensor and teniperatL~re.Equation (1.8) is said to provide a neralized Newtonian’ description of the fluid behaviour. Analogous to connt viscosity Newtoiiian flow, this equation is used to derive the ‘generalized N ~ v i e r - ~ t o ~ equations es’ via the substitution of the ex stress in the equation of motion in terms of viscosity and velocity gradients. nce, the only requirement for the solution of these equations i s the determination of the appare~~t fluid viscosity. Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-d~~en$ional, constant shear stress). The rate of defor~atlolitensor in a simple shear flow is given as
+
where i s a scalar called the shear rate. Consequently in this case an experimental flow curve which relates the shear stress to the shear rate (called a rlic~gram)can be used to calculate the fluid viscosity as
where ‘12 i s the shear stress. Tn practice, it is very difficult to establish a simple shear flow and instead ‘viscometric’ regimes are used to determin~p a r a ~ e t e r s such as viscosity. In a viscometric flow, a fluid elenieiit deformation observed in a frame of reference which translates and rotates with that*element, will be identical to a siinple shear system. In addition l o the apparent viscosity two other material parameters can be obtained using simple shear flow viscometry. These are primary and secondary normal stress coefficients expressed. respectively. as @I2
and
=
7-11
-
7-22
r2
(11.11)
THE BhSlC EQUATIONS OF NON-NEWTONlAN FLUID MECHANICS ?1123 = 7-22 - 7-33
i’2
(1.12)
Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the m~crostructureof long-chain polymers subjected to high shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus tlie primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. utnerous examples of polymer flow models based on generalized N e w t ~ ~ i ~ d n behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the tiinc-independent generalized Newtonian fluids are divided into three groups. These are Bingha rn plastics, p s e ~ i ~ o p ~ afluids s t ~ c and dilatant fluids.
ingharn plastics ingham plastics are fluids which remain rigid under the application of shear stresses less than a yield stress, rJ1b like a simple Newtonian fluid once d shear exceeds this value. of fluids were developed (1947) and Casson (1959).
uids have no yield stress threshold and in these fluids the ratio of shear stress to the rate of shear generally falls coii~ii~uo~sly and rapidly with increase in tlie shear rate. Very low and very high shear regions are the excep, where the flow curve is almost ~ o r ~ z 0(Figure ~ t ~ l 1.1). c o m m ~ nclioice of functional rclatioiishi~between shear viscosity and rate, that usually gives a good prediction for the sliear doplastic pluicls, i s the power Law model proposed by de wald (1925). This model i s written as the ~ollo~/iiig equat (1.13)
where the pa~ameters,qo and n, are called the consistency coefficient and the power law index, respectively, It is clear that a uid with power law index of unity will be a pureIy Newtonian fluid. It is also corninonly accepted that the nonew~0nianbehaviour of fluids become more pronounced as their correspondiiig
CLASSIFICATION OF TNE1,ASTIC T T M ~ - l N D E P FLUIDS E ~ ~ ~ ~ ~7 ~
i---+--
. .............................
Shear rate i, (s-9
~~~1~~
1.1 Shear thinning behaviour of pseudoplastic fluids
power law index shows greater departure from unity. The consisteixy coef~cient is fluid viscosity at zero shear and it has a higher value for more viscous fluids. A modi~edversion of the power law inodel which can represent very low shear regions has also been proposed ( ~ i d d ~ e ~1977). a n , Hn some cascs it may be more realistic to apply a segmented form of this model in which different values of the p a r a ~ e t e r sover different ranges of the shear rate are used. The f o l l o ~ n g tem~crat~~re-dependent form of the power law equation, based on the forinula for thermal effccts, is the niost frequently used version o f this p o l ~ ~flow i e ~niodcls ( ittrnan and Nakazawa, 1984)
where h is called the temperature dependency coefficient and T,..t is a referenc~ n addition to the power law inodel a plethora of other relasenting the constitutive behaviour of pseudoplastic fluids can also be found in the literature. For example, the f ~ ~ ~ o wequation iii~ by Carreau (1968) has gained wides~~ead application in polymer prQ~essing analysis
(1.15)
where r/o and vx, are zero and infinite shear rate 'constant viscosities', respectively, X is a material time constant and y2 is the power law index.
THE BASIC EQUATIONS OF N 0 ~ " ~ E ~ T O ~FI,IJII) I A NM
~
~
~
~
N
l
~
Shear stress T~~(Pa)
Pseudoplastic
I-----
~~~~~
1.2 Comparison of the rheological behaviour of Newtoriian arid typical gencralized Ncwtoaian fluids
ilatant fluids (also known as shear thickening fluids) show an increase in viscosity with an increase in shear rate. Such art increase in viscosity may, or may not, be acco easurable charige in the volume of the fluid wer law-type rheological equa~iollswith y1 > 1 are usually used to model this type of fluids. Typical rheograins representing the behaviour of various types of geiier~lize~ ~ ~ w t o n i afluids n are shown in Figure 1.2.
Under the application of a steady rate of shear, the viscosity of some types of n ~ n ~ ~ e wfluids t o ~changes ~ ~ nwith time. ~ i i ~ e - d e ~ e n dfluids, e n t which show an increase in viscosity as time passes, are called 'rheopectic'. Fluids showing the opposite effect of decreasing viscosity are called 'thixotropic'. thixo~ropyare complex phenomena resulting from transient changes of the molecular structure of time-dependent fluids under an applied shear stress. In general, it i s extremely difficult to introduce molecular effects of this kind into the con~titutiveequations of non-~ewtonianfluids. Thus the proposed constitutive models for these fluids are based on inany simplifying assumptions (Slibar IJaslay, 1959). In cases where the elastic effects shown by a t i ~ e - ~ e p c n d e n t are negligible then, basically, a mathemdtical model similar to the generalThe constitutiv4: ized Newtoniaii fluids can be used to represent their fl dency of the fluid equation in such a model must, however, reflect the time d ~
S
ViSCOELASl'IC PL U iDS
viscosity. The construction of flow models for time-dependent fluids often requires the use of kinetical relations. These relations represent molecular nomenon such as polymer degradation as a fhction o f time ( ~ e m b ~ o w s l ~ i etera, 1981).
Apart from the prediction of a variable viscosity, generalized N e w t ~ n i con~i~~ stitutive models cannot explain other phenomena such as recoil, stress relaxation, stress overshoot and extrudate swell wliich are commonly observed in polymer processing flows. These effects have a significant impact on the product quality in polymer processing and they should not be ignored. ~ ~ c o r e t i ~ all all~, of* these phenomena can be considered as the result of the material having a Combination of the properties of elastic solids and viscous fluids. ~ ~ t h e n i a t i cmodelling a~ of polymer processing flows should, ideally. be based on the use of viscoelastic constitutive equatioiis. Formulation of the constitutive equations for viscoelastic fluid s been the subject of a considerable amou~itof research over many decades. ails of the derivation of the viscoelastic constitutive equations and their tion are covered in many textbo review papers (see Tanner, d rt al., 1977; it soul is, 1990). these efforts and the prolifer oposed viscoelastic constitut~vee in recent years, the problem of selecting one wliich can yield v a fluid under all types of flow conditions i s still wiresolved practice, therefore, the reinaini~igoption is to choose a constitutive viscoelastic model that can predi the most dominant features of the fluid behaviour for a given flow situatio~i. should also be mentioned here that the use of a com~ u t a ~ i ~ ~ ncostly a l l y and complex viscoelastic model in situations that are different from those assumed in the formulation of that model will in general yield unreli~~ble ~ r e d i c t i ~ nand s should be avoided.
atcrial paraiiiete~§,such as relaxation time, elongational viscosity and normal ess coefficients, are essentially used as coiivenieiil means of ~ ~ t ~ o d L i c various aspects of viscoelastic fluid behaviour into a constitutive e ~ ~ ~ aThe ~ i o ~ exact d e ~ n ~ ~ of i o these n parameters under general conditions i s ~ i ~ ~and c u i ~ hence they are regarded as empirical parameters in viscoelastic models. These parameters can, ever, be regarded as compatible with ~ / c l l - ~ ~ ~ fphysical ined f~nctionsunder le flow conditions. Thus, analogous to the fluid viscosity in ~ e w t ~ ~flows, ~ ~ athei imaterial ammeters in a viscoelastic rnodel are found via rheonietr~ce~perir~ents conducted using simple flow regimes.
THE BASIC EQUATIONS OF NON-NEWTONIAN FL,UID MECHANLCS
tress rclaxation time, obtained from rheograms based on viscometric flows, i s used to define a dimensionless parameter called the ‘ eborah number’, which
quantifies the elastic character of a fluid
(1.16) where A is the relaxation time (characteristic of polymer chains relaxation) and 0 is an appropriate duration of the deformation time. The magnitude of the Deborah number is used as a measure for deciding whether viscoelastic effects in a certain flow problem are significant or not. An alternative definition based on the dimensioiiless ‘Weissenberg number’ is also used to provide a quantitative measure for viscoelasticity of non-Newtonian fluids ( ~ i ~ ~ ~1977). ~ e ~ Thea n , eissenberg number is defined as
w,
=;
x-Y H
(1.17)
where Y and N are the characteristic process velocity arid process length, respectively. The rate of ~ e f o ~ ~ a t itensor o i i in a pure e~ongationalflow has the fol~ow~ng form
(1.18)
where P is a positive scalar called the principal extension rate. ~longa~ional Wow the elongational viscosity o f a fluid is defined as (1.19) where 7-11 and 5-22 are the normal stress co~poiients.In practice, it i s very difficult to set up a controlled, pure elongationd flow and the measurement of the elo~gationalviscosity of fluids is not a trivial matter. For a ~ e ~ ~ t o n fluid i~ii the e ~ ~ i i ~ a t i o viscosity nal is three times the value of ,U ( ~ t e v e n ~1972). o ~ ~ In ~ vjscoelastic fluids the ratio of elongational viscosity to shear viscosity can be much higher than three. The elongation viscosity defined by Equation (1.19) represents a uni-axial extension. Elongational flows based on biaxial extensions can also be considered. In an equi-biaxial extension the rate of d e f ~ ~ ~ t itensor o I i is defined as
1
VlSCOELAS’TlC FLUIDS
(I 20)
where iA i s a positive scalar called the biaxial elongation rate. The biaxial viscosity is defined as TB = 711 - 7-33 - 7-22
gr3
- r33
(1.21)
ependiiig on the method of analysis, constitutive models of viscoelastic fluids can be formulated as differential or integral equations. In the differential models stress components, and their niaterial derivatives, arc related to the rate of strain components and their material derivatives. Idroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950)
(1.22) d A are material parameters and the time derivative (A,,h,c
(1 2.3)
is the unit tensor, = $[ov+(ov)~‘I and CI) = &[OV -(vv)~ 1 are the symmetric and antisymmetric parts of the velocity gradient tensor. Equation (1.23) gives the most general definition of a time derivative of any second-order tensor and it contains the local, convective, rotational and strain related changes of the tensor with respect to the time variable. In practice special cases are considered. The Jaumann or co-rotational time derivative is defined as a case where the parameters (1, b and c are zero. the upper-convected (or codeforrnationali) Oldroyd derivative, a = -1 a 0 and c are zero. The lowcrdroyd derivative is defined using a = + I and b and c as zeros. ell class of viscoelastic constitutive equations are described by a in which 12 = 0. For example, the upper-
THE BASIC EQUATIONS OF NON-NEWTONIAN FLUID MECHANICS
+A-
A-,ooz = At 2rl
ther combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman ( I 977) have proposed the following equation (1.25)
where -,c is a parameter between 0 and 2. A frequently used example of Oldroyd-type constitutive equations is the model. The Oldroyd- model can be t~oL~ght of as a des the ~ o ~ i s t i t u behavio ~~ve of a fluid made by tlie dissolution of a (UC a Newtonian ‘solvent’. ere, the parameter 11, called the ‘retardati defined as A X (risl(q + qs), wlie rls is the viscosity o f t solvent. Hence thc model is made up of axwell and solvent extra stress tensor in the (41 itutive equation i s writteii as contri~utions.The Oldroyd (1.24) axwell or Oldroyd models do not give realistic ~redictionsof the flow and deformation behaviour of poly ric fluids. In particular, in cases w h ~ r ethe €low regime is cliara~terized e l o n g a t ~ o ndefoi~atioiis ~~ these models are found to give very poor predictions, There luve been many attempts to derive constitutive models that incorporate both shear and ~ ~ o n g a ~ i o i i a l b e ~ a v i ~ uofr viscoelastic fluids. ~han-Thieriand Tanner (1 977) ~ o ~ . ~ u al ~ ~ c astic model based on the network theory for macromolecules. This model 11 shown to give relatively good results for elon~ationalflows (Tanner, haIi-~hieii~Taniier equation is expressed as (1.27)
ii where E: i s defined as a characteristic elongational parameter. In ~ ~ u a t i o(1.27) ammeters E and c (0 5 c 5 2) are representative of the elongatio~aland ~eliavioL~r of the fluid, respectively. As it can be seen the insertion of E = 0 han-~liien/Tannerequation to the ~ o l i i i s o ~ ~ e gna ~l ~o d ~ en ~ ~ All of the described differential viscoelastic coiistitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtoniaii flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved ‘sii~ultaiieous~y’ with the governing flow equations.
VISCOELASTIG FLIJIDS
13
In integral niodels, stress c o ~ p o n e n t sare obtained by i~itegratinga p p r o ~ r ~ ~ t e functions, representing the amount of deformation, ovcr the strain history of the h i d . The simplest integral constitutive model for r~ibber-like fluids was proposed by Lodge (1964). ther equations, belonging to this category of by Jolinson and Segalman (1977) and stitutive models, have been dev er, the most frequently used singleand Edwards (1978, 1979). integral constitutive model for viscoelastic fluids is the K in~epeiidentlyproposed by aye (1962) and ernstein, Kearsley aiid Zapas (1963). The generic form of the integral constitutive equations for isotropic fluids i s written as t
M(t - t’){@l(llJ2)[
ere z(t) is the stress at a fluid particle given by an integral of d e f o ~ ~ a t i o ~ i istory along the fluid particle trajectory between a deformed configuration at time t’ and the current reference time 1 . In Equation (128) f~inctionM(t - t’) is the tinie-dependent niemory f u n ~ t ~ o n and Q2 and are the funcof linear viscoelasticity, non-dimensional scalars tions of the first invariant of the ( t t‘), which arc, rcspectiv~~y, (called the Finger strain tensor) right Gauchy Green tensor itsoulis, 1990). The memory fUiictioa is usually expressed as (1.29)
where ,riA(lc = 1, n) are the viscosity coefficients and Xk(k = I , n) are the relaxation times. The general Equation (1 28) can be used to derive various single-iii~e~~ai viscoelastic constitutive models for incompress~b~e fluids, For example, by setting Q, = 1 and @, = 0 the model developed by Lodge (196 derived. This model can be shown to be equivalent to the up~~er-coi~v axwell equation described in the previous section. To obtain the the scalar fixnclions in the strain-dependent kernel of the integral in ~ q u a t i o n (1.28) are chosen as
(1.30) where 11 and tr( ) and W(I,, 12)is a potential for functions and (15., As it can ial case of the K in which W = 1,. modcl proposed has found widespread application in the modelling o f viscoelastic flows lley and Coates, 1997).
THE BASIC EQLJATIONS OF NON-NEWTONIAN FLUID MECHANICS
Integral models have the apparent advantage of giving the extra stress tensor exp~~citly and thercfore they can be used to find the stress in a separate step to other field unknowns. owever, the integral models are mathematically difficult to handle and, in general, should be solved in Eagrangian frameworks. The high ~ o ~ p u t ~ t i ocosts n a t of adaptive meshing required in Lagrangian systems and problems arising from the calculation of functions which are dependent on in history are regarded as the set backs for these models. ome of the integral or differential constitutive equations presented in this revious section have an exact equivalent in the other group. There are, however, equations in both groups that have no equivalent in the other category.
The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Siniilar to the geiieralized Newtonian approach, under steady-state viscornetric flow conditions cornponents o f the extra stress in the (CEF) model are given as explicit relationships in terms of the components of the rate of deformation tensor. However, in the model stress components are 'corrected' to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dii~ensionalplanar coordinate system the components of extra stress in the (CEF) niodel are written as (1.31) where D,, etc. are the components of the rate of deformation (strain) tensor and %!12 and 9 2 3 are the primary and secondary normal stress coefficients, respectively. An analogous set of relationships which reflect eloiigational behaviour of polymeric liquids has also been proposed ( itsoulis, 1990) as
where 71, and qs are elongational and shear viscosity, respectively. low computational costs of using the (CEF) model, this approach has been advocated as an attractive alternative to more complex viscoelastic equations in that do not significantly deviate from the iiiodelling of poly v i s c ~ ~ e t rcondition ic
REFERENCES
1
REFERENCES ., 1989. Vectors, Tensors arid the Basic Equations
if
Fluid Mechanics, Dover
. and Zapas, LA.>1963. A study of stress relaxation with finite
strain. Tram, Soc. RReoI. 7, 391-410. , a., 1977. Dynamics of Polymeric Fluidsq Vol. Bird, R.B., Armstrong, 1 . Fluid ~ ~ ~ e c h a n i c s ~ ., 1960. Tvanspovt Phenomena. Wiley, New York. Casson, N., 1959. In: Mill. C. G . (ed.), Rheology of Disperse Systems, Pergamon Londo11. Carreau, P. J., 1968. PhD thesis, Department of Chemical Engineering, University of Wisconsin, Wisconsin. Criminale, W. 0. Jr, Ericksen, J. L. and Filby, 6.L. Jr., 1958. Steady shear flow of nonNewtoiiiaii fluids. Arch. Rat. Mech. Anal. 1, 410-417. de Wade, A., 1923. See Bird, R.B., Armstrong, R. C. and Hassager, 0. 1977. Dynamics of Polymeric Fluids, Yol. I : Fluid Mechanics, Wiley, New York. M. and Edwards, S.F., 1978. Dynamics of concentrated polymer systems: 1. rowniaii motion in equilibrium state, 2. Molecular motion under flow, 3. Constitutive equation and 4. Rheological properties. J. Ckmz. Soc., Faraday Tvans. 2 7 1802, 1818-1832. . and Edwards, S. I:.. 1979. Dynamics of concentrated polymer systenis: 1. Brownian motion in equilibrium state, 2. Molecular molioii under flow, 3 , Constitutive equalion and 4. Rheological properties. J. Chem. Soc., Faraday Tmns. 2 7 38 -54. . and Bulkley, R..1927. See Rudraiah, N. and Kaloni, P.N, 1990. Flow of non-Newtonian fluids. In: Encyclopaedia OJ Fhid Mechanics, Vol. 9, Chapter 1,
n, D., 1977. A model for viscoelastic fluid behaviour which l ~Uccli.2, 255- 270. allows non-affiiie deformation. J. Non-Ntwtoniu Raye, A., 1962. ~ o ~ - ~ e ~ Flow ~ t ~in) Irzcompr ~ ~ i ~ r i Fluids, CoA Note No. 134,
College of Aeronautics, Cranfield. Keiiiblowslti, Z. and Petera, J.. 1981. Memory effects during the flow of thixotropic fluids in pipes. Rheol. Acfa 20, 31 1-323. Lodge, A. S., 1964. Elastic Liqiiids, Academic Press, London. iour or concentrated dilatant susvier Processing, McGraw-Hill, New York. Mitsoulis, E., 1986. The nun1 ger fluids: a viscometric approxiiiiation approach. Polym. itsoulis, E., 1990. Nunieri oelastic Fluids. In: Encyclopueidiu of Fluid Mechanics, Vol. 9, Chapter 21, Gulf Publishers, Houston. Oldroyd, J. G., 1947. A ratioiial formulati of the equations of plastic flow for a Bingliam solid. Yroc. Cumb. Philos. ,%c. ." 1950. On the formulation of rheological equations oC state. Pro(, Roy.
THE BASIC bQUATIONS OF NON-NEWTQNIAN FLUID MECHANICS
Olley, P. and Coates, P. D., 1997. An approximation to the Ic. KZ conskitutive equation. J. A l o r z - ~ e ~ ~ ~Fluid t o n ~Mech. f~~ ~ s t ~ aW., l ~2925. , See Bird, R.B., h i s t r o n g , R. C. and Hassager, 0. 1977. Dynnmics of Polynierir FluidJ, Vol. 1: Fkid Mechanics, Wiley, Ncw York. earson, J. R. A., 1994. Report on University of Wales Institute of Won-Newtonian echanics Mini Symposium on Continuum and Mi Coinputational Rheology. J. Non-Nt.wtoniuw Fluid M d z . Phan-Thien, N. and Tanncr, R.T.,1977. A new constituti network ttieory. .I. ~on-Ne~vtoniun Fluid Mech. 2, 353-365. ittcnan, J. F. ‘r.and Nakazawa, S., 1984. Finite element analysis of polymer processing ations. In: Pittman, J. F. T., Zienkiewicz, 0 ,C., Wood, R. D. and Alexander, . (eds), Niimericul Apm1pi.s of Forming Prnressm, Wiley, Chichester. Slibar, A. and Paslay. P. R., 1959. Retarded flow of’ Bingham inaterialli. T r a m ASME 26, 107-113. Stevenson, J. F., 1972. Elongational flow of polymer melts. AIChE. J . L Tadrnor, Z. and Gogos, C. G., 1979. Priuc@les of Polymer Prucemin,o, Wiley, New York. Tanner, R. I., 1985. Engineering Rheology, Clarendon Press, Oxford. Wagner. M.N., 1979. Towards a network theory for polymer melts. Rlzeol. A c ~ a .18, 33 50.
As described in Chapter 1, mathematical models that represent polymer flow systems are, in general, based on non-linear partial differential equatioiis and cannol be solved by aiialytical techniques. Therefore, in general, these equations are solved using numerical methods. Nuiiierical solutions of the ~ ~ i f f e r e i ~ t i a ~ equations arising in engineeriiig problems are usually based oii finite difference, finite element, boundary element or finite volume schemes. Other nuinerical techniques such as the spectral expansions or newly emerged mesh inde~eiident metliods may also be used to solve governing equations of specific types of e i i ~ ~ i e eproblems, ~ i i ~ Numerous examples of the successful ap~licatioriof these methods in the computer modelling o f realistic field problems can be found in the literature. All of these methods have strengths aiid weaknesses and a number of factors should be considered before deciding in favour of the applicatio~of a particular method to the modelling of a process. The most important factors in this respect, are: type of the governing equations of the process, geometry of the process domain, nature of the boundary conditions, required accuracy of the calculations and compLitatioiia~cost. In general, the finite element method has a greater geoi~etriea~ f l ~ x i b i ~tb~li it~ other currently available numerical tecliniques. It can also cope very e~fect~vely with various types of boundary conditions. The most significant setback for this method i s the high coniputational cost of three-dimens~onalfinite el models. In practice, rational a p p ~ o x i ~ a t i o nare s often used to obtain simulations for realjstic problems without f i l l three-dimensional aiialysis. In the nite element modelling of polymeric flows the following a p ~ r o a c hcan ~ ~be adopted to achieve c o n i ~ ueconomy: ~i~~~
‘rwo-dimensioiialmodels can be used to provide effective a ~ ~ p r o x i n ~ a in ~~ons the modelling of polyiiier processes if the flow field variations in the remaining (third) direction are sinall. In particular, in a x i s ~ n m ~ t rdoina~ns ic it may be possible to ignore the c i ~ c u ~ f ~ r e variations i l ~ ~ A ~ of the field u~i~~now and n s analytically integrate the flow equations iri that directio reduce the n u ~ e r ~ cmodel al to a t w o - ~ i ~ e n s i o nform. al
WEIGHTED RESIDXJAL FINITE ELEMENT METHODS - AN OUTLINE
Process characteristics may justify the use OC simplifying assumptions such as the ‘lubrication approximation’ which may be applied to represent creeping flow in narrow gaps. Components of the governing equations of the process can be decoupled to develop a solution scheme for a three-din~eiisioiialproblem by combining one- and two-dimensional analyses. Examples of polymeric flow models where the above simp~i~cations have been successfully used are presented in Chapter 5. Finite elenient modelling of engineering processes can be based on different inet~o~o~o~ For i e sexample, . the preferred method in structural analyses is the i is placement method’ which is based on the minimization of a variational statei~entthat represents the state of equilibrium in a structure ( and Taylor, 1994). Engineering fluid flow processes, on the other hand, cannot be usually expressed in ternis of variational principles. Therefore, the mathematical ~ o d ~ ~ l iof i i flnid g dynamical problems is mainly depeiident on the solution of partial differential equations derived from the laws of conservation of inass, momentum and energy and constitutive equations. Weighted residual methods, such as the Galerkin, least square and collocation techniques provide theoretical basis for the numerical solution of partial differential equations. owever, the direct application of these techniques to engine~~ing problems usually not practical and they need to be combined with finite clement oximatioii procedures to develop robust practical schemes. oiily adopted approach in computer modelliiig of flow processes in polymer engiiieeri~igoperati oiis is the application of weighted residual finite elenient methods. The main concepts of the finite element a p p r o x ~ ~ a t ~and o n the general outline of the weighted residual methods are briefly explained in this chapter. These concepts provide the necessary background for the development of tbe w o ~ ~equations i i ~ ~ of the iiuinerical schemes used In the s ~ ~ u l a t i oofn polymer processing operations. In-depth analyses of the mathematical theorems underpinning finite element approximations and weighted residual methods arc outside the scope of this book. The re, in this chapter, mainly descriptive outlines of these topics are given. ailed explanations of the theoretical aspects of the solution of partial rential equations by the weighted residual finite element methods can be found in many textbooks dedicated to these subjects, FOP.example, see Mitchell and Wait (1977), Johnsoiz (1987), renncr and Scott (1994) and, specifically, for the solution of incompress~ble Navier-Stokes equations see Girault and Raviart ( 1986) and Pironneau a t h e ~ ~ t i c derivations al presented in the following sections are, occasionally, given in the context of one- or two-di~ie~sional Cartesian coordiii~~te systems. These derivations can, however, be readily generalized and the adopted style is to make the explanations as simple as possible.
FINITE ELEMENT APPROXIMATION
19
The first step in the formulation of a finite element approximation for a field problem is to divide the problem domain into a number of smaller sub-regions without leaving any gaps or overlappiiig between them. This process is called ‘Domain Discrctization’. An individual sub-region in a discretized domain is called a ‘finite element’ and collectively, the finite elements providc a ‘finite element mesh’ for the discretized domain. In general, the elements in a finite eleinent mesh inay have different sizes but all of them usually have a common basic shape (e.g. they are all triangular or quadrilateral) and an equal number oC nodes. The nodes are the sampling points in an element where the numerical values o f the ~ i n k n o ~ are n s to be calculated. All types of finite elements should have some nodes located on their boundary lines. Some of the commonly used finite elements also have interior nodes. Boundary nodes of the individual finite eleinents appear as the junction points between the elements in a finite elemcnt mesh. In the finite element modelling of flow processes the elements in the cornputational mesh are geometrical sub-regions of the flow domain and they do not represent parts o f the body of the fluid. In most engineering problems the boundary of the problem domain includes curved sections. The discretization of domains with curved boundaries using meshes that consist of elements with straight sides inevilably involves some error. This type of discretization error can obviously be reduced by mesh owever, in general, it cannot be entirely eliminated unless finite elements which themselves have curved sides are used. The discretization of a problem domain into a finite element mesh consisting of randomly sized triangular elements is shown in Figure 2.1. In the coarse mesh shown there are relatively largc gaps between the actual domain boundary and the boundary of the mesh and hence the overall discretization error is expected to be large. The main consequence of the discretization of a problem domain into finitc elements is that within each element, unknown functions can be approximated using inter~olationprocedures. Y
X
~i~~~ 2.1 Problem domain discretization
WElGHTED RESIDUAL FINITE ELEMENT METHODS
-
AN OUTLINE
be a well-defined finite element, i.e. its shape, size and the iiuniber and locations of its nodes are known. We seek to define the variations o f a real valued continuous function, such as f, over this element in terms of appropriate geometrical functions. I f it can be assumed that the values o f f on the nodes of Qe are known, then in any other point within this element we can find an approximate value for j’using an interpolation method. For example, consider a one-dimensional two-node (linear) element of length E with its nodes located at points A(x* = 0) and B(XB= I ) as is shown in Figure 2.2. ( ,
A(xA== 0)
B(XB =
I)
Figure 2.2 A one-dimensional linear element
Using a simple interpolation procedure variations of a continuous function such as f along the element can be shown, approximately, as
-
.fx
t-x = -“-,fA
x
7.h
Equation (2.1) provides an approxiinate interpolated value for f a t position x in t e r m o f its nodal values and two geometrical functions. The geometrical [unctions in ~ q u a t i Q(2.1) n are called the ‘shape’ functions. A simple inspection shows that: (a) each function is equal to 1 at its associated node and is 0 at the other node, and (b) the sum of the shape functions is equal to I . These fun~tioiis,shown in Figure 2.3, are written according to their associated nodes as NA and Wu. 1
0 A
B X
~~~~r~ 2.3 Linear Lagrange interpolation functions
FINITE ELEMENT APPROXIMAL‘ION
21
Analogous interpolation procedures involving higher numbers of samplling points than the two ends used in the above example provide higher-order approximations for unknown functions over one-dimensional elements. The method can also he extended to two- and three-dimensional elements. In general, an inte~olatedfunction over a multi-dimensional element expressed as P I-:
1
where (.%Irepresents the coordinates of the point in SZ, on which we wish to find an approxiiiiate (interpolated) value for the function5 i is tlie node index, p is the total number of nodes in 52, and Nl(X) is the shape function associated with imilar to the one-dimensional example the shape functions in multi~ ~ ~ e n s i o elements iial should also satisfy the following conditions
(2.3) where j r , i = 1, . . .p are tlie nodal values of the function j’ (called the nodal degrees of freedom). Nodal degrees of freedom appearing in elemental interpolations (i.e. fi, i = 1,. . . p ) are the field unknowns that will be found during the finite element solution procedure. The general form of the shape functions associated with a finite element depends on its shape and the number of its nodes. In most types of commonly used finite elements these functions are low degree polynomials. In general, if the degrees of freedom in a finite element are all given as nodal values of unknown functions (i.e. function derivatives are excluded) then the element is said to belong to the Lagrange family of ele~ents. However, some authors use the term ‘Lagrange element’ exclusively for those elements whose associated shape functions are specifically based on ~ a g r a n g e interpolation polynomials or their products (Gerald and ~ h e a t ~ e 1984). y, Hermite interpolation models involving the derivatives of field variables (Ciarlet, 1978; Lapidus and Pinder, 1982) can also be used to construct function a ~ p r o x i ~ a t i o nover s finite elements. Consider a two-node one-dimensional element, as is shown in Figure 2.4, in which the degrees of freedom are nodal values and slopes of unknown functions. Therefore the expression defining the approximate value of a function f a t a point in the interior of this element should include both its nodal values and slopes. Let this be written as
22
WEIGHTED RESIDUAL FINITE ELEMENT METHODS
-
AN OUTLINE
X
Figure 2.4 A one-dimensional Werniite element
where N&) and Nll(x) are polynomial expansions of equal order. As it can be seen in this case each node is associated with two shape functions. At the ends of the Line element Equation (2.4) must give the function values and slopes shown in Figure 2.4, therefore: NOI(X) must be 1 at node number I and 0 at the other node, Nb,(x)must be 0 at both nodes, and
Nlr(x) must be 0 at both nodes and N i f (x) must be 1 at node I and 0 at the other node.
A simple inspection shows that cubic functions (splines) shown graphically in Figure 2.5 satisfy the above conditions.
igure 2.5 One-dimensional Herniite interpolation functions
Inherent in the development of approximations by the described interpolation models is to assign polynomial variations for function expansions over finite elements. Therefore the shape functions in a given finite element correspond to a
FINITE ELEMENT APPROXIMATION
particular approximating polynomial. owever, finite element approximations may not represent complete polynomials of any given degree.
2. Standard procedures for the derivation of the shape functions of cominon types of finite elements can be illustrated in the context of two-dimensional triangular and rectangular elements. Let us, first, consider a triangular element having three nodes located at its vertices as is shown in Figure 2.6.
Figure 2.6 A linear trimgular element
Variations of a continuous function over this element can be represented by a complete first-order (linear) polytiomial as
By the insertion of the nodal coordinates into Equation (2.5) nodal values o f f can be found. This is shown as
where xE,y,, i = 1,3 are the nodal coordinates andj; i = 3,3 are the nodal degrees of freedoin (Le. function values). Using matrix notation Equation (2.6) is written as
WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE
hence
-
I
e
(2.8)
Equation (2.5) can be written as
(2.9) ornparing Equations (2.2) and (2.9) we have
f=Np
(2.10)
where N is the set of shape functions written as (2.1 1)
In the outlined procedure the derivation of the shape functions of a threenoded (linear) triangular element requires the solution o f a set of algebraic equations, generally shown as Equation (2.7). hape functions of the described triangular element are hence found on the basis of Equation (2.1 1) as (2.12) (2.13) (2.14)
can be readily shown that these geometric functions satisfy the condilions seribed by Equation (2.3). Shape functions of a quadratic triangular element, with six associated nodes located at its vertices and mid-sides, can be derived by a similar procedure using a complete second order polynomial. Similarly it cm be shown that a complete cubic polynomial corresponds to a triangular el t with 10 nodes and so on. The ar~angernentshown in Figure 2.7 (called 1 cal triangle) represents the teiins required to construct coinplete polynomials of any given degree, p , in two v~~riables x and y . The number of terms of a complete polynoniiai o f any given degree will hence correspond to the number of nodes in a triangular element belonging to this family. An analogous tetrahedral family of finite elements that correspo~dsto complete polynomials in terms of three spatial variables can also be constructed for three-diinensional analysis.
FINITE ELEMENT A ~ ~ ~ ( ~ X l M A 25 ~ ~ O N
p=o
1
x 2 xy
x3 x*y xy2
x4 x3y
p=2
y2
p=3
y3
x y xy3
y4
p”4
.............................................................................................................
.......................... lire 2.7 Pascal triangle
The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom i s tedious and becomes impractical for higher-order elenients. Furthermore, the existence of a solution for these equations (i.e. existence o f an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these faniilies. A very convenient ‘indirect’ procedure for the derivation of shape functions in rectangular elements is to use the ‘tensor products’ of one-dimensional interpolation functions. This can be readily explained considering the four-node rectangular element shown in Figure 2.8.
e 2.8 Bi-linear rectangular element
The interpolation model in this element i s expressed as
7= + azx 4- q y -ta4xy U1
(2.15)
26
WEIGHTED RESIDUAL FINITE ELEMENT METHODS
-
AN OXJTLINE
The polynomial expansion used in this equation does not include all or the terms of a complete quadratic expansioii (i.e. six terms corresponding to p = 2 in the Pascal triangle) and, therefore, the four-node rectaiigular element shown in igure 2.8 is not a quadratic element. The right-hand side of Equation (2.15) can, however, be written as the product of two first-order polynomials in tenns of x and y variables as
.r = (htx + b2) (b3.y -tb4) *
(2.16)
Therelore an obvious procedure for the generation of the shape functions of the element shown in Figure 2.8 is to obtain the products of linear interpolation functions in the x and y directions. The four-noded rectangular element constructed in this way is called a bi-linear element. Higher order members of this family are also readily generated using the tensor products of higher order one-dimcnsional interpolation functions. For example, the second member of this group is the nine-noded bi-quadratic rectangular element, shown in T'g '1 ure 2.9, whose shape functions are formulated as the products of quadratic Lagrange polynomials in the x and y directions. A similar procedure is used to generate 'tensor-product' three-dimensional elements, such as the 27-node tri-quadratic element. The shape functions in twoor three-dimensional tensor product elements are always incomplele polynomials.
Figure 2.9 Bi-quadratic rectangular element
nalogous to tensor product Lagrange elements, tensor produc ents can also be generated. The rectangular element developed by ul. (1965) is an example of this group. This element is shown in Figure 2.10 and involves a total of 16 degrees of freedom per single variable. The associated shape functions of this element are found as the tensor products of the cubic polynomkals in x and y (see Figure 2.5).
FINITE ELEMENT APFROXIMATION
27
Another important group of finite elements whose shape functions are not complete polynomials i s the ‘serendipity’ family. An eight-noded rectangular element which has four corner nodes and four mid-side nodes is an exampje of this family. Shape fuiictiorls of serendipity elenients cannot be generated by the tensor product of one-dimensional Lagrange interpolation functions (except for the four-node rectangular element which is the same in both families). these functions are found by an alternative method based on using products of selected ~ o ~ y ~ o m ithat a l s give desired function variations on element edges (Reddy, 1993). Y
I
* x
~ i ~ M 2.10 ~ eA rectangular Hermite eleinent
ts Finite element families described in the previous section are used to obtain standard discretizatioiis in a wide range of different engineering problems. In addition to these families, other eleineiit groups that provide specific types of approximations have also been developed. In this section a number of ‘iionstandard’ elements that are widely used to model polymeric flow regimes are ood elements are among the earliest examples of this group designed for the solution of incoinpressible flow problems. elements interpolation of pressure is always based on a lower-order polynomial than the polynomials used to interpol~tevelocity components (Taylor and Hood, 1973). The rectangular element, shown in Figure 2.1 1, is an example of this family.
WEIGHTED RESIDUAL FINITE ELEMENT METHODS - A N OUTLINE
Y
~ i g 2.1~1 1~ Nine e node Taylor-Wood element
In this element the velocity and pressure fields are approximated using biquadratic and bi-hear shape functions, respectively, allis corresponds to a total of 22 degrees of freedom consisting of 18 nodal velocity c o ~ ~ p ~ ~ i(corner, ents mid-side and cciitre nodes) and four nodal pressures (corner nodes). Croweix-Raviart elements are another group of finite elements that provide different interpolations for pressure and velocity in a flow domain (Grouaeix aviart, 1973). The main characteristic of these clenients is to make the e on the element bouiidarics discontinuous. For example, the cornbinalion of quadratic shape f~nctionsfor the o x ~ ~ n ~ of t ~ vo~n~ o c i t(correy sponding to a six-node triangle) with a const essure, (given at a single node inside the triangle), can be considered. Another ieinber of this family is the rectangular element shown in Figure 2.12, in which the a ~ ~ ~ o x i ~QEa t i ~ velocity is based on bi-quadratic shape functions while pressure i s approximated linearly using three internal sampl g nodes. This element usually provides a greater ~ e x ~ b i than ~ ~ t the y Taylor- ood element, shown in Figure 2.11, in the modelling of incompressible flom7 problems.
Pressure
MR
2.12 An element be1
FINITE ELEMENT A P P ~ ~ X I ~ A ~ I O 2 N
The global Cartesian framework used so far is not a convenient coordinate system for the generation of functioii approximations over different elements in a mesh. Elemental shape functions defined in t e r m of global nodal coordinates will not remain invariant and instead will appear as polynomials of similar degree having different coefficicnts at each element. This inconvenient situation is readily avoided by using an ap~ropriatelocal coordinate system to define elemental shape functions. If required, interpolated functions expressed in terms of local coordinates can be transformed to the global coordinate system at a later stage. Shape functions written in terms of local variables will always be the same for a particular finite element no matter what type of global coordinate system is used. Finite element approximation of unknown f u i i ~ t i ~ in n sterms of locally defined shape functions can be written as
where x* represents local coordinates andf; are nodal degrees of freedom. shown in Figure 2.13 a local Celrtesian coordinate system with its origin located at the centre of the element is the natural choice for rectangular e ~ e ~ e n t s ~ rl
3 Local coordinate system in rectangular elements
Using this coordinate system the shape functions for the first two ~ e ~ b eofr s the tensor p r ~ d u c ~t a g r a ~ eg lee ~ i e family ~t are expressed as Four-node bi-linear element
30
WELGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE
Nine-node bi-quadratic element
3
1 6
(2.19)
2
the local Cartesian coordiiiate system used in rectangular elements its not a suitable choice for triangular elements. A natural local coordinate system fur the triangular elements can be developed using area coordinates. Consider a triangular element as is shown in Figure 2.14 divided into three suh-areas of A I , A2 and AR.The area coordinates of L,, Lz and L3 for the point B inside this triangle are defined as A1
=A
L1
Az A A3 L3 = A
(2.20)
L2 = -
3
A, is the total area of the triangle. 1:;
1
1
X
Area coordinates in triangular elements
FINITE ELEMENT ~ ~ ~ O X l ~ A T l31( ~ N
It can be readily shown that L,,i = 1,3 satisfy the requirements for shape functions (as stated in Equation 2.3) associated with the triangular element. The arca of a triangle in terms of the Cartesian coordinates of its vertices is written as
(2.21) Therefore the area coordinates defiacd by Equation (2.20) in a global Cartesian coordinate system are expressed as
(2.22)
Therefore
(2.23)
The expansion of Equation (2.23) gives the transformation between thc local arca coordinates and the global Cartesiaii system (x, y ) for t ~ i a n ~elements. ~ar This transforrnation also coilfirms that in a global Cartesian coordiiiate system the shape functions o f a linear triangular element should be exprcssed as ~ q ~ a t (2.121, i o ~ (2.13) ~ and (2.14). Using the area coordinates the shape Eiinctions for the first two meinbcrs or the triangular finite elements are given as Three-node linear triangular elcrneiit 3
1
2
32
WEIGHTED RESIDLJAL FINITE ELEMENT METHODS
-
AN OUTLINE
Six-node quadratic triangular element
(2.25)
.5 A general requirement in most finite element discretizations is to maintain the compatibility of field variables {or functions) across the boundaries of the neig~ibouringelements. Finite elements that generate uniquely defined function approxiinations along their sides (boundaries) satisfy this condition. For example, in a mesh consisting of three-node triangular elements with nodes at its vertices, linear interpolation used to derive the element shape functions gives a unique variation for functions along each side of the element. Therefore, field variables or functions on the nodes o f this element are uniquely defined. This example can be contrasted with a three node triangular element in which the nodes (i.e. sampling points for interpolation) are located at inid-points of the triangle, as is shown in Figure 2.15. Clearly it will not be possible to obtain unique linear variations for functions along the sides of the triangular element shown in the figure.
2
.I5 A iiodal arrangement that cannot provide inter-elemcnt compatibility of functions
Finite elements that maintain inter-element compatibility o f functions are ‘ e o n f ~ r ~elements’. i~g Finite elemeiits that do not have this property are d to as the ‘non-conforming elements’. Under certain conditions nonconforming elements can lead to accurate solutions and are inore advantageous to use. The order of continuity of a conforming finite element that only ensures the c o ~ ~ ~ ~ i of b ifunctions ~ i t y across its bouildaries is said to be Co. Finite eleme~i~s that ensure the inter-element compatibility of functions and their derivatives gher order of continuity than Co. For example, the gure 2.4 which guarantees the ~ o ~ p a t i b i l i of t y function values and
FINITE ELEMENT AP~ROX~MATION 33
slopes at its ends is C' continuous. According to this definition, the nonconfo~i~ triangular g element shown in Figure 2.15 is said to be a C continuous element. The order of continuity and the degree of highest order complete polynomial obtainable in an elemental interpolation are used to identify various finite elements. For example, the three-node linear triangle, shown in Figure 2.6, and the four-node bi-linear rectangie, shown in Figure 2.8, are both rercrred to as P'C* elements. Similarly, according to this convention, the nine-node biquadratic rectangle, shown in Figure 2.9, is said to be a P2Cuelement and so on.
'
All numerical computations incvitably involve round-off errors. This error increases as the number o f calculations in the solution procedure is incre~ised. Therefore, in practice, successive mesh refinenieiits that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. Hn this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P G n element with sufficient orders of interpolation and continuity, at the Jirnit (Le. when element d~~iiens~ons tend to zcro), an exact solution should be obtained. This has been shown to be true for h e a r elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following coiid~~io~is are satisfied: e r a ~ should be small, aspect ratio of ~ u a d r ~ ~ a telements internal angles of elements should not be near 0 " or 180", the exact soliition should be sufficiently smooth and must not include si~igular~t~es, robleni domain should be convex, and elemental calculations (i.e. evaluation of integrals etc.) inust be s u f ~ c i ~ ~ i t l y accurate. and Theoretical analysis of convergence in non-linear probleins i s ~nco~iplete in most instalices does not yield clear results. Conclusioiis drawn from the analyses of linear elliptic problems, however, provide basic guidelines for solving n o n ~ l i n ~ or a r non-e~lipticequations.
WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE
~lexibilityto cope with irregular domain geometry in a straightforward and systematic manner is one of the most important characteristics of the finite element method. Irregular domains that do not include any curved boundary sections can be accurately discretized using triangular elements. In most engineering processes, however, the elimination of discretization error requires the use of finite elements which themselves have curved sides. It is obvious that randomly shaped curved elements cannot be developed in an ad hoc manner and a general approach that is applicable in all situations must be sought. The required generali~ationis obtained usiiig a two step procedure as follows: a regular element called the ‘master element’ is selected and a local finite element approximation based on the shape functioiis of this element is established, and the master element is mapped into the global coordinates to generate the required distorted elements.
A graphical rep~esentationof this process is shown in Figure 2.16. Y
E
X
iwe 2.16 Mapping between a master element and elements in il global mesh
In the figure operation (M) represents a one-to-one t ~ ~ i i s f o r i ~ abetween t i ~ n the local and global coordinate systems. This in general can be shown as
(2.26) The one-to-one transformation between the global and local coordinate systems can be established using a variety of techniques ( ~ i e n k i e ~ 7 iand c~
FINITE ELEMENT APPROXIMA rZON
35
1983). The most general method is a form of ‘parametric mapping’ in which the transformation functions, x(&q) and y((,q) in Equation (2.261, are polynoi~ials based on the element shape functions. Three different forms of this technique have been developed: Subparametric transformations: shape functions used in the mapping functions are lower-order polynomials than the shape functions used to obtain finite element approximation of functions. upe~arametrictransformations: shape fimctions used in the map functions are higher-order polynomials than the shape functions used to obtain finite clement approximation of functions. Isoparametric transformations: shape functions used in the mapping functions are identical lo the shape functions used to obtain finite element a p p r o x i ~ a ~ i oofn functions. is the most commonly used form of the described para~ s o ~ a r ~ m e tmapping ric metric t r a ~ i s f o ~ a t i o Figurc n. 2.17 shows a schematic example of isoparai~etric tramformation between an irregular element and its corresponding regular (master) element. Shape functions along the sides of the master element shown in this example are linear in atid q and consequently they can only generate irregular elements with straight sides. In contrast the master element shown in Figure 2.18 can be mapped into a global element with curved sides. rl
3
Y 1 X
gure 2.17 Isoparametric mapping of an Jrregular quadl-ilrtteral element with sti-aigh-ht sides
n general, elements with curved sides can only be generated using quadratic or higher-order rnasler elements. Isopaaametric transformation functions between a global coordinate system and local coord~natesare, in general, written as
36
WEIGHTED RESIDIJAL FINITE ELEMENT METHODS - AN OUTLPNE
(2.27)
5
(--I 4
(-1 ,--I)
3 X
lsoparametric inapping of an irregular quactrilatcral element with curved sides
where x, and y E are the nodal coordinates in the global system. The shape functions in Equation (2.27) are given in terms of local variables defined by the natural coordiiiate system in the master element. Tsopararnetricmapping can also enerate triangular elements with curved sides, As already ~ x ~ ~ a i n e the local variables in triangular elements are giveii as area coordinates and lience ~ s o ~ a ~ a mmapping e t ~ c functions for triangular elements are expres§ed as
(2.28)
The most convenient coordinate system for a triangular inaster element is based on a ‘natural’ system similar to the one shown in Figure 2.19, where L1 = I - 4 - 71, Lz = and L1 = 7.
<
(090) tire
W)
2.19 Local natural coordinates in a master triangular elcinent
FTNJTE ELEMENT A ~ ~ R O X I ~ A T I 37 O~
In addition to the stated condition of one-to-one correspondence between local and global coordinates the transformation must preserve the geometr~~al conformity and continuity of the mesh in a way that no gaps or overlapphg can occur between the elements. To satisfy this, irregular elements ia a mesh should be gerrerated by mappings from a master element that has an app~opriateorder of continuity. For example, C" continuous distorted elcments should be generated from CO continuous parent elements. in conjunction with the usc of isoparainetric elements it i s necessary to express the derivatives of nodal fLinctions in tertns of local coordinates. This is a straightforward procedure for elements with CO contiiiuity and can be described as follows: Using the chain rule for differentiation of functions of multiple variables, the dcrivative of a hnction in tertns of local variables ([, q) can be expressed as
and (2.30)
Using matrix notations
(2.31)
(2.32)
is the Jacobiaii of coordinate transformations. Therefore
(2.33)
WEIGHTED RESIDUAL FINIlE ELEMENT METHODS - AN OUTLINE
lobal derivatives of functions can now be related to the locally defined finite element approximation, given by Equation (2.17), as I’ i= 1
(2.34)
bviously the described transformation depends on the existence of aii inverse for the Jacobian matrix (i.e. det J must always be non-zero). ~ifferentiationof locally defined shape functions appearing in Equation (2.34) is a trivial matter, in addition, in isoparametric elements members of the Jacobian matrix are given in terms of locally defined der~vativesand known global coordinates of the nodes (Equation 2.27). Consequently, computation of the inverse of the Jacobian matrix shown in Equation (2.34) is usually straigh~forward. It should be emphasized at this point that the isoparametric mapping of regular elerneiits into curved shapes inevitably generates a degree of approxim ~ ~ t ~The 5 n .magiiitudc of the error in such approximations directly depends on the degree of the irregularity of the elements being mapped. In general, mappings involving badly ~ ~ i s ~ oelements r t ~ d in coarse meshes should be avoided. In extreme situations the sign of the Jacobian changes during the transfor~na~ion an illogical element that folds upon itself is generated. certain types of finite element computations the app~icationof isoparametric ~ ~ p p i may n g require t r a n s f o ~ ~ a tof i osecond-order ~ as well as the firstorder derivatives. Tsoparametric transformation of second (or higher)-order derivatives is not s t r a i ~ h t f o r w and ~ r ~ requires ~ lengthy ~lgebraic~ a ~ i p u l a t i Q n s . etails of a convenient procedure for the isoparainetri~trans~or~atioii of cond-order derivatives are given by Petera et al. (1993). ~ s o ~ a r a ~ ernappiiig ~ r i c reiiioves the g e @ ~ e t r i ~iiiflexibility al of r e c t a n ~ l a r elements and therefore they can be used to solve many types of problems. For example, the isoparametric C’ continuous rectangular providcs useful discretizatioiis in the solution of v i s c o ~ ~ ~ sflow tic S.
