Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations

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Author: Michael Kleiber

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{ T ) { l )

l

1

dh

()
0 + —

<™>

i
+

da l

dh

dp

^ 0 "K hAA

w

OCT

dh

d <
dh

v

(7.68)

Sensitivity in Buckling and Post-Buckling Problems • 231 T h e sensitivities d a j dh and d ' p / d f o are known from t h e previous s t e p , while t h e t e r m d a* jdh has to be c o m p u t e d from t h e sensitivity solution of t h e linear 'reference load problem l

l

1

*K

q* =

(T)

(

Q*

(7.69)

by using t h e relations, cf. also Eq. (2.6)

dh

dh

dh

da

dh

dp

dh

y

a* = ' C B ' q *

(7.71)

l

*

A

. dh* dC ' C B - dh 7 - + —dhB ' q *

irr

l

n

dh

f

m

(7.72)

w h e n ' C is t h e current tangent c o n s t i t u t i v e m a t r i x . It is observed that Eq.(7.68) is t h e direct c o u n t e r p a r t of t h e c o n t i n u u m - b a s e d relationship (7.107). For large displacement elastic problems K reduces to t h e elastic stiffness matrix K and Eq. (7.68) becomes 4

T ( C )

(cl)

dA'A

/d K^ ... \ dh

d'K^d'q + — dq

l

T V

dh

d'K^dW , d *K. rr+—s T , - 1 A ' A - •;dh da dh da

{a)

M

d a* \ -— v dh x

(7.73)

d'K^/dh while t h e a s s u m p t i o n s of changes in t h e external load, as indicated in Eq. (7.58), and changes in t h e stress, according to a

=

\{T)**

(7.7-1)

allow us to simplify Eq. (7.73) further to yield dX c

T

/d'K

d*K \

( d )

w

in which t h e buckling load factor is a s s u m e d to have been c o m p u t e d from an eigenproblem of t h e form [K ">(q(/0; h) + X(h) *K >(er*{h))]

v(h)

= 0

(7.76)

with t h e o r t h o n o r m a l i t y condition (7.66) a d o p t e d . lem (7.76) is taken in t h e form, [140]

S o m e t i m e s t h e eigenprob-

l

K with

(el)

c

{&

(q(M;^) + ^ ' K ( W M ) ] v ( / i ) = 0 w

(7.77)

232 • Buckling and Post-Buckling and t h e n t h e buckling load sensitivity results as (7.79) Finally, for elastic p r o b l e m s t h a t are linear prior to reaching t h e buckling point, Eq. (7.75) b e c o m e s (7.80) since in this case K does not d e p e n d on q. Even if in t h e o r y Eqs. (7.68) or (7.75) lend themselves to effective i m p l e m e n t a t i o n , s o m e difficulties with using t h e m are clearly visible: (el)

(a) T h e formulation requires c o m p u t a t i o n of t h e d i s p l a c e m e n t sensitivities d q / d / i (which are needed to find der/dh and d p / d / i ) at points closely approaching t h e critical s t a t e at which K becomes singular. Even though it is unlikely to hit exactly t h e singular point in t h e course of t h e incremental F E M analysis, t h e sensitivity gradients m a y grow significantly in t h e vicinity of t h e critical point. m

1

(b) C o m p u t a t i o n of t h e conditional derivatives of t h e t a n g e n t stiffness m a t r i x may be troublesome. W i t h regard to t h e first of t h e above d r a w b a c k s , t h e r e exists an a l t e r n a t i v e , [137], valid only in t h e elastic range, which allows us to bypass t h e c o m p u t a t i o n of d q / d / i . To derive t h e a p p r o p r i a t e sensitivity expression within such an adjoint a p p r o a c h we consider an a u g m e n t e d e q u a t i o n at a s t a t e c close to t h e critical s t a t e as 1

v (/i) T

[ K < T ' q ( / i ) ; h) + '\(h) e

c

* K < « V ( / » ) ) ] v(A) (7.81)

in which / i is a Lagrangian multiplier vector enforcing e q u i l i b r i u m at t h e s t a t e c \ Differentiating Eq. (7.81) with respect to h yields

/ d <'F •1d q C

T

r)q d 'A

dh

3

C

'F

8h

d 'A C

dh

c

dh

d 'X

(v *K T

, C T )

v + ^

T

Q*) +

c

dh

(l + ,i Q*)+v' T

= 0

Sensitivity in Buckling and Post-Buckling Problems • 233 w h e r e t h e following a s s u m p t i o n was m a d e : Ad (

dh

( vT C• K m ' " v )) ' T

v

K

*

v

v

^ - ^ v dh

T

V

on account of a nearly zero vector d 'A

(

c

(

'd

C

'K

( T )

-aft-

v

v

+

K

c

"

(7.83) m

v . Therefore, we have

^3

R +

" ) Q

(,

'

81)

provided /x is such t h a t i) ' ' K

: /

^ T C

dq

H =

C

K

(

( T )

K

( T )

v

v

(7.^-.

=

j

T h e key t o t h e p o t e n t i a l efficiency of this formulation lies in only explicit derivatives on t h e r i g h t - h a n d side of Eq. (7.84): its d r a w b a c k s a r e t h e necessity of c o m p u t i n g t h e conditional derivatives a n d t h e inversion of t h e almost singular tangent stiffness m a t r i x at s t a t e c'.

7.3.3

Post-Critical Analysis

We shall begin this section by looking into a formula which uses t h e e x t e n d e d syst e m (7.23) with t h e constraint / = 0 given in t h e form (7.43). By differentiating t h e s t a t e e q u a t i o n (7.44) with respect to h we obtain Eq. (7.55), i.e. (

+

A

^

K

m ( T )

—

a x Q* , ' Q = A A —-• dh • 4 dh

i

d

d

-— dh -

d'+*'F dh

dAA , + -r7- Q (

Aq^Aq(/i.)

(7-87

dh

while t h e c o n s t r a i n t e q u a t i o n (7.43) becomes dAq dh

, dAA 4- AA — dh— = 0 x

(7.88!

The s y s t e m of (N 4- 1) e q u a t i o n s (7.87) a n d (7.88) has to b e solved for t h e ( A 4 - 1) u n k n o w n s d A q / d / i a n d d A A / d / / . As is usual in sensitivity analysis, t h e i n c r e m e n t a l e q u i l i b r i u m p r o b l e m is a s s u m e d to have been solved already so that A q . A A . ' ' ( T a n d * ' p a r e known. T h e values of d < q / M 'X/dh, d W/dh a n d d ' p / d A a r e known from t h e previous step of t h e sensitivity analysis. To solve Eqs. (7.87) a n d (7.88) let us i n t r o d u c e t w o auxiliary vectors q, and d A q „ / d / ? such t h a t r

A t

t + A t

+ A

K

{ T i

q

l

d

=

Q*

l

(7.89) dh

dh

dh

dh

Aq^Aq(/ ) t

234 • Buckling a n d Post-Buckling Since t h e r i g h t - h a n d sides are known, Eqs. (7.89) can he solved lor q, and dAq,,/el//. F u r t h e r m o r e , we have, cf. E q . ( 7 . 8 7 ) dAq

_

dAA

~7i/T ~ I T T

dAq,, q

'

7.90)

dh.

+

and E q . ( 7 . 8 8 ) can he rewritt.en as .

,,. d A q

dAA

. . dAA

dh

Aq'q.-f

dh

A A J ^ dh

+ Aq

0

^

1

(7.91

which yields dAA

. ... d A q - A q ' ^ ( A q el// <

v

dh

l

q ,

+

A A

7.92!

Relationships (7.90) and (7.92) are t h e solution to Eqs. (7.87) a n d (7.88). In other words. Eqs. (7.90) and (7.92) allow us to find t h e sensitivity of t h e solution ( A q . AA) ( a n d . consequently, ( q , A)) for any nonlinear problem analysed with the aid of t h e arc-length constraint of t h e hyper-spherical t y p e (7.13). For nonlinear elastic s y s t e m s , Eq. (7.92) b e c o m e s , cf. Eqs. (7.58) a n d (7.59) 1

t+At

dAq^

dh

dh.

A

J

a dh

Aq

q,

- f

AA

7.93)

ith

±K

q, = , 1 A c

l"

;

Q" 7.91)

+A,rJm(»+An^Sl_ l\ i)

t

'

dh

dh J

E q u a t i o n (7.93) was used for t h e sensitivity analysis of elastic buckling and post-buckling p r o b l e m s in [1 10]; t h e observations m a d e t h e r e apply to t h e inclastic formula (7.92) as well, and m a y be generalized as follows: (a) Since no buckling condition has been imposed in deriving Eq. (7.92), even it a critical point has been a p p r o x i m a t e l y (or exactly) d e t e r m i n e d during t h e e q u i l i b r i u m analysis, t h e r e is no g u a r a n t e e that t h e load sensitivity c o m p u t e d according to Eq. (7.92) will bo a good a p p r o x i m a t i o n of t h e real buckling load sensitivity. 1

(b) T h e advantage of t h e formula is that no buckling mode is needed to effectively u s e Eq. (7.92). Moreover, t h e formula is applicable to I>ot 1 I t h e buckling a n d limit load points. 1

1

1

1

(c) i W e ' e l l e s s to say. the formula, is applicable to any point along t h e equilibrium p a t h , except for t h e critical points themselves, which are unlikely to be precisely e n c o u n t e r o d . 1

1

Sensitivity in Buckling and Post-Buckling Problems • 235

£ (0

Displacement q Figure 7.1

Exact and numerically approximate critical points

O b s e r v a t i o n (a) ran be further s u b s t a n t i a t e d with reference to Fig. 7.1 by considering t h e exact and numerically a p p r o x i m a t e critical points along t h e ideal syst e m e q u i l i b r i u m p a t h , points B(h) and B'(h), respectively, and a point B'(h,-\-Ah) o b t a i n e d as t h e intersection of t h e p e r t u r b e d s y s t e m e q u i l i b r i u m p a t h with t h e constraint condition (7.13); point B(h 4- Ah) is t h e h y p o t h e t i c a l exact critical point on t h e p e r t u r b e d p a t h . T h e error in buckling load sensitivity assessment s t e m s from two i n d e p e n d e n t sources: (i) point B'(h) is d e t e r m i n e d only approxim a t e l y in t h e course of t h e incremental analysis (no eigenproblem is solved), and (ii) point B'{h 4- Ah) is not t h e p e r t u r b e d s y s t e m bifurcation point because t h e buckling constraint is not enforced on t h e p e r t u r b e d deformation process. Clearly, t h e approximate p e r t u r b a t i o n of t h e critical load factor A/, A'. ( t h e i n c r e m e n t A/, refers to a change in h r a t h e r t h a n to a change in A in p s e u d o - t i m e ) o b t a i n e d using t h e t e c h n i q u e proposed will generally be different from t h e u n k n o w n exact, p e r t u r b a t i o n A; A . 4

4

r

4

t

cr

An i m p r o v e m e n t to (a.) m a y possibly be achieved by replacing the hyper-spherical constraint (7.43) by t h e so-called hyper-cylindrical constraint, of t h e form

4

dAA _ _.,. r d dll

L

t + A t

dh

F

Q )] Aq^Aq(h)

in which t h e following n o t a t i o n is used: v -

M

" 'K A

/ :

Aq

(v 'Q*)~' r

(7.9b)

236 • Buckling and Post-Buckling A very i n t e r e s t i n g f e a t u r e of Eq. (7.96) is t h a t it lends itself nicely to a comparison with Eq. (7.56) derived earlier. In fact b o t h formulae are a l m o s t identical, t h e only difference being t h e vector v used here instead of t h e eigenvector v used in E q . ( 7 . 5 6 ) . However, A q at a point close to critical is a r e a s o n a b l e a p p r o x i m a t i o n to t h e buckling m o d e v , while Eq. (7.97) can b e t r e a t e d as a first i t e r a t i o n in t h e e i g e n p r o b l e m inverse i t e r a t i o n p r o c e d u r e , which m a k e s v even m o r e similar to v . W i t h all t h e a d v a n t a g e s m e n t i o n e d a b o v e , formula (7.96) is t h u s formally very similar to t h a t given by Eq. (7.56). For elastic s y s t e m s with t h e reference load being i n d e p e n d e n t of t h e design p a r a m e t e r h, Eq. (7.96) b e c o m e s dA

7.4

„ 8F /

„

C o n t i n u u m - B a s e d Approach

P r o b l e m s of response sensitivity at critical a n d post-critical p o i n t s have so far been considered in t h e framework of a discrete (finite-dimensional) formalism. T h e reason for doing this has been t h e relative simplicity of t h e a p p r o a c h and convenience in discussing p r o g r a m m i n g a s p e c t s . To i l l u s t r a t e how a c o r r e s p o n d i n g c o n t i n u u m - b a s e d analysis can be performed we shall consider below t h e p r o b l e m of load sensitivity at bifurcation p o i n t s in t h e non-discretized f o r m a l i s m . We shall base our derivations on t h e v i r t u a l i n c r e m e n t a l work e q u a t i o n of t h e form (2.-T2). A s s u m e t h a t t h e i n c r e m e n t a l load changes p r o p o r t i o n a l l y to a scalar load p a r a m e t e r A A, A A £ [0,1]. We have then / A f AAu dil Jn< =

AA

f f A t AAu Jdix f

Jn

1

d(dtt) + f

'rSAudfl

't* AAu

JdQ$

d(dQ)

7.99)

where f and t * a r e pre-defined reference load vectors. T h e v o l u m e t r i c and surface load vectors at t h e critical point are t h u s given as (

"f -

(

'f-t-A'Vf* .

c

t =

t

t + A A't* c

(7.100)

F u r t h e r m o r e , let t h e response b e linear in t h e interval considered (again, much m o r e s o p h i s t i c a t e d a s s u m p t i o n s can be m a d e r e g a r d i n g t h e i n c r e m e n t a l b e h a v i o u r at t h e s t e p p r e c e d i n g t h e critical point, cf. [101, 122, 181], b u t t h e simplistic approach will suffice to highlight t h e basic m e t h o d o l o g y of t h e critical-point analysis) so t h a t t h e v i r t u a l work e q u a t i o n b e c o m e s

J

[ A ( AA A u ) • C • A AAu + W • AA( AA A u , AAu) + AA Acr • 1(AA

A u , SAu)] J

dft = AA [ / lJn

(

'f* AAu dft + / 't* AAu d(dSl) Jam 7.101)

Continuum-Based Approach • 237 in which we have shown t h e way in which t h e p a r t i c u l a r functions d e p e n d on AA. From Eq. (7.101) we o b t a i n [AA ( A A u ) • C • ASAu

/

+ AA W •

2A(Au, <5Au) VSAudn

+ ( A A ) Vr*• 2A(Au,<5Au)ldft = AA 2

r

un

J

(

+

'i" Jan*

6Aud{dQ) (7.102)

where cr" - Acr is t h e i n c r e m e n t a l stress corresponding linearly to t h e incremental reference load given by f W-ASAudn

= f

'f <SAudft + / ' U A u d ( t f f l )

(7.103)

w i t h o u t t h e factor AA. E q u a t i o n (7.102) finally becomes

J

[(AAu) • C • (AAu) + W •

2A(Au,SAu)]dft

-F AA / V-2A(Au,£Au)dft = / ' f * £ A u d f t + / 't* Jn< in Jan* 1

6A11

d(#ft) (7.104)

For AA to a s s u m e a critical value A A such t h a t t h e left-hand side of Eq. (7.104) a d m i t s m o r e t h a n one solution, we enforce t h e variational condition C

/ [ A v • C A SAu -F W 2A(v,

<5Au)ldft + A A / W- 2A(v, C

SAu) dft = 0 (7.105)

in which t h e a r g u m e n t in t h e form of t h e i n c r e m e n t a l d i s p l a c e m e n t s A u have been replaced with t h e e i g e n m o d e s v . E q u a t i o n (7.105) is t h e generalized eigenproblem in which t h e smallest AA is sought such t h a t for any kinematically admissible SAu, Eq. (7.105) has a non-zero solution for v. Because of t h e original nonlinearity of t h e i n c r e m e n t a l solution and t h e linearization a d o p t e d for it in deriving Eq. (7.101), t h e factor A'A cannot he too large, otherwise t h e equilibrium i n c r e m e n t a l p r o b l e m has to be solved and upd a t e d to t h e load level at t i m e t + At a n d t h e eigensolution o b t a i n e d again. R e t u r n i n g to t h e sensitivity p r o b l e m , Eq. (7.105) is differentiated with respect to t h e design variable h yielding / [ A ( ^ V C - A £ A u + V - 2 l f ^ , e A u ) dft Jet' I V dh / \dh /. dC — dW — (Av) — • A SAu + — - • 2 A ( v , SAu)dft dh dh Ar\ AA 2A(v,cAu)dft + ACA / •2A(v.^Au)dft d/T JQ' dh

/

+

+ AA C

JQ

1

in which SAu

2A\dhf ^ ^ A u "J )

dft = 0

is a s s u m e d not to be a function of h.

(7.106)

238 • Buckling and Post-Buckling Hv now selecting" e A u = v . v being found from t h e oigenproblem Eq. (7.105), and using t h e a p p r o p r i a t e s y m m e t r y in t h e integrand expression, we observe that, t h e first and last integrals in Eq. (7.10b) vanish on account of condition (7.105), whereas t h e rest of Eq. (7.10b) implies t h e relationship 1

— =-{/ dh

XJiV + AA

;av)

dC — .(Av)+ dh

d cr — dh l

• 2 A ( v , v ; dil

r

~dh

2A(v. ; v

J ><2 A ( v , v ) d i )

(7.107

This e q u a t i o n can be used to e v a l u a t e t h e buckling load factor sensitivity provided t h a t t h e buckling oigenproblem has been solved for ( A A , v ) .

7.5

Numerical Illustrations

Many of t h e most essential aspects of sensitivity analysis for buckling-prono structures can be d e m o n s t r a t e d numerically with reference to t h e 31) truss spring s y s t e m shown in Fig. 7.2. T h e two truss m e m b e r s are a s s u m e d to be of equal

Figure 7.2

Truss spring system

length and their individual ( F u l e r ) buckling b e h a v i o u r is excluded; t h e structure is cotifine'd to deform s y m m e t r i c a l l y in the plane of Fig. 7.2(a). On the' othe'r h a n d , theTe is a pre-set imperfection assumed zero in some' analyses below but non-zero in o t h e r s . For e/" = 0 t h e structure is allowed to buckle (bifurcate ) from the vertical position as shown in Fig. 7.2(b). T h e secondary post-buckling path as wedl as the e q u i l i b r i u m path for t h e imperfect structure are t h u s complex cemibinations of a s n a p - t h r o u g h - p r o n e vertical motion represented by t h e displaccMnent.
1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Numerical Illustrations • T h e i n p u t d a t a are taken as follows (length unit /' = 2.188618, h

[m], force unit

239

[kN]):

= 0.22, A = 6 x 1 0 ~ , Q" = 1, h = 40. E = 2.1 x 10 . 4

0

elasto-plastic m a t e r i a l with k i n e m a t i c h a r d e n i n g is a s s u m e d with E /E

= 0.01,

vn

a = 3 . 5 x 10 ; E and E

An

8

d e n o t e Young's and tangent m o d u l i , respectively: a

4

rn

is

y

t h e initial yield limit of t h e m a t e r i a l . Figures 7.3(a). ( H ) , ( C ) , (d) d e m o n s t r a t e t h e highly nonlinear elastic behaviour of t h e system presented in t e r m s of t h e curves of A and elA/d.l versus non-dimonsionalized q (Figs. 7.3(a), (c)) and non-dimensionalized (//, (Figs. 7.3(b). ( d ) ) , respectively. T h e elastic solutions of t h e ideal s y s t e m in t h e full range of A and d A / d . l are a d d i t i o n a l l y shown in T H E small windenv in Figs. 7.3(a) and 7.3(c). Figure 7.4 has a similar make up with T H E axial FORCE A and its sensitivity d.V/d. 1 represented on t h e ve^rtical axis; Fig. 7.4(b) d e m e ) N S T rates T H E sensitivity versus (non-elime nsionalize el) (/,, E'urves, while Fig. 7.4(d) the sensitivity versus
4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

(a) T h e design sensitivity of a q u a n t i t y ( d i s p l a c e m e n t c o m p o n e n t , stress r o m p o n e n t , etc..) D O P E M I D S basically on t h e process E*ontrol for instance , T H E load factor will be elesign-insensil,ive for load-controlleel p r o b l e m s and generally design-sensitive tor elisplace meiit-EE)nt rolle el pre)BLE ms. The basie E)Bservat ion resulting fre>m the analysis performed! is T H A T A S leuig A S A spevifie d c o m p o n e n t in T H E (.V+l )-dimensional loael facte)r D E N I A L elisplace mt nt space is not fixed at the step considered, its sensitivity will generally be neui-ZEM'E) even if it, enters the c o n s t r a i n t of the t y p e (7.43) or (7.95). Moieener, its n i m u l a t i v e ( t i m e - i n t e g r a t e d ) sensitivity is indepenele nt of tlie specific form E>f the c o n s t r a i n t used in t h e analysis provided the same d e f o r m a t i o n p a t t e r n is g e n e r a t e d t h r o u g h o u t . 4

4

4

4

4

4

4

4

-

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

More precisely, the above conclusion aj)plies strictly te> nonline ar tda.st.ic problems- for path-elepenelent pre)ble ms the accuracy m a y be ne gativelv influenced by the pat.h-elepenele ncy within E ach i n c i e m e n t a ! s t e p so t h a t slightly different deTormatiem history m a y be ge nerateel AT any stt p B Y elifferent j)roc(ss control c o n s t r a i n t s . It is believed t h a t this slu)uld not leally be a p r o b l e m in practical larg( -scale compul,atie)ns, thtnigh. 4

4

4

4

4

4

4

4

4

4

4

4

4

(b) Kcgartlless of the significance E)f having the se nsit ivity ELE termine el AT E A E H ])se uele)-fime i n s t a n t in the pren e s s , E)iu E)f the maje)r O B J E C T ives E)f T H E apP R E M E ' h was to pre)j)t)se a useful TEE"hniqu< TE> e v a l u a t e B U E kling LE>ael SE iisitivity. To see that we have achieved this goal we first O B S E T V E that 1 HEfirst B U E ' k l i n g point along t h e equilibrium p a t h for t h e ide al edastie' strucT U R E can be eennputeel by using the ' e x a c t ' analytieal censiele rat IE)ns (i.e\ by solving T H E a p p i o p r i a t e E igenprobl(>m AT E aeh s t e p TE> fine! the smallest eigenvalue suffieuMit ly clejse to ze re>) T O yie ld in our ease Q?" = 8.4873 ANEL q -h = 7.822 x 1 0 , cf. [113]. T h e s e are t h e c o o r d i n a t e s of the point H(h) as shown on Fig. 7.1. Analytical design differential ie>n of Qf" with respeeM. to A gives dQ^/dA = 1.4135 x 10' . 4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

A1

_ : J

4

v

Al

1

240

• Buckling a n d P o s t - B u c k l i n g (a)

(b)

Load parameter A 10 t r-— i i i i T

:

\

|i

\ i

5

Load parameter A, f • 10

1SII]

-501

NX

::•.:/1

P 1 2; r":""tVi"/i

1!

i

j

n |i

ii....

\

i i -i

0.0

1

r-

1

0.5

• • i t•

1.0

\\

-5

\-l

ii

-10

J

ii

Vertical displacement

T

-10

•. 1.5 I . . • 2.0 .ij

'

'

0.5

Horizontal displacement

(

(c)

1

0.0

(q /h ) v

'

1.0 (q /h ) h

0

(d)

Sensitivity dA/dA (rrr 1.5E+4 -n

Sensitivity dX/dA(m )

2

2

j i 1.0E+4

7.5E+3 -!|

- I

5.0E+3 -

1

!\

I j

0.0E+0

\^

0.0E+0 -7.5E+3 -5.0E+3 • i : I

-1.5E+4

-1.0E+4 0.5

1.0

1.5

2.0

Vertical displacement

{q /h ) v

0

0.0

0.5

Horizontal displacement

1.0

(q /h )

imperfect system ideal system, fundamental path ideal system, post-bifurcation path F i g u r e 7.3 Elastic equilibrium and sensitivity of the load parameter

h

0

Numerical Illustrations

(b)

(a) Axial force N (kN)

Axial force N (kN)

0

-

H— -200 -400

\r \

-40

-4-'—i—.—i—

i

0.5

» •t

0

1 1 1 1 1 1

J

-15

1

1 1 1 1 » 1 1 1 1 1 1 1 1

-30 i v

r ——,—r- i- .

1.0

1 I 1 1

/

-6QQ

-20

0.0

• 241

-45 1.5

2.0

Vertical d i s p l a c e m e n t

{g /h ) v

•'

i

-

0.0

0.5

••"

'

1

1.0

(q /h )

Horizontal displacement

0

(c)

X.X. X X

h

0

(d)

Sensitivity d/V/dA (kN/m )

Sensitivity d/V/dA (kN/m )

2

2

3E+4 - i

7

j

4E+4

0E+0

-3E+4 -f

0E+0

-6E+4 -

-4E+4

I 1

Jf 'J I

-8E+4

-9E+4 0.0

0.5

1.0

1.5

Vertical displacement

2.0

(q /h ) v

Q

.

0.0

.

.

1

0.5

Horizontal displacement

1.0

(q /h )

imperfect s y s t e m ideal s y s t e m , fundamental path ideal system, post-bifurcation path F i g u r e 7 . 4 Elastic equilibrium and sensitivity of t h e m e m b e r axial force

h

0

242 • Buckling a n d Post-Buckling Clearly, t h e s a m e values can be o b t a i n e d by using t h e finite difference c o m p u t a tions with t h e a n a l y t i c a l expression for Q^ r a t h e r t h a n t h e differentiation above. T h e s e values can now to be c o m p a r e d with t h e a p p r o x i m a t e ones taken from t h e solution curves given in Fig. 7.3. R e t u r n i n g to t h e results presented therein, we j u s t consider an a p p r o x i m a t i o n to t h e buckling point (B'(h) in t h e notation of Fig. 7.1) as d e t e r m i n e d a p p r o x i m a t e l y d u r i n g t h e step-by-stop analysis the value of t h e load at this point reads Q = 8.4809 with a relative error of niil

l min

1

h

•buckling load error = 1

*

J

h

= 0.075%

b

T h e c o r r e s p o n d i n g value of the buckling load sensitivity is taken (rom t h e sensitivity solution as 1

1

l/)nuni

dA with a relative error of

' b u c k l i n g load sensitivity e r r o r = 1

d

A

d

A

}

= 0.65%

dA It is seen that t h e m e t h o d provides excellent results and t h u s it. p r e s u m a b l y has great potential in dealing with t h e buckling sensitivity p r o b l e m s considered. (Nevertheless, further testing is necessary!) T h e r e m a i n i n g numerical results r e p o r t e d in Fig. 7.5 d u p l i c a t e t h e ones just discussed above for the load p a r a m e t e r response , t h e only difference being t h e elastic- plastic k i n e m a t i c h a r d e n i n g m a t e r i a l p r o p e r t i e s a s s u m e d in t h e mode !. An i m p o r t a n t feature whieh should be pointed out is that at elasto-plastic transition points sensitivity curves exhibit large discontinuity. Figure 7.5 shows t h e elasto-plastic solutions ol t h e load p a r a m e t e r response a n d load p a r a m e t e r sensitivity with respect to t h e m e m b e r cross-sectional area. We observe t h a t j u m p s occur at t h e yield points A and C , and a b r e a k p o i n t at t h e u n l o a d i n g point B: j u m p i n g is considerably larger when t h e s t r u c t u r a l response is c u r r e n t l y unstable, i.e. at point C. For t h e effectively one-dimensional elasto-plastic p r o b l e m at hand t h e results m a y be verified analytically. 1

1

1

Y

7.6

S u m m a r y of the Chapter

In this c h a p t e r a c o m p u t a t i o n a l a l g o r i t h m for t h e sensitivity analysis of nonlinear mechanical s y s t e m s has been presented. T h e a l g o r i t h m can be used for sensitivity evaluation at critical points a n d along t h e post-critical p a t h ; it exploits in a n a t u r a l way all t h e a d v a n t a g e s offered by t h e ' a r c - l e n g t h t e c h n i q u e s . N u m e r i c a l illustrations confirm t h e applicability of t h e a l g o r i t h m no o t h e r technique a p p e a r s c u r r e n t l y c o m p e t i t i v e for solving the class of p r o b l e m s addressed in t h e c h a p t e r . 1

1

5

Summary of the Chapter •

imperfect system ideal system, fundamental path ideal system, post-bifurcation path

Figure 7.5 Elasto-plastic equilibrium and sensitivity of the load parameter

243

244 • Buckling and Post-Buckling T h e arc-length in t h e load d i s p l a c e m e n t space has been chosen as t h e controlling variable. Even if for obvious reasons t h e choice of control m a y influence t h e sensitivity of p a r t i c u l a r variables, it has been d e m o n s t r a t e d numerically that reliable results are o b t a i n e d as long as t h e q u a n t i t y in question is not e n t i r e l y ' fixed. In o t h e r words, even if a variable e n t e r s t h e control c o n s t r a i n t its sensitivity m a y b e c o m p u t e d by t h e present m e t h o d . 4

P a r a m e t e r sensitivity of "ideal' (buckling-prone) s y s t e m s a p p e a r s much more p r o n o u n c e d t h a n t h a t of t h e ' i m p e r f e c t ' s y s t e m s . T h e sensitivity plots exhibit a highly complex b e h a v i o u r even for the relatively s i m p l e s t r u c t u r a l configurations discussed in t h e c h a p t e r t h e complexity will clearly have a p r o n o u n c e d influence on any sensitivity-based o p t i m i z a t i o n , reliability or identification m e t h o d o l o g y employed for nonlinear s y s t e m s .

4

4

Chapter 8 Nonlinear Dynamics 8.1

P r o b l e m Statement

In c o n t r a s t with t h e extensive l i t e r a t u r e dealing with nonlinear s t a t i c s sensitivity f o r m u l a t i o n s on t h e one h a n d and d y n a m i c control on t h e o t h e r , t h e d y n a m i c sensitivity p r o b l e m s for nonlinear deformable solids have a t t r a c t e d relatively little a t t e n t i o n , [71, 8 1 , 83, 172]. T h e subject is t a k e n u p in this c h a p t e r in t h e h o p e to at least set a uniform b a c k g r o u n d for those i n t e r e s t e d in n o n l i n e a r d y n a m i c s a p p l i c a t i o n s of t h e sensitivity concepts. T h e a t t e n t i o n in t h i s c h a p t e r will b e focused on semi-discretized formulations, with q (r,h), a = 1 , 2 , . . . , A , being t h e vector of generalized c o o r d i n a t e s . r

a

As usual, t h e first step in developing sensitivity t e c h n i q u e s is to define t h e class of c o n s t r a i n t functionals to be considered. Most of t h e c o n s t r a i n t s e n c o u n t e r e d in p r a c t i c e m a y be classified as follows: (a) T h e p e r f o r m a n c e b o u n d c o n s t r a i n t 1 < 0

for every r 6 [0, T]

(8.1

w i t h G\ being a pre-set admissible value, T a fixed t e r m i n a l t i m e a n d t h e a r g u m e n t q i n d i c a t i n g t h a t any generalized c o o r d i n a t e s m a y be involved. a

(b) T h e critical point (critical t i m e , interior p o i n t ) c o n s t r a i n t G {q {i, 2

a

h); h) = 0

for every 0 < i < T

(8.2)

(c) T h e t e r m i n a l t i m e c o n s t r a i n t G (q (T(h),hy,T(h);h) 3

a

with T = T(h) equation.

= 0

(8.3)

either given or a s s u m e d to b e found from an additional

246 • Nonlinear Dynamics (d) T h e interval p e r f o r m a n c e c o n s t r a i n t ,T(M

J with T(h)

G,(q (r,h);h)dr

= 0

Q

(8.4)

given e i t h e r way i n d i c a t e d above.

A n y of t h e s e c o n s t r a i n t s can b e considered as a special case of t h e following unified, or canonical, form: Q(h)

= G[q (T,h)',h]

-

a

g(q (t,h),q (t,h);h) Q

+ [ g(q (r,

a

h); h) dr

Q

(8.5)

Jo

in which G s t a n d s for a given functional of q (r,h) while g a n d g are explicitly given functions of their a r g u m e n t s ; t h e constraint characteristic time t typically t a k e s on t h e value t — T unless t h e critical point c o n s t r a i n t ( 8 . 2 ) is involved. T h e t e r m i n a l t i m e T = T(h) m a y be defined in an implicit way by t h e c o n d i t i o n a

T(q(T(h);h),q(T(hy,h);h)

= 0

(8.6)

We n o t e t h a t t h e c o n s t r a i n t ( 8 . 2 ) m a y be w r i t t e n as a specific case of Eq. ( 8 . 5 ) e i t h e r as Q(h)

= g(q (i,h);h)

i

G{h)

= g(q (t,h);h)

U I'G (q (r,

a

G (q (t,h)-h) 2

(8.7)

a

or as a

A

a

h)-h) 8(1 - r ) d r ,

i 6 [0, T]

Jo (8.8)

w h e r e 6(T) is t h e Dirac d e l t a d i s t r i b u t i o n . T h e form ( 8 . 8 ) will t u r n o u t m o r e convenient in s o m e of t h e derivations below. It is n o t e d t h a t t h e functions g and g m a y be re-defined as integrals over s p a t i a l d o m a i n s , in which case Eq. ( 8 . 5 ) would include t h e p e r f o r m a n c e b o u n d c o n s t r a i n t ( 8 . 1 ) as a special case. To d e r i v e sensitivity expressions t h e p e r f o r m a n c e functional ( 8 . 5 ) a n d t h e t e r m i n a l t i m e condition ( 8 . 6 ) have to be c o m p l e m e n t e d with t h e c o n s t r a i n t composed of t h e vector q in t h e form of t h e semi-discretized e q u a t i o n s of m o t i o n a

R

= Qa{T,h)-

a

F {q (T,h)ih)-[M (h)q(T,h) a

0

a,/?=l,2,...,JV, which, cf.

Eq.

at

the

t i m e interval

+ C (h)q(T,h)]

aP

= 0

a0

9a(0,/0

[£,£-|-A2] for

=
q (v,h)

= q°

a

path-dependent

a

(8.9)

situations,

reads

0

(8.10)

(2.206)

-\M (h) q(T,h) t+At

Q0

+ C (h) q(T,h)\ t+At

a0

=

Problem Statement • 247 As before, R is t h e residual vector r e d u c e d t o ( a l m o s t ) zero in t h e process of i t e r a t i o n , Q a n d F a r e t h e vectors of e x t e r n a l nodal loads a n d (possibly p a t h - d e p e n d e n t ) internal nodal forces, respectively, while M p a n d C p are t h e mass a n d d a m p i n g m a t r i c e s , respectively. a

a

a

a

a

T h e form of Eq. (8.10) is based on t h e a s s u m p t i o n t h a t t h e i n c r e m e n t a l stresses needed t o c o m p u t e AF have been d e t e r m i n e d from t h e r a t e - t y p e cons t i t u t i v e law using an a p p r o x i m a t e t i m e i t e r a t i o n a l g o r i t h m ; cf. C h a p t e r 5. Because AF is generally a nonlinear function of A q , E q . (8.10) is a nonlinear differential e q u a t i o n for t h e u n k n o w n vector function q ( r ) . T h e basic steps in t h e solution p r o c e d u r e s as described in C h a p t e r 2 a r e : a

a

(i) an i t e r a t i v e t e c h n i q u e is employed which, in case of t h e N e w t o n - R a p h s o n m e t h o d , a m o u n t s t o solving t h e following linear differential e q u a t i o n for t h e correction "fAq : a

M

t a

+

0

% ^

+ C p

t + A t

a

q^ +K

a

f

A q ^

n

=

t + A t

Q

a

-

t + A t

F^

(8.11)

<+ AtD(w)

where, cf. Eq. (2.179) W:

t+A

=

+1)

t

+

A

W + >yAq£

(8-12)

+1)

(ii) each of t h e above e q u a t i o n s for u> = 0 , 1 , . . . , u is i n t e g r a t e d in t i m e by using a finite difference s c h e m e reducing t h e differential e q u a t i o n s t o a s y s t e m of linear algebraic e q u a t i o n s . c o n v

We shall b e considering below two sensitivity p r o b l e m s : t h e first o n e is called t h e time-interval sensitivity problem a n d is defined by Eqs. (8.6) a n d (8.10) a n d the constraint Q(h)

=

G[q (T,h)',h] a

=

g(q (T(h),h)MT{h),h)-,h) a

T(h)

f

+

/

Jo

g(q (T,h);h)dT

w h e r e a s t h e second o n e is called t h e critical-time instead of E q . (8.13) t h e c o n s t r a i n t G(h)=

f

T

Jo

g(q (T,h);h)6(i-T)dT a

(8.13)

Q

,

sensitivity

problem

t £ [0, T]

a n d involves

(8.14)

We a s s u m e t h a t Q is continuously differentiable with respect t o h . T h e objective is t o e v a l u a t e t h e first-order design sensitivity of Q, i.e. t h e design g r a d i e n t coefficients dQ/dh. T w o m e t h o d s which can b e used t o achieve t h a t goal are discussed in t h e s u b s e q u e n t sections.

248 • Nonlinear Dynamics

8.2

T h e Direct Differentiation M e t h o d

T h e chain rule of differentiation allows us to c o m p u t e t h e design variation of t h e functional (8.13) as IQ = d

+

T 9

q

+ q ST) T -~{~S dq

T

dq

T

a

K

q

d

a

a

;

+ q

T

ST)

T

a

v

a

a

+ g~6T+

'

f(d g+^-S q )dr dq /

1

T

T

a

Jo \

a

= d 9 + | ^ oW

| ^ o

T

+ C7oT + fid

g +

r

o ^

T

dr

(8.15)

with t h e t i m e derivative of G t a k e n at r = T defined as G

T

= ! ^

a

^ ' ?

+

+ 'S

a

(8.16)

To assess t h e first-order change in t h e t e r m i n a l - t i m e function T(h) due to variations in /i, let us first express <5T in t e r m s of t h e variations in t h e nodal disp l a c e m e n t s a n d their t i m e derivatives. T h e design variation of t h e t e r m i n a l - t i m e condition (8.6) yields ST

= aT + _ 6

r 9

dq

a

+ — 6

T 9

dq

a

a

+

—

+ —

r 9

\dq

a

a

r 9

dq

a

o

) 5 T

= 0

J

a

(8.17)

Using t h e necessary condition for t h e t e r m i n a l t i m e T to be d e t e r m i n e d uniquely f

= ^ q dq

+ ^ - % ^ 0 dq

T

a

a

(8.18)

a

t h e expression for t h e design variation in T can b e w r i t t e n as ST

= dT + ST

= - + d T - ^ ( ? - 8 T T\dq

T

q

+

a

a

—S q ) J

(8.19)

T

a

dq

a

which, s u b s t i t u t e d into Eq. (8.15), leads to SQ = dG + SG

(8.20)

where TQ

dQ = d g-

T

-^dJ

T

(d g T

+J

a

(8.21)

T

GdT\-

(d g

T dq J

\ dq

T

V dq

d gdr GdT\~

T

a

[T g-

T

T

T

J

a

dq J a

d

Jo

dq

n

Since t h e variations Sh are a r b i t r a r y , Eq. (8.20) can be r e w r i t t e n in t e r m s of t h e sensitivity g r a d i e n t as dG

dg r

d gd q T

T

d g d q T

a

T

a

The Direct Differentiation Method For t h e case of t h e t i m e - i n s t a n t

• 249

sensitivity, E q . (8.23) t a k e s t h e form,

cf. E q . (8.14) tfd g

d£

8g

T

= / ( dh

dh

d^q

T

|

a

dq

dh

a

dg

r)dr =

8(t-

dh

dq

dq

dh J =

a

JO \

T= t

dg

|

(8.24;

c

T

T h e u n k n o w n d i s p l a c e m e n t a n d velocity sensitivities d q /dh a n d d q /dh involved in E q s . (8.23) a n d (8.24) are t o b e d e t e r m i n e d from E q . (8.10). Taking t h e design variation of E q . (8.10) a n d observing t h a t it should vanish at b o t h t h e n o m i n a l a n d p e r t u r b e d designs, yields T

T

a

8

R

t + A t

a

(8.25;

= 0

Q

which, in view of t h e relations 6

R

t + A t

= - { M ^ l ^ ^ A r C ^ S ^ ^

a

+

K^SAq^

(8.26)

with dAF

(T)

a

K

dAqp d

t + A t

R

d

a

Q

t + A t

dF

a

a

+AF ) a

dcr d

t + A t

F

t + A t aP

dW

l

0(
- dM

a

q

- dC p

p

^dAF

S j

dfr

a

dh

V dodtr

a

t + A t

a

[

dh

q

p

-

dAF

a

dAF

d
dp

dh

a

]

s

h

'8.21

Sh

dh

Aq ^Aq {h) a

a

becomes Af,

d

t + A t

a/3

q

-( T) dAqp

0

dh d

' ~^

t + A t

Q

a

dh dM +At a0t

dh

dh

' "

qa

a

dh

P

dC

d F t+Ai

a

P

t+At

dh

qa

n

dh

(8.28) Aq ^Aq {h) a

a

T h e s y s t e m of A^ linear differential e q u a t i o n s (8.28) has to b e solved t o g e t h e r with t h e h o m o g e n e o u s initial c o n d i t i o n s , cf. (8.9) 1

d

°q

dh

a

dh

0 ,

d°q

a

dq°

dh

dh

a

= 0

(8.29;

Details of t h e solution a l g o r i t h m a r e p o s t p o n e d until Section 8.4. Now we only m e n t i o n t h a t t h e differential o p e r a t o r s on t h e left-hand side of E q s . (8.11) and (8.28) a r e precisely t h e s a m e — a fact of f u n d a m e n t a l a l g o r i t h m i c significance. For by e m p l o y i n g a finite difference s c h e m e to solve E q . (8.11) we m a y 'The initial conditions in the primal system (8.9) in which the right-hand sides are assumed independent of h are not restrictive since any non-homogeneous system of equations may he converted into a homogeneous one by using an appropriate variable transformation, cf. [102].

250 • Nonlinear Dynamics t a k e t h e effective consistent stiffness m a t r i x K \ cf. E q . (2.215) used in t h e last ' c o n v e r g e d ' i t e r a t i o n u ; in t h e d y n a m i c e q u i l i b r i u m p r o b l e m , a n d employ it t o find dAq /dh with t h e only additional c o m p u t a t i o n s needed t o form t h e r i g h t - h a n d - s i d e vector in Eq. (8.28). T h e way t o c o m p u t e it, i.e. effectively t o upd a t e t h e vector dF /dh does not differ from w h a t we have already learned in discussing s t a t i c sensitivity p r o b l e m s . From t h e viewpoint of t h e overall s t e p - b y - s t e p p r o c e d u r e t h e solution t h u s proceeds ' i n p a r a l l e l ' — a t each t i m e s t e p t h e i t e r a t i v e d y n a m i c e q u i l i b r i u m solution is followed by j u s t one sensitivity solution. e

a 0

c o n v

a

a

}

T h e above e q u a t i o n s get significantly simplified in t h e case of nonlinear elastic s y s t e m s . In such a case Eq. (8.10) reads t + A t

R

=

a

Qa{h)

-

t+At

-

F { q {h);

t+At

h)

t+At

a

a

[M (h) q(r,h)

+ C (h) q(T,h)]

t+At

=

t+At

aP

a0

0

(8.30)

in which t h e i n t e r n a l nodal force vector is c o m p u t e d from Eq. (2.198). T h i s vector is a nonlinear function of t h e c u r r e n t nodal d i s p l a c e m e n t vector q ', t o solve Eq. (8.30) for q t h e i t e r a t i v e s c h e m e given in C h a p t e r 2 m a y b e employed in which t h e correction 7 + g(^+') \ c o m p u t e d from t+Ai

a

t + A t

a

(

At

s

M

+C

%"

t+A a0

+1)

t + A t a0

q^

l )

+ K% 7

t

+

%

u

+

i

)

=

t + A t

tt:

}

(8.31)

where

and q = q : + iq : (8-33) Since n o t i m e i n t e g r a t i o n of t h e c o n s t i t u t i v e e q u a t i o n is now involved, no consist e n t t a n g e n t m a t r i x concept needs t o b e invoked a t all. (We m a y a l t e r n a t i v e l y say t h a t t h e t a n g e n t stiffness m a t r i x K J is always consistent.) T h e governing direct differentiation m e t h o d e q u a t i o n (8.28) simplifies t o t h e form (

M

(

]

{ +l)

a

(

0

d qp ^ - j ^+C dh t+At

M

a0

M

r

+

C

a

a0 p

d^ ,j ^, dh

d<+ q ^ dh At

) -( +T K$

0

_ d *+"R = ~ dh

(8.34;

c

in which

dk

-

~

h

dh~

q

f

,

- ^ h ~

q

a

~

(

8

J

5

'

The Adjoint System Method • 251

8.3

T h e Adjoint S y s t e m M e t h o d

In a c c o r d a n c e with our early conclusions regarding t h e use of t h e ASM for p a t h - d e p e n d e n t p r o b l e m s , only nonlinear elastic s y s t e m s are dealt with in this section. T h e variational A S M t h e o r e m presented in C h a p t e r s 3 a n d 4 will be employed; because some additional c o m m e n t s regarding its use a p p e a r in place, a derivation m o r e detailed t h a n j u s t a plain application of t h e t h e o r e m will be given. Let us s t a r t by defining t h e a u g m e n t e d functional G*[q {r, h), A ( r ; h)\h] a

a

T(h)

F

= G[q {r,

h); h] + / Jo

a

A ( r , h)R {q {r, Q

a

h); h)dr

p

(8.36)

in which t h e adjoint vector \ is a s s u m e d to be k i n e m a t i c a l l y admissible and chosen to m a k e G* s t a t i o n a r y with respect to q , i.e. a

a

dC*

SG* =

o 8q

= 0

a

dq

(8.37)

a

We note t h a t q and q are t r e a t e d here as ' d e p e n d e n t on q so t h a t their variations Sq and Sq are expressible in t e r m s of 8q . By observing t h a t 1

a

n

a

n

Q

a

dG"

6G- =

a G

dG*

- + SG- = —+

—

dq

a

^L

( 8

and c o m b i n i n g it with Eq. (8.37) in which 8q arrive at

.

3 8 )

are m u t u a l l y i n d e p e n d e n t , we

a

SG* = dG"

(8.39)

However, we also have SG* = 6Q + £ ( 6 \

R

a

Q

+ X 8R )dr a

a

= 8G

(8.40)

because we m a y use t h e conditions R = 0 and 8R — 0, t h e l a t t e r on account of t h e d y n a m i c e q u i l i b r i u m condition holding t r u e at b o t h t h e n o m i n a l a n d pert u r b e d designs. T h u s , Eqs. (8.39) and (8.40) give a

a

SG = dG*

(8.41)

or, in t e r m s of t h e design derivatives r a t h e r t h a n variations dG

dG*

d/T = ~K

^

E q u a t i o n (8.42) is t h e f u n d a m e n t a l ASM relationship for direct evaluation of t h e sensitivity g r a d i e n t — t h e condition being t h a t t h e r i g h t - h a n d side of it is easily c o m p u t a b l e . It has t h e form SG =

d g-^-dT T

T

T

+ J {d g+ \ [d Q -d F T

Q

T

T

a

T

a

a

(dM

q + dC

T aP

p

qp))} d r

T aP

(8.43)

252 • Nonlinear Dynamics which in t e r m s of t h e design derivatives r a t h e r t h a n variations reads dg

dG

_G_dT_

T

T

~T~dh

~dh ~

+f

d Q«

dT«

dh

dh

T

+ A T

dh

Q

fdM .. (dM V dh

„

a0T a0

dC dC . dh

t

T

a0T T

a0

\]\

{-jh-^ -mr^)\\ +'

1M

-

dT

(8 44)

Jo To establish t h e adjoint e q u a t i o n for A t h e s t a t i o n a r i t y condition (8.37) is invoked to read, cf. Eq. (8.38) a

rT

SG* = 8G + [ \ 6 Jo fd g G dT

R

T

T

—

a

T

(

CT

1 0°

:

dq

V

T dq

a

)

a

+ / ^ 6 Jo L dq

q * -

T

dr

T

a

, l

9

T

T

v

a

0 6 1

Q

d

T

TT^~

T

a

h ( M

T

d

(/a <7a t+ I — \ dq

dq

\ C T) 0
a

+ C 6%

a

K™6 q ) d r '

+

aP

T

(8.45)

T

a

a

By i n t e g r a t i n g t h e inertial a n d d a m p i n g t e r m s by p a r t s we arrive at T

(— V

GdT

T

9 dq

SG*

+ M

%

a0

- C

%

a0

8 q T

a

+

f

T

'd

GdT

T

9 \ dq djj a

r

dq

a

8 q dr T

a

. dq

= 0

(8.46)

a

T h e above condition for t h e vector function A will be satisfied if A is a solution to t h e (linear!) t e r m i n a l - v a l u e p r o b l e m consisting of t h e s y s t e m of second-order o r d i n a r y differential e q u a t i o n s a

M

t

+

a0

A

% - C

t

+

a0

A

% + "K§ t+

t

+

A

%

Q

=

(8.41

dq

a

with t h e t e r m i n a l conditions at r = T in t h e form

X a

Aa

~ -

M

a

d

\ d q

M [-

f

0

--h

a0

dq )

(8.48)

0

YG^q

-3T-

> \dq

T

J

0

dq

B

T h e r i g h t - h a n d side of Eq. (8.48) (as well as of Eq. (8.47)) can easily be c o m p u t e d once t h e original initial-value p r o b l e m has been solved. For t h e critical-point sensitivity p r o b l e m defined by t h e functional (8.14) Eq. (8.44) takes t h e form dG

d/T

dj_ dh

U

d F (dM .. [-&r - ~dT - \-dir \d Q T

fi Aw

dC

T

a

a

a0

T

qfl

a0

+

\1

~dh~ )\ qp

dT

-

(8 49)

The Adjoint System Method •

253

By observing t h a t g(r)S(t-r)

= g(t)Ht-r)

(8.50)

and following t h e s a m e line of reasoning as for t h e time-interval sensitivity we o b t a i n t h e t e r m i n a l - v a l u e p r o b l e m as M

^0 — C

T

a0

+K

\(3

T

a0

r E [0, t] ,

\p

T a0

dg

—

, J

{

t

-

T

(8.5,)

}

i E [0, oo)

a n d t h e t e r m i n a l conditions at t h e critical t i m e r = t as X (i)

= 0 ,

a

\ (t) a

= 0

(8.52)

We shall c o m p l e t e this section by m a k i n g t h e following c o m m e n t s : (i) T h e differential o p e r a t o r on t h e left-hand side of Eqs. (8.47) a n d (8.51) is not typical of t h e d y n a m i c s e q u a t i o n s b e c a u s e t h e m i n u s sign occurs at t h e d a m p i n g t e r m . As a m a t t e r of fact, however, b o t h t h e p r i m a l e q u a t i o n (8.9) a n d t h e adjoint e q u a t i o n (8.47) m a y be i n t e g r a t e d in t i m e by using t h e s a m e p r o c e d u r e . For Eq. (8.47) m a y be replaced by an e q u a t i o n with t h e s a m e differential o p e r a t o r as in Eq. (8.9) only if we e m p l o y t h e t i m e t r a n s f o r m a t i o n T' = T — r , cf. C h a p t e r 3. In fact, such a t r a n s f o r m a t i o n t u r n s t h e t e r m i n a l - v a l u e p r o b l e m , Eqs. (8.47) a n d (8.48), into t h e c o r r e s p o n d i n g initial-value p r o b l e m . (ii) By i n t e g r a t i n g by p a r t s t h e last two t e r m s in p a r e n t h e s e s on t h e r i g h t - h a n d side of Eq. (8.43) we o b t a i n SG = 8 g + ( % 8M T

+j

[d g T

+ \ (d Q r

a

q

-

T

a0

p

T

\

8M

a

q

T a0

0

dC y ]

- dE )-(\ dM

T

a

- X 8G ) T

a0

a

a

a

nfi

a0

q0

dr

(8.53)

a n d , consequently, a n o t h e r form of Eq. (8.44) as dC7

d g

(-

T

dg T

H

dM

T

a0

fd Q T

i T I

dC

Tx

a0

dF\ T

a

l ~ d h + H ^ h

a

\ _ T

/..

d h ~ ) ~ \

dM

A

a0

dM T

a0

X

a

~ m r

T

, :

8V

T

+

.

a

d

h

y

^ j d r

(8.54)

K

It is observed t h a t t h e analytically fully equivalent Eqs. (8.44) and (8.54) m a y b e considerably different in t e r m s of c o m p u t a t i o n a l efficiency. For t h e relationship (8.54) does not require t h e p r i m a l velocity a n d acceleration vectors to be stored at each forward integration t i m e s t e p , whereas t h e adjoint velocities and accelerations are o b t a i n e d at each b a c k w a r d integration time step. (iii) T h e linear s y s t e m (8.51) with an i m p u l s e - t y p e load m a y be a l t e r n a t i v e l y solved for A by i n t e g r a t i n g an equivalent, h o m o g e n e o u s s y s t e m excited by an initial velocity, cf. [102], for i n s t a n c e . Q

254 • Nonlinear Dynamics

8.4

Computational Issues

To discuss t h e effectiveness of t h e direct differentiation m e t h o d ( D D M ) and t h e adjoint s y s t e m m e t h o d ( A S M ) in nonlinear e l a s t o - d y n a m i c s , we first follow t h e flow-charts of specific a l g o r i t h m s to find t h e sensitivity g r a d i e n t coefficients. In t h e following c h a r t s A , D, C and L d e n o t e t h e t o t a l n u m b e r of s y s t e m degrees of freedom, design variables, response functionals a n d critical p o i n t s , respectively, and t h e t e r m i n a l t i m e is a s s u m e d known. r

Time-interval sensitivity by the DDM At each time instant r = At, 2At,. .., T (a) integrate (iteratively) the primal ODEs of motion to get Aq => q , ft = 1.2 V d' F i (b) calculate , a = 1, 2, . .., N, g = 1, 2 D Cihe l A q < , # A q ( h « ) ' to form D right-hand sides of the displacement sensitivity ODEs dAq d q (c) integrate D additional linear systems of the ODEs to get => dhc dh Compute and sum up C sensitivity gradients over the time interval [0,7] t + At

a

n

+ A t

a

t+At

a

n

e

Critical-point sensitivity by the DDM At each time instant r = At, 2At, . . . ,t

L

(a) integrate (iteratively) the primal ODEs of motion to get Aq , o = l,2,...,A d F l (b) calculate —, ft = 1, 2, . . ., A\ g = 1, 2

q,

t+At

(

(

.

r

( + A t

dh

e

D

lAq ^Aq„(he)' a

to form D right-hand sides of the displacement sensitivity ODEs (c) integrate D additional linear systems of the ODEs to get :—d/jf dhe (d) at each r = f , I — 1,2, . .., L, compute and sum up C sensitivity gradients a

Time-interval sensitivity bv the ASM Over the whole time interval [0, T] (a) integrate (iteratively) the primal ODEs of motion from 0 to T to get Aq q , ft = l , 2 , . . . , i V , T = At, 2 At, . . ., T (forward integration); at each time step save information required at step (b) a

t + A t

a

(b) calculate the terminal conditions \ , <7 , « = 1, 2, . .., N; integrate C adjoint linear systems of the ODEs from T to 0 to get A , r = T — At, T— 2At, . . ., 0 (backward integration) T

T

a

a

a

T

a

(c) compute and sum up C sensitivity gradients over the time interval [0,T]

Computational Issues • 255 Critical-point sensitivity by the ASM Over the whole time interval [0,t ] L

(a) integrate (iteratively) the primal ODEs of motion from 0 to i to get Aq q « = 1,2, r = At, 2At,..., t (forward integration); at each time step save information required at the next stage L

a

t + A t

L

a )

For each critical time t , I = 1, 2, . .., L 1

(b) integrate C adjoint linear systems of the ODEs from f to 0 to get T — V — At,t - 2At,. . .,0 (back ward integration)

X,

T

Q

l

(c) sum up C sensitivity gradients at the time instant V

T h e differences between t h e D D M a n d t h e ASM are as follows: (i) t h e n u m ber of a d d i t i o n a l e q u a t i o n s y s t e m s in t h e D D M d e p e n d s on t h e t o t a l n u m b e r of design variables, while t h e n u m b e r of adjoint s y s t e m s is equal t o t h e n u m b e r of response functionals, and (ii) by using t h e D D M t h e i n t e g r a t i o n of b o t h t h e p r i m a l a n d a d d i t i o n a l (sensitivity) e q u a t i o n s can b e performed at each t i m e s t e p , whereas t h e A S M requires t h a t forward i n t e g r a t i o n of t h e p r i m a l s y s t e m from r = 0 to r = T a n d b a c k w a r d i n t e g r a t i o n of t h e adjoint s y s t e m s from T to 0 m u s t generally b e performed s e p a r a t e l y b e c a u s e of t h e changing stiffness m a t r i x . T h e l a t t e r difference is non-existent only for t h e case of linear elastic s y s t e m s in which t h e s y s t e m m a t r i c e s are c o n s t a n t ; t h e p r i m a l a n d adjoint s y s t e m s m a y t h e n b e s i m u l t a n e o u s l y i n t e g r a t e d with t h e h o m o g e n e o u s i n i t i a l - t e r m i n a l conditions by using t h e b a c k w a r d - t i m e variable a n d t h e i n d e p e n d e n t variable t r a n s f o r m a t i o n . Therefore, for linear s y s t e m s t h e A S M is m o r e c o m m o n l y used since it can h a n d l e a larger n u m b e r of design variables t h a n t h e D D M . However, w h e n using t h e A S M for sensitivity assessment in nonlinear d y n a m i c s a lot of i n f o r m a t i o n (displacem e n t , velocity a n d acceleration vectors, d e c o m p o s e d stiffness m a t r i x a n d p a r t i a l derivatives of t h e i n t e r n a l force vector w i t h respect t o t h e design variables) m u s t b e saved after each t i m e step d u r i n g forward i n t e g r a t i o n of t h e p r i m a l e q u a t i o n s to b e retrieved at t h e corresponding step of t h e b a c k w a r d i n t e g r a t i o n of t h e adjoint e q u a t i o n s . Moreover, it is difficult to e s t i m a t e how t h e error in t h e forward i n t e g r a t i o n will influence t h e solution d u r i n g t h e b a c k w a r d i n t e g r a t i o n to follow because t h e l a t t e r can only be s t a r t e d after t h e former is c o m p l e t e d . As p o i n t e d out before, t h e effectiveness of t h e sensitivity analysis d e p e n d s to a large e x t e n t on t h e way in which dF /dh is c o m p u t e d . Besides t h e finite difference t e c h n i q u e frequently used, this m a y t u r n out t o b e s t r a i g h t f o r w a r d for m a n y nonlinear elastic p r o b l e m s for which we can form t h e c o n s t i t u t i v e secant m a t r i x K Jp = K p(q^(h); h) so t h a t a

{

{ S a

dF

dK 3 {

a

( ( v

_£ elements

/

f

dC\ ~ ^ - B s)

i

i

a

B

k

W

^ q

0

(8.55)

256 • Nonlinear Dynamics where C\^ is t h e c o n s t i t u t i v e secant m a t r i x explicitly design d e p e n d e n t , while B{j is t h e F E M s t r a i n - n o d a l d i s p l a c e m e n t o p e r a t o r a s s u m e d to be design indep e n d e n t . (In t h e case when B is design d e p e n d e n t , Eq. (8.55) b e c o m e s m o r e c o m p l i c a t e d , cf. [34].) We also n o t e t h a t in c o m p u t a t i o n a l p r a c t i c e t h e n u m b e r of non-zero entries in these derivative m a t r i c e s is small since as a rule specific design variables affect only a very few stiffness c o m p o n e n t s , which in t u r n are r e l a t e d to only a few d i s p l a c e m e n t c o m p o n e n t s . As a result, m a n y i n t e r n a l force derivative entries are zero. Moreover, t h e non-zero entries of this vector can be c o m p u t e d at t h e e l e m e n t level. ]

kl

a

l j a

8.5

Numerical Illustrations

Let us consider t h e sensitivity response of a portal frame s u b j e c t e d to a t i m e - d e p e n d e n t c o n c e n t r a t e d load at t h e apex A as shown in Fig. 8.1. T h e

geometrically nonlinear F E M model includes 100 b e a m e l e m e n t s a n d 297 degrees of freedom; t h e vertical a n d oblique b e a m s are discretized by 25 equal-length elements each. T h e s t r u c t u r a l p e r f o r m a n c e functional is defined as G = ^

v v

(

( ?

/

(a))2

- 1 < 0

(8.56)

-

V

/

where q d e n o t e s t h e vertical d i s p l a c e m e n t at t h e apex of t h e frame and q is an a d m i s s i b l e value. T h e element cross-sectional areas A , g = 1 , 2 , . . . , 100. are a s s u m e d to be t h e design variables. T h e t i m e - d e p e n d e n t g r a d i e n t s of t h e functional Q with respect to A can be i n t e r p r e t e d as t h e sensitivities of t h e a p e x ' s vertical a m p l i t u d e with respect to t h e change in t h e cross-sectional areas at any point of t h e frame. T h e following d a t a are a s s u m e d : base length / = 200, height h\ = 100, height hi — 50, Young's m o d u l u s E — 2.0 x 1 0 , Poisson's ratio v — 0.2, m a s s density g = 0.001 a n d d a m p i n g factor £ = 0.05. All t h e cross-sectional areas are a s s u m e d equal to 10. T h e cross-section's m o m e n t of inertia is a s s u m e d to b e p r o p o r t i o n a l to t h e cross-sectional a r e a s q u a r e d , j . e . / = a A , with a = 1/6. {A}

y

e

e

7

2

Numerical Illustrations • 257 T h e e q u a t i o n of m o t i o n is solved for t h e p r i m a l d i s p l a c e m e n t s e m p l o y i n g t h e constant-stiffness i t e r a t i o n s c h e m e . T h e e q u a t i o n s are i n t e g r a t e d step by step (forwards a n d b a c k w a r d s ) using 500 equal t i m e i n c r e m e n t s with At — 0.001. Figure 8.2(a) shows t h e t i m e b e h a v i o u r of t h e vertical d i s p l a c e m e n t q (r) at t h e apex A. In Figs. 8.2(b,c) t h e t i m e - d e p e n d e n t sensitivities of functional (8.56) with respect to t h e cross-sectional areas at t h e apex A and t h e corner B are given. y

0

100

200

300

400 500 m it e step number

0

100

200

300

400 500 m it e step number

0

100

200

300

400 500 m it e step number

Figure 8.2 (a) time-dependent vertical displacement at the apex point A; (b) sensitivities with respect to cross-sectional areas at the apex point A; (c) sensitivities with respect to cross-sectional areas at the corner point B

nonn ilear elasticity linear elasticity

258 • Nonlinear Dynamics I n the second example we consider a spherical shell subject to a t i m e dependent concentrated load at the apex as shown i n F i g . 8.3.

T h e problem

i s to assess the static and dynamic ( t i m e - i n s t a n t ) s e n s i t i v i t y response of the apex deflection q w i t h respect to the s h e l l ' s thickness h at the clamped edge.

Q

Figure 8.3 Spherical shell. Geometry and load function Q(r) T h e following data are assumed: thickness h = 1.576 x 1 0

- 2

r = 4.76, H

= 8.59 x 1 0

- 2

, 0 =

10.9°,

, Y o u n g ' s m o d u l u s E = 1 x 1 0 , P o i s s o n ' s ratio v = 0.3, 7

yield s t r e s s crj = 4 x 1 0 , E /E 4

T

= 0.02, mass density g = 2.45 x 1 0

- 4

; damping

effects are ignored. T h e nonlinear F E M model includes 20 8-node a x i s y m m e t r i c isoparametric elements and 197 degrees of freedom. T h e p r i m a l and the D D M displacement s e n s i t i v i t y equations are integrated step by step by the ^ - W i l s o n t i m e integration scheme u s i n g 200 equal t i m e increments of At

= 1 x 1 0 ~ ; the 5

N e w t o n - R a p h s o n e q u i l i b r i u m iteration i s employed at each step. F i g u r e 8.4 show the nonlinear behaviour of the s y s t e m loaded alternatively statically and dynamically; the static load increases f r o m 0 to 2 w i t h 200 incremental load steps. F i g u r e s 8.4(a,b) are presented i n t e r m s of the apex deflection and i t s s e n s i t i v i t y versus load step number curves; the s o l u t i o n s of the corresponding s y s t e m modelled w i t h s m a l l deformations are also given for comparison. T h e dynamic s y s t e m behaviour i s shown i n F i g s . 8.4(c,d) which have a s i m i l a r make up as F i g s . 8.4(a,b) w i t h the apex deflection and deflection s e n s i t i v i t y w i t h respect to the edge thickness represented on the vertical axes but the t i m e step n u m ber on the h o r i z o n t a l ones. I t i s observed that the amplitude ratio i

is much smaller than the s e n s i t i v i t y amplitude ratio T h e fact that the values of d<7

static

q™*

x

. /o" "

^dynamic/

1

^static

(dq™**z /dh)/(dq™** /dh). mic

c

/d/i are always negative d u r i n g deformation

agrees w i t h engineering i n t u i t i o n (increasing the edge thickness i m p l i e s decreasing the apex deflection). T h e s i t u a t i o n of dynamic s e n s i t i v i t y i s different, w i t h the positive values of d g

dynamic

/ d / i periodically generated d u r i n g the t i m e response

due to i n e r t i a l effects. T h e t h i r d example concerns a perforated s t r i p under tensile load u n i f o r m l y d i s t r i b u t e d over the s t r i p edges, F i g . 8.5. W e evaluate the dynamic s e n s i t i v i t y response of ( i ) vertical displacement at point A and ( i i ) tensile s t r e s s at B w i t h respect to a change i n Y o u n g ' s m o d u l u s E i n the whole s t r u c t u r e .

Numerical Illustrations • 259

-

elasto-plasticity

-

nonlinear elasticity

-

linear elasticity

F i g u r e 8.4 Spherical shell. Time-dependent deflection and deflection sensitivity

260 • Nonlinear Dynamics

600 unit length

Q

r=2xio-

0

3

v »—I Figure 8.5 Perforated strip. Geometry and load function Q(r) T h e following d a t a are a s s u m e d : r = 5, 6 = 20, h — 32, thickness i = 0.5, Young's m o d u l u s E — 2 x 1 0 , Poisson's ratio u — 0.3, yield stress er° = 4 x 1 0 , E jE = 0.01, m a s s density g = 2 x 1 0 ; t h e d a m p i n g effects are ignored. T h e large deformation F E M model includes 33 8-node plane-stress i s o p a r a m e t r i c finite e l e m e n t s resulting in 234 degrees of freedom. T h e p r i m a l a n d t h e D D M sensitivity e q u a t i o n s a r e i n t e g r a t e d step by s t e p by using t h e ?^-Wilson t i m e integration s c h e m e with 200 equal t i m e i n c r e m e n t s of At — 1 x 1 0 ~ ; t h e N e w t o n R a p h s o n equilibrium i t e r a t i o n is employed at each s t e p . Figure 8.6 shows t h e t i m e response of t h e d i s p l a c e m e n t at point A and its sensitivity, a n d t h e tensile stress at point B a n d its sensitivity, respectively. As in t h e previous example t h e r e exist s o m e positive sensitivity values on t h e sensitivity versus t i m e curves. From t h e d y n a m i c s point of view this m e a n s t h a t increasing t h e s t r u c t u r a l stiffness (Young's m o d u l u s ) does not always imply decreasing t h e vertical d i s p l a c e m e n t at A and tensile stress at B , which is in c o n t r a s t to t h e s t a t i c b e h a v i o u r . We also observe t h a t t h e spectral response of t h e sensitivity response plots includes c o m p o n e n t s of higher-order frequencies t h a n those of t h e e q u i l i b r i u m response, since t h e d i s p l a c e m e n t sensitivity is a nonlinear function of d i s p l a c e m e n t s . 5

3

- 5

T

5

1

8.6

S u m m a r y of the Chapter

In this c h a p t e r we have derived t h e e q u a t i o n s t h a t m a k e it possible to effectively o b t a i n sensitivity solutions for p r o b l e m s of nonlinear d y n a m i c s . B o t h t h e D D M and t h e ASM t e c h n i q u e s have been discussed, t h e l a t t e r for nonlinear elastic application only. O n t h e basis of our numerical e x p e r i e n c e we c o n c l u d e t h a t : • T h e d y n a m i c sensitivity response exhibits highly c o m p l e x b e h a v i o u r even for t h e relatively simple s t r u c t u r a l configurations discussed in this c h a p t e r -the complexity will clearly have a crucial influence on any g r a d i e n t - b a s e d o p t i m i z a t i o n , reliability, control or identification m e t h o d ology e m p l o y e d for nonlinear d y n a m i c a l s y s t e m s .

Summary of the Chapter • 261

-3 I 0

i

ii

i

| 5 0

i

i

i

i

j

1 0 0

i

i

i

i

|

1 5 0

t

i

i

i| 2 0 0

-0.75

-[ i i i i | i i o

time step number

5 0

i

ijiiii| iiii| 1 0 0

1 5 0

2 0 0

time step number

elasto-plasticity nonlinear elasticity linear elasticity

Figure 8.6 Perforated strip. Time-dependent displacement and stress and their sensitivities

262 • Nonlinear Dynamics • For nonlinear d y n a m i c p r o b l e m s t h e D D M a p p e a r s to b e m o r e cost-effective a n d a c c u r a t e t h a n t h e A S M . T h e m e t h o d s are c o m p e t i t i v e for linear systems, though. • A c c u r a c y r e q u i r e m e n t s i m p o s e m o r e severe t i m e step l i m i t a t i o n s in t h e sensitivity analyses t h a n in e q u i l i b r i u m analyses. For a c c u r a t e sensitivity evaluation t h e e q u i l i b r i u m iteration is essential at each step of t h e numerical i n t e g r a t i o n of t h e p r i m a l s y s t e m . • T h e t i m e (or time-like) p a r a m e t e r is used in b o t h nonlinear statics and d y n a m i c s ; t h e only difference consists in a d d i n g t h e inertial and d a m p i n g t e r m s to t h e d y n a m i c p r o b l e m s . T h i s enables us to unify t h e s t a t i c and d y n a m i c sensitivity analyses a n d to use a single numerical code for both classes of p r o b l e m s in a F E M - b a s e d package. It should be n o t e d t h a t for p r o p r i e t a r y codes sensitivity i m p l e m e n t a t i o n in t h e form of post-processing is possible but will typically be m u c h less efficient.

Chapter 9 Metal Forming Using the Flow Approach 9.1

Introductory C o m m e n t s

A n effective way to develop computational models for the problem of large deformation of metals is to treat the material as an incompressible viscous fluid of a non-Newtonian kind. This approach, formally identical to that typical of the theory of rigid-viscoplasticity and known as the flow approach, has been used extensively in recent years in conjunction with the finite element method to solve a variety of bulk and sheet metal forming problems, [5, 114, 145, 196, 197]. T h e flow approach has already been briefly described in Section 2.3.2. Much of its success is due to its simplicity. In particular, if inertia terms are neglected, then the flow equations for the von Mises material are analogous to those of classical non-linear incompressible elasticity with velocities, strain rates and non-linear viscosity playing the role of displacements, strains and shear modulus of the corresponding elasticity model, respectively. This allows finite element computer programs for metal forming analysis to be directly obtained by a simple modification of standard finite element codes written for 2D and 3D elasticity applications. This chapter is devoted to the sensitivity analysis of different metal forming processes described in the framework of the finite element flow approach. T h e literature on the subject appears to be very scarce, [7-9,16,51,104,112,118,165]. We shall begin this chapter with a discussion of the finite element model to be used for the analysis of flow problems, Section 9.2. The formulation of metal forming sensitivity with respect to material parameters entering the constitutive model considered, such as the yield limit <J , fluidity 7, exponent of the viscoplastic flow law n, etc. follows in Section 9.3. Some specific computational studies presented therein concern a direct extrusion process and a cutting problem. N e x t , the problem of shape sensitivity analysis for a class of metal forming processes is discussed, Section 9.4. The procedure is illustrated by calculating 0

264 • Metal Forming Using the Flow Approach the sensitivities of some independent and dependent variables with respect to the die angle in an extrusion problem and to the roller radius in a plane rolling process. Finally, the influence of frictional contact effects on variations of the nonlinear behaviour in deep drawing simulations is studied in Section 9.5. An advanced numerical illustration is given to emphasize the practical usefulness of the approach proposed.

9.2 9.2.1

T h e Discretized Formulation Equations of Flow Approach

In the following sections we shall consider the problem of evaluating the sensitivities of primary and secondary variables describing metal forming problems. In view of the practical aspects pointed out in the previous section and outlined earlier in Section 2.3.2, we shall develop sensitivity equations in the framework of the flow approach. Keeping in mind the main objective of computing sensitivities as a byproduct of the equilibrium solution, we shall first discuss a discretized formulation describing the latter problem. Metal forming processes typically involve very large deformations. This allows us to neglect the elasticity of the material and to adopt the rigid-viscoplastic (or rigid-plastic) material model. If necessary, the elastic recovery can be approximately calculated in the final step of the analysis by removing the loads. In some situations this may result in relatively large displacements, significantly affecting the final shape of the workpiece being manufactured. This happens especially in sheet forming and is much less noticeable in bulk forming; the effect is referred to in the literature as elastic springback. Additional features of metal forming processes are that (i) in many processes material particles follow similar paths along streamlines so that we can basically speak of stationarity conditions prevailing during the process, and (ii) dynamic terms are rarely significant. The first observation suggests that the Eulerian approach is more naturally suited to describe the material motion than the Lagrangian one, and that velocities give more useful information about the material state than displacements, which if needed can be calculated by integration of the velocities for given material points. Furthermore, as has been pointed out in Section 9.1 the material is considered to be a non-Newtonian fluid for which inertia terms can be neglected since for a typical process the Reynolds number is of the order 1 0 ~ . This justifies the second observation above and leads to a formal analogy between the visco-plastic flow and nonlinear incompressible elasticity referred to in Section 2.3.2. On the basis of these facts the flow approach seems best suited for an analysis of steady-state processes. Nevertheless, non-stationary problems can also be solved using velocity-based updating of the configuration, or by some other alternative techniques (for example, by considering a fixed mesh in which the metal flows together with a fictitious material, and the interface between them tracked by solving an advective transport equation 9

The Discretized Formulation •

265

[6, 168]). Transient solutions can be obtained in most cases by using either an explicit or implicit time integration scheme. To solve the 3D equilibrium problem describing a stationary metal forming operation the equilibrium equation in absolute notation (cf. Section 2.5) is taken in the weak form as, cf. Eq. (2.28) / a • be dn = / f bv dft + /

JQ

JQ

JdQo

t Sv d(dfl)

(9.1)

where be is the virtual plastic deformation rate compatible with the virtual velocity field
a

ij -

a

- Phi = J"* £fj -pSij 2

(9.2)

where, cf. Eq. (2.161) C £

\

r.

' 7

(9.3)

is the equivalent viscosity which is a function of the equivalent strain rate, defined in terms of the strain rate tensor i{j as

e = \fliijiij

(9.4)

iij is computed in terms of velocities t>, as 1 / dv{

dvj \

" = 2l^ ^)

£

+

-

r X

(9 5)

and p = —\cra is the pressure (assumed positive in compression). No strain hardening material is assumed for simplicity. We should point out that because of the nonlinearity entering in a complex manner through the viscosity /u*, there is no known functional yielding the solution by making its first variation vanish. Instead, either the principle of virtual velocities or the weighted residual method must be used to obtain the set of equations suitable for discretization. Another important feature of the problem which has to be accounted for is that the flow is incompressible. The condition is expressed by requiring the rate of work generated during material compression/expansion to vanish

L

in 8pdU

= Q

(9.6)

n

The index 'p' used so far to indicate the plastic (or visco-plastic) part of the process is dropped here as no other (i.e. elastic) effects are accounted for in the flow approach. 1

266 • Metal Forming Using the Flow Approach T h e system consisting of Eqs. (9.1) and (9.6) can be used to derive the finite element equations. We assume the velocity v and pressure p to be functions of their nodal values by using their respective expansions

p = 4P

v = ^q,

(9.7)

T h e (3 x N) matrix collects shape functions interpolating the velocity field in the whole region considered in terms of the (TV x 1) nodal velocity vector q, N being the overall number of kinematic degrees of freedom in the discretized system. Specification of the equations to the plane strain case considered in the examples is straightforward. For 5 nodes in the system and no kinematic boundary conditions accounted for so far we clearly have N — 3 5 . The components in the vector q are ordered by the node numbers and, for each node, by the component numbers. The shape s function matrix $ consists of 5 matrices s = 1 , 2 , . . . , 5 , of dimension 3 x 3 each

^ \

(S)

(9.8)

with

o

3x3

0

0

4

o

S)

1,2,...,5

(9.9)

o o i

9)

T h e buildup of the vector q is of the form {q

q^xi

( 1 )

q

( 2 )

q

( S )

(9.10)

}

with s -

1,2,...,5

(9.11)

being the vector of the sth node velocities. 3 Most frequently the diagonal elements will be the same for 1,2,3; generally, different components may have different shape functions. Since the pressure is a scalar quantity and are assumed to be defined at the 2 system nodes, the matrix is of dimension 1 x 5 (i.e. it is a row vector) with the <^*)'s possibly different from the ^ ^ ' s

\ ^

p

i=

pi,p ,...

(9.12) B y differentiating the shape functions with respect to the space coordinates we obtain (cf. Eq. (9.7)i) the strain rate vector e expressed in terms of the nodal velocities q as BexN

q^xi

(9.13)

The Discretized Formulation • B being the (6 x N) B

matrix (9.14)

„ = [ B ^ B W . . . B< > 5

6 X

267

defined in terms of the components of the matrix as 0

0

0 2<4'l 0 '2,2

4:1 04 }24$ 0 o 4*1 43 4% o 4 1 0

= 2-

6X3

(9.15)

s

S

L

Quite similarly the virtual velocities, rate of deformation and pressures are expressed in the form

Sv = ^ 6 q Si = BSq Sp = BSp

(9.16)

where the row vector was renamed for notational consistency as

=

B

(9.17)

B y substituting Eqs. (9.7), (9.13) and (9.16) into Eq. (9.1) and observing that the last one holds for any virtual velocity, we obtain / B
(9.18)

= Q

T

Jn

Q=

[ F do+

/

Jn

FT d(an)

+

(9.19)

p

Jdn

t

Here, Q , f, t and P are vectors with the same component ordering as employed in defining the vector q, and P is the vector of concentrated loads applied at the nodes. B y substituting Eqs. (9.2) and (9.3) into Eq. (9.18) we get

(jf

dft)

2p* B B T

B rB T

q -

dft)

p = Q

(9.20)

i.e. K

( M )

q + K[ p = Q

(9.21)

p )

where K u) (

=

/

Jn

f

k k

Jn

dft = / 2p* B B dft T

0

( p )

Jn

D

Q

= -

I Jn

(9.22) B

T

r

6

dft

268 • Metal Forming Using the Flow Approach the column vector T — { 1 , 1 , 1 , 0 , 0 , 0 } converting the total strain into its volumetric components. T h e equilibrium equation (9.1) has to be solved together with the incompressibility condition (9.6) written in the weak form as 6xl

/ feu dft =

0

(9.23)

JQ (9.24)

which, after discretization, reads or, more ( j f compactly B T B dft)q = 0 T

K

( p )

T

q = 0

(9.25)

T h e fundamental set of equations for the (N+S) "

-

K?P>1

( M )

K

K

0

(p)

unknowns q and p thus becomes

q

(9.26)

.0.