The finite element solution of differential equations r e ~ ~ i i rfe~s ~ c t i integ~ation on over element domains. Evaluation of integrals over elemental domailis by analytical methods can be tedious and i ~ p r a c t i c aand ~ i s not a t t e ~ p t ein~
FINITE ELEMENT APPROXIMKIION
39
general. Furthermore, in the majority of problems isoparametric mapping is used to generate meshes involving irregular and curved elements and hence the analytical evaluation of elemental integrals is practically impossible. Therefore, in finite element computations integrals given over elemental domains are found by numerical integration (quadrature) techniques. A commonly used quadrature method is the application of the Gauss-legendre formula for the evaluation of definite integrals between limits of -1 and +l. This procedure i s summarized using the following example: Consider the integration of a function.f(’(xl,x2) over a quadrilateral element in a fmite element mesh expressed as
(2.35) where x1 and x2 are the global coordinates and al etc. are constants. The t r a n s f o ~ a t i o nbetween the global and local domains gives 1
J
(2.36) 1-1
where the limits of integration are defined by the local coordinates and the integration measure is transformed as (2.37)
After algebraic i ~ a ~ ~ i p u ~ awet i can o ~ swrite 1
1
(2.38) -1 -1
Using the ~ ~ u s s - L e g e n dqua~~rature, r~ I is found as
(2.39) rjl are the quadra ture point coordinates, corresponding weight factors and M and N are the number of quadratwe poiiits in each summation. The number of quadrature points in these s u ~ ~ ~ a t i o n s depends on the order of the p o ~ y ~ o ~ ifunction ;al in the integral. In onedimensional problems this quadrature yields an exact result for a polynomial of degree 272- 1 (or less) using rz points. In finite element coniputations~integ~aiic~s
0
WEIGHTED RESIDUAL FINITE ELEMENT METHODS
AN OUTLINE
in elemental equations are based on shape functions, which are low-order polynomials, therefore the number of required quadrature points is low (usually n = 2 or 3). Table 2.1 shows the coordinates of quadralure points and their associatcd weighting factors for M = 3 in the Gauss Legendre formula. Table 2.1
= Ik’ 1 2 3
wr 2 1 0.5555555556 0.8888888889
[I,
VJ
0 I 0.5773502692 k 0.7745966692
0
As already mentioned, the local coordinate system in triangular elements i s defined in terms of area coordinates. Therefore, in these elements the i~itegration lim~tswill also be given in terms of area coordinates precluding their evaluation by the described Gauss-Legendre quadrature. For details of quadra ture techniq~es,sampling points and weighting factors for triangL~larelenients see ~ ~ ~ n k i e wand i c zTaylor (1994).
FG
etd
The sta~idardtechnique for improving the accuracy of finite element approximations is to refine the computational grid in order to use a denser mesh consisting of smaller size elements. This also provides a practical method for testing the convergence in the solution of non-linear problems tlirou~hthe coniparison of the results obtained on successively refined meshes. In the %-version’ of the liiiite clement method the element selected for rho domain discretization r e ~ n a ~ ~ i s unclianged while the number and size of the elements vary with each level of nicsli refinemerit. Alternatively, the accuracy of the finite element discretizations can be e ~ ~ a using n c ~higher-order ~ elements whilst the basic mesh constant. For example, after obtaining a solution for a problem on a mesh consist in^ of bi-linear elements another solution is generated via hi-quadratic e l e ~ ~ e nwhile t s keeping the number, size and shape o f the elements in the mesh uncliaii~e~. In this case the number of thc nodes and, consequently, the node-to~ ~ ~ i nratio e n t in the mesh will increase and a better accuracy will be o b ~ a i ~ ~ e ~ roach is commonly called the ‘p-version’ of the finite element m ~ ~ h o ~ . mily of ‘hierarchical’elemeiits are specifically designed to minimize the of repeated coniputations in the ‘p-version’ of the finite ienkiewicz and Taylor, 1994). Successive a p ~ r o x ~ ~ a t i o n s based on hierarchical elements utilize the derivations of a lower step to generate tlrc solution for a higher-order ap~roxiinatioii.This can significan~~y reduce the
D ~ ~ ~ E ~ E N TEQUATIONS IAL BY THE WEIGHTED RESIDUAL METHOD
cost of repeated computations in p-version finite element simulations. hierarchical elements lack the flexibility of ordinary clements and their application is restricted to specific probleins (Bathe, 1996).
The weighted residual method provides a flexible mathematical framework for tlie construction of a variety of iiuinerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followbig section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem r~~reseiited by the following inathcinatical model S[u(x)j = g in 52
(2.40)
subject to -- 11 on 1'
) . ( U
where I'r i s a linear differential operator, u(x) is tlie unknown variable (function o f i ~ ~ d e p c n ~ spatial ent variables), g is a sourcelsink term, Q is a suffic~eiitly smooth closed doinain surrounded by a continuous boundary I' and 11 is the specified value of u(x) at r. hi the absence of an exact analytical solutioti for Equation (2.40) we seek to represent the field variable u(x> a p p r o x ~ ~ a t eas ly
/=l
where a, ( i = I ,nz) are a set o f coefficients (constants) and ( i = 1 , ~r~present ) a set of geometrical functions called busis Junctioizs. To satisfy the b o ~ ~ i ~ d a conditions the defined approximation should have the followiiig property i2Ir = = v whatever the values chosen for the constants. Therefore the basis functions should be selected in a way that @, I r = 0 for all i. Accuracy coizvergence o f tlie defined ~ ~ p ~ r o x i ~will a ~ idepend on on the selecte~b fun~tionsand as a f~Iiidam~nta1 rule these functions should be chosen in a way e approxiin~tionbecomes more accurate as rn increases. ~ ~ ~ ~ s t i t ~ i t quatisii (2.41) into Equation (2.40) gives
a f 0 is the residual which will inevitably appear through tlie ~ n s e r t i o ~ of an ~ ~ p r o x ~ ~ instead ~ t i o iofi an exact solution for the field variable into the d i ~ ~ e ~ e n et iqaul ~ ~ t i o ~qi~. ~ a t i o(2.42) c ~ is written as
2
WEIGHTED RESIDUAL FINITE ELEMENT METHODS
-
AN OU'I'LINE
(2.43) The residual, Rsk, is a function of position in R. The weighted residual method i s based on the elimination of this residual, in some overall manner, over the entire domain. To achieve this the residual is weighted by an appropriate number of position dependent functions and a s ~ ~ m a t is ~ carried on out. This is written as W&&2
=0
.j = 1,2,3, . . . rn
(2.44)
a are linearly independent weight functions and dS2 is an appropriate integration measure. Substitution of RS2from Equation (2.42) into Equation (2.44) gives (2.45) Equation (2.45) represents the 'weighted residual statement' of the original different~alequation. Theoretically, this equation provides a system of I?? simultaiieous linear equations, with coefficients a,, i = 1,. . . tn, as unknowns, that can be solved to obtain the unknown coefficients in Equation (2.41). Therefore, the required approximation (i.e. the discrete solution) of the field variable becomes dete~nined. Despite the simplicity of the outlined weighted residual method, its application to the solution of practical probleins is not straightforward. The main difficulty arises from the lack of any systeniatic procedure that can be used to select appropriate basis and weight functions in a problem. The combination of finite element approxiinatioii procedures with weighted residual methods resolves this p r o b l e ~This . is explained briefly in the forthcoming section.
As already discussed, variations of a field unltnown within a finite element is approxi~atedby the shape functions. Therefore finite element discreti~ation provides a natural method for the construction of piecewise ~~pproxim~tions for the u~iknownfunctions in problems formiilated in a global domain. This is readily ascertained considering the mathematical model represen~~d by Equation (2.40). After the discretization of a into a mesh of finite elenients weighted residual statement of Equation (2.40), within the space of a finite elenient Q,, is written as
DIFFERENTIAL EQUATIONS BY THE WEIGHTED RESIDUAl
(2.46) where E
Sz, =
E
=
total number of elements
e =l
In Equation (2.46) the unknown function zb is approximated using the shape functions o f Sz, in the usual manner. It is important to note at this stage that the elemental discretization used to obtain Equation (2.46) should guarantee the existence-of a finite integral for all of the terins in the weighted residual statement. Therefore the selection of appropriate elements for a problem primarily depends on the nature of the differential operator S in the original model equations. For example, in the solution of second order differential equations it appears that elemental discretizations that provide inter-element continuity at least up to the first-order derivatives should be used. This restriction can, however, be eased to a large degree by the application of Green’s theorem (iiit~~ration by parts) to the second-order derivatives to reduce the order of differentiations to one. The weakening of the continuity requirement achieved by this operation is an important aspect of many practical schemes used in engineering problems. Further discussion of the forniuiation of ‘weak statements’ in the finite element modelling of polymer flow problems can be found in Chapter 3 . For each Wl, Equation (2.46) generates a corresponding equation and collectively these equations can be shown using matrix notation as
where [A] and ( b ) are called the elemental ‘stiffness matrix’ and ‘load vector’, epeated application of the method to all elements in the finite element mesh and subsequent assembly of elemental stiffness matrices and load vectors over the common nodes yiclds a global set of algebraic equations in terins of the nodal unknowns. After the prescription of the boundary conditions Ihe assembled global set of equations becomes determinate and can be solved. An important property of the finite element method i s that it always produces a sparse banded set of global equations. This is an inherent property of the method and should hence be used to achieve computing economy.
In the standard Galerkin method (also called the Bubnov-Galerkin method) weight functions in the weighted ~ e s i d u astatements ~ are selected to be identi~al
44
WELCHTED RESIDIJAL FINITE ELEMENT MEl'HODS - AN OUTLINE
to the basis functions (Zienkiewicz and Morgan, 1983). Therefore the standard Galerkin representation of Equation (2.46) is given as dS2,=O
(2.48)
j = I,.,.,p
In conjunction with Equation (2.48) the prescribed boundary conditions at the boundaries of the solution domain should also be satisfied. The simplicity gained by choosiiig identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. ecause of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example.
ite ele As an illustrative example we consider the Galerkin finite element solution of the f ~ l ~ o w i ndifferential g equation in domain a,as shown in Figure 2.20. d2T -+T=O
in
dX=
I;L (2.49)
subject to: TA = 0, r, = I
B
A xA= 0
~
X
x*= 2
i 2.20~ One-dimensional ~ ~ e problem domain
Step 1: discretizat~onof the problem domain is discretized into a mesh of five unequal size liriear finite elements, as is shown in Figure 2.21.
~ I ~ F E R E N T IEQUATIONS A~ BY TEE WEIGHTED RESIDUAL ~
2
e,
1 I 0.00
e2
E
T
H
~
3 e3 4 e4 5
I
I
I
I
I .oo
1.50
1.95
2.00
x
Figure 2.21 Discretization of the problem domain
S l q 2: ~ ~ ~ p r o x iusing ~ ~ tshape i o ~functions
ithin the space of finite elements the unknown function is a p p r o x ~ ~ ~using ted shape functions corresponding to the two-noded (linear) Lagrange elentents as L
(2.50) 1=1
where N,(n), i = 1,2 are the shape functions and T,, i = 1,2 are the nodal degrees of fseedont (i.e. nodal unknowns).
Step 3: ~ a l e r ~ i n - ~ ~ e iresidual g h t e ~ statement The residual obtained via the insertion of into the differential equation i s weighted and integrated over each element as
(2.51) where w is a weighting function. In the standard Galeskin method the selected weight functions are identical to the shape hnctions and hence Equation (2.51) is written as
(2.52) J
Step 4: integration by parts (Green’s theorem) At this stage the formulated Galerltin-weighted residual Equation (2.52) contains second-order derivatives. Therefore CO elements caiiitot generate an acce~tablesolution for this equation (using C* elements the first derivat~veof
~
~
6
W E I ~ ~ ~ TRESIDUAL ED FINITE ELEMENT METHODS - AN OUTLINE
the shape functions will be discontinuous across element boundaries and the integral of their second derivative will tend to infinity). To solve this difficulty the second-order derivative in Equation (2.52) is integrated by parts to obtain the ‘weak’ form of the weightcd residual statement as
where Fe represents an element boundary (for simplicity sumniation limits are not written).
Step 5: formulation of the elemental stiffness equations The weight function used in the Galerkin formulation can be identical to either of the shape functions of a two-node liriear element, therefore, for each weight function an equation corresponding to the weak statement (2.53) is derived
where 0, represents the boundary line term. XJsing matrix notation, Equation (2.54’)is written as
Although the elemental stiffness Equation (2.55) has a common form for all of the elements in the mesh, its utilization based on the shape functions defined in the global coordinate system is not convenicnt. This is readily ascertained considering that shape runclions defined in the global system have different coefficients in each element. For example
~ ~ ~ F E R ~~QN U A~ TII OA~BY S ~ THE WEIGHTED RESIDUAL METHOD
and NI
1.5 x 0.5
=: ______
x- 1 and N ~ = I - in e2 0.5
Furthermore, in a global systeni limits of definite integrals in the coefficient matrix will be different for each element. This difficulty is readily resolved using a local coordinate system (shown as x) to define the clemental shape functions as
(2.56)
where t, is the element length. Thereforc Equation (2.55) is written as
Substitution for the shape functions from Equation (2.56) into Equation (2.57) gives
(- 1
+ !:- 2 l , -t-~ X2)dx
1
(2.58) After the evaluation of the definite integrals in the coefficient matrix and the boundary line terns in the right-hand side, Equation (2.58) gives
(2.59)
WEIGHTED RESIDIJAIs FINITE ELEMENT ~ E T H Q -~ AN S OTJTLINE
Therefore for el, C, = 1 (2.60)
For e2, t, = 112 (2.61)
For e3, le = 114 (2.62)
For e4, l, = 114
[ -47/12
97/24
97/24 -47/12]{
g} { -$}
(2.63)
=
Step 6: nssemb1.y of the elemental stijjness equations &to a l ~ e b r a ~equations c
LI
global system of
~lementalstiffness equations are assembled over their common nodes to yield -2/3 7/6 0 0 0
716 -1516 25/12 0 0
0 25/12 -69/12 97/24 0
0 0 97/24 -4716 97/24
0 0 0
97/24 -47/12
ql
0
0 0
(2.64)
-(a5
Note that in equation system (2.64) the coefficients matrix is symmetric, sparse ( i s aa significant number o f its members are zero) and banded. The symmetry of the coefficients matrix in the global finite elemeiil equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be s ~ n ~ e t r i cHowever, ). the finite element method always yields sparse and banded sets o f equations. This property should be utilized to minimize con]puting costs in complex probletns.
Step 7: ~ ~ p o ~ s ioft ithe o ~ boundary conditions rescribed values of the unknown function at the boundaries of f2 (i.e. Tl = 0, T5= 1) are inserted into the system of algebraic equations (2.64) and redundant
DIFFERENTIAL EQUATIQNS BY THE WEIGHTED RESIDLTAL METHOD
equations corresponding to the boundary nodes eliininated from the set. After algebraic inanipulations the following set of equations is obtained -15/6 25/32 0
25/12 49/12 97/24
0 0 -97/24
97/24 -47/6
(2.65)
Step 8: solution of the algebraic equations Equatioii set (2.65) is a determinate system and its solution gives 7’2 = 0.9011 T3 = 1.0813 T4 = 1.0738 At this point by repealing the above solution on a grid constructed by the division of the domain given iii Figure 2.20 into 10 elements of equal size the effect of mesh refinement on the accuracy of the slandard Galerkiii procedure can be demonstrated. The gencral elemental stiffness equation is identical to Equation (2.59) which after the insertion of an equal element length o f t , = 0.2 gives the stiffness coefficient matrix for all o f the elements as -74J15 151/30
151J30 -74/15
Therefore the assemblcd global stiffness matrix in this case is written as -74/15 15I /3n 0 0 0 0 0 0
isi/jo
0
0
0
0
n n
0
0
0
148/15 151/30 0
0 0
0 0 0 151/30 -1481 I5 isi/30 151/30 -148/15 0 151/30 0 0
0
0
n
0
0
0 0
o
n
0 151130 -148/15 151/31) 0 0
a
0 0
0
0
a 0 0 0 151/30 148/15
151:30 0 0 0 0
0 0 0
0
0
0 0
0
0
0
0 0 0
1s1,’no -14&/15
0
0
151/30 -148/15 151/30 0
0 151/30
0 If
0
0
0
n n
0
0
0
0
n
151/30 148,’15 151/30 0 0
n
0
151/30
0 0
0
-148/15 151/30
0
0
0 0
0 lil/30 -74/15
After the insertioii of the boundary conditions the solution of the system of algebraic equations in this case gives the required nodal values of T (i.e. T2 to TId as
o
WEIGHTED RESIDUAL FINITE ELEMENT ME r
m m - AN OUTLINE
T2 = 0.2178 T3 = 0.4259 T, = 0.6191 T5 = 0.7867 T6 = 0.9230 T7 = 1.0227 T8 =z 1.0817 T9 = 1.0977 Tlo = 1.0709
The analytical solution of Equation (2.49) is T = 1.09975 sin(xj
(2.66)
Figure 2.22 shows the comparison of the analytical solution with the Galerkiii finite element (FE) results obtained using the 4 and 10 element grids.
~
~ 2.22 ~ Comparison M r ~ of the analytical and finite elcnient solutions
In the finite element solution of engiiieering problems the global set of equations obtained after the assembly of elemental contributions will be very large (usually consisting of several thousand algebraic equations). They may also be
51
D I F F E R E N T I ~EQUATIONS ~ AY THE WEIGHTED RESIDUAL METHOD
ill-conditioned (Gerald and heatley, 1984). Therefore the solution of the global system of equations is regarded as one of tbe most important steps in the finite element modelling of realistic problems. Various ‘direct’ elimination techniques, such as ‘frontal solution’ or ‘LU decomposition’, and ‘iterative’ procedures, such as the ‘preconditioned conjugate gradient method’, are used as equation solvers in the finite element programs. Computjng economy, speed and the required accuracy of the solutions are the most iniportant factors that should be taken into account in selecting solver routines for finite element escriptions and full listings of computer codcs based on these techniques can be found in the literature (e.g. see Forsythe and Hood, 1976; Ninton and Qwen, 1977; Irons and Ahmad, 1980). Further explanations about the solution of the systems o f algebraic equations arising in finite element computations are given in Chapter 6.
Coordinate t r a n s ~ ~ ~ m between a ~ ~ olocal ~ and globul systems
-
~apping
lsoparainetric mapping described in Section 1.7 for generating curved and distorted elements i s not, in general, relevant to one-dimensional problems. ever, the problem solved in this section provides a simple example for the illustration of important aspects o f this procedure. Consider a ‘master’ element as is shown in Figure 2.23. The shape functions associated with this element are
(2.67)
Figure 2.23 Isopamnetric inaster element
Therefore the approximate form of the unknown function within this element i s written as (2.68)
52
WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE
In order to establish an isoparametric mapping between the master element shown in Figure 2.23 and the elements in the global domain (Figure 2.20) we use the elemental shape functions to formulate a transformation function as (2.69) hence +
.e, dx =: -dE 2
(2.70)
wliere X I and XI^ and are the nodal coordinate values in the global system and 4, represents the length of a linear element. Thus the derivatives of the shape functions in the global system are found as
(2.71)
Note that in the one-dimensional problem illustrated here the Jacobian of coordinatc transformation is simply expressed as dx/dJ' and therefore (2.72)
After the substitutioii for f froin Equation (2.68), dx froin Equation (2.70) and global derivatives of sliape functions froin Equation (2.71) into the elemental stiffness equation (2.55) we obtain, for the equation corresponding to NI
-1
-1
(2.73)
(2.74)
DlPFEREWrIhL EQUATlONS BY THE WEIGHTED RESlDUAL METHOD
and
Similarly for the second equation in set (2.55) using N f l as the weight function
And the elemental stiffness equation is written as
and
(2.78)
--3
!,I
As it can be seen equation sets of (2.78) and (2.59) are identical. Note that the definite integrals in the members of the elemental stiffiiess matrix in Equation (2.77) are given, uniformly, between the limits of - 1 and -t1 . This provides an important facility for the evaluation of the members of the elemental matrices in finite element computations by a systematic n~imerical integration procedure (see Section 1.8).
elrovThe standard Galerkin technique provides a flexible and powerful method for the solution of problenis in areas such as solid mechanics and heat coilduction where the model equations arc o f elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of
WElCllTED RESTDUAL FINITE ELEMENT METHODS - AN OUTLINE
continuity and inotion in creeping (very low Reynolds number) inconipressible flow regimes encountered in many types of polymer processing systems. Nowever, partial differential equations describing convection dominated phenomena, such as high Peclet number heat transfer or viscoelastic constitutive behaviour are of hyperbolic type and cannot be solved by this method. Application of the Galerkin method to these problems gives unstable and oscillatory results unless proper procedures are adopted to stabilize the solution. evelopment of weighted residual finite element schemes that can yield stable solutions for hyperbolic partial differential equations has been the subject of a considerable amount of research. The most slxccessfd outcome of these attempts is the development of the streamline upwinding te~~inique by Brooks and Hughes (1982). The basic concept in the streamline upwinding is to modify the weighting function in the Galerkin scheme as (2.79) are a shape function and its derivative, respectively, is the where NI and NI,? velocity vector and a is a coefficient called the upwinding parameter (in Equation (2.79) summation over the repeated index is assumed). of the appropriate upwinding parameter in multi-dimensional straightforward. In general, the analysis dcscribed by Brooks an for the derivation of the optimum upwiiidiiig parameters in problems is used, heuristically, to define the upwinding parameters in multidimensional problems (Pittman and Nakazawa, 1984). In practice, numerical tests based on trial and error may be needed to find the appropriate level of upwinding required in a problem. T n the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection terms (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called ‘selective’or ‘inconsistent’ upwinding. Selective upwinding can be interpreted as the introduction of an ‘artificial diffusion’ in addition to the physical diffusion to the weighted residual statement of the differenlial equation. This improves the stability of the scheme but the accuracy of the solution declines. ~ ~ r ~upwinding ~ ~ i schemes n e can also be compared with the Petrov-~alerl~in methods. Petrov- Galerkin techniques are a class of Galerkin~weiglitedresidual methods in which the weight functions are not identical to the basis fknctions (i.e. shape functions in the finite element context). These schemes offer greater flexibility than the standard Galerkin method in dealing with problems such as the solution of hyperbolic partial differential equations. Therefore it is natural to amline upwind Galerkin schemes as a type of Petrov ever, in contrast to the selective upwinding, in streamli Petrov-Galerkin schemes the modified weight function (here shown as Eqraation (2.79)) is applied consistently to all terns in the weighted residual statement. $9
DIFFERENTIAL EQUATIONS BY THE WEIGHTED RESIDUAL METHOD
55
To illustrate the basic coiicepts described in this section we consider the following worked example.
e consider the finite element solution of the following differential equation in domain 52, as shown in Figure 2.24 (an identical domain to problem 2.2.2* is used)
xA=0
- x
xg= 2
The problem domain
d2T dT -+a-+T==O dx2 dx subject to: TA= 0, TB 1
(2.80)
=I
Let us first consider the standard Galerkin solution of Equation (2.80) obtained using the previously described steps. Following the discretization of the solution domain f;z (i.e. line AB) into twonode Lagrange elements, and representation of T as T = )-JV,(x)T,) in t e r m of shape functions N2(.x), i = 1,2 within the space of a finite element Q,,, the elemental Galerkin-weighted residual statement of the differential equation is written as
(2.81)
After the application of Green’s theorem to the second order term in Equation (2.81) we get the weak form o f the residual statement as
(2.82)
56
WEIGHTED RESIDUAL FINITE ELBMENT METHODS - AN OUTLINE
where Ti. represents an element boundary. mental stiffness equation is formulated as
ased on Equation (2.82) the ele-
where 4 represents the boundary line term. Using matrix notation Equation (2.83) is written as
(2.84) Note that in contrast to the example shown in Section 2.2.2" the element stiffness equation obtained for this problem i s not symmetric. After the substitution for the shape functions and algebraic mani~u~ations
After the evahation of the integrals iiithe terms of the coefficient matrix, we have
~ ~ ~ F E R E N EQUATIONS TI~L BY THE WEIGHTED RESIDUAL METIlOD -1
a
P,
3-1
n
e,
4r
-
(2.86) --
4II
Choosing a domain discretization based on 10 elements of equal size (4, = 0.21, we have
(5.033 + n / 2 )
(-4.934 - a/2) (5.033
-
42)
(-4.934
(2.87)
+4 2 )
After the assembly and insertion of the boundary conditions the following set of global stiffness equations is derived -d c s d
0 0 0 0 0 c 0 0 0 0 0 s d c 0 0 0 0 0 s d c 0 0 0 0 0 s d c 0
0 0 0 0 0
‘0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 Q s d c 0 0 0 0 0 0 0 s a! c 0 0 0 0 0 0 0 s d c
O
~
O
O
O
O
O
s
d
(2.88)
-C
where d = -9.868, c = 5.033 + (a/2)and s = 5.033 - (a/2). the solution for this problem and compare it with the analytical result. The analytical solution of Equation (2.80)with the given bouiidary conditions for a = 1 is
fi
T = 2.754e-iX sin-x
2
(2.89)
The comparison between the finite element and analytical solutions for a relatively small value of n = 1 i s shown in Figure 2.25. As can be seen the standard Galerkin inethod has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solutioii is expected to improve even further with mesh refinement. As Figure 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh o f 10 elements, for larger values of this coefficient the numerical solution produced by the standard
WIGHTED RESTDUAL FINITE ELEMENT METHODS - AN OUTLINE
Galerkin method becomes unstable and useless, It can also be seen that these oscillations become more intensified as a becomes larger (note that the factor affecting the stability is the magnitude of a and oscillatory solutions will also result using large negative coefficients).
1.40
1.20 1.oo
0.80
T
0.60 0.40
0.20 0.00 0.00
0.20
0.40
0.60
0.80
1.00 X
1.20
1.40 1.60 1.80 2.00
c 2.25 Comparison of the analytical and finite element results for a low Peclet number problem
The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustrates the inability o f the standard Galerkin method to produce meaningful results for convectiondominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (conve~tion-do~inated) equations, upwindetrov-Galerkin methods are employed. To demonstrate the application o f upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. The weighted residual statement corresponding to Equation (2.80) is hence written as
59
DTFFERENTIAL EQUATIONS BY THE WEIGHTED RESIDUAL METHOD
2.00 1.a0
1
1-60 1.40
1.20
T 1.00 0.80
0.60 0.40 0.20 0.00 0.00
___-.~
0.20
0.40
-~-
0.60
0.80
,
1.00
1.20
1.40
1.60
---
1.80
2.00
F i ~ M ~ ~ Comparison of the finite element solution of low and high Peclet number problems
Integration by parts (Green’s tlieorern) of the second order tcrni in Equation (2.90) gives the weak Tom of the problem as
(2.91) 9,
Using two-noded Lagrangian elements the shape fuiictions are given as
NI =
a, - x ~
.ee
and
x
NI[ =-
.e,
therefore
In the simple one-dimensional example considered here the upwinded weight function found using Equation (2.89) is reduced to W = N -C ~?(d~/~x), Therefore, the modified weight fundions applied to the first order derivative term in Equation (2.91) c a be ~ written as
WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE
(2.92)
The general elerneiital stiffness equation can thus be written as
(2.93) Li~s~itLi~ion of the shape functions gives
fter i ~ i t e ~ r a t i o ~ i
esh of 10 elements of equal size we have
i
(-4.933 i .5.0a[;1- n / 2 )
(5.033 + a / 2 - 5.On,L?)
4I
(2.96)
..-
_I
(5.033 - M / 2
-
5.0aP)
(-4.933 I- 5.0np t M / 2 )
-411
As an example we consider the solution of Eqiiatioii (2.80) with a value of n = SO,in this case the general form of the elemental stiffness ~ ~ ~ ~is written a t i as ~ n
DI~~ERE~T EQUATIONS IA~ BY THE WEIGITTED RESlDUhL METHOD
(-29.933 (-19.967
+ 250.0,liI) - 250.0p)
(30.033 - 250.0sfi) (20.067
+ 250.0p)
4I
(2.97)
__ -41r
After the assembly of elemental equations into a global set and iniposi~~on of the boundary conditions the final solution of the original differential equation with respect to various vnlucs of upwinding parameter ~3can be found. The analytical solution of Equation (2.80) with LZ = 50 is fouiid as
The finite element results obtained for various values of p are compared with the analytical solution in Figure 2.27. As can be seen using a value of /? = 8.5 a stable numerical ution is obtained. ever, this solution is over-damped i s to find a value of u p w i n ~ ~ n ~ refore the main p and inaccurate. parameter that eli~iiiatesoscillations without generating o v e r - d a ~ ~ results. pe~ To illustrate this concept let us consider the following convection-diff~isioii equation
(2.99)
where the constants a and v represent diffusivity and velocity, respectively. Note that the s e ~ ~ c t of i ~an ‘liard’ downstrea~boundary condition is inteIitiona1 to make the iiuinerical solution prone to oscillations, with use of a ’soft’ boundary ~ o i i ~ i t such i o ~ as x = L, (dTldx) = 0 stable results may be obtained without difficulty. At the niost basic level u ~ w i n d i ncan ~ be viewed as the iiitro~uctio~i of a certain amount o f artificial diffusivity into the problem so that instead of yuation (2.99) the following equation is solved d’T dx2
(aty)--v-=O
dl’ ds
Tlaeorerical analysis by of
rooks (1979) has shown that using a value
(2.101)
62
WEIGHTED RESIDUAL FINITE ELEMENT METHODS -- AN OUTLINE
-
T analytical FE 10 elements a = 50 C @ = 0.05 + FE 10 elements Q = 50 & p = 0.1 -y FE 10 elements a = 50 C = 0.5 +-
1
where h is the length of individual linear elements in the computational mesh E = (vhla) is the mesh Peclet number (a dimeiisionless number represenof the relative dominance of convection over diffusion in energy transport), the standard Galerkin solution of Equation (2.100) will be exactly the same as the analytical solution of Equation (2.99). Extendjtig this approach to the solution of Equation (2.80), with the c~efficien~ of the first-order ~~erivative teiin a = 50, the mesh Peclet nutnber corresponding to a discrctization of the problem domain into 10 linear elements of equal size i s calculated. Insertion of this number into Equation (2.101) yields the artificial diff~isivity,y = 4.0. After addition of this value to the coefficient of the second-order term the standard Galerkin solution of Equation (2.80) can be found. s t r ~ a i ~ l i nupwindii~g e method the calculated value of y is taken to be the ~ ~ ~ parameter, w i i.e. ~ Q~ in ~Eqiiatioii ~ ~(2.79). g This gives the ‘appropriate value’ of the upwinding efficient in the modified weight function W = N C P(dlVld~)as /3 = -0.08. epeating the procedure outlined through Equation (2.90) to (2.97) the streamline upwind solution of Equation (2.80) can now he found. As c o ~ p ~ ~ of i ~this o i solution i with the ai~a~ytical result, given in Figure
DIFFERENTIAL EQUATIONS BY THE WEIGHTED RESIDUAL METHOD
63
T 0.60 0.40
FE 10 element a = 50, using optimal upwinding I
0.20
x
0.00 0.00
0.20
0.40
0.60
0.80
I
1.00
1.20
1.40
1.60
1.80
2.00
X ~~~~~~~
2.28 Comparison of the aiialytical solution with the finite element result obtained using optimal upwiiiding
2.28, shows the numerical solution is not affected by any cross-wind diffusion and hence i s very accurate; hence optimal ~tpwindi~g has generated a ‘superconvergent’ solution which coincides with the analytical result. (1979) extended the approach to ~iulti~dimeIisiona~ proble ~ s t r e a ~ u~ ~n~ev i n ~ i itechnique ig’ in which the modified weight functions are adjusted to introduce the necessary dissipation only along the streamlines in a flow field aiid avoid excessive ovcr-da~pingby ‘cross-wind’ diffusion. In multirns, however, it i s impossible to develop a theoretical analysis act value for the required artificial diffusivity a n a l o g o ~ ~ ~ (2.101) for the one-dimensional case. ased on the conce ing modified weight functions such as (2.102) where 0 yc 2 1 is a constant, v1 and ‘1’2 are velocity components and hl and IQ are characteristic element lengths, can be used to obtain stable solutions for twodinieiisio~i~l problems. In general, however, because of the inipossibi~ityof selection of an o ~ ~value i for ~ yc u that ~ can guarantee the e ~ i ~ i n a t ~ ofoall n spurious cross-wind difftision and indeed the uncertainty of definition of element length in a two-diinens~onaldomain the nuinerical solution is e ~ p e c t e ~ to involve some degree of upwindi~i~ error.
WEIGHTED RESIDLJAL FINITE ELEMENT METHODS - AN OUTLINE
The standard least-squares approach provides an alternative to the ~ a l e ~ ~ method in the development of finite elenient solution schemes for differential to belong to the class of weighted ver, it can also be s equations. an, 1983). In the least-squares finite es (Zienkiewicz and residual te element method the sum of the squares of the residuals, generated via the substitL~t~on of the unknown functions by finite element ap ed and subsequeiitly minimized to obtain the working scheme. The procedure can be illustrated by the following example, consider 3(u)--g--OonW
(2.103)
For ~ i m p ~ ~ cwe i t yassume that "s: = (d2/dx2) + a(d/dx), where N is a ~onstant. The basic steps in the least squares scheme are: (a) formulation o f a funclioiial using the squares of the residual obtained by the substitution o f the unknown as P 1
in the original equation, thus
and (b) minjmiza~ionof the derived fuiictional as
(2.105) where rn is the total number of unknowns in the problem. Equation (2.105) can be ~ ~ g a r d as e da weighted residual ~tateinentin which the weigh~iiigunction is in terms of the derivatives of shape functions in a Corm that reflects the original differ~nti~l equation. Combination of this equation with the wei s t a t e n i e ~derived ~ from the standard ~ a ~ e rmethod ~ i n (~quation ~trov-~alerkiii formLil~tio~ for the original equation. Further explan~tioii~ about this point are given in Chapter 4.
Important classes of polymeric flow processes are described by time-dependent differential equations. The most convenient method for solution of the tirne-
~ I ~ ~ E R ~ EQUATIONS N T I A ~ ~BY THE WEIGHTED KESIDUAL METHOD
65
dependent differential equations by finite element procedures is the 'partial discretization' technique. In this technique, the space-time doinaiii is not discretized as a whole and instead, time derivatives are treated separately. In addition, finite element discretization of temporal derivatives is usually avoided and inslead morc direct methods are used. The 8 time-stepping technique and the Tay~or-~alerkiii method are the most frequently used partial discretization procedures for the solution of transient problems by weighted residual finite elemeiit schemes.
In this technique, initially, the time derivatives in a differential equation are kept iinchanged and the spatial discretization is carried out to form a weighted r e s i ~ ustatement ~l in the usual manner. Therefore after the spatial ~ ~ s c r e ~ i ~ a ~ instead of a set of algebraic equations which are normally derived for steadystale problems, a system of ordinary differential equations in ternis of time d e r ~ ~ a ~ i are v e sgenerated. n general, for the class of single step 8 m e t ~ this ~ ~ s system is shown using matrix notation as i~.~06)
where the subscript 8 indicates that the weighted residuals s t a t ~ ~ n eis n tderived at time level 0 (0 5 fI 5 I), as is shown in Figure 2.29, The teniporal derivative term in Equation (2.106) is a p ~ r o x i m a tby ~ ~a
(2.107) The reni~iningterms in Equation (2.106) are approximated using a linear inte~polationas
Therefore
from Equations (2.1071, (2.108) and (2.109-2.110) into ~ q u a t i o n bstitu~~on 106) and carrying out algebraic inanipulations gives
((1-
w%+
e{%L,)
(2.111)
66
WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE t
Time stepping scheme
The algebraic system given as Equation (2.1 11) represents the working equation of the 0 method. On the basis of this equation a global set is derived aild solved to obtain the unknowns at time level n + 1 using the known values at time level n. The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 8 = 0.5, the mcthod yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for siinple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems; Zienkiewicz and Taylor, 1994). bviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of Q and time increment At is usually based on trial and error. Factors such as the nature of non-linea~tyof physical parameters and the type of elements used in the spatial discretization usually inhencc the selection of the values of B and At in a problem. ~ q u a t i o n(2.106) gives rise to an implicit scheme except for Q = 0. The application of implicit schemes for transient problems yields a set of simultaneous equatioiis for the field Linknown at the new time level n $. 1. As can be seen from Equation (2.11 1) some of the terms in the coefficient matrix should also bc evaluated at the new time level. Therefore app~icationof the described scheme requires the use of iterative a l g o r i t l ~ sVarious , techniques for enhancing the speed of convergence in these algorithms can be found in the literature ( ~ i t t i ~ a 1989). n,
a partial differeiitial equation, representing a time dependent given as
DIFFERENTIAL EQUATIONS BY TIIE WEIGHTED RESIDlJAL METHOD BU(X, t)
at
+ S [ u ( x ,t ) ]- g = 0
67
(2.112)
where S is a linear differential operator with respect to the spatial variables x. Taylor series expansion of the field unknown, u(x,l), in Equation (2.112) with respect to the time increment, gives
(2.113)
Time derivatives in expansion (2.1 13) can now be substituted using the differential equation (2. I 12) ( onea, 1984). The first order time derivative in expansion (2.113) is substituted using Equation (2.112) as DU(X, --
at
t ) - --%[u(x,r)]
+g
(2.114)
o i i14) with respect to the time variable epeated diffcrcntiation of ~ q u ~ ~ t i(2.1 also gives the higher-order time derivatives of the unknown, €or example
(2.115)
Aiiy first-order temporal derivative of u(x,t) appearing on the right-hand side of Equation (2.11s) can again be substituted from Equation (2.11 temporal derivatives in expansion (2.1 13) can in principle be found using the original differential equation. This provides a differential equation that is exclusively in terms of the spatial variables. This equation can be ~ ~ s c r e t iand ze~ solved by the Galerkin method in the usual maiiner. In practice, Taylor series expansion of the field variable is truncated and usually only the first few terms are kept. ~ c c u r a c yof the time-stepping scheme in the Taylor- Galerkiii method is therefore dependent on the highest order of the time derivative rcmaiiiing in the e x p a ~ s i oafter ~ its truncation. or example, a second-or involve the second-order time derivative of the field unknow differ~ntiatioiiand substitution of the tcmporal derivatives ~if~erential equations of complex field problems may prove to be difficult. A c o ~ p u t a t i o n a ~more ~ y efficient fomi of the second-order T a y ~ o r - ~ a ~ e r k i n ieme based on a time-split procedure has been developed by Townsend and bster (1987) which can resolve this difficulty.
WEIGHTED RESIDUAL FINITE ELEMENT METHODS
AN OUTLINE
athe, K. J., 1996. Finire Element Procedures, Preniice Hall, Englewood Cliffs, NJ, Breimer, S. C. and Scott, L. R., 1994. The Mathemuticul Tlwory qf Finite E h e i i t ~ ~ e tSprin~er-Verla~, ~ ~ o ~ ~ New , York. ., Fox, F.L. and Schmit, L.A., 1965. The generation of interelementcoinpatible stiffiiess and inass matrices by the use of interpolation formulae. Proc. Conf oi? Matrix Methods in Strwcturd Mechunicss,Air E:orce Institute of Technology, ~ ~ ~ h t - ~ a t t eAr Fs oBase, n OH. Brooks, A. N.and Hughes, T. J. R., 1982. Strean~line-upwindl~e.lrov Galerkin foi~iiulalions for convection dominated flows with particular emphasis on the incompressible Navier Stokes equations. Compzit. Methods Appl. Mcch. Eng. 3 Ciaslel, P. 6., 1978. The Fikite Element Method for Elliptic Problem, Nortl2-Holiand, Ai~st~rdam. Crouzeix, 142. and Raviart, I?. A., 1973. Coilforming and non-conforming finite elements for solviiig the stationary Navies- Stokes equations. RAIRO, &ric Rouge 3, 33 -76. onea, J., 1984. A Taylor-Calerkin method for convective transport problems. Int. J. Num. ~ e t ~ i Eng. ~ j ~ s . and Meler, C. 13.. 1967. Conprter Solution of Lineur AIgehraic Sj?slcrns, II, Engiewood Cliffs, NJ. nd Wheatley, P. O., 1984. Applied jVumerical Anuljw’s, 3rd cdn. AddisoiiWesley, Reading, MA. Girault, V. and Kaviart, 986. Finite Elemmt Met1iod.r. for ~ u i ~ i e r - S t o E ~ ep .a~tions, ~ p s i ~ g e r ~ v ~ ~ l a g , Hinton, E. and Owen, D. R. J., 1977. Finite Ehnent Progrunir.pzirzg, Academic Press, London. 1976. Frontal solution program for unsynxnetric matrices. Int. J Nunzer. 399. A. N ., 1979. A multidiniensional upwind scheme wiih iio ughes, T. J. K. (ed.), Finite Elemenf I ~ ~ e t Coitvectiopi ~ o ~ ~ ~ ~ nuted Flows, AMD Vol. 34, ASME. New York. . and Ahmad, S.. 1980. Techniques of Filzite ~ l e ~ e ncb. ~ .13, ~ , Ellis ey, Chichesier, pp. 215 244. 1987. Numerirrd Solution of Parlid Dqjerentiul Eqiiutioizs by the Finite ge University Press, Cambridge. , 1982. Numerical Solution oj Purtial , Wiley, New York. , 1977. D i e Finite Element Method h Purtiul ~ ~ ~ e ~ e n t i a ~ ~ L i u ~ i Wiley, ~ ~ n ~ rLondon. ., Petera, J., Nassehi, V. and Pittnian, J. F.T., 1989. Petrov-Gderkin methods nu isoparametric bilinear aiid biqriadratic elements les d for a scalar conveciion-diffus~on Metlz. Neat Fluid Flow problem. Int. .?. NZ~YEZCT. Pironaeau. Cl., 1989. Finite eleineiit meiliods for fluids. Wiley, Cliichestcr.. ~ i t t i ~ aJ.nF.T., ~ 1989. Finite elements deling for Polymer Pro 237-33 L ittman, J. F. T. and Nakaziwa, S., 1984. Finite element analysis of polymer processing .1
REFERENCES
operations. In: Pittman, J. F. T., Zienkiewicz, 0.C., Wood, R. D. and Alsxandcr, J. M. (eds), N ~ ~ ~ e rAnalysis ~ c a l of Formivig Processes, Wiley, Chichester. ~ to the Finite Eft7ment Method, 2nd edn, ,McGrawReddy, 3. N., 1933. A F fntrohiction Hill, New York. Slrang, G . and Fix, 6.J., 1973. An Ana1jwi.s of llze Finite Element Merhocl, Prentice Hall, Englewood Cli€fs, NJ. ood. I?., 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Ctmput. Fluids 1 , 73-100. Townsend, P. and Webster, M.F., 1987. An algorithm for the three dimensional transient simulation of non-Newtonian fluid flow. Tn: Pande, G. N. and ~ i ~ ~ l e tJ.o n , (eds), Transient Dynamic Ancllysis arid Constitutive L n ~ )Jor s Engineering ~ ~ ( ~ ~ e r i c ~ l Vol. 2 , 7'12, Nijhoff-Holland, Swansea, pp. 1-11. Zienkiewicz, 0.C. and Morgan, K., 1983. Finite Elements und Approxim&vz. Wiley, New York. Zienkiewicz, 0.C. and Taylor, R. L., 1994. 'The Finite Element Method, 4th edn, Vols I and 2, McGraw-Hill, London.
This Page Intentionally Left Blank
Weighted residual finite element methods described in Chapter 2 provide effective solution schemes for incompressible flow problems. The main characteristics of these schemes and their application to polymer flow models are described in the present chapter. As already discussed, in general, polynier flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows i s a separate scheme for its solution required. Equations of continuity and inotion in a flow model are int~insically connected and their solution should be described simultaiieoi~siy.Solution of the energy and viscoelastic constitutive equations can be considered independently.
N
~pplicationo f the weighted residual method to the solution of i ~ i c o ~ p r ~ s s j b l e no^-Newtonian equations of contiiiuity and motion can be based on a variety of different schemes. In what follows general outlines and the formulation of the working equatio~isof these schemes are explained. In these formulations Cauchy’s e~L~ation of motion, which includes the extra stress derivatives (Equa>,is used to preserve the generality of the derivations. velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Gauchy’s equation of motion is either substituted in terms o f velocity gradients or citlculated via a viscoelastic constitutive equation in a separate step. The convection term in the equation of motion is kept for c ~ ~ i p ~ e t e nof e sthe s derivations. In the majority of low Reynolds number polymer flow models this term can be neglected.
72
FlNITE ELEMENT MODELLING OF POLYMERIC FLOW ~
R
~
U-V-P schemes belong to the general category of mixed finite element tecliniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible n ~ i i - ~ e w t o n ~ ~ n flow (Equations (1.1) and (1.4), Chapter 1) expressed as 77.v = 0 av pat
+pv.vv = --vp
(3.1)
where v, p, -r and g represent velocity, fluid density, pressure, extra stress and body force, respectively, and 6 is the onecker delta. In the U-Vweighted residual statements of the above equations over elenient SZ,, in the discretized domain are formulated as
where an over bar indicates the approximated (i.e. el~inentallyinterpolated~ variables, ML and NJ are appropriate weight functions, V" is assumed to represent the velocity found at a previous iteration step to linearise the convectioii term in the discretized equation of motion. Equation set (3.2) is based 011 the general equation (2.46) representing a weighted residual iinite element statement. The most immediate r e q u i r e ~ e nin~the application of the U-Vto the modelling of inc ressible flow regimes is the satisfaction of condition known as th 1974). Using a siinyle said that this r e q ~ t j r e ~ c narises t from the absence of a pressure term in the inco~press~ble continuity equation. Tn the ~ p ~ ~ i c a t iofo nthe finite element tec~niqueto inco~pressib~e flow it was found that the U-Vconjunction with elements generating identical in tei-~olationsfor velocity and pressure yields inaccurate and oscillatory results. These oscillatioiis were shown od or Crouzeixwhen elements belonging to the Taylorrovide unequal order interpolations for ocities and press (~ien~iewicz et d., 1986). Therefore the d i s c r c ~ ~ ~~elocity ed and pressure shown in Equation (3.2) are expressed as
73
SOLUTION OF TXIE EQUATIONS OF COhTINUITY AND MOTION
Further details of the , sometimes referred to as Ladyzhenskayacondition and its importance in the numerical solution of incomequations can be found in textbooks dealing with the theoretical aspects of thc ~ n i elcrnent ~ e m ~ t h o d(e.g. see ~iistabili~y (or ~hecker~boardiiig~ of pressure i avoided using a variety of strategies. The main strateg~esfor obtai~iingstable results by the U -V ~ncoxn~r~ssible flow are as follows: use n ~ n ~ s ~ a ncl~me~its dar~ onging to the Taylor ood or Crouzeixviart groups that satisfy the condition. Examples useful ~ l e m e ~int s this category are given in Table 3.1, for further explanations about the erties of these elements see Pittman (1989).
To use an element that, although does not satisfy the ~apab~ ofefiltc~~ng out arrisilic oscillations (
Element
Interpolation ~-
Triangular Taylor Hood
ectai~gLi~a~ Taylor-
Triangihr Cromzcur-Raviart
Velocity
Pressure
Quadratic
Lmear
Bi-quadratic
uadratic
Bi-linear
Coustxnt
Number of nodes and order of continuity Velocity
Pressure
6 Vertices and mid-sides CO 9 Corners, mid-sides and centre
3 Vertices
C'O
c70
6 Vertices a n d mid-sides
1 Centre
co
Rec~anyularCrouieix-
i-quadratic
Linear
c" 4 Corners
9 Corners, mid-sides and ccntre C"
c-l
1 Centre
e' ~
FINITE ELEMENT MODELLlNG OF POLYMEKIC FLOW PROCESSES
To use a technique that can circumvent the necessity for satisfaction of the condition. Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et aE. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible AQW. These schemes are shown to be the spccial cases of the method described in the following section developed by Zienkiewicz and Wu (199 1).
sc
As already explained the necessity to satisfy the stability condition restricts the types of available elements in the modelling of incompressible flow problems by the U-V -P method. To eliminate this restriction the continuity equation re~r~senting the incompressible flow is replaced by an equation corresponding to slightly compressible fluids, given as
where c is the speed of sound in the fluid. In this case the discretized form of the governing Bow equations are forrnulatcd as
Using different types of time-stepping techniques Zienkiewicz and showed that equation set (3.5) generates naturally stable schemes for incornpre~sibleflows. This resolves the problem of mixed interpolation in the U-V-I’ formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-s~atesolutions are also obtainable from this schcme using iteration cycles. This may, however, increase coinputational cost of the solutions in comparison to direct simulation of ste~dy-st~te problems.
SOLUTION OF THE EQUATIONS OF CONTTNUITY AND MOTION
75
The penalty method is based on the expression of pressure in terms of the incompressibility condition (i.e. the continuity equation) as p = -X(V.Y)
(3.6)
where X is a very large number called the penalty parameter. Equation (3.6), which represents a perturbed form of the continuity equation, is used to substitute the pressure in the equation of motion in terms of velocity gradients. This can, in physical terns, be interpreted as easing of the incompressibility constraint to consider instead, a slightly compressible flow regime. ~epending whether the described substitution of pressure is carried out bef'ore or after the discretization of the governing equations, two different types o f the penalty method are dcveloped. Regardless of' which formulation is used the elimination of pressure as a prime unknown yields a more coinpact set of working equations and hence the total computational cost of the penalty methods is less than comparable U-V-P schemes. The main drawback of the penalty method is the generation of' ill-conditioned equations that result from the multiplication of some of the terms in the stiffness matrix by a large number. Iterative methods designed to cope with ill-conditioned equations may be needed to improve the performance of the penalty schemes (Zienkiewicz et al., 1985). In general, in the application of the penalty schemes to the polymer flow problems the following points should be considered: Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dorninated by the ~ n c o ~ ~ r e s s i b iconstraint ~ity and may become s i n ~ ~ aOn r . the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Netvtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as X = Xoq Wakazawa et al., 1982) whcre Xo is a large diinensionless parameter and J? is the local viscosity. The recommended value for Xo in typical polymer flow problems is about 108. By using a variable penalty parameter related to local element size, roundoff error in the solution of ill-conditioned finite element equations obtained in the penalty schemes can be reduced (Kheshgi and Scriven, 1985). Eliniiiiation of the pressure term from the equation of motion does not automatically yield a robust scheme for incompressible flow and it is still necessary to satisfy the BB stability condition by a suitable technique in both forms of the penalty method.