P

which can be written in a residual form as R = Q - Kf q = 0

(9.27)

5)

where q =

q

"

LPJ

-

0

( M )

K

K

(p)

Q

(9.28)

It can be easily recognized that Eq. (9.26) corresponds to the equations describing the nonlinear Stokes flow. T h e solution to Eq. (9.27) can be found by either direct iterations, for which only a quasi-linear rate of convergence can be achieved, or by the quadratically convergent N e w t o n - R a p h s o n method, for which a certain minimum level of strain rate hardening is required to converge [5]. In the latter case by defining the residual after the u>th iteration as R

( « ) = Q _ K(s)<«> q<»>

u, = 0 , l , . . .

(9.29)

we obtain the solution correction <5>q by setting the residual after the o;th iteration equal to 0. Employing the first-order Taylor expansion around q , this condition reads (u,)

(a,)

dR

( w )

R 0 - + D ^ R(-) + r _ _

7 q

(-+i)

dq

=

o

(9.30)

T h e solution correction is computed as 3R<«) 0q

v

i

R

(")

(9.31)

The Discretized Formulation • 269 so that the (cu+l)th approximation to the solution vector q reads j(«+D

=

(«)

q

+

7 q

«-+D

(9.32)

where dO = - I C

(9.33)

T )

K is the (unsymmetric) tangent stiffness matrix, the term the relations dfi* de dq dfi*

di

t

de diij dq

a

Q

=

9

, o

+

dfi*/dq

is computed using

n - i ) ( i n J \7

(9.34)

0

and k

0

=

k 0

0

0 0

(9.35)

T h e matrix ko depends only on the problem geometry; in order to obtain K ^ ) , k is multiplied by the viscosity (i* and integrated element-wise (Eq. (9.22)i). The formulation of the boundary value problem is completed by a suitable specification of the boundary conditions, which can be either of the sticking or the no-friction slipping-type, or of an intermediate type defined by the tangential force applied to the material boundary and computed according to a given friction law. The basic flow formulation can be extended to better account for real situations encountered in metal forming practice: effects such as heat generation, material damage, free surfaces, boundary friction and contact can then be included, [4, 5, 7, 145]. 0

In fact, complexity of most real metal forming problems exceeds the modelling possibilities of the flow approach in its standard version. Moreover, particular features requiring special attention, or at least a careful numerical treatment, are different for different processes. A very general and instructive example of metal forming processes is offered by considering the cutting problem. Similar, but typically much simpler situations may be found in the simulation of other forming processes. A typical cutting process can be modelled in a simplified way by the movement of a rigid tool parallel to the surface of a semi-infinite domain representing the metal matrix at a given cutting depth d. Equivalently the tool may be fixed with the metal moving in the opposite direction; incidentally, such a situation is more convenient for numerical calculations. A metal chip is separated from the workpiece after it has been in contact with the tool. The metal chip has two free surfaces: one is the continuation, after a sharp change in direction at the shear deformation band, of the original free surface of the workpiece. The

270 • Metal Forming Using the Flow Approach other side of the chip, beginning at the tool edge, is first in contact with the tool but after a certain distance it separates, generating the second free surface. Both of t h e m must be found from the condition that for plane strain idealization they should coincide with streamlines. In addition, the former free surface representation must allow for the occurrence of sharp angles corresponding to the discontinuity in the velocity field for the perfectly plastic case. The latter free surface involves a contact problem because the point of the chip separation from the tool is unknown a priori. Finally, for the parts of the boundary in contact with the solid surfaces friction forces should be calculated. We note again in passing that once we know the way to model the cutting problem nearly all other metal forming problems can also be reliably addressed.

9.2.2

Boundary Friction

A simple way to model the boundary friction, suggested in [195], is to insert a thin layer of elements connecting the fixed boundary with the material. According to the Coulomb friction law, a yield stress proportional to the pressure is assigned to the elements so that the material is acted upon by a tangential force proportional to the force normal to the boundary and directed opposite to the material movement. If the pressure happens to be zero, then a residual viscosity is imposed in order to avoid numerical singularity of the matrices. The viscosity is given by = |

(9.36)

with o~f defined as

i {^ll)

0

a

if if if

n

=
i = \ pv

p v > o

v

v

0 < pu < a p < 0

vp

(9-37)

where v is the friction coefficient in the Coulomb friction law and K is a small positive number. In this way the more compressed the material against the surface, the bigger the shear stress between the friction layer and the rest of the material. For the friction elements the derivatives (9.34) take the form 2pu l3 ijBija £

df*

J 1

9e

-7^eijB 9e

ija

II

0 < pV <

if

p < 0

CT

vp

and

-fJh

9

dp

l f

I 0

if

° < ^ < ^ p < 0

( 9

.39)

The Discretized Formulation • 271

9.2.3

Contact and Friction

A more effective and rational alternative is to handle boundary friction by using friction-contact elements in which both effects (contact and, if required, friction) are treated at the same time. Before discussing such a formulation another issue requires additional comments. The contact description for steady-state flow differs somewhat from that known in transient, displacement-based situations. The difference comes from the way the free surfaces are handled. A non-steady process may (and usually will) have free surfaces with non-zero normal velocities. New locations of the surface are found by node updating in terms of their velocities and further corrections for the nodes which come in contact with a solid surface (representing the tool). Then they remain in contact (with the velocity component normal to the boundary set to zero) until there is a traction force between the solid boundary and the workpiece. On the other hand, in a steady-state process the problem consists in finding a stationary surface (or a stationary line, in 2D problems) to which the velocity is always tangent. If some parts of the surface are likely to interact with a solid boundary, then the two conditions must be satisfied simultaneously: no penetration of the flow into the solid boundary (i.e. the imposed zero normal velocity) and no traction forces between the workpiece and the tool. These conditions allow for specification of a criterion for determining the point at which the material separates from (or comes into contact with) the solid boundary. According to the flow direction with respect to the solid boundary two similar problems may be considered in the form of looking for either the so-called separation point or the point where the free surface comes into contact with the solid boundary. Although in our analysis we shall only consider the former case, the latter one can be treated similarly. Along the surface, which may be either free or in contact with the solid boundary, normal velocities are constrained to zero if the material tries to cross the solid boundary, and they are set free if any traction exerted on the flow by the wall is found. In other words, a fixed node of the contact surface is released if the pressure is negative, and a free node is fixed if it has a penetrating normal velocity. The changes are performed one node at a time in the neighbourhood of the present limit of the contact zone, keeping the remaining nodes in the previous state. In this way, the separation point is adjusted until the free boundary originating from it becomes a streamline. It is worth pointing out that points on the free part of the surface may have a normal component of the velocity directed towards the solid boundary. However, this does not necessarily mean that the flow gets through the wall. The imposition of optional restrictions on the velocity component normal to the wall is achieved by four noded contact elements which connect two nodes of the material with two auxiliary nodes fixed to the solid boundary (velocities and pressure are constrained to zero). At the beginning both sides are coincident. If the element is on the free surface, then its stiffness matrix is assumed as the null matrix and

272 • Metal Forming Using the Flow Approach no restriction is imposed on the fluid degrees of freedom. However, whenever the contact condition is satisfied, the normal velocities are constrained at the nodes in the fluid by equating them to their counterparts on the solid boundary (which are equal to zero) by means of a penalty parameter. According to the element numbering given in Fig. 9.1, the equations a(vi

= 0,

-v ) 4

a(v

- v) = 0

2

(9.40)

3

are added to the global system in the respective equations for i> and v through the element matrix, a being the penalty parameter. Since v\ and t? are set to zero, the fluid normal velocities v and v will also be zero. In Fig. 9.1 the nodal velocities are given in the local coordinate system. If the local axes are not coincident with the global ones, Eqs. (9.40) are appropriately transformed so that the penalty parameter will then affect the two equations of the global system corresponding to both the velocity components at each node in contact with the solid boundary, thus ensuring the zero velocity across it. 3

4

2

3

4

h=0

k

4 4

— k\\

— —k \

&33 = ^22 —

4

—

— —k\ '• 4

&32 — — ^23 —

a 0

if pressure at 4 > 0 otherwise

a 0

if pressure at 3 > 0 otherwise

Figure 9.1 Contact element Upon updating the free surface (see Section 9.2.5) no change will result in the nodes defining the surface whenever the normal velocity is zero. This may happen either because the free surface is already a streamline, or because it is in contact with the solid boundary. On the other hand, a different updated configuration will be obtained if there is a non-zero normal velocity component. T h e friction conditions can be imposed in terms of the force vector using the same contact elements. According to the Coulomb friction law, the part of the boundary with the active contact condition is assumed to be acted upon by a distributed tangential friction force proportional to the pressure and opposite in direction to the velocity. (Other friction laws may also be considered and easily implemented since all the necessary information is available at the element level.)

The Discretized Formulation • If p

3

and p

4

273

are pressure values at nodes 3 and 4, respectively, cf. Fig. 9.1, then

the distributed friction force along the element side 34 will be

= -7rn[^

f

+ p

4

(9-4i)

( i - ^

where ip = (x — X 4 ) / ( x — x ) and X4 < x < x is the local coordinate along the side 34, 0 < i/> < 1 and v is again the Coulomb friction coefficient. The equivalent nodal friction forces become 3

f

3

=

-

(P3

M ^

4

3

, P4\ , +

l|v|| 6 v

V

T > " - M 3 '

f

t ( 9

4 2 )

||v||

where ft is the contact element length. The friction forces calculated in this way may lead to unrealistic situations during the solution process and even cause the algorithm to diverge: a high pressure value may produce a non-physical opposite-to-the-flow friction force so large that it will produce a boundary flow moving against the main flow. In the iteration process the friction force may be changing sign from one iteration to another; if additionally its absolute value grows, then the solution algorithm will clearly not converge. For these practical reasons it has been found convenient to bound the friction force derived from Eq. (9.42) by two extreme values: one corresponding to the 'traction case (the contact pressure less than, or equal to zero), in which the friction force vanishes, and another corresponding to the 'sticking' case, in which the tangent velocity is constrained and a m a x i m u m in the allowed friction force absolute value, say f?, is obtained. Hence, the actual module of the friction force for use in the model is 1

||?|| for /OH?! It'll = Hfl. 0
for

t

for

/ <0

(9.43)

t

In comparison with the previous friction element this one is more effective computationally since: (i) it needs no numerical integration and has fewer degrees of freedom, and (ii) it is more accurate as it does not require a 'thin' layer of elements (which in addition may introduce numerical errors because of their high l e n g t h - w i d t h ratio) instead of just a contact surface. On the other hand, the friction force must be bounded either by imposing a no-counter-flow condition or the previously calculated tangential reactions for the sticking type.

9.2.4

Pressure Stabilization

T h e application of any algorithm based on the pressure values at the boundary requires an accurate enough pressure field. It is a known fact that for nonlinear problems (unlike the linear case, [55]) the so-called Brezzi-Babuska condition is

274 • Metal Forming Using the Flow Approach not satisfied even by the Q 2 / Q 1 elements ). As a consequence, spurious pressure modes appear. These pressure peaks, ranging alternately from negative to positive values, seriously disturb the solution and, consequently, any further procedure based on the values of the pressure field. A m e t h o d for pressure stabilization circumventing the Brezzi-Babuska condition has been proposed in [86]. According to it, the equilibrium equation suitably weighted by the gradient of the shape functions is added to the pressure equation in order to obtain the negative Laplacian in the diagonal block for the pressure, which is responsible for the stabilizing effect. The additional equation should vanish when the converged solution is reached. By applying this technique to the flow formulation for metal forming problems we add to the incompressibility condition (the second row of Eq. (9.26)) the equation 1

where a is a stabilization coefficient, 0 < c t < l , / i i s a geometric characteristic (the element size parameter) and
(K

( p )

where L

(9.45)

+ L ) q 4- M p = H

=

[U

L ] 2

(9.46)

and k is a pressure scaling factor [5] assumed proportional to the largest value of the equivalent viscosity /i* while <j> = d/dx{. The element matrices will in general be unsymmetric except for linear interpolations, in which case the second derivatives of the shape functions vanish. The submatrix M connecting pressures at different points is the negative discrete Laplacian which is responsible for the stabilizing effect. The formulation has been implemented for Q l / Q l shape functions (hence <)>=) within the developed finite element code for metal forming analysis and is used for the present calculations [3]. p

i

The term 'Q2/Q1 element' refers to the Lagrangian isoparametric quadrilateral element with biquadratic and bilinear shape functions for velocity and pressure interpolation, respectively, and with nine nodes for the velocity and four for the pressure. x

The Discretized Formulation • 275

9.2.5

Free Surfaces

To reach the final configuration as part of the solution a particularly important feature of the formulation at hand is the way to handle free surfaces. Since steady-state velocities are tangent to the free surfaces (which are streamlines in the 2D description), the updated coordinates can be calculated by integration from a fixed point. The most straightforward way of performing the integration, cf. [197], is by updating one of the coordinate components while leaving the other unchanged for each point defining the free surface, starting from a known point

(9.47) where v and v are the velocity components in the global coordinates. According to Eq. (9.47), the nodal coordinates in one direction (i.e. x) are kept fixed and the free surface is followed by changing the coordinates in the other direction (i.e. y ) . This scheme is convenient for its simplicity provided the free surfaces are roughly parallel to the coordinate axes and it is known in advance that the variations from an initial, estimated configuration will not be very large. However, in cases where the free surfaces are expected to have more significant variations or even change the orientation with respect to the global coordinate system, a more general algorithm is required. For such cases we impose the free surface condition in the form x

y

(9.48) where dfl is the unknown free surface, and v is the velocity component normal t o the surface. B y calculating this integral in the discretized model we find the configuration at the ( o ; + l ) t h iteration on the basis of the velocities and configuration at the iteration u. T h e fulfilment of Eq. (9.48) will require the rotation of each segment ( a , a + l ) , to ensure that the velocity is tangent to it, Fig. 9.2. The condition of a constant length of each segment must be included because Eq. (9.48) leaves this magnitude undetermined. F

n

As the next step a node rearrangement within the whole domain is performed in order to reduce the mesh distortion. From Eq. (9.48) we get the corrections to the boundary nodal coordinates which define the free surface for the new solution. Taking these corrections, denoted by for each coordinate direction (i = 1,2), as imposed values in separate auxiliary problems in which the boundaries not belonging to any free surface are assumed fixed, we solve for each Wi the Laplace (pure diffusion) equation

Vw 2

{

= 0

(9.49)

to obtain the field of the nodal boundary corrections desired. These two linear problems can also be solved by the finite element method, making use of the

276 • Metal Forming Using the Flow Approach

Figure 9.2 Free surface updating by curvilinear integration same discretization as for the main problem. It is assumed that each such free surface update is small enough to justify dealing with the particular components as separate and independent problems. In any case the present discussion aims only at obtaining a more regular mesh resulting from the free surface updating. Even if the updating values were significant the mesh obtained would still be acceptable.

9.3

Material Parameter Sensitivity

9.3.1

Formulation

In this section we shall derive equations for the sensitivity analysis of metal forming processes with respect to parameters defining the process through the constitutive equation (9.2) and other parameters such as the friction coefficient or an imposed velocity. T h e information resulting from this analysis can be very useful for deciding how to modify an already existing technology in order to improve its performance according to a given set of criteria. For example, we may want to know to what extent an increase in the ram velocity in an extrusion matrix will affect the dissipation energy rate of plastic deformation. Let us consider the general form of the sensitivity function as Q(h) =

G(
(9.50)

D

We wish t o calculate the sensitivity of G with respect to a material parameter h entering the theory, i.e. n, 7 or a , cf. Eq. (9.3). B y following the direct differentiation approach ( D D M ) we calculate the design gradient 0

dG _ dG dh ~ dh

dG d
dG dq

D

+

d
+

dq~dfc

(9.51)

Material Parameter Sensitivity • 277 in which, given the solution to the equilibrium problem, dG/dh, dG/dq and dGJdo® are known (or can be routinely obtained) while da /dh can be written as D

d
da

D

dq

v

(9.52)

dq dh

dA where d
d
D

dq

d
D

D

L dq

dp

According to Eqs. (9.18), (9.20) and (9.2) d
D

dq J exN rd
=

2(B/i*+Bq^ V dq

(9.53)

D

= 0

dp J

By substituting Eq. (9.52) into Eq. (9.51) we arrive at dG

dG

dG_do°

dh

Oh

dcr dq D

dG\ +

dq

(9.54)

~dq~) dh

It is seen that in order to effectively compute dG/dh only dq/dA has to be obtained from additional calculations. By differentiating the equilibrium equation (9.27) with respect to h we have d_

(9.55)

dh

dh where

dK > dh s

In

In

dp*dq

dh k ° ^

r

dQ

dq~dq dh^°

(9.56)

B y combining Eqs. (9.55) and (9.56) we arrive at (9.57) i.e. at, cf. Eq. (9.33) (9.58) This equation can be used to find the nodal velocity and pressure design gradients d q / d / i provided the vector q has been solved for from the equilibrium problem and substituted in the right-hand side of Eq. (9.58). We can equivalently solve the sensitivity problem by the adjoint system m e t h o d ( A S M ) , cf. Sections 3.4.3 and 4.1.2, for which we construct an extended function Q* = G*(cr (h),q(h); X(h); h) as D

G* = G + A R T

(9.59)

278 • Metal Forming Using the Flow Approach in which the state equation R = 0 is the same as before, cf. Eq. (9.27), and A (ft) is a (N+^-dimensional kinematically admissible adjoint vector (cf. Eq. (9.7) ) such that G* is stationary with respect to q, i.e.

dG*

dq

= 0

(9.60)

We have dG*

dG

dA

-dT=dft ^T +

. dR

T

T

R

+

A

dft

< - ' 9

6 1

which simplifies to dG* dft

dG dft

(9.62)

because the state equation holds true for both the nominal and perturbed values of ft. Furthermore, by noting t h a t 1

m m*f m =

dq

.

+

da

D

dq

(9 63) dq

V

;

with d o - / d q given by Eq. (9.53), we have D

d C ^ ^

dh

dh

d^dq +

d q dft

[

'

which by Eqs. (9.60) and (9.62) becomes dft dG

(9.65)

dh dG*

and can be directly used for the computation of the sensitivity dG/dft required. T h e vector A is computed in a non-iterative way from Eq. (9.60) written as

dG* _ dG . .„ fdK _ + a ( — q dq dq \ dq s)

% t

.

+ K

s )

)

J

dG + A K dq

= ^

T

T )

= 0

(9.66)

i.e. as

fdGV K-A = -^j

(9.67)

We observe that the system matrix in Eq. (9.67) is the transpose of the tangent stiffness matrix of the equilibrium problem. Usually the ASM makes use of symmetry of this matrix to directly solve the sensitivity problem after finding the solution to the equilibrium problem without further manipulations. However, As observed in Section 3.1, this notation becomes slightly awkward when (•) stands for a function of
D

Material Parameter Sensitivity •

279

in this case we see from Eqs. (9.45), (9.46) and (9.33) that the tangent stiffness matrix K< is unsymmetric. Therefore, it is the transpose of the inverted system matrix that must be used to obtain the vector A. Experience with solving the metal forming problems described by Eq. (9.27) and reported in [3, 5, 95], for instance, shows that the Newton-Raphson scheme, Eq. (9.30), is applicable (i.e. it converges) only for markedly rate-dependent materials, like steel in hot working conditions. This roughly corresponds to the exponent values n < 2 in the model (9.3), cf. [95]. Thus, for a material behaviour closer to perfect plasticity direct iterations involving the secant matrix K\ cf. Eq. (9.28), i.e. based on r)

s

Kf >(q< >)q<«* > = Q s

w

(9.68)

+1

must be performed instead. This iterative scheme is always convergent, although only linearly. Procedures for accelerating convergence were analysed in [5]. In each specific problem the crucial question thus is: Which of the matrices (JC versus K* ) should be used for calculating the sensitivity? In the latter case an iterative version of the D D M results in the form (cf. Eq. (9.58)) T)

55

- e r - a ^ ^ ^ - i x ^ ^ H e r <»•»> Similarly, for the ASM the iterative form equivalent to Eq. (9.67) results as

To compare both versions of the solution algorithm (tangent versus secant stiffness) for sensitivity analysis we have to keep in mind that frequently, as has been stated above, the equilibrium problem is solved by using the secant stiffness matrix so that its inverse is available at the end of that stage. If the equilibrium problem could be solved by using the tangent stiffness matrix, then the decision that this matrix should also be used for sensitivity is straightforward. Unfortunately, the N e w t o n - R a p h s o n method (which involves the use of the tangent stiffness matrix) is virtually always divergent in situations of most practical importance regarding sensitivity analysis, namely when no non-zero velocities are prescribed. Therefore, it is important from a practical point of view to critically assess both versions of the solution algorithm. The tangent stiffness option with a well-conditioned matrix yields the sensi tivity solution in one step. On the other hand, whenever the equilibrium problem cannot be solved by the use of the tangent stiffness matrix, a solution to the new set of equations is required. In contrast, the secant stiffness option requires iterations but no new solution of the equations system. Moreover, when the material behaviour is close to ideal plasticity, the numerical inverse of the matrix K< does not yield the solution in one step: it may then be necessary to iterate even with the tangent stiffness matrix just to improve the numerically unreliable one-pass T)

280 • Metal Forming Using the Flow Approach solution. And it sometimes happens that this iterative procedure is divergent so that the solution using the secant stiffness matrix as given by Eqs. (9.69) and (9.70) is anyhow compulsory. T h e question of superiority of one approach over the other is thus strongly dependent on the material behaviour, the way the loads are applied and the boundary conditions. This is further illustrated in Section 9.3.4. In particular, for a perfectly plastic material the tangent matrix is singular and the solution becomes infinitely sensitive to changes in the physical properties if only the loads are held constant. As we already know, both the sensitivity methodologies, i.e. the D D M and the ASM, are theoretically equivalent (although they are by no means so in terms of the computational time required by each of t h e m ) . In the present context their equivalence m a y be most easily shown by considering the case of G = q^ (a being a specified component number), i.e. when the sensitivity function coincides with a given nodal velocity. B y Eq. (9.51) we have for the D D M

where the flth component of the vector \(a) is defined as i^=6?, 1 , 2 , . . . , 7 V + S . For the A S M , Eqs. (9.65) and (9.67) assume the form

13=

and T

K

( 7 ,

A

( < I )

=

(9.73)

where A(<j) is the adjoint vector corresponding t o G = By substituting A ( « ) from Eq. (9.73) into (9.72) we obtain Eq. (9.71). Moreover, taking the set of sensitivity functionals as G = = {qa} = q, Eq. (9.67) takes the form K

( T )

A = - I

(9.74)

where the a t h column of the matrix A is the adjoint vector A ( « ) corresponding to the constraint G = qs,- From Eq. (9.65) we substitute A in the expression for the sensitivities obtained by replacing in Eq. (9.65) G by q, A by A and G* as given by Eq. (9.59), t o get T

which is equivalent to Eq. (9.58). The forming processes are usually conceived of as controlled (typically holding constant) by either the imposed velocity or the applied force. In the numerical model, as a result of the elimination during the solution procedure of rows and columns corresponding to the fixed degrees of freedom, the former case is equiv alent t o the application of external forces which are proportional t o the viscosity.

Material Parameter Sensitivity • 281 Through this dependence the applied forces are then also functions of the design parameters. It turns out from the sensitivity analysis of problems with constant imposed velocities that the velocity field exhibits very low sensitivity to material parameters whereas the pressure is sensitive to them. On the other hand, when fixed loads are imposed it is the velocity field that exhibits a strong dependence on the design parameters while the pressure is almost constant. For this reason, the latter case seems to be more interesting and is thus discussed in the applications below. To calculate the sensitivity of the solution to the material parameters entering the viscoplastic law we have to find, by Eq. (9.58), the partial derivative of the viscosity with respect to a given parameter. Equation (9.3) suggests the static yield stress o~ , the fluidity 7 and the exponent n of the viscoplastic law as possible parameters with respect to which we can analyse the sensitivity of the solution. We have for each of them 0

dfi* _ _1_ dcro

Se

dfi*

1

/ e

#7

3£7ra \ 7

dn

Zen

2

(9.76)

V7

B y appropriately differentiating Eq. (9.36) we can find sensitivity to the friction coefficient. This derivative will be non-zero only for some of the friction elements (those which are referred to in Eq. (9.37)2). It has the form °-f

dv

= f 3e

y

(9-77) '

In the case of friction modelled by the contact-friction elements, sensitivity to the friction coefficient is calculated by simply differentiating the friction force, cf. Eq. (9.43), with respect to u to obtain

eft

0

v

0

for

/oUtfH

for

0
for

/,• < 0

(9.78)

It is worth pointing out that in the sticking case these derivatives are zero, i.e. the solution is not sensitive to variations of the friction coefficient at all. The next issue of potentially great practical significance is an attempt to find solution sensitivity with respect to imposed velocities. The velocity-type parameters may be singled out, for instance, in the form of the ram or forging hammer velocity, or the tangential velocity of the roll for extrusion, forging and rolling operations, respectively. In such a case only the loads depend on the design parameter through the viscosity. Thus, if the tool action is modelled by prescribing for all workpiece degrees of freedom whose numbers belong to a

282 • Metal Forming Using the Flow Approach subset A = {1,2,... ,a,..., the tool velocity v

N A ] G { 1 , 2 , . . . , N} a velocity value proportional to

0

q =a vo, Q

a GA

a

(9.79)

then a vector 7 = { 7 ^ } , a = 1 , 2 , . . . , NA, is defined such that f — 7d = < "0 \ 0

if

a e A . otherwise

, (9.80) x

We have 75 = 0 if either no value is imposed at the d t h degree of freedom or this unknown is set to zero or a velocity is imposed independently of v . In such a case, by recalling the sensitivity functional Eq. (9.50) we have 0

dG T~

dvo

dG =

T~

dqs,

dG 7a =

-r^7

( 9

dq

where dG/dqs, is the sensitivity of G with respect to the velocity q^ (i.e. taken as the design parameter).

9.3.2

8 1

)

is

Material Model at Low Effective Strain Rates

A critical look at the use of the tangent stiffness matrix as the system matrix for the sensitivity problem demonstrates a possible lack of accuracy of the standard flow approach implementation. To briefly discuss it let us return for a moment to the material model. It is widely known that the flow approach leads to numerical problems when the velocity field tends to be uniform, making the effective strain rate e approach zero and the viscosity fi* approach infinity, cf. Eq. (9.2). The traditional way out of this difficulty (cf. [195]) is to define a cutoff value e for the effective strain rate so that /1* = fi* for every e < e, . In addition, by the so-called pressure scaling a uniform order of magnitude can be obtained for the stiffness matrix entries thus allowing a very small value for the effective strain rate limit. Nevertheless, in some parts of the domain an 'approximate' value of the viscosity will always be obtained. For a mechanical solution this is quite acceptable since the strain rate-viscosity curve is continuous. However, if we consider the tangent stiffness matrix (which is also the system matrix for the sensitivity problem as demonstrated in Eqs. (9.58) and (9.67)), we notice that the elements of the partial derivative dl£ /dq are proportional to £ in the viscoplastic range but are null in the 'rigid flow' limit when a constant cutoff value is adopted, cf. (9.34)i, for e < £u . This means that if the cutoff value for the effective strain rate is, say, 1 0 , then there is a discontinuity of the order 1 0 in the effective strain rate limit. The j u m p clearly affects both the convergence of the N e w t o n - R a p h s o n scheme and, as will be seen later on, coincidence of sensitivities calculated by one of the analytical methods and by finite differences. lim

Um

im

s)

m

- 6

1 8

Material Parameter Sensitivity • 283 However, the difficulty can be overcome in a simple way by replacing (i* = fi* in the interval 0 < e < e by

h]

lim

(t-t )-^ lim

(9.82)

m

i.e. by requiring the derivative of the equivalent viscosity in the effective strain rate interval considered be equal to the derivative at the strain rate limit.

9.3.3

Analytical Illustration

9.3.3.1

P u r e Shear Strain

Let us consider the configuration shown in Fig. 9.3. The rectangular domain is

Figure 9.3 Sensitivity analysis of pure shear strain fixed along the lower edge while a horizontal uniform velocity vi is imposed along its upper boundary. Consequently, the velocity in the body can be represented as

V

x

V

=

-

y

l

~H '

v = 0

(9.83)

y

The strain rate tensor components result as dv

x

dx

= 0,

y

ay"

0 .

while the effective strain rate becomes = const

(9.85)

from which the viscosity is calculated by using Eq. (9.3) as ft

=

H

CO +

(9.86)

284 • Metal Forming Using the Flow Approach B y using the constitutive equation (9.2) we calculate the shear stress as

*•

- 9 0~xy ~

-

JL

&xy —

(9.87)

y/3Hj

T h e sensitivity coefficients result in the form

dv dv' » - O ~dh ~dh de xy di di dh dh dh x

=

x

y

«

=

dh

(9.88)

= 0

0

while for the viscosity we have

d/i* _ dfi*

dfi* dk _ dfi*

(9.89)

~dh ~ ~dh ~dFdh ~ ~dh +

T h e specific expressions for the partial derivatives with respect to the design parameters, o"o, 7, and n are given in Eqs. (9.76). If we take as the constraint functional the deformation energy rate G = / (Tij iij dfi = / 2a Jq

Jq

xy

i

dO = 4ft fi* i

(no sum on x , y)

2

xy

(9.90)

then its sensitivity coefficient with respect to the generic design parameter h is

dh ~ 9.3.3.2

4 i

dh

(9.91)

£ x y

Free Forging

Let us now consider the free forging operation defined in Fig. 9.4.

A uniform

y

L

Figure 9.4 Sensitivity analysis of free forging distributed compression force t = F/L

is applied to the upper and lower faces

Material Parameter Sensitivity • 285 of the workpiece which is assumed to undergo a plane strain deformation. By elementary equilibrium considerations the stress field is described in this case by = 0 a = t

<7

X

(9.92)

y

= 0

0~ y X

from which the pressure and stress deviator components result as 0~

x

p=-cr

=

m

a-

x =

<

(

=

T

x

+

p

-

+ Oy

t

2

2

= - - t

( 9

9 3 )

+P= o

°y

o° = (T = 0 xy

xy

T h e velocity field is unknown a priori. B y symmetry considerations we can expect v to have a linear variation along the y direction for all points of the domain so that denoting the velocity on the upper edge by v we arrive at y

y

Vy = ^ V y

(9.94)

Since there is no friction the domain preserves its rectangular shape in the process but the aspect ratio changes. Therefore, v can be assumed constant along each of the left and right boundaries, i.e. for x = L/2 and x = —L/2, respectively. B y the incompressibility condition and symmetry considerations we can write x

vH x

+ vL

= 0,

y

i

x

+ i

y

= 0,

i

xy

= 0

(9.95)

from which we have that

2x v

= —v

x

(9.96)

x

T h e components of the rate of deformation tensor are

—v = T

L

H

v,, v

y ~ 2 jj y

£

v

(9.97)

while the effective strain rate becomes .__

4 £ =

v

2i

v

v

Vf# vl =

(9

-

98)

B y recalling that < = 2/i*c

tf

(9.99)

286 • Metal Forming Using the Flow Approach we arrive at | = -

( ^

7

+ a„)

(9.100)

from which =

Vy

-7

(V- °)V ff

(9

-

l0l)

Finally, we calculate the sensitivity coefficients with respect to the parameters of the constitutive equations for the viscoplastic material model. From Eqs. (9.92) and (9.93) we see that the pressure and stress deviator do not depend on the material parameters, i.e.

dp

dcfi,

dH'

-df=°

(9102)

From Eq. (9.102)2 and the constitutive equations (9.2) and (9.3), we have + ^ = 0 (9.103) while, by differentiating Eq. (9.101) and then inserting the result into Eqs. (9.94) and (9.97), we get

dv

_

^

(y/Zt ^ _

=

_

=

_ ( v ^

y

d7

7

T

V 2

dv _ _ \ ~~2 dn y

Y'

1

+

„ j 0

V3# _ _

yyl£ /

( 9

.

1 0

4)

4

°) —

(—

ln

+

°)

a

T h e sensitivity coefficients of the velocity and strain rate components with respect to any of the design parameters are then arrived at in the form 2y

dVy

~

dh d i

~dh d i

~~

dh

y

dv

y

H~dh

2

y

dttxy dh

2

x

dv

~H~dh

~

(9.105)

dv

y

~H~dh

= 0

T h e viscosity sensitivity coefficients du*/dh can be obtained in different ways. For example, we can substitute Eqs. (9.97)i and (9.105)i (or ( 9 . 9 7 ) and ( 9 . 1 0 5 ) ) into Eq. (9.103) to obtain 2

2

Material Parameter Sensitivity • 287 Finally, the sensitivity coefficient for the deformation energy functional reduces to dG

[ di

29,dv

y

d i = L JZ°»

y

D

A

= T

I

1

(

9

1

0

7

)

B y comparing both the examples an interesting conclusion of practical importance can be re-iterated: the sensitivity coefficients depend strongly on the way the boundary conditions are imposed. In the first case a constant imposed velocity implies a non-constant load which should also be design differentiated to derive the sensitivity expressions. As a result of the analysis an insensitive velocity field is obtained in the first example. The conclusion is that in any sensitivity analysis the description of the boundary conditions strongly affects the results.

9.3.4

Computational Illustrations

9.3.4.1

Direct Extrusion

Figure 9.5 Layout of the extrusion process and mesh 2 As a first computational illustration of the sensitivity analysis for a forming process, an extrusion matrix is considered as shown in Fig. 9.5, with the given load applied at the left boundary representing the ram. Sticking friction conditions are assumed. The material model described by Eq. (9.3) is adopted with O~Q = 200, 7 = 0.1 and n = 1.5. Free surface conditions prevail when the material gets out of the extrusion die. Sensitivity of the nodal variables v and p with respect to the static yield stress CTQ is calculated. Sensitivity isocurves for the velocity module are plotted in Fig. 9.6. The character of the graph is very similar to that for the isocurves of the velocity module. Both the D D M and the ASM have been employed yielding the same results, which are also in full agreement with the finite difference approach—no noticeable differences appear in the graphs.

288 • Metal Forming Using the Flow Approach

Figure 9.6 Sensitivity of the velocity module to the static yield stress, d||v||/d<7o (a)

(b) pressure at selected points for two meshes

100

1

-B-—B

1

B

B~

Y

1

1

•«-.-B.,,-».^—B—-g--H

-200

pressure sensitivity at selected points for two meshei 1I

1

A ml

B. ml D. ml E. ml F. ml A, m2 B. m2 D. m2 E. m2 F. m2

-300

1

S

r—

*—i

1

—r-*

0.5

k

—*—

— e — —k-0 — b —

--•--A-©--

Ti

—H 194

196

200

M

X-

200

202

204

206

Figure 9.7 Pressure and pressure sensitivity with respect to static yield stress versus static yield stress at different points

T h e case of the extrusion force held constant is considered next. In Figs. 9.7 and 9.8 the pressure and velocity solutions and their respective sensitivities for the points indicated as a function of the static yield stress are displayed. In each case the results for two different meshes are shown: mesh 2, defined in Fig. 9.5, and mesh 1 such that every element contains four elements of mesh 2. In both the cases the results are in very good agreement with the sensitivities calculated by the finite difference approach with the perturbation taken as Acr = 0

Material Parameter Sensitivity •

289

Figure 9.8 Velocity components and velocity sensitivity with respect to static yield stress versus static yield stress at point D

0.005<7o = 1. T h e plots are virtually identical for any of the methods employed and the same mesh. Pressure results appear to be roughly mesh-independent while relatively significant discrepancies are observed for the pressure sensitivity. On the other hand, the equilibrium and sensitivity results exhibit a similar mesh dependency in the case of velocity solutions: for small relative values of a given velocity component a more noticeable mesh dependency is obtained than for large ones. As expected, velocities are more sensitive to changes in the static yield stress as the cross-section becomes smaller. Also, for constant loads the sensitivity of the velocity component in the direction at which the tool acts is negative, since an increase in the yield stress results in a slower process. In addition, nodal pressures are almost constant, as they should be in the case of

290 • Metal Forming Using the Flow Approach

01

01 (c)

Y

Y

sensitivity dv/dy versus y

sensitivity dp/dy versus y

-i

-B

r

B

B

B-

A. DDM B. DDM D. DDM E. DDM F. DDM A. FDM B, FDM D. FDM E. FDM F. FDM

^5

0.06

0 08

0 1 Y

Figure 9.9 Velocity and pressure and their sensitivity with respect to the fluidity coefficient versus the fluidity coefficient at different points applied loads. If fixed velocities are imposed, then the load happens to be nearly proportional to the viscosity. In such a case the velocities are virtually insensitive to variations in the yield stress whereas pressure sensitivity is positive and tends to be linearly proportional to changes in the yield stress for materials exhibiting behaviour close to perfect plasticity. A similar analysis is performed with the other parameters entering the constitutive model (fluidity 7, exponent of the viscoplastic law n, cf. Eq. (9.3)), and the friction coefficient v, cf. Eq. (9.37), as design parameters. The results for the chosen set of points are shown in Figs. 9.9 - 9.11, where the velocity, pressure and their design derivatives calculated by direct differentiation (full line) and by finite differences (dotted line) are plotted. It turns out

from the sensitivity analysis accounting for the

different

Material Parameter Sensitivity •

291

sensitivity dp/dn versus n

Figure 9.10 Velocity and pressure and their sensitivity with respect to the parameter n versus n at different points p a r a m e t e r s e n t e r i n g t h e c o n s t i t u t i v e m o d e l t h a t t h e fluidity 7 a n d t h e e x p o n e n t n h a v e a negligible effect on t h e pressure w h e n loads are held c o n s t a n t . It should b e n o t e d t h a t t h e discrepancies ( t h e D D M versus t h e F D M ) shown in Fig. 9.9(d) for t h e p r e s s u r e sensitivities w i t h respect t o t h e fluidity 7 are negligible w h e n c o m p a r e d w i t h t h e pressure discrepancies. Irregular results are o b t a i n e d b y t h e F D M , which s e e m s t o b e d u e t o insufficient precision while solving t h e p e r t u r b e d p r o b l e m a n d by t h e t r u n c a t i o n errors i n h e r e n t when t h e finite length of t h e pert u r b a t i o n A 7 is t a k e n ( A 7 = 0.01 in this case). Similarly, irregular sensitivity results for t h e F D M in t h e u p p e r r a n g e of t h e considered values of n suggest similar errors in t h e application of this m e t h o d . T h e analysis of sensitivity t o t h e

292 • Metal Forming Using the Flow Approach (a)

7

v ' 6

0 0

^ v, versus friction coefficient v

(b) o

f i s s u r e p versus friction coefficient v

Figure 9.11 Velocity and pressure and their sensitivity with respect to the friction coefficient v versus u at different points

friction coefficient u leads to a different situation. Unlike the previous cases, the sensitivity coefficients for the problem variables with respect to the friction coefficient calculated by the F D M are very sensitive to the size of the perturbation Au used to calculate the numerical derivatives. In fact, no range of Au could be found for which a stable value of sensitivities resulted. Considering the friction phenomenon, we can expect a non-smooth transition between the two different friction types: sticking (with zero tangential velocity) and sliding friction. In terms of the friction elements used for this analysis, we may find points alternating between both types of friction along with the friction coefficient changing according to the local pressure value. The specific friction model we deal with has two aspects which seem to explain the failure of the sensitivity analysis in this case: the non-smoothness of the friction law leading to a discontinuous derivative,

Material Parameter Sensitivity • 293 and the treatment of the surface contact phenomenon as a volumetric one, which implies numerical integration over an essentially discontinuous zone. This means that the elements which can provide an adequate friction description in the equilibrium problem may turn out to be unsuitable for sensitivity analysis. In view of the above we have performed the sensitivity analysis using the D D M without confronting the results against those by the FDM. The D D M results are plotted in Fig. 9.11, in which important response changes are obtained until a certain value of the friction coefficient is reached (for a constant value of the imposed extrusion load), starting at which all the boundary exhibits sticking friction and therefore no sensitivity to the friction coefficient appears. In this example the equilibrium problem has been solved by direct iterations so that the secant stiffness matrix was available at the end of this step. However, since the material properties were typical of a material with a relatively high rate dependence, the sensitivity problem converged in a few iterations only when the tangent stiffness matrix I C was used. Therefore, the K option was preferred, even though it required an additional solution of the equations system at the sensitivity stage. T)

9.3.4.2

l T )

Cutting

Another experience resulting from the previous example is that for low material rate sensitivity the direct solution method based on Eq. (9.58) or (9.67) is not possible, because the tangent matrix is nearly singular and cannot be used to obtain an exact solution in one iteration. In spite of this, the matrix, when applied iteratively to find the solution, usually yields convergence much faster than the direct iterative scheme defined by Eqs. (9.69) and (9.70), cf. [7, 104]. However, in some extreme cases the K scheme is not convergent and the solution is only possible by direct iterations. In addition, convergence is very slow and it may take thousands of iterations. ( T )

This problem may be illustrated by the following example in which a 2D cutting problem is simulated by placing a cutting tool at a given depth of an incoming metal flow causing a flow separation—a chip is separated from the main flow. Figure 9.12(a) shows the initial finite element mesh used. Constant loads are imposed on the upper boundary of the body. The metal properties are those of a nearly perfectly plastic material (n = 40 in Eq. (9.3)). A major part of the domain undergoes rigid body flow, as almost all the plastic deformation is concentrated in the vicinity of the flow separation region. The problem has two free surfaces which are iteratively modified to make the velocity tangent to them. For the lower surface the contact elements impose a penalty-based restriction on the normal velocity when a no-penetration condition requires so, and release it if normal traction is obtained in the fixed part of the curve. Figure 9.12(b) shows the final mesh after the solution obtained with the contact and free surface conditions have been satisfied. In the above example the iterative scheme of Eq. (9.69) had to be employed.

294 • Metal Forming Using the Flow Approach

(a)

(b)

Figure 9.12 Cutting: (a) initial mesh, (b) final mesh

In Fig. 9.13 sensitivity results for t h e velocity m o d u l e w i t h respect t o t h e viscop l a s t i c law i n d e x n o b t a i n e d by t h e central F D M a n d t h e D D M are c o m p a r e d . B o t h m e t h o d s give basically t h e s a m e solution which consists, similarly t o t h e velocity field, of s u b d o m a i n s with c o n s t a n t velocity sensitivity s e p a r a t e d by a n a r r o w b a n d of t h e shear deformation r a t e sensitivity coefficients (i.e. a b a n d w i t h high values of d e ^ / d n ) corresponding t o t h e plastic deformation zone. Difficulties in using t h e t a n g e n t m a t r i x have a physical i n t e r p r e t a t i o n in t h e fact t h a t a perfectly plastic m a t e r i a l u n d e r constant loads is infinitely sensitive t o variations in t h e m a t e r i a l p r o p e r t i e s . T h e whole process is numerically stable if t h e b o u n d a r y conditions consist of non-zero i m p o s e d velocities; it is t h e n seen t h a t t h e velocity sensitivities are virtually zero. As a m a t t e r of fact, in t h i s case t h e variations in t h e applied loads are p r o p o r t i o n a l t o t h e variations of t h e m a t e r i a l p a r a m e t e r s . Hence, no response change is observed.

Material Parameter Sensitivity •

295

Finite Difference Method 0.009 0.018 0.027 0.036 0.045 0.054 0.063

Direct Differentiation Method 0.009 0.018 0.027 0.036 0.045 0.054 0.063

Figure 9.13 Velocity sensitivity with respect to the viscoplastic index n by the FDM and the DDM

296 • Metal Forming Using the Flow Approach

9.4

Shape Sensitivity for Steady-State Problems

9.4.1

General Comments

Having successively presented the numerical model of metal forming processes and the way to assess its sensitivity with respect to material parameters, we now proceed in this section by deriving the corresponding equations for shape sensitivity. For the specific case of metal forming processes the results will have a great significance for optimization of the forming tools or, more generally, the process itself. Solving the equilibrium and sensitivity problems allows for the evaluation of cost functionals and their derivatives with respect to shape variables. B y combining the procedures with an optimization algorithm the cost functionals may by minimized with respect to the shape parameters subject to given design restrictions. In this way the optimal shape for a given forming tool or for intermediate shapes in a multistep forming process may be achieved. As we already know quite well, different sensitivity methodologies can be adopted. Having such a wide range of options one should recall that the equations resulting from different approaches are theoretically equivalent but may have quite different numerical implementations. Taking into account all the particular features previously discussed and our personal experience we have opted for the direct differentiation method ( D D M ) based on the continuum approach (discretization carried out after design differentiation) combined with the domain parameterization approach ( D P A ) .

9.4.2

The Domain Parameterization Approach

Let us consider a generic response functional Q=

/ f(«r,e,v;h)dft+ /

Jn

f l

+ /

g (v,t; {v)

Jan

h) d(dO ) v

v

<7 (v,t;h) ( a )

d(0OO

(9.108)

to be the subject of sensitivity analysis, where v and t are the specified velocity and traction vectors, imposed on the boundaries dQ and respectively. The integrands in Eq. (9.108) are assumed to be known functions of their indicated arguments. T h e general form (9.108) may be specified to describe selected point or (sub)domain design constraints or cost functions to be minimized during optimization. These conditions may be imposed on stress, strain and/or displacement components at selected points or averaged over some domains, or may concern such quantities as the rate of dissipation energy, heat generated, etc. The problem in the shape sensitivity analysis is to find the derivatives of the response functionals with respect to shape parameters such as one of the coordinates of a v

Shape Sensitivity for Steady-State Problems • 297 given point, the radius of a roller or punch edge, the extrusion ratio, the angle of a die, etc. The design sensitivity expression in this case reads 6G = 6 [ gdtt+sf

g

d(dSl )

{v)

Jq

+ sf

y

JdQ

g

d{dtt )

(o)

(9.109)

a

JdQff

v

in which the integration domain can clearly be design-dependent. Equation (9.109) involves expressions containing the design derivatives of the problem's basic variables (velocities, possibly pressure), dependent quantities (strains, stresses) and geometric characteristics (coordinates, Jacobian, area metrics). The design variations of the problem's variables are derived in Section 9.4.2.2 below. As we already know from Chapter 6, regardless of which method (the D D M or the ASM) is used to compute the sensitivity expression (9.109), there still exist different possibilities to evaluate the shape design sensitivities. Among them, the D P A is by far the most popular technique in the numerical context, as explained in Chapter 6; it will also be adopted below. Hence, the methodology to be employed can be symbolically denoted as D D M / D P A (the adjoint system method could also be used with no difficulty). 9.4.2.1

Transformation to Reference Configuration

To express the 'old' reference coordinates in terms of the design variables and the 'new reference coordinates x we define the mapping x = x ( x ; hj) which enables us to calculate the Jacobian transformation matrix F = F{j = dxi/d Xj and to relate the volume and surface differential elements between the actual and the reference configurations as 1

r

r

r

dfl d(dn)

= =

dWJ d{dn )j r

( 9 e

-

1 1 0 )

where J — | | F | | , J = | | F n | | and n is the unit outward normal to the 'old' reference boundary. Likewise, all the spatial derivatives in Eqs. (9.1) and (9.6) can be written in terms of the reference coordinates. In this way we obtain the equilibrium equation _ T

9

1

/ (a 6i jQ T

- f 8v)J

dir -

T

r

/ t 6vJ JdQ t r

d(dCl )

= 0

r

a

(9.111)

r

where .

1 / dvi

d Xk

dvj

dx

2 \d Xk

dxj

d Xk

dxi

r

(£ij( k))

6

v

= eij(6v ) k

r

= - Hb-^-r— + 2 \

d Xk r

dxj

T

k

r

d Xk r

(9.112) dxi

A simplified notation J = , etc. is used here since no confusion is likely to arise below regarding the definition of either C or C . l

r

%

298 • Metal Forming Using the Flow Approach 9.4.2.2

Design Variation of Problem's Variables

Upon t h e introduction of design variations in Eqs. (9.1) and (9.6) the problem's variables (including now the geometric ones) undergo variations which can be expressed in terms of implicit and explicit components as 8F = dF,

(5x = Ox, 8p = 8p,

8v = 8v,

8(8v) = d(8v),

(9.113)

8(8p) = d{6p)

Furthermore, by using Eqs. (9.112)i and (9.113)3 we have the implicit variation of t h e strain rate given as dvj \ k i

> = 2

dx J r

dxj

k

_ 1 (d(8v )d x

dx{

2 V dx

r

\d x J r

d Xk

k

t

r

k

r

k

dxj

d{8v )d x 3

dx r

k

r

k

dx

l

(9.114) so that Siij =

iij{8v)

(9.115)

provided t h e variation is smooth enough to allow the permutation of the differentials. T h e explicit strain rate variation reads

* =\

dvi

T

dx r

fd x

a

dii

k

dvj

|

\ dxj )

k

dx T

k

d

fd x r

k

\ dxi

(9.116)

since, by Eq. (9.113)2, 8(d x /dxj) = d(d x /dxj). In Eq. (9.116) the expression for the design variation of the inverse Jacobian has to be developed on the basis of that for the Jacobian itself. From the formula for the derivative of a matrix in terms of the derivative of its inverse, cf. Eq. (5.195)i r

r

k

k

(9.117)

dh

dhd

c

we have d Xi

d Xj

dx,

dx

T

d

d xi^/dx

r

k

r

k

dxj

(9.118)

\d xi r

In t h e case of the viscosity parameter, using Eqs. (9.3) and (9.4) results in 8p* = ——;— 8i{j de deij

(9.119)

de de^ and in t h e case of the stress tensor, by Eq. (9.2), in la = 2(8p*)i +2p*8e -T da = 2(dp*)e +2p*de

8p

(9.120)

Shape Sensitivity for Steady-State Problems • 299 Analogous t o Eqs. (9.114) and (9.116) we can also write d{6vj)}

M

dx r

d(Siij)

=

r

J

k

k

|

fd(Sv )\

E

dx r

j

dxj

dx r

dx

jd x

Sensitivity

dx

\ dxj

k

r

k

dxi r

Vj

-

)

k

d(S ) r

9.4.2.3

dx

(9.121)

k

V dxi

k

Expressions

We make use of the preceding relations to obtain the design variations of the equilibrium equation (9.111) in the form / 8(a 8eJ)

<m -

T

f

r

JQ

8(rSxJ)dfl -[

8(?8vJ )d(dn )

r

(9.122)

JdQ\

JQ

R

= 0

r

a

R

which, by Eqs. (9.113), results in / T[8
JQ

T

+ a 8edJ]

T

-I

[d(t J )8v r

+

e

dfi r

T

t J 8{8v)] T

/

[d(f J)8v

+ f J6{8v)\

T

JQ

T

T

d(dft ) = 0

r

(9.123)

r

e

dQ

JdQ t r

or, separating the implicit from the explicit variations, in (/
r

+ 1 T [6a 8eJ+

JQ

-f

* d(6e)J+

T

T

d(f J)8v dW T

I

-

T

* 6idJ] T

dO

d(t J )8vd{dfl ) = T

d{dW)\

d

r

r

a

0

(9.124)

We assume the arbitrary variation 8(8v) to be a kinematically admissible velocity field satisfying t h e smoothness requirements needed to ensure continuous derivatives. Since 8(8i) is assumed to be compatible with 8(8v) we get from Eq. (9.121)i_ that 8(8e) = e(6{8v)). Therefore, 8{8e) is compatible with 8{8v) and, since 8(8v) is arbitrary, the term in braces in Eq. (9.124) represents the equilibrium equation (9.1) and thus vanishes. B y separating t h e stress variations and using also Eq. (9.113)6 we obtain / 8a 8eJdn =

JQ

T

f d(f J)8v

r

JQ -

d(T + /

T

JSQ

/ [da 8iJ

JQ

T

d{t J )8v

+ a d(8e)J T

T

d{dtt ) r

a

+ a 8edJ]

dQ

T

r

(9.125)

Similarly, we perform the design variation on the incompressibility condition (9.6) to get J

(8e Y T

8pJ + e T8{8p)J

+ i T 8pdj)

T

r

dfi = 0 r

(9.126)

in which t h e second term vanishes by Eq. (9.6). B y applying Eqs. (9.113)2 and (9.115), Eq. (9.126) yields / Te (8v)T

JQ

T

8pJ dW = -

[

de T 8pJ d l T -

JQ*

T

/

JQ*

e T SpdJ dW T

(9.127)

300 • Metal Forming Using the Flow Approach

9.4.3

Discretization

We now proceed to evaluate different terms resulting from the combined solution of Eqs. (9.125) and (9.127) after the finite element expansion. It can be a matter of discussion whether to attempt analytical differentiation of each term (and thus, the matrices), or to use a central difference approach as suggested in a different context in [14], at least for those terms which may involve differentiation of complicated expressions. In the cited paper, although only linear problems were under discussion, the central difference scheme for the evaluation of the equivalents of dQ and d K ^ q (cf. Eq. (9.27), which correspond to the right-hand side of the equation system (9.125) and (9.127)) was preferred. In our case, however, we see no reason to avoid analytical differentiation of the expressions, either in the discretized or in the continuous form. It is a known fact that one of the main drawbacks of the finite difference approach is the requirement to properly choose the size of the adopted perturbation, which should meet two opposite error criteria of being neither too large nor too small, in order to avoid numerical differentiation and truncation errors, respectively. This holds true if applied 'externally', that is by solving twice a perturbed problem and calculating afterwards a finite difference approximation to the derivative, as well as in the case described in the quoted paper, where only some terms are numerically differentiated, the rest being differentiated analytically. It is also necessary to keep track of the mapping to get the perturbed terms from a perturbation in the design parameter. This means no saving concerning the necessary information that must be available, in comparison with the case where the explicit variations are analytically calculated. Moreover, as shown in what follows, analytical differentiation leads to matrices built in analogy with those of the analysis phase. Thus, the approach is straightforward to implement. Having discretized the analysis problem we have the variables expressed by Eqs. (9.7). B y using Eqs. (9.115) and (9.119)i together with the constitutive law (9.2) we write Eq. (9.120)i in the discrete form as

8a = 6{2p*Bq) + 8pT = 28p*B q + 2/i*B<5q + 8pT T

r

(9.128) Moreover, we have from the discretization assumptions, cf. Eqs. (9.7) and (9.13)

6v = 8i = Sp =

8q B6q B8p

j>

which, inserted into Eqs. (9.125) and (9.127), yield

(9.129)

Shape Sensitivity for Steady-State Problems • 301

- J

2 B [{dfi*)B + fi*dB] qJ dW

- (

2u*(dB) BqJ

T

T

dfT -

/

+ k

8J]

2//* B B q dJ dfT

(9.130)

T

and K 6q

=-f

{p)

[(dk )J

Q f

ip)

(p)

dWq

(9.131)

In the above system we recognize the tangent matrix I C of the equilibrium analysis problem (Eqs. (9.30), (9.33) and (9.35)). Moreover, the right-hand side can be computed by using products of the already calculated matrices with the new matrices dB and <9k formed analogously to the submatrices entering K , so that T )

t T )

(p)

de = dk

=

{p)

(dB)q

(9.132)

BT dB T

In comparison with the matrices B and k , cf. Eqs. (9.13) and ( 9 . 2 2 ) , respectively, the components of the inverse Jacobian matrix have now been replaced by their explicit variation obtained from Eq. (9.118). The terms containing explicit variations of the external forces or surface tractions are to be calculated from the appropriate expressions defining them. T h e last step to complete the derivation of the expression for 6q is to define the mapping x = x ( £ , h ) which enables us to calculate cbt = (dx/dh)6h, from which the variations of any other geometric quantity can be obtained. There are different ways to solve this problem. Probably the most natural one is to divide the domain into as many so-called master elements as to completely define the discretized domain and the possible design variations. These elements, which may contain from one to several finite elements each, connect the master nodes. The mapping can be defined by the use of shape functions. By means of the inverse transformation we find the native coordinates £ of every node in each parent element, and after checking that the point belongs to the element we evaluate the shape functions and recover the nodal coordinates as a linear combination of the master node coordinates. Denoting by X , the vector containing the ith coordinate of the master nodes we have (p)

2

1

x = ^ ( C ) X = <^ (C(x))X M

t

(s)

t

s t

(sum on s)

(9.133)

with 5 = 1 , 2 , . . . , S , S being the number of master nodes and ^ the interpolating functions of the parent element. The explicit variation of the jth nodal coordinate then becomes M

M

d

Xj

=

M

dXiSij

(9.134)

We denote by £ the parent element coordinates in the isoparametric transformation. They are here equivalent to the more general concept of the reference coordinates x used so far. 1

r

302 • Metal Forming Using the Flow Approach

Figure 9.14 Nodal coordinates of the roller surface as a function of the roller radius N e x t , we have to find an expression for the explicit variations 6*X, in terms of the design parameter variations. The actual form of this relation will depend on the design parameter, which may affect the coordinates of either one master point only or several of them (such as an angle, etc.). The simplest situation is when the design parameter coincides with just one nodal coordinate, say the fcth, of the tth. master node, h = so that dX* = 8{k8 8h. We may also have a more elaborate relation. For example, if the roller radius p is assumed to be the design parameter in a plane rolling operation (as it will be in the second example below), and the imposed reduction in thickness is kept constant, then the master nodes are defined as remaining on the roller surface under the design variations, with an additional constraint, e.g. the requirement that they have only horizontal, vertical or radial coordinate variation (dX = 0, dX[ = 0, 69 = 0, respectively, where 9 — a r c s i n ( ( X * — x\ )/p) and s = 1,2, . . . , 5 , see Fig. 9.14). Because of the restriction of the constant reduction in thickness, assuming that x\ is the rolling direction, the coordinate of the centre of the roller will have the variations s

s

2

s

c

M

dxic = 0 dx — 6h

(9.135)

2c

For the master nodes placed on the roller surface we may calculate the variations dX' by using a transformation to polar coordinates. Considering the case of 8X2 = 0, we have this condition ensured by the relation pcos9° + 8p = (p + 6p) c o s ( 0 + 80 ), s

s = 1,2,..., 5,

s

(9.136)

from which we obtain 89 . Then, the horizontal variation of the nodal coordinates reads s

dX[

= (p + Bp) s i n ( 0 + 69 ) - p sin 9 s

s

s

(9.137)

Shape Sensitivity for Steady-State Problems • 303

For s = s such that 0* = 0, we get, by Eq. (9.136), 60 - = 0. Then, from Eq. (9.137) there results the condition dXf = 0. T h e design element shape functions may be the same as those of the isoparametric finite elements. However, some implementational problems may arise for non-rectangular elements, in which straight lines are not mapped into straight lines so that it may be difficult to determine whether a node belongs to an element or not. For this reason we propose another set of shape functions which preserve straight lines during the transformation. Such a mapping is illustrated in Fig. 9.15, where A, B, C and D are the middle points of the element sides. Every point in the quadrilateral has local coordinates in the interval ( — 1,1) while those located on any side have one coordinate constant equal to either 1 or —1. s

9.4.4

Analytical Illustrations

Before we continue with computational illustrations, we shall discuss two simple analytical examples of shape sensitivity analysis. First, we recall the results obtained for non-shape parameter sensitivity (see Section 9.3.3 for the pure shear deformation and free forging cases, respectively). In the first example we consider the effect of changing the workpiece height H while keeping the imposed velocity vi constant. Making use of Eqs. (9.83)-(9.87) we compute the derivatives with respect t o the design variable H as de™

de

v-i

304 • Metal Forming Using the Flow Approach whereas AO

Q = HL,

—

an

= L

(9.140)

T h e above relationships allow us to calculate the sensitivity of the deformation energy rate as

< ' 9

1 4 1

>

Free forging solution sensitivity with respect to the billet height H is considered next. B y combining Eqs. (9.97) and (9.101) we observe that

§ =°

< > 9142

because, as results from Eq. (9.101), for the constant load imposed the punch velocity depends linearly on the billet height H. This means that the strain rate will also be constant. We similarly find that the viscosity (i* is not sensitive to changes in the billet height i / , which, together with the previous result, confirms the validity of Eq. (9.102) also for this design variable, since by Eq. (9.92) the stresses do not depend on H. Finally, the expression for the deformation energy reads G=

!

aiJ y y

dft

(9.143)

0

where J = Q/f&o = H/Ho is the Jacobian transforming the actual to the reference configuration. In Eq. (9.143), this is the only quantity that depends on i / , and hence the deformation energy functional sensitivity is given only by the change in the integration area, i.e. . dj „

dG

/>/3

\ V3 n

d r ^ j j j ^ - M T ' ^ J i

1

( 9 1 4 4 )

If we consider, instead, the billet width as the design variable, then we have, for a constant load, t = F/L. Combining Eqs. (9.100) and (9.98) we have

£

»

=

-

7

(9145)

T ( ^ T H

Then the sensitivity reads

di

y

3

(y/ZF

y-

1

F

From Eq. (9.103) we find the derivative of the viscosity

Shape Sensitivity for Steady-State Problems • 305 Unlike the previous (cf. Eq. (9.92)) **L = ~

dL

example,

we obtain

a

non-zero

derivative

for o~

y

(9.148)

L

2

'

K

while the other components remain identically equal to zero. T h e Jacobian now reads J = £- = T

(9149)

So the deformation energy functional sensitivity reads

dG —

dL

da . di , -j^eyjn -\-(T ^JQo v

=

dL

.

v

0

yT

aL

+

dJ_

(T £y—no dL y

(9.150)

which, after replacing the corresponding expressions, simplifies to

£=".*»( v£) i +

(9

-

15I)

It is interesting to note that the first and third term (stress and domain variations) cancel out so that only the term corresponding to strain rate variations remains.

9.4.5

Computational Illustrations

The procedure for shape sensitivity analysis is first illustrated with an extrusion problem in which the horizontal coordinate of the die corner (point P i n Fig. 9.5) is subject to changes, thus causing a variation of the die angle. (The segment PC is, by definition, a straight line.) Next, a plane rolling problem is presented, in which the effect of changing the roller diameter while keeping the reduction in thickness unchanged is shown. 9.4.5.1

Extrusion

In the first example we consider an extrusion process in which the material is forced to pass through the extrusion die by the action of the ram, on which a constant force F acts. T h e layout is the same as presented in Fig. 9.5; the horizontal coordinate x\ of point P is subject to design variations. (An alternative analysis can be performed by considering a constant imposed ram velocity—the applied force has then t o be regarded as a function of the design variable.) T h e velocities obtained appear to be almost insensitive to the design parameter, whereas the pressures depend on it in an almost linear fashion. Figures 9.16 and 9.17 show the sensitivity coefficients of the velocity module and pressure at selected points indicated in Fig. 9.5. This is followed by the contours of the effective strain rate, Fig. 9.18(a), and its sensitivity with respect to the ^-coordinate of point P, Fig. 9.18(b), as well as of the effective stress, Fig. 9.19(a), and its sensitivity, Fig. 9.18(b). T h e results are shown for the primary variables, Figs. 9.16 and 9.17, in

306 • Metal Forming Using the Flow Approach (a) v

'

velocity v versus h j

(b)

sensitivity of velocity versus h

A. DDM B. DDM D. DDM E. DDM F. DDM A. FDM B, FDM D. FDM E. FDM F. FDM

-0 25 -0.2 -0 1 5 -0 1 -0 05

0

0 0 5 0 1 0 1 5 0 2 0 25

-025 -02 -0 15 -0.1 -005

0

0.05

0.1

—*- -+-

-

---x-Q-

-

0 15 0.2

0 25

h h Figure 9.16 Variation of velocity at different points with h: (a) v versus h, (b) dv/dh versus h

sensitivity of pressure versus h

-0.24

-0 16

-008

008

0 16

024

Figure 9.17 Variation of pressure at different points with h: (a) p versus h, (b) dp/dh versus h

terms of the design parameter h, where + h, and x® is the given value of the coordinate Xi of point P at the initial configuration from which other configurations corresponding to different values of the design parameter are obtained. T h e results obtained by the D D M (full lines) are compared with the solution obtained by the central difference-based FDM (dotted lines). It is seen that the results compare very well. The sensitivity contours obtained by both methods show no noticeable difference. Comparing the respective maxima, we have for the strain rate and its sensitivity that they do not coincide in the same place, but the one for the strain rate sensitivity is in correspondence with the maxim u m strain rate gradient, which is concentrated in a small area and farther from the exit with respect to the m a x i m u m strain rate. Similar considerations can be

Shape Sensitivity for Steady-State Problems • 307

(a)

(b) ; |

1

H

;!

u

IV

!' \ \ rD ' \ < \ ' \ \ \

^

\

/

\

0057 0562 1119 1676 2233 2791 3348

•. \

\ \

j

Figure 9.18 (a) Effective strain rate, (b) Effective strain rate sensitivity with respect to the x-coordinate of P

Figure 9.19 (a) Effective stress, (b) Effective stress sensitivity with respect to the x-coordinate of P

Shape Sensitivity for Steady-State Problems •

309

stated for the effective strain rate and its sensitivity. However, it was necessary to reproduce very precisely the mapping of the design elements when the perturbed meshes were considered in the F D M . Small errors in this calculation have in this case a large effect on the results because different rules to generate the mesh following the variations of the design parameters are adopted in different parts of the analysis. 9.4.5.2

Rolling

A slab undergoing a reduction in thickness due to the action of two rollers shown in Fig. 9.20 is considered. The rollers impose a prescribed tangential velocity t o the material in contact with them. The roller radius is taken as the design parameter. Either a constant angular velocity or a constant tangential velocity may be assumed in the design process. In any case we obtain an explicit variation of the velocity in terms of the design parameter, which will be reflected in the expressions for the explicit variation of the strain rate tensor and, therefore, of the viscosity and stress tensor. Thus, in Eq. (9.114) tiv should be written instead of 6 v , and Eqs. ( 9 . 1 1 3 ) and (9.116) read 3

6v = 6v + dv while dvi d i

»

=

2

d x r

k

^(d x r

k

\

dx

dvj

-xfd Xk

r

dx{

+ d Xk

1

r

\

+

2

d(dvi)d Xk

^

r

d Xk r

dxj

d(dvj) d Xk r

d Xk T

dxi

From the algorithmic point of view, the resulting terms can be added when the imposed boundary conditions change in magnitude and/or direction during the

Figure 9.20 Layout of the rolling process

310 • Metal Forming Using the Flow Approach —»*"

Figure 9.21 Equivalent stress and sensitivity design process. Alternatively, the explicit variation of the velocity can be shown to be equivalent to a non-zero 'imposed sensitivity' of the velocity. In other words, a preset value is assigned to it on the surface in contact with the roller, as happens with the velocity in the analysis problem. This imposed sensitivity is calculated from geometrical considerations taking into account the mapping with which the mesh is modified as a function of the design parameter variations. It is adopted in the present case that the roller has a constant tangential velocity (therefore variations in the roller radius imply variations in its angular velocity) and the reduction in thickness (rolling ratio) is also kept constant. Considering the generality required by the FEM analysis programs it is possible to foresee the most c o m m o n situations encountered in practice and to introduce for them the necessary parameters to define the geometry and the variations. Additionally, it may be preferable to define some general functions by a set of parameters and differentiate t h e m in order to get the explicit variations. T h e equivalent stresses and their sensitivities are plotted in Fig. 9.21. The isocurves show no noticeable difference whether calculated by direct differentiation or by the finite difference approach. Figures 9.22-9.24 show some analysed functionals and their sensitivities versus the roller radius. We have in turn considered the overall distortion, the total deformation energy, the average deviation of the effective strain rate defined as where

(9.152)

and the local m a x i m a of the distortion rate, the effective strain rate and deformation energy, and finally the material flow rate. Only some of the plots are shown, since the remaining have a similar pattern to that of the corresponding global or local quantity, or of another measure of the same type (e.g. components of the strain rate). It can be seen that the variation is smooth for the range of the roller radius r considered in the figures, and that better conditions are obtained when a bigger roller is used: less deformation energy, the material is less distorted, etc.

Shape Sensitivity for Steady-State Problems • 311 (a)

deformation energy £, versus r

sensitivity of deformation energy versus r

Figure 9.22 Deformation energy rate and sensitivity The improvement is seen for both global and local quantities. We can explain this fact by noting that the deformation is distributed over an area which grows with the radius. The larger the area, the more homogeneous the process. We can also note that the sensitivity of the deformation energy is calculated as the sum of three terms (see Fig. 9.22). While the total value is negative, the explicit term (i.e. that due to domain variations) is positive since an increment in the roller radius makes the integration area bigger so that the variation of the inverse Jacobian is positive, whereas the product of strain times stress (which are kept constant for this term) is also positive. In the sensitivity plots we also see the results of the central finite differences approach (denoted FDM in the figures). They show some minor disagreements with the DDM solution. These differences are reduced when smaller increments are considered for the numerical derivative, which can be seen near the value of r = 5, where Ar = 0.004 (instead of 0.04) was taken for the finite difference verification. In addition, it is seen that the results calculated by finite differences match very well the analytical ones if they are considered as forward, rather than central differences. In fact, the meshes following each increment in the roller radius are generated in terms of the instantaneous mesh velocity of the present configuration multiplied by the roller radius increment. However, as can be seen from Eqs. (9.136) and (9.137), the mesh variations are a nonlinear function of the radius increment. The meshes are explicitly generated and this causes the differences between the finite difference and direct differentiation solutions. Some final comments can be made about the numerical performance of the shape sensitivity analysis presented in this section. They are valid for both examples. Calculations have been carried out on a Sparc Station-2 (33 MHz). For a problem with 2100 degrees of freedom, as in the first example, each iteration of the analysis problem takes 57 s, of which 8 s correspond to the generation

312 • Metal Forming Using the Flow Approach (a)

(b) maximal effective strain rate versus r

2.19

sensitivity of the maximal effective strain rate versus r •0.102

2.185 2.18

1

2.175 2.17

| 2.165 2.16 2.155 2.15 2.145

4.8

4.85

4.9

4.95

5

5.05

5.1

5.15

0.112

5.2

4.8

4.85

4.9

4.95

5

5.05

5.1

5 15

5.2

Figure 9.23 Maximal effective strain rate and sensitivity (a)

sensitivity of material flow versus r

(b)

material flow Q versus r

2.185

0.015

2.184

2.183

2.182

2.181

2.18 -

2.179

4.8

4.85

4.9

4.95

5

5.05

5.1

5.15

5.2

0.0125 4.8

4.85

4.9

4.95

5

5.05

5.1

5.15

5.2

Figure 9.24 Material flow and sensitivity of the element matrices for the direct iteration scheme. However, when the N e w t o n - R a p h s o n method is applied, this time grows to 14 s. The system of equations is solved by the frontal method [89]. For a selected set of material parameters, which allows a fast convergence at the analysis stage (a = 2 0 0 , 7 0.1, n = 1.5, see Eq. (9.3) and cf. [5, 95]), the whole problem is solved in 718 s, of which only 15.5 correspond to the sensitivity stage (namely, the right-hand-side evaluation and its multiplication by the inverse matrix). The relative low cost of the sensitivity analysis is even more spectacular if materials with a lower strain rate hardening effect are considered. In such cases the number of iterations needed in the analysis problem is greatly increased and the N e w t o n - R a p h s o n =

0

Deep Drawing Problems •

313

m e t h o d frequently does not converge. Therefore, direct i t e r a t i o n s are performed, but still t h e t a n g e n t m a t r i x can be used at t h e sensitivity stage, t h u s requiring an additional solution of t h e s y s t e m of e q u a t i o n s . In very e x t r e m e cases t h e i t e r a t i v e s c h e m e is t h e only possible solution also for t h e sensitivity analysis. A detailed discussion of this aspect is beyond t h e scope of t h e present work.

9.5 9.5.1

D e e p Drawing Problems Formulation

T h e very essence of t h e flow a p p r o a c h u n d e r consideration in this c h a p t e r is t h e formal analogy of t h e rigid-viscoplastic flow e q u a t i o n s with t h o s e of linear elasticity. T h e objective of this section is to use a similar m e t h o d o l o g y for solving t h e deep d r a w i n g p r o b l e m of m e t a l sheets. It should b e e m p h a s i z e d t h a t t h e deep d r a w i n g p r o b l e m s are i n h e r e n t l y t r a n s i e n t and application of t h e flow a p p r o a c h , best suited for s t e a d y - s t a t e s i t u a t i o n s , requires a p p r o p r i a t e r e - i n t e r p r e t a t i o n of t h e e q u a t i o n s a n d a careful numerical t r e a t m e n t , [144] [146]. An a d d i t i o n a l difficulty is t h a t realistic modelling of virtually every sheet m e t a l forming p r o b l e m of technological significance involves complex c o n t a c t / f r i c t i o n p h e n o m e n a . 1

T h e class of p r o b l e m s to b e discussed is limited to a x i s y m m e t r i c d e f o r m a t i o n s . T h e use of an a x i s y m m e t r i c shell t h e o r y for m e t a l forming a p p l i c a t i o n s was first employed in [146]—the a p p r o a c h will be followed below w i t h o u t r e p e a t i n g t h e ' g e o m e t r i c ' details of it for brevity of t h e p r e s e n t a t i o n . Any o t h e r shell theory can b e e m p l o y e d in this context as well. T h e description of t h e c o n t a c t / f r i c t i o n effects follows t h a t given in [186. 187] as it is widely considered most a c c u r a t e a n d effective. Before we s t a r t we should point out t h a t no fully satisfactory sensitivity form u l a t i o n exists in t h e l i t e r a t u r e for p r o b l e m s with unilateral c o n s t r a i n t s . T h e inherent feature in such p r o b l e m s is non-differentiability of t h e response at some p o i n t s along t h e deformation p a t h — a m o n g very scarce a t t e m p t s to deal with this class of p r o b l e m s we m a y m e n t i o n t h e theoretical discussion given in [20] a n d t h e p r e s e n t a t i o n of a p p r o x i m a t e a p p r o a c h e s t o g e t h e r w i t h s o m e a l g o r i t h m i c issues and n u m e r i c a l results in [17, 30, 112, 165]. We shall base our analysis on t h e following simplifying a s s u m p t i o n s : (i) 2D p r o b l e m s are considered only, (ii) one of t h e c o n t a c t i n g bodies is rigid resulting in unilateral c o n s t r a i n t s imposed on t h e m o t i o n of t h e m e t a l sheet considered, (iii) t h e b o u n d a r y of t h e rigid b o d y is a p p r o x i m a t e d by a collection of straight linear s e g m e n t s , and (iv) a linear C o u l o m b friction law is a s s u m e d . Let us consider a slave n o d e \ s ' of t h e deforming b o d y t h a t is likely to penet r a t e a m a s t e r segment of t h e rigid b o d y defined by t h e nodes ' m i ' , 'm-l, Fig. 9.25. 'This section (including all figures) is to a large extent based on the authors' article [112] published in Computational Mechanics by Springer-Verlag. The permission of the Publisher to reproduce the material is acknowledged.

314 • Metal Forming Using the Flow Approach

9y

s

',2

+•

Figure 9.25 Geometry of the slave nodes with a master segment m\-m2 T h e so-called gap function is defined at the point V as g

= (x, -

a

x

) n

(9.153)

T

m i

where n denotes the unit normal to the master segment, x is the current position of the slave node ' 5 ' and x is the (constant) position of the master node 'mi'. T h e inequality s

m i

(9.154)

9s > 0

has to be checked for all candidate contact nodes V from the finite set 5", 5 6 S. In general, S will contain all the boundary nodes of the deforming body. For g < 0 the constraint equation for the node ' 5 ' becomes active (s G SU); otherwise, the constraint is inactive (s € 57). The length of the master segment is defined as a

^—

I|^m2

(9.155)

"mi

while the unit vector tangent to the segment is defined as (9.156) We are looking for a solution to the problem at a fixed time instant t -\- At assuming that the solution up to the preceding time instant t inclusive has already been obtained. T h e just completed iteration number at the time step considered is denoted by to, as before. T h e contact constraint condition (9.154) at time t + At and iteration 10 reads T A +

V / )

=

(

.

+

A

. ( ,_ x

r

X

m

i

jr > n

(9.157)

0

which at the ( a ; + l ) t h iteration becomes

n

t+At< ) u

w ( + l)

+ 19.

(9.158)

Deep Drawing Problems • 315 where t h e index H + AV has been (and will be below) dropped in all symbols denoting variations t o simplify notation. To effectively use Eq. (9.158) in the framework of the flow approach the change in the gap function has to be expressed in terms of the nodal velocities by using a t i m e integration scheme, for instance t+At (w x

+ 1 )

t+At^.)

=

0A* q

+

"

(

7

s

+ 1 )

,

0 < 6 < 1

(9.159)

i.e. ju[

= 0At~fq["

u+1)

(9.160)

+1)

where 0 is a parameter of the integration scheme while

is t h e 'next' iterative correction t o the velocity * q of the node V . T h e expression (9.161) is a straightforward consequence of the relationship < + A

s

t - r A ^ a )

=

+

%

A

t

t^

0

+

A

t

t At^

(

+

+ 1 )

= * x + (1 - 9)At
s

_ ^ < + A t q

^

}

+ 1

)

(9.162)

The so-called perturbed Lagrangian function is adopted in the discretized form for one 'e'th finite element with 7V degrees of freedom and just one node V in contact as e

n (

< + A T

£

( W + 1 ) Q

/

+ A T

A

( W + 1 )

= n('+*«q<»+*>) +

)

< + A

'A ; (

+ 1 )

t

+

A t

^+ ) - 1 (A ; 1

(

+ 1 )

)

(9.163)

2

in which e is a parameter, i t has a simple interpretation in terms of the element potential energy associated with the velocity vector + q ( + ) t

pj^+Atq(*-n)^ _

I

t+At

? (w+i) t + A t

jgs)(«)

A t

<+A
w

1

t + A t q(u>) t + A t ( o , + i ) q

(9.164) and A is the contact force normal component at node V (due to the assumption of frictionless contact only t h e normal component of t h e force appears in t h e formulation). T h e velocity vector q is assumed t o be related t o the vector of the element nodal velocities by means of the Boolean transformation s

s

^2x1

— ^s\xN

e

^ ' X '

(9.165)

The velocity at t h e node V has to satisfy the constraint t + A i ^ + i )

=

y + T0

g

n

A<7q

(" D +

>

0

( ) 9166

316 • Metal Forming Using the Flow Approach Stationary conditions for the functional

n

e

dq

(9.167)

OA

yield for any admissible 8 q , 6X the equations (

" 0q,

V

+ «+^A<-+ >-

^ — Uq = 0

1

6>q

a

(9.168)

/ (9.169)

which read more explicitly

(9.170) t+AttfD^t+At^-H) _

t+At^

+ 1

) _

I

<+At^

(t+At M A

+

7A

M

1

+

(9.171)

7

A

(« i) J

By linearizing the equations

+ A<;+ >)

(l+At («)

+

o

=

(9.172)

at ( * + A t q ( « + i M + A t A

(

w+1) s

)

and noting

that,

cf. Eqs. (9.29) and (9.161) f+AtT^X)(u>) t + A t • (u,+i)

t+Atp|(«)

_

t + A j j ^ r ) ( « ; + l ) ^ ( « + l) _

_

t+AtTWT)(u,)

-(w+i) _

t + A t Q ( w ) _ | _ t + A t jgT)(u,) t + At q(<-)

(9.173)

t+Atp(u,)

we obtain t+At

n A[ 0At T

-\

e)

7A ; (

R (

.)