46
FINlTE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES
The cuntiiiuozis penalty technique In the continuous peiialty technique prior to the discretization of the governing equations, the pressure in the equation of motion is substituted froin Equation (3.6) to obtain i3V
pat
+ p v . v v = -8(-XV.V)6
4-Y7.z + pg
(3.7)
The discretization of Equation (3.7) gives
(3.8) Equation (3.8) is the basic working equation of the continuous penalty method. As already explained, in order to enforce the incompressibility constraint some of the terms in the stiffiiess equation in the penalty method are multiplied by a very large parameter. In general, this yields an equation which is overwhelmed by its penalty terms and can only generate a trivial solution. To solve this problem, the penalty sub-matrix in the eleniental coefficient matrix is forced to become singular. In practice, the penalty terms in the elemental coefficient matrix are calculated using a ‘reduced integration’ to achieve the required singularity (Zienkiewicz and Taylor, 1994). For example, in a two-dimensional problem all of the terms in the elemental stiffness matrix can be calculated using a 3 x 3 Gauss-Legendre quadrature except for the penalty terms that are found by a 2 x 2 procedure. This is said to be equivalent to using a lower order interpolation for pressure than the one used for velocity.
The discrete penalty technique The use of selectively reduced integration to obtain accurate non-trivial solutions for incon~pressib~e flow problems by the continuous penalty method i s not robust aiid failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motioii is slnbslituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is iii contrast to tbe continuous penalty method in w~i~cli the satisfaction of the stability condition is achieved indirectly through
SOLU’MON OF THE EQUATIONS OF CONTINUITY AND MOTlON
77
the reduced integration of the penalty terms. The discrete penalty method combines the advantages of computational economy with the robustness of a numerical sclienie in which the B condition is directly satisfied.
The well-known inaccuracy of numerical differentiation precludes the direct calculation of pressure by the insertion of the computed velocity field into Equation (3.6). This problem is, however, very effectively resolved using the following ‘variational recovery’ method: Consider the discretized Eorm of Equation (3.6) given as (3.9) Q,
Q,
where 1.1; is an appropriate weight function and the over bar means that the unknowns are approximated using the finite element shape functions in the usual manner. Equation (3.9j represents a variational statement corresponding to the penalty relation and its utilizatioii in conjunction with the Galerhin finite element discretization yields a scheme for the calculation of nodal pressures. This is shown as (3.10) Qe
where the penalty parameter in Equation (3.10) is substituted using ~ q u a t i o n t is important to note that the integrals on the right-hand sides of the equations arising from Equation (3.10) should be found at the reduced integration points. The coefficient matrix on the left-hand side of Equation (3.10) is the ‘mass matrix’ given as (3.11) $2,
This matrix is usually diagonali~edusing a simple mass lumping technique (Pittman and Nakazawa, 1984) to minimize the computational cost of pressure calculations in this method.
Normally, tlie extra stress in the equation of motion is substituted in terms of velocity gradients and hence this equation includes second order derivatives of
78
FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES
velocity. Substitution of the pressure term via the penalty relation also gives rise to second-order derivatives of velocity. Therefore it appears that only finite elements whose shape functions guarantee inter-clement continuity up to the first-order derivatives, at least, can be used to discretize the equation of motion. As explained in Chapter 2 the restriction on the order of continuity of permissible finite elements is readily relaxed (i.e. continuity requirement is weakened) by the application of Green’s theorem to the second order derivatives. The general form of this application in the context of the described finite element schemes is as follows: Consider an integral as (3.12) Using integration by parts
I’
V.(M)Vf)dS2 -
D
f
s2
(3.13)
According to Green’s theorem (Aris, 1989) we have (3.14)
r
D
where r i s the boundary surrounding the domain D and Z is the unit vector noimal to I‘ in the outward direction. Therefore
(3.15)
r refore the second-order derivative off appearing in the original form of I is aced by a term involving first-order derivatives of w andfplus a boundary term. The boundary terms are, normally, cancelled out through the assembly of the elemental sliffness equations over the conmon nodes on the shared interior element sides and only appear on the outside boundaries of the solution domain. However, as is shown later in this chapter, the a p p r ~ p r i a ~treatment c of these integrals along the outside boundaries of the flow domain depends on the prescribed boundary conditions. In practice, in order to maintain the symmetry of elemental coefficient matrices, soim OC the first order derivatives in the discretized equations may also bc integrated by parts. The described application of Green’s theorem which results in the derivation of the ‘weak’ statements is an essential step in the fomiulation of robust U-Vpenalty schemes for non-Newtonian flow problems.
MODELLING OF VISCOELASTIC FLOW
4
The basic procedure for the derivation of a least squares finite element scheme is described in Chapter 2, Section 2.4. Using this procedure the working equations of the least-squares finite element schemc for an incompressible flow are derived as follows: Field unknowns in the governing flow equations are substituted using finite element approxima~io~ls in the usual manner to form a set of residual statements. These statements are used to formulate a functional as
where an over bar represents approximation over an element .Qe. The constant k is used to make the functional dimensioiially consistent. Minimization of functional (3.16), with respect to field variables (i.e. velocity components etc.) leads to generation of the working equations of the least-squares scheme. As can be ascertained from flmctional (3.16) the working equations of this scheme will inevitably include second-order derivatives. Therefore only finite elements that can generate a sufficiently high-order discretization to cope with second-order derivatives can be used in conjunction with this scheme. This is a severe set back for this method that can restrict its general applicability in flow ~ o d e ~ ~ i ~ i g owever. the least-squares method has been advocated as a powerful technique that provides better nunierical stability than either U-V-P or penalty schemes in a wide range of fluid dynamical problems (Bell and Surana, 1994).
In a significant number of polymer processes the influence of fluid elasticity on the flow behaviour is small and hence it is reasonable to use the generalized Newtonian approach to analyse the flow regime. In generalized Newtonian fluids the extra stress is explicitly expressed in terms of velocity gradients and viscosity and can be eliminated from the equation of motion. This results in the derivation of Navier-Stokes equations with velocity and pressure as the only prime field unknowns. Solution of Navier-Stokes equations (or Stokes equation for creeping flow) by the finite element schemes is the basis of com modelling of non-elastic polymer flow regimes. In contrast, in viscoelastic flow models the extra stress can only be given through implicit relationships with the rate of strain, and hence remains as a prime field unknown in the governing equations. In this case therefore, in conjunction with the governing equations of continuity and momentum (generally given as Cauchy’s equation o f motion) an appropriate constitutive equation must be solved. Numerical solution of viscoelastic constitutive equations has been the subject of a considerable amount of research in the last two decades. This has given rise to a plethora of me~hods
0
FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES
that apply to various types of differential or integral constitutive equations. pite significant achievements of the last two decades a universal tnodelling h o d o l o ~ ,that can generate stable and accurate results for the range of lssenberg numbers observed in practical flow processes, has not been developed. It is generally accepted that the complex non-linear nature of viscoelastic fluids precludes the development of a universal method for all situations and efforts should be focused on solving individual problems. The main categories of numerical algorithms used to solve viscoelastic flow problems are summarized by Keuniiigs (1 989). These schemes are priinarily divided into ‘coupled’ and ’decoupled’ techniques. In coupled methods, the governing equations of contiiiuity, motion and rheology for a viscoelastic flow regime are solved simultaneously to obtain the whole set of velocity, pressure and stress variables as the prime field unknowns. These techniques usually use an iterative procedure such as Newton’s method. In contrast, in the decoupled methods the equations of continuity and motion are solved separately from the rheological equation. The main procedure in these techniques is to start from a ‘known’ flow kinematics to calculate the viscoelastic extra stress by solving the rheological equation. The calculated stress field is then inserted into the equation of niotion and the velocity and pressure fields are found. This process is iterated until a converged solution is obtained. The coupled techniques are more readily applicable to the ~ i f f e r e ~ tviscoja~ elastic constitutive equations and their extension to the integral models is not comnion. The main advantage of the coupled methods is that, poteiitially, they can benefit from the quadratic rate of convergence of Newton’s iteration coupled methods, on the other hand, make more efficient use of the mputer core capacity. However, the rate of convergence of the decoupled methods is generally slow. The i ortant point to note is that both techniques require considerable computer U times if solutions that reflect realistic conditions are attempted. Numer exanipies of both categories of coupled and decoupled methods in conjunction with various types of finite e l e ~ e n ft o r ~ ~ a t i o and n s ~ultiplicityof viscoelastic constitutive models can be found in the literature. However, it has not been possible to establish the accuracy, or clearly demonstrate the advantageous, of some o f the elaborate schemes that have been reported. A thorough description or critical evaluatioii of various classes of viscoelastic flow models is beyond the scope of the present book and the interested reader should refer to scientific papers and books published about this topic, In general, the utilization of ‘integral models’ requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under ‘general’ conditions. culties involved in the solutioii of viscoelastic constitutive equations have prevented the development of a modelling methodology with general applicability for these regim Using specially modified Petrov-Galerkin, techniques (Hughes et al.. 1986; ughes, 19871, numerically stable results for these
MODELLlNG OF VlSCOELASTIC FLOW
equations can be generated. However, it i s not possible to demonstrate the experimental accuracy of inany of these solutions. Further research is still required to develop more efficient and reliable finite element schemes for the modelling of viscoelastic Bow regimes under the conditions that arise in areas such as polymer processing. In the following section representative examples of the development of finite element schemes for most coininonly used differential and integral viscoelastic models are described.
3.
6
The first finite element schemes for differential viscoelastic models that yielded nunierieally stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for sonie benchmark viscoelastic problems, such as flow through a two-dimeiisiona~ section with an abrupt contraction (usually a width reduction of four to one), they can generate simula tions that were ‘qualitatively comparable’ with the experimental evidence. A no table example was the coupled scheme developed by archal and Crochet (1987) for the solutioii of Maxwell and Qldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwiiiding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive ~ q u a t i o ~for i a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid beliaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the BOW field was updated. These authors also used inconsistent streamline upwindiiig to maintain the stability of the scheme. In this section the discretization of upper-convected Maxwell and ~ ~ d r o y d models by a modified version of the Luo and Tanner scheme is outlined. This scheme wses the subdivision of elements suggested by Marcllal and Crochet (1987) to generate smooth stress fields (Swarbrick and Nassehi, 1992a). In the absence of body force, the dimensionless form o f the governing model equations for two-dimensional steady-state incompressible creeping flow o f a ~ ~ s c o e ~ afluid s ~ i are c written as (3.17)
In Equation (3.17), the Cauchy stress is defined as
(3.18)
82
FlNITE HLEMEXT MODELLING OF POLYMERIC FLOW PROCESSES
where p is pressure, 6 is the Kronecker delta, T~ and zc are, respectively, the viscous and elastic parts of the extra stress and v is velocity. The viscous part o f the extra stress i s given in terns of the rate of deformation as 2, = r ( V v iOvT), where r is a dimensionless material parameter defined as
where qm is the viscosity of the viscoelastic fluid and qs is the viscosity of the 'Newtonian solverit' (see Chapter 1). The elastic part of the extra stress for upper-convected Maxwell (UCIM) and Ruids is written as (see Chapter 1)
A-iooz At
z , i - W s L = ( I - r)('Ov 4-VvT)
(3.20)
where Ws is the Weisseiiberg number. The governing equations for the upperaxwell flow are obtained by putting r = 0, while the Oldroydis represented by Y = 0.11 (Crochet, 1982). After the substitution of Caucliy stress via Equation (3.20) and the viscous parr of the extra stress in terms of rate of deformation, the equation of motion is written as
- YV (Vv + VVT) = 0.7,
(3.21)
a
V
For s ~ p l i c i t ywe , define T = ze and (Q-,,,z,lAt). As explained by Luo and Tanner (1 989), the decoupled iiiethod requires a suitable variable transfornm lion in the governing equations (3.20) and (3.21). This is to ensure that thc discrete momentum equations always contain the real viscous term required to recover the Newtonian velocity-pressure fomiulation when K s approaches zero. This is achieved by decomposing the extra stress
-s+
(3.22)
where (3.23) Therefore Equations (3.20) and (3.21) are written in terms of (3.24) and
(3.25)
MOIIEL1,ING OF VISCOELAS’VIC FLOW
~ e ~ i ~ t ~oJt ithe o nworking equations The weighted residual statement of Equation (3.25) over an element domain a,, is
)]NdO,=
I.
(V.
(3.26)
i2,
Re
where N is a weighting function. The iiitegrand on the left-hand side or Equation (3.26) involves second-order derivatives of velocity and to preserve the inter-element continuity the velocity field needs to be approximated by appropriately high order interpolation functions. To weaken this requirement it is necessary to apply Green’s theorem on Equation (3.26). This gives
N dT,
(3.27)
re is the unit outward normal to the boundary re. The weighted residual statements of Equations (3.17) and (3.25) are also formulated as (3.28)
(3.29) Q#
Q,
epending on the type of elements used appropriate i n t e r p o ~ ~ ~ fi o~n i ~ t ~ ~ are used to obtain the elemental discretizations of the unknown variables, In the present derivation a mixed formulation consisting of nine-node bi-quadratic shape functions for velocity and the corresponding bi-linear interpolation for the pressure is adopted. To approximate stresses a 3 x 3 subdivision of the velocity-pressure element is considered and within these sub-elements the stresses are interpolated using bi-linear shape functions. This arrangement is shown in Figure 3.1. The approximate forms of U, 17 and T are hence given as
FINITE ELEMENT MODELLING OF POLYMElZIC k LOW PROCESSES
where the letters N a i d M represent bi-quadratic and bi-linear shape functions, respectively. The subscripts I, K and L represent the nodes inyolved in the described approximations. After the substitutiol? of the field unknowns in terms o f their corresponding shape functions, from Equation (3.30), in the weighted residual statements of the governing equations and application of the Galerkin weighting the working equations of the scheme (i.e. the elemental stiffness equation) are foixiukdted. The mo~eiitL~m and continuity equations give rise to a 22 X 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and d represent the nodes in the bi-quadratic element for velocity and K and L thc four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions, Nr and AdL, are bi-quadratic and bi-linear, respectively. The jth component of velocity at node .J is shown as tJ$. Summation convention on repeated indices is assumed. The dis~ret~zatioi~ of the continuity and momentum equations is hence based on the l J V P scheme in
Velocity, Pressure, Stress
~
for the discretization of viscoelastic extra-stress ~ 3.1 g Subdivided ~ ~ element c
(3.31)
MODELLING OF VTSCOELASTIC FLOW
In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle i s developed in which iii each iterative loop the stress fields are computed after the velocity field. The viscous stress (Equation (3.23)) is calculated by the 'variational recovery' procedure described in Section 1.4. The elastic stress then Gomputed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32).
(3.32) The integrals in Equation (3.32) are found using a quadrature over the element domain a,. The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection teniis in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding) (3.33)
I is the magnitude of a, 4) is a scaling factor and he is a characteristic element size defined as (3.34) where (E, T)) are local coordinates. The basic iterative procedure for the described decoupled scheme i s given in the following cizart.
FINI r E ELEMENT MODELLIXC OF POLYMERIC FLOW PROCESSES
Read input data and initialize variables
Using working equations (3.31) Ibrm the elemental stiffness equations, assemble the global system, apply boundary conditions and solve 10 obtain velocity and pressure fields
Using working equations (3.32) form the elemental stiffness equations corresponding to the constitutive equation, assemblc the global system, apply the boundary conditions and solve to obtain the stress field
Using newly found stress update the RHS o f flow equations and recalculate velocities and pressure
Check convcryence (stress) I
The outlined scheme is shown to yield stable solutions for non-zero number Bows in a number of benchmark problems (Swarbric 1992b). However, the extension of this scheme Lo more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather tb bi-linear sub-elements. It should also be noted that satisfaction of the 13 condition in viscoelastic flow simulations that use mixed formulations is not as clear as the casc of purely viscous regimes.
As mentioned in Chapter 1, in general, the solution of the integral viscoelastic models should be based on Lagrangian frameworks. In certain types of flow
MODELLING OF VISCOELASTTC FLOW
regimes, however, the solution of these equations can be based on an Eulerian approach. To give a brief description of the use of an Eulerian framework in the solution of integral constitutive models we consider the motion and straining of a fluid particle in a contracting domain as shown in Figure 3.2.
Figure 3.2 Deformation of a fluid particle along its trajectory in a contracting flow
If the position of the fluid particle at current time t is given by x(t) then its motion is defined using the following position vuctor
(3.35) where t‘ i s a past time between --3o and t, hcnce the above vector gives the position of the fluid particle at historical time t’. The deformation gradient for this particle can now be defined as (3.36) The right Cauchy-Green strain tensor corresponding to this deformation gradient is thus expressed as (3.37) The inverse of the Cauchy-Green tensor, C, I , is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress zv for a fluid particle as a time iiitegral of the deformation history, i.e. r
m(t - t’)S,(t’) dt’
(3.38)
-M
where the deformation-depeiident tensor S, and its coefficient (the memory function) are generally defined by Equations (1.28) and (1.29), as described in Chapter 1. To develop an Eulerian solution scheme for the integral models, the deformation gradient defined by Equation (3.36), should be found explicitly in terms of the velocity componeiits. To obtain such a relatioiiship the following time derivative of the deformation gradient, describing the kinematics of the fluid particle
FINlTE ELEMENT MODELLING OE POLYMERIC FLOW PROCESSES
(3.39) is i~tegratedbackward in time along the fluid particle path using the present state of the deformation as the initial condition Although a general solution for E q ~ a t ~ o(3.39) Ii is not available for certain classes of flow regimes it can be integrated to obtain the required explicit re~ationshipbetween the Eulerian velocity field and the strain history. As an example, we consider thc solution d~velopedby Adachi (1983) for the class of two-dii~~ensio~iaI steady-state Aow s y s t e ~ whose s streamlines can be represented by single-valued functions of only one of the spatial coordinates. Streamlines in a flow field represent the contour lines of the stream function .J/ which is defined by the following relations~ips
(3.40)
where x and y are the spatial coordinates in a Cartesiari coordinate system. now define a Protean Coordinate system as (Adachi, 1983) (3.41) In this system the coordinate x2 along a streamline is constant. Therefore tra~kingof a fluid particle in this system is significantly simplified. This allows a closed-form solution of the basic kinematic equation (3.39) in the defined rotean coordinates. Using the coordinate transformation relationships given by Equation (3.41) the closed-form solution of the kineinatical equation can be transformed into the Cartesian system in which the Eulerian velocity field i s . For example the component of the Finger strain tensor is
where I is an integral evaluated along the streamline as f
(3.43) 1’
where thc particle travel time along the streamline and the coordinate x are related on a one-to-one basis and
MODELLING OF VISCOELASTIC FLOW
89
(3.44) Therefore the Eulerian description of the Finger strain tensor, given in terrns of the psesent and past position vectors x and x' of the fluid particle as C (:; can now be expressed as
(3.45) The integral 1 corresponding to Equation (3.45) is defined as Y
(3.46) which is evaluated along the streamline passing througb the current position. The Eulerian form of the generic single-integral constitutive Equation (3.38) can now be defined as
(3.47) Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evaluated upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (i987) to develop a finite element scheme for viscoelastic flow modelling. As nientioned earlier, the Eulerian solution of the single-integral viscoelastic constitutive equations can only be obtained for special classes of flow regimes and under general conditions a Lagrangian framework should be used. The T,agrsngian systems developed for this purpose are similar to those used to track moving boundaries or free surfaces in a flow field. The development of a Lagrangian system for tracking fluid particle trajectories is discussed in a separate section later in this chapter.
The theoretical description of a non-isothermal viscoelastic flow presents a conceptual difficulty. 'Ta give a brief explanation of this problem we note that in a non-isothermal flow field the evolution of stresses will be affected by the
FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES
temperature distribution. Let us now consider a viscoelastic fluid undergoing stress relaxation. The rate of stress relaxation is determined internally within the fluid, according to an internal timescale or ‘clock’. From an independent observer’s point of view, as the temperature rises the molecular motion within the fluid in a unit of time increases. Thus the internal timescale of the fluid has become shorter to allow a faster rate of relaxation, To evaluate the temperature dependency of material parameters a nicthod called ‘time-temperature shifting’ is used that takes into account the variations of the internal timescale of the fluid. Through the application of this technique the fluid properties at a ternperature T are found on the basis of a master curve given at a reference temperature of To. owever, in a flow regime a fluid particle experiences various states of temperature and hence for the iiicoiporation of time-temperature shifting into a material property model a systematic approach is needed. Such an approach has been developed for specific situations. For example, for linear viscoelastic bouiidary value problems a method called the Morland-Lee hypothesis is used wanner, 2000). According to this hypothesis, a pseudo-time based on the time measured by the internal clock of the fluid is defined. The relationship between the pseudo-time given by and the observer’s time t is expressed as
(3.48) where a17x( T ) is the time-shifting factor. Integration of Equation (3.48) gives t
E - \~;‘[T(t’)]dt’
(3.49)
0
which defines the relationship between the particle’s time and the t h e of the observation. The pseudo-time concept is utilized to obtain the n o n ~ i s o t h e ~ a l forms o f the viscoelastic constitutive equations. This kind of variable transformation is more convenient using the integral constitutive equations, however, differciitial non-isothermal models have also been developed.
The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers; the Peclet number is defined as pcUllk which rcpresents the ratio of convective to conductive energy transport). As emphasized before. numerical simulation of convection-domiiia.ed transport phenomena by the standard Galerkin method in a fixed (i.e. Euleriartj framework gives unstable and oscillatory results and cannot be used.
SOLLJTION OF THE ENERGY EQUATION
Streamline upwinding is the most commonly used technique in the development of stable finite element schemes for the solution of the energy eq polymer flow models. In particular, consistent ‘Streamline upwind Calerkin’ schemes (see Chapter 2, Section 2.3) have gained widespre cations in many practical problems (Choreishy and Nassehi, 199’7). It is also possible to use a moving (i.e. Lagrangian) framework to eliminate the convection term from the governing energy equation and gener stable solutions for heat transport in polymer processes without upwinding. wever, application of Lagrangian schemes in practical problems should be based on algorithms such as adaptive regeneration of the finite element mesh (Morton, 1996) to prevent unacceptable element distortions and in general is more difficult than the use of upwindiiig. Derivation of the working equations of upwinded schemes for heat transport in a, polymeric flow is similar to the previously described weighted residual Petrov-Galerkin finite element method. In this section a basic outline of this derivation is given using a steady-state heat balance equation as an example. ssuming constant physical coefficients for simplicity, the steady-state energy equation is expressed as
(3.50) where summation conveiition over the repeated index i is used. In Equation (3.50), p, c and k are constant density, specific heat and conductivity, respectively, v, represents the components of the velocity field, Tis temperature, 4 = qY2 is viscous heat dissipation rate and 6,is the Kronecker delta. Starting with the usual discrelization step the field unknowns in Equation (3.50) are replaced by their a p p ~ o x ~ ~ interpolated ate r e p r e s e ~ ~ a t i(e.g. o~~ 7; = NITIj over elemental domain (a,).The formulated residual statement is weighted and integrated to give
where WJ and WfJ are weight functions. After the application of Green’s theorem to its second-term Equation (3.5 1) yields
where reis the elemental domain boundary and ni is the component of oatward unit vector normal to the boundary line. The weight functions in the inconsistent strearnlinc upwind method are given as
FINI'PE ELEMENT MODELLlNG OF POLYMERIC FLOW PROCESSES
(3.53) and WJJ= NJ
(3.54)
where NJ ( J = I, . . . ,I)) represents elemental shape functions. In the consistent streamline upwind etrov-Galerkin (SUPG) scheme all of terms in Equation (3.52) are weighted using the function defined by ation (3.53) and hence dl/l,l = WJ. is evident that in the consistent ~~pwinding method the weight function ied to tlie second-order (conduction) term in the energy equation involves der~~atives of tlie shape functions. Therefore in this case the weight fun~tion correspo~idingto the conduction term in the weak form of the discretized energy eq~iationincludes second-order derivatives o f shape functions. C o n s e ~ ~ ~ e ~ini t l y , the ~ o n s i s ~ ~ ~ i t schemes the ~ ~ of the shape ~ frrnctions ~ at ea ~ ~ mc ~ n ~n~egration po o d d be calculated. This i s not a trivial calculation and, in partic~~lar, in discretizations t t use i s o ~ a r a ~ e tniapping r~c requires consi able ~ l ~ e b r anm i c nipul at ions tera et al., 1993). In m ~ ~ t i - d i ~ ~ i e n sprobl i o n a ~ the ~ ~ ~ w i nparameter, di~g a (Equation (3.53)), fiiicd through a rigorous r e ~ a t i Q n s hUsing ~ ~ . h ~ ~ r i se~xi tce n ~ i of ~~s ensional optimal streamline u p w i ~ ~ ~~nogr r n ~al ~number ~, of n7ulti-dixilensior?a1domains have been derived ( ~ i e n k ~ and ~~cz r, 1994). All of these relationships inevitably involve c ~ e f ~ c ~that e n tdo ~ r physical defi~iitions.Jn practice, therefore, the selection of the ameter for a multi-dimensional heat transport problem in should be based 011 ~ ~ m e r i expe~niei~ts ca~ involving trial and error p~oce~~iires. It is reconirnended to niaintain the level of up wind in^ just above the threshold o f instabi ~ ~ t e n sofi othe ~ strea~ljne alerkin method to ~ r a n s i e heat ~ ~ t transo ~ l e by ~ sa space-time least-squares ~rocedu nen (1984). The close relationship between SU nite e ~ e ~ e discreti~atioI~s nt is discussed in C ~ ~ p4.t An ~ rawa ~ p w ~ nscliienic, ~ i n ~based on the previously described B time-ste can also be developed (~ienkiewic~ and Taylor, 1994). ral, thermal properties of polymers are t ~ ~ i ~ e r ~ t u r e e sigiiificant~~ during a flow process. Therefore in or the accura~yo f the finite elemeiit solution of the energy e necessary to include a procedure in the scheme that allows and conductivity k , to vary with tein rature. In practical therinal coef~cientsin the energy equ' n are repeated~yU the latest ~ ~ ternperature m field, ~ andu the solution ~ i~ ~ ~ p ~ of ~these a parameters ~ i ~ by~ a simple itera tive process i n ~ ~ o ~ v ienvga l ~ a t ~ oof n elemental stiffliess
~ O U N ~ A RCONDITIONS Y IN POLYMER PROCESSING ~
~
~
solution of the global system is repeated in each cycle is not compLit~tional~y efficient. Tn addition to the large number of calculations involved in such a cycle, tlie rate of convergence of the iterations caii also be very low. To achieve computing efficiency the required iteration loop should hence be modified. The product ap~roximatiorimethod developed by Christie et al. (1 98 1) provides an for the reduction of computational time in the finitc element solution of non-linear problems. Using this method, thermal coefficients in the energy equation are discretized in conjunction with temperature. Tliercfore in tlie global set they appear as nodal values inultiplied by a constant and can be easily updated according to the correspoiidiii~ nodal temperatures. techniques have also been developed to avoid repeated evaluatiori and ass of the right-hand si in the solution of the energy equation (e. and. ~delsohn,1986 our-Clmid, 1987).
~ d e ~ ~ i ~and c ~prescription ~ ~ i o n of appropriate boundary conditions is a crucial step in the deve~opmeiitof finite element models for engineering problems. nLi~erica1s ~ m u l a t i oor~ polymer Bow processes, these conditions niay i n ~ l u ~ ~ e velocity, stress (surface force coniponents), temperature and a datum for pressure. It is clear that in the case of viscoelastic models, where the flow inside a in is iiiflue~ccdby thc history of fluid deformation before it enters th an inlet, all ambiguity about the boundary coiidilioiis cannot be resolved. Therefore, iii practical (e~i~inee~ing) simulations of viscoelastic flow a set of c o n ~ ~ t i o nthat s can be shown to make the best possible physical sense, under the given conditions, is usually used. Complete mathematical evaluations of such boundary condition^ are, in general, not possible. ~ e i i e r a l i ~ eNewd tonian behaviour i s ortea coiisidered as a useful limiting case that indication for the general validity of tlie iniposed boundary conditions in polymer processing models. In this section comnion types of boundary conditions used in these models are described.
ts
Consider the weiglited residual statement of the equation of motion in a steady tokes flow model, expressed as (3.55)
~
94
FINITE ELEMENT MODELLING OF II'OLYMERlC FLOW PROCESSES
where NJ is a weight fitnctioii, 5 is the Cauchy stress (the over bar represents approximation within SZ) and pg is the body force. Application of Green's theorem to Equation (3.55) gives
s
J'
VNj * iT dS2 - A'jiT. /a dS2 - Njpg dSZ = 0
r
(3.56)
Q
where r is the boundary surrounding f2 and is the unit vector normal to r in the outward direction. Substituting for tr in the first term of Equation (3.56) via Equation (1.3) gives
r
52
G?
where z is the extra stress, p is pressure and roiieclter delta. In a gene~dli~ed Newtonian flow the extra stress is given in terms of the rate of deformation tensor as z = v(Vv -t. VvT), where rl is fluid viscosity. Therefore the weighted residual statement of the described flow model consistiiig of the equations of continuity and motion is written as
where ML is a weight function. Equation (3.58) provides the basis for the ~ o r m ~ ~ a of t ~ elemeiital on equations in the previously described finite element schemes. Imposition of boundary coiiditions in incompressible flow models is now described using this equation without any significant loss of generality. ethods that may be used to prescribe the, tally unknown, boundary conditions at the entry to a domain in viscoelastic models are also briefly discussed. Consider the two-dimensional domain shown in Figure 3.3 where r = rl CJ rzU r3U r4. 1;
(Solid wall)
(Exit) 1;
l-4
...............(............... *
~
*
"*......." .........
..............I..
"I............
(Line of symmetry) I'2
e 3.3 Boundary lines in a flow domain
BOUNDARY CONDITIONS IN POLYMER PROCESSING MODELS
9
Typically velocity components along the inlet are given as essential (also called irich1et)-type boundary conditions. For example, for a flow entering the domain shown in Figure 3.3 they can be given as
{
lil
=0
vx = % ( Y )
on I ' l . To impose this type of boundary condition the specified values arc inserted into the equations where they appear and rows corresponding to the given degrees of freedom are eliminated from the global set of finite element equintions. In general it is beneficial to use finer niesh divisions near a solid wall. In viscoelastic models in addition to the described conditions, stresses at the inlet should be given. As already mentioned there is no universal method to define such conditions, however, the following options may be considered (Tanner, 2000): Adding tirne-dependent terms to the equations the simulation is treated as an initial value problem in which at a given reference time all stresses are zero; the steady-state solution can be found iteratively. Adding sections before the actual domain inlet arbitrary condi~ion~ are imposed at the artificial starting point and then iterated to obtain required stresses. Using a known solution at the inlet. To provide an example for this option, let us consider the h i t e element scheme described in Section 2.1. Assunling a 'fully developed' flow at the inlet to the domain shown in Figure 3.3, v y , (Dvy/3y) = 0 and by the iiicompressibility condition (Avx/8.x) = 0, .x derivatives of all stress components are also zero. Therefore at the inlet the c o ~ ~ o ~ iof~ the n t equation s of motion (3.25) are reduccd to
c p is-%, - as,y,, dx ay2 ay ap as,i, -ay - -&-
-
i
~
_
_
_
_
.
hencc the constitutive equation (3.24) is reduced to
s,, = 0
96
FINITE ELEMENT MODELLING OF POLYMEKIG FLOW PROCESSES
Also Equation (3.23) yields R,, = 0 Kyy = 0
3V x: R,, = (1 - r ) y
CfJ’
at the inlet. Therefore S., = 0 and S,, = 2V3(1 r) (3vY/3y)*,where the I cornpoiient of velocity is usually giveii as a parabolic function in terms of y .
Line of synirnetry The normal component of velocity and tangential component of surface force are set to zero along a line of symmetry. For the domain shown in these are expressed as V],
=0
x=
0
on Tz. Impositioii of the first condition is identical to the procedure used for pre~cribiii~ inlet velocity components and the second condition is simply satisfied by setting the boundary line integral in the discretized equation of dT in Equation (3.58) to zero. Note that in this example it vector iiormal to the line of s y i n ~ e t r yare (0,- 1) and containing a, in the boundary line integral is multiplied by zero and elirniiiated. The remaining term, i.e. the shearing force component, is -crAY = - T , ~ = -y[(3~,/8~y) + (3vY/3x)]which is reduced to -rl(avY/8y) along erefore irnpositioii of ,= 0 is equivalent to prescribing a natural (von am)-type boundary condition for the axial component of the velocity. n general, lines of symmetry withiii a flow field may not be parallel to the coordinate axes. oundary conditions along these lines will still be physica~~y similar to the conditions used in the simple example given here, but their in~~osition will not be trivial. In this case bounda~yinteg~alsco~respondi~ig to these lines should be evaluated and additional algebraic relationsl~ips repr e s ~ n tzero i ~ ~normal velocity conditions (i.e. v.m = 0 across a h e of s ~ ~ e t r y ) should be incorporated into the global set of equations.
no-slip walls zero velocity conipone~tscan be resdily imposed as the uired boundary conditions (v., = ijY = 0 QII I‘3 in the domain shown in Figure etails of the imposition of slip-wall boundary conditions are explained later in Section 4.2.
BOUXDARY CONDITIONS IN POLYMER PROCESSING MODELS
97
Exit conditions Typically the exit velocity in a flow domain i s unknown and hence the prescripirichlet-type bwndaty conditions at the outlet is not possible. However, a t the outlet of su€ficiently long domains fully developed flow conditions may he imposed. In the example considered here these citn be written as vv = 0 =0
position of the first cond~tioiiis as discussed before and the second coiid~tion the boundary line integral (i.e. Jr NJ . i s again satisfied by set note that along the exit line component Equation (3.58) to zero.
(1,O) and hence in this case the normal component of the surface force, expressed as n,, = -p -+ 277(3v,/dx) will be set to zero. In addition, along the exit line (&,,/ay) = 0, which using the incompressible continuity equatioii gives rcforc setting CT,=,0 implies a pressure ~ a t ofu zero ~ ~at ~the is also shows that if the normal component of surface force is not given at any part of the domain boundary then zero pressure at a si node may be imposed as a datum. The setting of pressure b o u n ~ condit ~~y is, in general, ~ n c o n s i s ~ with ~ n t tlie incompressibility constraint and should be avoided. In the specific case wliere the prcssure in a single node is prescribe^, the continuity equation re~atingto that node should be removed from tlie set of discretized equations to avoid algebraic inconsistency. The validity of imposing fully developed exit boundary conditions in a flow model depends on factors such as the type of elements used, number of element layers between the inlet 1, idet and wall boundary conditions and generally on fluid viscosity. riate iniposition of developed flow conditions at a domain outlet reduces the accuracy of the solution ancl may give rise to spurious oscil~at~ons (Gresho et al., 1980). In the flow domains that are not considered to be long c i i o u ~to~ iiiipose developed flow conditions, stress-free condi~ionsat the doniaiii outlet may be used. In this case, both shear and nornial coinpon the surface forces at the exit are set to Tero, This is again satisfied by sc the b o ~ n d a ~int~g~A1 y along the exit line to zero. To avoid imposition of unrealistic exit boundary conditions in flow models Taylor et al. (1985) developcd a method called ‘traction bound~~ry conditions’. 111 this ~ e t ~ starting o d from an initial guess, outflow c o i i ~ ~ t i oi snupda iterative procedure which eiisures its consistency with the flow regime ately upstream. This method is successf~illyapplied to solve a nunibe~of turbulent flow problems. ~ a p ~ ~ a scft a~1~s (I~992) ~ ~ suggested that in order to generate realistic vier-Stokes equations the exit conditions should onditions shorrlcl be imposed). In this approach reen’s theorem to the ecluativns corresponding to thc exit boundary nodes is av~ided.This is equiva~eii~ to ~ ~ ~ o‘no s ~ exiti conditions’ ~ g if ~ ~ e ~ wit^ ~ e n ~
98
FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES
shape futictioiis of the order at least equal to that of the differential equation are . In discretizations usiiig lower-order elements a boundary condition based on setting highest order differentials to zero is imposed (Sani and Gresho, 1994). Renardy (199’7) has shown that the ’free’ or ’no outlet’ method provides a welldefined problem in the context of discretized governing equations.
Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This implies that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses tlie critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier’s slip condition, which is a relationship between the tangentid component of the momentum flux at the wall and tlie local slip velocity (Silliman and Scriven, n a two-dimensional domain this relationship is expressed as
+
[,8r.n (v - Vb)].tT = 0
(3.59)
are unit vectors tangent and normal to the boundary, extra stress tensor, /? is a slip coefficient, v is Lbe fluid velocity vector and vh is the velocity of the solid wall. Equation (3.59) together with the following equation which represents no flow through a solid wall, are used to impose slipwall boundary conditions. v.12
(3.60)
=0
Consider a solid wall section as i s shown in Figure 3.4. The follow in^ relationships between the components of unit outward normal and tangential vectors are true at all points ty = +nx
t , -- -?Z,
(3.61)
Equations (3.59) and (3.60) are recast in terns of their components and solved together. After algebraic manipulations and making use of relations (3.61) slipwall velocity components are found as (3.62)
(3.63)
BOUNDARY CONDITIONS IN POLYMER PROCESSING MODELS
Figure 3.4
99
Slip at a solid wall
The slip coefficient r’J’ is defined as
(3.64) where PO i s the initial slip coefficient and 1 is a characteristic flow domain dimension. The limit of /3 + 0 corresponds to no-slip (U = U!,, U = uh) and the limit of ,8-i 00 gives the perfect slip condition. In general, the coefficient /3 depends on the invariants of the stress tensor and surface rouglmess. Navier’s slip condition can be discretized in a manner similar to the main flow equations and directly incorporated into the finite element working equations (Ghoreishy and Nassehi, 1997) to impose a slip-wall condition.
3.4. Normally, in the finite element solution of the energy equation, t e ~ ~ ~ a t uatr e s the inlet and solid walls of a flow domain and zero thermal stresses along the Pines of symmetry and exit are specified as the boundary conditions. Imposition of these conditions is s t r a i g h t f o ~ a rand ~ ~ is carried out according to the procedures explained for the prescription of similar types of boundary conditions in the solution of flow equations. In problems where the temperature at a downstream boundary is specified, i.e. a ‘hard’ exit boundary condition i s to be imposed, preferential mesh refinement near the outlet is often necessary to maintain accuracy. This typc of ‘hardness’ may also be eased if the ~ r o b isl ~ ~ recast in a different frame that provides a simpler formulation for the model equations. For example, a cylindrical (r,O) coordinate system provides a more natural framework than a Cartesian coordinate systeni to model the heat transfer associated with a tangential viscous flow in an annulus. Easing of the boundary conditions by using more appropriate frameworks can lead to more accurate solutions or betta computing economy.
FINITE ELEMENT MODELLING OF PO1,YMERIC FLOW PROCESSES
In the finite element solution of the energy equation it is sometimes necessary to impose heat transfer across a section of the domain wall as a boundary coiiditioii in the process model. This type of convection ( obins) boundary condition is given as
where q h is the boun~aryheat flux, h is the heat traiiskr coefficient, T i s the boirndary surface temperature and 7;, is the outside temperature. In the discretized energy equation (e.g. see Equation (3.52)) boundary heat flux is represented by the boundary line integrals. Therefore at the bo~iid~iries repr~senti~ig heat transfer surfiices these integrals do not vanish and should be evaliiated using ~ q u a ~ i o(3.65). n In most instances it is simpler to avoid the evaluation of these integrals and impose the required condition by a technique based on the use of ‘virtual’ elements. In this technique a thin layer of elements is attached to the outside of heat transfer boundaries. These elements are not part of the p l i y s i ~ ~ ~ d~~~~~~ and hence referred to as the virtual elements. eat flux t h r o u ~ hthese elements is assumed to be in the direction nornid to the surface (see Figure 3.5). ~ e i ~ p e r a t uvariation re across the virtual e l e ~ e n t is s also a s ~ u ~ to e dc o ~ e s ~ o i i ~ to an analytic stea~~y-state profile. These assuT~ptio~s depend on using very thin ~ ~ e i ~and e ~setting ~ t s of the virtual spwific heat to zero (Pittman, 1989). Hear ~ o i ~ ~ L ~t ~h t ri o~ nthe u ~layer ~ of virtual e l e ~ e n t scan now be ~ e ~to~the~ rcal t ~ d surface heat flux as Layer of virtual elenients (tl1iCkneSS = d,)
I
Physical&
~ ~ 3.5 ~ MVirtual ~ e element layer for the imposition of boundary heat flux
(3.46) where k, and d, are the conductivity and thickness of the virtual elements, respectively. The virtual conductivity is hence expressed as
surfiice the virtual conductivity is found a.nalogously as
FREE SURFACE AWL) MOVPNC BOUNDARY PROBLEMS
1
(3.68) are the local radius of curvature of the boundary surfae and ius of the boundary and virtual surfaces, respectively. In practice, therefore, the outside temperature is set as an essential condition at the virtual element boundaries and the evaluation of line integrals at the real boundary is avoided. scrip~ioiisgiven in Section of this chapter about the imposition of ary conditions are mainly i context of finite eleinent models that use CO elements. In models that use ite elements derivatives of field variable should also be included in the set of required boundary conditions. Tn these oblems it is necessary to ensure tliat appropriate ‘normality’ and ‘tangenns along the boundaries of the domain are satisfied (Petera and
In these problems, Bow geometry is riot known a priori and some sections of the domain boundary may change with flow. This situation arises in a variety of olyrner proccsses such as injection moulding and mixing in ambers. Free surface Bow regimes are also encountered in extr coating opei-ations where die swell i s a coininon phenomenon. Various techniques for the modelling of free boundary flow regimes have been ~ e v e l o ~ e the last two decades. Some of these methods are process specific, or they were developed in conjunction with particular numerical sclienies and cannot be ed as general simulation tools. delling of steady-state free surface flow corresponds to the solution of a boundary value probleni while moving boundary tracking is, in general, viewed as an initial value problem. Therefore, classification of existing methods on the basis of their suitability for boundary value or initial value problems has also been advocated. The general class of free boundary flow problems can, however, be modelled using thc volume of tluid (VOF) approach (Nichols et al., 1980). concept in this technique is to solve, simultaneously with the governing flow equations, an additional equation that represents the unknown boundary. Three different versions of this method are described in the following sections.
ethod in ‘Eulera’an’fr
In a fixed two-dimensional Cartesian coordinate system, the continuity equation €or a free boundary is expressed as
FINITE ELEMENT MODELLING OF POLYMERIC FLOW ~
R
~
~
~
(3.69) where vx and v,. are the flow velocity components and F is defined as a material density function. This function has a value of unity in filled sections of the flow domain and is zero outside of the free boundary. At the free boundary itself this function has a value between 0 and 1, (usually 0.4-0.5; Petera and 1996). The hyperbolic partial differential Equation (3.69) describes convection of a free boundary in a flow field. In iiiodels based on stationary (Eulerian) frameworks the solution of this equation usually requires stre~nline up~?~nd~n~. ~ t r ~ ~ g h t f o application r~~rd of the VQF method in domains that include irregular and curved boundaries is rather complicated. To resolve this problem a more flexible version of the original eth hod has been developed by T h o ~ p s o n (1986). In this technique the free boundary flow regime is treated as a two-phase system. The phases are assumed to consist of the fluid filled and void regions, respectively. The free boundary is regarded as the interface separating these phases. To model the flow field in this manner, voids are assumed to contain a virtual fluid represented by a set of virtual physical properties. 1 t ~ c h n i ~ uise referred to as the ‘pseudo-density’ method. In practice, physical coefficients in the governing flow equations are expressed as X = = X f F + X , ? ( l 4)
(3.70)
where x is a given physical parameter (e.g. density) and xfand xv are the values of this parameter in the fluid filled and void regions, respectively. The free surface density function is 1 in fluid filled sections and is 0 in the voids. Hence using ~ q u a t i o n(3.70) appropriate coefficients are automatically inserted into the working equations and the solution for the entire domain is readily obtained, the free boundary itself this equation yields the interpolated values of the sical coefficients. The solution scheme starts from a known ~ i s t ~ i ~ u t of ion the surface fundion values. At the end of each time step new values of F are und and the position of the interface between the phases (i.e. the free oundary) is updated. Despite its simplicity the effectiveness of the ‘pseudodensity’ method is restricted by the necessity to use artificial physical pa~ainete~$ in parts of the flow domain.
3.5.
~ - E ~ l e r ~ aframe n’
The geoin~tricalflexibility of the VQ scheme can be significantly i ~ p r o v ife ~ in its for~ulation,instead of using a fixed framework, a ~ o n ~ ~ i n a of t~o an Zagranfiiaii-Eulerian approach is adopted. The most comnon approach to develop such a combined framework is the application of the A r b ~ t r a ~ y
~
FREE SURFACE AND MOVING BOUNDARY PROBLEMS
1
kagrangian-Eulerian (ALE) method, In the ACE technique lhe finite element niesh used in the slanulation is moved, in each tinie step, according to a pred e t e ~ i n e dpattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape andlor position of the elements change from one time step to the next. Tlierefore the solution mesh appears to move with a velocity which is different from the flow velocity. Componeiits of the mesh velocity are time derivatives of nodal coordinate disptacenients expressed in a two-~imensionalCartesian system as
overning flow equations, originally written in an Eulerian framework, should hence be modified to take into account the movement of the mesh. The time derivative o f a variable f in a moving franiework is found as
and E refer to moving and fixed frameworks, respecl~~~ely. Substitution of the Eulerian time derivatives via relationships based on Equation (3.72) gives the modified form of the governing equations required in the ALE scheme. The free boundary equation i s therefore expressed as
di”’
-+ (v, - V m x ) at
3F
i(vJ’ 3x
--
3F ay
VHZY) =‘I
0
(3.73)
The s t r e a ~ ~ upw~nding ne rvelliod is usually employed to obtain the dis~r~tized form of Equation (3.73). The solution algorithm in the ALE technique is similar to the procedure used for a fixed VOF method. In this technique, however, the solution found at the end o f the nth time step, based on mesh number YI,is used as the initial condition in a new mesh (i.e. mesh number n -t- 1). In order to niinimize the error introduced by this approximation the diffcrence between the mesh c o n ~ ~ L ~ r a t iato isuccessive i~ computations should be as small as possible. Therefore the time increment should be small. In general, adaptive or remeshing algorithins are employed to construct the required finite element mesh in successive ste s of an ALE procedure ( onea, 1992). In some instances it i s possible to ge rate the finite elemcnt esh required in each step of the c o ~ p u t a t i o nin advance, and store them in a file accessible to the c o ~ ~ p u t e r program. This can significantly reduce the CPU lime required for the simu~atio~i assehi and. Ghoreishy, 1998). An exaniple in which this approach is used is given in Chapter 5.
1
FINTTE ELEMENT MODELLLNG OF ~ O L Y M ER ICFLOW PROCESSES
.3
F
11 a L ~ g r a n ~ i aframework ti the coordinate system in whicli the governing flow equations are formulated moves with the flow field. Therefore flow equations written in such a system do not include any convection terms. In part~cular,the free surface continuity equation (Equation (3.69)) is reduced to a simple time derivative expressed as (aF13t)= 0. However, the integratioii of this derivative is not trivial because in a free boundary domain it should be evaluated between variable (i.e. time-dependent) limits. In addition, a method should be adopted to prevent computational mesh distortions that will naturally occur in a ~ ~ ~ g ~ a Iramcwork. n ~ i a n Various methods based on adaptive r e - ~ e s l i i i iare ~ used to achieve this objective (e.g. see orton, 1996). However, some of these techn~qLi~s lack geometrical flexibility g. they are only suitable for domains that do not include curved boundaries) or they require high CPU times. A robust and geometrically flexible method with imylenientatioii that requires relatively moderate CPU times is described in this section. This method is based on the tracking of fluid particle trajectories passing through each node in the c ~ m p ~ t ~ t i o idomain ial and provides an effective technique for regeneration o f esh at the end of each time step (Petera and assehi, 1996). The free surface contin~iityequation can be readily integrated using this approach. Consider the position of a material point in a flow field described by the followi~igposition vector
re p i s tlic position at the current time t. t a given reference time, t r , the tick occupies a specific position defined as
(3.75) iven material point can only occupy a single position at a time and hence Equation (3.75) can be used to find the position of the point as (3.76)
'The substitut~onfor p from Equation (3.76) into Equation (3.74) givcs
(3.77) f the reference time t r and the current time t coincide then the reference an current positions will also coincide and the rigl~t-l~and side of ~ ~ u a t j o( n can be replaced by the reference position defined as n in ~ q u a t i o n(3.76). velocity field given as .a = ~ ( x , t)' the motion of a material point can be described as
FREE SURFACE AND MOVING BOUNDARY ~
R
~
~ 5 L
(3.78)
at
The distance covered by a fluid particle in this flow field in a time iiiterval of hi, = - t,, can be fourid by integrating Equation (3.78) as (3.79) 1"
TJsing the mean value theorem for definite integrals (3.80)
where B is a parar~eterbetween zero and one. Using Taylor series ex velocity function in the neighbourhood of the point (x,t,) is given as
The second and third terms on the right-hand side of Equation (3.81) are o ~ i ~using ~ t known ~ d values at a b a c k ~ a r dtime step as
(3.82) (3.83)
fter the substitL~ti~11 from Equations (3.82), (3.83) and (3.84) into (3.81) and in turn s ~ ~ s t i t u tfrom i i ~ ~the resultant relationship into (3.80) and r c a r r ~ n ~ i nthe g following equation describing the trajector particles is found
he solutio~3algorithm based on consists of the lollowing steps:
A domain that can be safely assumed to represent the entire and discretized into a 'fixed' mesh o f finite elements. The uid i s called the 'current' mesh. Nodes within the current mesh
~
FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES
~~~
are regarded as 'active' nodes. At the active nodes located inside the current mesh the surface density function F =, I. This function for the nodes on the free boundary is normally given as 0.5.
step 2 Location arrnys showing the numbers of elements that contain each given node in the fixed mesh and its boundary are prepared and stored in a file.
Step 3 A ~ ~ r o pinitial r i ~ conditions ~ ~ on all nodes are given.
Step 4 Time variablc is increniented.
step 5 overning flo~7equations are solved with respect to the current domain.