-t+A^'A^ntfA*

+ 1 )

I t+A< («) _ A

B y solving the second equation for 7 A 7

A

(

W + 1 ) 3

= e (0<"> + n A ^ q " T

(

7

+ 1 )

(

t + A t (w)

(9.174)

U,+1) 5

n0Ar.) -

t + A t

A ; (

)

(9.175)

and substituting it in the first equation, we obtain t+Atj^DM

X ( ; ) n n A ^ ( ^ ) ] iq T

+

e

2

(w+1)

= ^R -tA^n'+^O t+

M

At (9.176)

Deep Drawing Problems • 317 or [< ftK< ><"> + +

K . ] q " '* =

T

<

( e )

c

+

, +

7

ftR

M

+ f4R :» ,+

(9.177)

(

c

T h e quantities T

()

n. . n

e

\ W

T

fflA/V

(9.178) = - AW n„..n>)0A/ » Af xl (e)RflAi are the contributions to the element tangent stiffness matrix and tangent residual force vector due t o contact. T h e expressions (9.178) have been derived relative to a single slave node ' 5 ' coming into contact with a master segment. T h e matrix K is symmetric. In forming the global stiffness matrix and the global residual force vector all the contributions due to contact have to be accounted for in the course of the usual assembly process. Observing the global ordering of nodes and interelement connection we may symbolically write f

c

e

C A

(9

k =X;V c

-

179)

s£S

A

while t h e global system of F E M equations becomes [K(r)(«)

+

K

(«)]

7 q

(. i) +

=

R

(

«)

+

R

(w)

(

9

1

8

0

)

If in t h e iteration scheme the secant K ' rather than the tangent stiffness matrix K is employed in Eq. (9.180), then only a linear convergence can be expected. On the other hand, the consistent way in which K and R have been obtained ensures quadratic convergence, provided the iteration is based on Eq. (9.180). So far, only frictionless contact effects have been discussed. Limiting ourselves to t h e linear Coulomb friction law we postulate the tangent force residual in the form, cf. Eq. ( 9 . 1 7 8 ) [ S ) ( A

)

: t ) ( u , )

c

c

2

* f.)ft/.*.x. +

=

Ail t'

-»*

x2

2xl

»"g

a

0 At

(9.181)

in which t = tsign(q.)= ij^jj-

(9.182)

,

taking t h e variation of Sq Kf T

Wg*!«q =

with respect to + q ( " + ) gives <

t

-,<0 AtSifly

(g*

+

At

1

t < ^ ) 6q

(9.183)

318 • Metal Forming Using the Flow Approach B y observing the relation, cf. [186]

Sq

| | q f l V dq

q

=

dq

jj^(I-nn -tf)A<;>«q

(9.184)

T

which vanishes for 2D problems, and the relation ^-6q dq

= 6Atn 6q T

= 6Atn A^6q

(9.185)

T

s

we obtain from Eq. (9.183) »£3ffi*2«q = «tfK,.«q dq

(9.186)

K,

(9.187)

where = -ue(6At) A^t'n A^ 2

f

T

is the unsymmetric contribution to the element stiffness matrix due to friction at the node ' 5 ' . As before in Eqs. (9.179)i, the global quantities due to friction result as

JL*

se

(9.188)

s£S

A

while the final system of F E M equations becomes (K<

T)

+ K + K/)

( w >

c

7

q

= ( R + Rc + R / )

( w + 1 )

(9.189)

W

This system of equations, after appropriate specification to the shell problem at hand, allows us to iteratively determine the unknown velocity field + q ( ^ + ) . If^ as observed before, the secant stiffness matrix K " rather than the tangent one K is to be employed, then Eq. (9.189) is used in the form t

cs)(l

At

1

)

CT)(u,)

(tf

s )

+ K + K ) f

c

( u ) 7

q

( w + 1 )

in which the unsymmetric cf. Eqs. (9.22i) and (9.33) Kf"> = / 2 B B q Jn T

(

^dq^

w

)

= (R + R + R / ) ^

(9.190)

c

contribution

dfl

has been neglected in the iteration matrix.

to the tangent

stiffness

matrix,

(9.191) '

Deep Drawing Problems •

319

Figure 9.26 Coulomb friction law: (a) classical, (b) regularized To improve the convergence properties of the iteration the neglected term can be recovered by replacing Eq. (9.190) by (tf

s )

+ K

+ K ) f

c

q " > = (R + R (

M 7

+ R )

+ 1

f

c

M

- Kf-^q^

(9.192)

If only the symmetric equation solver is available, then Eq. (9.190) can be further modified to read + K ) C

M

q " (

7

= (R + R

+ 1 )

c

+

Rf f

]

- (K +

KJ f

u) 7

q

w

(9.193)

A simple procedure to deal with friction can be based on the iterative adjustment of nodal reactions in contact nodes at the blank-tool interfaces until they satisfy a Coulomb type of friction law, cf. [132], for instance. Convergence and other problems appear when relatively small strain velocity increments with different signs for neighbouring nodal points are observed. As can be seen from Fig. 9.26, in which the classical and a regularized Coulomb rule are presented, the Coulomb law may give very different friction forces for nodes sliding over rigid surfaces with almost the same velocity. Even for nodes with a 'proper' sliding direction, different friction tractions result from different slip values and 'artificial wrinkling' is produced, [112]. T h e Coulomb friction law is usually expressed as °iip_/

A l l

0

-\-v\\t \\ t

where A u scalar, f

n

s l i p

i f

if

ll «ll
||ft|| =

(9.194)

Hlf»ll

denotes relative sliding between two contact points, v is a positive

is the normal contact force and f< is the tangent contact force which may

320 • Metal Forming Using the Flow Approach be treated as a reaction t o sliding. Condition (9.194)i corresponds to sticking (sliding impossible) while condition (9.194) corresponds to possible sliding. Additional regularization of the Coulomb law is necessary to avoid 'artificial wrinkling'. Algorithmic issues relevant to the computer code used in this study were thoroughly explained in [164]. In particular, the slip condition for those neighbouring nodal points at which small strain velocity increments with different signs are observed is replaced by the sticking condition. 2

9.5.2

Parameter Sensitivity Analysis

As we know by now very well, assessment of the response first-order sensitivity to parameter variations means finding the response gradient with respect to the parameter. However, for complex problems involving inequality constraints (such as those due to contact and friction) the sensitivity information carried by the gradient is limited. In fact, this class of problems is inherently nonlinear and non-differentiable, and a more sophisticated mathematics leading to Gateaux differentials (directional derivatives) is needed to properly address the sensitivity problem. Here, just one parameter (the coefficient of friction) is considered so that the sensitivity results based on the gradient information are believed to be useful provided they are interpreted with sufficient care, as explained later in this section. With h as any parameter entering the theory we consider a function, whose sensitivity with respect to the parameter is to be assessed, in the form

G =

G{
(9.195)

D

where the dependence of G on its arguments (but not of the stress deviator a° and velocity q ) on h assumed known. In the present discretized 'non-shape' sensitivity considerations h m a y be taken as one of the following parameters: n, 7, cr , v or e at some (possibly all) integration points in the region analysed. For brevity, we confine ourselves to deriving sensitivity equations for just one parameter h—a more general case of sensitivity defined as the directional derivative with respect to a vector of the parameters directly follows the pattern set by the simple discussion. 0

Employing the direct differentiation method we differentiate Eq. (9.195) with respect to h to get the response sensitivity as dG

dG

dG

d
dh

dh

da

dh

D

D

dG dq

(9.196)

dq dh

in which dG/dh, dG/d
D

da dh D

dd
dq

dq

dh

D

= 2//*B

dq

dh

D

(9.197)

Deep Drawing Problems • 321 From Eqs. (9.196) and (9.197) we have that only dq/dh needs to be obtained from additional considerations in order to determine dG/dh. To this end we consider first the non-contact problem, i.e. K \f**(q,h))q(h)

= Q(h)

s

(9.198)

This formula can be extended by allowing the right-hand-side vectors to depend additionally on q(h) which has not been done here in the interest of brevity only. Differentiating Eq. (9.198) with respect to h gives dKS,

dh

q+K

,isdh 49dh

.

=

(9 199)

Since dK*>

dh

dK^d^

=

du* dh

dX^_ (d(S

du* \ dh

d^dq\

dq dh J

[

'

'

Eq. (9.199) becomes

R

,

dK<

+

5)

^d/i* ^ dq )M

dh

3

-

dh

^dfi* qdh

(9.201)

or

dh

dh

d/i* dh

B y using the definition of the residual given in Eq. (9.29) we obtain dq

dR

dh

dh

(9.203)

in which the right-hand side vector can be computed provided the primal (equilibrium) problem has been solved; the velocity sensitivity vector dq/dh then follows by solving Eq. (9.203) which is linear and does not require iteration. As emphasized many times before, the latter property has fundamental significance in terms of computational efficiency. T h e right-hand side vector in Eq. (9.203) can be presented more explicitly by noting that

dK = / 2 B B dft du* Jn s)

T

(9.204)

T h e DDM-based sensitivity equation (9.203) can be extended to account for the contact and friction effects by observing Eq. (9.189). Before doing this we have to point out again that an obvious difficulty in effective computation of the sensitivity gradient in such a case is that problems with unilateral constraints are generally Frechet non-differentiable and only directional (Gateaux) derivatives exist. This apparently invalidates the standard sensitivity technique based upon

322 • Metal Forming Using the Flow Approach the design differentiation of the primal equations. However, as is clear from the theoretical discussion of [20] the variables in the formulation are not differentiable with respect to the parameter only if some active constraints are associated with zero reaction forces—a situation which can hardly be expected in finite element analysis of large-scale problems. Thus, we effectively assume differentiability of the response—even if it is theoretically not quite satisfactory, the results are believed to be correct within general finite element accuracy. B y analogy to Eq. (9.203) the sensitivity equation reads now ( ^

+

K

e

_dq _ _d_ K )-=S = - ^ ( R dA dA

+

/

(9.205)

R. + R )

+

/

which can be effectively used to compute the sensitivity of any constraint via Eq. (9.196). In the above equation the following notation is used, cf. Eqs. ( 9 . 1 7 8 ) 2 , (9.179)i, (9.181) and (9.188)r.

R =

-tA ng 0At

E

c

{ e) s

s

(9.206)

s E S a T

R/ = £

-^A

(

/ e )

3

x

t'<7 0Ar. s

ses

A

In the context of sensitivity analysis the question arises of whether the actual tangent stiffness matrix K or the stiffness matrix JC \ Eq. (9.22), should be used to find d q / d A by Eq. (9.203). Using Eq. (9.205) has the great advantage of resulting in the non-iterative solution for dq/dA. However, as indicated in Section 9.5.1, K may quite often be unavailable because of the equilibrium iteration algorithm based on K (K in practical applications is frequently almost singular). To overcome this, Eq. (9.205) should then be rewritten as, cf. Eq. (9.190) T )

S

T )

s )

(Kf

s)

+ K

c

( T )

+ K/) I S = — (R + R dA dA

-K4T

+ R/)

c

(9-207)

for which the direct iteration scheme reads V dA /

dA

Q*Q(M We thus obtain the sensitivity equation which is an alternative to Eq. (9.205). It removes some drawbacks of Eq. (9.205) at the cost of having to iterate to find dq/dA. B y specifying the parameter A we can readily derive explicit expressions for the right-hand side of Eq. (9.205) or Eq. (9.206). Observing that, cf. Eq. (9.162) t+At

d c dh t+At

d

t+At

gs

d

Xs

t+Ae

=

n

^

X

'

+

.

(

/,

1

_ ,

m A ,

)

A

(

Q*Q(fc)

.(e)_ 5 q] W d <

A

( 9

. o ) 2

9

Deep Drawing Problems • 323 we obtain the respective non-zero terms as follows: for h = u dR

- E- M N(N F-^ + E

r

du d R / »+ q*qM

E)

T

„r^.

At

l>

~ E

du

t

(l-^A(;^L}^^

du

du

1)

£A t'g 9At { e) s

s

(9.210) d

t

X

s

du

ses

+(l-0)AtA^

d

t

* du]

A

9 At

for h = e dR, t + A t

E

q*q(0

A ^ n ^ A * — de

dR,

+ ( 1 - 0 A*A , (

v

el

M

de

6>A/

(9.2ir

= E -^/V^a*

- G ^ { „ ^

+

( i - , ) A A « ; . ^ ] } 0At (

for h = 7 dR

d7 for h = n

7JQ 2B Bq-4-(^)" Sney 7 ' T

q^qh)

_1

dftlq J

v

dR

/«

dn q^q(n)

2BrB

i©" (J) ln

(9.212)

(9.213)

dn

The assumption made above that each of the parameters is the same at every integration point or node can obviously be removed so that sensitivity can easily be analysed with respect to a parameter defined at just one point in the system, for instance. We also note that in accordance with the assumption of Section 9.5.1 the vectors n and t' are considered independent of h. Having solved Eq. (9.205) or Eq. (9.208) for d d x / d / i follows according to

t + A l

q / d / * the updating of s

s

d*+ : A(

dh

dx f

s

dh

+ At

(1-9)

d'q, dh

t+At,

+

dh

(9.214)

Both d q /dh and d x /dh have to be stored for the next step computations. Owing to technological reasons an important response characteristic (of which the sensitivity assessment appears definitely useful) is the blank thickness. The t+At

t+At

s

s

324 • Metal Forming Using the Flow Approach importance of such information originates in the broadly accepted conviction that the changing thickness distribution in the blank is a major factor in evaluating the quality of the forming process at hand. The way to evaluate the thickness sensitivity with respect to the friction coefficient is described below. T h e thickness of the blank is calculated using the incompressibility condition. For an incompressible material we assume e

= -(e

3

1 + £ 2

)

(9.215)

T h e thickness of the blank for element V (the index is later dropped for brevity) is calculated as a function of the initial thickness °6 and strain component in the thickness direction calculated at time t -{- At rt+At •t+At

k }y

/ Jo I

— dt = I b Jo

rt+At

i

3

dt

(9.216)

with U+At

«+ V n b \, e. = = I\nb t + A At

t+At

L

= \n—-^

+ A t

3

lo

(9.217)

Ob

or t + A t

b = °bexp( e )

(9.218)

t+At

3

T h e strain vector at time t -f At can be obtained by using the update rule in the form <+*< =

« .+

£t

t + A t

£|

i

t

At = tei + Bia'+^qaAt,

i = l,2,3

(9.219)

Using Eqs. (9.215) and (9.219), Eq. (9.218) becomes <+ b = ° 6 e x p { - [*£i + At

(B q )

At + e

t+At

la

l

Q

2

+

(B q )

At] }

t+At

2o

a

(9.220)

Thickness sensitivity with respect to friction can be obtained under the assumption that the sensitivity of the velocity field with respect to friction is known. We obtain d-<6 dv

=

0

r

^

L exp { -

9.5.3

+

d

^

A

du

J

+

^

du + (B q )At t+At

la

a

B

+

J _ ^

dv

dv

+ e + (B q )At] l

t+At

2

M

2a

Q

}

(9.221)

Computational Illustrations

T h e stretching of a thin, circular, isotropic sheet with a hemispherical punch is considered. T h e geometrical configuration of the problem, deformed sheet shape at a punch vertical displacement of 1.17 in. and its node numbers are shown in Fig. 9.27. Limits of the no-contact area are marked in this figure as well. T h e blank has an initial overall radius of 2.2 in. T h e coefficient of friction is v = 0.04. Fifty uniformly distributed linear axisymmetric shell elements are used for the

Deep Drawing Problems •

325

analysis, cf. [144]. The finite element program M F P 2 D described in [144] and extended as discussed in [164] is employed. T h e uniaxial stress-strain curve for the matrix material is given by a = 5.4 + 27.8 £

0 5 0 4

T/in ,

for

e < 0.36

a = 5.4 + 24.4 e

0 5 0 4

T/in ,

for

e > 0.36

2

2

[

'

In Fig. 9.28(c) the sensitivity of the horizontal nodal velocities with respect to the friction coefficient calculated by the D D M for a punch vertical displacement of qi = 1.17 in. is presented. Sensitivity values corresponding to the sheet node numbers given in the previous figure are shown. One can observe that the m a x i m u m sensitivity of q^. with respect to friction corresponds to the nodal point number 7 being in contact with the punch surface. Also a negative sensitivity area in the no-contact zone can be observed. Increasing the value of the friction coefficient results in decreasing the horizontal velocity in this area as compensation for the opposite effect occurring over the punch surface. A remarkable fact is that no finite difference sensitivity solution (out of many a t t e m p t e d ) has resulted in a reliable approximation of the sensitivity values. Small parameter perturbations have resulted in excessive round-off errors, as the velocity fields for basic and disturbed solutions, respectively, differ very little, while larger perturbations have induced unacceptable truncation errors. Thus, once again, and as in the sensitivity analysis of linear systems, [64], the so-called analytical approach to sensitivity appears not only more cost-effective but the only reliable way of assessing sensitivity. y

In Fig. 9.28 the following additional information is presented: (a) deformed shape of the shell, (b) sensitivity of the nodal vertical velocities with respect to the friction coefficient, (d) sensitivity of the nodal rotational velocities with respect to the friction coefficient, all values taken at q\ — 1.17 in. y

Figure 9.27 Hemispherical punch stretching problem: geometry of the tools, deformed shape and nodal points numbers

326 • Metal Forming Using the Flow Approach MODEL SBN381

MODEL: SEN381I

W2: M u t t i v i t y

STHft 11TMB: 1.17 VARIATION ALONG A LINE M

42

4 8 SI

1.5 MODEL SBN381 WlMMhMty STEP: 1 TIME: 1.17 VARIATION ALONG A I J N B

©

-

@

2.0

Distance

MODEL: SEN3S1 W2:««kMfy STEP: 1 TIME: 1.17 VARIATION ALONG A LINE

-points

of sensitivity graph correspondng to node numbers

1.5

2.0

Distance

(from symmetry axis)

Figure 9.28 Hemispherical punch stretching DDM-based sensitivity: (a) deformed shape of the shell, (b) sensitivity of the nodal vertical velocities with respect to the friction coefficient, (c) sensitivity of the nodal horizontal velocities with respect to the friction coefficient, (d) sensitivity of the nodal rotational velocities with respect to the friction coefficient; all values taken at q\ = 1.17 in. y

Sensitivity of the horizontal nodal velocities with respect to the friction coefficient calculated by the D D M for three increasing punch vertical displacements q is presented in Fig. 9.29(a). As expected, the results indicate that the response sensitivity decreases as the process develops. The greatest sensitivity is observed at the beginning of the process. T h e sensitivity solution has been obtained using t h e tangent rather than secant stiffness matrix as described in Section 9.5.2. T h e solution, generally less stable than that obtained with the secant stiffness matrix, has not converged at early stages of the process so that the latter had to be employed initially. iy

A useful way of presenting the results is by re-defining the sensitivity variable to a non-dimensional form as (v / q )dq / dv, Fig. 9.29(b). Such values provide interesting information indicating the relative function (i.e. the appropriate velocity component) change due to the relative change of the system parameter. T h e thickness sensitivity was obtained according to the formulae given in Section 9.5.1. T h e evolution of friction only due to the changing area of contact x

x

Deep Drawing Problems •

327

Figure 9.29 Hemispherical punch stretching: DDM-based sensitivity of the horizontal velocity component versus friction; (a) sensitivity for different punch vertical displacements, (b) re-defined sensitivity values

was accounted for. (Any friction evolution due to friction parameter changes can be easily included as well.) Sensitivity values corresponding to the sheet node numbers given in the previous figure are shown for four different punch vertical displacements q in Fig. 9.30. It can be observed that the m a x i m u m sensitivity of thickness with respect t o friction corresponds to the points being in contact with the punch or die surface, depending on punch vertical displacement. Also, a negative sensitivity area in the no-contact zone can be observed. The results for the advanced process (punch vertical displacements of 0.927 and 1.227 in.,

328 • Metal Forming Using the Flow Approach

thickness sensitivity - secant matrix

..vor. I — i (1

,1

' 10

•——• 1,1 20

" 2.1

* — i 3(1 .1.1

> -1(1

1 -11

1

.10

Node number

Figure 9.30 Hemispherical punch stretching. DDM-based sensitivity of the thickness of the blank versus friction for different punch vertical displacements respectively) indicate that the response sensitivity decreases with the process development. At the very beginning of the process (punch vertical displacement q = 0.327 in.), very low sensitivity of thickness with respect to friction is observed, which is due to the prevailing sticking condition on the rigid tools (punch and die) surfaces. All solutions have been obtained using the secant stiffness matrix and have converged very well at all stages.

9.6

Summary of the Chapter

In this chapter sensitivity evaluation techniques were employed to study the different metal forming processes in the framework of the flow approach. First, the constitutive parameters were taken as the design variables. The methodology developed turned out generally reliable and effective. Problems involving the friction model described in Section 9.2.2 proved to be very sensitive to changes in the friction coefficient; serious difficulties in finding the size of the parameter perturbations for calculating the sensitivity gradients by the finite difference approach were encountered. On the contrary, the techniques yielding the sensitivity information as a by-product of the equilibrium analysis were demonstrated to work well even for this challenging class of problems. An iterative form of the sensitivity formulation based on the secant stiffness matrix was also presented. An example was shown in which the tangent matrix could not be employed at all, implying the importance of the iterative scheme. Similar situations will typically arise in solving problems for which at the same time (i) the material model exhibits no or very little hardening and

Summary of the Chapter • 329 (ii) no non-zero boundary velocities are prescribed. Techniques developed for shape sensitivity analysis of metal forming processes offer an effective way to generate valuable information about the influence of the tool shape modification on different forming process characteristics. Tool shape optimization is thus an immediate and natural application of the results. As before, serious drawbacks of using the finite difference method for sensitivity assessment were demonstrated. On the other hand, the direct differentiation m e t h o d combined with the domain parameterization approach made it possible to obtain solution to the problem at hand in an elegant and straightforward way. Towards the end of the chapter a very complex problem of non-stationary sheet metal forming with contact and friction was considered. The study had a preliminary nature and was aimed at developing a practical approach to sensitivity evaluation for such problems merely to get more theoretical and computational insight necessary before more solid formulations will emerge. These, together with effective sensitivity techniques for other non-differentiable problems for which only directional derivatives exist, definitely deserve special interest in the near future.

Chapter 10 Nonlinear Thermal Systems 10.1

Introductory Remarks

Having efficient c o m p u t a t i o n a l m e a n s for t h e realistic assessment of nonlinear h e a t transfer p r o p e r t i e s in bodies subject t o t h e r m a l loadings is crucial for solving very m a n y engineering p r o b l e m s . Sophisticated F E M - b a s e d a l g o r i t h m s have been developed for this p u r p o s e , cf. [19, 110], for instance. A n a t u r a l e x t e n s i o n of t h e analysis capabilities are d e v e l o p m e n t s in t h e area of p a r a m e t e r sensitivity for t h e r m a l p r o b l e m s . T h e sensitivity information so o b t a i n e d m a y be used to assess t h e effect of u n c e r t a i n t i e s in t h e t h e r m a l s y s t e m m a t h e m a t i c a l m o d e l , to p r e d i c t t h e response changes d u e to a change in model p a r a m e t e r s or to o p t i m i z e t h e s y s t e m by using a p p r o p r i a t e o p t i m i z a t i o n t e c h n i q u e s , to n a m e j u s t a few sensitivity application areas. U n f o r t u n a t e l y , even if in t h e last decades design sensitivity analysis has att r a c t e d considerable a t t e n t i o n in t h e field of solid and s t r u c t u r a l m e c h a n i c s (which we have so frequently e m p h a s i z e d t h r o u g h o u t t h e b o o k ) , t h e work published on sensitivity for nonlinear t h e r m a l s y s t e m s has been r a t h e r scarce. T h e subject of t h e design sensitivity analysis in t h e specific context of t h e r m a l s y s t e m s was a p p a r e n t l y first u n d e r t a k e n in [60]. T h e s t e a d y - s t a t e and transient linear a n d nonlinear p r o b l e m s were discussed t h e r e in t h e discrete F E M formalism, and a I D illustration was provided. T h e two basic methodologies for solving sensitivity p r o b l e m s , later called t h e direct differentiation m e t h o d ( D D M ) and t h e adjoint s y s t e m m e t h o d ( A S M ) , were identified in t h a t p a p e r . T h i s enlightening c o n t r i b u t i o n was followed by o t h e r articles, [38, 39, 129, 149]. In t h e t w o - p a r t p a p e r [38, 39] t h e a u t h o r put forward t h e basic c o n t i n u u m formulation for DSA of linear isotropic t h e r m a l s y s t e m s . Almost every aspect relevant to DSA for such a class of p r o b l e m s was included in t h e formulation: n o n - s h a p e a n d s h a p e sensitivity t e c h n i q u e s , t h e D D M and t h e A S M , s t e a d y - s t a t e and t r a n sient p r o b l e m s , first- and second-order sensitivities, etc. T h e s h a p e sensitivity a p p r o a c h was based on t h e so-called m a t e r i a l derivative concept while t h e ASM took a d v a n t a g e of t h e Lagrange multiplier technique resulting in a t e r m i n a l - v a l u e

332 • Nonlinear Thermal Systems p r o b l e m for t h e adjoint t e m p e r a t u r e field. A t i m e m a p p i n g was i n t r o d u c e d to t r a n s f o r m t h e t e r m i n a l - v a l u e p r o b l e m into an initial-value p r o b l e m . In t h e papers [129, 149] a formalism was developed for solving t h e r m a l sensitivity p r o b l e m s by t h e b o u n d a r y e l e m e n t m e t h o d . In [149] a nonlinear s t e a d y - s t a t e p r o b l e m was considered a n d e q u a t i o n s describing s h a p e design sensitivity and s h a p e o p t i m i z a tion were given using t h e m a t e r i a l derivative concept within t h e adjoint s y s t e m methodology. F i r s t - o r d e r design sensitivity for transient conduction p r o b l e m s was analysed in [169] by t h e adjoint m e t h o d . T h e a p p r o a c h was based on t h e m u t u a l energy principles with a t e r m i n a l - v a l u e p r o b l e m resulting for t h e adjoint sensitivity equation. The so-called d o m a i n p a r a m e t e r i z a t i o n m e t h o d was used for t h e analysis of s h a p e sensitivities. Both c o n t i n u u m and discretized formalisms were p r e s e n t e d . S h a p e and n o n - s h a p e sensitivity formulations for t r a n s i e n t nonlinear t h e r m a l p r o b l e m s were subject to scrutiny in [171]. T h e ASM involving t h e Lagrange multiplier t e c h n i q u e was used with convolution replacing m u l t i p l i c a t i o n as t h e o p e r a t o r b e t w e e n t h e multipliers and t h e governing e q u a t i o n s . T h i s resulted in an initial-value adjoint p r o b l e m with no need for additional t i m e m a p p i n g s . B o t h c o n t i n u u m a n d F E M formulations were employed. T h e use of t h e so-called Kirchhoff t r a n s f o r m a t i o n for s h a p e sensitivity analysis in nonlinear heat conduction p r o b l e m s was proposed in [163]. In some m o r e recent l i t e r a t u r e [71, 170] t h e t h e r m a l sensitivity analysis is considered as a p a r t of m o r e general t h e r m o - e l a s t i c formulations; a review of t h e work is beyond t h e scope of t h e book, t h o u g h . T h e objective of this c h a p t e r is a unifying presentation of t h e c o n t i n u u m a n d discretized, D D M - and ASM-based sensitivity formulations valid for fully nonlinear s t e a d y - s t a t e and transient heat conduction problems with anisotropic m a t e r i a l p r o p e r t i e s and various b o u n d a r y conditions.

10.2

Continuum-Based Formulation for Transient Problems

10.2.1

On Solving Nonlinear Heat Transfer Problems

T h e design of t h e r m a l processing s y s t e m s requires selecting of a design p a r a m e ter h. Such p a r a m e t e r s describe t h e m a t e r i a l properties or b o u n d a r y conditions a n d can also represent t h e g e o m e t r y (i.e. shape) of t h e d o m a i n ii in which t h e p r o b l e m is defined. Only non-shape design p r o b l e m s are considered in this section, t h e m o r e complex s h a p e sensitivity formulation being t r e a t e d in Sections 10.4 a n d 10.5. As we already know q u i t e well, t h e n a t u r e of t h e design d e p e n d e n c e of particular functions is in t h e sensitivity analysis of crucial significance. Therefore, t h e design h will be explicitly indicated as t h e function a r g u m e n t below when formulating t h e heat transfer i n i t i a l - b o u n d a r y value p r o b l e m .

Continuum-Based Formulation for Transient Problems •

333

T h e governing e q u a t i o n s of nonlinear t r a n s i e n t heat transfer in a t h e r m a l l y a n i s o t r o p i c 3D d o m a i n ft can be w r i t t e n in differential form as follows — 1,2,3): dO + Q = c— , or

(kjOjli

X i

€ ft; r e [0,/]; h G R

(10.1)

with t h e b o u n d a r y conditions imposed on t h e b o u n d a r y surface t e m p e r a t u r e 0 = 0,

Xi € dQ ;

r £ [0,t]; h G R

9

(10.2)

a n d t h e b o u n d a r y surface heat flux - kij rij 0,- = q ,

Xie

dft ; q

T e [0, /]; h € R

(10.3)

T h e initial condition imposed on t h e initial t e m p e r a t u r e d i s t r i b u t i o n has t h e form 0 = 0°,

G 0 ; r = 0; h e R

(10,1)

T h e following n o t a t i o n is used above: 0 = 0(X ,T; h) — body temperature Xk, T; h), x ; h) — thermal conductivity tensor Q = Q(0(x{, T; h), 0 i(x{, r ; h), x , r ; h) — rate of heat generated per unit 1

k

t

t

volume c = c(0(xi, T; h), Xi; h) = pc — material heat capacity p = p (6(xi,r; h), X{; h) — material density c = c ( # ( x , , r ; h), Xi; h) — material specific heat 6 = 0(xi, T; h) — known temperature on the boundary surface dft 0

q = q(0(x , t

T; h), x , r; h) — t

heat flux input on the complementary boundary surface

dft

q

0° = 0°(x ,0;h) rii = rii(xj) l

— initial temperature — unit outward-drawn vector normal to the boundary surface dft x — position vector which identifies the material particle in the domain Q T — time variable t

T h e n o t a t i o n (•),, s t a n d s for t h e p a r t i a l derivative with respect to t h e coordin a t e Xi, i.e. d(-)/dx . t

T h e Fourier

constitutive

relation

reads

m = -knOj, q being t h e heat flux vector. t

E q u a t i o n (10.3) m a y be specified to include:

(io.5)

334 • Nonlinear Thermal Systems (a) convection b o u n d a r y conditions on a p a r t 8Q

£ 8W

q

- k

nj 0,,- = ( {6

tJ

- 0^)

{c)

,

{c)

x £ dni ; t

c)

r £ [0, T)

where £( ) is t h e (possibly t e m p e r a t u r e - d e p e n d e n t ) convection and c

(b) r a d i a t i o n b o u n d a r y conditions on a p a r t 8iV 3

= £

ti

(V - 0 ) = 4

(r)

( r)

in which t h e coefficient £ Z r =
(r)

+

)

x

coefficient,

£ <9Q

9

u)

- kij n 0

(10.6)

(0 - 0 )

(10.7)

(r)

is c o m p u t e d as 1

(10.8)

and

where a is t h e Stefan B o l t z m a n n c o n s t a n t , 0

{l)

is t h e t e m p e r a t u r e of a

known e x t e r n a l radiation source, V is t h e radiation view factor, e is t h e surface emissivity and e

(r)

is t h e emissivity of t h e r a d i a t i o n source.

T h e following a s s u m p t i o n s have to be observed while using t h e above equations: (i) t h e m a t e r i a l particles are at rest so t h a t t h e heat transfer in solids ( r a t h e r t h a n in a moving fluid) can only be analyzed, (ii) all t h e m e c h a n i c a l couplings are neglected (in p a r t i c u l a r , no heat g e n e r a t e d by t h e d e f o r m a t i o n is i n c l u d e d ) , and (iii) no phase change and latent heat effects are a c c o u n t e d for. To find an a p p r o x i m a t e t e m p e r a t u r e solution to t h e above i n i t i a l - b o u n d a r y value p r o b l e m we u s u a l l y form t h e residuals (generally non-zero if 0 is only approximate) 1

86

a n d t h e n solve t h e problem (10.1) (10.4) by d e t e r m i n i n g a t e m p e r a t u r e field 0 satisfying t h e t e m p e r a t u r e b o u n d a r y condition and zeroing t h e following weighted residual: R=

[ r d><m+

Jn

[

n

Jdn?

r d(dVl)

= 0

mq

(10.11)

for all ( s q u a r e intcgrable) weighting functions 4>{x ) t h a t vanish on 8i\°. By using t h e divergence t h e o r e m for t h e first t e r m on t h e r i g h t - h a n d side of Eq. (10.11) t h e residual R can be t r a n s f o r m e d as t

R

_

r r Jn

j

Jn

,. - {k

tJ

k 0 i ij

j

'

„ , , / .8$ 0 ),i + (c —- - Q)J \ 8T / J tj

'

80 + (c—-Q)\dn+[ V 8T

/

J

dQ + / {q + Jan
q(j>d{dn)

Jan?

nj 0,) d{dQ)

= 0

(10.12)

An alternative approach would be the use of the principle of virtual temperatures, [19].

Continuum-Based Formulation for Transient Problems • 335 E q u a t i o n (10.12) is nonlinear with respect to the u n k n o w n t e m p e r a t u r e 0. To solve it for 9 at any given t i m e r we m a y use t h e iterative t e c h n i q u e of N e w t o n R a p h s o n which is based on zeroing t h e ' n e x t ' ( u ; + l ) t h residual, written as in which R = R^' (9(X T; h),x r; h) corresponds to t h e Mast u;th approxim a t i o n to t h e t e m p e r a t u r e field 9 = 9 assumed known, 7 # is t h e iterative correction to be d e t e r m i n e d from Eq. (10.13) such t h a t (u,)

)

1

11

l:

{w)

a n d R '^^

= dR /dO

t a n g e n t o p e r a t o r defined by

dk d —^- 0,4) i + k„6; d6 ' dxj

j

(T)

(lo.M)

is t h e uth

{uj)

R

,JV

JQ

1)

+ ^+V

e^

=

1

(u,+

+ JDQI

3

d

h (j>[ cV dr

d~cd0_

dQ

dQ__d_

dO dr

dO

d0

ti

^

dr,

(10.15;

DU

T h e compact operator notation i ? ° 7$ was introduced earlier and should be clear from t h e present context with ' o ' indicating t h a t -)0 multiplies t h e a p p r o p r i a t e i n t e g r a n d r a t h e r t h a n t h e whole integral expression t h a t defines R . The operator R is nonlinear in linear in <j> and o p e r a t e s linearly on j9. We shall use t h e n o t a t i o n ( T )

{T)

{T)

R {9;)O 9 (T)

1

=

i?

( T )

(0;^ 0) 7

dk

=J

L dO

0 Oi "j0 -f k <j> ej

^ V

dr dq

13

JQQ

+

+ JDQI

tJ

XJ

tn

dQ dO dr

1

86

dQ 1

dOj

'V.

dn (10.16)

d6

or, in brief R {9; f>)O 9

K ^{9-

(T)

(

{T)

1

6) -f C{9; ) — ~i0 dr.

) + K (9; (T)(2)

A^(0;0) + C ( M )

dr\

~f0

(10.17

where Km K(T)(i) 0

+ JDQI /^(T)(2)

, dc d9

_

JQ C

=

d9

d9 dr

"

d9

dO.dxJ (10.18)

dQ

JQ T h e i t e r a t i v e p r o c e d u r e can be seen m o r e clearly for a time-discretized form u l a t i o n in which we a s s u m e t h a t : (i) a solution u p to a typical t i m e i n s t a n t /

336 • Nonlinear Thermal Systems has been o b t a i n e d , (ii) a solution 0 a t t i m e t + At is sought, a n d (iii) a finite difference s c h e m e in t i m e such as t h e one-step b a c k w a r d Euler s c h e m e is employed so t h a t t + A t

1

(10.19)

At and consequently t+

1

AtQ(u)

At t + AtQ(w)

70«"> d dr

7

0"

dr

_

(10.20)

Al-y'+^e^ At

1

At

1

Expiation (10.16), w r i t t e n at r = / -f- At for t h e (u;-f-l)th i t e r a t i o n , b e c o m e s

<9

R

At

Q t 00

+

d0

1 ( 0 M - '0) At

dQ

dQ

d

30

d0,i dx{.

70 " dn (

+1)

9<7 Jon*

(10.21)

du

where all t h e functions (except for 0) are u n d e r s t o o d to be c o m p u t e d at t i m e t + At a n d t h e t e m p e r a t u r e value T h e o p e r a t o r e q u a t i o n (10.13) with the term / ? o 0 ( " > given as in Eq. (10.21) can be solved for 7 0 by any of t h e known t e c h n i q u e s in use for solving P D E s with respect to space variables. It should be noted in this context t h a t even t h o u g h t h e o p e r a t o r R in Eq. (10.12) g e n e r a t e s t h e s y m m e t r i c finite element ' s e c a n t ' stiffness m a t r i x ( d e p e n d e n t on 0). t h e s p a t i a l discretization applied to Eq. (10.13) with (10.21) results in an u n s y m m e t r i c t a n g e n t stiffness m a t r i x which may unfavourably influence t h e efficiency of t h e solution p r o c e d u r e typically based on s y m m e t r i c linear e q u a t i o n solvers. T h e r e f o r e , different s y m m e t r i c a p p r o x i m a t i o n s to t h e u n s y m m e t r i c tangent stiffness are used in p r a c t i c e with t h e n o n - s y m m e t r y effects a c c o u n t e d for in an i t e r a t i v e fashion; we shall c o m m e n t further on this aspect in Section 10.3. l

( r ) ( u / )

+1

( w + , )

7

10.2.2

The Direct Differentiation Method

Let us now consider t h e functional Q(h) = f Jt

0

\ j Un

+ /

g{0(x ,T;h),0, {x ,T;h),x T\h)dn j

j

i

j}

g {0(x ,T;h) q(x ,T]h),x ,r;h)d{dn) d r (q)

l

:

l

l

(10.22)

JdW>

whose design g r a d i e n t dQ/dh is of concern in our sensitivity analysis, g and are a s s u m e d to be known functions of their i n d i c a t e d a r g u m e n t s and q — q n, t

g

(q)

(10.23)

Continuum-Based Formulation for Transient Problems •

337

338 • Nonlinear Thermal Systems If we now observe t h a t t h e last two integrals in Eq. (10.28) can easily be c o m p u t e d (at a given t i m e s t e p , after a solution to t h e p r i m a l p r o b l e m has been o b t a i n e d ) and c o m p a r e Eq. (10.28) with Eq. (10.16), we conclude t h a t Eq. (10.28) can be c o m p a c t l y r e w r i t t e n as a linear in 0,h equation of t h e form R o0>

0i

A'< (M) + C ( M ) ; f -

iT)

r )

OT

_

_

f

JQ

\dk

0 dh dh L

J

\i + [

dcdO

dQ\

\ dh dr

dh ) \

dft

or

L

dQi

(10.29)

dh

dR

tf

e, = -~dh

o

T)

(10.30)

since t h e r i g h t - h a n d side of Eq. (10.29) is just t h e partial (explicit) derivative of t h e residual (10.12). E q u a t i o n (10.29) is a result of f u n d a m e n t a l significance. It essentially says t h a t having at a given t i m e i n s t a n t t h e t e m p e r a t u r e d i s t r i b u t i o n 0 and t h e t a n g e n t operator R we m a y solve Eq. (10.29) for O.h at t h a t i n s t a n t , which is t h e only u n k n o w n needed to c o m p u t e t h e sensitivity function (10.24). Moreover, while t h e p r i m a l p r o b l e m requires iterations according to Eq. (10.13), t h e sensitivity g r a d i e n t O.h can be found directly (i.e. with no iterations) from Eq. (10.29) in which we j u s t need to use t h e last available value of t h e t a n g e n t o p e r a t o r R . T h e above characteristics of t h e sensitivity problem have a crucial significance for t h e efficiency of t h e a p p r o p r i a t e c o m p u t a t i o n a l process. {T)

{T)

10.2.3

The Adjoint System Method

An a l t e r n a t i v e a p p r o a c h , known as t h e adjoint system m e t h o d ( A S M ) , a t t e m p t s to c o m p u t e directly (i.e. not by t h e way of t h e t e m p e r a t u r e sensitivities 0/J t h e functional sensitivity by eliminating 0_h t h r o u g h t h e use of L a g r a n g e m u l t i p l i e r s . To do so we consider an a u g m e n t e d functional

dQ/dh

G* = Q + R o

(10.31)

(a)

in which Q is given in Eq. (10.22) while R ° 0

(A}

*-'- = i ' U ' - [ - ( ^

is defined as

dO

(^-o) dft

+

(10.32) We observe t h a t t h e integral of t h e t i m e integration is j u s t t h e residual (10.11) with t h e weighting function <j> replaced by t h e adjoint t e m p e r a t u r e # ; we require 9 t h a t t h e l a t t e r vanishes on b u t it is a r b i t r a r y otherwise. E q u a t i o n (10.32) becomes (a)

dft

o

R

QV-' dO dr { JQ I q 0 d{dft)\ dr 9

It'to 0

(a)

J8Q0

J

0

(10.33)

Continuum-Based Formulation for Transient Problems • 339 T h e variational formalism for t h e adjoint system m e t h o d follows here t h e m e t h o d o l o g y described earlier in Sections 3.4 and 4.2. In it, t h e adjoint variable # is d e t e r m i n e d from the r e q u i r e m e n t t h a t t h e a u g m e n t e d functional (10.31) be s t a t i o n a r y with respect t o t h e p r i m a r y variable 0, i.e. (a)

6Q* = ^ -

° 60 = 0

(10.34)

which, by considering O.h as a possible specific case of 66, becomes dQ"

o6

d = 1,2,. . . , £ >

= 0

h

86

(10.35)

According to t h e ASM variational methodology, t h e s u b s t i t u t i o n into Eq. (10.31) of t h e adjoint variable 0 c o m p u t e d from Eq. (10.35) implies t h a t t h e desired response functional sensitivity can be c o m p u t e d as (a)

dg dh

_ _

qq*_

(10.36)

dh

in which t h e right-hand side involves only t h e explicit design differentiation of Q*, which is straightforward to c o m p u t e as n f ^DO+ Jt 1 Jn dh

dh

/ Jdui

0

dm)

d g i q )

dh dft

+X[^ (i£-!2)<+

+ 1 Idni dh

~j-6^d(dn)\dr

(10.37)

T h e above result has been obtained by using a c o m b i n a t i o n of Eqs. (10.22), (10.31) a n d (10.33), which yields dft 'to

+J

(g

{q)

+ q6 ^

d(dft)} d r

(

(10.38)

T h e condition (10.35) is rewritten as

dQ^_

dk

dOj

Jo

dx

t

d6

dO "-

00,- dxi

h

fdc

d6 6.h -

o

+j

dg

.