Using: coordinates of the nodes in the current domain (i.e. x) velocity field found at step 5 (i.e. U,), and old time step velocity values (i.e.
coordinates o f the feet of the fluid particle trajectories (i.e. nodal points are fouiid via Equation (3.85).
) passing through
Step 7 For each active node in the current mesh the corresponding location array is searched to find inside which element the foot of the trajectory currently passing through that node is located. This search is based on the solution o f the following set of non-linear algebraic equations
FREE SURFACE AND MOVING BOUNDARY PROBLEMS
107
I
The unknowns in this equation are the local coordinates of the foot (i.e. and 7). After insertion of the global coordinates of the foot found at step 6 in the left-hand side, and the global coordinates of the nodal points in a given element in the right-hand side of this equation, it is solved using the Newton-Raphson method. If the foot is actually inside the selected element then for a quadrilateral element its local coordinates must be between -1 and + I (a suitable criteria should be used in other types of elements). If the search is not successful then another element is selected and the procedure is repeated. In the example shown in Figure 3.6, thc trajectory passing through point A at the ‘current time’ is found to originate from the inside of element (e) at the previous time step.
Current time
F i ~ ~ 3.6 r e Determination of fluid particle trajectories
Step 8 After identification of the elements that contain feet of particle trajectories the old time step values of F at the feet are found by interpohting (or extrapolating for boundary nodes) its old time step iiodal values. In the example shown in Figure 3.6 the old time value of I;at the foot ofthe trajectory passing through A is found by interpolating its old nodal values within element (e).
Step 9 Free surface density functions calculated at step 8 are used as the initial conditions to update the current position of the surface using the following integration
FlNlTE ELEMENT MODELT,TNG OF POLY MEKlC FLOW PROCESSES
(3.86) QI
where 9,is a moving solution domain. plsiiig Reynolds transport theorem (Aris, 1989) Equation (3.86) is expressed as
The right-hand side of Equation (3.87) is set to zero considering that FlDt and the divergence of the velocity field in incompressible fluids are all equal to zero. Therefore, after integration Equation (3-87) yields (3.88)
Steps 4 to 9 are repeated until the end of required sii~~ulat~on the. An example describing the application of this algorithm to the finite element modelling of free surlace flow of a Maxwell fluid is given in Chapter S.
Adachi, K., 1983. Calculation of strain histories in Protean coordinate systems. RIieo2. Acta 22, 326-335. Ark, R., 1989. Vectors, Tensors and the Busk Equations of Fluid Mechanics, Dover Publications, New York. abuska, B., 1971. Error bounds for finite element method. Nurner. MPthotl's 1 333. Bell, B.C . and Surana, K. S . , 1994. p-version least squares finite element formulations for two-dimensional, incompressible, non-Newtonian isothermal and non-isothermal fluid flow. h i t . J. Numer. Methods Fluicls 18, 127- 162. rezzi, F., 1974. On the existence, uniqueness and approximatio problems arising with Lagrange multipliers. RAIRO, Serie Rouge Carclona, A. and Idelsohn, S., 1986. Solution of non-linear thermal transient problems by a reduction method. Int. J. Numer. Methods Eng. 23, 1023-1042. Christie, I. er nE., 1981. Product approximation for non-linear problems in the finite element method. I M A J. Numur. Anal. 1, 253-266. Crochet, M. J ., 1982. Numerical simulation of die-entry and die-exit fIow of a viscoelastic fluid. In: - ~ u ~ ~ ~ Method7 e r i c ~ ~inl Forwing Processes, Pineridge Press, Swansea. de Sampaio, P.A. B., 1991. A Petrov-Galerkin formulation for the incompressiblc Navier-Stokes equations using equal order interpolation for velocity and pressure. Int. J. Nutwer. Methods Eng. 31, 1135-1 149. onea, J., 1992. Arbitrary Lagrangian-Eulerian finite element methods. In: Belytschko, T. and Hughes, T. J. R. {eds), Compututional Methods for Transient Analysis, Elsevier Science, Amsterdam.
REFERENCES
10
Ghoreisl-ry, M.H.R. and Nassehi, V., 1997. Modelling the transient flow of' rubber compounds in the dispersive section of' an internal mixer with slip stick boundary conditions. Adv. Poly Tech. 16, 45-68. Gsesho, P. M., Lee, R. L. and Sani, R. L., 1980. On the time-dependent solution of the incompressible Navier Stokcs equations in two and three dimensions. In: Recertf Advances in Nurnericcil Methods in Fluids, Ch. 2, Pineridge Press, Swansea, pp. 27-75. Hughes, T ,J.R.,1987. Recent progress in the development and understanding of SUPG methods. int. J. Nmvwr. Methods Eng. 7,1261- 1275. Hughes, T. J. R., Franca, L. P. and Balestra, M.,1986. A new finite-element formulation for computational fluid dynamics. 5. Circumventing the Babuska-Brezzi condition a stable Petrov-Galerkin formulation of the Stokes problem accommodating eqiial order interpolations. Cornput. Methods Appl. Mech. Eng. 59, 85-99. Hughes, T. J . R., Mallei, M. and Mizulwni, A., 1986. A new finite element formulatioii for computational fluid dynamics: 11. Beyond SUPG. Comput. Methods Appl. Mech.
-
Simulation of' viscoelastic fluid flow. Tn: Tucker, C. L. I11 (ed.), Comyuler Modeling Jor Polymer Processing, Chapter 9, l1anser Publishers, Munich, pp. 403-469. Kheshgi, H. S. and Scriven, L. E., 1985. Varmble penalty method for finite element analysis of incompressible flow. Int. J. Nwner. Methodr Eluidr 5, 785-803. Lee, R.L., Gresho, P.M. and Sani, R.L., 1979. Smootliing techniques for certain primitive variable solutions of the Navier-Stokes equations. Int. J. hTzinir7r.Methods
Luo, X. L. and Tanner, K.I., 1989. A decoupled finite element streamline-upwind scheme for viscoelastic flow problems. f. Non-hlewtonmm Fluid Mech. 31, 143-162. Marchal, J.M. and Crochet, M.J., 1987. A new mixed finite element for calculating viscoelastic flow. J. Non-Newtonian E k i d Mech. 20, 77-1 14. Morton, K. W., 1996. Numerical Solution of Convection DifJusion Problep~~. Chapmall & Iiall, London, Nakamwa, S., Pittman, J . F. T. and Zienkiewicz, 0.C., 1982. Numerical solution of flow and heat transfer in polymer melts. In: Gallagher, R. H. et al. (eds), Finite Elements in Huids, Vol. 4, Ch. 13, Wiley, Chichester, pp. 251-283. Nguen, N. and Reynen, J., 1984. A space-time least-squares finite element scheme for advection-dif'fusioii equations. Lhwiput. Methods Appl. Mech. Eng. 42, 33 1 342. Nichols, B.D., Hirt, C. W. and Hitchkiss, R. S . , 1980. SOLA-VOF: a solution algorithm for transient fluid flow with multiple free surface boundaries. Los Alamos Scientific Laboratories Report No. La-83S5, Los Alamos, NM. Nour-Omid, B., 1987. Lanczos method for heat conduction analysis. Xnt. J. Numer. Methods Eng. 24, 251-262. Papanastasiou, T. C., Malarnataris, N. and Ellwood, lC.,1992. A new outflow boundary condition. hit. J. Nunrer. Methods Fluids 14, 587 608. Papanastasiou, T. C., Scriven, 1,. E. and Macoski, G. W., 1987. A finite element method for liquid with memory. J. Non-Newtonimi Fluid Mech 22, 271-288. Petera, f . and Wassehi, V., 1996. Finite element modelling of free surface viscoelastic flows with particular application to rubber mixing. Int. J. Nunzer. Methods Fluidv 23, 1 117- 1 132. Petera, J., Nessehi, V. and Pittman, J. F.T., 1993. Petrov-Galerkin methods on
110
FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES
isoparanietric bilinear and biquadratic elements tested for a scalar convection-. diffusioii problem. Int. J. Nzwner. Methollj. Heat Fluid Flow 3, 205-222. Petera, J. and Pittman, J.FT., 1994. Isoparametric Hermite elements. Int. J. Numer. Methods Eng. 37, 3489-3519. Pittman, J. F. T., 1989. Finite elements for field problems. In: Tucker, C. L. III (ea.), Computer Modeiing ,for Polymer Proces,ving, Chapter 6, Hanser Publishers, Munich, pp. 237- 331. Pittman, J. F. T. and Nakazawa, S.,1984. Finite element analysis of polymer processing operations. In: Pittman, J. F.T., Zienkiewicz, 0.C., Wood, R. D. and Alexander, J. M. (eds), Numerical Analysis qf Forming Processes, Chapter 6, Wiley, Chichester, pp. 165-218. Reddy, J. N., 1986. Applied Functional Analysis nnd Voriutionul Methods in Engineering, cGraw-Hill, New York. Kenardy, M., 1997. Imposing ‘no’ boundary condition at outflow: why does it work? 1nf. J. Numer. Methods Fluids 24, 413-41 7. Sani, R. L. and Grcsho, P.M., 1994. Resume and remarks on the Open Coridition Mini-symposium. Int. J. Numer. Fluid7 18, 983-1008. Silliman, W. J. and Scriven, L. E., 1980. Separating flow I a static contact line: slip at a wall and shape of a free surface. .II Cornput. PIiys. Swarbrick, S. J. and Nassehi. V., 1992a. A new decoupled finite element algorithm for Part 1: numerical algorithm and sample results. Int. J. A‘unzer. Swarbrick, S. J. and Nassehi, V., 1992b. A new decoupled finite element algorithm for art 2: convergence propcrtics of thc algorithm. Int. J. Mumer. heology, 2nd edn, Oxford University Press, Oxford. Taylor, C . , Rance, J. and Medwell, J.O., 1985. A note on the imposition of traction boundary conditions when using FEM for solving incompressible Bow problcms. Cumuuzun. Appl. Numcr. Metfrods 1, 11 3- I 2 I . Thompson. E., 1986. Usc of pseudo-concentration to follow creeping flows during transient analysis. Inl. J . Nzitner. A4ethod.s Fluids 6 , 749 -761. Zienkiewicz, 0.C. and Taylor, R. L., 1994. The Finite Element Method, 4th edn, Vols 1 and 2, McGraw-Hill, London. Zienkiewicz, 0,C. et al., 1986. The patch test for mixed formulations. h t . J. Mumer. Merhods &g. 23, 1873-1883. Zienkiewicz, 0.C. et al., 1985. Iterative method for constrained and mixed approxi~ o d s expensive improvement to f . e m performance. Compuf. ~ ~ t ~ Appl. Zienkiewicz, 0.C . and Wn, J., 1991. Incompressibility without tears - how to avoid restrictions on mixed formulation. lnt. 9; Numer. Methods Eng. 32, 1189-1203.
In this chapter derivation of the working equations o f various finite element schemes, described in Chapter 3, are explained. These schemes provide the basis for construction of finite element algorithms for simulation of polymeri~flow regimes. The working equations of the most commonly used schemes are given in Cartesian, polar and axisymmetric coordinate systems to simplify their incorporation into the finite element program listed in Chapter 7. An example d e ~ o n s t r a t i ~the g utilization of these equations for modification (or further development) of this program is also included in Chapter 7.
4.1
ES
FA
The majority of polymer flow processes are characterized as low nuniber Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison ihhe body forces acting upon them are small and can be safely ignored. In this section the governing Stokes flow equations in Cartesian, polar and axisymrnetric coordinate systems are presented. The equations given in twodimensio~alCartesian coordinate systems are used to outline the derivation of the elemental stiffness equations (i.e. the working equations) of various finite element schemes.
nsional Cartesian
COW
Tn the absencc of body force the equations of continuity and motion representing Stokes flow in a two-dimensional Cartesian system are written, on the basis of Eqwations (1.1) and (1.4), as
I I2
WORKING EQLJATIONS OF THE FlNlTE ELEMENT SCHEMES
au av +-= 0 @X
aJT
(continuity)
where U and v are the x and y components of velocity, respectively, and
(motion)
where p is pressure and r,, etc. are the components of the extra stress tensor. For a generalized Wewtoiiian fluid using the constitutive equation (1.8)
(4.3)
where r) is fluid viscosity. Substitution from Equation (4.3) into Equalion (4.2) gives
Equations (4.1) and (4.4) are the governing flow equations.
Similarly in the absence of body forces the Stokes flow equations for a generalized Newtonian fluid in a two-dimensional (Y, t?) coordinate system are written as
STOKES FLOW OF A GENERALIZED N E ~ r O N I AFLUlD ~
113
and
T n an axisymmetric flow regime all o f the field variables remail? constant in the c i r c ~ ~ f e r e n t idirection al around an axis of symmetry. Therefore the gov~r~iiiig flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation €or an incoinpressible fluid written in a cylindrical (r, 8, z ) coordinate system as
13 dr
--(rV,.j I
13 r 8B
+--(v~)
3
+--(v~) = O
02
(4.7)
and
In an axisymmetric flow regime there will be no variation in the circumferential (i.e. 19)direction and the second term of'the integrand in Equation (4.8) can be eliminated. After integration with respect to B between the limits of 0 -27r Equation (4.8) yields
WORKING EQUATTONS OF THE FINITE ELEMENT SCHEMES
Therefore the continuity equation for an incompressible axisyininetiic flow is written as
(4.10) Similarly the components of the equation of motion for an axisymmetric Stokes flow of a generalized Newtoiiian fluid are written as
(4.11)
Working equations of the U-V-P
erne in Gartesiian COQT
Following the procedure described in Chapter 3 , Section 1 .I the Galerkinweighted residual statements corresponding to Equations (4.4) and (4.1) are written as
(4.12)
In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. N j j = 1,n and Ml I = 1,m where, in general, men). The weight function used in the continuity equation is selected as --Ml to retain the symmetry of the djscretized equations. After application of Green's theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the forinulation) and algebraic manipulations the working equations of the XJ-V-H4 scheme are obtained as
STOKES FLOW OF A GENERALIZED NEWTONIAN FLUID
115
(4.13)
where (4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
AF=O
(4.22)
(4.23)
(4.24)
l3; = 0
(4.25)
PP6
WORKING EQUATIONS OF THE FINITE ELbMENT SCHEMES
where reis the element boundary and rz, and q,are the components of unit outward vector normal to re.A superscript e is used to indicate eleinental M p l ) throughout this clvapter. discretization (e.g. pe =
orking equations of the UUsing a procedure similar to the derivation of Equation (4.13) the working equations of the U-V-P scheme for steady-state Stokes flow in a polar (Y,6) coordinate system are obtained on the basis of Equations (4.5) and (4.6) as
(4.26)
where (4.27)
(4.28)
(4.29)
(4.30)
(4.32)
(4.33)
(4.34)
A33 - 0
I1 -
(4.35)
S'TOKbS FLOW OF A GENERALIZED NEWTONIAN FLUID
117
(4.36)
B;
=;
0
(4.38)
xisy Using a procedure similar to the formulation of two-dimensional forms the working equations of the U-V P scheme in axisymmetric coordinate systems are derived on the basis of Equations (4.10) and (4.11) as
(4.39)
where (4.40)
(4.41)
(4.42)
(4.43)
(4.44)
(4.45)
WORKING EQUATIONS OF 'THE FINITE ELEMENT SCHEMES
(4.48)
(4.49)
(4.50)
(4.51)
.1.7
In the continuous penalty method prior to the discretization or the flow equations the pressure term in the equation of niotion is subs~itutedby the penalty relationship, given as Equation (3.6). Therefore using Equations (4.4) and (4.1j, we have
(4.52)
Following the procedure described in the continuous penalty technique subsection in Chapter 3 the Calerkin-weighted residual statements corresponding to Equation (4.52) arc written as
STOKES FLOW OF A CENERAIXZED NEWONIAN FLUID
119
+
After application of Green's theorem to the second-order velocity derivatives (to reducc inter-element continuity requirement) and algebraic manipulations the working equations of the continuous penalty scheme are obtained as
(4.54) where
(4.55)
(4.56)
(4.57)
(4.58)
1120
WORKING EQUATIONS OF THE FINITE ELEMENT SCHEMES
As it can be seen the working equations of the penalty scheme are more compact than their counterparts obtained for the U-VIn some applications it may be necessary to prescribe a pressure datum at a node at the domain boundary, Although pressure has been eliminated from the working equations in the penalty scheme it can be reintroduced through the penalty terms appearing in the boundary line integrals.
.I.
alty sc
After the substitution of pressure via the penalty relationship the flow equations in a polar coordinate system are written as
Using a procedure similar to the derivation of Equation (4.53) the working equations of the ~ontinuouspenalty scheme for steady-slate Stokes flow in a polar (r, 0) coordinate system are obtained as (4.62)
where
STOKES FLOW OF A GENERALIZED NEWTONTAN FLUID
124
(4.64)
2--
""91
r 80
rdrdB
(4.65)
(4.68)
After the substitution of pressure via the penalty ~elationsh~p the Bow equations in an axisyminetric coordinate system are written as
122
{
WORMING EQUATIONS OF THE FTNJTE ELEMENT SCHEMES
(4.69)
Using a procedure similar to the derivation of Equation (4.53) the working equations of the continuous penalty scheme for steady-state Stokes flow in an axisyrnmetric coordinate system are obtained as (4.70) where
(4.71)
(4.72)
(4.73)
(4.74)
(4.75)
STOKES FLOW OF A GENERALIZED NEWTONLAN FLUID
123
As described in the discrete penalty technique subsection in Chapter 3 in the discrete peiialty method components of the equation of motion and the penalty relationship (i.e. the modified equation of continuity) are discretized separately and then med to eliminate the pressure term from the equation of motion. In order to illustrate this procedure we consider the following penalty relationship (4.77) Discretization of Equation (4.77) gives
where nCr, is a weight function, identical to the shape functions Ad,il k = 1,m which are used for the discretization of the pressure, and NI, j = 1,n are the shape functions used to discretize the velocity components. Using matrix notation Equation (4.78) is written as
Therefore
I
WORKING EQUATIONS OF THE FINlTE ELEMENT SCHEMES
After substitution of pressure in the equation of motion using Equatioii (4.80) and the application of Green’s theorem to the second-order derivatives the working equations of the discrete penalty method are obtained as (4.81) where
(4.84)
(4.86)
STOKES FLOW OF A GENERALIZED NEWTONIAN FLUID
(4.87)
In conjunction with the discrete penalty schemes elements belonging to the aviart group arc usually used. As explained in Chapter 2, these elements generate discontinuous pressure variation across the inter-element boundaries in a mesh and, hence, the required niatrix inversion in the working equations of this scheme can be carried out at the elemental level with ~ i n i ~ u m computational cost.
Following the procedure described in Chapter 3 , Section 1.6, after the discretization of Equations (4.1) and (4.4) the generated residuals are used to fori~ulate a functional as
(4.88) where fits,A.:, and A!&represent the obtained residuals corresponding to u and v components of the equation o f motion and the equation of continuity, respectively. The constant k in functiorial (4.88) is used to make the functional dimensiomlly consistent. The magnitude of this constant depends on fluid viscosity and usually a large number (comparable to thc parameter used in the penalty schemes) i s selected. Utilizing equal order elemental i n ~ ~ r p o ~ ~for t i othe ~s velocity and pressure, functional (4.88) i s written as
(4.89)
126
WORKING EQUATIONS OF THE FINITE ELEMENT SCHEMES
where NI,.j = 1 ,n are shape functions and IZ is the number of nodes per element. inimization of functional (4.89) with respect to pressure and velocity components yields
(4.90) It is evident that application of Green's theorem cannot eliminate second-order derivatives of the shape finctions in the set of working equations of the leastsquares scheme. Therefore, direct application of these equatioiis s h o u ~ ~ in, general, be in conjunction with C' contiiiuous ermite elements (Petera and Nasselii, 1993; Petera and Pittman, 1994). However, various techniques are available that make the use of CO elements in these schemes possible. For exaniple, Bell and Surana (1994) developed a method in which the flow model equations are cast into a set of auxiliary first-order differential equations. They used this approach to construct a least-squares scheme for non-Newtonian flow equations based on equal-order CO continuous, p-version hierarchical elements.
F
'U
Incorporation of viscosity variations in non-elastic generalized Newtoiiian flow models is based 011 using empirical rheological relatioiiships such as the power law or Carreau equation, described in Chapter 1. In thesc relationships fluid viscosity is given as a fuiictioii of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rate i s defined as the second invariant of the rate of de~ormation
(4.91)
Equation (4.91) is written using the components of the rate of deforiiiation
MODELLING OF STEADY-STATE VISCOMETRIC FLOW
127
in Cartesian (x,y ) coordinate system (4.92) in polar (r, 6)) coordinate system (4.93)
in axisynimetric (r, z ) coordinate system (4.94)
The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in ternis of rate of strain components. For example, assuming a zero second normal stress difference for very slow flow regimes such relationships are written as ( itsoulis et al., 1985) (for flow dominantly in the x direction)
where qu is the fluid consisteiicy coefficient, rl is shear-dependent viscosity and Yf1 is a inaterial parameler representing the first normal stress difference. This p a r ~ m e t ~can r be defined empirica~~y as
128
WORIUNG EQtJATIONS OF THE FINITE ELEMENT SCHEMES
where A and b and are characteristic material constants. After substitution of the extra stress term in the equation of motion through the relationships given above and discretization of the resulting equations according to the previously described continuous penalty method the working equations of this scheme are obtained as
(4.97) where
(4.98)
(4.100)
(4.10'1)
(4.102)
(4.103)
where an over bar indicates a value found at the last iteration step.
.4
NER
ANC
The majority of polymer flow processes involve significant heat dissipation and should be regarded as non-isothermal regimes. Therefore in the finite element ~ ~ d e ~ofl polymeric i n ~ flow, in conjunction with thc equations of continuity
MODELLING OF THERMAL ENERGY BALANCE
and motion, an appropriate heat balance equation should be solved. temperature dependency of the fluid viscosity, coupling of the flow and energy equations introduces an additional form of non-linearity into the nonNewtonian polynier flow models. The degree of this non-linearity and interdependency between the equatioiis of motion and energy is determined by the rnagnitudes of the ~ ~ h m e - ~ r i f fnumber ith and the Pearson number in the flow system. These dimensionless numbers are defined as
Nu =-P 2 J k
(4.104)
Pn = (QT),,[
(4.105)
where p, and U are characteristic viscosity aiid velocity in the flow domain, ( is the t e ~ p e r a t ~ idependency re coefficient of viscosity (parameter 27 in Equation (1.14)), k is fluid conductivity and (AT),, is the operationally imposed te~nperat~re difference across the boundaries. Physically, the Nahme-Griffith number gives a Iy~easureof the ratio of viscous heat dissipation to the heat r e ~ i ~ i ~tor ~ d significantly alter the fluid viscosity while the earsoii number provides a measure for the influence on viscosity of the imposed ernperature difference throu flow domain boundaries. Analysis of these factors has shown that, in gene temperature dcperidency of viscosity in polymeric flow processes cannot be neglected (Pearson, 1979; Pittnian and Nakamwa, 1984). In practice. however, ling of the flow and energy equations can be based on an iterativc a l g o r ~ ~ h ~ . ils of such an algorithm are described in Chapter 7.
~ o l l o w i i the i ~ procedure described in in Chapter 3 , Section 3 the st~eainli~iedupwind ‘weighted residual’ statement o f the energy equation is formulated as
130
WORKING EQUATIONS OF THE PINI'T'E ELEMENT SCHEMES
where N, is the weight function (identical to NJ elemental shape functions) and N; is the upwinded weight functions defined according to the streamline upwind formula for a rectangular element as (4.107)
where 0 < E 5 1 is the upwinding constant and characteristic element dimensions are defined as
where 6 and 17 and are the elemental coordinates. As mentioned before the definition of tlic upwinding parameter (i.e. coefficient multiplied by the derivatives o f the shape functions in Equation (4.107)) is based on a heuristic analogy with one-dimensional analysis and other forms are also used (e.g. see Equation (3.33)). fter the application of Green's theorem to the second-order derivatives of temperature in Equation (4.106) the working equation of this scheme is obtained as (4.109) where in a Cartesian coordinate system
(4.1 11)
In a polar coordinate system the steady-state energy equation is written as
Using a siinilar discretization the terms of the stiffness and load matrices correspond~n~ to Equation (4.112) are written in this scheme as
MODELLING OF THERMAL ENERGY BALANCE
13
(4.114)
re Similarly in an axisymmetric coordinate system the terms of stiffness and load matrices corresponding to the governing energy equation written as a,
are derived as
(4.1 17)
The inconsistent streamline upwind scheme described in the last section is formulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In this section we consider the d e v e l o p m ~ ~oft weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as
After application of the 8 time-stepping method (see Chapter 2, Section 2.5) and fo~lowingthe procedure outlined in Chapter 2, Section 2.4, a funct~onal represe~itingthe sum of the squares of the approximation error generated by the finite element discretizaiion of Equation (4.1 18) is formulated as NJ
qn
WORKING EQUATIONS OF THE FTNTTE ELEMENT SCHEMES
where n is the time level, 4 ,,j = 1, p are the shape functions and p, c, Y, k and 4 are all given in SE,. Minimization of functional (4.119) with respect to nodal temperatL~resgives
(4.120) ~ s s u m i n gconstant density and specific heat (i.e. pc = constant) Equation (4.120) i s written as
(NJ -t SAt,v.VN,
- SAtnV.@Vlv,)[~~s]dS2, =0
(4.121)
where 0 = klpc is the thermal diffusivity. Equation (4.121) represents a weighted residual statement where the weighting function is given as
The foI~owingoptions can now be considered: If the second and third terms in the weight function are negkctecl the alerkin scheme will be obtained. If only the third term in the weight f~inctionis neglected a first-order lerkin scheme correspond in^ to the method will be nconsistent up~~in4~ing will be a special which the second term in the weight function is only retained for the weighting of the convection terms.
inin^ in^
all of the terms in the weight function a least-squares scheme ~ o r r ~ s p o ~ ~ to d i nag second-order ~ e ~ r o v - ~ a ~ e~~ok~iLni l a t i owill i ~ be obtairied. In steady-state problems 0 I, = 1 and the time-de~endentt e r n in the residual is eliminated. The steady-state scheme will hence be equivalent to the combination of Galerkin and least-squares methods.
t y motion representn the absence of body force the equations of c ~ n t ~ n u iand ing transient Stokes flow of a generalized Newtonian fluid in a t w o - d i ~ ~ ~ e n s i o i ~ ~ ~ ~ t e system s ~ d nare written, on the basis of E ~ u ~ t (1~ o1 ) nand ~ (1.4), as
1
STOKES FLOW OF N~WTONIANAND NON-NEWTONTAN FLUIDS
au av + -= 0 ax ay
(continuity)
-
where U and
1)
(4.123)
are the .x and y compoiicnts of velocity, respectively, a i d
where p is pressure. Followiiig the partial discrctization technique, described in Chapter 2, Section 2.5, various finite element schemes for this problem can be s an example, we consider derivation of the working equations of us penalty scheme in conjunction with the 0 time-stepping procedure. After substitution of pressure in equation set (4.124) via the rela~ionship~ arid spatial discretization o f the resulting system, in a manlier siniilar to the procedure described in Section 1.7, we obtain (4.125) where
Q,
The remaining terms in equation set (4.125) are identical to their counterparts derived for the steady-state case (given as Equations (4.55) to (4.60)). application of the 6’ time-stepping method, described in Chapter 2, Section to the set of first-order ordinary dif€crenrial equations (4.125) the working equations of the solution scheme are obtained. The general form o f these equations will be identical to Equalion (2.1 l l} in Chapter 2. The described continuous penalty0 time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical st ility under a wide range o f c o ~ ~ i t i o This ~~s. scheme is based on the XJ-V method for the slightly compressible continuity equation, described in Chapter 3 , Section 1.2, in conjunction with the TaylorGalerkin time-step pin^ (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows 3 ap 8 u av --+-+-=U pc2 at ax ay
(continuity equation, based on Equation ( 3 .
(4.127)
134
WORKING EQTJATIOKS OF THE FINITE ELEMENT SCHEMES
(4.128)
(equatioii of motion for Stokes flow based on Equation (1.4)) As explaiiied in Chapter 3, it is possible to use eqLd order interpolation models for the spatial discretization of velocity and pressure in a U-V-P scheme based quations (4.127) and (4.128) without violating the stability c o n ~ i ~ i o n . To develop the scheme we start with the normalization of the governing equations by letting
p=E F
(4.129)
Thus we have I d V au a v cz at +-+-=o ax ay
I -
(4.130)
and
(4.131)
Following the procedure described in Chatper 2, Section 2.5 the Taylor series expaiision of the field unknowns at a time level equal to FI + aAt, where 0 5 5 I, are obtaincd as (4.132)
(4.133)
(4.1 34)
The selection of a time iiicremcnt dependent on parameter (U (i.e. carrying out Taylor series expansion at a level between successive time steps of rz and n + l ) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as
(4.135)
(4.136)
(4.137) The second-ordcr derivatives of the variables are now found as
The extra stress is proportional to the derivatives of velocity comporients and consequently the order of velocity derivatives in terms arising from
will be higher than the tern1
1
WORKlNG EQUAriOhS OF THE FTNITE ELEMENT SCHEMES
hence the stress terms in Equation (4.138) can be ignored to obtain (4.139)
(4.140) Note that in polar and axisymrnetric coordinate systems the stress term will include some lower-order terms that should be iiicluded in the f o ~ u l ~ t i o ~ s . Using a similar procedure the second order time derivative of pressure is found as
(4.14 I) After substitution of the first- aiid second-order time derivatives of the u ~ l ~ ~ o in w nEquations s (4.132) to (4.134) from Equations (4.139) to (4.141) and spatial discretization of the resulting equations in the usual manner the working equations of the scheme are derived. In these equations, functions given at time t can be interpolated as
In generalized Newtoiiian fluids, before derivation of the filial set of the working e ~ u ~ ~ ithe o nextra ~ , stress in the expaiicfed equat~oiisshould be replaced using the co~ponentsof the rate of strain tensor (note that the viscosity should also be normalized as i j = rl/p). In contrast, in the modelling o f viscoelaslic fluids, stress c ~ n i ~ o i i e nare t s found at a separate step through the solut~onof a constitutive equation. This allows the development of a robust Taylor Galerkin/ scheme on the basis o f the described procedure in which the stress c ~ ~ p ~ ~are ~ all e nfound t $ at time level n. The final working e q ~ ~ a t i oof~ this i sclieme can be expressed as
STOKES FLOW OF N ~ ~ ~ OAXD ~ INON-NEWTONIAN A ~ ' FLlJIDS
137
(4.143)
(4.144)
(4.145)
(4.150)
(4.152)
(4.153)
WORKING EQUATIONS OF THE FINITE ELEMENT SCHEMES
(4.154)
(4.155)
(4.156)
(4.157)
(4.158)
(4.159)
(4.160)
(4.161j
(4.162)
(4.163)
The last teim i n the right-hand side of Equation (4.143) represents boundary line integrals. These result from the application of Green's theorem to second-order derivatives of velocity first-order derivatives of pressure first-order derivatives of stress
I
(in equations ~ o r r e s p o n ~ i ~ g to U and V )
STOKES FLOW OF NEWTQNIAN AND NON-NEWTONTAN FLUIDS
139
and second-order derivatives of pressure second-order derivatives of stress
(in equation corresponding to P )
Therefore
(4.165)
(4.166)
The described scheme can also be incorporated into iterative algorithms and used to solve steady-state flow problenis (Zienkiewicz and Wu, 1991).
.C . and Surana, K. S., 1994. p-version least squares finite element formulations for two-dimensional, incompressible, non-Newtonian isotliemal and non-isotliermal fluid flow. Int. J. Numer. Methods Fluids 18, 127-162. Bird, R. B., Arnistrong, R. C. and Hassager, O., 1977. Dynamics of Polymeric Fluids, Vol, 1: Fluid Mechanics, 2nd edn, Wiley, New York. Mitsoulis, E., Valchopoulos, J. and Mirza, F. A., 1985. A numerical study of the effect of normal stresses and elongationai viscosity on entry vortex growth and extrudate swell. I’oly. Eng. Sci. 25, 677 -669. Pearson, J. R. A., 1979. Polymer flows doiiiinated by high heat generation and low lieeat transfer. Polym. Eng. Scsci. 18, 1148-1 154. Petera, J. and Nassehi, V., 1993. Flow modelling using isoparametric Hermite elements. In: Taylor C. (ed.), Nurnericnl method.^ in Lnminur und Turbulent Flow, Vol. VHI, Part 2, Pineridge Press, Swansea.
WORKING EQUATIONS OF THE FINITE ELEMEh'T SCIIEMES
Peters, J. and Pittmaii. J. F. T 1994. Isoparametric Hermite elements', bit. J. Nuin. ~ ~ ~Eng. l 37, ~ 3489 o ~3519. ~ s Pittman, J, F. T. and Naltazawa, S., 1984. Finite element analysis of polyiner processing operations. In: Pittinan, J. F. T., Zienkiewicz, 0.C., Wood, R. D. and Alexander, J. M. (eds), Nurtterical Analysis of Forming Proccsscs, Wiley, Chichester. Tanner, R. I., 1985. Engineering Rheology, Clarendoii Press, Oxford. Zienkiewicz, O . C . and Wu, J., 1991. Incompressibility without tears - how to avoid restrictions on mixed formulations. hit. J. i\lumer. Methods Eng. 32, 1189--1203.
In this chapter, selected examples of the application of weighted residual finite element schemes to the solution 01polymer flow problems are presented. These examples provide a relevant background for description of the rational approxima lions that may be used to obtain realistic computer simulations for industrial polymer flow processes. Tn general, extension of the finite element techniques, described in the previous chapters, to a direct three-dimensional solution of nonNewtonian Bow problems does not introduce any fundamental ~ifficultics, because of high cost of the required computations (or excessive length of execution times) a practical difficulty arises. Therefore rational approximatio~s that reduce the ‘size’ of a probleni without neglecting its core characteristics are of utmost iniportance in the development of practical computer models. The main categories of approxiniat~onsused in the modelling of polymer processes are listed in Chapter 2. The techniques of implementation of these approximations are discussed in the following sections.
Straiglitforwai-done- or two-dimensional modelling of polymeric flow systems that correspond to a linear or planar geometry often yields useful results and is considered to be a permissible approximation in most cases. Important examples of such systems are found in fibre spinning and film casting which cm be satisfactorily defined using one- or two-dimensional frameworks. Therefore the modelling of these processes is usually based on one- or two-dimensional schemes (e.g. see Andre et al., 1998; eaulne and Mitsoulis, 1999). Similarly if the process of interest, within a three-dimensional domain, is confined to a dominant direction (or cross-section) a reduction in model dimensionality may also be considered. ispersive mixing of rubberlcarbon compounds in partially filled internal mixers provides such an example in which the process of interest (i.e. breakdown of filler agglomerates) occurs predominantly in the crosssectional plane of the rotor blades. Simulation of this process is considered next.
RATIONAL APPROXIMATIONS A N D ILLIJSTRATIVE EXAMPLES
A full account of finite element modelling of dispersive mixing can be found in the literature (see Nassehi and Choreishy, 1997; Nassehi et ul., 1997; Nassehi and Choreishy, 1998; Nassehi and Choreishy, 2001) and i s not repeated in this section. The main focus here is to show that, despite the complexity of the process, two-dimensional models that incorporate its significant features can generate useful predictive results. atch internal mixers are used extensively by industry to mix polymers with other materid to produce coniposites with desirable properties. For exanple. mixing of rubber with carbon black is almost exclusively carried out in partially filled internal mixers. Essentially this process consists of three stages of incorporation, dispersion and distribution. During the incorporation stage pelletized carbon black and rubber arc brought together, allowing the diffusion of macromolecular chains of the polymer into the void spaces inside the filler agglomerates. After this stage, dispersive mixing starts in which, through the imposition of an uneven stress field by the action of rotor blades, carbon black agglomerates are broken into sinaller aggregates and dispersed within the matrix. Finally, the compound is distributed within the chamber to achieve uniformity. The described incorporation and dispersion phases mainly take place in the rotor blade cross-sectional pjane and are stress dowinated. In contrast, distribution of the material inside the mixer is pressure dominated and mainly takes place along the rotor axes (Clrtrli-e and Freakley, 1995). Therefore in the analysis of rubber mixing in internal mixers, it is reasonable to develop separate models for each stage of the process. In addition, the stress field that gives rise to the incorporation and dispersion of the phases, is obtainable from the simutntion of the two-di~ensionalflow in the plane of the rotor blades cross-section. Therefore the developiiient of two-dimensional models for this process is explained in the following section.
Flow simulation iE n single blade pmtially filled mixer The plane of the rotor blade cross-section representing the Bow field configuration at the start of mixing in a partially filled single-blade mixer is shown in Figure 5.1. Initial distribution of the compound inside the mixer chamber corresponds to a fill factor of 71 per cent and is chosen arbitrarily. It is evident that the flow field within this domain should be modelled as a free surface regime with random moving boundaries. Available options for the modelling of such a flow regime are explained in Chapter 3, Section 5, In this exaniple, utilization of the volume of fluid (VOF) approach based on an Eulerian framework is described. To inaiiitain simplicity we neglect elastic effects and the variations of coinpound viscosity with mixing, and focus on the simulation of the flow corresponding to a generalized Newtonian fluid. In the VOF approach
MODELS BASED ON SIMPLIFIED DOMAIN GEOMETRY
I
~ i ~ 5.1 u ~Initial e configuration in a partially filled single blade internal niixer
the flow regime i s treated as a multi-phase system in which compound- and airfilled regioiis are each assumed to represent a different phase. Similar to the coinpound-filled sections, the flow regime in the air-filled regions is considered to be incompressible. The exception is tke narrow gap between the blade tip and the chamber wall, where an air bubble may become compressed if pressure exceeds beyond a given threshold. This is a realistic assumption because the chamber in an industrial mixer is not airtight and only air pockets trapped inside the compound matrix are expected to be compressed under high pressure. Utilizing the domain symmetry of a single-blade mixer, it can be assumed that the blade remains stationary throughout the mixing and the flow is generated by the rotation of the chamber wall. This is readily achieved by the imposition of the appropriate boundary conditions arid results in a significant simplification in the inoclelliiig by allowing the use of a constant finite element mesh for the entire simulation. Figure 5.2 shows the finite element mesh corresponding to the configuration shown in Figure 5.1. This mesh consists of 225 nine-node bi-quadratic elements and its utilization in the present model i s based on the application of isoparametric mapping, described iii Chapter 2. The governing equations used in this case are identical to Equations (4.1) and (4.4) describing the creeping flow of an incompressible generalized Newtonian fluid. In the air-filled sectioiis if the pressure exceeds a given threshold the equations should be switched to the following set describing a compressible flow (continuity)
1
RATIQNAL ~ P K O X I M A T I O N $AND I ~ ~ U S ~ I ~ A T EXAMPLES IVE
ignre 5.2 The mesh used in this example consisting of 225 nine-node bi-quadratic elements
(motion)
where
where and k are shear and volume (bulk) viscosity of the fluid, respectively. The shear-dependent viscosity of the compound is found using the temperature-dependent form of the Carreau equation, described in Chapter 1, given as
MODELS BASED ON SIMPLIFIED DOMAIN GEOMETRY
1
Temperature variations are found by the solution of the energy equation. ?‘lie finite elctnent scheme used in this example is based on the implicit 8 timesteppinglcoiitinuous penalty scheme described in detail in Chapter 4, Section 5. As described in Chapter 3, Section 5.1 the application of the VOF scheme in an Eulerian framework depends on the solution of the continuity equation for the free boundary (Equation (3.69)) with the niodel equations. The developed algorithm for the solution of the described model equations and updating of the free surface boL~ndariesis as follows: Step 1 - the domain of interest is discretized into a mesh of finite elements. Step 2 an initial configuration representing the partially filled discretized domain is considered and an array consisting of the appropriate values of P 1, 0.5 and 0 for nodes containing fluid, free surface boundary and air, respectjvely, is prepared. The sets of initial values for the nodal velocity, pressure and temperature fields in the solution donlain are assumed and stored as input arrays. An array containing the boundary conditions along the external boundaries of the solution domain is prepared and stored. =I
Step 3 - the time variable is updated. incrementing it by At. To maintain the accuracy a small time step of 0.01 s is used. Step 4 - it is initially assumed that the flow field in the entire domain is incompressible and using the initial and boundary conditions the corresponding flow equations are solved to obtain the velocity and pressure distributions. Values of the material parameters at different regions of the domain are found via Equation (3.70) using the ‘pseudo-density’ method described in Chapter 3, Section 5.1. Step 5 -~ the obtained velocity field is used to solve the energy equation (see Chapter 3, Section 3). Step 6
-
compound viscosity is updated via Equation (5.4).
Step 7 stcps 4 to 6 are iterated until the solution is converged. Air-filled regions where the calculated pressure i s more than 1 Pa are iclcntified. Following the identification of high-pressure air-filled regions the solution is repeated by treating the air within these areas as compressible. Note that in a domain divided into compressible and incompressible parts the boundary line integrals, obtained by the application of Green’s theorem to the second order derivatives in the equation of motion, will not be compatible at all inter-element boundaries. Therefore it cannot be assumed that all such terms will be aulomatically cancelled during the assembly of elemental stiffness equations. To avoid this difficulty an approximation based on the use of old time step values of the field variables in the line integral terms along the ~
146
IIATIONATdA ~ P R ~ ~ I M A T IAND O N ~ILLUSTRATIVE EXAMPLES
shared boundary of the compressible and incompressible parts is introduced. To mainlain accuracy of the solutions in the VOF method small time steps should be used, hence the effcct of this approximation is insignificant. Step 8 - the new values of the nodal velocities found at the end of step 7 are used as input and the free surface equation is solved. Step 9 - using updated values of the free surface function tlie location of the free surfaces are identified and the positions of each phase in tlie current flow domain are marked accordingly. Step 10 - steps 3 to 9 are repeated and the solutioii is advanced in time until the required end of the simulation. The predicted free boundary distributions within a chamber of 0.05 rn radius for a fluid with the following physical parameters, rlo = ]OS,IZ = 0.25, b = 0.014, TECf = 373, X = 1 .S, p = 1055, c' = 1255, k = 0.13 (SI units), used in the Carreau, flow and energy equations, and penalty parameter of 109 after 10, 30, 50 and 100 time steps are shown in Figures 5.3a to 5.Jd.
Flow , ~ i ~ ~ ini a~partially ~ ~ i ~filled n twin blade mixer Simpli~cationachieved by using a constant mesh in the modelling of the flow field in a single-blade mixer is not appiicable to twin-blade mixers. Although the model equations in both simulations are identical the solution algorithm for twinblade mixers cannot be based on the VOF method on a fixed domain and instead the Arbitrary Lagrangiaii-Eulerian (ALE) approach, described in Chapter 3, Section 5.2, should be used. However, thc overall geometry of the plane of the rotors blades cross-section is known and all of the required mesh configurations can be generated in advance and stored in a file to speed up the calculations. Figure 5.4 shows the finite element mesh corresponding to 19 successive time steps from the start of the simulation in a typical twin-blade tangential rotor mixer. The finite element mesh configurations correspond to counter-rotating bladcs with unequal rotational velocities set to generate an uneven stress field for enhancing dispersive mixing efficiency. Calculation of mesh velocity, required for modification of the free surface equation (see Equation (3.73)) at each time step, is based on the following equations (Ghoreishy, 1997)
MODELS BASED ON SIMPLIFIED DOMAIN GEOMETRY
~
1
~ 5.3 ~ (a-d)u Simulated r ~ material distribution within the single-blade mixer after 10, 30, 50 and 100 time steps
where (x);etc. are the coordinates of node rt at time t. As already mentioned the time increment used in the VOF method should be small and hence Equation (5.5), which represents linear nodal movements between successive mesh configurations, does riot generate inaccurate results. In this example, the solution algorithm starts by inserting zero for mesh velocity in the governing equations. These equations are then solved in the first mesh corresponding to the initial configuration inside the mixer. After the convergence of the iterations this solution yields nodal velocities (and other field variables) at the end of the first time step. After this stage, mesh velocities are found using Equation (5.5) and the solution proceeds to the next step. The procedure is repeated until a complete set of field data for all of the configurations correspoiiding to different relative positions o f rotor blades is found. Initial distribution and the predicted free surface boundaries within the twinblade mixer represented by the mesh configurations shown in Figure 5.4, after 30, 60 and 90 * rotatioii of the left blade are presented in Figures 5.5a to 5.§cl, respectively. Samples of the predicted velocity fields after 30 and 45 O rotation of the left rotor are shown in Figures 5.6a to 5.6b, respectively. Thc linite element
RATIONAL A ~ P R O X I M A T I O ~AND S ILLUSTRATIVE EXAMPLES
The finite element mesh configurations in the Arbitrary Lagrangan-Eulerian scheme
MODELS BASED ON SIMPLIFIED DOMAIN GEOMETRY
1
Figure 5.5 (a-d) lnitial coiifiguration and simulated material distribution within the twinblade mixer after 30, 60 and 90" rotation of the left blade
@I (a,b) The predicted velocity fields after 30 and 45 rotation of the left blade in the partially filled twin-blade mixer
150
RATIONAL A P P R O X I ~ A T I OAND ~ ~ I~LUSTRATIVEEXAMPLES
scheme used in this simulation is based on thc iinplicit Blcontinuous penalty method and the physical parameters are equal to the values given in the previous exarnplc.
As discussed in the previous chapters, utilization of viscoelastic constitutive equations in the finite element schemes requires a significantly higher computational effort than the generalized Newtonian approach. Therefore an irnportant simplification in the model development is achieved if the elastic effects in a flow system can be ignored. However, almost all types of polymeric fluids exhibit some degree of viscoelastic behaviour during their How and deformation. Hence the neglect of these effects, without a sound evaluation of the flow regime chtzracteristics, which may not allow such a simplification. can yield inaccurate results. In the following sections modelling o f the free surface Bow of silicon rubber in a two-dimensional domain using both generalized Newtonian and viscoelastic constitutive equations is presented. Compming the results of these simulations, the effects of assuming a simplified generalized Newtonian behaviour against using a viscoelastic constitutive equation for this Buid are evaluated. This evaluation illustrates the conditions under which the described simplification leads to useful predictions and can hence be regarded as acceptable. Thc design and operation of a flow visualization system for highly viscous fluids, such as silicon rubber, has been reported by Ghafouri and Freakley (1994). This system consists mainly of a rotating roll and fixed-blade assembly, as is shown in Figure 5.7, and can be used to generate and maintain, essentially,
(Blade
1
Fluid flowing towards the
the gap betwe& I blade and roll surfaces II
Free surface
1 I
~~~~e 5.7 Schematic: diagram of the rnoctelled flow visualization rig
MODELS RASED ON SIMPLIFIED GOVERNING EQUATIONS
151
two-dimensional Couette flow regimes. To generate a Couette flow, a given mass of fluid is placed (or banded) in front o f the fixed blade. As the roll starts to rotate the fluid is dragged through the gap between the blade and roll surfaces and a transient reservoir in front of the blade is formed. The main flow domain is hence defined as a converging channel connecting two free surface flow regions on its sides through a narrow gap. Published flow visualizatioii results showing the evolution of the free surface boundaries at the sides of this channel provide experimental data for evaluation of the accuracy of the simulations obtained using various constitutive equations.
Assuming a generalized Newtonian bchaviour for silicon rubber, the Couette flow established in the described flow visualization experiment can be modelled using the VOF scheme based on a fixed (Eulerian) framework. The governing equations aiid finite element discretization in this model are identical to the implicit ~/c~ntinuous penalty scheme explained in Chapter 4, Section 5. This scheme is used in conjunction with the VOF procedure described in Chapter 3, Section 5.1. Figure 5.8 shows the finite element mesh corresponding to the maximum extent that the fluid can be expected to reach during the simulation time of 60 s for a roll velocity of 1 rpm. Predicted free surface boundaries at two different times are compared with the experimental data (Nassehi and Ghoreishy, 1997) in Figures 5.9a to 5.9b. The close comparison between the simulation resuXls and the experimental data points to the conclusioii that despite ignoring the elastic effects the simulated flow domain is realistic. The diminished extent of swelling at the exit from the narrow gap is attributed to the reduction in the influence of normal stresses in the present shear-induced flow regime.
re 5.8 The finite element mesh used to model free surface flow in example 5.21
I
RATIONAL APPROXIMATIONS AND KLUSTRATlVE E X A ~ P I , ~ ~
(b)
~ ~ g 5.9 ~ r (a,b) e Coinparisori of the simulated and observed free surface positions
Keeping all of the flow reginie conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic Quid whose constitutive behaviour is defined by the following upper-
where T,, etc. are the components of Maxwell stress defined as
(5.7) where 7 is viscosity, A i s relaxation time and is the Kroiiecker delta. The upper-convected Maxwell constitutive model is written as Equation (5.6) for the advantage that the simplified representation of the right-hand side provides in a numerical solution. Substitution of Equation (5.7) into Equation (5.6) results in the more familiar form of the axwell model given in Chapter 1.
MODELS BASED ON S I M ~ L I ~ GOVERNING I~D ~ Q ~ J A ~ I O1 ~ S
Equation (5.6) is a hyperbolic partial differential equation and its finite element solutioii in a fixed coordinate system requires the use of an ~ i p w ~ n d ~ n g scheme (see Chapter 3, Section 2). owcver, numerical damping generated by upwinding may adversely affect the accuracy of the solution. Therefore it is preferable to use a Lagrangian framework for the solution of free surface and constitutive eqi'.atkm in this problem. Equations of continuity and motion, on the other hand, can be solved using a fixed coordinate system. An optimum solution algorithm based on a decoupled procedure, in which flow equations are solved after the updating of the free surface and the calculation of stress components, is thus developed. The Taylor-Galerkin/U-V-P scheme corresponding to a slightly compi-essible continuity equation for viscoelastic fluids, described in Chapter 4, Section 5, is used to solve the flow part of the model. The constitutive and free surface equations are solved at a different step, using the VOF method in a Lagrangian framework. Derails of the algorithm for the updating of Free boundaries is given in Chapter 3, Section 5.3, here we focus on the sohition of the constitutive equation in a moving coordinate system. Similar to the solution of the free surface equation, the moving system in this problem i s constructed along the Auid particle trajectories. This framework can be defined as a general curvilinear system in which a line element is given by the "metric form' (Spiegel, 19741, as
where s u m m a over ~ ~ ~repeated ~ indices is assumed and
are the components of the metric of the reference coordinate system where gLWL (in a Cartesiaii coordinate system g'"" = bk'"j. components of the stress tensor are written as
In the moving coordinate system the constitutive equation (i.e. Equation (5.6)) can be written as I (5.1 1) Note that in a Lagrangian system the convection ternis in Equation (5.11) vanish. Application of the previously described Q time-stepping method to Equation (5.11) gives
154
RATIONAL APPROXIMATIONS AND ILLLJSTRATIVE EXAMPLES
After rearranging, the Galerkin-weighted residual statement arising from Equation (5.12) can be written as
trt Q,
where an over bar rneaiis elemental discretization in the usual manner. The working equation yielding the stress components in the current time level of (n + 1) can now be constructed, combining Equations (5.12) and (5.13), as
where
(5.15)
and
+
(1 - 0) NfXYPqd12,,+, 17
L
NIz,PqdL!,
(5.16)
a,, Note that the shape functions used in the above discretization preserve their originally defined forms. This is in contrast to the Lagrangian formulations in which the shape functions need to be modified (Donea and Quartapelle, 1992). In Equation (5.14), is found by interpolating existing nodal values at the old time step and then transforming the found value to the convected coordinate system. Calculation of the components of yIJqand (B$q)" depends on the evaluation of first-order derivatives of the transformed coordinates (e.g. as seen in Equation (5.9). This gives the measure of deformation experienced by the Buid between lime steps of n and n + 1 . Using the time-independent local coordinates of a fluid particle (E, q) we have Q X k l
(ny)"
MODELS BASED ON SIMPLIFIED GOVERNIKG EQUATIONS
5
(5.1 7) Smoothness of the transformation between the fixed and mobiiig systems depends on the inter-element continuity of first-order derivatives of coordinates. This can be achieved by using G' continuous Hermite elements. As for all types of finite elenients, applicability of undistorted Hermite elements in practical problems is very limited. Therefore a procedure for the construction of isoparametric C" continuous Hermite elements is incorporated in the present solution scheme. Details of a numerical technique for the construction of isoparametric C' continuous Hemite elements can be found in the literature (e.g. see ittmaii, 1994). This method is used to obtain a discretization for the working equation (5.14) based on the isoparanietric form of the rectangular Bogner-Pox-fchmit element, shown in Figure 2.10. Results obtained using the described viscoelastic model arc only marginally different from the results shown in Figures 5.9a and 5.9b (Retera and Nassehi, 1996). Although this confirms the overall validity of the utilization of a simplified constitutive equation for this flow process, detailed and direct comparison of the two sets of results is needed to obtain a quantitative evaluation of the effects of the adopted approximation. To carry out such an evaluation let us consider the simulated velocity fields at the exit section of a domain analagous to the narrow gap in the converging Bow channel between the roll and blade shown in Figure 5.7. Figures 5.10a and 5. I0b show the velocity field corresponding to the generalized Newtoniaii and
.I0 (a,b) Comparison of the siinulated die swell in a Couette How for the powerlaw and uppcr-convected Maxwell fluids
viscoelastic models, respectively. All other data used to obtain these results are identical. As this comparison shows the overall velocity fields in the two cases are not s~gnificantlydifferent. owevzr, the distinct 'localized' swelling, obtained using the viscoelastic constitutive equation, has not been generated by the generaliLed Newtonian model. It is therefore evident that the latter model has a restricted applicability and should only be used in situations where localized effects resulting from the viscoelasticity of the fluid calli he ignored.
Under certain conditions it may be ap ropriate to focus on the modelling of a s e ~ ~ eof~ ia t larger domain in orde to obtain detailed results within that section, while maintaining computing ewnomy. TQ develop such a model, first tlie entire flow domain is sirnulated usin a relatively coarse finite element mesh. The results generatec~at the end of this stage are used to define the boun conditions at the borders of the selectcd part. This section is then mod L i t ~ ~ a~ refined ~ ~ ~ i mesh g to obtain detailed predictioiis about phenomena of int~restin the flow field. The procedure can be repeated using a step-~y-step ~ ~ ~ p iin- wliicli o a ~ the ~ zooming 011 a segment within a very large domain is hsough successive reduction of the size of the segments at each step of In rhc ~ ~ l l sections o ~ ~examples ~ ~ g of (he ayplicatio of this procedure to tlie a i i ~ ~ y sof i s specific ~henomenasuch as wall slip an stress overshoot which affect p o ~ y ~ ~ eflow r i c processes are illustrated.
rs established that during the flow of 1 ~ ~ n g " c hpa~i ~ y m estr~sses may rise signi~cant~y (overshoot) at certain locations in the e~omenoiiis more clearly observe^ in cont ves from wider to narrower sections of t of a polymeric fluid in a two-~imensional c~oss-sectio~al plane of a s ~ ~ escrew t rand ~ a ~c ical outer tube. ~ n ~ t i a ~ l y the entire domain is discretized into a mesh consis f 5 12 hi-quadratic finite e ~ e i ~ i e ~asi t shown s~ in Figure 5.11. The flow inside this domain is ~ e i i e ~ aby te~~ rotat~onof the screw in the anticloc~wise~ ~ ~ r e c t i osin^ n . the s i i ~ u l a t i oresults ~ obtained on this mesh, the boundary conditions for a repres~~~tative section of o i ~ on^^^^ ~ i ~ between two fli d mesh consisting o f 256 hi-qu structed, Considering the symmetry of the domain, simulation o f stress field in this section shoul provide an insight for the entire domain,
MODELS R E ~ R E S ~ N ~ lSELECTED NG SEGMENTS OF A LARGE DOMAIN
Fi
159
.I1 Finite element discretization of the symmetric doiaaiii in example 5.3.1 Flow direction
Fi
Schematic didgrain of the predicted norrnal stress contoiirs in a typical section of the syinmetric domain shown in Figure 5.1 1
stress contours corresponding to a peak at the entrance to the narrow gap between the flight and the outside wall arc shown, schematically, in Figure 5.12. Solution of the flow equations has been based on the a~plicationof the implicit 0 time-steppinglcontinuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Plian-Thie~/~anner equation for viscoelastic fluids given as Equation (127) in Chapter 1. Details of the finite element solutioii of this e ~ ~ ~ a t are i o n~ublishedelsewhere and not repeated here (I’iou and ~ ~ ~ s e h i , 2001). The predicted normal stress profiles along the line A (see Figure S. 12) at five successive time steps are shown in Figure 5.13. The predicted pattern is expected to be repeated t h r o ~ g ~ o the u t entire domain.
1
RATIONAL ~ P K O X I ~ A r l O NAND § ILLUSTRATIVE EXAMPLES
Stress
Time step
~
~ 5.13~ The u prcdicted r ~ pattern of the stress overshoot in example 5.3.1
Fluid slippage at solid wall surfaces is a common phenomenon affecting polymer flow regimes. In particular in processes involving long-chain elastomers, such as natural or synthetic rubbers, a noticeable percentage of the energy input may be wasted through wall slip in the domain. As explained in Chapter 3, Section 4.2 the irn osition of wall slip boundary conditions in a finite element model should be based on Navier’s slip condition. A straightforward method for osition of these boundary conditions is to modify the original working equations in a flow model via the incorporation of the discretized ~ e l a t i o ~ s l ~ i p s arising from the components of the Navier’s slip condition (e.g. Equations (3.62) and (3.63)). This is shown as: odified weighted residual statement = Original weighted residual statement +G [slip-wall boundary conditions]. G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick, Therefore imposition of wall slip introduces a form of ‘non-Linearity’ into the flow model which should be handled via an iterative loop, The slip coefficient (i.e. I? in the Navier’s slip c o ~ ~ i t i ogiven n as Equation (3.59) is defined as
MODELS R
~
~
~ SELECTED ~ S SEGMENTS ~ ~ ~OF AI LARGE N DOMAIN ~
o x - -1
'
1
(5.18)
k,
where k, is the experimentally determined coefficient of sliding friction between the fluid and a solid surface. The iteration loop in the imposition of slip condition may start by assuming /3 = PO = Ilk, and subsequently U coefficieiit as
(5.19) where V is a ch~racteristicBow velocity and &(cl) is the second invariant of the rate o f ~eformatioiitensor. Equation (5.19) sbould be regarded as a heuristically defined relationship between the extent of wall slip at a given stage of flow and the state of stress at that time. In practice, comparison with experimentid data may be needed to verify the accuracy of model predictions. redicted velocity fields in a segment adjacent to the tip of the blade irz the single-blade mixer, described in a previous sub-section, before and after irnposition of the wall slip are shown in Figures S.14a and S.l4b, respectively. As expected, momentum transfer from the rotating wall to the fluid is significantly affected by the imposition of the wall slip. In Figures 5.15a and S.15b temperature contours corresponding to these velocity fields are shown.
/
Rotating wall (no-slip wall)
Rotating wall (wall slip i~posed) (a>
(b)
(a) Tlie predicted velocity field corresponding to no-slip wall boundary conditions. (b) Tlie predicted velocity field corresponding to partial slip boundary condilions
As comparison of the simulated temperature fields sfiows, fluid slippage results in t e ~ ~ e r a t u peak r e shifting from a location furthest away from the
R A ~ I O ~ A~ L ~ ' K O X I M A T I O NAND S ILLIJSTR ATIVE EXAMPLES
Figure 5-15 (a) The predicted temperature distribution conesponding to the no-slip conditions. (b) The predicted temperature distribution corresponding to the partial slip conditions
cooling walls to a layer next to the rotating surface. This provides an indication that because of the wall slip the energy input is converted to heat instead of generating flow.
In Chapter 4 the developmelit of axisymiietric models in which the radial and axial compoi~entsof Bow field variables remain constant in the circ~imfere~itial direction is discussed. In situations where deviation froin such a perfect symmetry is sinall it may still be possible to decouple components of the equation of motion and analyse the flow regime as a co~ibinationof one- an dimensional systems. To provide an illustrative exanlple for this t this section we consider the modelling of the flow field inside are among the most commonly used rheometry sentially consist of a steel cone whic generating a Couette flow regime. same fundamental concept various types of single and double cone devices are dev~loped.The scheindtic diagram of a double cone visconieter i s shown in
MODELS BASED ON DECOUPLED FLOW EQUATIONS
161
Chamber
.16 Schematic diagam of a bi-conical cone-and-plate viscometei
assunied that by using an exactly symmetric cone a shear rate eh is very nearly 11 within the equilibrium (i.e. can be generated r, 1985). Therefore in this t torque reqiiired for the steady rotation of the cone is shearing stress 0x1 its surhee by a sirnpli equat~ongiven as
+ihc are the e ~ ~ e r i m e iand ~ ~ atheoretical l torque, res The shear rate i s given by
(5.21) where w i s the a ~ ~ uvelo~ity ~ a r of rotation. The geometrical paramet~rsgiven as cto are tiic cone radius and coize angle ~iiegl~eting thc shaft t h i c ~ n ~ s s ) , respectively. ‘The inaterial parameters calculzateed on the basis of ~easiiredby cone-aidplate rheometry are only true if the stress on surface and the temperature field in the flow doinain remain exactly during the experiment. In-d analysis, however, reveals that distinctly iion‘f0;Os-m stress field iiiside these devices ( ~ h a t u r a i i and ~ this, deviation froin ~~~ifoi-mity is that ~ s ~ ~ n i 1990). an, the outer wall of the chamber is not exactly u ~ i i d ~ ~ e c tand io~i~~ it has been shown that the maxiximum error from this source is less than five percent (Kaye el al., 1968). ore important reasoiis for the deviation of the ~ o ~ ~ - a n ~ x- p~elr a~ t~from ~e n t a theorctical ‘visco to the action of c ~ ~ t r ~forces, f u ~ whicli a ~ cause secoiidary ~ ~ n ~ f oof~ viscous ~ ~ i t heat y g e ~ e r ~ t within i o ~ the flow dom
RATIONAL APPROXIMATIONS AND KLUSTRATTVE EXAMPLES
involving highly viscous fluids the n o n - ~ ~ ~ ~ f o r of m ithe t y rate of internal heat generation gives rise to a non-uniform stress distribution on the cone surface. This effect is more noticeable for Newtonian fluids. In relatively low viscosity fluids the secondary flows are the main cause of the stress non-uniformity in the cone-and-plate flow domain. This type of non-uniformity is more significant in experiments involving non-Newtonian fluids. Unlike the wall effects, the experimental errors arising from these types of flow field non-uniformity cannot be readily estimated. In the absence of direct experimental measurements, finite element modelling provides a reliable method for the evaluation of the errors resulting from the described stress non-uniformity in the cone-and-plate flow domain. Despite the deviation from an exactly viscornetric regime, it is still possible to assume that the flow ihside the cone-and-plate rheometer is very ncarly axisymnetric. Therefore for generalized Newtonian fluids, the circumferential component of the equation of motion can be decoupled from the other two components without neglecting centrifugal and Coriolis forces acting upon the flow field. In viscoelastic fluids, however, even after assuming an axisymmetric flow regime inside the domain, sonie of the terms in the circumferential component of the equation o f motion remain dependent on the variables in the radial and axial directions. In the following sections the goveriiing equations and finite element modelling of the flow field inside a cone-and-plate viscometer are described.
For e q u i l i b r i ~(i.e. steady) flow in a cone-and-plate domain with perfect s y m ~ e t r yit is evident that any changes in the circumferential (i.e. 8) direction can be neglected. In this case the governing conservation equations describing the flow can be considerably simplified by omitting all tenns containing derivatives with respect to the independent variable, 0. Note that the approach adopted for decoupling of the components of the governing equations in the present nearly axisymmetric flow is different from the procedure used for perfectly axisymmetric conditions. As described in Chapter 4, Section 1.3, in the latter case the governing equations can be analytically integrated with respect to 8. After omitting the derivatives with respect to 0 the resulting system of the model equations in the (r, 0, z) coordinate system is written as: Continuity
(5.22)
163
MODELS BASED O N DECOUPLED FLOW EQUATIONS
Component of the equation of motion in the predominant, i.e. 8 direction (5.23) where v,., $7, and vo are the components of the velocity vector and qr and qz are the extra stress components. Coinponeiits of the equation of motion in the ( r , z ) plane are given as
where p is pressure, g is acceleration due to gravity and are the components of the extra stress tensor.
T,,.,
r r Z rgn s
and
T~~
Energy equation
e , the specific heat capacity, k is the thermal where T i s t e ~ p e ~ ~ t ucr is coi~~~uctivity and ( 7 1 i i 2 ) is the viscous beat generation. The shear rate, 9, is expressed as
In order to account for the heat loss through the mcPallic body of the cone, a heat conduction equation, obtained by the elimination of the convection and source terms in Equation (5.25), should 'also be incorporated in the governing equations. In the Couettc flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer i s due to conduction. For very large conductivity c o ~ ~ ~ c i ethe n t heat s conduction will be very fast and the temperature profile will
1
RATIONAL APPROXIMATIONS AND TLLUSTRATIVE EXAMPLES
ost polymers, however, have very small coiiductivities and thc t e ~ ~ e r a t u profile re will be parabolic with a maximum between moving and stat~oiia~-y surfaces (Bird et al., 1959). This effect will be more significant in the bi-conical devices than a single cone rheorneter hecause most of the heat transfer is expected to be through the body of the cone. This analysis shows that heat transpo~twithin the flow domain and the cone body can be t n o ~ ~ ~ l e d si~ult~neou§~y: ~ o i i s t i t ~ t i equation. ve For a generalized Newtonian fluid the components of the extra seress and rate of d e f o r ~ ~ t i otensors n in this domain are related
(5.27)
and
e r ~ v ~ of t i an ~ ~appro i riate con§ti~utiv~ equation fo flow field can be sed on the u ~ ~ ~ r - c o i i ~ e ~ t e d ~ ~ q ~ a ~( I i,241. o t i owever, a crucial p ~ i n to t note is tb the flow regime ~ e n e r a ~by~the d rotatioii of a very small angle cone is only urbed. Therefore the coupling (i.e. de endency) of tlie radial and tial var~%bles is weak. varia~lesan the circuni xwell equatio~ican be e the c o n i ~ o ~ e not fstlie ion ~ x p a ~ s i o(Nayfeh, n 1993) and only keep e In~del.~ o l l o w i nthis ~ a ~ ~ r o a cthe h, a ~ ~ r o e ~ u ~ t ini othe~ cone-and-plate domain is found by of the c o ~ ~ ~ a i i eof~the i t sMaxwell equation, origina cat coordi~ate~ ~ ~witht res e cct ~ to, the small con . For a steady-state flow, k e e ~ i nrip~ to eh Cook, sions, these components are written in ter in the
MODELS BASED ON DECOUPLED FLOW EQUATIONS
165
(5.29)
8w A
2
a W
= 2Dc-n
8h
an + 2y + cr2De - U , an - vdr ah
[
where
and De = tref(w), where rotatioiial speed.
is the
axwell relaxatioii time and w is the
A similar a ~ p r o x i ~ a t i oshould n be applied to the components of the equat~on of motion and the significant terms (with respect to n) consistent with the expanded constitutive equation identified. This analysis sliows that orily U and A appear in the zero-order terms and heiice should be evaluated up to the second order. Furtherniore, nil of the remaining terms in Equation (5.29), except for E, appear only in second-order terms of the approximate equations of motion and only their leading zero-order ternis need to be evaluated to preserve the consistency of the governing equations. The term C, which only appears in the hig~er-orde~ ternis of the expanded equations of motion, can be evaluated a p p r o x i ~ a t ~ using l y only the viscous teims. Therefore the final set o f the extra stress ~ ~ m p o n e nused t s in conjunction with thc components of the equation or inotion are
(5.30)
The required working cquatioiis are derived by applicatioii of the following finite element scliemes to the described governing model: Standard Calcrkin procedure - to discretize the circumferential component o f the equation of motion, Equation (5.23), for the calculation of 1'0. Continuous penalty method - to discrctize the continuity and (r, z) components of the equation of motion, Equations (5.22) and (5.24)) for the calculation o f v, and v,. Pressure is computed via the variational recovery procedure (Chapter 3, Section 4). alerkin scheme ~ a ~ ~ u ~ofaT. tio~
-
to discretize the energy Equation (5.25) for the
Elemental stiffness equations (i.e. the working equations) resulting from the described discrcti~ationsare in general written as
(5.31) where, from Equation (5.23) for
vg
MODELS BASED ON DECOUPLED FLOW EQUATIONS
1
For generalized Newtonian fluids the load vector (i.e. the right-hand side in Equation (5.31) is expressed as I
=
J' iVt(~gPn,i-
(5.33)
7ijznz)rdT,
re where n, and n, are the components of the unit vector normal to rp.Note that for small cone angles IZ, is at least one order of magnitude smaller than n, and 7-o2 is one order of magnitude bigger than TO?. For viscoelastic fluids this term is given as
where Top = ror - rip
TgZ = Toz - T&
From the set of equations (5.22) and (5.24) (compact form based on the continuous penalty mcthod) for v, and v, Mij =
(5.35) where .A is the penalty parameter. The penalty terms in Equation (5.35) should be found using reduced integration. For generalized Newtonian Auids
1
RATIONAL A P P ~ ~ X I ~ A T AND I O ~ ISL L ~ S T R A T IEXAMPLES ~~
(5.36)
For the viscoelastic fluids
(5.37)
where Toe = roe - r&
From Equation (5.25) for T
where a1 and
a2
arc upwinding parameters ( etera er al., 1993).
The derived working equations are solved using the following soltrtioii a ~ ~ ~ r i t h ~ :
ASED ON DECOUPLED FLOW ~ Q U A ~ ~ ~I N S
Step I start with v, = v, = 0.0 and 77 = 710 constant. Step 2 - using appropriate working equations calculated vg. tep 3 - calculate i using Equation (5.26). tep 4 - update the value of viscosity (7)) using an appropriate r ~ e o l o ~ i c a l equation (e.g. te~~peraturc-dependent form of the Carreau model given as Eyuation (5.4)). Step 5 - using updated values of viscosity and calculated 110 calculate v,, vz arid p . For viscoelastic fluids also calciilate the ~ ~ d i t i o nstress al co~po~eiits at this step. Step 6 - update ?/ and 71 an Step 7 - if the ~ o l ~ t ihas oi~ p otherwise go to the next step. ate viscosity and elastic stresses and go to step 2. ~
Using the described a l g o r i ~ hthe ~ Bow domain inside the coiie-and-plate viscometer is simulated. 111 Figure 5.17 the predicted velocity field in the (r7 z ) plane ( s e c o ~ d afiow ~ ~ regime) establis~iedinside a bi-conical rheoaneter for a ewtonian fluid is shown.
.17 The prcdicted sccondary flow Geld in the bi-conical viscometer
The stress field c ~ r r e s p o n d to ~ ~this i ~ regime is shown in Figure 5.18. figure shows the ~ e a s u r surface i ~ ~ of the cone i s affected by these secondary stresses and hence not all f the measured torque is spent on generation of the p ~ ~ ~(i.e. ~ aviscome r y w in the c i r c ~ ~ f e r e i i td~ ai lr ~ c ~ o ~ .
ure
The predicted stresses arising from the secondary flow in the hi-conical
viscometer
RATIONAL APPROXIMATlONS AND ILLUSTRATIVE EXAMPLES
This simulation provides the quantitative measures required for evaluation of the extent of deviation from a perfect viscometric flow. Specifically, the finite element model results can be used to calculate the torque corresponding to a given set of experimentally determined material parameters as (5.40)
r where r represents the rotating cone surface. Through the comparison of this value with the experimentally measured torque a parameter estimation procedure can he developed which provides a strategy €or the improvement of the viscometry results in this type of rheometer (Petera and Nassehi, 1995).
D
R
N
There are many instances in polymer-forming processes where the flow i s confined to a thin layer between relatively large surfaces. For example, in calendering, injection moulding and film blowing the flow geometry is, in gcneral, viewed as a slowly varying thin film. The common approach adopted in the modelling of these flow systems is the utilization of the ‘lubrication approximation’ or its generalizatio~s.The original lubrication approximation, proposed by Reynolds, is based on the following assumptions: ~ a ldomiTlie Aaw regime is steady creeping (i.e. inertia less), i 3 o t ~ e ~and nated by viscous shear forces. The fluid is incompressible and Newtonian.
ne of the dimensions of the flow domain is very sinall in comparison to the other two dimensions. The sinall dimension (e.g. height) varies very slowly with respect to the geometrical variables in the other two directions (i.e. i%/c%x, ahlt3y << 1). There is no slip at the solid surfaces. Let us consider the flow in a narrow gap between two large flat plates, as shown in Figure 5.19, where L is a characteristic length in the s and y directions atid h is the characteristic gap height so that h << L. It is reasonable to assume that in this flow field vz < vZ, v?. Therefore for an inconipressible Newtonian fluid with a constant viscosity of p, components of the equation of motion are reduced ( ~ i d d l e ~ a 1977), n, as
171
MODELS BASED ON THIN TAYER APPROX1MATIC)N
(5.41)
. + h
’ I
X
e 5.19 Schematic representation of a thin-layer domain between flat surfaces
According to equation set (5.41) the pressure in this flow domain is a function of x and y only and the pressure gradient normal to the parallel walls is zero. The described approximations are equivalent to the assumption of a fully developed flow between two large plates, narrowly separated with a local gap height of Iz = h(x, y). The selection of appropriate boundary conditions for equation set (5.41) depends on the type of flow regime within the domain. For example, in the standard lubrication approximation model, the established flow in thc narrow gap is considered to be a Couette regime generated by the stcady motion of the top plate (Lee and Castro, 1989). In polymer processing, depending on the operating conditions, the established flow in a narrow gap may be a shear (i.e. Couette) or a pressure driven (i.e. Poiseuille) regime or a combination of both of these meclianisms. As an example, we consider a Poiseuille-type flow, characteristic of injection moulding, within the narrow gap shown in Figure 5.19. The boundary conditions corresponding to this flow are expressed as vx
=0
vu = 0
at z = 0 and z = h
(5.42)
Pressure gradients in the x and y directions do not depend on z, hence the integration of the components of the equation of motion, subject to conditions (5.42), yields
1'72
RATIONAL A P P R O X I ~ A T AND I ~ ~ I~ ~ L ~ ~ S T R~ X AATM~P L~E S
(5.43)
Integration of the velocity components, given by Equation (5.43), with respect to gap-wise direction (i.e. 2) yields the volumetric flow rates per unit width in the x and ,y directions as
(5.44)
We now obtain the integral of the coii~inuityequation for iiic~mpressibl~ fluids with respect to the Local gap height in this flow domain
or
(5.46)
The last term in Equation (5.44) vanishes because Therefore
llz
=L
0 at z = 0 and z = h.
h
(5.47) 0
(5.44) for the integrals in Equation (5.47) gives u ~ s t i ~ ~from i t iEquation ~~
(5.48) or
MODELS BASED ON THlN LAYER A P P ~ O X I M ~ T I ~ N1173
(5.49) Equatioii (5.49) derived for isothermal Newtonian flow in thin cavities, is called the ‘pressure potential’ or Hele-Shaw equation. Analogous equations in terms of pressure gradients can be obtained using other types of boundary conditioiis in the integration of components of the equation of motion given as Equation (5.41). The lubrication approximation approach has also been generalized to obtain solutions for non-isothermal generalized Newtonian flow in thin layers. The generalized ele-Shaw equation for non-isothermal generalized Newtonian fluids is used e nsively to model narrow gap flow regimes in injection and compression moulding (Hieber and Shen, 1980; Lee et al., 1984). Other generalized equations, derived on the basis of the lubrication approximation, are used to model larninar Row in calendering, coating and other processes where the ry allows utilization of this approach (Soh and Chang, 1986; The generalized Hele-§haw equation i s expressed as (5.50) where p(x, y ) is pressure, QO is a source term and S is called flow conductivity which for a non-elastic generalized Newtoniao fluid is given (Sch~ichting,1968) as
0
where 71 i s the apparent viscosity o f a generalized Newtonian fluid. For isothcrinal flow of a power law fluid the flow conductivity S can be found analytically. For the non-isothermal case and when q is temperature dependcnt, Equation (5.50) should be solved in conjunction with the energy equation yielding the gap-wise temperature distribution, and S has to be obtained by numerical integration. To deinonstrate application of the described approximation in polymer processing, in the following section modelling of flow distribution in an extrusion die is discussed. ite e
ie
e consider a co-extrusion die consisting of an outer circular distribution channel o f rectangular cross-section, connected to an extrusion slot, which is a slowly tapering narrow passage between tn70 flat, non-parallel plates. The polymer melt is fed through an inlet into the distribution channel and flows into
RATlONAL APPROXIMATIONS AND ILLUSTRATIVE EXAMPLES
the tapered slot to exit from its circular outlet. The objective of simulation is to find tapering in thc extrusion slot that can result in a uniform flow distribution at the die outlet. To find flow distribution the generalized ~ele-Shawequation is solved to obtain the pressure field within the domain. The pressure gradients are then used to calculate the mean velocity across the gap (or the flow rate per unit width, perpendicular to flow direction) from the following equations, derived analogous to Equations (5.43)
(5.52)
Thus (5.53) and Vv are the components of mean velocity, where and Qy are the flow rate per unit width in the x and y directions, respcctively, h is the gap height and S is defined as in Equation (5.50). To obtain the desired tapering the gap height is defined as
rrZ = a + by
(5.54)
where U and b are constants that should be determined through this simulation. The finite element discretization o f the generalized Hele-Shaw equation (Equation (5.50)) used in this example is based on the sLandard ~ a ~ c rscheme. k i ~ The velocity field corresponding to the simulated pressure distribution at each step is found by application of the variational rccovery method (see Chapter 3 , Section 1.4) to equation set (5.52). The developed solution algorithm is as follows: Step 1 - assume a Newtoniari flow and obtain the nodal pressures. Step 2 - use the calculated pressures and find the velocity components by variational recovery. Step 3 - using the calculated velocity field, find the shear rate and update viscosity using the power law model. tep 4 - calculate flow conductivity, 3, using updated viscosity and repeat the solution until convergeiice. The described algorithm inay not yield a converged solution; in particular for values of power taw index less than 0.5. To ensure convergence, in the iteration cycle {n -+ 1) for updating of the nodal pressures, an initial value found by
MODELS BASED ON THIN LAYEK A ~ ~ ~ O X I M A T I O N
interpolating the values ofp at cycles n and ( E - 1) should be used. Alternatively, the geometric mean values of the pressure gradients are found at each step and used in the iteration cycle to ensure convergence (Sander, 1994). Boundary conditions for the solutioll of Equation (5.50) are zero pressure on the exit and zero norrnal pressure gradient at the inlet. The exit condition can be imposed directly; to impose the inlet conditions a layer of virtual elements i s attached to the outer edges of the elements at thc inlet section. A suitable value for the source term in Equation (5.50) is chosen in these elements. The sourcc term in all other elements is set to zero. All of the boundary line integrals, appearing after the application of Green's theorem to the discretized model equation, are also set to zero. The described bouiidary conditions ensure, respectively, that melt leaving the extrusion slot at the exit is flowing in the radkal direction, and there is no fluid flow from the outer edges of the virtual elements. A series of simulations has been carried out, varying the parameters U and h in Equation (S.S4), this is equivalent to altering the extrusion slot height and taper. The results of successive simulations are compared until a die geometry that yields uniform exit flow i s obtained. In Figures 5.20a and 5.20b the exit flow unifomiity, indicated by the length of the predicted velocity vectors, for sets U € (a = 0.0011; h = -0.0005) and (a = 1.3; b = -0.001), respectively, is shown. As shown in Figure 5.20b, use of the second set of parameters generates an effectively uniform exit flow. This set has been found by trial and error after 10 runs (Nassehi and Pittman, 1989).
ne of the main restrictions preventing wider application of the thin-layer approach, described previously, to industrial polymer flow processes is that these models can only be used for narrow gaps between flat surfaces. Therefore generalization of the method to thin-layer flow between curved surfaces, which removes this limitation, can significantly enhance its ap~licabilityto realistic problems. To obtain such a generalization an asymptotic expansion scheme corresponding to the geometry of the thi layer domains defined ill a curvilinear coordinate system is used (Pearson and trie, 197Qa,b;Pearson, 1985). Let us consider a thin layer between two cur non-parallel surfaces as shown in igure 5.21: The ratio of characteristic gap height H t o lateral di~ensionsL is E = (h/L)--+ 0. In the general orthogonal curvilinear coordinate system ( 2 )defined for the thin domain the governing equations o f the steady, creeping Bow of a power law fluid are written as continuity
(5.55)
RATIONAL ~ ~ ' ~ O X l AND ~ A ILLUSTRATlVE T I ~ ~ ~ EXAMPL~~
Virtual elements used to
2 0 (a) The predicted non-uniform exit Row distribution in a ca-extrusion die. (b) The predicted uniform exit flow distribution obtaincd after altering the gap height and tapering in the co-extrusion die
I Figure
z
x3
Thin layer between curved surfaces and the general curvilinear coordinate system
MODELS BASED ON THIN LAYER A P ~ R O ~ ~ M A177~ I ~ ~
motion
(5.56) where in Equations (5.55) and (5,56), h, are the scale factors which represent the ratio of an infinitesimal displacement of a point corresponding to incremental changes of its coordinates (Ark, 1989), v ( i ) are the physical components of the velocity and p i s the pressure. In the equation of motion ~ ( i jj ,) is expressed in t e r m of the coniponents of the stress tensor as
(5.57) The Christoffel symbols (Spiegel, 1974), appearing in the second term on the right-hand side of Equation (5.57), are defined as
ere I, r and s are unequal integers in the set { 1, 2, 3 ) . As already mentioned, in the thin-layer approach the fluid i s assumed to be non-elastic and hence the stress tensor here is given in t e r m of the rate of deformation tensor as: ~ ( t j =) *t$)(g), where, in the present analysis, viscosity 77 is defined using the power law equation. The model equations are non-dimensionalized using
Non-dimensionalization of the stress is achieved via the components of the rate o f def’ormation tensor which depend on the defined non-~~ii~eiisional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stress. After substitution of the n o ~ - d ~ ~ e n $variables i o ~ ~ l into the equation of continuity it can be d i v i ~ ~ e ~ through by (&L2U).Note that in the following for simplicity of writing the broken over bar on the non-dimensional variables is dropped.
Let So be a surface located at mid-channel between two smooth surfaces separated by a narrow gap. The curvilinear coordinate system, cor~espoiidi~ig to this
RATlQNAT, APPROXIMATIONS AND lLLUSTR ATTVE EXAMPIBS
geometry, is constructed as (x'+x2) forming a set of orthogonal lilies on $0 whilst x3 is perpendicular to this surface. The length along the x 3 coordinate line from a point on SOto the top and the bottom surfaces of the channel is equal and is half the local chaniiel height. Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a tzniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as /L) 3 0. Coordinate scale factors A,, as well as dynamic variables are represented by a power series in 5. It i s expected that the scale factor h3, in the direction normal to the layer, is O(E) while hl and IQ, are O(L). It is also anticipated that the leading terns in the expansion of h, are independent of the coordinate x3.Similarly, the physical velocity components, v1 and 1'2, are O(U), where U is a characteristic layer wise velocity, while v3, the component perpeadicular to the layer, is @&U). Therefore we have
and
and
Following the procedure for obtaining the required equations corresponding to the leading terms in the asymptotic expansions, series (5.58) to (5.60) are substituted into the governing equations. This operation i s straightforward but rather lciigthy, as an example, the leading terms corresponding to the components of the shear stress are derived as
After substit~itionof the leading terms of the expanded variables into the model equations and equatiiig coefficients of equal powers of E from their sides, they are divided by common factors to obtain the followjng set: ~ontinuityequation corresponding to the first-order ternis
MODELS BASED ON THIN LAYEK APPROXIMAT~ON
d
'
-(Vihh
dx
a
a
f s ( h V 2 h 3 ) f-(hlh2v3) 8x3
17
(5.61)
=0
Components of he equation of motion
(5.62)
Equations (5.61) and (5.42) can be used to derive a 'pressure potential' equation applicable to thin-layer flow between curved surfaces using the following procedure. In a tliin-layer flow, the following velocity boundary conditions are prescribed: Lower surface, x3 = xL? (solid wall) -+ v1 = 1~2= v 3 = 0
av1 = --a v 2 iddle surface, x3 = 0 (surface of symmetry) -, -0, TJX-3
8x3
v3=0
Upper surface, x3 = x-2: (solid wall) --+ v1 = v 2 = v3 = 0 Integration of the first-order continuity equation between the limits of the thin layer gives
(5.63)
First-order t e r m of hi are independent of x3,hence
+ [hih2,';];!= 0
(5.64)
After the imposition of no-slip wall boundary conditions the last term in Equation (5.64) vanishes. Therefore
1 8 ~ RATIONAL A P ~ R O X ~ ~ A T ~AND O N SILLUSTRATIVE EXAMPLES
a ~
(3x1
(5.65)
Considering that the pressure is iiidepeiideiit of x3,integratioii of the xi and x2 components o f the first-order equation of motion from 0 to x3 gives
(5.66)
After the imposition o f the boundaiy conditions at x3 = 0 we have
(5.67)
Secoiid integration of components of the equation of motioii yields
(5.6%)
and
(S.69)
~ ~ ~ o s ioft the i o boundary ~ conditions at x3 = xi. gives
MODEI,S RASED ON ‘THIN LAY ER A P F R ~ X I M A T I O ~ 1
(5.70)
where
Assuming symmetry o f v1 and v2 with respect to mid-surface at (x3 = O), the velocity coniponeiit§, given by Equation (5.70), are integrated to obtain flow rates in the lateral directions within the limits of the thin layer as
dX3
(5.71)
dx3 x;,
Let
U
=
dx3)/r/, dv = dx’. The dllrmny variable x3 is the lower limit of
, J( x 3
U,
\’
therefore we have: du = -(x3 dx’)/q, v = x3. Taking terms independent of outside of the integration signs in the first equation of set (5.71)
X’
(5.72) The first tern1 on the right-hand side o f Equation (5.72) is zero, and
(5,734
Similarly
2
RATIONAL APPROXIMATIONS AND ILLlJSTR ATIVE EXAMPLES
(5.73bj
Substituting from Equations (5.73a) and (5.73b) into the continuity equation (5.65) yields
(x3)>” dx3
(x3)’dx3
71
TI
-- 0
(5.74)
After the substitution for A, and A2 into Equation (5.74) the pressure potential equation corresponding to creeping flow of a power law fluid in a thin curved layer is derived as
(5.75) where the flow conductivity coefficients are defined as
0
(5.76)
The scale factors given in the above expressions depend on the curvilinear coordinate system adopted to model a thin-layer flow. For example, the scale factors for a cylindrical coordinate system of (r, 8, z) are h, = 1, ho = r and hz = 1, and for a spherical coordinate system of (R, 0, 4) they are ht( = 1, h e = R and sin 8 (see Appendix and Spiegel, 1974). comparison of flow conductivity coefficients obtained from (5.76) with their counterparts, found assuming flat boundary surfaces in a thinlayer flow, provides a quantitative estimate for the error involved in ignoring the curvature of the layer. For highly viscous flows. the derived pressure potential equation sl.rould be solved in conjunction with an energy equation, obtained using an asymptotic expansion similar to the outlined procedure. This derivation i s routine and to avoid repetition is not given here.
STIFFNESS ANALYSIS OF SOLID POLYMERIC MATERIALS
1
The focus of discussions presented so far in this publication has been on the finite element modelling of polymers as liquids. This approach is justified considering that the majority of polymer-fonning operations are associated with temperatures that are above the melting points of these materials. However, solid state processing of polymers is not uncommon, furthermore, after the processing stage most polymeric materials are used as solid products. In particular, fibre- or particulate-reinforced polymers are major new material resources increasingly used by modern industry. Therefore analysis of the mechanical behaviour of solid polymers, which provides quantitative data required for their design aiid manufacture, i s a significant aspect of the modelling of these materials. In this section, a ~ a l e r k i ~ finite i element scheme based on the continuous penalty method for elasticity analyses of different types of polymer composites is described. To develop this scheme the mathematical similarity between the Stokes flow equations for incompressible fluids and the equations of linear elasticity is utilized. We start with the governing equations of the Stokes flow of incompressible Newtonian fluids. Using an axisymmetric (r, z) coordinate system the components of the equation of motion are hence obtained by substituting the sheacdependent viscosity in Equations (4.11) with a constant viscosity p , as
(5.77)
And the continuity equation which is identical to Equation (4.10)
av, v, 3v, --+-+--=o ar r 8 z
(5.78)
In the continuous penalty scheme used here the penalty parameter is defined as A.=-
2vp (1 - 2v)
(5.79)
where v it; Poisson's ratio which is equal to 0.5 for incompressible material. As seen in Equation (5.79) the present model cannot be applied to analyse perfectly incompressible materials. The working equation of the scheme is identical to Equation (4.70). by comparison it can be shown that the working equations in this scheme are identical to their counterparts obtained by the commonly used equilibrium finite
9
RATIONAL APPROXIMATIONS AND ~ L L U S l ~ A T I VEXAMPLES E
element approach for elasticity analysis, provided that p is defined as the shear modulus and X as the bulk modulus of the matcrial, respectively ( Note that in this case the main field variables should be regarded as displacements instead of velocity components. This similarity provides an important flexibility to switch the model from the analysis of fluid flow to solid material def~rmationunder applied loads. The majority of polymeric materials are processed as liquids and used as solid products. ence, the described approach has the advantagc that it can predict matcrial behaviour in both liquid and solid states using the same computer program. A full account of utilization o€ the present model in the analysis of polymeric composites has been published previously (Ghasse~nieh and Nassehi, 200 1 a) and here only an illustrative a p ~ ~ i c a ~ is i odiscussed. n
ulk mechanical properties of polymeric composites, such as their modulus, depend on the properties of their constituent materials, filler/matrix volume fraction and geoinetrical distribution of the filler phase inside the matrix. C ~ a ~ s i ~ c aoft ipolymer o~~ composites is based on the shape of the ~ ~ ~ n f o r c i i i g phase and hence they are grouped as particulate, continuous or short-fibre coinposites, each associatcd with distinct mechanical properties and used for a dif€~rentpurpose, Therefore in the modelling of ch class of polynier composites a different set of criteria should be considered. owever, the present model has the flexibility to be used under diffcrent conditions for a wide Tariety of polymeric materials (Nassehi et al.. 1993a,b; Ghassemieh and Nassehi, 2QQlb,c), In the following micro-mechanical analysis, it is assumed that the domain of interest for a polymer composite filled with spherical particles can be represented by a unit cell as shown in Figure 5.22. When this unit cell IS rotated 360" around the axis AD, a hemisphere embedded in a cylinder is produced, The inter-particle spacing corresponding to this geometry is equal to 2(rl - r2). Therefore the volume fraction of tlie filler can be calculated from the ratio r z / r l . For a square or cubic array arrangement the relationship between the filler volume fraction and this ratio is giveii as (5.80) While for a hexagonal array tlie same relationship is expressed as Vf =
5):(
3
(5.81)
It should be noted that the described axisymmetric unit cells do not represent actual repetitive sections of a material but their dimensions are related to the
STIFFNESS ANALYSIS OF SOLID POLYMERIC M A T ~ K I A ~ S 1
B
A
b*
c
Y
e 5.22 Problcin doniain in the micro-mechanical analysis o i the particnlatc polymer composite
inter-particle spacing. The theoretical maximum filler volume fractions corresponding to square and hexagonal armys are found when the ratio r2/rl in Equations (5.80) and (5.81) is equal to 1 . These volume fractions are hence found as 0.52 and 0.74 for square and hexagonal arrays, respectively. 111 practice, however, filler volume fractions in polymer composites are much lower than these theoretical values. To simulate tensile loading of the domain shown in Figure 5.22 the following boundary conditions are imposed v- = 0
vz =
v
v, = U vr
=U
along AB (2 = 0) along CD ( z r= bo) along BC ( r = ~ 1 )
(5.824 (5.82b) (5.82~) (S.82d)
where V is a prescribed constant and U is determined from the condition of vanishing average lateral traction rate, defined as
(5.83) 0
In addition to the boundary conditions (5.82a-5.82d), it i s required that the displacement components should vanish on the surface of a rigid filler. The highly constrained boundary conditions shown in Equations (5.82a) to (5.82d) can be relaxed via replacing conditions (5.82~)and (5.82d) by orr= 0 on P‘ = r l which is the stress-free condition along BC. Using this set of ‘unconstraint’ conditions the outer side wall of the cell does not remain straight and
186
RATIONAL APPROXIMATIONS AND 11,LUSTRATIVE E X A ~ ~ L ~ S
vertical under loading. The described relaxation of the boundary conditions permits consequences of deviations from lighly constrained filler distribution to be investigated. esults obtained using relations (5.82a) to (5.82d) are referred to as the ‘with constraint’ analysis. Calculation of the field variables in the ‘unconstraint’ analysis is straightforward, however, to model tlie ‘with constraint’ case, the following procedure is used: (1) The displacements and stress distribution are found imposing conditions (5.82~~) to (5.82d). Note that the displacements arc found using the working equations of the scheme; stresses are found via the variational recovery method.
(2) The field variables are recalculated replacing (5.82b) with v, = 0 along C ( 3 ) The first and second set of resuits are superimposed, therefore the frnal set of results are found as v =V]
+
k
~
2
(5.84)
and = 01
+ kcrz
(5.85)
where k i s determined such that the net force in the r direction along BC is zero, therefore (5.8G) where k+)
BC
(5.87)
Thus the predicted stress along AB is (5.88) And because ( v , ~ ) A=~ 0 the displacement along this direction is expressed as (5.89)
STIFFNESS ANALYSIS OF SOLID POLYMERIC MATERIALS
1
Let us now consider the modulus of the composite defined as the ratio of stress over strain, i.e. E = (cTJE~). The strain in this example is found using the specified boundary condition as
Cl). The average stress along A
ET, = A
is also calculated as
(5.90)
~
A
where A is the area of the top surface of a cylindrical unit cell in the finite element model. Using the results obtained by finite element simulatioris the modulus of a particula~~-filled polymer composite can heiicc be found. i n Figurc 5.23 the finite element model predictions based on ‘with constraint’ and ‘uiiconstrained’ boundary conditions for the modulus of a glasshpoxy resin composite for various filler volume fractions are shown.
20 composite modulus (Pa) x i09
15
10 5
0
10
20
30
40
50
$0
Filler volume fraction (%)
e 5.23 Coniposite modulus obtained using constrained and unconstrained boundary conditions
In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under ‘uncoiistrained’ conditions for both phases are sllown, The material parameters used in this calculation are: elasticity modulus and Poisson’s ratio of Pa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, ccording to this result the position of maxirriiun stress concentration i s almost ~ i r e c ~ above ~ y the pole of the spherical particle. Therefore for a
1
RATIONAL A ~ P R O ~ ~ ~ AND A r IL1,USTRATIVE I ~ ~ S ~ ~ A M P L ~ ~
glass sphere, well bonded to the epoxy resin, the cracks in this mateiial should start from this position and grow in the direction o f direct stress. This has been ~ x p ~ r i ~ i e n t aconfiriiied l~y using scanning electron xnicroscopy ( ~ l ~ a ~ ~ e ~ i e 1998).
0.9 0.8 0.7 0.6 0.5
Position
0.4 0.3 0.2
10
15
20
25
30
35
40
Filler volume fraction (77)
Figure
The predicted direct stress concentration at different locations within the cloiiiaiii
Andre. J. M. ci al., 1998. Numerical modelling of the polymer film blowing procew. In,. 1.Forming P r o c ~ , w s ancl Busic Equations o j Fluid 1WeriEiaiiics, Dover E. 1999. Nunierical simulation of the filni casting process. ssager, O., 1977. Dyrzurriics of Poljmer Fluids, Vol. tfoot, E. N., 1959. Trrrmport PherioPwenu, Wiley,
Chaturani, P. and Narasimman, S., 1990. Flow of power-law fluids in cone-plate viscometer. Acta Mcchmica Clarke, .1. and Freaklcy, P. K., of dispersive mixing and filler agglomerate sire distri~utionsin rubber compounds. Plu.,~. Xithber cbvnycix Process. Appl. 2 241-266. onea, 1. and Quartapclle, L., 1992. An i n t r ~ d u c ~to ~ ~finite i i element methods for transient advection problems. Coinput. Methods AppL Mech. B i g . Chafouri, S.N. and Freakley, P. K., A new method of flow visualisation for rubber mixing. Pol. Test. 13, 171-379,
REFERENCES
1
Ghassernieh, E., 1998. P1i Thesis, Department of Chemical Engineering, Loughborough University, Loughborough. Ghassemieh, E. and Nassehi, V., 200 la. Stiffness analysis of polymeric composites Using the finite element method. Adv. Poly. Tkh. 20, 42-51. Ghassemieh, E. and Nassehi, V.,2001 b. rediction of failure and fracture ~iiechatlismsof polymeric composites ng finite element analysis. Part 1: particulate filled composites. P01.v. Coinpas. Ghassemieh, E. and Nassehi, V., 2001~.Prediction of failure and fracture mechanisms o f polyriieric coniposites using finite element analysis. Part 2: fiber reinforced composites. Poly. Compos. 22, 542- 554. Ghorcishy, M. M, R., 1997. PhD Thesis, Department of Chemical ~ n~ itieering, Loughborough University, Loughborough. Ilannart, B. and Hopfinger, E.J., 1998. Laminar flow in a rectangular diffuser near dimensional numexical simulation. In: Nele-Shaw conditions - a . (eds), Flow ~ ~ u f l ein~~l n~d n~ ~ t rPri u l Lewis, B. A. and Warren, orwood, Chichestcr, pp. 110- 118. Hieber, C. A. and Shen, S.F., 1980. A finite eiemcn~~fini~e differcncc simulation of the injection-moulding filling process. J. Non--~eivtonianFluid Mech. 7, 1-32, Hughes, T. J. R.,1987. TXe Finite Element Method, Prenticc-Hall, Eiiglewood Cliffs NJ. ress - effective flow in rubber mixing. termination of normal stress difference
plifications. In: Tucker, CL.111 (ed.), CoPnpuZer ibfodelng for Polymer Procctr.sifig,cl?a;pter 3, Haiiscr Publishers, Munich, pp. 7-112. Lcc, G. C., Folgar, F. and Tucker, C. E., 1984. Simulation of coiiiprcssion molding for fiber-reinforced thevinosetting polymers. J. Eng. Ind. 10 , 1977. Fi~n~~z~ri~~i~~1.s o j Polyrner Processing, McGrawhillon, J. and MasL-a, L., 19%. Finite ele mechanics of interlayered polynierlfibre ColnpoSit : a study of the i i i t ~ ~ ~ t c ~ i o ~ between thc reinfo s. Cor~ipos.Sci. Tech. Nassehi, V., Kinsella, ascia, I..,1993b. liini delling of the stress distribution in polymer composites with coated fibre interlayers. J. Compos. Mnter. ., 1997. Simulation of free surface flow in partially
ng of flow distribution in ~ o ~ ~ e linl Iyz~z~,st~iul i~g Processes, Chapter 8, Ellis Horwood, Chichester.
190
RATlONAL APPROXIMATIONS AND ILLU STKATIVE EXAMPLES
Olagitju, D.O. and Cook, L.P., 1993. Secondary flows in cone and plate flow of an Oldroyd-B fluid. J. Non-Newtoninn Fluid Mech. 46, 29 -47. Pearson, d. R. A., 1985. Mechanics of Pol~imerProcessing, Applied Science Publishers, Barkings, Essex, UK. Pearson, J. K.A. and Petrie, C . J. S . , 1970a. The flow of a tabular film, part 1: fornial mathematical representation. J. Fluid Merh. 40, I - 19. Pearson, J. R. A. and Petrie, C. J. S., 1970b. The flow of a tabular film, part 2: interpretation of tlie model and discussion of solutions. J. Fluid Mech. Petera, J. and Nassehi. V., 1995. Use of the finite element modelling technique for tlie improvement of viscometry results obtained by cone-and-plate rheometers, J. 1VoizNewtonian Fluid Mecli. 58, 1-24. Petera, 1. and Nassehi, V.. 1996. Finite element modelling of free surfacc viscselastic flows with particular application to rubber mixing. h t . J. Numer. Methods Fluids 23, 1117-1132. Petera, J., Nassehi, V. and Pittman, J. F. T., 1993. Petrov CMerkin methods on isoparametric bilincar and biquadratic elements tested for a scalar convection-diffusion problem. Int. J. Numer. Methods Heat Fluid Flow 3, 205-222. Petera, J. and Pittinan, J. F. T., 1994. Tsoparametric Hermite elements. Ink. J. ATuiner. Methods Eng. 37, 3489-3519. Sander, R., 1994. PhD ‘Thesis. Cheniical Engineering Department, University College of Swansea, Swansea. Schlichtirrg, H., 1968. BouPidury-Luyer Theory. McGraw-Hill, New York. Soli, S. K. and Chang, G. J., 1986. Boundary conditions in the niodeling of iiqection mold-filling of thin cavities. Polynz. Eng. Sci. 26, 393-399. Spiegel, M.R., 1974. Vector Analysis, Scliawn’s outline series. McGraw-Hill, New York. Tanncr, R. I., 1985. Eagineering Rheology, Clarendon Press, Oxford.