M

do 3(a)

0

dtk

dQ

dQ

dr

dO

<90, dx

n

dg h

^ ^

n

d(AN)}dr

i

h

dn

t

(dq,

M

d0.

o':

3

{ ^

dq

x

6

h

+

d0

h

d0-^x(10.39)

340 • Nonlinear Thermal Systems which, by i n t e g r a t i n g by p a r t s in t i m e t h e t e r m c9^9 ^ becomes (

dQ* 89

It'to 0

(IQ

(dg

dg

I \dg JdQi L w-

+ ^

/;

00<->

d9

h

^dx

dxj

ij

89.

h

(dq,

dgW

[q)

h

7;

and re-ordering t e r m s ,

]

A w

n

dk

"h

d9

x

+

h

dO 0 0

tJ

r - ^d97 T

dq 0

d{dQ)

w,dx( a )

dx,

dx.j

h

d~c 80 (- — V — d9 ' dr h

dft

- c —— 9 - c9 dr h

JAP.-* da

J

UP.

(10.40) to

Consequently, by noting additionally t h e notation i n t r o d u c e d in Eq. (10.16) a n d t h e condition (10.41)

0 ('o) = O (a)

we require t h a t A ' ( 0 ; 0 ) - C(9; 9^) lT)

- C(0; 0 )

(i)

o0

(a)

J

fc

dG = - _ o ^ dO

(10.42)

with o^{i)

(10.43)

0

=

in which we used t h e following p r o p e r t y of t h e o p e r a t o r C: (10.44) T h e adjoint e q u a t i o n (10.42) together with the t e r m i n a l t i m e condition (10.43) is t h e f u n d a m e n t a l s y s t e m for solving t h e nonlinear, n o n - s t a t i o n a r y t h e r m a l sensitivity p r o b l e m by t h e A S M . E q u a t i o n (10.42) is a variational e q u a t i o n for the adjoint t e m p e r a t u r e field 0 ( x ) , which has to be satisfied for any 0^- T h e partial differential e q u a t i o n generated by this equation has to be solved b a c k w a r d s in t i m e ; t h e t i m e t r a n s f o r m a t i o n ( a )

t

' = t - r

(10.45)

T

allows us to replace it by t h e initial-boundary value problem of t h e form ^

( T ) ( 1 )

(0;0

( i )

)-A

r ( T ) ( 2 )

( 0 ; 0 \ ) + C ( 0 ; 0 O 4 - C ( 0 ; 0 ) ] ^ . / , = —QJf° -h ( i

(a

(a)

e

( - ) 10 46

with 0

( i )

Uo = 0

in which all t h e t i m e derivatives are taken with respect to t i m e r ' .

(10.47

Semi-Discretized Formulation • W e m a y observe t h a t for c = const, which implies C = 0 and A ' we h a v e [ / r < 7 > < i > (

M

( . > )

c(d-J )} M

+

o

e

h

=

~

o

( r ) ( 2 )

341 = 0,

(io.48)

0,

Identifying t h e field 6,h with t h e weighting function 0 of Eq. (18), we o b t a i n

w h e r e t h e following n o t a t i o n is A'

(T)(1)

( 0 ; 0 ) o 0<*> = A '

(T)(1)

dG i n t r o d'u c e d~~de : { )=

(0; 0

( a )

'

0
(10 49)

)°0

(10.50)

T h e r e m a r k a b l e formal similarity of t h e left-hand sides of t h e A S M a n d t h e D D M governing e q u a t i o n s , (10.49) and (10.30), respectively, is e m p h a s i z e d .

10.3

Semi-Discretized Formulation

10.3.1

Finite Element Equations

By replacing in Eq. (13) t h e weighting function
0(x,T\h)

= (f) {x)d {T-h) a

,

a

a = 1,2,...,AT

(10.51)

w h e r e <j) are t h e s h a p e functions, d are t h e nodal t e m p e r a t u r e s a n d N is t h e t o t a l n u m b e r of degrees of freedom in t h e discretized s y s t e m , we o b t a i n from E q . (10.11) a

a

/ (kij <j) pdp - Q<j> ) dft + / Jn ' JdQi ayi

a

3

q<j> d(<9ft) Q

a

v

+

£ ( c ) O W 0 - t f o o W a d ( 0 f i )

/

x{W0-*(T))ad(dn) = O

+ /

in which <9ft = dW \ (<9ft 9

9(1)

U <9ft )

(10.53)

9(2)

Denoting Kap

k

-

i3

p dft a
}J

(10.54)

J d Q q ( 2 )

Cp

=

a

/ c <^dft Jn Q

Q = [ Qtadn - [. q d{dn) a

a

J^l Qa =

f

JdUi {(c) 4 o

<j>a

d{dVt)

+

/

X

^(r)^d(0ft)

(10.5

342 • Nonlinear Thermal Systems we o b t a i n

C ti a0

g

+ K ;l

+ ( K§

+ K§

{

)

= ( ^ - H ^

(10.55)

or, in brief C fy a0

+ K ti a0

= Q

0

(10.56)

a

which is t h e f u n d a m e n t a l e q u a t i o n describing t h e nonlinear t r a n s i e n t heat transfer in t h e discretized s y s t e m . T h e m a t r i c e s A'^g , and A ' ^ m a y be briefly called t h e c o n d u c t i v i t y , convection and r a d i a t i o n stiffness m a t r i c e s , respectively. It should be n o t e d , however, t h a t only in t h e linear analysis do these n a m e s (in fact, only t h e first two since A ' ^ = 0 t h e n ) a p p e a r fully l e g i t i m a t e . T h e a p p a r e n t l y elegant form of t h e ordinary differential e q u a t i o n (10.56) is s o m e w h a t misleading since t h e m a t r i c e s C p and K p (in fact, all t h r e e t e r m s in t h e l a t t e r ) a n d t h e vector Q m a y generally d e p e n d on t h e t e m p e r a t u r e t ? . Therefore, Eq. (10.56) first has to be linearized with respect to d and only then solved by using an iteration scheme. We also n o t e t h a t t h e m a t r i c e s C (3 a n d K p are s y m m e t r i c , which is essential for efficiency of t h e c o m p u t a t i o n a l p r o c e d u r e s . G u i d e d by t h e linear heat transfer e q u a t i o n s we a d o p t t h e following linearization a s s u m p t i o n s : a

a

Q

a

a

a

t + Atfc. . t + Atfl + At£t+Atfl

^

tfc

^

t£t + Atfl

a

t+Atfl.

We n o t e t h a t (i) t h e above a s s u m p t i o n s are a u t o m a t i c a l l y satisfied in t h e linear analysis in which £( ) = const, k = const and c = const, and (ii) i t e r a t i v e solution schemes m a y fully restore neglected t e r m s in t h e nonlinear analysis. By w r i t i n g out Eq. (10.55) for t h e t i m e i n s t a n t t + At and using t h e linearization (10.57) we arrive at c

C *%

l

t+

aP

+

' A V

tJ

+

A

t

t y

=

"Qa

t+

(10.58)

T h e r i g h t - h a n d - s i d e vector is a s s u m e d here as given at t + At—if e i t h e r Q or q d e p e n d in a given way on t h e t e m p e r a t u r e d , t h e n some resulting c o n t r i b u t i o n will a p p e a r on t h e left-hand side of Eq. (10.58). Using a t i m e integration a l g o r i t h m , say t h e one-step b a c k w a r d s Euler s c h e m e of t h e form Q

Semi-Discretized Formulation • 343 we m a y solve Eq. (72) for i) in t e r m s of all t h e q u a n t i t i e s defined at t i m e t and t h u s a s s u m e d known using t h e equation i+At

_L_ t n

i

\

t+&tA

_

t+At

Q

1

+

a

.At

(10.60)

^ C p i)p t

t

t

a

or, briefly 1

(10.61)

T h e effective 'stiffness' m a t r i x s y m m e t r i c . T h e linearization errors can b e removed by using an iteration a l g o r i t h m t h a t restores t h e nonlinear heat flow e q u i l i b r i u m . A c o m m o n l y employed iteration s c h e m e is based on using t h e c o n s t a n t m a t r i x K 'g t o solve t h e following equation for t h e (u;-f 1 )th correction t o t h e t e m p e r a t u r e at / -f At: ] S

t

)

Q

(10.62)

to = 1 , 2 , . . . w h e r e t h e coth residual readss t+ A t ^

t + A t

=

Q

a

_

( t + A t

C

a

0

t + A t ^

t + A t j ^ ^ + A t ^

+

(10.63)

while t+At^)

_

At

(10.64) etc.

and t+Atfl(*

+ i)

_

t+At^)

_^ ^ ( " + 1 )

(10.65)

T h e value of d ^ is t h e one o b t a i n e d from Eq. (10.61). T h e o p e r a t o r form of Eq. (10.62) reads t + A t

= 1,2,...

(10.66)

At^+i)

(10.67)

because

7e = 7e - 7 +i)

+i)

t + At^+i)

_

a

tj

t+

<J

A*

A*

so t h a t (10.68) A faster convergence can theoretically be achieved by t h e full N e w t o n - R a p h son a l g o r i t h m , cf. Eq. (10.13). Let as rewrite Eq. (10.56) in t h e residual form as, cf. Eq. (10.63)

Ra = Qa~ (C p ~ + K p^J dp a

a

(10.69)

344 • Nonlinear Thermal Systems a n d use it to c o m p u t e t h e t a n g e n t o p e r a t o r R \

cf. Eq. (10.15)

{T

/C'

5|

=

00.70)

as

(io.7i)

«S = - ( ^ | : + /C ) )

in which

Kl'J = C ,

1% + K

ai p

+ K

a0

aitP

^ - Q,

(10.72)

a p

(if Q does not d e p e n d on $ , as a s s u m e d previously, then clearly Q ,p — 0). We n o t e t h a t for s o m e m i n o r technical reasons t h e discretized and c o n t i n u o u s residuals in our formulation differ in sign and, consequently, t h e adjoint variables a n d 0 * in b o t h formulations will have o p p o s i t e signs as well. We also n o t e t h a t even t h o u g h t h e m a t r i c e s C p and K p are s y m m e t r i c , t h e m a t r i x K Jp is not. T h i s is a serious d r a w b a c k when it comes to i t e r a t i v e corrections of t h e solution at a given t i m e step using t h e full N e w t o n - R a p h s o n s c h e m e . We have in such a case a

Q

(

a

]

(

a

t + A t ^

t + Mft*)

=

«+A

+

t f l

a

(r)(»)

7 l ?

(« .) +

=

0

j

(

0

7

3

)

5

)

i.e.

>

„

'

t+Atn(T)(u.')

where t+A t ^

t At

=

+

_ (t + t t c y

Qa

9_

+

t At +

f

<

^

t A t ^ +

(

1

0

7

E q u a t i o n (10.74) should be c o m p a r e d with Eq. (10.66). T h e o p e r a t o r R ^0 in Eq. (10.74) g u a r a n t e e s a faster ( q u a d r a t i c ) convergence of t h e i t e r a t i o n t h a n t h e o p e r a t o r in Eq. (10.66). However, t h e latter is kept c o n s t a n t for every i t e r a t i o n a n d it g e n e r a t e s t h e s y m m e t r i c iteration m a t r i x - factors which m a y c o m p e n s a t e for a slower convergence r a t e . {

F u r t h e r m o r e , in view of our f u n d a m e n t a l observation m a d e with respect to t h e D D M sensitivity e q u a t i o n (10.29), in which precisely t h e s a m e t a n g e n t ope r a t o r has to be used (in a n o n - i t e r a t i v e way, t h o u g h ) , t h e n o n - s y m m e t r y of t h e t a n g e n t m a t r i x presents a similar c o m p u t a t i o n a l d i s a d v a n t a g e . By differentiating Eq. (10.58) with respect to the design p a r a m e t e r h we o b t a i n

{

C

a

0

f r

°

t

+

+

A

dh

h

t

Q

a

* ° p ) - i i r

_ *' o0 dh dt+

Atj

C

d

t+

'

t + A t

dh

K

a ( i t + A t

'

v

'

Semi-Discretized Formulation •

345

which m a y be presented as, cf. Eqs. (10.69) and (10.71) d ^

ti ! ~dh

_

t + A t

^

!

l

p

d R t+At

Q

^

=

~

(,0.77

dh

T h i s e q u a t i o n is linear in d dp/dh with its r i g h t - h a n d side known provided t h e p r i m a l p r o b l e m has been solved in t h e t i m e interval considered. E q u a t i o n (10.77) can be solved either directly (i.e. using t h e n o n - s y m m e t r i c e q u a t i o n solver), in which case no iterations are required, or it can be t r a n s f o r m e d t o t h e form t+At,

I

° > » ;

c

+

^ ) ^ i r - ~ ~ W ~

h

'K~dh~

h

( 1 Q

-'

8 )

where I<:

=

t+At

0

Equation d i)p/dh

I<$

-

t+At

Kp

=

t+At

a

C

t + A t

Q 1

, ^ +

K ,pi),

-

t+At

p

ai

Q ,p

i+At

a

(10.79)

(10.78) can be solved for t h e nodal t e m p e r a t u r e sensitivity vector at t h e desired t i m e i n s t a n t by using t h e direct iteration according to

t+At

( t + A t

°

r

, t+At

\

K

-Z dr

^ap "

V

P

a

v

l^aP "J

1'

d

dh

P

+ At

-

dh

t

+

A

t

IC "

h

„ dh

a a ( 5

,

0

a; = 0 , 1 , . . .

(10.80)

w i t h d dp /dh a s s u m e d zero, or t h e last value available (i.e. d d /dh). Which of t h e a b o v e a p p r o a c h e s should be used for a specific p r o b l e m d e p e n d s to a large e x t e n t on t h e solution a l g o r i t h m employed for solving t h e p r i m a l p r o b l e m . t+At(

0)

t

a

Using t h e b a c k w a r d s Euler t i m e i n t e g r a t i o n s c h e m e for t h e sensitivity equation (10.80)

^

r

=

^

\

-

^

r

'

^

k

)

<

1 0

-

8 I )

we o b t a i n "

dh

e

~

dh

"

dh

B

+ i r ' C ^

(10.82)

E q u a t i o n (10.82) can be solved iteratively for d ' * $ / d / ? . It requires s t o r a g e of t h e previous ( c o m p u t e d at r = t) sensitivity vector d d /dh but no such vectors at any earlier t i m e i n s t a n t s . For p r o b l e m s with t h e t e m p e r a t u r e sensitivity slowly changing over t i m e t h e following n o n - i t e r a t i v e a p p r o x i m a t i o n to Eq. (10.82) m a y t u r n out r e a s o n a b l e : + A

a

i

n

t+ A t

R

>

Kf

^ )

d

t

+

^p

At

J+1)

dh

_ d

t

"

R

+ At

a

(t + At *

1

K

~ {

°*

K

t+ A t

" ~Kt

r

C a 0

\

J

d

t

^P

~dh~

n

(

)

o

°°-

M 8 3 )

which c o m b i n e s t h e a d v a n t a g e s of t h e s y m m e t r i c governing m a t r i x and lack of i t e r a t i o n at t h e cost of decreased accuracy.

346 • Nonlinear Thermal Systems Having an effective technique to c o m p u t e t h e t e m p e r a t u r e sensitivities at each t i m e i n s t a n t m a k e s it possible to d e t e r m i n e t h e sensitivity of any functional such as, q u i t e generally Q(h)

=

f

G{tf {r;h);h)dT

+ g(ti(i;h);h)

a

(10.84)

T h i s functional can be specified t o describe t h e so-called critical time sensitivity (in which t h e p r o b l e m is to find t h e design gradient of a functional at a prc-set t i m e i n s t a n t ) or t h e so-called t i m e interval sensitivity (in which t h e sensitivity gradient is o b t a i n e d as t h e t i m e integral over a selected t i m e interval). In fact, b o t h these s i t u a t i o n s could be described just by t h e first t e r m in Eq. ( 1 0 . 8 4 ) ; to be m o r e explicit we have directly included there t h e t e r m g which is local in t i m e . Differentiation of Eq. ( 1 0 . 8 4 ) with respect to t h e p a r a m e t e r h yields

dQ _ f'fdGd*. Jt<9iL \~

dh

n

0

dG . dh

dh

dx)

a

(10.85)

dh

dh

which clearly indicates t h e algorithm for c o m p u t i n g dQ/dh in t e r m s of d\9 /dh d e t e r m i n e d at all t h e discrete t i m e i n s t a n t s r £ [£rj?^]Before m o v i n g on t o discuss t h e adjoint system m e t h o d we should m a k e ano t h e r general c o m m e n t . In discretized nonlinear heat transfer p r o b l e m s t h e "sec a n t ' formulation is given in t e r m s of t h e s y m m e t r i c m a t r i c e s C p a n d of Eq. ( 1 0 . 5 6 ) ; t h e t a n g e n t formulation involves K p\ which is u n s y m m e t r i c , cf. Eq. ( 1 0 . 7 2 ) . In typical nonlinear solid mechanics problems t h e secant formulation is governed by some n o n - s y m m e t r i c matrices while t h e tangent formulation involves s y m m e t r i c m a t r i c e s (problems with configuration-dependent loadings, b o u n d a r y friction effects or non-associative plastic flow rules are n o t a b l e exceptions in this r e g a r d ) . Since t h e D D M approach is inherently based on t h e t a n g e n t m a t r i c e s , its use in nonlinear solid mechanics a p p e a r s m o r e straightforward t h a n in heat transfer p r o b l e m s . Let us now consider t h e a l t e r n a t i v e way of finding dQ/dh by using t h e A S M . We could proceed by simply discretizing t h e ASM e q u a t i o n s o b t a i n e d in Section 1 0 . 2 . 3 . However, we find it more instructive to go through t h e necessary derivations once more. We again employ t h e variational t e c h n i q u e of C h a p t e r 3 which states that n

a

T

n

^ dh

= ^

(10.86) dh

provided G* = G +

A tf Q

ft

(10.87)

and A , a = 1, 2 , . . . , j V , are such as to m a k e G* s t a t i o n a r y with respect to i) , i.e. a

n

(10.88)

Semi-Discretized Formulation •

347

Let us first form t h e e x t e n d e d functional Q* as G*(h)

f

g(h)+

=

,

X (T)R (T)dr Q

a

o=l,2,...,JV

(10.89)

in which t h e residuals R arc given by Eq. (10.69). Considering in Eq. (10.88) only t h e design variations Q

(10.90)

ah.

we c o m p u t e

+

/[

—r~ ^ « + L 0w J

/[

L

=

J

Q

1

h~ <

-^o

Mydr

K

= 0

(10.91)

which, by i n t e g r a t i n g by p a r t s in t i m e t h e third t e r m in p a r e n t h e s e s and o b s e r v i n g Eq. (10.72), b e c o m e s 6ti„

0d

—

=

A

Q

Cp a

6i9p

Oil

a

n

tfi^dr

= 0

(10.92)

<^dr

= 0

(10.93)

or ( n o t e t h a t 6 i ? ( / ) = 0) Q

OQ Oxh

8il

0

d

g

\

r

+ A [—K^Q Q

+

Cq/j)

+ A

FT

C,

T h e adjoint e q u a t i o n is thus implied to be Cp a

Xp ~ (A'J/j + C p) n

Xp =

,

(10.94)

with t h e t e r m i n a l condition at r = t resulting as t h e n a t u r a l b o u n d a r y condition in t h e form C dXp n

=

dg_

T = <

(10.95)

It is e m p h a s i z e d t h a t Eqs. (10.94) and (10.95) form a t e r m i n a l - v a l u e p r o b l e m which is to be i n t e g r a t e d b a c k w a r d s in t i m e once t h e p r i m a l p r o b l e m has been solved forwards in t i m e . T h e adjoint problem can be formally replaced by an

348 • Nonlinear Thermal Systems initial-value p r o b l e m only if we employ t h e t i m e t r a n s f o r m a t i o n r ' = t — r, in which case Eqs. (10.94) and (10.95) b e c o m e CaP h +

+ Cap) \P = | £ , a" C /?A^ - — ,

- K$™

£ [0, *-*„]

T'

(10.96) r = 0

Q

where t h e dot now s t a n d s for differentiation with respect to t h e variable r ' . T h e basic ASM sensitivity relationship, Eq. (10.85), can b e r e w r i t t e n as dQ

dg

dh

dh

l

dG [ J h

l

Jto

fdQa dK p dCaP X { c J h - - - ^ ^ - ~ d J r ^ a

+

X

a

g

dr

(10.97)

which defines t h e effective way to d e t e r m i n e dQ/dh once A is known at every discrete r £ [t , i]. F u r t h e r m o r e , we observe t h a t for c = const, which implies C p = 0 a n d K Jp = 0, we o b t a i n from Eq. (10.96) a

0

a

{

dG C \p

+ K%™\ =j±

a0

(10.98)

0

which clearly is t h e discretized c o u n t e r p a r t of Eq. (10.49) derived in Section 10.2. T h i s fact has t r e m e n d o u s c o m p u t a t i o n a l significance. By specifying Eq. (10.96) for t h e t i m e r = t + At we arrive at Cp

t + At

%

t

+ A

Q

+

=

(

(

+

A

7 ^

)

(

1

t+

-

)

"K™

+

C p)

t + At

%

t+

A

a

(10.99)

dv

a

T h e r i g h t - h a n d - s i d e vector is a s s u m e d given at t i m e t -f At. By using t h e t i m e i n t e g r a t i o n s c h e m e in t h e form of t h e backwards Euler a l g o r i t h m =

*

0

we can solve Eq. (10.99) for (~

Cp

t+At

+

a

= ~*r

(10.100)

A p

(

+

A

( + A (

7 i ™ >

A in t e r m s of all t h e q u a n t i t i e s defined at t i m e t

-

A - r '

+

t+A

«A'&

)(2)

+ C p) t+At

a

t

+

A

%

° ^

c

(

U

U

0

1

)

It requires storage of t h e previous adjoint t e m p e r a t u r e vector \p. E q u a t i o n (10.101) can be w r i t t e n in a shorter form as l

A^( ,(e

( +

a

f

f

,

tA +

%

=

+

_Lt

+

A t

C

a

p

%

(

1

Q

w h e r e +^ J\ ^ \ t h e effective stiffness m a t r i x for t h e adjoint p r o b l e m . m a t r i x is not s y m m e t r i c , which implies c o m p u t a t i o n a l d r a w b a c k s . t

t

i

{etl)

s

1

Q

2

)

This

Semi-Discretized Formulation • 349 E q u a t i o n (10.102) can b e solved either directly by using t h e n o n - s y m m e t r i c e q u a t i o n solver or it can b e transformed t o t h e form (note t h e s y m m e t r y of K p) a

1

t+At si

I (t+At

^aP

d G t+At

I t+At A

K

y

At +

t\ p

+

Q

+ ^7 Cp At

t+At

Q

a

^ap)

% + ( Kp

t+At

dd

\

K pT')

t+Ai

-

T )(2) a

[

%

t+A

(10-103)

where

E q u a t i o n (10.103) can b e solved for t h e nodal adjoint t e m p e r a t u r e vector ' at t h e desired t i m e i n s t a n t by using t h e direct iteration according to 1 ^

t+Ats^

j _ f t+At is | t + At sy \ + { AQP + l^ap)

nt + Atfi

-

t+

+ A

'A

At^(w+i)

i

+ ^

A

C

t

a

p %

4- (""K™

-

K^

t + A t

l )

')

%"

t+A

(10-105)

]

w i t h A ^ a s s u m e d zero, or t h e last value available (i.e. Ap). D e n o t i n g f

* »? ~h ° (

t+

tK

=

t+At

ap+

^

t+AtRa0

-

+ t+Atdap

(10 106)

and i

+

A^(T, 3) (

=

t+At m K

W

t+At m .

_

K

(10J07)

{i)

E q . (10.105) b e c o m e s flt+Atn

t+At ( p K :

t + A t ^

=

i +

i _ t+At

CQ0

%

t *t mm t+At y +

+

K

{ l o m )

x

E q u a t i o n (10.108) can b e iteratively solved for \ p . It requires storage of t h e previous adjoint t e m p e r a t u r e vector \ p . For p r o b l e m s w i t h t h e t e m p e r a t u r e sensitivity changing slowly over t i m e t h e following n o n - i t e r a t i v e a p p r o x i m a t i o n t o Eq. (10.108) m a y s o m e t i m e s b e used: t

+

A

t

{

+

+

A

1

)

l

t + A t j ^

t+A

%

=

+

^

t

+

M

c

^

+

t

t

K

^

%

(

1

Q

1

Q

9

)

We n o t e in closing t h a t c o n t r a r y t o typical nonlinear solid m e c h a n i c s p r o b l e m s the tangent matrix K in nonlinear heat transfer p r o b l e m s is u n s y m m e t r i c , a n d it is K t r a n s p o s e d which a p p e a r s in t h e adjoint formulation. Nevertheless, once t h e p r i m a l response is known, t h e adjoint response can b e o b t a i n e d numerically in an efficient m a n n e r . T h i s is p a r t i c u l a r l y so when sufficient storage is available a n d t h e d e c o m p o s e d stiffness m a t r i x from t h e p r i m a l forward-in-time analysis at each discrete t i m e i n s t a n t r can b e stored a n d later utilized t o d e t e r m i n e t h e adjoint response at t i m e i—r during t h e b a c k w a r d - i n - t i m e n o n - i t e r a t i v e analysis. ( r )

( T )

350 • Nonlinear Thermal Systems Such an a l g o r i t h m requires neither additional stiffness m a t r i x assemblies nor d e c o m p o s i t i o n s for t h e adjoint problem; only t h e right-hand-side assemblies and back s u b s t i t u t i o n s are needed to o b t a i n t h e adjoint solution. Consequently, t h e c o m p u t a t i o n a l cost of t h e adjoint analysis is much smaller t h a n t h a t of t h e primal analysis which m a y generally require several iterations to converge at each s t e p . 1

10.3.2

Computational Illustration

T h e direct differentiation m e t h o d has been employed for to t h e sensitivity analysis in which t h e angular speed is considered as t h e design p a r a m e t e r in t h e problem of friction welding. Finite element modelling of t h e friction welding process was u n d e r t a k e n in [109]. Friction welding is a process in which t h e heat for welding is p r o d u c e d by t h e direct conversion of mechanical energy to t h e r m a l energy at t h e interface of t h e workpieces without t h e application of electrical energy, or h e a t from o t h e r sources, to t h e workpieces. Friction welds are m a d e by holding a n o n - r o t a t i n g workpiece in contact with a r o t a t i n g workpiece under c o n s t a n t or g r a d u a l l y increasing pressure until t h e interface reaches welding t e m p e r a t u r e , a n d then s t o p p i n g r o t a t i o n to c o m p l e t e t h e weld. T h e frictional heat developed at t h e interface rapidly raises t h e t e m p e r a t u r e of t h e workpieces over a very short axial d i s t a n c e to a value a p p r o a c h i n g from below t h e m e l t i n g point, and welding occurs u n d e r t h e influence of a pressure t h a t is applied while t h e h e a t e d zone is in t h e plastic range. Friction welding is classified as a solid-state welding process in which joining occurs at a t e m p e r a t u r e below t h e melting point of t h e work m e t a l . In our e x a m p l e we will analyse t h e t e m p e r a t u r e sensitivity to t h e variation of t h e a n g u l a r speed of t h e workpieces. T h i s problem has a clear practical significance in t h e welding o p t i m i z a t i o n . T w o rods, shown schematically in Fig. 10.1, illustrate [mm]

B

3

Figure 10.1

A

z[mm]

20

Geometry and finite element mesh of the workpieces

t h e workpieces in friction welding. T h e p r o b l e m can be considered as a x i s y m m e t r i c . T h e simple t r i a n g u l a r finite e l e m e n t s with linear s h a p e functions in t h e radial cross-section are used. At t h e place of a b u t m e n t t h e heat source is given by t h e following e q u a t i o n : Q = / au u r dft

(10.110)

Shape Sensitivity for Transient Problems • 351 Table 10.1 Temperature sensitivity to variation of the angular speed in friction welding Time

1

2

3

Point

Temperature d

Sensitiv ity dd/dw

Linear analysis

Nonlinear analysis

Linear analysis

Nonlinear analysis

A

256

254

2.328

2.291

B

534

531

5.742

5.352

C

371

369

4.217

3.927

A

424

421

3.259

3.127

B

815

811

8.038

7.814

C

658

654

5.903

5.643

A

582

578

3.957

3.804

B

1034

1029

9.763

8.901

C

893

889

7.169

6.825

where a is t h e n o r m a l stress at t h e place of c o n t a c t , // is t h e coefficient of friction, u is t h e a n g u l a r speed, r is t h e radius a n d Q is t h e surface area u p o n which t h e h e a t acts. Steel rods of d i a m e t e r (f> = 12 m m are considered. In our analysis era is a s s u m e d to be equal to 3 x 1 0 J / m a n d to to 100 s . T h e m a t e r i a l p r o p e r t i e s are a s s u m e d as follows: t h e r m a l c o n d u c t i v i t y k = 5 0 ( 1 + 0 . 0 0 h ? ) W / m K , specific heat c = 5 1 0 ( 1 + 0 . 0 2 t ) ) J k g / K , t h e r m a l convection coefficient 0.25 W / m , initial t e m p e r a t u r e 0 ° C , s u r r o u n d i n g t e m p e r a t u r e 0°C. T h e derivatives dk/di). dc/dd a n d dQ/duj a r e constant t h r o u g h o u t t h e whole history of t h e process. T h e results of t h e sensitivity analysis dd/dco at t h e chosen points / \ , B and C at subsequent t i m e i n s t a n t s t\ = 1 s, t = 2 s a n d / = 3 s are presented in Table 10.1. In order to c o m p a r e t h e results with t h e sensitivity analysis for t h e corresponding linear p r o b l e m t h e l a t t e r results are also displayed in Table 10.1 to i l l u s t r a t e t h e influence of nonlinearity on t h e sensitivity process. 7

3

- 1

2

2

10.4 10.4.1

3

Shape Sensitivity for Transient Problems Introduction

W e have already pointed out in C h a p t e r 6 t h a t in recent years considerable research activity have been observed in t h e field of s h a p e design sensitivity analysis in solid a n d s t r u c t u r a l mechanics. T h e r m a l problems have been addressed in t h a t c o n t e x t only in t h e last decade. In this section we shall briefly recall t h e existing l i t e r a t u r e on t h e l a t t e r topic. T h e s h a p e design sensitivity analysis of t h e r m a l s y s t e m s was a p p a r e n t l y und e r t a k e n for t h e first t i m e in [39], where t h e basic c o n t i n u u m formulation for linear isotropic t h e r m a l s y s t e m s was p u t forward. T h e s h a p e sensitivity approach was based on t h e m a t e r i a l derivative concept. In [129, 149] t h e r m a l s h a p e

352 • Nonlinear Thermal Systems sensitivity problems were addressed using t h e b o u n d a r y element m e t h o d . In [149] t h e nonlinear s t e a d y - s t a t e p r o b l e m was considered with e q u a t i o n s describing s h a p e sensitivity a n d o p t i m i z a t i o n derived by m e a n s of t h e m a t e r i a l derivative a p p r o a c h c o m b i n e d with t h e adjoint system methodology. T h e s h a p e design sensitivity for t r a n s i e n t conduction problems was analysed in [169] using t h e adjoint m e t h o d . T h e a p p r o a c h was based on t h e m u t u a l energy principles with a t e r m i n a l - v a l u e problem resulting for t h e adjoint sensitivity e q u a t i o n . S h a p e and n o n - s h a p e sensitivity formulation for transient nonlinear t h e r m a l p r o b l e m s were presented in [171] using t h e convolution t e c h n i q u e . A general overview of t h e p a r a m e t e r sensitivity problem for nonlinear transient t h e r m a l s y s t e m s was presented in [111], whereas t h e s a m e a u t h o r s proposed in [163] t h e use of t h e KirchhofF t r a n s f o r m a t i o n for t h e s h a p e sensitivity analysis in nonlinear s t e a d y - s t a t e heat conduction p r o b l e m s . In fact, t h e last two articles form t h e basis for t h e c u r r e n t and next sections of t h e book, respectively. T h e p u r p o s e of this section is to apply t h e two basic s h a p e sensitivity m e t h o d ologies to t h e r m a l problems: t h e m a t e r i a l derivative approach ( M D A ) , in which design variations are considered over a variable d o m a i n , and t h e d o m a i n par a m e t e r i z a t i o n a p p r o a c h ( D P A ) based on transforming t h e problem to a fixed reference d o m a i n before t a k i n g any design variations.

10.4.2 The Material Derivative A p p r o a c h — M D A / D D M T h e first m e t h o d to be discussed is t h e material derivative approach. Even if m a n y versions of t h e t e c h n i q u e have been reported in t h e l i t e r a t u r e [2, 27, 40, 7 1 , 80, 190, 191], t h e approach developed here is believed to be very r e p r e s e n t a t i v e of t h e basic MDA concept. After a brief exposition of t h e main idea u n d e r l y i n g t h e m e t h o d we shall continue t h e discussion by considering t h e M D A in c o m b i n a t i o n with t h e direct differentiation m e t h o d ( D D M ) ; t h e resulting t e c h n i q u e will consequently be referred to in short as t h e M D A / D D M . O u t of m a n y possible ways of presenting t h e MDA we shall follow t h e one similar to t h a t presented in [13]. Let, for simplicity of t h e general discussion, just a single p a r a m e t e r h be a s s u m e d as t h e s h a p e design variable. Owing to a design modification defined by h = h -f Sh. t h e undeformed domain Q in assumed to be p e r t u r b e d to a new configuration i}. T h e a p p r o p r i a t e one-to-one m a p p i n g from ft to ft is d e n o t e d by r, G 0 ; x

%

E ft

(10.111)

which at t h e nominal design h = h becomes t h e identity t r a n s f o r m a t i o n •>•,(•'•.,: h) = Xi ,

Xi

e ft

(10.112)

E x p a n d i n g ,r (.r ; h) at h = h into Taylor series and retaining only t h e linear t e r m results in t h e following relation: (

;

Shape Sensitivity for Transient Problems •

353

or x (xj

\ h) = X{ -\- Vi(xj \ h) Sh

t

(10.114)

where dx •

^

Vi

^

Xj;

=

~dt

(10.115) h-h

is often referred t o as t h e shape velocity field associated with t h e s h a p e p a r a m e t e r h t r e a t e d formally as a time-like variable. O n c e v is specified at t h e current s h a p e (i.e. n o m i n a l design), any new p e r t u r b e d d o m a i n defined by t h e value of Sh can be found from Eq. (10.114). x

Let us consider any scalar function f(x ;h) ( t h e t e m p e r a t u r e field, for ins t a n c e ) whose s h a p e sensitivity, i.e. t h e total derivative with respect to h at h = h, is sought. Because of t h e possible and appealing i n t e r p r e t a t i o n of x as t h e m a t e r i a l (or Lagrangian) c o o r d i n a t e s of a m a t e r i a l p a r t i c l e and v as t h e ' d e f o r m a t i o n ' velocity, t h e t o t a l derivative df /dh m a y also be referred to as t h e m a t e r i a l derivative. T h i s derivative is c o m p u t e d as x

t

t

df(Xi]

h)

dh

lim h= h

f{xi(xj\h);

h) - f(xj] Sh

Sh^o

dh

h)

h=h

+

df

dx '

dx

dh

x

%

h=h

dh

dx.

(10.116) h-h

in which t h e velocity t>, has been defined by Eq. (10.115). Using t h e n o t a t i o n /.*,

dh

=S,H,

^ = / ,

TH

(10.11,

Eq. (10.116) b e c o m e s f.h{xj]

h) = f,h{xj]

h) + Vi(xj] h) f,t{xj;

(10.118)

h)

E q u a t i o n (10.118) indicates t h a t t h e t o t a l derivative /./, of any field / with respect t o t h e s h a p e p a r a m e t e r h consists of two p a r t s : (a) t h e i n s t a n t a n e o u s r a t e of c h a n g e in / observed while s t a n d i n g at a fixed point x ( t h e so-called local derivative of / ) , and (b) t h e r a t e of change in / d u e to t h e changing location of t h e p o i n t x which is m o v i n g at t h e velocity v ( t h e so-called connective derivative of / ) . T h e effective use of Eq. (10.118) in s h a p e sensitivity calculations requires at least t h r e e a d d i t i o n a l c o m m e n t s : t

x

x

(i) T h e m a t e r i a l derivative is m o r e complicated to h a n d l e m a t h e m a t i c a l l y t h a n t h e local derivative. For i n s t a n c e , t h e l a t t e r is c o m m u t a t i v e with respect to c o o r d i n a t e differentation, i.e.