In the finite element solution of engineering problems the main tasks of mesh generation, processiiig (calculations) and graphical representation of results are usually assigned to independent computer programs. These programs can either be embedded under a common shell (or interface) to enable the user to interact with a41 three parts in a single environment, or they can be implemented as separate sections of a software package. Development and ~ r g a n i ~ a t i oof n graphics programs requires expertise in areas of computer science and software design which are outside the scope of a text dealing with h i t e element techniques and hence are not discussed in the present chapter. Detailed explan~tio~i of mesh generation techniques and mathematical background of the available methods - although of general importance in numerical computations - are also not related to the main theme of the present book. Tn the following sections of this cliapter therefore, after a brief description of the main aspects of mesh generation, other topics that are of central importance in the finite element olyrner processes are discussed.
As disc~isse~ in the previous chapters, discretization of the solution doinaiii into an appropriate computational mesh is the first step in the finite element simuain factors in the selection of a particular mesh design for a problem are domain geometry, type of the finite elements used in the discretization, required accuracy and cost o f computations. In this respect, the accuracy of c o ~ p ~ t a t i o ndepends s on factors such as: consistency of the adopted mesh with the problem domain g~onietry, nature of the solution sought, and total number, size, aspect ratio and type of elements in the mesh.
FINITE ELEMENT SOFTWARE
MAIN COMPONENTS
As a gcneral rule simulations obtained on coarse grids consisting of deformed elements of high aspect ratio are expected to have poor accuracy and should be avoided. In order to iiicrease the accuracy of the solutions however, the mesh refinement shottld be based on a systematic approach which takes into account featL~resof the physical phenomenon being analysed. Therefore it is necessary to use all available knowledge about the nature of the problem as a guide to optiinize the mesh design and refinement.
es Finite element solution of eiigineering problerns inay be based on a ‘structured’ or an ‘unstructured’ mesh. In a structured mesh the form of the elements and local organization of the nodes (i.e. the order of nodal connections) are indep e ~ ~ e of n ttheir position and are defined by a general rule. In an unstructured mesh the connectioii between neighbouring nodes varies from point to point. Therefore using a structured mesh the nodal connectivity can be i ~ ~ p l ~ c i t l y d e ~ and ~ explicit ~ e ~ inclusion of the connectivity in the input mesh data is not bviously this will not be possible in an unstructured mesh arid nodal ut the computational grid must be s aified as part of the grids tant, however, to note that s t r ~ ~ c ~ uco~pL~lationa1 re lack flexibility and hence are not s ble for engineering problems which, in general, involve complex geometrics. cretization of domains with comi~licate~~ ~oundariesusing structured grids is y to result in badly distorted elements, thus pr g robust and accurate numerical solutions. Using an unstructured mesh, rical complexities can be handled in a more natural manner allowing for local adaptation. variable element conc ation and p r c f c r e ~ i t ~ ~ ~ lution of selected parts of the problem domain. wever, because o f the rent complexity of data handling in unstructu mesh gene ratio^ this oach requires special progranis for the organization and recor~ingof nodes, nl edges, surfaces, etc. which involve extra memory reqLiir~ni~nt.In p ~ r ~ ~any c ~increase ~ ~ r in , the number of e ~ e ~ c n during ts mesh r e f i n ~ ~ e n t requires rapidly rising computational efforts. A further drawback Cor uris~ructuredgrids is the difficulty o f handling moviiig boundaries in a purely angian approach in the siiiiL~lationof flow problen~s. o resolve the problems associated with structured and LinstrL~ctL~~~d grids, these fundame~tal~y different approaches may be combined to generate mesh types which partially posses the properties of both catcgoi-ies. This gives rise to ‘block-~~ructLi~~d’, ‘overset’ and ‘hybrid’ mesh types which under certain conditions may lead to more efficie simulatioiis than the either class o f purely structurcd or u ~ i s t ~ ~ c t ugrids. red etailed discussions related to the properties sses of coinputational grids can be found in specialized textbooks eikin, 1999) and only brief de~nitionsare given here.
CONSIDERATTONS RELATED TO FINITE. ELEMENT MESH GENERATION
Tn this approach the domain of the solution is first divided into a number of large sub-domains without leaving any gaps or overlapping. This division provides a very coarse unstructured mesh which is used as the basis for the generation of structured grids in each of its zones. The union of these local grids gives a computational grid for the entire domain, called a block-structured (or a m ~ i ~ ~ i - b l omesh, c k ~ The ~ e x i ~ igained ~ i ~ yby this approach can be used to haiidle complicated doniains having multiply connected boundaries, problems involving heterogeneous physical phenoniena and matheintttical non-uniformity. Figure 6.1 shows representative examples of block-structured grids with diffe~entforms of linking or ‘comniunication interface’ between adjacent sub-regions.
(discontinuous)
I
(non-smooth)
igure 6.1 Types of interface between blocks in block-structured grids
It is important to note that finite element computations on multi-block grids iiiv~lvinga dj$conti~uo~is interface are not straightf~rwardand special a ~ ~ a ~ g e inents for tlic transformation of nodal data across the internal boundaries are required.
verset grids ]in these girds the sub-regions or blocks are allowed to overlap and therefore the formation of a global grid is based on the assembly of i n ~ ~ v i ~ gu ean~~~r ayt e ~
I.
FlNlTE ELEMENT SOFTWARE
- MAIN COMPONENTS
structured mesh for sub-sections of the problem domain. To preserve the consistency of finite element discretizalioiis coirimunication between the overset regions should be based on systematic data traiisfo~matio~ using appro~riate interpolation procedures over overlapping areas of the computational grid. Figure 6.2 shows an example of this type o f computational mesh.
~~~~~e6.2 Representative fragment of an overset mesh
hrid grids ybrid grids are used for very complex geoinetries where combination of structured mesh segments joined by zones of unstructured mesh can provide the best approach for discretization of the problem domain. The flexibility gained by combining structured aiid unstructured mesh segments also provides a facility to improve accuracy of the numerical solutions for field problems of a c o n i p ~ ~ ~ anature. ted Figure 6.3 shows an example of this type of computational mesh.
e 6.3 Representative fragment of a hybrid mesh
CONSI~E~TIO KELATED ~S TO FINITE ELEMENT MESH GENERATION
It should be emphasized at this point that the basic requirements of cornpatibility and consistency of finite elements used in the discretkation of the domain in a field problem cannot be arbitrarily violated. Therefore, application of the previously described classes of computational grids requires systematic data transfo~nationproccdures across interfaces involving discontinuity or overlapping. For example, by the use of specially designed ‘mortar elements’ necessary coinmuiiicatiovl between illcompatible sections of a finite element grid can be established ( aday et al., 1989).
era The most common approaches for the generation of structured grids are as follows: Algebraic niethods - in these techniques calculation of grid coordinates i s based on the use of interpolation formulas. The algebraic methods are fast and relatively simple but can only be used in domains with smooth and regular boundaries. Differential methods - in tliese techniques the internal grid coordinates are found via the solution of appropriate elliptic, parabolic or hyperbolic partial differential equations. Using these procedures it is always possible to generate smooth internal divisions. Therefore they offer the advantage of preventing the extension of the exterior boundary discontinuities to the inside of the problem doinain. Variational methods - theoretically the variational approach offers the most powerful procedure for the generation of a computational grid subjcct to a multiplicity of constraints such as smoothness, uniformity, adaptivity, etc. which cannot be achieved using the simpler algebraic or d~fferen~ial techniques. However, the development of practical variational mesh generation techniq~tesis complicated and a universally applicable procedure i s not yet available. The most c o n ~ o approaches n for the generation of u n s t r u c t ~ ~grids r e ~ are as follows: ctree method - this method consists of two stages. In the first stage the problem domain is covered by a Cartesian grid and during the second stage this grid i s r ~ c ~ r s i ~subdivided. e~y owever, in this technique the computational domain bouii~aryis effectively constructed by combi~ingsides of grid elements and may not exactly match the prescribed problem domain bo~~tidary .
FINITE ELEMENT SOFTWARE - MAIN COMPONENTS
Delatinay method - in this method the computational grid is essentially constructed by connecting a specified set of points in the problem domain. The connection of these points should, however, be based on specific rules to avoid unacceptable discretizations. To avoid breakthrough of the doniain ~ o ~ ~ ~itdmay a r be y necessary to adjust (e.g. add) boundary points (Liseikin, 1999). dvanc~ngfront method - grid generation in this method starts from the boundary and is progressively moved towards the interior by the successive co~inect~on of new points appearing in front of tlie moving front until the entire doinain is meshed into elements. At the closing stages of the procedure the advancing front should be defined in a way that i t does not fold on itself. The selection of advancing step size should also be based on careful consideration and made to vary with the size of remaining uniiieshed space. In most types o f unstructured grid generation a secondary ~ n i o o t h i ~is~ g required to improve tlie mesh properties. In the majority of practical finite element s i ~ u l a t i o ithe i ~ mesh generation is c o ~ ~ ~ iinc conjunction tc~ with an interactive graphics tool to allow feedback and continuous monitoring of the c o ~ p u t a t i o n grid. ~~l The development of more robust, accurate, flexible and versatile mesh generation methods for facilitating the application of niodern coinp~tatioiial schemes is an area of active research.
typical finite element processor (sometimes called the ‘number ~ r u n ~ ~ i e r ’ ) program consists of the following blocks: Input and output subroutines to read and echo print data, allocate and initialize working arrays, and output the final results generally in a form that ost processor can use for graphical represe~tations~ Finite element library subroutines containing shape functions and their derivatives in terms of local coordinates. uxiliary subroutines for handling coordinate transfo~atioiibetween local and global systems, quadrature, convergence checking and updating of physical parameters in non-linear calculations.
MAIN COMPONENTS OF FINITE ELEMENT PROCESSOK PRO^^^^^^^
7
The main subroutine for evaluatioii of the elemeiital stiffness equations and load vectors. 0
Solver subroutines dealing with the assembly of elemental matrices and solution of the global set of algebraic equations.
Families of finite. elements and thcir corresponding shape functions, schemes for derivatiori of the elemental stiffness equations (i.e. the working e ~ ~ a ~ i ~ n s and updating of non-linear physical parameters in polymer processing simulations have been discussed in previous chapters. However, except for a brief explanation in the worked examples in Chapter 2, any detailed discL~ssion of the n u m ~ r i ~ solution al of the global sct of algebraic equations has, SQ far, been avoided. We now turn our atteution to this iniportant topic. Let us first consider the assembly of elemental stiffness equa tiolns in the simple example shown in Figure 6.4.
7
8
Global node numbering in a simple mesh consisting of bi-linear elements
e assume an e~einen~al node nu nib er^^^ order in the clockwise ~ ~ ~ e c tas ~on. shown in Figure 6.5.
.S
Local order of node nuinbering
ith respect to the selected e ~ e ~ e n and t a ~global orders of node ~ i ~ ~ ~ the elemental stiffness equations for elements el, et[ and e l in ~ Figure ~ G. expressed as
FINITE ELEMENT SOFTWARE
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for
- MAIN C Q M F O N E N ~ ~
e1
for ell
uring the assembly process the coefficients correspondiiig to the same degrees of freedom (i.e. the unknowns with equal indices) appearing in the elemcntal stiffness equations should be added together. Therefore the a ~ s e n ~ ~ l ~ global system for this example, arranged in ascending order of the unknowns, is
As thc number of elements in the mesh increases the sparse banded nature of the global set of equations becomes increasingly more apparent. However, as Equation (6.4) shows, unlike the one-dimensional examples given in Chapter 2, the b ~ ~ n d w ~ in d t hthe coefficient matrix in multi-dimensio~alproblenis is not constant and the main band may iriclude zeros in its interior terms. It is of course desirable to minimize the bandwidth and, as far as possible, prevent the a p ~ e ~ ~ r a nofc ezeros inside the band. The order of node numbering during
NUMERICAL SOLUTION OF ALGEBRAIC EQUATlONS
the mesh generation stage directly affects both of these objectives and should therefore be optiinized. In general, the imposition of boundary conditions is a part of the assembly process. A simple procedure for this is to assign a code of say 0 for an unknown degree of freedom and 1 to those that are specified as the boundary conditions. Rows and columns corresponding to the degrees of freedom marked by code 1 are eliminated from the assemblcd set and the other rows that contain them are modified via transfer of the product of the specified value by its correspo~~din~ coefficient to the right-hand side. The system of equations obtained after this operation i s determinate and its solution yields the required results.
Obviously selection of the most efficient elements in conjunction with the most appropriate finite element scheme is of the outmost importance in any given owever, satisfaction of the criteria set by these considerations cannot guarantee or even determine the overall accuracy, cost and general efficiency oi‘ the finite element siniulations, which dcpcnd more than any other factor on the algorithm used to solve the global equations. To achieve a high level of accuracy in the simulation of realistic problems usually a refined mesh consisting of hundrcds or even thousands of elements is used. In comparison to lime spent on the solution of the global set, the time required for evaluation and assembly of elemental stiffness equations is small. Therefore, as the number of’ equations in the global set grows larger by mesh refinement the computational time (and hence cost) becomes more and more dependent on the effectiveness and speed of the solver routine. The development of fast and accurate co~putational procedures for the solution of algebraic sets of equations has been an active area of research for many decades and a number of very efficient algorithms are iiow available. s menti~nedin Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of ‘direct’ or ‘iterative’ procedures. In the finite clement modelling of polymer processing problems the most frequently used methods are the direct methods. Iterative solution inethods are more effective for problems arising in solid mechanics and are not a commoti feature of the finite element ~ o d c l l of i~~ polymer processes. wever, under certain conditions they inay provide better computer economy n direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing an are ‘poteiitial~y’more suitable for three-dimensional flow modelling. chapter we focus on the direct methods commonly used in Aow simulation models.
FINITE ELEMENT SOFTWARE
0
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MAIN COMPONENTS
.I The most important direct solution algorithms used in finite element computations are based on the Gaussiaii eliniination method. To describe the basic concept of the Gaussian elimination eth hod we consider the following system of simultaneous algebraic equations
Let LN suppose that we can convert the n x system (6.5) into an upper triangular form as
1.1
coefficient matrix in e
where all o f the elements in the coefficient matrix which are below the main d i a g o ~ aare ~ zero. The superscripts (*) in the non-zero terms of the coefficient i ~ a ~ rand i x the r i g ~ t - ~ ~ aside n d of ~ ~ L ~ ~(6.6) t i osigiiify n the change in the values ts during the conversion of the original system to the u is evident that xn can be found i~mediatelyfro tern and the solution can hence progress by substit imatc equation to find q - 1 and so on. iiiatio~~ e t h ~or ~o v i ~ ae s$ystein~tica ~ ~ forri ~ ~p ~ de - ~ ~ e ~ t a t of ~ othen described 'forward reduction' and %back sL~bstit~~tion' processes for large systems o f algebraic equations.
ivotiF'ER It is readily recognized that in order to generate zeros in the columns of the c ~ ~ f ~matrix ~ i einnan ~ n X n system of algebraic equations ~nultiplesof successive rows should be subtracted from the rows which are above t e x a m p l ~after ~ multiplication o f the second row in Equation (6.5) by a1 m the first row the first c o ~ ~ o n e of' n t the new owever, in a large set of e~uatiollswith widely ration may give rise to .c/ery large or very small causing overflow or underflow in the c o ~ p u t e rsystem operat~on§ the solution impossible or inaccurate. There i s also the creation of zeros in the main diagonal of the system of equations that makes the ~ n t ~set r e s i n g u ~ and ~ r hence not solvable. To avoid t~iese~roblemsa p r o ~ ~ d u r e
NUMERICAL SOTLJTION OF AL,CEBRAIC EQUATIONS
1
known as pivoting IS used. This is described as rearranging the orderkg of the equations in the set, in every step of the foiward reduction, so that the coefficient of largest magnitude is located on the main diagonal. This can be achieved by interchanging rows and columns in the set. If both row and colurm,. interchanges are carried out the process is called jki6 pivoting. However, this is seldom necessary as the coefficient of largest magnitude can usually be placed on the main diagonal by p m t b l pivoting which only requires row interchange. t is evident that mLiltiplicatioii of the sides of an equation in a system by a large number will affect the pivot selection. In particular if the scaling of thc equations in f d l pivoting is ignored equations with larger positioned at rows above those having smaller coefficients. process during the reduction stage niny lead to a situation in which the coeffcients of equations located at the bottom rows are insignificant in comparison to the values of the terms at the top rows. This can become a sourcc of unacce able computational errors. To avoid dependence of pivoting on the scaling the equations, the system ould be normalized to make the largest ~ o e r ~ cini e ~ ~ ess et al., 1987). each row equal to unity The use of a uniform scale in partial pivoting can also significantly redrice round off errors (Gerald and Wheatley, 1984).
The sotution of linear algebraic equations by this method is based on the following steps: Step I - thc n x n coefficient matrix is augmented u7ith the load vector on the right-hand side to form an PZ x (n +- 1) matrix. Step 2 - interchanging rows the value of all is made to be the coefficient o f largest magnitude in the first column. Step 3 - subtracting a,llall times the first row from the ith row, the coefficients in the first coluinii from the second through nth rows are made equal to zero. The multiplier uZ1/a1 I is stored in u r I . i = 2, . ,n. I
(.
- steps 2 and 3 are repeated for thc second through the (n - 1)st rows, placing the coefficient of largest magnitude on the diagonal by interchanging rows (for only TOWS j to PZ)and subtracting times the jth row from the it11 row to create zeros in all positions o f tliejth column below the diagonal. The multiplier o,,/oJJi s stored in a,, i = .j + 1, . . . ,iz (note that U,/ aiid uu used during this step are different from their initial values given in the original set of equations). At the end of this step the forward reduction processes are carried to completion and the original PZ x n system is converted into an upper t ~ j a ~ ~ ~ form. ilar
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FINITE ELEMENT SOFTWARE - MAIN CONI'ONENTS
Step 5
-
the last equation is solved to give xn = a*,,,+~/a",,,
tep 6 - the remaining unknowns are found by back substitution using the following formula from the (n - I)st to the first equation in turn
Number of operalions in the Guussian elimination method To estimate the computational time required in a Gaussian elimiiiatioii procedure we need to evaluate the number of arithmetic Operations during the forward reduction and back substitution processes. bviously multiplica~ion and division take much longer time than addition and subtraction and hence the total time required for the latter operations, especially in large systems of eq~ations,is relatively small and can be ignored. Let us Gonsider a system of simultaneous algebraic equations, the representative calculation for forward re~uctionat stage i s expressed as k - k- 1
llZi - ai~
&l zk 1 L . k - l lcJ kk
k = 1, . . . , (12- 1)
where
j - (k+ I ) , . . . , ( U + 1) i = ( k t l ) , . . , (a)
(6.7)
The a ~ ~ g n i ~ ncoefficiellt ted matrix at this stage can be shown as
,
Rows reduced to upper triangular form
............................. tn-k)
...........................................................
I
{
ectangular sub-matrix
As ~ q u a t ~ o(6.7) i i shows in each of (a - k) rows in the rectangular sub-matrix we need to evaIuate (n - k ) multipliers and carry out (n - k + I) multiplications, therefore the total number of operations required is calculated as 91-1
[ ( n - k ) ( n - k + I ) + ( n - k ) ] ;=: I<= 1 n
.I
[(n- s ) ( n- k
k=l
For large n, n3 >> pz and S, = 113 n3.
+ I ) + (?I
--
k)]dk=
-n
(6.8)
SOIdUTIIOW ALGORITHMS: GAUSSIAN ELIMXWATTON METE-TOD
The described ‘operations count’ provides a guide to estimate the comp~rtational time required for reduction of a full n x n matrix to upper triangular owever, the global set of equations in finite element analysis will always be represented by a sparse banded (it may also be symmetric) coefficient matrix. It is therefore natural to consider ways for the exclusion o f zero terms from arithmetic operations during computer implementation of the Gaussian elimination method. An additional advantage of modification of the basic procedure to enable the forward reduction to be applied only to the non-zero terms is to reduce the storage (i.e. core) requirement. To take full advantage of this possibiIity it is important to optimize the global node numbering in the finite element mesh in a way that tlie inaxiinurn bandwidth of non-zero terms remains as small as possible and creation of zeros in the interior elements of the band i s avoided. Efficient band solver procedures such as the active coZurnn or skyline reduction method are now available ( athe, 1996) which provide maximum computer economy by restricting the number of operations and high-speed storage requirement.
The most frequently used modifications of the basic Gaussian eliininalion method in finite element analysis are the ‘ L U decomposition’ and “Jontcil solution’ techniques.
is This techiii¶ue (also known as the Grout reduction or Cholesky fa~tori~ation) based on the transformation of the matrix of coefficients in a system of algebraic equations into the product of lower and upper triangular matrices as
Therefore a l l = l l l x l = 111, a21 = Z2,, etc. (elements in the first coluima o f a are the same as the elements in the first columii of E); similarly multiplying rows of I by columns of U and equating the result with the corresponding element of a all of tlie clernents of lower and upper triaiigular matrices are found, The general formula for obtaining elements of I and U can be expressed as
2
FINITE ELEMENT SOFTWARE
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MAIN COMPONENTS
After obtaining the described decomposition the set of equations can be readily solved. This is because all of the information required for traiisforniation of the coefficient matrix to an upper triangular form is essentially recorded in the lower triangle. Therefore inodification of the right-hand side is quite straightforward and can be achieved using the lower triangular matrix as
(6.11)
ence the solution is found by back substitution based on X,
= b,*
(6.12)
In some applications the diagonal elements o f the upper triangular matrix are not predetermined to be unity. The formula used for the LU ~ecompos~tiol~ procedure in these applications is slightly different from those given in Equations (6.10) to {6.12), (Press et al., 1987). The LU decomposition procedure used in c o n ~ ~ n c t ~with o n partial pivoting provides a very efficient method for the solution of systems of algebraic equations. ‘The main advantage of this approach over the basic Gaussian elimination method is that once the coefficient matrix is decomposed into a product of lower and upper triangular matrices it remains the same while the right-hand side can be changed. Therefore different solutions for a set of algebraic equations with different right-hand sides can be found rapidly. Tn practice this property can be used to investigate the effect o f altering boundary conditions in a field problein with maximum computing economy. This property of the LU decomposition procedure can also be utilized to minimize the computational cost of iterative improvement of the accuracy of the solution of systems of linear equations (see Section 5.2).
SOLUTION ALGORITHMS: GAUSSTAN ELIMINATION METHOD
Computer implemen~ationsof band solver routines based on methods such as k U decomposition essentially depend on the “in core’ handling of the totally assembled e~ementalstiffness equations. This may prevent the use of small PCs (or even medium-size workstations) for simulation of realistic ei~ginee~ing problems, which require a relatively refined finite element mesh. The fronlal solution procedure, originally developed by Irons (1970), avoids this p~oblemby piecemeal reduction of the total matrix (or non-zero band) in a Gaussian elimination procedure. The original routine handled the solution of symmetric positive-definite matrix equations, however. in many problems (especially in the finite elemeiit simulation of flow processes) the equations to be solved are nonsymmetric. Therefore, in flow modelling a non-synmietric version of the original algorithm, developed by ood (1976), is usually used. The basic concept of the frontal solution strategy A work array of limited size (say d x J , where J is called the front width) is selected as the pre-assigned core area for the assembly, pivoting and reduction of elemental stiffness equations. Using a loop elemental stiffness matrices are assembled until the work array i s filled. n tliis limited area of the total matrix pivoting is ~ ~ p ~ e r n eand ~ t eforward d reduction is carried out. After e ~ i i ~ i i ~ a t i o ~ of a sufficientriumber of coefficients in the work array further assembly becomes possible and the cycle can be repeated. The progress of assembly and elimination and position of the front in a finite element mesh is shown, schematica~~y, in Figure 6.6. The active front shown in the figure implies that, at this stage of the process, coefficients of elemental stiffness equations for elcments 1 to 4 corresponding to variables which are not on the front have already been fully assembled and reduced.
ent next in line for assembly
Figure 6.6 Frontal solution scheme
Frontal solution requires very intricate bookkeeping for tracking coefficients and making sure that all of the stiffness equations have been assembled and fully reduced. The process time requirement in frontal solvers is hence larger than a straightforward band solver for equal size problems. Another consequence of using this strategy is that, unlike band solver routines, global node iiumbering in frontal solvers may be done in a completely arbitrary manner. Howwer, better computer economy is achieved if an eleinent nL~mberingwhich minimizes front width is used. In general, m a ~ i p u l a t ~ofo ~
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FINITE ELEMENT SOFTWARE - MAIN COMPONENTS
element numbering in a global domain is mucli simpler than nodes, consequently mesh design optimization in programs using frontal solvers i s simpler than those based on band solver routines.
NA
S
As mentioned earlier, overall accuracy of finite element computations is directly determined by the accuracy of the inethod employed to obtain the numerical solution of the global system of algebraic equations. In practical simul~tjoiis, therefore, computational errors which are liable to affect the solution of global stiffhess equations should be carefully analysed.
All coinputer systems operate under a predetermined floating-point word length which automatically imposes the rounding-off of digits beyond a given limit in all calculations. It is of course possible to use double-precision arithmetic to increase this range but the restriction cannot be removed completely. Therefore through accumulation of round-off errors, especially in simulations involving large numbers of calculations, serious computational problems may arise. In the solution of simultaneous algebraic equations, severity of pathological situations related to round-off errors depends on the conditioning of the coefficient matrix in the system. Conditioning of a matrix depends on: how sinall (i.e. near zero) i s its smallest eigenvalue and, more importantly how large is the ratio of the largest to the smallest eigenvalue. he degree of conditioning of a matrix is determined by the ‘ c ~ n ~ nurnbrr’ ~~fi~n fined as (Fox and (6.13) where A, and A1 are the largest and smallest eigenvalues of A, respectively. In practice the condition number is found approximately using the upper bound of A, as :A = IlAlI, where l(All represents any matrix norm, and a lower bound for A,, using the method of inverse itcration for the calculation of eigenvectors athe, 1996). Therefore cond ( A ) M
-
(6.14)
A1
A matrix with a large condition number is commonly referred to as ‘illconditioned’ and particularly vulnerable to round-off errors. Special techniques,
REFERENCES
2
generally known as preconditioning (Gluob and Van Loan, 1984), are used to overcome problenis associated with the numerical solution of ill~conditioned matrices.
Consider the solution of a set of linear algebraic equations given as (6.15) Let us assume that a numerical solution for this set is found as [.x)+(S.x), where {Sxl is iin unknown error. Therefore insertion of this result into the original equation set should give a right-hand side which is different from the true ( h f . Thus
]{.x + SX}
= {h
+ 6b}
(6.16)
Subtraction of Equation (6.15) from Equation (6.16) gives (6.17) ~ e p l a c ~ for n g (Sb} from Equation (6.16) in Equation (6.17) gives
[A]{ SX} = [A]{ x + SX}
- {b}
(6.18)
The right-hand side in Equation (6.18) is known and hence its solution yields the error {6x} in the original solution. The procedure can be iterated to improve the solution step-by-step. Note that implementation of this algorithm in the context o f finite element c~mputationsniay be very expensive. A significant advantage of the LU decomposition technique now becomes clear, because using this technique [A] can be decomposed only once and stored. Therefore in the quation (6.18) only the right-hand side needs to be calculated.
Bathe, K. J., 1996. Finite Element Procedure, Prentice Hall, Englewood Cliff, NJ. Fox, L. and Msyers, D.F., 1977. Cofltputing Methods for Scientists urad Engineers, Clarendon Press, Oxford. Gerald, G. F. and Wheatley, P. O., 1984. Applied Numerical Aiialysis, 3rd edn, AddisonWesley, Reading, MA. Gluob, G .H. and Van Loan, C . F., 1984. Mutrix Computations, Johns University Press, Baltimore.
FINITE ELEMENT SOFTWARE - MAIN COMPONEN'TS
Hood, P., 1976. Frontal solution program for unsyrnmetric matrices. Int. .I. Xumeu. Lh4!elhods Eng. 10, 379 399. 970. A frontal solution for finite element analysis. hi.J . n'zmzer. Methods 1999. Grid Gerzeration ilethods, Springer-Verlag, Berlin. vripilis, C. aiid Petera, A. T., 1980. Nonconforming mortar clciiicnt methods: application to spectral discretizations. In: Chan, T. F., Glowinski, R., Periaux, 3 . and Widlund, 0.B. (eds), Dowtuitz Decomposition ~ ~ ~ h SIAM, o d s ~ Philadelphia, pp. 392 418. Press, W. H. et ul., 1987. Numericul RecQes - The Art qf ScimfiJic Computing, Cambridge University Press, Cambridge,
In this chapter details of a computer program entitled PPVN.f are described. This prograiii is based on application of tlie Galerkin finite element method to the solutioii of the governing equations of generalized Newtoniaii Row. Flow equations are solved by the continuous penalty scheme and streamline upwinding is applied to solve the coiivection-dominated energy equation. Nodal values of pressure (and stress components) are found by the variational recovery method. In the last section of this chapter the program source code is listed which includes basic subroutines required in the finite element simulation of non-isothermal incompressible generalized Newtonian flow regimes. The central concern in the development of PPVN.f has been to adopt a programmi~igstyle that makes the modification or extension of the code as convenient as possible for the user. As an example, the modifications required for extension of the program to axisymetric domains are discussed. It is shown that by comparing the working equations used in PPW.f, with their counterparts for axisymmetric domains, necessary mod~ficationsfor this extension can be readily i ~ ~ ~ t i fand ied implemented. In the following sections, the solution algorithm used in P and the function of each subroutine in the code are explained. A manual describing the structure of the input data file is also presented. Finally, a simple example of the application of the program is included which shows tlie simulation of non-isothermal flow of a power law fluid in a two-dimensional domain.
7.
IT
PPVN.f is a FORTRAN program for the solution of steady state, generalized Navier Stokes arid energy equations in two-dimensional planar domains. This program uses a decoupled algorithm to solve the flow and energy equations iteratively. The steps used in this algorithm are shown in the following chart:
COMPUTER SIMULATIONS - FINITE ELEMENT PROGRAM
SOLVE CREEPING ISOTHERMAL ~ FLOW EQUATIONS
W
T
~
~
I
A
~
TO CALCULATE SECOND INVARIANT OF THE RATE OF
SCALAR INVAlUANT FOUND IN THE LAST STEP IS
THE SHEAR RATE (i.e. 9) USED IN POWER-LAW EQUATION. INSERT THIS VALUE TO UPDATE THE
I
FOR C O W E R
S T ~ S USING S VARIATIONAL
.f consists of a main module and 24 subroutines. These subroutines and their assigned tasks are described in this section. Variables in these subrou~ines are all defined using 'comment' statements inserted in the listing of the program code.
PROGRAM SPECIFICATIONS
AIN: The tasks allocated to the main module are: (i)
efines and allocates memory spaces and initializes variable arrays used in the program.
(ii)
Opens formatted file channels for input and output and the scratch files that are required during the calculations.
(iii)
Asks the user to give the input data file name (interactively).
(iv)
eads the input data file. The ‘read’ statements are either included in the main module or in the subroutines that are designed to read specific parts of tlie data file.
(v?
Checks the acceptability o f the input data
(vi?
Calls subroutines that prepare work arrays and specify positions in the global system of equations where the prescribed boundary conditians shouId be inserted.
(vii)
tarts the ‘solution’ loop.
(viii)
repares the output files.
~ A ~ S SGives F . Gauss point coordinates and weights required in the numerical integration of the members of tbe elemental stiffness equations.
S H A ~Gives ~ . the shape fuiictions in terms of local coordiiiates lor bi-linear or bi-qundratic qu~drilateralelements. DERIV. Calculates the inverse of the Jacobian matrix used in isoparanietric transformations. SECINV. Calculates the second invariant of the rate of deformation tensor at the integration points within the elements. FLOW. Calculates members of the elemental stiffness matrix corresponding to the flow model.
ENERGY. Calculates members of the elemental stiffness matrix corresponding to the energy equations. TRESS. Applies the variational recovery method to calculate nodal values o f re and, components o f the stress. A mass lumping routine is called by SS to diagoiialize the coefficient matrix in the equations to e~iminatethe
1’2
~
~ SIMULATIONS P ~ ~ - FINITE E R ELEMENT PROGRAM
~
reduction stage of thz direct solution. Prints out nodal pressure and stress components. These results should be stored in a suitable file for post-processing.
PM: Diagonalizes a square m&trix. ~ ~ Assembles V elemental ~ stiffness ~ :equations into a banded global matrix, imposes boundary conditions and solves the set of banded equations using the LU decomposition method (Gerald and Wheatley, 1984). S LVER calls the following 4 subroutines: - reads and assembles the elemental stiffness equations in a handed form, as is illustrated in Figure 7.1 to minimize computer memory requjrcnieiits.
~~~~~
7.1
Arrangement of the global matrix in the band-solver subroutine
- calculates the maximum bandwidth of non-zero terns in tlie AN coefficient matrix.
DIFY - addressing of members of the coefficient matrix are adjusted to allocate their row and column index in the banded matrix.
SOLVE - inserts the prescribed boundary conditions and uses an LU decomposition niethod to solve the assembled equations. eads and echo prints nodal coordinates; formatting should match ETN the output generated by the pre-processor. eads and echo prints element connectivity; formatting should match the output generated by the pre-processor. eads and echo prints prescribed boundary conditions, formatting should match the output generated by the pre-processor,
INPUT DATA FILE
CV: Inserts the prescribed velocity boundary values at the allocated place in the vector oC unknowns for flow equations. CT: Inserts the prescribed temperature boundary values at the allocated place in the vector of unknowns for the energy equation.
CLEAPU': Cleans used arrays at the end of each segment and prepares them to be used by the next component of the program. : Rearranges numbers of nodal degrees of freedom to make them compatible with the velocity components at each node. For example, in a ninenoded element allocated degree of frecdom numbers for v I and v2 at node n are X , and Xn+3, respectively.
AT: Reads aiid echo prints physical and rheological parameters and the penalty parame ter used in the simulation. VTSCA: Calculates shear dependent viscosity using the power law model.
L: Calculates ratio of the difference of the Euclidean norm ( L a ~ ~ d u s and Pinder, 1982) between successive iterations to the norm of the solution, a5
where r is the number of the iteratioii cycle, N is the tolal nuniber of degrees of' om and E is the convergence tolerance value. Note that criterion (7.1) is used for both velocity coi~iponentsand temperature in separate calculations. A converged solution is obtained when both sets of results satisfy this criterion. OUTPUT: Prints out the computed velocity and temperature fields. For postprocessing the user should store the output in suitable files
Structure of the input data file required for the running of P Line 1 Title format(A40)
COMPUTER SIMULATIONS - FINITE ELEMEKT PROGRAM
asic control variables
format(Ji5, 2f10.0) ncn ngaus mgaus alpha beta
= = = = =
number of nodes per element number of full integration points number of reduced integration points parameter to mark shear terms in the Bow equations parameter to mark penalty terms in the flow equations
Line 3 Mesh data format(Ji5) iinp nel nbc
= total iiuinber o f nodes
= total number of elements = total number of boundary conditions
Line 4 ~onvergeiicetolerance format(2f LOS)
tolv tolt
= tolerance parameter for the convergence of velocity calculations
= tolerance parameter for the convergence of te~iiperature calc~i~ations
Line 5 Rheological aiid physical parameters format(7flO.O) wise power tref tbco roden CP condk
= coiisistency coefficient in the power law model = power law index
= reference temperature in tem~erature~~ependent power law model = temperature dependency coefficient in the power lam7 inadel = fluid density = specific heat capacity of the fluid = conductivity coefficient
Line 6 Penalty paraineter format(fl5.0)
rplarn = penalty parameter
EX~ENSIONOF w v N . r TO AXISYMME'I'RICP K O B L ~ M S
A pre-processor program should usually be used to generate the following data lines
Lines 7 to ip = 7 + nnp
nodal coordinates format(i5,2el5.8)
cord (maxnp, ndim) = coordinates of the nodal points Lines ip to ie = ip
-t nel
element connectivity format(1Oi5)
node (iel, icn) = array consisting of element numbers and nodal connectivity Lines ie to ib = ie
+ nbc
boundary conditions format(2i5,flO.O)
numbers of nodes where a boundary condition is given index to identify the prescribed degree of freedom (e.g. jbc = 1 the first component of velocity is given etc.) vbc ( ~ a x b ~ value ) of the prescribed boundary conditions
ibc (maxbc) jbc (maxbc)
=I
I-;.
As already mentioned, the present code corresponds to the solution of steadystate i i o n - ~ s o t h e ~ Navier-Stokes ~al equations in two-dimensional Cartesian domains by the continuous penalty method. As an example, we consider modifications required to extend the program to the solution of creeping (Stokes) non-isothermal flow in axisymmetric domains: To solve a Stokes flow problem by this program the inertia term in the elemental stiffness matrix should be eliminated. Multiplication of the density variable by zero enforces this conversion (this variable is identified in the program listing).
General structure of stiffness matrices derived for the model equations of Stokes Aow in (x,y ) and (r, z ) formulations (see Chapter 4) are compared.
~
~
~ SIMIJIATIONS ~ W T
-
EFINITE ~ ELEMENT PROGRAM
~ t l f f n e simtrix ~ corresponding to flow equations in (x,y ) formulation
to Aow equrztioiis in (r, z ) formulation ~ t i ~ f matrix n e ~ ~cor~espon~ing
paring systems (7.2) and (7.3) additional terms in the members of the stiffness matrix correspond~n~ to the axisymi~etric€ormu~dtionare idcntified. Note that the measure of integration in these teims is (r drdz).
edification of subroutine S ~ C ~ ~ V :
efine (r) in the program and find its value at the integration points. Find radial component of velocity (v,.) at the integration points.
CIRCULA'TORY FLOW IN A RECTANGULAR DOMALN
217
Calculate 1/2(v,2/r2)and add this vdue to the previously calculated value of the shear rate. tel,
odification of subroutine FLOW:
cfiiie (r) in the program and find its value at the integration points. The measure of integration should be multiplied by this factor. After evaluation of tlie terms of the stiffness matrix modify them according to the additional terms shown in system (7.3). Modification of subroutine ENERGY: The only requirement is to modify the measure of integratio~lsimilar to subroutine FLOW. Other temis remain unchanged (sec Chapter 4 for derivation of tlie working equations of the scheme). odification of subroutine ST Find radial component of velocity (vr> and ( r ) at the reduced integration points and calculate ijy/r. Include this teim in the calculation of pressure via the penalty relation. Modify the measure of integration multiplying it to (T). All ofthe described modifications are shown in the program listing. The rest of the subroutines will rernaiii the same and no other modification is necessary. owever, in practice a switch parameter can be defined in the program which by s value from 0 to 1 allows the user to select planar or axisymmetric rite formats used in the echo printing of input data and output files can also be modified to represent nodal coordinates, velocity and stress components iii an (r, z) system instead of the planar forms given in the program listing.
As an cxample of the application of PPVN,f we consider simulation of the circulatory flow of low-density polyethylene melt i II a rectangular domain of
0.05 in length and 0.01 rn width. The flow regime is generated by the imposition of a steady motion at the top surface. The prescribed boundary coliditions in this problem are as shown in Figure 7.2. Therefore at the side and bottom walls irichlet) boundary conditions are given, no temperature condit~onat the top wall is equivalent to setting zero thennal stress (i.e. teniperature ~radients)at this boundary.
218
COMPUTER SIMULATIONS - FINITE ELEMENT PROGRAM
vx= I , v,=o
vx= -1, v,,=o
v,=v,=o
vx=v>=o
T = 400 K
T=400K v,=v,=O; T=400K X
Figure 7.2 Boundary conditions
111
the sample simulation
Table 7.1 shows the structure of the input data file that is prepared according to the foiinat described in the previous section. Figures 7.3 and 7.4 show, respectively, the computed velocity and temperature fields, generated by PPVNf for this example.
Figure 7.3 The simulated velocity field
F i g ~ e7.4 The siniuhlcd temperature contours
CIRCULATORY FLOW IN A RECTANGULAR DOMAIN
219
TaMe 7.1 a
1 0.00000000E+00 0.00000000E+00 2 O.OOOOOOOOE+OO 0.625000000-03 3 0.000000003+00 0.125OOOOOE-02 4 0.00000000E+00 0.18750OOOE-02
7
NODAL COORDTNATES
"_,"_l_l__-_l,"____-_-ll_____ll I__---__l_---l-___-__--l---I-----.-
693 694 695 696
0.500000013-01 0.75000000E-C2 0 , 5 0 0 0 0 0 0 1 E - 0 1 0.8125OOOOE-02 0.500000013--01 0.875000003-02 0.50000001E-010.937500003-02
( m lines) ~
661 663
660 662
677 679
BOUNDARY CONDITIONS
NCOD = 3 corm
(NBC
lines)
puter using the Unix operating system PROGRAM PPVN
c C
I
14
I I
USED AS A WORK FILE TN THE SOLVER ROUTINE (SCRA'I'CHFILE)
50
I I
IMPUT DATA CHANNEL
c C C C
C C
C
1 OUTPUT FILE FOR UOCJMENTATIOZL I 15 I STORES SHAPE FUNC'TIONS 24" THEIR DERIVATIVES AT I 'FULL' INTEGRATION POINTS(SCRATCI3 FILE)
60
1
C C
16
I I
STORES SHAPE FUNCTIONS AND THE19 DERIVATIVES AT 'REDUCED'INTEGRATTON POINTS(SCRATCH FILE)
c
C C C FWLN VARIAbLES -- _ _C -- --_ C NOUh (MAXhL, 18) ELEMENT rOrSNECTTVITY ARRAY C CORD (MAXPJP, NDIM) NODAL COORDINATES ARRAY C AA { 18, 18) ELEMENT COEFFICfENl NATRICES; FLOW EQUArlONS C iin ( i8) ELEMENT LOAD VECTOR; FLOW EQUATIONS AE ( 9 , 9) ELEMENT COEFFICIENT MATRICES; ENERGY EOUATION C C RE ( 9) ELEMENT LOAD VECTOR, ENERGY EQUATION C VEL (MAXD>) NODAL VELOCITIES C NODAL TFMPARRTURES TEMP {MAXNP) STTFF (MAXAR) GLOBAL STIFFNESS MATRSX C P ( 9) SHAPE FLWCTIONS C C DEL ( % , 9 ) 1,OCAL DERIVA'IIVFS OF SHAPE FUNCTIONS C R ( 2, 9 ) GLOBAT, DERTVATIVES OF SHAPF FUNCTTONS C BC (MnxDF) BOTJNDARY CONDITIONS 4 R M Y VHEAT GENERATED VISCOS HEAT C ALPhA FACTOR TOR THE SELECTION OF SWEIiE 'TEEMS IN AA C EACT'OR FOR THE SELECTIOICI OF PENALTY IEFClYS IN AA c BETA NN'1 TOTAL NUMBER OF N o n u PoIwrs C TOTAL NUMBER OF ELEMENTS c NEL I_
SOURCE CODE OF PPVN.f C C
NBC NDIM NDF TOLV TOLT NJM FMAT1 RMAT2
C C C C C C
c
TOTAL NUNBER OF BOUNDARY CONDITIONS DIMENSIONS OF THE SOLUTION DOMAIN DEGREE O F FREEDOM PER NODE COiWERGENCF TOX,CF!ANc'E PARAKETER FOR VELOCITIES CONVERGENCE TOLERANCE PARAMETFR FOR TEMPERATURE NUhIBER OF 1NTEGRA"TON POTNTS PER ELEMENT MA'IEEKlAL PAWIETERS AT F'IJLL INTCGRATTON 2OTNTS MATERlAL PARAMETERS AT REDUCED INTEGRRTION POTNTS
~ ~ * * ~ * h h * * * , * X ~ X * * h k * * ) h * * h * h * * * * * k * * k h * A * * * ~ ~ ~ * ~ * * * * * * * * * * * ~ * * k * * ~ * * *
C C * * * STORAGE AIJJOCATTON C
PA~ETER(~EL=200,MAXNP=800,FIAXBN=1200,~BC=300) PAFUWIETER(MAXDF=2000,MAXST=18) C PARAMETERS SHOULD MATCH WJMBER OF ELEMENTS, XOUES, ETC. USED IN A PROBLEM C IMPTtTCIT DOUBLE PRECISION (A-H,O-Z) DINENSTON TITLE ( 80) DIMENSTON NODE (MAXEL, 18) ,PMIIT (MAXEL, 8 ) DIMENSION CORD (MAXNP, 2) DIMENSION NCOU (MHXDF ,BC (MAXDF ) DIMENSION IBC (MAXBC ) ,JRC (MAXBC ) ,VBC ( W B C DIMENSION VEL (MAXDF ) ,R1 (MAXDF ) ,TENP (MAXXP ) DIM&NSIOl' CLUMP (MAXNP ) , STRCS (MAXNP, 4) DIMENSION VET ( W D I ' ,TET (MAXNP DIMENSION SINV (MAXEIe, 13) ,NOPD (MAXRL, IS) ,RRSS (MAXbL.' ) DIMENSION AA ( 18, 18) 18) DIMENSION A E i 18, 18) ,RE ( DIMENSION XG i 3) ,CG ( 3) DIMENSlON P ( 3 ) , D E L ( 2, 9) , B ( 2 , 9) DIMENSION W4AT1 (MAXEL, 1 3 ) ,RMIIT2 (MAXEL, 1 3 )
C***
c'
CHARACTER*?O FILNAM
C
c * * * GLORAL
STLFFNESS MATRTX COMMON/ONE/ STiFF(20110,300)
C
DATA FILE'
1 C OPEN(UNT"=hO,FTZE='SO1;.OUT',i.ORM='FORMATTED',STATUS='NhW') C
c ***
INITIALIZE THE a K i i A Y s
C
OO 9111 ITL = 1,MAXEL DO 3111 IVL = 1,18 KODF (I'I'L,IVLI= NOPD (lTL,IVL) = 9111 CONTINUE DO 9 > 1 2 ITL = 1,l'UXXEL DO 9112 IVZ = 1.8 FMAT (ITL,TVL) = 9112 CONTINUE DO 9113 ITL = 1,MAXNP DO 9113 IVL 1,2 CORD (ITL.lVL)=
0 0
C.O
0.0
THE TNITIALIZATION STA TEMENI S MAY HE REDUNDANT IN SOME SYSTEMS
222
COMPlSTER SIMULATLQNS FINITE ELEMENT PROGRAM
9113 CONTINUE DO 9114 ITL = 1,MAXNP DO 9114 I V L = 1 , 4 STRES(I'TL,1VL)- 0.0 9114 CONTINUE DO 9 1 1 5 ITL = 1,MAXhL DO 91 15 IVL = 1'13 SlNV (ITL,IVL)=0.0 KMATl (ITL,IVL)= 0.0 RMAT2 (ITL,IVL)=0.0 9115 CONTlNUE DO 9116 ITL = 1,MAXDB NCOD (ITL) = 0 VEL ( I T L ) = 0.0 VET (I'I'L) = 0.0 R1 (ITL) 0.0 ac (ITL) = 0.0 YRRSS (ITL) = 0.0 9116 C O N T I N W DO 9117 ITL = 1,MAXNP CLUMP (TTL) = 0.0 TCT (ITL) = 0.0 TEMP J I T L ) = 0.0 91 17 CONTINUE DO 9 1 1 9 I T L = 1,MAXDl' 30 9119 LVL 1,MAXBN STIFF ( I W , , I T L ) = 0.0 9119 CONTI-WE DO 9121 ITL= 1,MAXBC (ITL) 0 IBC JBC (ITL) = 0 VBC (ITL) = 0.0 9121 CONTINUE I
C C'**
R.?R?kY
SUBSCRIPTS AND THEIR ULTIMATE LIMITS
C NDLM = 2
NUM
13
c (1
* * X * X * * * * * * k * h X * * * * * * * * * * * * X * * * * * * * * h * * * * * * * ~ * * * ~ ~ * * * * * * * * * * ~ * k * k ~ * * ~ * * ~ *
c C
c C C C
c
SET CONFROL P A W E T E R S (DEFAULlT VALUES ARE OVERWRITTEN BY INPUT DATA IF SPECTFKED) XCN NUMBER OF NODES PER ELEMENT NGAUS NUMBER. OF FULL I N T E G R A T l O N POlNTS MGAUS NUMBER OF KEDUCED INTEGRATlON POINTS NTFR MZiXIIWM NUMBER OF INTEGRATIONS FOR XON-NEWTONTAN CASE
C
c
* * * * X * X * * * * k X h * , * h * X x h * h h * * * * * * * * * * * * ~ * * * ~ * k * ~ * * * ~ * * * * * * ~ * * * * * * * ~ * * * * * * * * *
C
NCN = 9 NGALS 3 MGAUS = 7 NTER = 6
c C
* * * * * * * * * * * * PARAMMTERS FOR THE IDENTIFICATION OF PFNALTY
C 1.0 BETA = 0.0
ALPIIA =
TERMS
******
SOURCE CODE OF PPV3I.f C READ (50,1000) TITLE WRITE(60,2000) TTTLE C
C * - * ELENENT DESCRIPTION DATA C READ ( 5 0 , 1010) NCNR ,NGAUSR ,MGAUSR , ALPHAR , BETAR IRED = ALPHAR IF(NCNR .NF,.O ) NCN = N C m IF(TRFD .EQ.0 ) NUM = 4 IF(NCN .NE.4 ) GO TO 4780 NGAUS = 2 MGAUS =
NUM
1
5 4780 CONTINUE IF(NGAUSR.NE.0 ) NCAUS = NGAUSR TF(MGAUSR.NE.0 ) MGAUS = MGAUSR IF(NCAUSR.NE.0 ) ALPHA = ALPIiAR IF (MGAUSR.N E . 0 ) BETA BETAR WRITE(60,2010) NCN ,NGAUS ,MGAUS ,ALPHA ,BETA C C * * * MESH DATA, ROTJNDARY CONDITIONS AND TOLERANCE PARAMETERS =
C
R
W (50,1020 NNP ,NEL ,NBC
C
IF (NNP EQ.0 .OR.NNP .GT.MAXNP) GO TO 8000 IF (fiJEL EQ.O .OR.NEL .GT.PIAXEL)GO TO 8000 IF (NBC EQ.0 .OR.NBC .GT.M?SBC) GO TO 8000 WRITE (GO,2020 NNP ,NEL ,NEC C R E M (50,1030) TOLV ,TOLT
C
c
* * h ~ * * * * h h * X * * * * * * * * * * X * X h * * h * h * * * h * * * * * * * ~ * * * . k ~ ~ ~ k * ~ * * * * ~ ~ ~ ~ * * * * * * * * * * * * * * * * * *
C 1000 1010 1020 103 0 2000 1'
FORMAT(80Al) FORMnT(3IS,%F10.0) FOiiMnT(715) FORMAT (2Fl0.0) FORMAT(' ',5(/),' ',20X,60('*'),/'',2OX,'*',58X,'*',/
',20X,'*','A 'TWO-DIMENSIONAL,FINITE ELEMENT MODEL OF A ' , ',ZOX,'*',' NON-NEWTONTAN, NON-ISOTHEFNAL FLOW USING ' , 3'REDUCED',9X,'*',/'',20X,'*',' INTEGHATION / PENALTY FUNCTION ' , 4'H9THOn.',18X,'*',/'',20X,'*',58X,'.*',/'',20X,60('*')///,' 520X,80('--'),/' ',208,80A1,/'' , Z O X , 8 0 ( ' - ' ) , / / / ) 2010 FORMAT(' ',20X#3('['),'ELEMENT PRESCRIPTION ',10('.'),/ 125X,' NO.OF NODES PER ELEMENT =',IIO,/ 225X,'NO.OB INTEGRATION P O I N T S (*FULL*) =',110,/ 325X, 'NO.OF INF'EGKATION POINTS (*REDUCED*) : : I , 110,/ 425X,'SHEAR TERMS INTEGRATION FACTOR =',F15.4,/ 525X,'PENALTY TEKMS 1NTiX;RATION FACTOR =/ rF15.4,///) 2020 F O m T ( ' 0 ' , 2 0 X , 3 ( ' [ ' ) , ' MESH DATA PRESCRIPTION ' , ; a ( ' . ' ) , / 1%5X,'NO.OPNODAL POINTS =',I10,/ 225X, NO.OF ELaEMENTS =',110,/ 3258,'NO.OF NODAL BOUNDARY CONUITIONS = ' , s10,/ / / I 29X,'*',/'
I ,
C
c:
~ t * * . , : * * X * * r h h h * h k h * * * * * * h * * * h X * i * * * * * * * * * * * * * % * * * * * ' h * ~ * * * * * * * * *
C C
C
READ INPUT DATA FROM MAIN DATA FILE AND PREPARE ARRAYS F O R S O L U T I O N PROCESS
c
C
ikh**k*IX*h***fh**k**XY*Xt***Xkt*X**X*h*~************~~*~***~*****~***
c CALL CETMAl "EL, PMAT,50,GO,MAXEL,, RTCM) CALI, GETNOU (NNP, CORD,5 0,60,MAXNP,NDTM ) CALL GETELM(NEL,NCN,NODE,50,60,MAXEL) CALL GFTBCD(NBC,IBC,JEC,VBC,50,60,MAXBC) C (-
* * * * t * * * * * * * X k * * * * * * * * * n * * h * * * h * * * * * * * * * * * * * * * * * * , ~ * * * * * ~ * * * ~ * * * * ~ * * * * * ~ * * * ~ *
C C *INITIALIZE TEMPERATURE & SECOND 1WmlANII OF RAZE OF DEFORMATSON TFNSOR C DO 9996 IEL = 1 , K U E L DO 9996 LG - 1, NUM SINV (IEL,LG)= 0.250 9996 CONTlNUE DO 9997 SVEL= 1,MAXDF VEL (IVCL) = 0.0 9997 CONTINUF DO 9998 ITEM= 1 , N A X N P TEMPjI'I'EM)
-
RThM
9998 CONTlNUE C C*** MAIN SOLUTION LOOP C
DO 9999 I T h K = 1 ,NTER P R I N T k , 'ITER=',ITER C 'CJnI'I'E ( 60,?800) ITER 2800 FORMhT ( / / / ' 3 ( ' 1 ' ) , 1 5 , ' -I'H l T I E M T I O N ' , - 0 ( ' . ' ) / / ) C C CALCULATE NODAL VELOCITIES * * * h X * X * * k k X h k X * * * * * * X X * X * * k * * * * X * * * x * * X * C REWIND 15 REWIND 16 I ,
fl
NDF
=
2
NTOV = NDF * NNP NTRIX = NDF NCN CALL, CLEAhT ( R I ,EC ,NCOD ,NTOV ,MAXDF,MAXBN) CALL SETPRM (NNP ,NXL ,NCN ,NODE ,NDF ,MAXFL,YXST) CALL PUTECV (NNP ,NBC ,IRC ,JRC ,VBC ,NCOD ,MAXBC,MAXDF,BC) C
DO 5001 IFL=l,NEL
c CALC FLOW
1 2
3
(NODE , CORD , PPMT ,NDF ,W.XBN,NCOD , BC ,VEL ,R1 ,RRSS , T E W , N U M ,IEL ,ITER ,NEL ,NCW , S S W #NGAUS,MGAUS,P ,L)EL ,B ,ALPhA,BETA ,NTRIXrE/IAXEJ~,MAXNP,NOPD ,MAXST,PIAXBC ,XG ,DA ,NTOV ,IRED ,IBC ,JBC ,VBC ,RMATI,RMAT7)
4
tMAXDF,NDSM , A A
5
,NRC
C 5001 CONTlNUE
C C
h * x * a *
CHECK FOR CO&&VERGENCEx X * * X k r X * * * * h * + * * * *
c C 4 L L COYTOL
1(VEL,TEMP,ITER,NTOV,NNP,~P,MAXDF,ERROV,E~ROT,VET,TE?)
C IF(ERROV.LJT.TOCV.AI\ID.FRROT.I~T.TOLT) GO TO 8 8 8 8 8
SOURCE CODE OF PPVN.f C GO T O 88880 C 88888 CALL OUTPUT (NNP ,VEL ,TEMP ,MAXDF,WNP) C
CALL CLEAN(Rl,BC,NCOD,NTOV,NAXDF,MAXBN) C
GO TO 9000 C 88880 CONTINUE C
c
*h******hh*k**f*X*)****kX*Xht*h*****X******~~********************************~*
C CALL S E C I N V 1 (NEL ,NNP ,NCN ,NGPJJS,MGAUS ,NODE , SlNT ,CORD ,P , B 2 DEL , n A ,VEL ,P ~ PMAXEL, , MAXST,NDIM , IRED ,mr.i) C
*
C
CALCULATE NODAL TEMpEMTURE,? .
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C REWIND 15 REWIND 16 NDI? = 1 NTOV = NDF * NNP NTRIX= NDF * NCN CALL CLEAN (RI ,BC ,NCOD ,NTOV ,1LIAXDF,MBN) CALL SETPRM(NNJP ,NEL ,NCN ,NODE ,NDF ,MAXEL,MHXST) CALL PTJTBCT(NBC ,IBC , J B C ,VBC ,NCOD ,BC ,MAXBc,MAXDF) C DO 5002 IEL=l,NEL
C CALL ENERGY 1 (NODE, ,CORD ,PMAT ,NDF ,IWBN,NCOD ,EC ,TEMP ,VET , R R S S 2 ,R1 , I R E D ,XG ,NDIM ,DA , I E L ,NEL ,NCN ,NTOV ,NTJM 3 , I T E K ,NGAUS,MGAUS,P ,DEL ,B ,SINV ,NTRJX,MAXEL,MAXNP 4 ,I.WXST,MAXDF,MAXBC,IBC , J B C ,VBC ,NBC ,AE ,RE ,NOPD) R
L
5002 CONTINUE
C 9999 CONTINUE C 9000 CONTINUE C C C
* * * CALCULATION OF THF NODAI, PRESSURE
&
STRESS USING VARIATIONAL RECOVERY
CALL LUPTPM
1 (CLUMP, NNP, MAXNP,NEL ,NGAUS,P ,DEL , B ,MAXST,NODE,MAXEL,NCN) c
CALL STRESS 1 (NEL ,NNP ,NCN ,NGAUS,MGAUS,NODE,CORD ,P ,B ,DEL 2 ,VEL ,MAXNP,MAXEL,I.IAXST,EMAT1,RMAT2, IRED , S?'RES,CLUMP) C CLOSE (UNIT=E;O) STOP 8 0 0 0 CONTINUE C
C C
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
COMPUTER SIMI31,ATIONS - FINlTE ELEMENT PROGRAM
WRITE ( 60,29 9 5 ) 2995 FORMAT('0',10('['),' IXPUT DATA UNACCEPTABLE ',10('[')///) STOP
SOURCE CODE OF PPVN.f DEL8(1.,3)= (l.+H)/4 D E L ( 1 , 4 ) = --(l.+H)/4 DEL(2,l) ZL -(l.-G)/4 DEL(2,2) = -(l.+G)/4 DEL(2,3) = ( 1 .+G)/4 DEL(2,d) = (l.-G)/4 GO TO 30 9 IF(NCN.NE.8) GO TO 3.0 GG = G*G HH L- H"H GGH = GG*E GHH = G * m
c*** C 2.BI-QUADRATIC C***
10 G1=.5'*G* (G-1.) G2=1.-G*G G 3 = . 5*G* (G+I.1 H1=.5*H* (H-1.) H2=1.-H*H H3=.5*H* (Hcl.) P (1)=Gl*Rl P (2) =G2"Hl P (3) =G3*Hl P ( 4)=G3*H2 P ( 5 ) =G3*H3 P ( 6 )=G2 *H3 P (7)=C1*H3 P (8)=Gl*H2 p(9)42*~2 3GP:G--. 5
DG2=-.2. *G DG3=G+. 5
mi=H-.5 DH2:-2 .*H DH3LYHI- .5 DEL(l,l)=DG1*HX UEL(1,2)=DG%*Hl DEL(1,3)=DG3*H1 =DG3*H2 DEL (1,4) DEL (1,5)=DG3*H3 DEL (1,6) =DG2*H3 DEL(L,7)=DGl*H3 DEL ( 1,8) =DGl*H2 DEL (1,9) =UG2%2 DEI,(2,L) =G1*UHl DEL (2,2) =G2"DHl DEL ( 2 , 3 1 =G3 "DSII. DEL ( 2 , 4 ) =G3* D I E DEL ( 2 , s )=G3*DH3 DEL (2,6)=G2*DH3 DEL (2,7)=G1 *DW3 DEL(2,8)=Gl*DH2 DEL (2,9) =G2*DH2 30 CONTTNUE C
RETURN END C
NINE-NODED QUADRILATERAL
8
COMPUTER ~ 1 M U L A ~ ~- ~FINITE N S ELEMENT PROGRAM
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C SUBROUTIYE DERIV 1 (TFT. , IG ,JG , P ,DEL , B ,NCN , DA 2 ,CORD ,YAXEL,MAXNP,MAXST)
,CG
,NOP
C JACOBIAN Ob COORDINATES TRANSFORMATION & DERIVATIVES OF' THE SHAPE GLOBAL VARIARTZS IMPLICIT DDUBLF PRECISTON(A-H,O-ZI DIMENSION P ( 9 ) , E ( 2 , 9 ) , D E L ( 2 , 9 ) , C G ( 3 ) , C J ( 2 , 7 ) , C J I ( 2 , 2 ) DIMENSTON NOP(MAXhL,~ST),CORD(MAXNP,Z)
C***
c FUNCTIONS w m
C
DO 22 J=1,2 DO 22 L-l,2 GLSH- 0 . DO 21 I < = l , N C N NN=IABS(NOP(IEL,IO) 21 GASM=GLSH+DEL(J,K)*CORD(NN,L) 22 CJ(J,L)=GLSH DETJ=CJ(1,l)*CJ(2,2) -- CJ(3,2)"CJ(2,l) I F (DETJ) 29,33,29 13 -m1'r~(60.34) 34 FORMAT (1H , 2 2 H DFTJ=O PROGRAM HALTED) STOP
c
*t*
29 CJI(1,l) = CJ(2,2) / DETJ CJI(1,2) =-CJ(1,2) / DETJ CJI(2,L) =-CJ(2,1) / DZTJ CJI(2,2) = CJ(1,l) / DETJ
C *** DO 40 J = 1 , 2 DO 40 L=l,NCN B (J,I,) = a . o DO 40 h-1,2 40 B ( d , L ) = E(J,L) i- CJi(J,K) * DEL(K,L) DA = DETJ*CC (IG)*CG (SG) C RETURN END C C
C
SUBROUTINE SECINV 1 (NEL ,NNP ,NCN ,NCAUS, MGAUS,XODE , STNV ,CORD , P , B 2 ,DEL , DA ,VEL WPJP,MAXEL, YAXST, NDIM , IRED ,NUM) I
c IMPLICIT DOUBLE PRECISION(A-H,0 - Z ) C
C FUNCTION C - - - - - --C FTNDS THE SECOND IWARIANT OF FATE OF DEFORMATION AT INTEG,PATION C POINTS C DIMENSION '$EL (NNP , N D I M ) ,CORD (MAXNP, N D I M J DIMENSlON NODE (MAXEL,MAXST),SINV (MAXEL,NUM) DIMENSION P ( 9) ,DEL ( 2 , 9) DIMENSION B ( 2 , 9 )
c C
SOURCE CODE OF PPVN.E C REWIND 15 REWlND 16 DO 5000 IEL = 1 , NEL
c c
3**
C
TF(IRED.EQ.0) GO TO 5001
c
e
X * k
FULL INTEGRA'TIONpOI&-TS
X * X * * * h A * x X * * * * * X h f i * * * * k * ~ ~ * ~ * k X ~ k ~ * X ~ ~ * X *
C LG = 0 DO 5010 IC = 1 ,NGAUS DO 5010 JG = 1 ,XIcAUS LG = LG+I REriD (15) IIEL,lIG,JJG,P,DEL,E,DA x1 0.0 U1 0.0 U11 = 0.0 U12 = 0.0 U21 = 0.0 U 2 3 = 0.0 30 5020 TCN = 1 , N ~ N JCN = TABS(NODS(iEL,ICN))
X I = X I + P(ICN)TORD(JCN.I) U1 = U1 + P ~ I C ~ ~ * C O ~ D ( ~ ~ ~ , ~ ~
DO LOOP
IN THIS c C C
* * * COMPONENTS
OF THE RATE OF DEFORMATION TENSOR
U11 = U11 + U12 = U12 + U21 = U21 + U22 = U22 + 5020
B(l,ICN)k VEL(JCX,l) B(2,ICN)" VEL(JCN,l) B(l,TCN)*VEL(JCN,2) B(2,1CN)* VEL(JCN,2)
CONTINUE
c C * * * SECOND SNVARJHNT OE THE RATE OF DEFORMATION TENSOR
c SlNV(IEL,LG) = 0.125*((U11+U11)*(Ull+Ull)t 1 (U12+U21)= ( ; S l / ' + U 2 1 ) + 2 (U21+U12)* (U21+Ul?)+ 3 (u22+u22j*(u77+u22)) C 5010 CON? iNUE C 5001 CONTINUE
c
C * * * RF,T)UCED INTEGRATION POINTS
* * * * * C h * R * * * X k h * * K X * * X ~ ~ * * * * ~ ~i l k * * * * *
C
I:F(IRED
E Q . 0 ) LG = 0
DO 6010 IG DO 6010 JG
rx;
1 ,MGAUS ,MGAUS = I + LG READ (16) IZEL,IIG,C J G , P,DEL,R , DA = I
X1, = 0.0 U1 = 0.0 U11 = 0.0 U12 = 0.0 U21 = 0.0 U22 = 0.0 DO 6023 ICN - 1 ,NCN JCN = IABS (NODE (IEL,ICN))
FOR R,Z OPTION H N D X1 & IJ1 AS X l = XI + P(ICN)*CORD(JCN,I) U1 = 171 + P(ICN)*CORD(JCN,l)
IN THIS C C
lr**
DO LOOP
COMPONENTS OF THE RATE OF DCFOF3VATION TENSOR
C 311 = U11 + B(l,ICN)* VEL(JCN,l) IT12 = U12 + B(2,TCN)* VEL(JCN,l)
U71 = U 2 1 + B(1,ICN)" VEL(JCN,2) U22 U22 + B(2,ICN)" VEL(JCN,2) 6023 COIQTINUE C C * * * SECOND TNVARIANT OF TIE RATE OF DEFORMAZ'ION TENSOR C SINV(IEL,I,G)= 0.125*((U11+U11)*IUll+U11)4 1 (U12+U21)* (U12+U21)+ 2 (U21kU12)* (IT21+U12j + 3 (U22+U22) * (U22+U22)j 6010 CONTJNUE C 5000 CONTINUE
r REl'URN
END C
c
* * * * k h * * i h * * * * * * h * * * ~ ~ ~ * * * * * * * * * * * * * * ~ * * * * * ~ * * * h * * * k * * ~ * * * * * ~ * * * * * * * *
n
L
SUBROUTINL FLOW 1 (NODE ,CORD ,PMAT , N D F ,MAXSN,NCOD , B C ,VEL ,R1 ,RRSS 2 ,TLMP NUM , IEL ,TTER ,NEL ,NCN , SINV ,NGAUS,MGAUS,P 3 ,DEL ,U ,ATfiPHA, BETA ,N'l'RIX,PIAXEL,MAXNP,NOPD,MAXS'T 4 ,MAXBC, MEXDF, N D I M AA ,XG ,DA ,NTOTI ,TRED ,TBC ,JBC 5 ,vE(C ,NBC ,RMFITl,RMAT2) I
I
c C"
* * SOLUTION OF THE GENblUl1,IZEONAVIER--S?OKES EQUATION
c JMPLICTT DOUBLE PKEClSION(A-H,O-Z)
c DIMENSION NODE (YAXEL,MAXST),PMAT (MAXEL, 8 ) DIMENSION COED (YUAXNP, NDTM) DIMENSION NCOD (I'CAXDF) ,BC (MAXDF) , S I N V (MAXEL,NUM) DTMENSION VFIi (MAXNP,N D I N ) ,K1 ( W O k ) ,TCIQ (MAXNP) DTMFNSION AA ( 18, i8),KR ( 18) DIMENSTON XG ( 3) ,CG ( 3 ) DIMhNSIOAV X ( 2) ,V ( 2)
SOURCE CODE OF PPVN.f DIMJ3NS ION BTCId (
7) ,HH ( 2 ) DIMENSION P ( 9 ) ,DEL ( 2 , 9 ) , B ( 2, 9) DIMENSION I B C (MAXBC) ,JBC (MAXBC) , VBC (MAXBC) DIMENSION RMATl(MAXEL, 13),RMATZ(MAXEL, 13) DIMENSION NOPD (MAXEL,MAXST),RRSS (MAXDF) COMMON/ONE/ STIFF(2000,300)
C
c
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C
5600
RVISC = PMAT'(IEL,l) RPLAM = PMAT(IEL,2) POWER = PMAT (IEL,3 ) RTEM = PMnT(IEL,4) TCO = PMAT(IEL,5) KODEN = PMAT (IEL,6) DO 5603 IDF = 1,NTRIX = 0.0 KR(IDF) DO 5 6 0 0 JDF = I,NTRIX AAJIDF,JDF)= 0 . 0 CONTTNUE
R
L
C
h**
C IF(IRED.EQ.0) GO TO 5700
c c ***
'FULL'
jrNTE.mrl(JN
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C CALL GAUSSP (NGAUS,XG,CG) LG=O DO 100 IG=1,NGAiTS G = XGIIG) DO 100 JG=l,NGAUS H = XG(JC) LC=LG+1 IF(ITEK.GT.1) GO TO 9996 CALL SFIAPE (G,H, P,DEL,NCN) CALL UER.IV 1 (IDL,IG,ZG, P IDEL,U,MCM, DA, CG,NODE,CORD,MAXEL,MAXNP, MAXST) WRITE(15) IEL ,IG ,JC ,P,DEL,U,DA GO I 0 9995 9996 READ (15) IIEL,IIG,JJG,P,DEL,B , DA C 9995 CONTINUE C START A LOOP AND C * * * UPDATING VISCOSI'I'Y EIND X I (SIMILAR C
STEMP = 0.0 DO 5333 IP = 1,NCN JP = .CABS(NODE(IEL,IP)) ST'EMP = STfTP + TEMP(JP) * P(1P) 5 3 3 3 CONTIPJUE EPSTI = 1.D-10 SRATE = SINV(IEL,LG) TF(SRATE.L'f.EPS1I)SRATE = EPSIT CA1218 VISCA(RVISC,POWER,\IlSC, SKATE,STENP,RTEM,TCO) C C * * * CALCULA'I'EVISCOSITY-DEPENDENT PENALTY PARKMEThK C PIAM = RPLAM * V I S C
C rnT1(lEL,LG) = VISC RIVLAT2(lELtLG)= PLAM
C C C
***
F.OM IXDEX
DO 1 9 T=I,NCN PICN = P (I) EICN(I)= R(1,I) BICN(2)= E 2 I PRVB Jll = I J12 = I + NCN C
C * * * COIsUMcJ INDEX C
DO 19 J=l,NCN J21 = J J22 = LT
+ NCN
c
c
A * +
STIFFNESS MKTRLX
*k****i*****h**hh*t**C********XX****************~~kk***
C AA(Jll,J21) = AA(J11,J21) 1 + ALPHA"PKV3" (V(1)*R ( J ,J)+ V ( 2 ) *B (2,J ) I DA 7 + ALPHAhVZSt*(2.0*B(I,~)~B(l,J)+E(2,I)*B(2,J))*DA 3 + BETA *PIAM* R(L,I)*H(i,J) 'DA AA(J11,522) = AA(Jll,J22) 1 + AT,PKA*VISC* B ( 2 , I ) *B(l,J) "DA 2 + BETA *PLAM* E(l,i)*B(2,J)*DA An(JL2,J21) = AA(512,321) 1 + ALPHA*VlSC* B(l,I)*B(2,J) *DA 2 + BETA *PIARM* B(2,I)*B(I,J) "UA BA(Jl2,522) = U(J12,J22) 1 + ALPIIA*PRVB* (V(1)k B ( l , J)iV(2)" B ( 2 , J) * DA + AIlPH9*VISC*(2.0*E( 2 , I)*E( 2 , J ) +B (1,I)" 3 (1,J)) *DA 2 3 + BETA *PLAM* E(2,I)*E(2,J)*DA 19
FOR R,ZOPTION MODIFY AS
CONTTWE
c
AA(J11.321) = AA(J11.521) + BETA * P L A M ~ ( [ ~ ( ~ , I ~ ~ P ( ~ ) 100 CONTINUE X(1) + B ( l , J ~ ~ P ( I ) / X (+l )P ( t ) * P ( ~ )I)*+2)DA ~( C AA(JI 1,522) = AA(JIIJ22) -t BETA8 P L A ~ * ~ ( l ) * ~ ( Z , J ) / 5700 CONTINUE X( l)*DA c AA(JI2,J21) = AA(J12J21) + BETA* Pl,A~*P(J)*B(Z,I)/ C * * * 'REDCCED' INTEGEW.TION * * X(l)*DA C
IF(1RED .EQ.O) LG = 0 CAM GAUSSP(MC-AUS,XG,CG) DO 200 IG=l ,MGAUS G - XG(IG) DO 200 JG=l,$lGAUS H = XG(JG) LG=LG
I
I
IF(lTER.G'l'.l)GO TO 9994 CALL SHAPE ( G , H,P IDEL,NCN) CALL DERIV NAXEL,MAXNP,MhXST) 1 (IEL,IG,LTG, P I DEL,R , NCIil, DA CG, NODE,CO-W, WRITE(16) IEL ,IG ,JG ,P,DEL,E,DA GO TO 9 9 9 3 9994 READ (16) I L E L , IIG,JJG,P I DEL,B,DA I
SOURCE CODE OF PPVN.f C 9993 CONTTNUE C C * * * UPDATING V I S C O S l ' T Y C STEMP = 0.0 DO 3334 TP = 1,NCN FOR R,% OPTION JP = TABS (NODE ( T E L , IF)) START A L,OOP A;\ID STEMP = STEMP + TEMP(5P) * P(IP) FIND X I (SIMILAR 3334 COWrINUE TO SECINV) THEN YPSII = l.D-10 MODIFY D A AS SRATE = SINV (TEL,LG) DA = DA"X(1) TF(SE?ATE.LT.SPSJI) SRATf = '3PSII CALL VISGA(RVISC,POWER,VISC, SRATE,STEMP,RTEM,TCO) C C **' CALCULA'I'E VISCOSITY-DEPEN3EN'I'PENALTY PARAMETER
C PLAM =
nmm *
VISC
C KMA'l'1 (IEL, LGI = VISC RMAT2 (IET,,IG] = PL,AM
c
SHOULD BE MULTIPLIED BY ZERO FOR STOKES FLOW CALCULATIONS
C * * * ROW INDEX
G DO 20 I-1,NCN PICN - P (I) BICN(lI= B(3,T) BICN(2)= B ( 2 . I ) \ PRVB = RODEN * PICN Jll = I J12 = 1 t MCN C
C * * * COLUMN LMDEX DO 20 J=1,NCN J21 = J
522 J+NCN C C * * * STIFFNESS MATRTX FOR 'REDUCED' INTEGRATION * * * * = * * * * * * + I * * * * * * * * * C AA(J11,321) = A A ( J 1 1 , J2l) 1 + ( 1 .&ALPHA) *PHVB* ( V ( 1 ) *B(1,J) +V(2)' R ( 2 , J ) ) * DA + (1.@-ALPHA)* V I S C ' (2.O"B(1,I)*B (1,J) +B (2,I)* B ( 2 , J) * DA 2 3 + (1.0-BETA)*PLAN* B(l,T)*B(l,J) "DA AA(J11,322) = AA(JII,J32) 1 i (l.O-ALPHAI*VISC*B(L,I)"B(l,J) *DA 2 + (1.0-BETA) *PLAN* B(l,I)*B(2,J) *DA AA(J12,;21) = AA(JlZ.J21/ 1 * (1.0-ACPIIA]*VXSC* B (1,I)* R ( 2 , J 1 *UA 2 t (1.0-BETA) *PLAM* 5(2,lI*B(l,Jl "DA A A ( J 1 2 , J22)=AA (Jl2, J221 + [ I.0-ALPHA)* PRVLI" (V( 1)* B ( 1,J)+V ( 2 * B ( 2,J)) * 1 DA + (l.O--ALPKA) *VISC* (2.O*B(2 T ] *R (2,J j + B ( 1,I)*B (1,J ) 1 *DA 2 3 + (1.0-BETA)*PLAMA B(2,3)*B(2,J)*DA I
20
200
CONTIIWE CONTTNUE
MODIFICATIONS AS THE FULL INTEGRATION IJSING (1.0-%TA)
COMPUTER SIMULRTIONS
-
FINITE ELEMENT PROGRAM
CAIJL SOLVER
1 (AA ,RR , IEL ,NODE ,NCN , IBC ,3BC , VBC ,NBC , RC , NTOV 2,NCOD ,NTKIX,NEL ,VEI, ,R1 ,MAXEL,MAXDF,MAXST,MAXBC,MflXBN,NDF 3 ,NOPD ,RRSS) C
RETUKN
END c
c
C C
***x******************************f*****************************~~~~**~**~
SUBROUTINE ENERGY 1 (NODE ,CORD ,PMAl' ,NDF ,MAXBN,NCOD ,BC ,TEMP ,YEL ,RRSS 2 , R.! , IRED ,XG , N D l M , DA , IEL ,NET,,NCN ,NTOV ,NUM 3 ,TTER ,NCAUS,MGAUS,P ,DEL ,B , S T W ,NTRIX,M?.XEL,MPXNP 4 ,MAXST,MAXDF,MAXBC,IBC,JBC ,VBC ,NBC ,AE ,RE ,NOPD) 1-
C*"*
SOLUTION OF THE ENERGY EQUATION
i
IMPLICIT DOUBLE PKEClSION(A-H,O-Z) DIMENSTON DIMENSION DIMENSION DIMENSION DIMENSION DIMFNSION DTIMENSION DIMENSION DIMENSTON DIMENSION DIMENSION
NODE (PAXEL,MAXST), CORD (MAXNP, NDIM) NCOD (MAXDF) ,BC (MAXDF) ,SINV (MAXEL, NIM) TEMP (MAXNP) ,R1 (MAXDF) .VEL (MAXNP, NDTM) AE ( 1 8 , 18),RE ( 18) XG ( 3) P ( 9) ,DEL ( 2, 9),B ( 2 , 9) X ( 2) ,V ( 2) BICh ( 2 ) ,BJCN ( 2 ) HH ( 2) ,HD!2) ,PMAT (MBXEL, 8 ) IBC (MnxBC) ,JBC (MAXBC) ,VBC (MAXBC) NOPD (MAXEL,MAXSTI, RRSS (MAXDF)
C COMMON/ONE/ STTFF(2000,300) c
L.
C C
****t****************************i*****~********~****~****************~~**
RVTSC = PMAT(TE1,l) QPLAM = PMAT (IEL,2) POWER = PMAT (?.EL,3 )
KTEM
-
PMAT(IEL,4)
= PMAT(TFL,5) RODEN = PMAT (TEL,6 ) TCO
C C * * * BASlC ELEMENT LOOP * * * * * * * * * * * * * * * * i* * ix * * " x * * * * * * * * x * * * * * * * * * * 9. * * .*
c DO 490C lTRIX = 1 ,NTKIX RE(ITR1X) = 0.0 DO 4900 JTRIX . 1 ,NTRIX AE (ITRIX, JTRIX) = 0.0 8900 C
CONTIXUE
* * * NUMERICAL INTEGRATION
C MO = IVGAUS ZFiIRED .EO.0 ) MO = MGAUS LG = 0 DO 5010 IG = 1 ,MO DO 5010 JG = 1 ,MO LG = LG +1
SOURCE CODE OF PPVN.f C C SHAPE FUNCTTONS & THEIR CAKT'ESIHPJ DERIVATIVES ARE READ FROM A
WORK FILE
C IF(IRED .EQ.1) READ (15) IIEL,IIG,JJG,P,DEL,B,DA 1F(IMD .EQ.O) READ (1.6) IIEL,IIG,JJG,P,DEL,B,DA C
C * * * UPDATING VISCOSITY C STEMP = 0.0 DO 3337 IP = 1,NCN JP = IABS (NODE(IEL,IP)) STEMP = STEMP + TEMP (JP) * P (IP) 3337 CONTINUE EPSIl = 1.0-10 SXATE = STNV(IEL,I-G) IF(SRATE.JdT.EPSII) SRATE = EPSII C CALL VISCA(RVISC,POWER,VISC,SRATE,STEMP,RTEM,TCO) CP =PMAT(IEL,7) CONDKzPMAT (TET,, 8) C C * * * CALCULATE VISCOUS HEAT DISSIPATION C VHEAT = 4. * 'JISC * SRATE C C * * * COEFFICIENTS EVALIJATED AT THE INTEGRATION POINTS C DO 5030 IDF = 1 , 2 =.: 0.0 X(IDF) V(1DF) = 0.0 HU(1DF) = 0.0 5030 CONTINUE DO 5040 ZCN .= 1 ,NCN JCN = IABS(NODE(TEL,ICN)1 DO 5040 IDF = 1 , 2 = X(1DF) + P(ICN)*COKD(JCN,IDF) X(LDF) = V(1DE') + P(ZCN)*VEL (JCN,IDF) V(1DF) DO 5040 JDF = 1 , 2 HD(1DF) = HD(IDF)+ 2.0*DEL(JDF,ICN)*CORL)(JCN,IDF) 5040 CONTINUE C * * * STREAMLINE UPWINDING - CALCULATION OF UPWINDING PARAMETER C HDD = SQRT(€iD(t)**2 + HD(2)**2)
c *** 1
AVV lP(AW RW
CONST C
= SQRT( V ( 1 ) * * 2 + V ( 2 ) ' * 2 )
.LT. 1.D-10) = l.D-10 = 0.5 * EDD / AT$
FIND (X1) AT THE INTEGRATION POINTS AND MODIFY DA (SIMILAR TO SUBROUTINE SLOW)
DO 6000 ICN = 1 ,NCN PLCN = P (ICN) BICN(1)= B(1,TCN) BICN ( 7 ) = B (2,ICN) C * * * CALCULATE UPWTNDED WEIGHT FUNCTION DO 4 8 4 0 JDF = 1 , 2 PICN = PICN + V(JDF)xCONST*BICN(JDF) 4850 CONTTNUE C
235
COMPUTER SIMLJLATIONS - FINITE ELEMENT PROGRAM
236
C ' * * ROW INDEX C
IR = ICN
c * * * SOURCE FUNCTION
C C
RE(1R)
= KE(IR)
+ P(ICN)*VHEAT*DA
C DO 6010 JCN = 1 ,NCN PJCN = P ( J C N ) BJCN(lJ= B(1,JCN) BJCN ( 2 ) = B ( 2 , JCN)
c C * * * COLUMN lNDEX
C IC = JCN C DO GO20 MDF
-
1 ,2
C
* * * DTAGONAL ENTRY * * * CONVECTION AND DIFFUSION TERMS
C
c AE (IR,IC) - AE(IR.IC) + RQDEN*CP*PICN*V(MDFJ*BJCN(MUF)*DA + CONDK*BICN(MDF)*BJCN(I%DL.') *DA 1
C CONTINUE
GO20
C CQNTTNUE
bOlO
6 O O C CONTINdE
C 5 0 1 0 C-ONTINUE
C C C
A * x
ASSEMBL>EAND SOLVE
CALL SOLVSR l(AE ,RE ,IEL ,NODE ,NCN ,IBC ,JBC ,VBC ,NBC ,BC ,NTOV 2,NCOD ,NTRIX,NEL ,TEMP ,RI , M A X E L , M A X U F , ~ S T , ~ L 3 C , M A X B N , N D F 3,NOPD , RRSS)
c
C * * * END OF BASIC ELEMENT Loop
h * * X * * k h h X * * X * X * h * * * * * * * X * * * * * * * * X * * * * X k *
C C
RETURN
END C C
* * * * * 1 * * * * + X f X X * * * * * h * * * * * ~ * * * h * * k * h * k k k * * ~ * * * * ~ k * ~ * ~ k * * * ~ * ~ * ~ * * * * * * * * *
C
SUBROUTTNE STRESS 1 (NEL ,NNP ,NCN ,NGAUS,MGAUS,NODE,CORD ,P ,B ,DEI 2 ,17EL .MAXNP,M?iXEL,MAXST,RMATl,RMAT2,IRED ,STRES,CLUMP) C IMPIjICIT DOUBIxE PRECISIONJA-H,OL)
C C FUNCTION
C - - -- - C CALCULATES PRESSURE AND STRESS COMPONENTS AT 'REDUCED', INTEGKATLON c POINTS AND WRITES INTO o w w r FILE. C VARIATIONAL RECOVERY OF PRESSURE, AND STRESS COMPONENTS AT NODES C I
SOURCE CODE OF PPVN,f NODE IMAXEL,MAXSI') ,CORD ( W N P , 2) RMAT1 (MAXEL, 13) , KMAT2 (MAXEL, 13 ) P ( 9) ,DEL ( 2, 9) R ( 2, 9) DIMIQ?STON STRES (NNP , 4 ) ,CLUMP(MAXNP)
DIMENSION DIMENSION DIMENSION DIMENSION C
c
*******************kr******ihXr*r*kXfi***~*~**************~*~********~**~*~***
C
no
REWIND 16 4990 INP = I , ~ P
DO 4990 1 C P = 1 , 4 STRES IINP,ICP) = 0.0 4990 CONTINUE DO 5000 IEL = 1 ,NEL NG = 0 DO 6010 iG - 1 ,MGAUS DO 6010 JG = 1 ,MGATJS NG = 1 + NG READ (16) JEL,KG,LG, P , DEL,B,DA IFG = NG +ZRF.D* (NGAUS*": RTJISC=RMATl(IEL,IFG) RPLAM=RbfAT2(IEL,IFG) xc1 = 0.0 XG2 = 0.0 FILE; IT IS ALREADY U11 0.0 MODIFIED IF YOU ARE IJSTNG R.2 U12 = 0.0 U21 = 0.0 O€'TIC)N U22 = 0.0 DO 6020 ICN = 1 ,MCN JCN = IABS(NODE(IEL,ICN)) XG- = X G I + l'(ICN)*COKD(JCN,1) XGZ = XG2 + P(ICN)*CORD(JCN,2) U11 = U11 + €3(1,ICN)*VEL(JCN,l) YJ12 = U12 1 B(2,ICN)*VEL(JCN,1) U21 = U21 + B(l,ICN)*VEL(JCN,2) L'22 = U22 4 B(2,ICNIXVEL(JCN,2) 6020 CONTINUE C C * * * CAQTESIAN COMPONENTS OF THE STRESS TENSOR 2
\
c PRES =:-RPLAM* (U11 + U22) SD11 = 2.0 *RVISC * U11 SD12 : : RVISC * (U12 4- U 2 1 ) SD22 -: 2.0 cRVISC U22 C
PRES = PRES
-
RPLAM&VR
S11 =-PRES i SD11 S12 = SD12 522 =-PRES + SD22 C C
*
CIGCLlLATh PRESSURE k STRESS AT NODAL POINTS
C
1 1
DO 6500 ICN = 1 ,NCN JCN = lABS(NODE(IEL,ICW)) STRES(JCN,l)- STRES(JCN,1) + P(TCW)*PRES *DA / CLUMP(JCN) STRES(JCN,2)=STRFS(JCU,2) + P(ICN)*S11 *DA / CLLJP(JCN)
* ( V A K I A T I O N A I ~RECOVhRY) *
237
COMPUTER SlMULATIONS - FINITE ELEMENT PROGRAM STRES(JCN,3)= STRES(JCN,3) i P(ICN)*S22 *DA / CLUMP(JCN) STRES(JCN,4)= STRES(JCN,&) 1 + P(ICN)*Si2 "DA / CLUMP(JCN) 6500 CONTINUE 6010 CONTINUE
1
C
5000 CONTUWE
c
kX************h***X*************hX********"*****k***"*~**"************
WRITE (60 2100) 2100 PORMAT('1',' ***VARLATIONAL RECOVERY***',/ I/' NODE',llX,'PRES',12X,'S11',12X,'S2%',12X.'S12') = 1,4),TNP= 1,NnrP) WRITE(60,2110) (INP,(STRES(INP,ICP),ICP 2110 FORMAT(15.4EL5.4) C
STORE 'THE RESULTS FOR POST-PROCESSING
RETURN END C
SUBROUTINE LUMPM 1(CLUMP,NNP ,MAXNP,NEL ,NGAUS,P ,DEL ,Y ,MAXST,NODG ,MAXEL,NCN) IMPLICIT DOUBLE PRECISION(A-H, 0 - 2 ) DIMENSION R ( 2, 9) ,DEL ( 2, 9) ,P DIMENSION CLUMP(MAXNP) DIMENSION NODE (MAXEL,MAXST) DO 5000 INP - 1 ,NNP CLUMP (INP)= 0.0 5000 CONTINUE REWIND 15
c
(
9)
****r***i**h****h*X*****X*Xh*r*rX******X*k~****~*kk*****~**~******~***
DO 5010 IEL 1 ,NEL DO 5020 IG = 1 ,NGAUS DO 5020 JG = 1 ,NGAUS REXR (15) JEL ,KG ,LG ,P ,DEL ,B ,DA DO 5030 ICN = 1 ,NCN ww = 0.0 DO 5040 JCN = 1 ,NCN bv%J = IW + P(1CN)*P(JCN)"DA 5040 CONTIKUE LNP = IABS(NODE(IEL,ICN)) =CLUMP(INP) + WW CLUMP ( INP) 5030 CONTTNUE 5020 CONTINUE 5010 CONTINUE RETURN END
C C n
* * * * * * * * * * * * * t * k * * k * * * * * * * X X * * * * * * * * * * X * * * * * * * * * ~ ~ ~ * * * * ~ * * * * * * * * * * * * * *
SOURCE CODE OF' PPVN.f IMPLICIT DOUBLE PRECISZON(A-H,O-Z) C C**+ THTS SUBROUTINE ASSEMBLES AND SOLVES GLOBAL STIFPNESS EQUATIONS C
c
x * * * h * k h * * h % i r * X k * * * * h x X * * * * * * * X * X * * * * * * * * * * ~ * k * * * * * * ~ ~ * * ~ * * ~ * * ~ * * * * * * * * * * *
C
ARGWIENTS
C
-
C C C C C
C C
C C C
C
C C C (2
---- - - -
RELST (MnxST, MAXST) ELEMENI' COEFFICIENT MATRICES ELEMENT LOAD VECTOR RELRH (MHXST) NOP (MAXEL,WZXST)ELEMENT CONNECTIVITY RSOLN (MAXDF) NODAL VELOCITIES RRHS (MAXDF) GLOBAL LOAD VECTOK STIFF (MAXAR) GLOBAL STIFFNESS MATRIX ARKAY FOR SORTING BOUNDARY CONDITIONS RBC (MAXDF) NE TOTAL NUMBER OF F'.I,EMENTS IN THE MESH TRC (MHXBC) ARRAY FOR BOUNDARY NODES JBC (MAXBC) ARRAY FOR DEGREES OE FREEDOM CORRESPONDING TO A BOUNDARY CONDITION VBC (MAXBC) ARRAY FOR BOUNDARY CONDITION VATJUES **************f*t***tr*****f************~*******~**********~**~~****~*~***
C DIMENSION DIMENSION DIMENSION DIMENSION DIMFNSTON DIMENSION DIMENSION DIMENSION DIMENSION DIMENSION
RELST (MAXST,KAXST), R E L R I I (MAST) NOP (WEL,MAXST) RRHS (MAXDF) RSOLN ( M n x D F ) NCOD (MAXDF) nBc ( MAXDF ) IBC (MAXBC) JBC (MAXBC) VRC (wax) NOPD (MAXEL,MAXST) , QRSS (NAXDF)
C COMMON/ONE/ STIFF(2000,300) C C
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C TF(MM .EQ. 1) REWIND
14
WRITE (14) KELRh C C I E ' ( M M .LT. NE) RETUXW
C DO 5000 I=l,NE DO 5000 J=l,NTKIX NOPD (I, J )=NOP(I,.T) 5000 CONTINUE C
REWIND 14 C
CALL MODIFY 1 (NOP ,NE ,NCN ,IBC ,JBC ,RW,NBC,VBC,NCOD,NTOV, W E L ,MAXDF,MAXST,MAXBC,NDF) 2 C
CALL BANDWE 1 (NOP ,NE
,IBAND,NTRIX,MAXST,MAXEL)
C DO 5010 IMM=l,NE
239
~ O ~ ~ U TSIMLJLATIQNS E R - FINITE EIXMENT PROGRAM C READ ( 14 ) RELRH n
L
CALL ASSEMB 1 (IBAND,NTRTX, IMN,NOP,NTOV,RELRH , RELST, Rfih'S,MAXEL,PTAXST) C
5010
CONTINUE
c CALL SOLVE 1 (RRHS,MAXBN,RSOLN, IB~UD,NTOV,NCOD,RBC,~DF,NDF, RRSS) C DO 5020 I=1,NE DO 5020 J=l,NTRTX NOP (I, J)=NOPD (I,J) 5020 CONTINUE
ASSEMBLES THE ELEMENTAL STIFFNESS MATRTCES DIMENSION F.ELAT (W?.WXST, MAXST) ,RELRH (MAXST) DIMENSION RRHS (NTOV ) DTMENSION NOP (MAXEL,MAXST) CONFIONIONEI STIFF (2000,300) t**************************************~**************~***~**
C c * * CALLCULIATEHALF BANDWIDTF PARAMETERS
C I H B W l = (1EAND-i 1)/ 2
t
LOOP THROUGH ROWS OF ELEMENT STIFFNESS MATRICES
C***
C
DO 5000 ITRIX=
1
,NTRIX ,ITRIX)
IROW =NOP(MM C C"**
ASSEMBLE RIGHT-HAND SIDE
c RRHS (IROW) = RRES (IROW) + PELRH ( ITRIX) C C***
IcOOP THROUGH COLUMNS OF ELEMENT STIFFNESS MATRICES
C DO 5000 J T R I X =
JCOLM=NOP(MFr
1
,NTRIX ,;TRIX)
C
C*** ASSEMBLE GLOBAL STIFFNESS MATRIX IN A BALVDhD FORM
c 3BAND=JCOLM-IKO~+IHBWl
SOIJRCE CODE OF 1 T V N . f STIFF(IROW,JISAND) = STIFF(IROW,JBAED) + RELST(ITRXX,JTRIX) 5000 CONTIWE
c RETURN END C *i********************h*i***********************************************
c SUBROUTINE BANDWD 1 (NOP ,NE ,IBAm,NTRTX,MAXST,MAXEL) C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
c FUNCTTON
C
c -- ---- --
FINDS THE ilW(IMUP1 BANDWIDTH IN THE ASSEBIBLED GLtOAZIT, MATRIX
C
C
DIMENSION NOP (MnxEL,MAXST) C
c
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C DO 5000 MM=l,NE NPMAX=NOP (MM,1) NPMIN=NOP(MM,l) DO 5001 JTRIX=2,NTRIX IF ( N O P (MM, JTRIX) . GT .NPMAX) N€'MAX=NOP (MM, J T R I X ) IF (NOP(MM, J T X I X ) .L?'.NPMIN) NPMIN=NOP(MM, JT'NIX) 5001 CONTINUE NDELTz:NPMAX-NPNTN MBAND=:2*NDELT+l IF(MX . E Q . I) IBAND=MBAN3 IF(IBAND .LT.MBAND) IBAND=MBAND 5000 CONTINUE WRTTF,(60,2000)IBAND 2000 FORPlnT(1H , 7 H IBMgD=,I5)
c RETURN
END C
c
***~hXc.~********hh**~**r,k***t*****h***********.k*****~*********~********,**.~***
C SUBR-OUTINE SOLVE
1
XBAIUD,NTOV,NCOD,RBC,MAXDF,NDF, RRSS) (RRHS,MAXBN,RSOLN,
c IMPLICIT DOUBLE PRECTSION(A-H,O-Z) C C
FUNCTIOX
c -- - - - - - C C
SOLVES THE GTJOBAL STIFFNESS MATRIX USING LU DECOMPOSITION DIMENSION DIMENSION DIMENSION DIMENSION BIMENSlON
r
RRHS (NTOV ) RSOLN (NTOV ) NCOD (YAXDF) RBC (MAXDF) RRSS (MAXDF')
1
2
COMPUTER SIMULATIONS
-
FIN1lE E1,EMENT PROGRAM
C
DO 1021 IMBO = 1,MAXDF RRSS (TMBO) 0.0 1021 CONTINUE
C C*
* * CALCLJLATE HALF BANDWIDTH PARAMETERS
c IHBW=(IBAND-1)/2 IHBW3=IHBWtl C
r*** BOUNDARY
CONDITIONS
DO 8995 ITOV=I,NTOV IF(NCOD(IT0V). N E . 1 ) GO TO 4994
c e h * * INSERT BOUNDARY CONDTTIONS C
STIFF(ITOV,IHBW')=STIFF(ITOV,.LHAW1)*2.D10
c C***
MODIFY RHS VECTOR
C RIiHS~TTOV)=RRHS(ITOV)+STiFF(ITOV,IHBW1)*RRC(ITOV) go to 4995 4994 if(ncod(itov).eq.O) go to 4995 4995 CONTINUE C C*** L'ii DECOMPOSITION C C**;' SET UP THE FIRST ROW. C DO 5 0 0 0 LFIRST=l,IHRW MF.KKST=LFIRST+l NY.LRST=1MBWI-LFIRST STIFF(MFIRST,NFIRST!=STIFF(M~lRST,MFIRST)/STIFF(1,IHBW1) 5000 CONTINUE C
C*** COMPIJWE LU DECOMPOSSTION FOR INTERlOR ELEMENTS C DO 5001 ITOV = 2 ,NTOV JTOV =-1 KTOV = IT'OV+IHBW -1 IF (KTOV.GT .NTOV) KTOV = NTOV LTOV = IZOV-THBW1 DO 5002 MTOV = ITOV,KTOV JTOV = JTOV+1 K1 = ITOV lHBW +JTOV IF(K1 . L T . 1) K1 = 1 LCOLM= MTOV-LTOV 111 - ITOV-1 DO 5002 KROW = K1 ,111 K2 = KROW-lHBW1 MCOLM= KROW-LTOV KCOLMz MTOV-K2 STIFF (ITOV,LCOLM) = 1 S1'IFE (ITOV,LCOLN) -STIFF(ITOV,MCOLM)*STTFF(ICROW,MCOLM) 5002 CONTINUE IF(STO7I.GE.NTOV) GO TO 5001 JTOV = 0 KTOV = KTOV+1
SOURCE CODE OF PPVN.f IF (KTOV.GT .NTOV) KTOV N'fOV 112 = ITOV+L DO 5003 I H O W = I12 ,KTOV JTOV = JTOV+1 ~1 = II'OV IHBW t j ~ o v IF(K1 . L T . 1) K1 - 1 J2 IKOW-IHBWl L = ITOV-J2 IF(JTOV.GE IIIBW) GO TO 8000 DO 5004 K = K1 ,111 K2 = K -1HEW1 M = K -52 N = ITOV-K2 STIFr(IKOW,Lt) =STIFF(IROW,L)-STIFF (IRObJ,Irl) "STIFF ( X , N ) 5004 CONTINUE 8000 CONTINUE STIFF(IROW,L)=STIFF(IROW,L)/STIFF(.CTOV,IHBW1) 5003 CONTiNUE 5001 CONTINUE
c FORWARD REDUCTION LY=F
C***
c 5005
I = 2 ,NTOV = I -1HBW IF(i1 .LT J ) I1 = 1 JZ = I -1HRW1 T T 1 = I -1
DO
11
5006 K = I1 ,Ii1 L = K -I2 RRHS(1)-RRHS(1)-STIFF(L,L)*RRHS(IC) 5006 CONTINUE 5005 CONTINUE DO
c C***
FIND THE SOLUTION VECTOR BY BACK SUBSTITUTION UX=Y
c RSOLN (NTOV) KRHS (NTOV)/STIFF (NTOV,IHBWl) DO 5007 I = 2 ,NTOV I1 = NTOV-I E l I1 - I1 +IHEW IF(I1 .GT.NTOV) I1 - NTOV I 2 - I1 -1HBWl I11 = I1 +1 DO 5008 K = T T Z ,Ii L - K -12 RRHS (11)=RRHS (IT)-STIFF (11, L)*RSOLN(I<) 5009 CONTlNUE RSOLN(II)=RRHS(II)/STIFF(II,IHBWl) 5007 COi\ITINLTC IF(NDF .EQ. 1) GO TO 6 0 0 8 C
C*** MODIFY THE SOLCTION VECTOR ACCORDING TO THE MAIN eh** PRINTING FORMAT' ' 1 1 C
DO 6005 I T O V = 1 ,NTOV RRSS (ITOV)=RSOLN (ITOV) RSOLN (TTOV)-0.0 6005 CONTJNUG
PROGW'S
2
GQMPIJTER SlMU LA'IIQNS - FINITE ELEMENT P R ~ ? ~ ~ A M LTOV = NTOV-1 6006 ITOV = 1 ,LTOTI , 1 JTOV = (ITOV+ 3 ) / Z RSOEN (JTOV)=ZRRSS(ITOV) 6006 CONTINUE DO 6007 ITOV = 2 ,NTOV ,2 JTOV = (NTOVr ITOV)/ 2 RSOLN (JT'OV) ERRSS ( I T O V ) 6007 CONTINUE
DO
L
6009
CONTINUE
C RETURN END C:
C
T * * X * * k * * h A X ~ * X * * * * * t * * * * * * h h h * * * * * h h h * * * ~ * * * * * * * ~ * * * * * * k ~ * * * * * * ~ ~ * * * * * * * * ~
c SUBROUTINE MODIFY 1 (NOP ,NE ,NCN ,IBC ,JBC ,RBC ,mr ,VBC ,NCOD , N T O V , 2 ?/IAXEL,MAXDF,MAXST,MAXBC,NDF ,MAXBN) C lMPCICIT DOUBLE PRECISION(A-H,O-Z)
c C FUNCTION
c C
C C
--
------
MODTFIES ELEMEVT CONNECTIVITY AND AUDRESSTNG OF THE POINTS TO MAKE THESE CONSISTANT WITH THE PROGRAM
DTMENSIOM NOS' DIMENSTON IBC DIMENSION RBC
(MAXEL,MAXST) (MAXBC) (MAXDF)
(MAXBC) DIMENSION VBC (MAXBC) DIMENSION NCOD (YinXDF) DINENSION INDX ( 9 ) RIBENSION JBC
c C * + * MODIFY ELEMENT CONNECTIVITY C DO 6 0 0 0 MM =1,NE DO 5000 I C N = l , N C ? J INDX(ICNJ=NOP(MM, TCN) $000 CONTINUE DO 5001 ICN=l,NCN I C N = IMnX ( ICN DO 5002 IDF=I,NDF IND=NCN*(IDF-l)+TCN JVD=NDF*(KCN-l)+IDF NOP (MM,IND)=JND >002 CONTINUE 5001 COWI'INUE 6000 CONTINUE C l E ( N D F . E Q . 1) GO TO 8000 C C * + * MODIFY ARRAY FOR ADDRESSING BOUNDARY DATA
c DO 5999 INP=l,NTOV RBC (INP) 0 NCOD ( INP)= 0
SOURCE CODE OF PPVpI.f 5 999 CONTINTdE
DO 6001 INP=Z,NRC IP(IBC(INP).EQ.O.OR.JBC(INP).EQ.3) GO TO 6003 ICOD-NDP"(lBC(INP) l)tJBC(INP) NCOD (TCOD)= I RBC ( TCOD)=VRC ( ZNP) 6001 CONTINLIE
C U 0 0 0 RETURN
END C ..........................................................................