354 • Nonlinear Thermal Systems while t h e former is n o t , i.e

±(2L

dxAdh

or

dh \ Ox,

f.h,i

7^

(10.120)

f,i.h

(See t h e proof below, Eq. (10.132).) T h u s , special care m u s t b e exercised while using t h e m a t e r i a l derivative in sensitivity derivations. T h e m a t e r i a l derivative o p e r a t o r can be applied to a scalar, a vector, or a tensor function of t h e spatial position x a n d design A, i.e. t

dfi ^

=

dfij

dh

fi.h

~

fi,h

T

fij.h

=

fij,h

+

Vj

for any

fij

= f {xj\

h)

{

(10.121) for any fl3

Vkfij,k

=

f (x \h) l3

k

iii) Generally, we deal with functions / t h a t are given implicitly in t e r m s of /i, i.e. with functions t h a t cannot be d e t e r m i n e d for any given h by merely a straightforward function evaluation. Therefore, while t h e second tern] on t h e right-hand side of Eq. (10.118) can easily be c o m p u t e d as an explicit calculation provided v is specified, this is not so with t h e first t e r m , which calls for an additional analysis. In other words, t h e n o t a t i o n = Of /dh should not b e taken here as suggesting an explicit d e p e n d e n c e of / on h, which is in contrast with w h a t we have been dealing with so far in t h e book. l

T h e J a c o b i a n m a t r i x of t h e t r a n s f o r m a t i o n is o b t a i n e d as ,

r

dxj(x ; k

r {x ; tJ

h)

h) = ——

k

(10.122)

= bij +bh v

itj

with its d e t e r m i n a n t denoted by J , J = d e t f F ^ ] . We recall t h e following results, cf. E q s . (5.189) (5.198) a n d [2,*33, 40, 160, 190, 199]: • t h e m a t e r i a l derivative of t h e differential surface area dH = d[J(x \

d(dfi)

\di1{xi\h)\

x

dh dJ{x ;h)

h)

h) dQ(x ; h)} t

dh

h= h

dO(.Tj;

h=h

dn{xi\h)J. dn

t

dh h-h JndSl = V • v d O =

h

(10.123)

vadQ

• t h e m a t e r i a l derivative of t h e differential surface area d(dU) = d(0fl(x ; 1

[d(on)(x h)}. i;

h

=

d[d(0Q)} dh

d[dJ(xi;h)d(dn{xi;h))] dh

h=h

d[0J{x ;h)) t

dh

h))

d{dn{xi]h)) h=h

= [ V - v - ( V v n ) - n ] d ( a n ) = (6 = [ V - v - V - ( n u ) + //t> ]d(0ft) n

n

(10.124)

S h a p e S e n s i t i v i t y for T r a n s i e n t P r o b l e m s

•

355

—V • n = - n , , , is t h e c u r v a t u r e of 0ft in / ? where V = \T~ ^- T~\, H ax\ 0x2 0x3 (and twice t h e m e a n c u r v a t u r e of 0ft in R ), and v — v • n = v n, is t h e normal c o m p o n e n t of t h e design velocity field v . As shown in [71, 199], for i n s t a n c e , only t h e n o r m a l c o m p o n e n t of t h e design velocity field m a y influence t h e variation of t h e differential surface element so t h a t t h e last equation becomes 2

1 1

A

n

r

x

[d(0Q)],

= Hv

k

d(<9ft) = -n

n

v n,

jtj

(10.125)

{

We also recall t h e expression dn

t

r>,.h = ("« n

~dh

- %)

3

(10.126)

nv k

ky

which will soon be m a d e use of since t h e normal to t h e b o u n d a r y will typically d e p e n d on t h e design in s h a p e sensitivity problems, n . = t>,(xi,. h ) . F u r t h e r m o r e , we recall t h e expressions for: • t h e m a t e r i a l derivative of any volume integral F(h) = JQ F

=

h

dh Jn

= J (%

h=h

+ K-)

n

d ( . dh Jn' Jn

/dft

f(x -h)J(x :h)dtt l

1

h=h

= j^-f.h + fr,.i)

m

f(x')dft(.r)

(10.127

dft

t h e m a t e r i a l derivative of any surface integral G(h) — f Q

g(x )d(0Q(x ))

d

= f-1 f . ah Jan

G.

h

g(x,)d(dQ)

dh Jan g(xi;h)dJ(xi;h)d{dn)

h=h

= f (g. + gllv ) Jan

= [ [g.h + g(Sij-n n )v , ]d(dQ) Jan ' ' i

j

!

i j

h

n

d(0ft)

t

h=h

(10.128)

Let us n o t e for future reference t h a t t h e m a t e r i a l derivative of t h e t e m p e r a t u r e field 0(x ; h) is given as l

O. = 0 + v O h

h

i

(10.129)

ii

while t h e m a t e r i a l derivative of t h e t e m p e r a t u r e gradient field 0 , = 00/dx given as

l

0M = {0,i),h + VJ (0,)

= 0,

}J

+ r, (),,,

ih

is

(10.130)

By differentiating Eq. (10.129) with respect to x we obtain t

0. ,i = 0 h

ihi

+

VJ

Oji + v

jti

0

tj

= 0

tih

+

VJ

0

tij

+ vjti

0j

(10.131)

which, by Eq. (10.130), yields O.uk = 0. .

h t

d f m_ dh V dx.

-r 0,j

(10.132)

JA

d (d0\ Ox,

\dh

dvj 00 0x Ox,j x

(10.133)

356 • Nonlinear Thermal Systems We now r e t u r n t o t h e sensitivity problem a n d c o m p u t e t h e m a t e r i a l derivative of t h e response functional of t h e t y p e (10.22) by using E q s . (10.127) a n d (10.128) as Q.h = f \ ( JO

{g.h + g v ) dfi + /

{ ™ + g Hv )

[JQ

dO

[q)

iti

g

n

(10.134)

dr

JdQi

where g.h

8g dO

dg d0

dOdh^

W ~dh

9,0 0-h dg

(q)

9,&,)0,i.h

dO

+ f l

dg

dq

r

, / i

dg dx~

V i

+

+

9,i

i

v

dq

dg

dh

dh

{q)

dO dh

g.h

dh

+

t

+

lq)

dg

tl

dg

iq)

(10.135)

(<j)

dxi

g,o v.h + .9;, q.h + .9,/, + «' 9,i v

-

By n o t i n g t h a t on dfi\

cf. E q s . (10.3), t h e r e holds

q = q m = -kij(xk]

h) n^Xk] h) 0 (x ;

t

tj

h) = q{x : h)

k

(10.136)

z

we o b t a i n by E q s . (10.118) a n d (10.132) q.h — —kij.h =

0,j — k{j rii.h 0j — kij n,

-k^

n,{().>,,j - vi

-k-j

0 (m n - S ) ni v

tJ

y]

0j) - (kijj ik

k

0 .h tJ

O.h + kij

th

+ vi k jj)

n, 6j

t

(10.137)

i k

S u b s t i t u t i n g this into Eq. (10.135)2 allows us t o express g f (similarly as is in t h e case of g) entirely in t e r m s of t h e p r i m a l p r o b l e m q u a n t i t i e s a n d t h e m a t e r i a l derivatives of t h e t e m p e r a t u r e O.h as well as its spatial derivatives 0.h • (Recall t h a t t h e t h e r m a l c o n d u c t i v i t y tensor k is an explicitly given function of its a r g u m e n t s so t h a t k h a n d k j can easily be d e t e r m i n e d . ) In o t h e r words, in order t o c o m p u t e t h e t h e r m a l response sensitivity dQ/dh by E q s . (10.134) a n d (10.135) we j u s t need t o solve t h e p r i m a l heat transfer p r o b l e m for 0(x ;h) a n d t h e n t o find a way t o d e t e r m i n e its m a t e r i a l derivative 0.h{x \ h). To o b t a i n e q u a t i o n s defining t h e first-order sensitivity field Vf we differentiate t h e heat transfer e q u a t i o n (10.1) with respect to h which yields {

tl

t J

tJ

lh

t

t

t

kj t

0.h,,j

+

0j - kij(20

k .h

- 0

vj

lJtl

tTn

m

c,,,.!.;) +

Q.h

=

('O.h

+

c.h

0

(10.138)

where we have used t h e s y m m e t r y of k j a n d t

kij,i.h

—

kijjfi

f'.h

-

cg

Qh

=

Q.O O.h + Q,(0.,) 0,i.h + Q.h + Q,m m

=

Q.O O.h + Q,{0 )

t

Oh O.h

k jih

+ c

t

ih

-f £

T k m

v

l

j

t

m

v

m

j^gj

m

v

it

(0.h,i — V ,i 0m) + Q,h + Q.m V m

m

Shape Sensitivity for Transient Problems •

357

E q u a t i o n (10.138) can now be r e w r i t t e n in t h e form + kjjo 0j 0. - d 0 e. + Q,e Q.h + Q,{o ) O.h.i h

~b

+ Q,h

<e

~h ^ij,tm

(kij,ih +

Q,m V

m

h

A

) @ ,j ~

(2.6

k{j

Q,(6, )

0,im

t

V

l

m

-

m

6

Vj m

{c,h. +

C

^'m.j'j')

m

U

m

)0 =

m

C

6^

(10.140)

or kij6. i

+ (k , g6j-c od

hi j

ij i

+ Q )e.

f

t0

h

+ Q e )0 ii

i

+ Q = c,6.

Mti

(10.141)

h

in which t h e ' s o u r c e ' t e r m Q is defined as

+ Q,h + @,m v

- Q,(e, ) 6,tm v

m

t

m

~ (c,h + c

m

v) 6

(10.142)

m

a n d can easily be c o m p u t e d provided t h e p r i m a l p r o b l e m has been solved for 9(x{, r ; h) a n d v(x ; h) is given. l

N e x t , by differentiating t h e b o u n d a r y condition (10.2) and t h e initial condition (10.4) we arrive at d6

d/i

dd

.

or

d/i

fl.fc

.

= e.

h

on aft" at any r

(10.143)

and d^

d0_ dh

The boundary

f

or

dA

T=0

condition

h

= 9\ +

i

n

ft

imposed on t h e surface heat

at r = 0

(10.144)

flow (10.3)

becomes,

cf. Eq. (10.137) — k{j rij 6. ,i - kijfi Hi 0j 6.h = q.h + h

+ u; k ) lhl

— kij(rii6 i-\-ni6 i)vij-\-qrijnjVj i t

t

n, 0j

on

t

dft

9

(10.145)

t h e r i g h t - h a n d side of which can be d e t e r m i n e d given t h e p r i m a l solution 6(x T\h). T h i s basically c o m p l e t e s t h e local formulation of t h e M D A : in order to find O.h one has to solve t h e i n i t i a l - b o u n d a r y value problem consisting of t h e field e q u a t i o n (10.141), b o u n d a r y conditions (10.143) and (10.145), and initial c o n d i t i o n (10.144). Having solved it for 0.h(x ,T\h) t h e response sensitivity follows by Eqs. (10.134)-(10.137). l:

t

To find e q u a t i o n s which can be used as t h e basis for an a p p r o x i m a t e analysis we m a y proceed e i t h e r by forming a residual expression for t h e p r o b l e m just f o r m u l a t e d (with 6.h as t h e u n k n o w n function) or by using t h e residual considered for t h e p r i m a r y p r o b l e m followed by its m a t e r i a l different at ion. For reasons of easier c o m p a r i s o n s of t h e DPA versus t h e MDA to be carried out later on t h e

358 • Nonlinear Thermal Systems l a t t e r a p p r o a c h is t a k e n . residual (10.12) t o o b t a i n

We t h u s perform t h e m a t e r i a l differentation of t h e

R

~ fjykvh

0 ,j, o . , 4 - k

O.j 0., \ k

rj

\(.

00

t

J)0

„

h

+f

M. o,

+ /

(q.h4> + q(p.h)d{0Q)

- r

(/OO

,

\

(<- Jf ~Q)o (

4

7

O^o.h.,

tJ

o,k

k j

\

, 1)

dQ

(0Q)j,

T

+ f

q<j>[d{dn)].

= 0

h

(10.M6)

which is equal t o zero on account of t h e imposed p r o p e r t y t h a t t h e residual vanishes at both t h e n o m i n a l h a n d p e r t u r b e d h + fih values of design. Moreover, t h e u n d e r l i n e d t e r m s in Eq. (10.146) vanish because they simply represent t h e residual (10.12) with t h e weighting function
J

R.h

k,

|

tj/

4" k

tJ

0 j o,, 0.

t-

h

[0.h,j ~ " m , j

k,

hk

v

k

I).,

4-

o . ,

k,jj,

0j

m ) (P,i ~ k{j 0j V ,i 0f,c k

kij Oj , • + ^c ^ - - Q^j 0 v

+

o . ,

t

k:k

Q,o O.h + (1(0,) 0 .h +

+ (c,o 0./, + c/, 4 - < " > ?.

+ Q,ih v ) <^> } dQ

yl

m

[('/.o 0./. + ih + f/,- ?',) 4> + q Uv 0l d ( f M ) = 0

+ /

(10.147

n

which further becomes

f = /

r

1

/1

U,j 0 , i

Ju L

T /

1

+ ^./'•*>., (

#

OXJ

h

0, (~ \

Or

0 '/,/ ^ . / i <1(M>) t / I A:

-

^ i j ( " m , j 0 . m 0.1

+

+

0

)

[c.h

+ /

4

C

> M

+

V

m

u

c7

-

+

(Q,/,

+

4-

~ 80 c,o ~

Or

^ ^ Q,o - Q,(6,) 0

Ox-iJ.

Y O.hdQ —

0 • o,, 4 - * 7 , / , 0., 0 , ,

;?>

kj 0J 0 . ,

" m . i 0,j 0 , m ) +

—

—

Q,

m

?>

m

+

) 4-

-

0

fC — T'fc.t' 0;fc

0

}

dQ

(10.148)

Using t h e c o m p a c t n o t a t i o n i n t r o d u c e d earlier in Section 10.2, cf. Eqs. (10.133) and (10.119). Eq. (10.148) is reduced to t h e form

/<*•'> o0 = - P h

(10.149)

Shape Sensitivity for Transient Problems •

359

w h e r e t h e r i g h t - h a n d - s i d e t e r m P is defined as P =

| kj,k Vk &,j ,i + k

tJ:h

Oj (j> i - kij [y j n

4-

0

mt

m

j 4 - v ^ Oj 0 , m

m

)

Vk,k

+ 4> (c,h

/

4-

09 4- c

m

t; ) m

—

-

4 - Q,

[Q^

h

[( + q Hv

m

i> )

+

m

Q,((5,) Vjt,t-0;jt

I

dft (10.150)

] d(<9ft)

n

E q u a t i o n (10.149) m a y be regarded as t h e final result of t h e M D A derivatives. It says t h a t t h e t e m p e r a t u r e s h a p e sensitivity field #/, can be c o m p u t e d from t h e linear differential e q u a t i o n (10.149) provided: (i) we have solved t h e p r i m a l problem for 9 which m a k e s it possible to e v a l u a t e t h e r i g h t - h a n d side of Eq. (10.149), and (ii) t h e s h a p e velocity field v is given. Having found 0 ^ t h e response s h a p e sensitivity follows from Eqs. (10.134)-( 10.137). It is left to t h e reader to check t h a t t h e a l t e r n a t i v e way of deriving t h e gove r n i n g M D A e q u a t i o n based on forming t h e integral residual for t h e local formulation (10.141) with t h e corresponding b o u n d a r y conditons result in Eq. (10.149) as well.

10.4.3

The Domain Parameterization Approach— DPA/DDM

We r e m e m b e r from t h e discussion in C h a p t e r 6 t h a t t h e d o m a i n p a r a m e t e r i z a tion a p p r o a c h consists in m a p p i n g all t h e m a t e r i a l configurations of t h e b o d y for different s h a p e design p a r a m e t e r s into t h e selected fixed d o m a i n called t h e reference s t a t e . All t h e field variables and integrals are t h e n t r a n s f o r m e d to this fixed reference d o m a i n . T h e a p p r o a c h is in fact q u i t e similar to t h e isoparam e t r i c a p p r o a c h widely employed in t h e finite element m e t h o d (with a different objective, though). Recalling t h e D P A t r a n s f o r m a t i o n , cf. Sections 3.5 and 6.1 Xi = Xi{ Xj\ h) ,

xi G ft ; Xi G

T

r

a n d t h e c o r r e s p o n d i n g J a c o b i a n s such t h a t dft = JdW

,

ft

r

(10.151)

1

d(dft) = J d(0Q )

(10.152)

r

d

t h e residual (10.12) is t r a n s f o r m e d to t h e reference d o m a i n to r e a d

R=

f

\kij0.yx j<j>.j + k

Jn l r

(c^-Q)]jdil + I r

\

OT

/

J

2

q
d

(10.153)

JdiY

Because of the assumption of no mechanical straining in the body, the indices indicating the configurations at hand {t, t 4 etc.) can be dropped without any danger of confusion. T h e weighting function is assumed for simplicity in the form 0 ( X j ) , which does not influence the final results. 1

2

r

360 • Nonlinear Thermal Systems all t h e functions being now d e p e n d e n t on x . becomes

T h e response functional (10.22)

r

t

Q{h) =

I J0

[

gJdW-r

g™J d(diV) d r

[

tJw

(10.154)

d

JdQ <J r

whose t o t a l design variation follows as, cf. Eq. (10.134) SQ = JT

J

(6g J + g dj)

dfl

+J

T

(tg J

4- g

iq)

iq)

d

dJ )

dr

d(dtt ) r

d

(10.155)

with 6g Sg^

4- g,(0 )( Xi,j

g,e SO

= gySO

4- d x

60

r

+ gySq

+

O.^j 4- g

r

;J

tl

ltJ

th

Sh

(10.156)

gySh

a n d dJ and dJa defined in Eq. (6.46) It is seen t h a t in order to c o m p u t e SQ we need to d e t e r m i n e t h e t e m p e r a t u r e s h a p e design variation SO. To o b t a i n t h e e q u a t i o n for SO we t a k e t h e design variation of t h e residual (10.153) leading to Eq. (10.28) s u p p l e m e n t e d by t h e t e r m s resulting from t h e variation of t h e J a c o b i a n s SR

-

d Kj 0,k ;x + kij i d x,

a

, , d

.

+ /

00

•E k,j

;

r

„

„

d

dn

1

SOJ

4>ie~60J d(dtt ) T

d

+ J

(dkij

r

x

k

o

d

+

+ / dqJ d{dtt ) Jdn
0.j (f>-i

x , ) k

3

4-

+[ \kij0, x j(f>,i^(c^ Jn'l

r

k

qd)dJ d{dn )

kf

dQJdW

[dc — -

~Q\AdJ d^ J .

T

T

d

r

T

\ or

= 0

r

d

(10.157)

T h e design variation SR is set equal to zero on account of t h e usual a r g u m e n t , cf. Eq. (10.25). Using t h e c o m p a c t n o t a t i o n , Eq. (10.157) becomes, cf. Eq. (10.30) R^

oS0=

-dR

(10.158)

in which f dR

r

=

(dk

tJ

+ [

Jdn i r

4- / Jaw

x

r

/

do 0. . 4 - <j> \ dc —

4 - k^ d x ^ T

kiJ

k

cj>dq J d(dW) d

q<j>dJ d{diY) d

}J

4- / Jn

r

= 0

n

{

\kij 0,j x r

L

\ JdO -dQJ

ko

cp.,

+

\

( c ^ - Q)J or

J

T

.

dJ

dn

r

(10.159)

Shape Sensitivity for Transient Problems •

361

E q u a t i o n (10.158) is t h e f u n d a m e n t a l D P A / D D M e q u a t i o n for t h e s h a p e sensitivity of t h e nonlinear t r a n s i e n t heat transfer problem; it can be used to determ i n e t h e t e m p e r a t u r e s h a p e sensitivity coefficient dQ/dh once t h e c o r r e s p o n d i n g h e a t transfer p r o b l e m has been solved for 6. T h e finite element formulation for t h e p r o b l e m (10.158) can be o b t a i n e d in a r o u t i n e m a n n e r following t h e p a t t e r n described in Section 10.3. T h e i m p l e m e n t a t i o n issues, t h e ' p a r a l l e l ' t i m e integration s c h e m e for t h e primal and sensitivity p r o b l e m s in p a r t i c u l a r , have also been a l r e a d y described and are therefore o m i t t e d here as well.

10.4.4

Comparison of Methods

It is q u i t e n a t u r a l to expect t h a t using any of t h e two m e t h o d s described in Sections 10.4.2 and 10.4.3 leads to t h e s a m e results (even t h o u g h their c o m p u t a t i o n a l performances very clearly differ). However, j u s t by looking at t h e final expressions derived in t h e framework of each m e t h o d it is difficult to confirm t h e i r equivalence. To check w h e t h e r t h e m e t h o d s are indeed identical, we first establish t h e following c o r r e s p o n d e n c e between t h e q u a n t i t i e s inherent in t h e D P A (i.e. S,d) and t h e M D A (i.e. g ^ g ^ ) , respectively: Sg = g.h,Sh,

dg = g^Sh

for any g

(10.160)

In view of t h e above relationships it is now u n d e r s t a n d a b l e why we h a v e used t h e s a m e s y m b o l s d/dh and d/dh to define t h e variations typical of t h e D P A

6g = ^ 6 h ,

d

g

=

A

9

S

°-

h

(1

161)

a n d t h e derivatives typical of t h e M D A

9

= dh'

h

<

9

h

=

dh

( 1

°-

1 6 2 )

Recall t h a t t h e D P A relies on m a p p i n g from a fixed reference d o m a i n to a d o m a i n occupied by t h e m a t e r i a l configuration whereas in t h e M D A t h e m a t e r i a l configu r a t i o n is m a p p e d to a configuration m o v i n g according to imposed s h a p e varia t i o n s . Therefore, t h e t r a n s f o r m a t i o n s inherent in b o t h techniques have in fact not m u c h in c o m m o n . However, let us recall t h a t , cf. Eqs. (10.123) a n d (10.125) J

'

h

=

ell

=

^

V i , i

'

'

Jd ,h

=

~oT

=

J &

H

V

(10.163)

n

a n d , since t h e derivatives are explicit dJ = J.h&h = Jvi^Sh

,

= JQ Hv 8h

dJd = Jd.h^h

(10.164)

n

Using E q s . (10.160) a n d (10.164) t h e total design variation (10.155) b e c o m e s (g. + g V i J 6h J dQ

8Q = f\( J0

r

h

lJQ

r

= f\l (9h + 9 Jo Un r

+ [

(g^

+ g

{q)

Hv )Sh n

J

d(dQ )

d(r?fT

dr

r

d

JdQ
dW + / (g^ JdQ i T

+ g«> Hv )Sh n

(10.165;

362 • Nonlinear Thermal Systems In view of t h e relationship

SQ =

Q.h

&h

(10.166)

t h e DPA e q u a t i o n (10.155) reduces t o t h e form derived by t h e M D A a n d given in Eq. (10.134). T h e proof of t h e equivalence of b o t h Eqs. (10.149) a n d (10.158) (from which we m a y alternatively solve for t h e t e m p e r a t u r e s h a p e sensitivity) follows similar lines. T h e t o t a l design variation of t h e residual R in t h e framework of t h e DPA reads, cf. Eq. (10.157)

SR

/

kj 9- x

\

09

4>- +yc — - Qj

r

k

k%J

t

kij 9. x ,j -,i 4- ^ c ~

+

-

r

k

+ /

k

\6(q<{>)J +

<j> dJ | dfT

q4>dJ }d(dn )

(10.167

r

d

d

Using Eqs. (10.160) a n d (10.164) this equation becomes

SR

=j

d

[ »

—

r

n

k

~ \

0

ShJ

- Q ) (p

ky

Sh

kij 9- x r

k

k<J

<j). -4-

\4rti) +

JdiYi

d

kij U. x j (p-i -ryc~

+ dh + /

( ~

1

an

x

^

- Q

qHv }6hd{dn

T

n

i>i,i

}

dfT

(10.168)

J

i.e.

6R =

R. Sh

(10.169)

h

In o t h e r words, we have just shown t h a t Eq. (10.149) a n d Eq. (10.158) a r e effec tively t h e s a m e e q u a t i o n , i.e.

dR=

PSh

(10.170)

We conclude this section be saying t h a t t h e DPA and t h e M D A a r e theoretically fully equivalent a l t h o u g h they may, and typically will, differ significantly in t e r m s of their c o m p u t a t i o n a l efficiency.

10.4.5

The Adjoint System Method

We have already learned in previous chapters how t o employ an a l t e r n a t i v e sensitivity t e c h n i q u e known as t h e adjoint system m e t h o d . It a t t e m p t s t o c o m p u t e t h e response sensitivity directly by eliminating t e m p e r a t u r e sensitivity t h r o u g h t h e use of Lagrange multipliers. To do so for nonlinear transient t h e r m a l p r o b l e m s we consider an a u g m e n t e d functional

g* = Q +

R

o

(10.171)

Shape Sensitivity for Transient Problems • in which Q is given as in Eq. (10.22) while t h e t e r m Ii °

363

is defined as

dO

+ I ^ " ( 9 + A:, n fl ,)cl(an ))d7 (10.172) JdQ i J We observe t h a t t h e i n t e g r a n d of t h e t i m e integration is just t h e residual (10.12) with t h e weighting function 0 replaced by t h e adjoint t e m p e r a t u r e 0 ; we require t h a t t h e l a t t e r vanishes on dVt b u t is a r b i t r a r y otherwise. E q u a t i o n (10.172) becomes r

J

J

;

r

(a)

e

+ / g9^d{dn )dT (10.173) JdQ i To derive t h e A S M e q u a t i o n we shall again use t h e variational principle discussed in Sections 3.4.2 and 4.2. Using it in t h e present context consists first in finding 0 from t h e r e q u i r e m e n t t h a t t h e a u g m e n t e d functional (10.171) be s t a t i o n a r y with respect to t h e p r i m a r y variable 0, i.e. T

r

( a )

dQ* 8 Q* = -Z- o 80 = 0 80

(10.174)

e e

T h e solution 0 ( x , , r ; h) o b t a i n e d from Eq. (10.174) and s u b s t i t u t e d in t h e expression (10.171) t h e n implies t h a t (a)

8Q = dQ*

(10.175)

in which t h e left-hand side is t h e q u a n t i t y sought in t h e sensitivity analysis while t h e r i g h t - h a n d side involves only explicit design derivatives and can be c o m p u t e d as dQ* = f { ( (dgJ Jo UQ

+ gdJ)dQ

+ J

+ / (dg JdU i

r

T

|\dkij

{q)

T

0, Of +

(dq <9 J W

{q)

dJ ) a

d{dtt ) r

{

K

+ q 6> dJ ) (a)

d

+ q

3

< d~c^--dQ^Q

3

^ 0 , 0 \ f ^ ( c ^ - Q ) 0 ^ J

J

d

dJ } d O d{dfi ) r

7

} dr

(10.176)

T h e above result has been o b t a i n e d by: (i) using t h e d o m a i n p a r a m e t e r i z a t i o n t e c h n i q u e according to which all functions and integrals have been t r a n s f o r m e d to t h e reference d o m a i n fT and (ii) c o m p u t i n g t h e explicit variation of Q* given by E q s . (10.171) and (10.173) in t h e form Q' = / ' { JQR [g + k o

+

{j

/

(g

{q)

o, o\f + ( c | £ +g0 * )d(dn ))dT {

)

r

g)

dfi

T

(10.177)

364 • Nonlinear Thermal Systems E v a l u a t i o n of t h e expression (10.176) is straightforward provided 9(x r\ h) and 0 ( x T ; h) needed in it have been d e t e r m i n e d from t h e p r i m a l h e a t transfer p r o b l e m and t h e condition (10.174), respectively. By recalling Eqs. (10.160) and (10.164) we m a y observe t h a t Eq. (10.176) can be t r a n s f o r m e d to t h e form which coincides with t h e m a t e r i a l derivative of t h e functional Q"\ t h u s b o t h t h e MDA and t h e DPA are also equivalent within t h e framework of t h e adjoint m e t h o d . tl

( a )

n

To c o m p u t e Eq. (10.174) is used. However, a detailed discussion of t h e p r o c e d u r e leading to an explicit form of t h e o p e r a t o r e q u a t i o n for has a l r e a d y been given in Sections 10.2.3 and 10.3.1, see Eqs. (10.48)-( 10.55) for t h e c o n t i n u u m and Eqs. (10.109)—(10.113) for t h e semi-discretized a p p r o a c h , respectively. Since t h e relevant derivations involve differentiation with respect to t h e t e m p e r a t u r e (and not t h e design p a r a m e t e r s ), t h e results o b t a i n e d t h e r e apply also to t h e present situation. All further c o m m e n t s regarding t h e solution p r o c e d u r e are now clearly valid as well.

10.5 10.5.1

Kirchhoff Transformation in Shape Sensitivity Steady-State Problems Formulation

As we have seen so far in this c h a p t e r , sensitivity formulations for t h e most general t r a n s i e n t nonlinear heat transfer p r o b l e m s are m a t h e m a t i c a l l y q u i t e c o m p l e x . T h e p u r p o s e of this section is to present an a l t e r n a t i v e for solving s h a p e (and n o n - s h a p e ) sensitivity problems describing isotropic heat c o n d u c t i n g solids with t e m p e r a t u r e d e p e n d e n t properties u n d e r s t e a d y - s t a t e conditions. T h e m e t h o d ology is q u i t e straightforward: (i) t h e nonlinear problem is reduced to a linear one by using t h e so-called Kirchhoff t r a n s f o r m a t i o n , (ii) t h e resulting problem is solved by using a 'linear general sensitivity techniques described above, and

1

version of one of t h e

(iii) t h e sensitivity solution so o b t a i n e d is transformed back to t h e original problem. T h e m e t h o d selected for (ii) is t h e adjoint system m e t h o d c o m b i n e d with t h e m a t e r i a l derivative a p p r o a c h ( A S M / M D A ) . Any other t e c h n i q u e could be used as well, t h o u g h . We consider t h e nonlinear s t e a d y - s t a t e heat conduction p r o b l e m described by, cf. Eqs. (10.1)-(10.3) with Q = 0 and DO/dr = 0 V{k(0)S79)

(k(0)0,ih

0 d9_ drii

9 ,

= 0 ,

Xi

G

ft

x, G 0ft',6 Xi

G

dtt

q

(10.178)

Kirchhoff Transformation in Shape Sensitivity Steady-State Problems • 365 To t r a n s f o r m Eq. (10.178) into t h e s t a n d a r d Laplace problem t h e so-called Kirchhoff t r a n s f o r m a t i o n is used. T h e t e m p e r a t u r e 0 is replaced by a new variable 0 = 0(0) such t h a t 0(0) =

1° k(0)d0 Je

k

0

(10.179)

0

i.e.

m = f

(10.180)

k =

(10.181)

ko 0

where 0

k(0 ) Q

By s u b s t i t u t i n g Eq. (10.180) into Eq. (10.178) we arrive at V 0 = 0 ,

(10.182)

e ft

2

whereas t h e b o u n d a r y conditions b e c o m e 1 0 = — I k{0)d0 d0_

9 6

9

9

80 dn,

dn.

_

= I ,

x e x

dn

6

(10.183) 9

6

k

x e

dn.

t

dn

q

E q u a t i o n s (10.182) and (10.183) form t h e s t a n d a r d linear Laplace b o u n d a r y - v a l u e p r o b l e m which can be solved by any m e t h o d discussed previously in which all t h e simplifications implied by t h e linearity should clearly be taken a d v a n t a g e of. T h e p r i m a r y variable 0 can be recovered using t h e inverse t r a n s f o r m a t i o n symbolized by 0 = 0(0)

(10.184)

For s o m e forms of t h e t e m p e r a t u r e d e p e n d e n t t h e r m a l c o n d u c t i v i t y k(0) relation (10.179) can easily be o b t a i n e d in a closed form. In particular, for a linear function k(0) of t h e form (used in t h e e x a m p l e below) k(0)

ko

(10.185;

1 + /?

we have 0(0)

vo_ 213

1 +

(10.186)

0

and 0(0)

0 o , j

2/30 k Go

^ / 1

2

(10.187)

0

O n c e t h e p a r a m e t e r sensitivity p r o b l e m has been solved for t h e sensitivity coefficient dO/dh, t h e sensitivity coefficient for t h e original variable dO/dh follows by differentiating Eq. (10.187) with respect to h.

366 • Nonlinear Thermal Systems

10.5.2

Computational Illustrations

As an e x a m p l e t h e t h e r m a l difFuser shown schematically in Fig. 10.2 is a n a l y s e d .

an 3

aa

o

4

i

x

Figure 10.2 Axisymmetric thermal diffuser scheme T h e p r o b l e m is a x i s y m m e t r i c with t h e X J - a x i s t a k e n as t h e axis of s y m m e t r y . T h e following b o u n d a r y - v a l u e problem is considered: V{kV0)

= 0

in Q

q = 0 on dft2 U dft q = 10 W r r f on dft 4

2

x

k = 2 0 0 ( 1 + 0.10) W m (x*,x*) = (0.1 m , 0.1 m ) {x?,x°)

_ 2 o

C

= (0.0m, 0.05m)

An i n t e r e s t i n g o b j e c t i v e of t h e considerations (not p u r s u e d here) could be to find t h e s h a p e of t h e d o m a i n ft which gives a uniform t e m p e r a t u r e d i s t r i b u t i o n on dft3 u n d e r t h e m a x i m u m area c o n s t r a i n t . As an i m p o r t a n t step toward t h e above goal t h e sensitivity of t h e following cost functional is considered: Q = f

(0-0) d(dft) 2

where 0 is t h e given t e m p e r a t u r e of 50°C. In t h e analysis dfli and dft are kept fixed, dfta r e m a i n s a straight segment but is allowed to move horizontally and dft2 is allowed to change s h a p e . T h e b o u n d a r y p a r t <90 is a p p r o x i m a t e d by t h e cubic function 4

2

3

X2 —

"

a

™

x

n=0

which has to be satisfied by t h e coordinates of t h e four points A,B,C and D such t h a t x f = 2 x ^ / 3 , x ^ = x ? / 3 and i f = 0 . Since x f = 0 , x f = 0 , x = 0 , x f = 0 and x%=0 are fixed, t h e design variable vector hd, rf=l,2,3, can be t a k e n in t h e form c

x

h =

{h

}

h

2

h} 3

=

{x

2

4

xf

x} 2

Kirchhoff Transformation in Shape Sensitivity Steady-State Problems • 367 T h e vertical c o o r d i n a t e of points located on t h e b o u n d a r y dQ

2

3

x {x ;h) 2

, 3

= 7^1 = 1

l

can be expressed as

Yl ^i d=l c

V

h + c

x

d

n 3

x"xf

while t h e s h a p e velocity vector field becomes, cf. Eq. (10.115) dx

:

v (x \h) 2d

x

dh

r

where

[Cnd]

1

x*

(xf)

2

(x?)

3

1

x?

(xf)

2

(xf)

3

1

xf

(xf)

2

(xf)

3

1

xf

(xf)

2

(xf)

3

-1

T h e new position x of t h e b o u n d a r y dQ its points) d u e to a design p e r t u r b a t i o n 2

x

=

2

+

x

2

(i-e. t h e new vertical c o o r d i n a t e s of is given by 7

2

2dt>h

v

d

d=i

O t h e r q u a n t i t i e s needed to c o m p u t e t h e sensitivity are given as dx

2

1 +

dxi

2n

1

n

2

d x

( \ dxi

2

H

dx

2

1 +

2

dx

2

dxi

A regular mesh of 48 linear triangular elements is used. A m o n g m a n y analyses performed t h e one corresponding to t h e nominal design taken as h° =

{ 0.1000

0.0883

0.0666 }

is considered. T h e results of t h e sensitivity analysis are given in Table 10.2. Defining t h e p e r t u r b e d design vectors as h° } 3

h( ) = {

h*

1.01 h°

h° )

h^

h°

h°

1.01^}

2

= {

2

2

3

we c o m p u t e d t h e t h r e e corresponding s h a p e sensitivity values by t h e finite difference m e t h o d (AQ/Ah) and t h e analytical t e c h n i q u e (dQ/dh) presented above. T h e results are c o m p a r e d in Table 10.2—the differences are a t t r i b u t e d to errors inherent in finite difference evaluation of sensitivity.

368 • Nonlinear T h e r m a l Systems Table 10.2 Shape sensitivity results for the thermal diffusion problem S(h°)

h° 0)

33.256

—

_.

—

—

—

32.799

-0.457

-0.511

10.6

-

30.828

2.428

2.540

4.4

31.717

-1.538

1.559

1.4

h

10.6

AC7/Ah

Design

h

(2)

h

(3)

dC7/dh

% Error

Summary of the Chapter

In this c h a p t e r general procedures for t h e design sensitivity analysis of a r b i t r a r y nonlinear t r a n s i e n t heat transfer problems have been developed. We have thoroughly discussed b o t h t h e c o n t i n u u m - b a s e d and discretized formulation in t h e framework of which t h e direct differentiation as well as t h e adjoint s y s t e m m e t h ods have been e m p l o y e d . For t h e s h a p e design sensitivity t h e material derivative and d o m a i n p a r a m eterization a p p r o a c h have been presented, analysed and c o m p a r e d . T h e equivalence of b o t h techniques has been d e m o n s t r a t e d . As a specific p r o b l e m we have considered t h e use of t h e so-called Kirchhoff t r a n s f o r m a t i o n to develop a s h a p e design sensitivity a l g o r i t h m for nonlinear s t e a d y - s t a t e heat conduction problems. It has been d e m o n s t r a t e d in every case discussed t h a t t h e sensitivity formulations described lend themselves to a straightforward c o m p u t e r i m p l e m e n t a t i o n .

Appendix A Some Mathematical Background A.l

Concepts from Variational Calculus

A. 1.1

Fundamental Problem

T h e m a t h e m a t i c a l p r o b l e m of minimizing (or m a x i m i z i n g ) an integral is dealt w i t h in a special b r a n c h of t h e calculus, called t h e calculus of variations. The m a t h e m a t i c a l t h e o r y shows t h a t t h e final result can be established w i t h o u t t a k i n g into a c c o u n t t h e infinity of t e n t a t i v e l y possible p a t h s . A t e n t a t i v e p a t h that differs from t h e a c t u a l p a t h in an a r b i t r a r y but infinitesimal degree is called a variation of t h e a c t u a l p a t h , and t h e calculus of variations investigates t h e changes in t h e value of an integral caused by such infinitesimal variations of the path. To be specific let us s t a r t with t h e simple case of (A.l) w h e r e J is t h e q u a n t i t y t h a t is to take on an e x t r e m e value, t h e variational i n t e g r a n d / is a known real-valued function of t h e t h r e e real variables indicated b u t t h e d e p e n d e n c e of u on x is not fixed; t h a t is, u(x) is u n k n o w n . In o t h e r words, we are to choose t h e p a t h u(x) passing t h r o u g h t h e points (x,,u^) and ( x , u ) , cf. Fig. A . l , which renders t h e functional J s t a t i o n a r y . In this book. 2

2

u(x)

(*l."l)

6-

X

F i g u r e A . l Optimum path of u(x) and the variation 6u(x) a n d a l m o s t always in cases of physical significance, t h e s t a t i o n a r y value of J be a minimum.

will

370 • Some Mathematical Background Let us a s s u m e t h a t an o p t i m u m p a t h exists and c o m p a r e t h e value of J for our ( u n k n o w n ) p a t h with t h a t o b t a i n e d from neighbouring p a t h s . T h e difference b e t w e e n t h e s e two for a given x is called t h e variation of u(x) and is d e n o t e d by 6u(x); for convenience t h e following n o t a t i o n is a d o p t e d : 8u(x]

(A.2)

rj{x)

w h e r e a scalar factor t gives t h e m a g n i t u d e of t h e variation. T h e function m u s t be d i f f e r e n t i a t e and such t h a t VM

=

= 0

y(x)

(A.3)

i.e. all varied p a t h s m u s t pass t h r o u g h t h e fixed e n d - p o i n t s . Let us further define u(.r: <) = u(x) + erf(x)

(A.4)

a n d a s s u m e t h a t u(x;0) = u(x) is t h e u n k n o w n p a t h t h a t will m i n i m i z e J. functional J m a y now be t r e a t e d as a function of t h e p a r a m e t e r c J(e)

(A.5)

f ^ . u ( x : < ) . - - - ) dx \ dx

= J 'x

{

The

a n d t h e condition for an e x t r e m e value is t h a t dJ« de

(A.6)

= 0 e=0

T h e d e p e n d e n c e of t h e integral on x is contained in u(x;e) therefore dJ(c\ dt

\ du de

Jx,

w h e r e u (x;e) x

du(x;e

= du(x; e)/dx.

du

e)/dx,

(A.7)

de J

T

a n d du(x;

Noting t h a t by Eq. (A.4) du (x; x

= V(x) ,

e)

de

dr/(x)

(A.8)

d.x

Eq. (A.7) reads dj(t)

f**\df

de

Jx,

.du

y(x)

4-

0/

drjjx]

du

dx

: —

x

(A.9)

dx

I n t e g r a t i o n of t h e second t e r m by p a r t s leads to t h e expression dj(e)

X2

. du

T](X]

(K

r){x)

dx\du

dx 4 -

df rj(x)du

(A.10)

which, by Eqs. (A.3) a n d (A.6), b e c o m e s

L

X-2

'dj__d_fdl_ . du

dx V du

rj(x) dx x

= 0

(A.11)

Concepts from Variational Calculus • 371 We n o t e t h a t in this form e has been set to zero and is no longer part of t h e p r o b l e m . Moreover, it is sometimes m o r e convenient to consider Eq. ( A . l 1) multiplied by e, which gives

.du

dj{i

t rj(x) dx

dx\du

8J

de

T

A.12)

0

w h e r e 8J is known as t h e first variation of t h e functional J\ its significance can also be seen by considering t h e Taylor expansion for t h e function AJ(c) = J ( e ) - J(0)

(A.13)

which has t h e form

l O

dj

8J + ^8 J e + =o' 2! d e For s o m e ' t e c h n i c a l ' reasons it is useful to note t h a t d{8t du AJ(e)

=

2

2

2

+

A.l

1)

e

dx

dx

A.

because

du \ dn = c dx~) dx

d(en)

d(8u)

dx

dx

is:

;A.I6)

T h a t is, t h e derivative of t h e variation is equal to t h e variation of t h e derivative. We can similarly prove t h a t variations performed on t h e s u m s , p r o d u c t s and ratios of two functions are defined quite analogously to t h e corresponding rules of differentiation so t h a t

u(x)

u\(x)

u(x)

ui(x)

u (x)

+

2

u (x)

+ 8u2(x) + U\(x)

8u (x)

8u\(x) u (x)

— u\(x)

8u (x)

2

2

ui(x)

u(x)

8u(x) = 8u\(x)

8u(x) = 8u\(x) u (x) 8u(x)

u (x)

2

A. 17

2

2

u\(x)

2

Also, t h e variation of a definite integral is equal to t h e definite integral of t h e variation, i.e. 8 j \(x)dx X

= P

8u(x)dx

(A.18)

Since rj(x) is a r b i t r a r y , condition (A.6) for the existence of a s t a t i o n a r y value of J can only be satisfied if t h e bracketed t e r m in Eq. ( A . l 1) is identically zero, i.e.

df

d f df

du

dx \ du

0

;A.I9)

T h i s is a p a r t i a l differential equation for t h e unknown function u(.r), referred to as t h e Euler equation for t h e problem ( A . l ) . It can readily be seen t h a t Eq. (A. 19) is t h e necessary condition for t h e integral ( A . l ) to be stationary. In o t h e r words: T h e necessary condition for t h e integral ( A . l ) to be stationary, with t h e b o u n d a r y conditions u(xi) = u\ and u(x ) = u , is t h a t t h e Euler e q u a t i o n (A.19) be satisfied. 2

2

372 • Some Mathematical Background Example A . l

For the integral

J=f

^[ku (r)

- rnu (r)} dr

2

Jo

(A.20)

2

with ii = du/dr and k and rn being constants we have / ( r , u,u) = \ [ku {r) - m u ( r ) ] 2

- *

2

(A.21)

U

df -rcr — —mu du and the Euler equation is

^--r-f^n au dr V au /

=

u + uj u 2

= 0

with

w = — m

(A.22)

2

The extremals (general solution) are thus u = /I cosur + B sinu>r

(A.23)

where A and B are arbitrary constants. (Physically, Eq. (A.23) describes the simple harmonic oscillation.) A particular extremal through any two end-points (ri,ui) and (t"2, u-i) with T\, r-i £ [0, T] is obtained by solving the two end-point conditions for A and B. For example, for the extreme curve passing through two points (0,0) and (l.sinw) we have ,4 = 0

and

B =\

(A.24)

that is u = sinu/r

(A.25)

T h e f u n d a m e n t a l variational problem ( A . l ) m a y be generalized in several respects. For i n s t a n c e , we m a y consider a functional typical of s t a t i c t h r e e dimensional solid mechanics applications, which has t h e general form

J=Jf(xi,Ui(x ),Ui j(x ))dn, k

i , j , f c = 1,2,3

k

t

(A.26)

w h e r e u, a n d x represent t h e vectors u = { u i , W 2 ? ^3} — { \) 2, ^3)1 respectively, u = dui/dxj a n d t h e b o u n d a r y conditions are given as a n a l

x

x

x

t

UJ

>)

=

Ui

,

x

k

E dO,

(A.27)

<9ft being t h e b o u n d a r y of a three-dimensional region Q. In o t h e r words, we now have t h e i n t e g r a n d in t h e form of a function of t h r e e i n d e p e n d e n t function variables Ui(x ), which involves t h e functions themselves a n d their first partial derivatives. W i t h o u t presenting tedious derivations which basically follow- those described above, we simply cite t h e final result in t e r m s of t h e Euler e q u a t i o n s as k

2L du,

± ( 1 L ) ~' dxj \ du ) x

=

0.

i

=

1,2,3

(A.28)

ltJ

Thus: Integral (A.26), with b o u n d a r y condition (A.27), will be s t a t i o n a r y if t h e set of t h r e e s i m u l t a n e o u s partial e q u a t i o n s of t h e second order (A.28) (i.e. t h e Euler e q u a t i o n s for t h e integral) a r e satisfied.

Concepts from Variational Calculus •

A. 1.2

373

Integral Form of the Euler Equation

In t h e derivations so far we have a s s u m e d t h a t t h e n(x)) was twice differentiable for all x £ [ x i , x ] so o b t a i n e d . T h e s i t u a t i o n is different for functions b u t are discontinuous at a finite n u m b e r of points corners t h e left- and right-sided derivatives of u(x) t h e existence of t h e derivative 2

unknown function u ( x ) (and t h a t t h e Euler e q u a t i o n was whose first derivatives exist in [ x ! , x ] , i.e. at each of t h e are u n e q u a l . T h i s is because 2

d x V du

x

c a n n o t b e a s s u m e d and Eq. (A.9) cannot be i n t e g r a t e d straightforwardly to obt a i n Eq. (A.10). However, we observe t h a t at a corner point x £ [,Ti, X ] a l t h o u g h t h e derivative u is discontinuous t h e integral 2

x

z{x)

= ££/(r,u(r), , (r))dr U

(A.29)

T

is c o n t i n u o u s ; a n d t h u s

z{x ) x

= 0 (A.30)

e x c e p t at t h e corner where t h e integrand in z(x) h a n d , we have

jfv( )

= z (x)rj(x)

x

=

x

Ou

dx

L

is discontinuous. On t h e o t h e r

z(x]

^-\z(x)n(x]

drj(x)

(A.31

dx

which, i n t r o d u c e d into Eq. (A.9), yields

,_.=f{£N*>'<*>] + ix)n(x)\

+ /

1L

— z(x

du

dx

x

—

~z{x)

drj{x) ~ A

D

L

dx d7?(x; f \ df f d . . , .\ . dx / / —J [ i ( J ? T\T) 1 d: dx Jxi ldu Jx du ' T h i s leads to t h e du Bois-Reymond equation of t h e form X2

x

c (

r

U

T

v

du

x

r

0

(A. 32'

7

x

x

df

U

d

A.33'

£!;/(T, (r),Mr))dr C U

+

w h e r e C is a c o n s t a n t . Since df/du is continuous at t h e corner, by using t h e n o t a t i o n u a n d u respectively for t h e values of t h e left- and right-sided derivatives of u(x) at t h e corner we o b t a i n l

x

r

x

x

A.34 du

u =v- . r

x

which is called t h e first

x

Weierstrass-Erdmann

corner

condition.

374 • Some Mathematical Background T h i s e q u a t i o n is sufficient to answer t h e question w h e t h e r an e x t r e m e curve (with a corner) exists, provided its solution gives u ^ iY \ if it is not t h e case, a further corner condition is needed. We consider now t h e class of admissible p a t h s given by a x-variation r a t h e r t h a n by t h e y-variation as described by Eqs. (A.3) (A.5), i.e. l

x

u

u (r)

=

e) =

x(T;

T

8x — C(x,)

e

4-

£(r)

({x )

=

2

o

.

d~r

=

du

_

dx

~~

A.35)

e£( ) r

=

and t h u s dx

x

• l +

X

,

cC(r)

. dr

A.36)

u(r) 1+

dx

c

((T]

T h e functional J now becomes J(c)

= / ^ ( r "'ri

\

+ ( ( r ) , u ( r ) , - ^ - ) [ l 1 -f e ((r) / t

+c((r)}dr

L

(A.37)

J

a n d , consequently dfdx

de

dx

df

du

du

dt

x

L

F J

dt

W

c

dx

du

x

(1 4-

dr

(

]

( l +

e

C ) 4 - / c } d r

;A.38)

( ) 2

c

X\

which, s e t t i n g t — 0, yields

iX\ [ dr

de

Of. ^

( ^

') ^

dr

(A.39)

with / = / ( r , u ( r ) , w ( r ) ) . Observing t h a t Eq. (A.39) has exactly t h e s a m e form as Eq. (A.9) with replaced by

du

f J

x

d

f

r»

du a n d following t h e s a m e lines as for t h e du B o i s - I l e y m o n d e q u a t i o n we arrive at df. ^

du

J

Xl

(A. 40)

d£

with C being a c o n s t a n t . As before, t h e integral on t h e right-hand side of Eq. (A.40) is continuous at t h e corner a n d so is t h e left-hand side. Hence, if t h e values of t h e left- and right-sided derivatives of u(x) at t h e corner are u and u , respectively, we o b t a i n l

r

x

f

du

T

which is t h e second

x

(A.41)

*

d

J

u

dU

T

Weierstrass-Erdmann

corner

condition.

Concepts from Variational Calculus • 375 Thus: For t h e existence of an e x t r e m e curve with a corner at (x, ?/), at which t h e left- and right-handed derivatives exist, Eqs. (A.34) and (A.41) m u s t have at (x,u) at least one solution (u ,ii* ) for which ii / u . l

l

x

A. 1.3

x

r

x

x

Variable End-Point Problem

W e now consider a m o r e general t y p e of variational problems in which t h e unknown function u(x) is subject to t h e s a m e b o u n d a r y condition u(x\) = u\ at x = x\ b u t is u n c o n s t r a i n e d at x — x . We have then 2

Su(xi)

= crj(x\)

= 0

Su(x )

= en(x )

—

2

2

unspecified

A.42'

a n d using Eq. (A. 10) t h e necessary conditions for s t a t i o n a r i t y b e c o m e df du

d f df dx V du •j J = u x

du

= 0 x

(A.43)

-x

2

T h e b o u n d a r y condition ( A . 4 3 ) 2 is considered as p a r t of t h e Euler e q u a t i o n s ; it follows from Eq. (A.10) since Su(x) (i.e. also n(x)) is a r b i t r a r y b o t h in ( x i , x ) a n d at x = x , and Su(x) in (a, b) is i n d e p e n d e n t of Su(b) (i.e. 8u satisfies different conditions inside t h e d o m a i n and on t h e b o u n d a r y ) . T h e b o u n d a r y conditions t h a t are o b t a i n e d as a Euler e q u a t i o n , like Eq. ( A . 4 3 ) 2 , are called t h e natural boundary conditions. T h e b o u n d a r y conditions t h a t are included in t h e specification of t h e admissible space for t h e variation 6u, like Eq. (A.42)i, are called t h e essential boundary conditions. Essential b o u n d a r y conditions are also known as Dirichlet b o u n d a r y conditions, while n a t u r a l b o u n d ary conditions are known as N e u m a n n b o u n d a r y conditions. In p r o b l e m s of solid m e c h a n i c s it is typical t h a t t h e k i n e m a t i c b o u n d a r y conditions enter t h e variational formulation as t h e n a t u r a l b o u n d a r y conditions while t h e s t a t i c b o u n d a r y conditions have t h e n a t u r e of t h e essential b o u n d a r y conditions. Clearly, leaving t h e b o u n d a r y condition at x = X \ , cf. Eq. (A.42)i, unspecified as well would g e n e r a t e t h e second n a t u r a l b o u n d a r y condition 2

2

| ^ on

x

= 0

(A.44)

x = X i

A n o t h e r i m p o r t a n t class of problems which m a y be t r e a t e d by variational calculus is t h a t consisting of problems with end-points changing their location according to some solution-dependent rule. To exemplify t h e variational proced u r e necessary in such a case let us consider again t h e functional ( A . l ) in which t h e left e n d - p o i n t x = xi is fixed so t h a t Eq. (A.42)i holds t r u e , while t h e location of t h e right end-point x = x is implicitly d e t e r m i n e d by t h e condition, cf. Fig. A.2 2

T (x,tx(x)) = 0

(A.45)

376 • Some Mathematical Background «(*)

T=0

Figure A.2 Family of u(x; i) with variable end-points on the curve T = 0 We t h u s consider a o n e - p a r a m e t e r family of functions u(x; e) originating at t h e point (xi, iii) and t e r m i n a t i n g at t h e point (x , u ) on t h e curve T = 0 for which t h e following condition holds: 2

T (x (c),u(x (e)-e);e)) 2

=

2

2

(A.46)

0

x being a yet u n k n o w n (and therefore d e p e n d e n t on t h e variation p a r a m e t e r e) value of t h e end-point coordinate along t h e rr-axis. In o t h e r words, we define, cf. Eq. (A.4) 2

u(x;t)

=

u(x)

-f

tr)(x)

A.47)

r}{xx) = 0

and a s s u m e t h a t u(x, 0) = u(x) and x (0) = x are t h e u n k n o w n p a t h a n d t h e end-point location t h a t minimize t h e functional J. T h e functional ( A . l ) now becomes 2

2

7

J{t) = J ,

f^

x

c) \ (x;c),—i-^-ij

J

X2

x

,

u

, du(x:

dx

(A.48)

By a s s u m i n g t h e continuity of all t h e derivatives considered and a p p l y i n g t h e rule for t h e p a r a m e t e r differentiation of an integral with t h e u p p e r limit as well as t h e i n t e g r a n d being functions of c, t h e condition for an e x t r e m e value can be w r i t t e n as dj(c)

f —j J2 d

dc

X

+h

2

de

f=0

f

df

x , df

(

JX

X

drj(x]

-\

7]{X)

. du

du

x

dx

0

(A.49)

dx

in which, as a convenient s h o r t h a n d , t h e suffix 2 is employed to d e n o t e values at points on t h e curve T ; t h u s f = f(x ,u(x ),u (x ))Aor\nst&nce. By i n t e g r a t i n g by p a r t s t h e last t e r m and observing t h e essential b o u n d a r y condition r)(x\) = 0 we arrive at 2

dx

2

de

df

2

ou

x

f

jx J x-\ x

2

2

d£_±fdf_^\ .du dx\

Since r)(x) is a r b i t r a r y in (x\,x ) 2

x

2

rj(x) d:

du

0

x

we o b t a i n t h e Euler equation

;A.50)

Concepts from Variational Calculus •

377

a n d t h e variable end-point condition 7

^

de

+

^

=

du

0

(A.52)

x

which, after i n t r o d u c i n g t h e relation, cf. Eq. du du -z— = dt dt 2

V2 =

du ~ dx

2

w r i t t e n out at

(A.47)i

x

2

dx

2

2

A.53

T—

A

dt

a n d m u l t i p l y i n g t h e resulting e q u a t i o n by de, becomes / , df du \ df [f - - 7 — ) dx +-—du = 0 A.54 \ du dx J 0u T h i s formula, called t h e tranversality condition, is known as t h e variable end-point theorem. It can b e used to find t h e location of t h e end-point (x ,u ), giving a non-zero e n d - p o i n t displacement (dx ,du ) from (x ,u ) along t h e curve T = 0. E q u a t i o n ( A . 5 4 ) has t o b e satisfied for any admissible dx and du . However, t h e s e two values are not i n d e p e n d e n t because of t h e relation T = 0, i.e. 2

2

2

2

2

2

x

x

2

2

2

2

2

2

2

2

c7T_

dx 4 — du = 0 (A.55) du X—X2 dx We n o t e t h a t t h e variable end-point t h e o r e m as expressed by Eq. ( A . 5 4 ) is in fact t h e condition t h a t t h e e x t r e m a l u(x) and t h e e n d - p o i n t curve T = 0 cut o r t h o g o n a l l y at (x ,u ). 2

2

2

2

E x a m p l e A . 2 Consider the motion of a particle described by the energy functional J -

I

2

(T-V)dr

(A.56)

where T = ^ m u ( r ) is the kinetic energy and V = mg u(r) the potential energy, with m,g,u and r being the mass, gravity acceleration, position of the particle and time variable, respectively. The problem is to find the minimizing curve for the integral (A.56) such that ( t i , U i ) = (0,0) and (^,1*2) satisfies the condition 2

T = r- T = 0

(A.57)

with T being a positive constant. By using Eq. (A.51) the Euler equation takes the form g + ^-(ii) = g + ii = 0 dr and the extremals for the integral (A.56) are

(A.58)

u = A + Br-^gr

(A.59)

2

with A and B being arbitrary constants. The fixed end-point condition u = 0 at r = 0 implies A — 0 so that the extreme family (A.59) reduces to u = Br-\gr

(A.60)

2

The variable end-point condition requires that the point [T , U ) lies on the extremals (A.60) and on the vertical straight line (A.57), i.e. 2

2

378 • Some Mathematical Background To complete the equations for the unknowns B,T (A.54) has to be satisfied at ( r u ) so that

and u the tranversality condition

2

2 l

J2 ~

- 3 -

Ou

u

1

2

dr + ^— du 2

2

2

2

Ou

= 0

(A.62)

or, from the integrand in Eq. (A.56) -(gu

+ \u\) dr + u du

2

2

2

= 0

2

(A.63)

At ( r , u ) the time derivative u is determined along the extremals by Eq. (A.60) 2

2

2

u

= B - gr

2

(A.64)

2

which, substituted into Eq. (A.63), yields [gu + \{B - gr f) 2

dr +(B - g T ) dr

2

2

2

2

= 0

(A.65)

The end-point displacement (dr , du ) is now obtained from the equation of the end-point straight line (A.57) 2

2

dr

2

= 0

du

2

= arbitrary constant

(A.66)

so that Eq. (A.65) becomes B-gr

(A.67)

= 0

2

and the equation system for {B,r ,u } Br -\grl - u 0 r - T = 0 B - gr — 0 2

2

is completed as, cf. Eq. (A.61)

2

2

(A.68)

2

2

Solving, we find B = gT r = T

(A.69)

2

Thus, the required extremal is the parabola »( ) r

= 9{T

(A.70)

-\T)T

joining the end-points (0,0) and (T, \gT ) 2

(0,0)

as illustrated in Fig. A.3.

T

Figure A.3 Extreme curve u(r) with the end-point on T = 0

Directional Derivatives • 379

A.2

Directional Derivatives

A.2.1

The Gateaux Derivative

In this book we are m a i n l y interested in deriving explicit expression for t h e design variation of a response functional £/(h), i.e. in finding dQ SQ = -^-Sh dh

(A.71)

w i t h h = {hi, h • • •, h } being a D-dimensional vector of design p a r a m e t e r s . T h e symbol 6 ( t h e b a r over 6) is m e a n t to indicate t h a t t h e variation is taken w i t h respect to t h e variables h; t h e functional Q will as a rule d e p e n d on m a n y o t h e r a r g u m e n t s (generalized coordinates, for instance) a n d a precise indication r e g a r d i n g t h e n a t u r e of t h e variation performed a p p e a r s useful. Every effort is m a d e in this text to r e m a i n within a conventional ' e n g i n e e r i n g a p p r o a c h deprived of any m o r e sophisticated m a t h e m a t i c s . However, t h e formal c o m p l e x i t y of some of t h e p r o b l e m s t o b e t r e a t e d (such as those involving ine q u a l i t y c o n s t r a i n t s typical of elastic-plastic a n d c o n t a c t / f r i c t i o n formulations) requires m o r e detailed c o m m e n t s on t h e n a t u r e of t h e m a t h e m a t i c a l o p e r a t i o n s involved in t h e c o m p u t a t i o n of t h e design variations. A full account of t h e problem can b e found in [71], for instance. 2l

D

1

Let us consider Q as a nonlinear m a p p i n g from a vector space / / of design vectors h, h £ H, into a vector space Y of response variables such as t h e disp l a c e m e n t or stress c o m p o n e n t , strain energy, e t c . Q : H -> Y

(A.72)

In o t h e r words, t h e m a p p i n g Q(h) defines a response variable y £ Y once a design vector h £ H is specified. T h e n o r m s in H a n d Y are d e n o t e d as || • \\ a n d || • | | , respectively. Typically, t h e space H m a y b e taken as a D-dimensional Euclidean space 3 t of D - t u p l e s of real n u m b e r s h = (hi, h , • • •, h ) with t h e n o r m H

y

D

2

||h||=

n

(h,h)?

(A.73)

a n d t h e scalar p r o d u c t of two vectors in 5 t defined by D

(h',h )= 2

h ' V

= £

h\hl

(A.74)

d=l

T h e space Y m a y b e t a k e n as, say, t h e space of Lebesgue s q u a r e displacements u L (0) = | u ( x ) , x £ f t 2

: J^u (x)dx 2

< oo J

integrate

(A.75)

w h e r e t h e integral over Q, is t h e Lebesgue integral (which coincides with t h e Riem a n n integral when t h e l a t t e r exists). A n a t u r a l n o r m on this space is defined as L2(Q)

( ' )Unr[L u u

J

u2{x)dx

-

(A 76)

380 • Some Mathematical Background with t h e scalar p r o d u c t defined as t h e integral of t h e p r o d u c t of two functions, i.e.

= [/-(.)•*<.)*

(A.77)

Let us further consider a nominal (given) design h and its variation defined as Sh = e h

(A.78)

where h a n d h are elements of H while e is a real n u m b e r . T h e Gateaux ential of t h e m a p p i n g Q is defined as t h e limit

d £ ( h , h ) = Jim l[0(h

differ-

(A.79)

+ eh)-0(h)

or, equivalently d£(h,h) = - £ ( h + eh) de

(A.80) e=0

A crucial observation is t h a t t h e G a t e a u x differential d ( ? ( h , h ) needs to be n e i t h e r continuous nor linear in h, although it is homogeneous of degree one in it. If d ( / ( h , h ) h a p p e n s to be linear in h, then we can set d£(h,h) =

£h(h)h

(A.81)

in which case £ / ( h ) is called t h e Gateaux derivative of t h e o p e r a t o r Q. T h e G a t e a u x derivative is not continuous generally. Using t h e definition (A.78) we o b t a i n from Eq. (A.81) t h e relation n

Sg(h,-8h) V e

/

= -d£(h,6h) = e e

-Q {h)Sh

;A.82)

h

i.e. after replacing c\Q and Q^ by t h e m o r e often used in this context symbols SQ a n d SQ/Sh, respectively, t h e relation 8Q(h,8h)

=

SQ TtSh Sh

(A.83)

with t h e right-hand side of it referred to as t h e first variation, or directional derivative of Q, cf. Eq. (A.71). It should be e m p h a s i z e d t h a t C/ (i.e. SQ/Sh) is a m a p p i n g which, for fixed h , operates linearly on t h e element Sh; a b e t t e r n o t a t i o n would t h u s be n

SQ(hJh)

= \g (h)\(8h) h

=

^h

( H )

(AM)

Directional Derivatives •

A.2.2

381

The Frechet Derivative

If, for t h e nonlinear o p e r a t o r (A.72) at some point h £ H t h e r e exists a linear o p e r a t o r D £ ( h , h ) such t h a t Q(h, 6h) - Q{h) w h e r e p(h,h)

-

DQ{h, h) 4- p(h, h)

(A.85)

is a r e m i n d e r , i.e.

lim M ^ llhll -0 ||h||»

= 0

(A.86)

H

t h e n D £ / ( h , h ) is called t h e Frechet differential of t h e m a p p i n g Q. It follows directly from t h e above definition t h a t D £ / ( h , h ) is linear in h so t h a t it can b e p r e s e n t e d as D£(h,h)

= £ (h)h

(A.87)

h

in which t h e s y m b o l £?h(h) s t a n d s now for t h e Frechet derivative. If 2 ( h ) is Frechet-differentiable at all points h £ H (i.e. its Frechet derivative exists at every point h £ H), t h e n 2h(h) is a continuous linear m a p p i n g on H. We observe t h a t t h e notion of t h e Frechet derivative has b e e n chosen in such a way as to m a k e all t h e classical rules of differentiation (such as t h e chain rule) available. T h e r e a d e r should not be confused by t h e use of t h e s a m e n o t a t i o n for G a t e a u x a n d Frechet derivatives, t h e m o r e so t h a t t h e definitions are not always (i.e. for any Q) equivalent: t h e existence of t h e Frechet derivative implies t h e e x i s t e n c e of t h e G a t e a u x derivative b u t t h e converse does not hold t r u e . However, if t h e G a t e a u x derivative is c o n t i n u o u s , t h e n it is also t h e Frechet derivative, i.e. d£(h,h)

-

D£(h,h) = £h(h)h

(A.88)

Very often in our general derivations we consider, for simplicity of n o t a t i o n , j u s t one scalar design p a r a m e t e r . T h e implicit a s s u m p t i o n b e h i n d such an analysis is t h a t t h e result SQ = G Sh

(A.89)

h

can b e generalized a n d applied to a m o r e general case of, say, D design p a r a m e t e r s h = {hi, h ,. • •, h }, h £ H, H — Hi x H x • • • x H , by using t h e following important properties: 2

D

2

D

(i) We a s s u m e t h a t t h e directional derivatives (or t h e so-called p a r t i a l G a t e a u x differentials) 6(0 Q(h\, hi; h \...;

h)

2

<$(2) Q(h\\h ,

= — Q(h

D

h ;. ..; h )

2

2

fi(D) Q(hi; h \...; 2

D

h

= -r- Q(h\\ h 4- eh \...;

h

x

2

h, h) D

+ eh ; h \...;

x

D

2

2

= — Q{h\ \ h ;...; 2

h

D

D

D

4- eh

D

c=o

dhi

1=0

dh

2

dh

D

(A.90)

382 • Some Mathematical Background exist and are continuous and linear in 6h\, 8h ,.. ., Sh , respectively; t h e n Q(h) is Frechet differentiable on Hi x H x • • • x H\ x H a n d 2

D

2

SQ — S

Q+ S

(1)

{2)

ohi

Q + • • •+ S

(D)

dh

D

Q oh

2

dh

D

'ii) T h e Frechet derivative (equal to t h e G a t e a u x derivative) dQ_ _

(dQ_

3Q_

dh

I dhi

dh

2

3Q__

•km)

dh

D

which is a continuous function of h = {h\, h . . . , h } contains in such a case c o m p l e t e information a b o u t t h e first-order sensitivity b e h a v i o u r of t h e m a p p i n g Q. 2 l

D

T h e above properties of functional derivatives have serious consequences in t e r m s of t h e p r a c t i c a l aspects of sensitivity evaluation. For it h a p p e n s not infreq u e n t l y in p r a c t i c e t h a t t h e response is not Frechet differentiable at s o m e p o i n t s a n d only directional derivatives exist. A m o n g n u m e r o u s e x a m p l e s we j u s t m e n tion p r o b l e m s with r e p e a t e d eigenvalues, c o n t a c t / f r i c t i o n formulations or theories of plasticity where t h e elastic range in which t h e design changes m a y result in t r a n s i t i o n s from elastic to plastic m a t e r i a l behaviour and vice versa. If c o n t i n u i t y of £ ( h ) ^ h c a n n o t be a s s u m e d , t h e n no single derivative vector £ h contains t h e c o m p l e t e first-order sensitivity information at h since has different values on different sides of t h e non-differentiability region. T h i s is highly inconvenient since, d e p e n d i n g on t h e direction of t h e design p e r t u r b a tion, completely different sensitivity p a t t e r n s m a y be g e n e r a t e d at a given design point h. An obvious and highly instructive implication is t h a t for p r o b l e m s to which Eq. (A.91) applies (i.e. for functionals whose G a t e a u x derivative SQ/8h is c o n t i n u o u s a n d linear in 6h and t h u s equal to t h e Frechet derivative dQ/dh) comp u t a t i o n of this derivative is sufficient to evaluate t h e change in t h e functional d u e to any change in t h e design Sh. For functionals whose G a t e a u x derivative is not c o n t i n u o u s , sensitivity c o m p u t a t i o n s have to be effectively carried out for s o m e specific classes of design p e r t u r b a t i o n vectors separately, t h u s greatly complicating t h e c o m p u t a t i o n a l a l g o r i t h m s . a

n

F o r t u n a t e l y , t h e above is very often a theoretical r a t h e r t h a n practical difficulty—for it is e x t r e m e l y unlikely t h a t t h e discontinuity point will be precisely e n c o u n t e r e d d u r i n g c o m p u t a t i o n . T h e reasons are due to: (i) limited n u m b e r s of so-called Gauss integration points and t i m e i n s t a n t s at which t h e processes a r e m o n i t o r e d in space a n d t i m e , and (ii) numerical precision. Certainly, t h e possibility of j u m p i n g over possible discontinuity points during n u m e r i c a l p r o c e d u r e s has to be carefully analysed to validate t h e results b u t , as it t u r n s o u t , such p o i n t s can very often simply be ignored and t h e analysis continued w i t h o u t any special t r e a t m e n t of t h e m .

Appendix B Stress and Stress-Rate Measures B.l

Stress Measures

Consider t h r e e locations of a m a t e r i a l b o d y referred to as t h e initial, reference a n d c u r r e n t configurations, respectively. Consider further a cross-section of t h e b o d y passing t h r o u g h a typical m a t e r i a l point P , Fig. B . l . T h e a r e a of an

Figure B . l Three configurations of a material body i m a g i n a r y infinitesimal cross-sectional surface element in each of t h e configurations is d e n o t e d by du; , d u / and du;', respectively, while t h e c o r r e s p o n d i n g unit n o r m a l vectors at P are d e n o t e d by n ., n and njj., k = 1, 2, 3. If t h e force vector t r a n s m i t t e d at t i m e r = t t h r o u g h t h e area d u / is given as df , t h e n t h e vector 0

0

r

k

l

k

A =

^4

(B.l)

dur

is called t h e stress vector at P on t h e surface s e p a r a t i n g two p o r t i o n s of t h e m e d i u m . T h e Cauchy ('classical') tensor a \ is defined by t h e relationship k

t

t

(B.2)

384 • Stress and Stress-Rate Measures ( S u m m a t i o n over r e p e a t e d indices except t is employed.) We observe t h a t t h e C a u c h y stress ' m e a s u r e s ' t h e stress with respect to t h e current configuration at t. Let us further i n t r o d u c e a fictitious load vector as

= gd//

i l l

(B.3)

a n d two new stress vectors as H

du

(B.4)

Clearly, definitions ( B . 3 ) and ( B . 4 ) take a d v a n t a g e of t h e a s s u m p t i o n t h a t t h e one-to-one m a p p i n g between t h e reference and current configuration x[ = xl(xlt)

(B.5)

is known. T h e stress tensors corresponding to t h e definition ( A . 2 3 ) can now be defined using t h e relationship _

(r)ft

( r ) l - t

i ~ k

(r)lt

kl

_

h

1

(r)~t

—

(B6) K^-VJ

V

u

a

i

n

T h e tensors ojt/ and < 7 ^ are called, respectively, t h e first and second PiolaKirchhoff stress tensors with respect to t h e reference configuration. T h e second Piola-Kirchhoff stress tensor is s y m m e t r i c , regardless of t h e choice of reference configuration. An i n t e r e s t i n g observation can b e m a d e by considering t h e reference configur a t i o n coinciding with t h e current configuration. We t h e n have t h a t (t)~t _ (t)i~t (r)I

( r )

a

k i

—

a

(B.7)

k i

For t h e case of r = t t h e simplified n o t a t i o n is i n t r o d u c e d to read (t)~t (

° k i

"kl

~t -

—

° k

a

;B.8) k l

It is a straightforward task to prove t h e following relationships b e t w e e n t h e c o m p o n e n t s of different stress tensors expressed in t h e fixed C a r t e s i a n reference frame: (a) T h e C a u c h y stress versus t h e second Piola-Kirchhoff stress based on t h e reference configuration V

4i =

det

dx:

A

A

d

. dx

d

dx\

dx)

13

( B > 9 ) K

'

or, in a b s o l u t e n o t a t i o n T

t 6

Q'

TT?t

(r)~t

TJT\t

(B.r

Stress Measures •

385

where =

det

Q

r

dxi

dxi

dx]]

dx]

(B.10)

g being the mass density. (b) The first Piola-KirchhofF stress versus the second Piola-KirchhofF stress, both based on the reference configuration V (no sum on t) dx

1

(r)I~

t

k

UX

_

(r)

or

(

B.ir

(c) T h e Cauchy stress versus the first Piola-Kirchhoff stress based on the reference configuration V (no sum on r) <9<

dx L

dxi

dxi

W

(B.12)

or

It is seen that for r = t we rediscover the properties (B.8) Another useful stress tensor is called the Kirchhoff r

stress tensor defined as (B.13)

=

It can be shown, [50, 53, 91, 100, 126] that the contravariant components of r in the convective coordinate system (coinciding with the fixed Cartesian system at 'time' r) can be related to the Cartesian coordinates of the Cauchy stress tensor through the relation dxS g dx r

T m

B.14]

dx\

We emphasize that r are the contravariant components of the Kirchhoff stress tensor in the convective coordinates. Relation (B.14) directly results from the fact that the covariant base vectors of such a convective coordinate system are related to the base vectors of the Cartesian system by a linear transformation involving the gradient dx jdx\. By comparing Eqs. (B.9) and (B.14) we may verify the interesting property that the components of d r in the Cartesian coordinate system are equal to the contravariant components of r in the convective system kl

k

(r)

t

M

(B.15)

This in no way implies that the Kirchhoff stress and the second Piola-Kirchhoff stress are equal. In fact, r ^ & since they do not have the same components in the same coordinate system. {r)

1

386 • Stress and Stress-Rate Measures

B.2

Stress-Rate Measures

Before precisely defining t h e tensorial measures of t h e stress i n c r e m e n t generally u n d e r s t o o d as (stress increment) = (stress at t+At)

- (stress at t)

we observe t h a t in order to c o m p a r e ( s u b t r a c t one from a n o t h e r ) two stress m e a s u r e s we have to m a k e sure t h a t they b o t h are referred to t h e s a m e reference s t a t e . In o t h e r words, in t h e definitions of t h e stress at t+At and t t h e s a m e surface e l e m e n t du; m u s t be employed since it is only u n d e r such c i r c u m s t a n c e s t h a t b o t h stresses have t h e correct physical i n t e r p r e t a t i o n of forces related to t h e s a m e reference area. Accordingly, t h e incremental C a u c h y stress defined by

4t '4i At

&°ki =

(B.16)

has no correct i n t e r p r e t a t i o n and c a n n o t be rationally used in t h e finite deformation analysis. O n t h e o t h e r h a n d , t h e incremental Piola-Kirchhoff stress tensors t a k e n with respect to any fixed reference configuration V and defined by

A">»i =

- ""si

1

• '

satisfy t h e above r e q u i r e m e n t and can therefore be used in such an analysis. We e m p h a s i z e t h a t in spite of condition (B.8) t h e i n c r e m e n t a l stresses for t h e reference s t a t e coinciding with t h e one at r = t are not equal to each o t h e r , i.e. ±

A"%

ACT*

± Acr \

(B.18)

A<«&i ^ Am^ k

In t h e case when infinitesimal r a t h e r t h a n finite stress i n c r e m e n t s in t i m e are n e e d e d , t h e stress rates d ,a and a can be defined by t a k i n g t h e limit of t h e c o r r e s p o n d i n g finitely i n c r e m e n t a l stresses with At t e n d i n g to zero. To satisfy t h e so-called objectivity r e q u i r e m e n t s imposed on stresses in any finite d e f o r m a t i o n process t h e objective, r a t h e r t h a n o r d i n a r y t i m e derivatives of t h e stress tensors should be employed. O n e of such widely used objective rates of t h e C a u c h y stress is t h e so-called Truesdell rate defined by l

kl

ir-trV

kl

-L<7 +

kl

crtrD

(B.19)

D = symL

(B.20)

where L = FF-' ,

T i m e differentiation of Eq. (B.9) and t h e p r o p e r t y t h a t f(4) dx

V

= t,D Q

)

r

(B.21)

=t

lead to t h e i m p o r t a n t relationship V

I

°kl = °~ki

B.22^

Stress-Rate Measures • 387 T h i s effectively says t h a t in t h e r a t e - t y p e c o n s t i t u t i v e e q u a t i o n s , in place of t h e objective C a u c h y stress r a t e a typically p o s t u l a t e d for large deformation analysis one m a y use t h e ordinary t i m e derivative of t h e second Piola- Kirchhoff stress tensor b \ based on t h e current configuration (i.e. taken with respect to t h e configuration at t i m e t). kl

k

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390 •

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Nonlinear Continuum Mechanics For Finite Element Analysis

Nonlinear Continuum Mechanics for Finite Element Analysis

Nonlinear continuum mechanics for finite element analysis

Nonlinear continuum mechanics for finite element analysis

Nonlinear continuum mechanics for finite element analysis

Nonlinear continuum mechanics for finite element analysis

Nonlinear Continuum Mechanics for Finite Element Analysis

Nonlinear Finite Element Methods

Nonlinear finite element methods

Finite Element Methods in Mechanics

Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd Edition

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Finite Element Based Design Sensitivity Analysis and Optimization

An introduction to nonlinear finite element analysis

An Introduction to Nonlinear Finite Element Analysis

Finite Element Methods for Nonlinear Optical Waveguides

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An Introduction to Nonlinear Finite Element Analysis

Shape sensitivity in elasto-plastic computations

Multiscale Finite Element Methods: Theory and Applications

Multiscale Finite Element Methods: Theory and Applications

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The finite element method. Solid mechanics

The finite element method. Solid mechanics

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Statistical Mechanics: Algorithms and Computations

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Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations

Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations

Nonlinear Continuum Mechanics For Finite Element Analysis

Nonlinear Continuum Mechanics for Finite Element Analysis

Nonlinear continuum mechanics for finite element analysis

Nonlinear continuum mechanics for finite element analysis

Nonlinear continuum mechanics for finite element analysis

Nonlinear continuum mechanics for finite element analysis

Nonlinear Continuum Mechanics for Finite Element Analysis

Nonlinear Finite Element Methods

Nonlinear finite element methods

Finite Element Methods in Mechanics

Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd Edition

Finite Element Based Design Sensitivity Analysis and Optimization

An introduction to nonlinear finite element analysis

Finite Element Based Design Sensitivity Analysis and Optimization

An introduction to nonlinear finite element analysis

An Introduction to Nonlinear Finite Element Analysis

Finite Element Methods for Nonlinear Optical Waveguides

Finite Element Methods for Nonlinear Optical Waveguides

An Introduction to Nonlinear Finite Element Analysis

Shape sensitivity in elasto-plastic computations

Multiscale Finite Element Methods: Theory and Applications

Multiscale Finite Element Methods: Theory and Applications

Multiscale Finite Element Methods-Theory and Applications

Multiscale finite element methods: Theory and applications

The finite element method. Solid mechanics

The finite element method. Solid mechanics

The finite element method. Solid mechanics

Nonlinear Parameter Estimation

Statistical Mechanics: Algorithms and Computations

Statistical mechanics: algorithms and computations

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