C SUBROUTTNE GETNO2 (NNP ,CORD , IDVl
, IDV2 ,MAXNP, NDIM)
_--_ -__ IMPLICIT DOUBLE PRECISION(A-h,O-L!
C C
IDVI INPUT DEVICE ID I D V 2 OUTPUT DEVICE ID
C C
c DIMF,NSIO>T CORD(MRXNP, NDIM) C READ (IDV1,lOOO) (IN13 , (COKU(INP,IDF),IDF=1,2! ,JNP=l,NNP) WRITE(IDV2,2000) WRITE (TDVZ,2010) ( J N P , (CORD ( J N P , IDF), IDF=l,2) ,JNP=1,NNP)
c RETURN C
1000 2000 1' 2010
FORMAT(Ii,lE15.8) FORMAT('l',///' ' , 2 0 ( ' * ' ) , ' NODAL COORDINATEd ' , 2 0 ( ' * ' ) , / / ',2(7X,'ID.',7X,'X-COORD',7X,'Y-COORD',20X)/) FORMAT( ' 110,2G35.5,20X, T10,2Gli.5) I ,
END C C C
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SUBROUTINE CETELM (NEL ,NON ,NODE ,IDV1 ,IDV2 ,MAXEL) C IMPLICIT DOUBLE PRECISION(A-H,O-Z)
C
DIMENSION NODE, ( M A x m , 18) C
DO 5 0 0 0 I E L = 1 ,NET, 5000 READ (IDV1,1000)MFX '(NODE IEL,I C N ) ,?CN=;,NCN) bJRlTE(IDV2,2000) no 5010 JEL = 1 ,NEL 50LO Wr;ITE(IDV2,2010) JEL , (NODE J R L ICN), lCN=1,NCN) C RETURN I
ZOC0 FOlWAT(1015) 2000 FOFNAT('l',///,' ',20('*'),' ELEMENT CONNECTIVITY '.20('*'!,// 1' ',7X,'ID.',5X,'N0 D A L -- P 0 L N T E N T R I E S',/! 2010 FORMAT(' ' , I l O , S X , l . O I 8 / , ' ',15X,1018/,'',15X,1018) t END
C
c
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
~ ~ M P SIMULATIONS ~ T E ~
FINITE ELEMENT PROGRAM
C SUBROUTINE GETBCD (NBC,IBC,JBC,VBC,TDVi,IDV2,MAXBC) C IMPLICIT DOUBLE PRECTSION(A-H,O-2) C
DIMENSION IBC (MAXRC) ,JBC (MAXBC),VW (MAXSC) C READ (lDVl,1000) (IBC(LND) ,J B C (IND) , VRC ( IND) ,IND=1,NBC) WRITE(IDV2,2000) WRITE(IDV2,%010) (lBC(IND) ,JBC(IND) ,VBC(IND) ,lND:L,NBC) c
RETURN C 1 000 FORMAT ( 2I5 F10.01 2000 FORMAT(‘ ’ , / / / , ‘ ‘,20(‘*‘),’ BOUNDAKY CONDITIONS ‘,20(’*’),// I
1’ ‘,2(7X,‘ID.’,2X,’DOF’,10X,‘VALUE’,lOX)/) 2010 FORMAT(’ ‘,5X,?T5,G15.5,15X,215,G15.5) C
END c
C
................................................................
n
i .
SUBROUTINE PUTBCV 1 (NNiiE’ ,NBC ,IBC , J R C ,VBC ,NCOD ,MAXEC,MAXDF,BC) C IMPLICIT DOUBLE PRECIS.LON(A-H,O-Z) C C C
NCOD ARRAY FOR IDENTIFICATION OF BOUNDARY NODES BC ARRAY FOR STORING BOTJNDARY CONDITION VALUES
C DIMEPaSION IBC (MAXRC) ,JBC (MAXBC) ,VBC (MAXEC) DIMENSION NCOD (MAXDF) ,BC (MAXDF) C
DO 5000 IND = 1 ,XBC IF(JBC(IND).GT.2) GO TO 5000 rnD lBC(IND)+(JBCIIND)-I)*NNP BC (JND) = VBC (IND) NCOD ( J N D ) = 1 5 0 0 0 CONTTNL‘E
SUBROUTINE PUTBCT 1 (NBC ,IBC ,JBC ,VBC ,NCOD ,BC ,MAXBC,MAXDF) IMPLICIT DOUBLE PRECISSON(A-H,O-Z) DIMENSION IBC (PIAXBC) ,JBC (MAXBC! ,VBC (MAXBC) DIMENSION NCOD (MAXDF) ,BC (MAXDF) DO 5000 IND = I
,NBC TF(JBC(JND) .NE.3) GO TO 5000 JNU = IBC(1ND) BC (JND!= VBC(IND) NCOD (JND)= 1
SOURCE CODE OF PPVN.f 5000 CONTINiiE C
RETUKN END
c C
......................................................................
C
SUBROUTINE CLEAN 1 (R1 , B C ,NCOD ,NTOV ,MAXDF, PWXBN! C
IMPLICIT' DOUBLE PRECISION(A-H,O-Z)
c DIMENSION R1 (MAXDF) DIMENSIDIT BC (MAXD?') DlCMENSlON NCOD ( W D F ) CO>lMON/ONE/ STIFF(2000,300) C C FUNCTION C _ _ _ _~ _ _ _ C CLEANS THE USED ARFLRYS AND PREPARES THEM FOR REUSE C
DO 5 0 0 0 I
= 1,NTOV Rl(1) = 0.0 RC(I) = 0.0 NCOD(I)= 0 5000 CONTINUE DO 5020 I = 1,600 DO 5020 J - 1,500 STIFF ( J ,T! = n. n 5020 CONTINUE
c RETIJRW END C C C
***h*f*fh**X*h**A***hXt**xX*t****"*Xh**************~****-*********~*~~**~~~~**
SUBHOUTINE SETPRM 1 (NNP ,NEK ,NCN ,NODE ,NDF ,XAXEL,MTLXST) C
IMPLICIT DOUaLE PRFCTSION(A-H,O-2) C
DIMENSION NODE (MAXEL,MAXjT) L
C FUNCTION C --_ ----C SETS THE LOCATION DATA FOR NODAL DEGREES OF FREEDOM C DO 5000 IEL - 1 ,NEL DO 5000 ICN - 1 ,NCN KCN =NODE(IEL,I C N ) JCN =ICN+(NDF-1)"NCN LCN =KCN+(NDF-1)"NNP
c NODE (IEL,JCN) = LCN C
5000 CONTINUE C
KETURN END
7
IMPLaICT'I 30URLE P R E O I S I O N ( A - H , 0 - Z )
c DIMENSION PMAT (EIIAXEL,
8)
C WRITE ( LDV2 2 0 0 0 ) I
C READ ( T D V 1 , l O O O ) RVISC ,POWER .TRFF ,TBCO ,RODEN , C P ,CONDK
READ ( I D V l , 1 0 1 0 ) RPLAM lFROM = 1 I T 0 = NEL I F ( T R E F . E Q . 0 . ) TREF = 0.00: DO 5 0 1 0 I E L = IFROM ,IT0 P M A T ( I E L , l ) = RVISC P M A T ( I E L , 2 ) = RPLAl4 PMAT ( I E L , 3 ) = POWER P M A T ( I F L , 4 ) = TREF PMAT(TEL,S) = TBCO PMnT ( I E L , 6) = RODEN PMAT(TEL,7) = CF PMAT ( I E L , 8 ) CONDK RTEM = TREF C
C C C C C C
* + * PARAMETERS OF 'SHE POWEK-LAW MUDEL
*** **k
*** *** ***
RVISC =
MEU NOUGHT, CONSlSTENCY C O E F F I C I E N T
RPLAM - PENALTY PARAMETER POWbR = POWER LAW TNDEX TRFF = REFERENCE TEMPERATURE TRCO = COEFFICIENT b I N THE POWFH LAW MODEL
C
c ***
w Y s I C a L PARAMETERS
C
C C
C
*** *** ***
RODEN = MATERIAI, DENSITY CP = S P E C T F I C HEAT CONDK = HEAT CONDUCTiVITY C O E F F I C I E N T
C
5 0 1 0 CONTINUE
c WTITE ( l D V 2 , 2 0 1 0 ) IFROM ,I T 0 , K'gISC ,KPLAM ,POWER WRITE ( I D V Z , 2 0 2 0 ) b J R I T R ( I D 7 J 7 , 2 0 3 0 ) TREF ,TBCO V\JRITE (IDVZ, 2 0 4 0 ) C 1 ; R I T E ( l D V 2 , 2 0 5 0 ) RODEN , C P ,CONDK 5 0 0 0 CONTINUE
C RETU RX L
1 0 0 0 FORMAT(7FlO.O)
1010 FORMAT'(F15.0)
SOURCE CODE OF FPVN.f 7000 rORMAT('U',//' ',35('*'),'IVIATERlAL PROPERTIES',35('*'),// I' ' ,7X,'ID.' , iX, 'EID.(FROM-TO)' ,7X,'CONSISTENCY COEYFICIEN'I'',8X, 2 ' PENALTY P A W E T E R ',R X , POWER LAW INDEX', / 1 2 010 FORlviAT ( ' ' ,I10,114,14,3G2 0.5 2020 FOXMAT(lOX,#**REFERENCE TEMPERHTURE * * * * COEFFICIENT h " " ' 1 2 0 3 0 FOflMAT(17X.G10.3,17X,Gl0.3) 2040 FORMAT 1(1OX,' * * DFNSITY * * * * SPECIFiC €IFAT * * * * CONDUCTIVITY * * ' ) 2 0 5 0 FORMAT(tdX,GlO.3,8X,Gl0.3,l?X.G1~1.3~ C
END
c (-
* * * * h * * x ~ % * k * * * * * * * % % * ~ * * r ~ * ~ * * * t * * h * * * * ~ % * * * * * * * * k ~ * * * * * * * * * * ~ * ~ * ~ % * * ~ ~ * * % ~
C
SUBROUTINE VISCA 1 (RL7ISC, POWER,VISC,SRFlTE, STmP,RTEM,TCO) C IMPLICIT DOUBLE PRECISION(A-Ii,O-Z) C C * * * CALCULATE SHEAR DEPENDENT VISCOSITY
c PTNDX = (POWER-L )/2
VISC
=
RVISCX((4.*SRATF,)**PINDX)*RXP(-TCO*(STEMP-RTEM))
C RETURN END C
c
i * * % * * * * * ~ * * * * * * * * f * * * * * * * * * * * * X * * * * ~ * ~ * * * * * * * * * * *
c SUBROUTINE COIVI'OL 1 (VEL,TEPIP, IThR,NTOV,~P,MnXNP,P.IAXDE, ERROV,ERROT,VET,TET) i
IMPLICI'IDOUBLE PRECISION(A-I-1,O-Z) c
L
DIMEi\ISIOMVFL (MAXDF) ,TEMP iK&XNP) DIMENSION VET (MAXDF),TET (MAXNP) C C
* * * CALCULATE DIFFERLNCE BETWEEN VELOCITIES IN CONSECUTIVE 1TERATIONS
C ERRV = 0.0
TORV = 0.0 ERRT = 0.0 TORT = 0.0 DO 1000 ICHECR = 1,NTOV IF ( I T E R . EQ .I) VET (ICFIECK) = 0.0 ERRV = ERRV + (VELIICHECR) VET(ICHECK)) + n 2 TORV = TORV + (VLL(ICHEC1C)) * * 2 C VEI'
(ICHECK) = VEL (TCWFCK)
C 1000
CONTTPJUE ERROV= ERRV/TORV
c
C * * * CALCULATE DIFFERENCE BETWEEN TEMPERATURES IN CONSECUTIVE C ITEKA'I'I ONS DO 2 0 0 0 TCHRCK = 1,NNP IF(ITER.EQ.1) TET(1CHECK) = 0.0 ERRT = ERRT t (TEMP(1CHECK) -TETIICHECK)) **2 TOR'I' = TORT + (TEMPJICHECIC))**2
COMPUTER SIMULATIONS - FINITE ELEMENT PROGRAM C
TET (ICHECK) = TEMP (ICHECK) C
2000
COI\I'I?INUE
ERROT- ERRT/TORT
C RETURN
END f
c ...................................................................... C SUBROUTINE OUTPUT 1 (NNP ,VET, ,TEMP ,MAXDF,MAXNP) C
IMPLnICIT DOUBLE PRECISLON(A-h,O - Z ) C
DIMENSION VEL (MAXDF),TEMP (MAXNP) C
WRTTE(G0.6000) 5999 FOPM?LT(' ID. UX
UY
T' / 1
C
E o w r ( - * * *NODAL
VELOCITIES AND NODAL TEMPERATURES * * * I / ) DO GO01 INP = 1,mP JXTP = I N P + NNP WRITE(G0,6002) 1 INP ,VEL(TNP),VEL(SNP) ,TEMP(IYP) 6002 FORMAT(J5,2E13.4,F13.4)
6000
6001 CONTINUE
c FETURN
END C
c
* * * * l * * * * * X * * * h * f * * * * * * * * * * * * * ~ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * % * * *
C
Gerald, C . F. and Wheatley, P. O., 1984. Applied Numerical Anulvsis, 3rd edn, AddisonEapidus, I,. and Pinder, G. F., 1982. Numerical Solution o j Partial Diyfirential Equations in Science and Engineering, Wiley, New York.
In a Cavtesiaii (planar) coordinate system of (x, y, z ) shown in Appendix Figure I the position vectors for points PI and P2 arc 8'wen as
are unit vectors in the x, y and z directions, respect~vely~ ~ ~ a g ~ i o~f line u d PIP2 e (i.e. the distance between points PIand P2)is thus found 8s
52
A P ~ ~ N ~ SUMMARY I X OF VECTOR AND TENSOR AKRLYSIIS
The ~ ~ ~ r e ccosines ~ i o n of the vector r = x i
-C J$
+z
are the n u ~ b e r sd c ~ ~ as ~ed
z cos y -- -
lel
where n, /j and y are the angles which the vector makes with the positive ~ i r e c t i o ~of§ the coordinate axis shown in App x Figure I and 1 vector ~ a ~ ~ i t found u d e as Irl i- y2 -i- 9” er. ~ o ~ iused ~ orthoo ~ l ~ onal ~ ~ o r systems d ~ are ~ acylindrid ~ ~ (polar) a pherical systems s k o ~ nas ~
7;:
d,.w2
or
x
Spherical (r, 0, 4)
a l t r a n s f o ~ ~ ~of i oanCartesian vector A wit system o f 0123, under rolation of thc coordiaate system to 01 2 3 is by the ~ o ~ ~ ~e qwuia~t ~i ~gn
3 -
AJ = &/Ar
(su~mationconvci~t~on over the repeated index I i s ~ ~ ~ l i e d )
where ,& denote the c ~ ~ ~ o n eof~the i t vector s in coordinate system 01 2 3 and dJI are the cosines of the angles between the old axis oi (i 1,231 and the new one c’j. Therefore I--
vector which remains unchan~edin such a transfo~iation(i.e. to be i n v ~ r i a ~ t .
-
) is said
are equal if they have the same ~ a ~ n iaiid t ~ ~ c position of their origin. ctor whose ~ a ~ n i ~ is u dequal e to the magnitude site direction is denoted by - A . The sum (resultant) of vectors ~ a ~ ~ ~ l llaw e ~~ ro a~p r~ ai c~shown ~ ~ l yas
PH
is a vector having the same
(commutative law for addition) (associative law for addition) (coi~mutativelaw for ~ ~ u l t i p ~ ~ c ~ ~ i ~ (associative law for n i ~ i l t i ~ l ~ ~ a t i o i i ~ (distribu~~ve law)
54
APPENDIX
-
SUMMARY OF VECTOR AND TENSOR ANALYSIS
The scalar (dot) product of two vectors is a iiutnber found as
1 cos0 where 0 < 8 < ?r is the angle between the directions of ~~tei-~iatively in terms of components of
= 1 then A and (commutative law for scalar product)
+ A.C
(distributive law for scalar product)
Vector (cross) produc with a ~ a ~ n i t u of de the directions o f A and B such that A , B and C make a right-handed system shown as
l t ~ r ~ a t i v ein~terns y of the components of can be expressed as the followitig determinant
, the vector p r ~ ~ uAc tx
~speci~cally i x i =j x j =
(distributive law)
SOME VECTOR CALCULUS RELATIONS
c
S
be a vector whose c o ~ p o n e ~are t s functions of a scalar variable (e.g. timedependent positioii vector of a point P in a three-dimensional domain) ( t )= x(t)i ty ( t ) j
+ L(t)k
then
If are ~ i f f ~ r e n t i ~ vector b ~ e functions of scalar t and # is a differentiable function of I then
xB
and
d -(A x
3t
The vector differential operator del (or mblu) written as 0 is defined by
are the unit vectors in a Cartesian coordinate system. The del operator has properties analogous to those of ordinary vectors. The ~~~~~e~~ of a scalar 4 (x, y , z> i s defined by
The urivergence of a vector V(x, y , z) i s defined by
A P ~ ~ N ~ SUMMAKY I X OF VECTOR AND ‘TENSOR ANALYSIS
The ctid of a vector
(x, y ,
z) is defined by
x A ) = V(V.A)- V2A
where
(in a Cartesiaxi system) is called the Lapucim operator. (1) =
P’Jt)i+ VJ(t)j-+ Vz(t)
If there exists a vector
then
is an arbitrary constant vector.
If V is the volume bounded by a closed surface S and ~ ~ ~with t icoi~ti~ o ~ uous derivatives, then
is a vector fbnction of
SOME VECTOR CALCULUS RELATIONS
dS =
.dS
S
If
Note that the surface integral surface S.
$&
denotes the flux of
Over the closed
oke Let S he an open, two-sided surface bour~de~l by a curve C, then the line inte of vector A (curve C i s traversed in the positive direction) is expressed as
C
S
s
Note that Green's theorem in the plane expressed as
is the special case of the toltes theorem. It should also be noted that 5. div~~gence theorem can.be obt ned by ~ c i ~ ~ ~ a l ~of~~d~t ie oe nntheor 's plane by replacing the region aiid i t s boundary curve C with a space region
corern provides 3 coiivcnieiit means for obtaining rate of change of vector field f u ~ c t over i ~ ~a volume V(t) as
where
is the i ~ a t e r ~ as u~ b s t a ~ ttime i~ld ~ e r i v ~ t i vand ~
1'
i s the velocity vector.
(i
A ~ P E -~S ~U ~ MX A ~OF Y VECTOR AND TENSOR ANALYSIS
8. These definitions arise from the transformation properties of vectors and can be s ~ ~ i i ~ i i i a r ias ~ efollows: d If in the tramforination of the coordinate system (XI, x2,. . . ,x") to another system (d,.Z*, , . . ,P)quantities AI, A2,. . .A, transform to AI, A*,. . . ,AlI, such that A,,p = 1 , 2 , . . . ,n
then A I A2,. . . ,A , arc the components of a covariant vector. A',A2,. . . ,A" are said to be components of a contravariant vector if in the tra~sformatio~i of the coordinate system ( x i ,x2,. . . ,xN) to anothcr system (Z',Z2,., . they transform according to
,a)
A Y , p I= 1,2,.. . , n
Field variables i ~ e n t i ~ ebyd their m a ~ i t u d eand two associated called s ~ c o n ~ - o r dtensors er (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order t~~so is rthe physical function stress which i s a surface force i d e n t ~ ~ eby d n i a ~ ~ t directioii ~ ~ e , and orientation of the surface upon which it is acting. Using a inathematical approach a second-order Cartesiaii tensor is defined as an nine components To, i, j = 1, 2, 3, in--_ the Cartesian coordinate system of 0123 which on rotation of the system to 01 2 3 become
where n'ip arid djq are the cosines of the mglcs betwecn the new and old conditions of these direction cosines the c o ~ r ~axis. ~ i ~y the ~ ~orthogo~ality e inverse of this transforniatioii is expressed as
in a t h r e c ~ d i ~ e n s ~ o nf raal I ~ ~of e onents of a second-order tensor reference are written as the following 3 x 3 matrix
TENSOR ALGEBRA
It follows that by using coinponent forms second-order tensors can also be i~anipulatedby rules of matrix analysis. A second-order tensor whose components satisfy TEI= T,, is called symmetric a i d has six distinct components. If T , = -Tiz then the tensor is said to aiitisymmet ric. To obtain the transpose of a tensor the indices of its components (originally given as TPq)are transposed such that
P
Y
where ap is the unit vector in direction p . Tensors of second rank are shown either using expanded or Dyadic notations. Dyadic forms are genesali~ationo f vectors and are sbown as two vectors together without brackets or rnultiplicatiorii symbols. For exam denotes the dyadic product of vectors A and which is a second-order tensor (a further generalizatio ading to ‘triads’ ms of its c o ~ p ~ n e n t s is shown in the denote third-order tensors). expanded form as
P
Y
(i.e. ii, j j , ij, ji,etc.) are red pairs of c o o r ~ ~ ~ i a t e directions. An isotropic tensor is one whose components are unchan~edby rot~~tion of the c o o r ~ i ~ asystem. te
The unit seco~id~order tensor is the given as the fol~ow~ng matrix
he sum of two tensors is found by adding their corsespon~ixigcomponents as
P
Y
2
A P P E -~SUMMARY ~ ~ OF VECTOR AND TENSOR ANALYSIS
The scalar (double-dot) product of two tensors is found as follows
T
using ~ y a ~~roducts ic
The tensor ( s i i ~ ~ l ~ - dproduct ot) of two tensors i s foun
-
i,e. yl of the
nt is Y
ot> p ~ o ~ of u a~ teiisor t with a vector i s foun
TENSOR ALGEBRA
The tensor (cross) product of a tensor with a vector i s found as follows
__ -
i.e. pi component __ Y
ere egkis the permutation symbol which is a third-order tensor de
0, if any two of i, j,k are the same 1 if ijk is an even per~utationof 1,2,3 - 1 if jjk is an odd permutation of 1,2,3
The n ~ ~ g n i t u dofe a tensor i s defiiicd as
Any tensor may be re a ~ ~ ~ s y ~ m part etric
as the sum of a sy~iiietricpart and an
The o p ~ r ~ ~ tofi o~i d~e n t i ~ y two i n ~ indices of a tensor arid so su thein is ~ ~ n o wasn c o n t ~ a c t i o T, ~ , = TIT $- T22 $- T33.
The fol~owingthree scalars r c ~ a i ni n ~ e ~ e ~ ~ of d ethe n t choice of c o o r ~ ~ n a t e are defined and hence are called t
The first invariant is the trace of the tensor, found as T,,
ZE
+
T22
+
T33
(sum of diagonal terms in the c o m ~ o ~ e n t s
~ - SUMMARY ~ OF ~ VECTOR N AND TENSOR ~ ANALYSIS ~
A
The second invariant is the trace of
LI = trace of T~ = tr
TzjT/i 1
1
It can heiicc: be seen that the magnitude of a sy~metrictensor its second invariant as
he third invariant is the trace of
Jia~o~ous to vector operations the tensorial form of the divergence theorem is
written as
A n a l o ~ o to ~ svector operations the tensorial form of tokes t l ~ e o r is e ~written as s Y
c
itrrilar to vectors, based on tlie tra~isfo~i~atioii p r ~ p e ~ t i of e s the second tensors the following three types of covariant, contravariant and mixed coiiiponents are defined
SOME TENSOR CALCULUS RELATIONS
JPr
3.P 3%' = .~
3xq dx'
coiitravariant
(summation convention is used)
Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for n o n - ~ e w t o n i afluids ~ will have different forms depending on whether covariant or contravariant components of these tensors are uscd. For cxainple, the convected time derivatives o f covariant and contravariant stress tensors are expressed as
(covariaiit tensor and
.T
(contravariant tensor
+ ( V V ) ~are ] the vorticity vector aiicl rate of ld, respectively. Note that the c the special case of the general time derivative the coinponelit forms the above time derivatives are written as o,
dVk
fit{+ T,n Chj
~ ~ o v a r icom~oneiits a~~t corresponding to lowerconvected d e r i ~ ~ ~ t i v c ~
and
In a Gartesian coordinate system the differential o f arc length of a line is defined as ds = ddx2 idy2 dz2 (and hence d? = dx2 + dy2 + dz2). After t r a n s f o ~ ~ a tion from the Cartesian system (x,y , z) to a general t h r e e - d ~ ~ e n s i ocurvi~~a~ linear coordina~esystcrn (E) this can be written as
+
2
A
~
~ - ~S U~~ ~ NA ROF ~ Y VECI'OR ~ X AND ~ ~ N ANALYSIS S O ~
or using s L ~ m ~ ~ t convention ioii
where the quaiitities gpyare elemenis of a matrix found as the dot ~ r o ~ L i cof ts the pairs of basic tangential vectors as p,q,k = I , ...,n The f o l ~ o w i nfigure ~ i s a two~di~iiensional example il~ustratin~ the synibols use in the ~ r ~ v r ~ l ao t i~o isi s h ~ ~ .
Y
E' x
The matrix gpcl represents the components o f a covariant ~ ~ c o i i d - oteiisor r~~~r called the 'metric tensor", ~ e c a u it s ~defines ~ ~ s t a ~~ ic ~c a s u ~with e i ~ rese ~ ~ inatcs E', , . . ,<". To illustrate the a lication of this d e ~ ~ i t i oinn the te ~ r a n s f o ~ ~ n ~we t i o nconsider th
For s ~ m ~ l i we c ~ assume ~y a t w o - ~ i ~ e n ~ i ocase, ~ i a tlicrefore I
SOME TENSOR CALCULUS RELATIONS
5
the Jacobian of the t r a n s f ~ ~ i n a t is i owritten ~ as
The t ~ * a ~ i ~ ~of oc~o ~~ ap it ~i ~ oe~f ~a t second-order s tensor, given as the ~o~lowiiig matrix in the coordinate system x
is based
011
where
r e ~ r e s e the ~ t corn oiients of gr,<[.Therefore
w~~~~A I 1 etc. are the c o ~ p ~ n e of n t ~in tlie new c ~ o r d ~ nsyste ~te
This Page Intentionally Left Blank
Page numbers
111
italic arc to bc found in the References sections
Adachi K 88; 108 Ahmad S 51; 68 Alexaiider JM 16. 69, I10 Andre JiM 141; I88 Aris R 2. 4, 78, 177; IT, 108, 188 Armstrong RC 15, 16, 139, 188 Babuska B 72; I08 Balestra M 109 Bathe KJ 41, 203, 206; 68,207 Beaulxie M 141; 188 Bell BC 79, 126: 108, 139 Belytshchko T 108 Beriistein B 13; 17. Bird R B 2, 9, 126, 153, 164; 15, 16. 139, 188 Bogner FK 26; 68 Rrenner SC 18; 68 Brooks AN 54, 61, 63; 48 Brezzi F 72; 108 Hulklcy K 6: 15 Cardona A 93; 108 Carreau PJ 7; I5 Casso11 N 6 , I5 Caxtro JM 171; 189 Chan TF 208 C'hang CJ 173; I90 Chatui-an1P 161; 190 Christie I 93; I08 Ciarlet PG 21; 68 Ciarke J 142; I88 Coates PD 13; 16 Cook LP 164; I90 Criminale WO 14; 15 Crochet MJ 81, 82; 108. I 0 9 Crouzeix M 28. 68
Dhillon J 189 Doi M 13; 15 Doiiea J 67. 103. 154; 68, 108, I88
Edwards SF 13; 15 Ericksen JI, 14; I5 Fillbey GL 1 4 15
Fix GJ 33: 69 Folgar F 189 Forsylhe GE 51: 68 Fox FL 68 Fox L 206;207 Franca LP 109 Freakley PK 142. 150; 188 Gallagher RH 109 Gerald CF 21, 51, 201, 212; 68, 207, 250 Ghafoun SN 150, 188 Ghassemteh E t84; 189 Ghoreishy MHK 91, 99, 103, 142, 146, 151; 109, 189 Girault V 18, 68 Glowinsk~R 208 Gluab GH 207; 207 Gogos CG 1; 16 Gresho RH 97, 98; 109. 110 Ilaniiart B 173; 189 Hassager 0 15, 16, 139. I88 Herscliel WH 6; 15 Hieber CA 173; l89 Hinton E 51: 68 Hirt CW 109 Hitchkiss RS 109 Hood P 27. 51, 205; 68, 69, 208 Hopfinger EJ 173; 189 Hou L 1.57; 208 Hughes TJK 54, 61, 63, 74. 80, 184; 68, 108, 109, I89 Tdetsohn S 93; 108 Irons B 51, 205; 68,208 Johiisoii C 18, 68 Johiison MW 12, 13; 15 Kalonr PN 15 Kaye A 13, 161; 15, 189 Kearsley EA 13, 15 Keinblowski Z 9; 15 ICeunings R 80, 88, I0Y Kheshgi HS 75: 109 Kinsella M 189
A U T ~ O lNDEX ~ L ~ ] x ~ uLs 21. 213, 68, 250 LCCCC 171, 173, 189 Lee RI, 73, I09 1.rghtfoot EM IS, 188 Liseikrn VD 192. 196. 208 Lodge AS 13, 15, I89 Luo XL 73, 109 Macoski CW I 0 9 Maday U 1%,208 Mdamatarrs N 109 Mallet M 109 Malchd JM 81, 109 Masaa L 189 Mawipilrs C 208 Mayerr DE 206: 207 Medwell JO 110 Meler CR 51, 68 Mclzerier AB 8, 15 Middleman S 2, 7, 10, 170, I $ , 189 Middleton J 69 Mvza FA 1.39 Miteliell AR 18, 68 Mrtsoulis E 9, 13. 14, 127, 141. 15, J j Y . I88 Mimkami A I 0 9 Morgan K '34, 44, 64;69 Mortoii KW 91, 104, 109 Nakazawa S 7, 54, 75, 77, 129, 16, 68, 109. 110, I40 Nassehr V 81, 86, 91, 99. 102, 103, 104, 126. 142, 151, 157, 155. 170, 175, 184, 68. 109, 110, 139, 189, I90 Narasii~imanS 161, 188 Nayfeh AH 164, 189 Nichols BD 101: 109 Nguen N 92, 109 Noui-Oniid B 93; 109
Olaguiiju DO 164, 190 Oldroyed JG 6, 11; I S Olley P 13; 16 Ostwald W 6, 16 Owen DRJ 51: 68 Pdnde GN 69 Papanastasiou TC 89, 97; 109 Paslay PR 8; 26 Pearson JRA 9. 123, 175; 16, 139, IYU Pcnaux J 208 Petera AT 208 Pctera .I 9, 38, 92, 101, 102, 104, 126, 155, 168, 170, 15, 68, 109, 110, 139, 140, 190 Petrie CJS 175; 190 Pinder CF 21, 213; 68, 250 Pitrnian JFT 7, 54, 66, 73, 77, 100, 101, 126, 123, 155. 175; Ih, 68, IVY, I I 0 , 140, 189, I90
Pironneau 0 18; 68 Phan-Tliren N 12; 16 Presa WH 201, 204; 205
Quartapellc L 154, ZA?? Rance J 110 Raviart Ph IS, 28: 68 Reddy JN 27, 73; 69, 110 Renardy M 98; 110 Reynen J 0% 109 Rudraieh N 15 de Saiiipaio PAB 74; I08 Sander R 175; 290 Sani RL 98; 109, 110 Schlichtiag H 173; I 9 0 Schmit LA 68 Scott LR 18; 68 Shen SF 173. I89 Scrrven LE 75, 98; 109, IIiJ Segalman D 12, 13; 15 Silhmdn WJ 98; I10 Slibai A 8: 16 Soh SK 173, I90 Spiegel MR 153, 182; 190 Stevenson J F 10; 16 Stewart WE 15. 188 Strang G 33, 68 Surana KS 79. 126; 108, 139 Saarbnck SJ 81, 86; I10 Tadmor 2 1, 16 Tanner RI 9, 12, 81, 82, 90, 95, 127, 161; 16. 110, 140,I90 Tayloi C 27, 07; 69, 110, 139 Tayloi RL 18, 40, 66, 72, 76, 92, 69, 110 Thonipion E 102; 110 Townsend P 67, 69 Tucker CL 110, 189 ValeDC 189 Vatchopoulos J 139 Van Loan CF 207; 207 de Wade A 6, 15 Wait R 18; 68 Wagner MN 13; I6 Wehster M F 67; 69 Wheatley PO 21, 51, 201, 212; 68, 207, 250 Whitlock M 8, 15 Widlung OB 208 Wood R D 16, llcl Wn J 74; 110, 140 Zapas Id 13, 15 Zienkrewicr OC 18, 34. 40, 44. 64, 66, 72, 74, 75, 76, 02, 16, 69, 109, II0, 140
Active front 205 Adaptive 103 meshng 14 rcgellcratlon 9 1 re-mechmg I04 Anisotropic 3 Aiiisotropy 6 Arb3trary Layranginn Eulerian franiework 102 method 103 scheme 148 Area coordinates 10-31 36. 40 Arrhenius formula 7 Assembly of elemeiital equations 61 of elemental matrices 197 of elemental stiffness equations 48, 78, 145, 197. 199 of elcmcntal stifXiiess mati ice?, 43 roll and fixed-bladc 150 Asymptotic expansion 178. 182 expansion scheme 175, 177 Auxiliary first-order differential equations 126 subroutines 196 Auisyimnetric coordinate system 1 11. 113, 117, 121, 127, 129 131, 136. 183 doniains 17, 209, 215 flow 113 114, 162 formulation 21 6 models 160 option 217 problciiis 215 systems 113 unit cells 184
criterion 72 stability condition 74-75. 134 Back substitution 200, 202. 204, 243 Band solver 203. 205-206 Bandwidth 198. 203, 212, 241, 242
Bi-linear elenient 29, 40 four-noded y uadriiateral 226 interpolation 73. 83-84 rectangle 33 rectimgular element 25 shape functions 28, 83 sub-elements 86 quadrilateral elements 21 1 Bingham plasticb 6, 8 Bi-quadratic ekmeni 30. 40, 84, 143, 156 interpolation 73 mne-noded qu~tdrilateral 127 rectangle 33 reclangular element 26 shape functions 83 quadfilsteral elctnents 21 1 Body force 2, 3, 72. 81. 94, 111-112, 132 pseudo- 81 Boundary curve 257 element 17 heat fiux 100 integrals 96 line 19, 91 line terms 46. 47 line integrals 96-97, 100, 120, 138, 145. 175
Boundary (cnndiiitms) Dirichlet 95, 97, 217 essential 95, 101, 217 llalural 96 Robins 100 V C ~ Netirnann I 96 Boundary value problem 41, 90. 101 tlubnov-Calerkin method 43 Bulk modulus 184 Bulk viscosity 144 Carreau equation 126, 144 model 169 Cauchy’s equation of motion 71. 70 Cauchy stress 2-3, 81-82, 94 Choleskv factorisation 203 Co-defoimatioual I I. 26.1
270
SLJBJECT INDEX
Condition number 206 Conforming clement 32 Consibtency coefficient 6-7, 127, 214, 248-249 Contravartant 258, 262-263 Convection dominated 54, 58, 90, 209 Co-rotational 11 Couette (flow) 151 152, 160, 163, 171 Cobanant 258, 262 263 Creeping (flow) 2, 18, 54, 79, 81, 143, 170, 175, 182, 210, 215 Crout reduction 203 Deborah numbcr 10 Diffusion; artificial 54 cross-wind 63 physical 54 Direct solution 200, 212 Discrete (also see discrete penalty) momentum equation 82 solution 42 Dissipation (heat, viscous heat) 3. 91; 128-129, 235 Eigenvalue 206 Elasticity; analysis 183-184 flllld 79 linear 183 modulus 187 Elliptic 33. 53, 195 Euclidian norin 21 3 Eulerian (&U see Framework) approach 87 coordinate system 2 description 89 solution scheme 87, 89 time derivative 103 velocity field 88 Finger strain teusor 13, 87- 89 First invariant 13, 261 Framework Arbitrary Lagrangian-Eulerian 102- 1.03, 146 Cartesian 29 Eulerian (fled) 87, 90, 101-1033 142, 145, 151 Lagrang1a.n 14, 86, 89, 91, 104, 153 Free boundary 101-104, 104, 106, 145 .. 146, 153 Free surFace 89, 101-102, 104, 107-108, 142, 145147, 150-151, 153 Freedom, degrees of 21, 23, 25-.-26,28-29.45,95, 198-199, 213, 215, 221, 239, 247 F’roiital solution 51, 203, 205-..-206 Frontal solver 205-206
Galerkin finik element procedure 44 finite element scheme 183
formulation 46 metliod 54 scheme 54 weighted residual equation 54 weighted residual method 54, 64, 67 weightcd residual statement 45, 55, 114, 118, 154 Gaussian elimination 201--204 Gauss-Legendre qtiadrature (,we quadrature) Green’s theorem (see theorem) Generalized Newtonian approach 14, 79 behaviour 6. 93, 150, 151 constitutive equations 6 constitutive models 9 flow 12, 94, 126, 209 tluids 5-6, 8, 79, 11-112, 114, 132, 136, 142-143, 162, 164, 167, 173 Global coordinate (system) 29, 31, 34.-35, 37~-39, 46, 107 Global (equations) 43, 48, SO, 95 -96, 197-199, 203, 206, 21 i I
Hcle-Shaw approach 175 equation 173-174 Hermite clement 26. 32, 38, 101, 126, 155 Hermite interpolation 21 Hierarclira1 elements 40, 126 h-version 40 Ill-conditioned equation 75 matrix 207 Incomprcssibility condition 75, 95 constraint 2; 75-76. 97 Inelastic time dependent (fluids) 4, 8 tiinc independent (fluids) 4 Initial value problems 101 Interpolation function 25-27, 83 iiivariants of a second order tensor 261 Isopwametric elcments 37- 38, 220 Iterative algorithm 66, 129, 139 cycle 85 improvement of accuracy 204 improvement of solution 207 loop 85, 92, 158 methods 75 proccdure 51, 80, 85, 97, 199 solution niethod 199 Jacobian of coordinate traiisfoniiation 37, 52, 228 Jacobian of’ the transformation 265 Jacobian matrix 38, 21 1 Jaumann or corotational time derivative 11 JohnsodSegaIman model 12
SUBJECT INDEX KDKZ mock1 13 Kronecker delta 3, 72, 82, 91, 94, 105, 152. 259 Lagrange eleinents 21. 26, 29, 45, 55 inierpoldtion 21 interpolation function 27 polynomials 26 Least squaiea 64, 79, 125-126, 131 Line of syinmetiy 94, 96 Load vector 43, 167, 197, 201, 220, 226, 239 Local coordinate 29-30, 34 -35, 37, 39-40,47, 85, 107, 154, 196, 211 Local area coordmates 31 Local Cdrtesian cooidinate system 29 30 Lowei and upper taanguldr niatriLes 203-204 Lowei triangle 204 Lower triangular matrix 204 Lubrication dpproximation 18, 170-171, 173 LU decomposition 51 203, 205, 207, 212 Mapping functions 35 ~sop~uametiic31 36. 38-39, 51-52, 92, I43 parametric 35 Mass lumping 77. 21 1 Mdss matrix 77 Master element 34-37, 51 Mdxwell fluid 108 Mdxwell stress 152 Mciiiory function 13, 87 Mesh refinement 40, 49, 57. 100, 192, 199 Mesh \elocity 146 147 Mctiic form 153 Metiic tensor 263 264 Mixed forninlation 83 86 Mixed mlerpolation 74 Morland-Lec hypothesis 90 Mo\ing boundary 101 Nahme-GnfEth number 129 Namer’s slip condition 98-99, 158 Ntvier-Stokes equations 4-5, 18, 79, 97, 209 210, 215, 230 Newton-raphson method 107 Non-isothermal now 89, 209, 215. 223 Oldroyd constitutive equation 8 1 derivative 11 models 12 type differential constitutive equations 11-12 Oldroyd-B model 12, 81 82 Optimal convergence 33 streainline upwinding formula 92 Pdrabolic 53, 96. 164, 195 Parametric mapping 35
Parcnt elements 37 Pal tial discretization technique 64 Pascal triangle 24 26 Pearson number 129 feclet number 54, 90 Penalty contiiiuous 76, 118-122, 127-128, 133. 115, 150-151, 157, 166 167, 183, 209, 215, 220 discrete 76, 123 125 method 75-77 paianieter 75, 77, 146, 167, 183, 213, 231. 233, 248 relation 77, 217 relationship 76, 118. 120-121, 123, 133 schemes 75, 77-79, 120. 125 sub-matiix 76 terms 76-77, 120, 167, 220, 222 Pcrniutation symbol 261 Petrov-Galerkm formulation 64, 132 method 54, 58 scheme 132, 166 Phaii-ThiedTanner equation 12, 157 Pivoting 200-201 Poiseuille (flow) 171 Poisson’s iatio 183, 187 Power-law equation 7, 126, 177, 210 fliud 173. 175, 182, 200 index 6-7, 114, 214, 249 model 6-7, 174, 213-214, 248 Preconditioned conjug& gradient method 51 Pre-processor, program 212, 21 5 Piessuie potential 173 Protean coordinate system 88 Pseudoplastic fluids 6-8 Pseudo-dcnsity method 102, 145 Pseudo-time 90 p-vcrsioii hierarchical clcineiits 126 of the finite element 40 Quadratuie 39, 85, 196 Guass-Lcgendie 39, 76 point coordinate? 39 points 40 Quadiilateral 19 elements 33, 39, 107. 211 isoparametric elemcnts 220 Rate of convergence 80. 93 Rate of deformation 3, 5, 82, 263 instantaneous 4 Rate of deformation tensor 4-5, 10, 12, 14, 94, 126, 164, 177, 229-230 second invariant of 126, 159, 210 211, 224, 228-230 Rate of strain 1, 4, 11. 79, 127 Rate of strain tensor 136, 263 Rate of stress relaxation 90
Reduced integration 76-77, 167, 214, 217, 220-222, 229, 232-233. 236 Relaxation lime 10, 13. 152, 165 Retardation time 12 Reynolds number 2, 54. 71. 11 I, 206 Rheopectic fluid 8 Right Cauchy-Green (strain) tensor 13, 87 Round-off error 33, 75, 201 Self-adjoint problems 66 Serendipity elements 27 Shape functions 20 29, 31-32, 34-35, 38, 40, 42 47, 5 1-52, 54-56, 59, 64. 74. 77. 83 84. 92, 114. 123, 126, 130. 132, 154, 196-197.211, 220, 226, 228 Shear thickening 8 Shear th~iiiniiig 6-7 Shear modulus 184 Skyline reduction method 203 Slip wall boundary conditions 96, 98 Spaise and banded 48 Sparse banded 43, 198, 203 Spatial discretizalion 6.5 66 Standaid Galerkia method 43 45, 49, 53-55. 57-58, 62, 64, 90 yroceduie 166 scheme 132, 174 Stiffness and load matrices 130- 131 Stiffness coefficicnt matrix 49 Slrffiiesv equation 46, 48-49, 52-51, 56-57, 60, 76, 78, 84-85, 86, 92, 111, 145, 166, 197-199, 205, 206, 211-212, 239 Stiffness matrix 43, 49, 53, 75-76, 197, 205, 211, 215-217, 220 221, 232-233, 239-241 Stokes equation 79 Stokes flow7 2, 93, 111-112, 114, 116, 120, 122, 134. 183. 215, 232 233 Stokesian Huid 4 Streamline Upwind Petl-ov-Galerkin nictliod 53 Slreamliiie Upwind Petrov-Galerkin scheme 91 Subparametric transforniations 35 Superconvergent solution 63 Superparanietric transformations 35
‘Taylor-Hood element 27-28, 84 family 72 recttaiigle 73 tridngle 73 Ta ylor-Galerkin method 65 67 aclieine 61 time stepping 133 U-V-P scheme 136, 153 Tayloi series 67, 105, 134-135 Tensile loading 185 Tenvor product elements 26
Heriruk elements 26 Lagrangc eletncnts 26, 29 Theta (0) time stepping 65, 92, 131, 133, 145, 153, 157 Theoi em Divergencc 262 Divergence (Guas5) 256 Green’s 43, 45, 55, 59. 77-78, 83, 91. 94 97, 114, 119, 126, 130, 138, 14.7, 175, 257 mean value 105 Reyiiolds transport 108, 257 Stokes 257, 262 Thernial cotiductiviiy 3, 90, 163 Thernial difiusivity 132 Theimal stress 99, 217 Third invnriaiit (or a Tecoiid ordei tensor) 262 Thixotropic 8 Time-temperaturc s h ~ f t ~ n 90 g Traction boundary condition 97 Transient Stokes flow 132 Tn-quadiatic 26 Turbulent flow 97 Upper convected Maxwell md Oldroyd-B Buid 82 Maxwell and Oldroyd-B models 81 Mdxwell constitutive model 152 Maxwell equation 13, 152, 164 Maxwell flow 82 Maxwell Huid 82 Maxwell model 11 Oldroyd derivatrve 11 Upper Inangular foim 200 204 ‘CJpwindiiig 58. 61, 91, 153 application of 55, 58 consrsteiit 92 constant 130 error 63 inconsistent 54, 8.5, 132 parameter 54, 61, 92, 130, 168, 235 scheme 92, 153 selective 54 stieamlme upwinding 81, 85, 91, 102-103. 209, 23s forinula 92 Petrov-Gdleikni scheme 91 scheme 54 technique 54, 63 Unconstraiiied boundary condition 187 Variational methods 195 principle 18 iecovery 77. 85, 166, 174, 186, 209 211, 225, 236, 237 238 statement 18, 77 Velocity gradients tensor 4, 1 I VII tual element 100-101, 175 fluid 102
SUBJECT INDEX Viscoelastic constitutive beiiavmir 54 constitutive equations 9. 12. 71. 79, 85, 89-90. 150, 155-156 Maxwell class of 11 constitutive niodcls 6, 13 effects 10 extra stress 80, 87. 89 flow 13, 38, 71, 80, 86, 89 tnodellmg of 79 models 79 non-isotherrmal 89 simulation of 93 tlutd 9-13, 80-81, 90, 136, 152, 153, 157, 162, 164, 167-169 model 9, t2-13, 81, 93-95, 152, 155 Viscoelarticity 10, I3 Viscometric flow 6, 10, 14. 127, 161, 169-170 rcgiinc 162
Weak form 46, 55, 59, 92 formulation 77 statement 43, 46, 78 vamiional slatement 58 Weight function 2. 43, 45 46, 53, 54, 58, 72, 77, 84, 91-92, 94, 114, 123, 130. 132 modified 59, 62-63, 74 upwinded 130, 235 Weighted residual finile elernenl method 17-1 8, 71 fiiiite eletiieiit scheme 54, 65, 141 inethod 18. 41, 42 Petrov-Galerkin h i t e element method 91 scheme 131 statement 42-43, 46, 54. 58. 64-65, 72, 83-84, 93-94, 131-132, 158 technique 64 Weisxenherg number 10, 80-82, 86
Wall slip 98. 156, 15X-160
Zero shear 7
