Some numerical methods in elastoplasticity

SOME NUMERICAL METHODS IN ELASTOPLASTICITY

A. BERMUDEZ - I. M. VIA~O (1)

ABSTRACT - In this paper we study some methods for the numerical resolution of the deformation equation in elasto-plastic media, subjected to Hencky's law.

We describe an iterative algorithm and show its convergence to the stress field. Then, its performances are given in a typical example with Von Mises plasticity

convex set. PfLagrange Finite Element scheme is implemented by using MODULEF'S Library for numerical computation.

1. Introduction.

In this paper, some numerical methods for solving the deformation equation in elasto-plastic media subjected to Hencky's Law and Von-Mises plasticity criterium, are given. Following Duvant-Lions [4] and Mercier [8], we consider a continuous body occupying an open bounded subset ~2 of R N (N<3) with smooth boundary P. Suppose J-2 fixed on F v c F and let f (x) and F (x) respectively be the body forces and the surface forces acting in ~ and on Fv=F--ieu.

If a denotes the stress field, we have the following equilibrium equations:

(1.1)

I ~ g- / i=0 in ~, i=1 . . . . . N. j

I ~, ~ j n i = F ~ o n I F , i = l . . . . . N. j=l

U = 0 on Fu.

where u is the displacement field.

Received September 24, 1981. (1) Departamento de Ecuaciones Funcionales Unlversidad de Santiago de Compostela

La Corufia (Spain).

336 A. BERMUDEZ - j. M. VIAfiO: Some numerical

Let e (u) be the linear strain tensor, i. e.

l [ au , + ()g' /, (1.2) eq (u)=T\D---~- i Ox, ) 1<i, ]<N,

then Hooke's law for elastic behaviour is

(1.3) Act (x)=s (u (x)),

where A denotes a symmetric automorphism in R m . However, in plastic regions the equation (1.3) is not satisfied and we suppose

that the body obeys Hencky's law. Let B be a closed convex subset of R N' and PB the orthogonal projection from R s' into B relative to the norm

(1.4) I rl = (At . ~)1/~.

Hencky's law states that

(1.5) cr (x)--ea (A -1 e (u (x))),

and therefore the problem consists in solving the two equations (1.1) and (1.5).

For this purpose the following functional framework is considered: Let H be the linear space defined by

(1.6) H--{ 'r=vqeL2(O), 7:ii=zji, l<_i,]<_N},

and K the closed convex subset of R m :

(1.7) K = { z e H : z (x)eB a. e. in s

We endow H with the norm associated with the inner product

(a, "r) = (Aa, r ) , (i .8)

v~here

(1.9) (e, ~')=f D

Let V be the linear space defined by:

(1.10)

e (x). �9 (x) dx.

V={ve(H' (n))~: v=0 on Fv},

methods in elastoplasticity 337

which is a Hilbert space with the classical inner product in (H a (s N. Thus, e may be considered a bounded linear operator from V into H. We

denote by e*eg2 (H,V') the operator given by

(1.11) (a,e(v))=(e* (a),V)v,v for all vei l , ~eH.

Finally, let L be the element of V' defined by

(1.12) L ( v ) = f ] ( x ) . v ( x ) d x + f F ( x ) . v ( x ) d F , g2 I'$,

where ] e (L 2 (s u and F e (L' (FF)) N. Taking into account the previous definitions it is easy to prove that the

set of the solutions of (1.1) is given by:

(1.13) E (L)={'ceH: (v, e ( v ) ) = L (v) for all veV} ,

and then, if a satisfies (1.1) and (1.5), o" is also the unique solution of the following <<stress problem>>:

Minimizing on K N E (L) the ]unctional

.

(1.14) l T i1 11 -

(see Mercier [8]). Numerical methods for solving (1.4) have been given by Mercier in [8]

by considering a dual problem (<<displacement problem>>), for which several algorithms as conjugate gradient, penalty-duality, etc. have been applied.

In this paper, after some background on maximal monotone operators, we recall Mercier's variation of the penalty-duality algorithm though stated in a different form.

Next, another algorithm for solving the ~< displacement problem >> is introduced and its convergence to the stress field is shown.

Their performances are compared in an example with Von Mises plasticity convex subset. We use a PrLagrange finite element method and we accomplish its implementation with routines of the MODULEF library.

2. Maximal monotone operators. (see Br~zis [3] or Pazy [9]).

Let A be a maximal monotone operator in a Hilbert space W. We denote by A ~' the perturbed operator A-ooI, oo being a positive real number.

338 A. BERMUDEZ - I. M. V ~ o : Some numerical

The Yosida approximation of A '~ is defined by

(2.1) A~=I--~]~ ~ ~ ,

where h ~ is the resoIvent operator of A ~, i. e.

(2.2) ]~'~ =- (I q- 2 A~) -1,

and 2 a positive real number such that 2go< 1. For go=0 we write Az instead of A~ ~ We have the following equality

(2.3) A~ '~ ( x ) - 1 - 2go A x 1 - 209 x.

On the other hand, the following result is due to Bermtidez-Moreno [ 1 ]:

(2.4) y e A ~ (x) if and only if y e A z ~ (x+2y)

with 0 < 2 < 1--- . go

EXAMPLE 2.1: Let ~b be a proper, convex, lower semi-continuous function from W into (--oo, ~o ]. It is known that 3~b, the subdifferential of ~b, is a maximal monotone operator. Moreover, the functional ~ba defined by:

(2.5) ~bz (x)= inf t~b (y)-t- ~ ]x--ylZw t, VE IV" t

is Fr6chet-differentiable and we have:

(2.6) (a~bh (x)=~bq (x) for all x e W .

In particular, if C denotes a closed convex subset of W, the face Fc of C defined in Zarantonello [I0] by

(2.7) Vc (x)={yeC: (x, y)w= sup (x, Z)w}, zoO'

is the subdifferential of the support ]unction of C given by

(2.8) ic (x)= sup {(x, z)w}. zEU

methods in elastoptasticity 339

Furthermore, a simple expression for the Yosida approximation of Fc can be obtained. In fact, we have

(2.9)

so that (2.6) implies

(2.10)

(Fc)z (x) ' -Pc(-~).

(ich" (x) = Pc (x).

Finally, observe that from (2.5) (2.8) the following equality ,holds:

1 (2.11) (jch (x) = (x, Pc (x))w----f IPc (x)12w.

3. Numerical algorithms.

In this section two iterative numerical algorithms for solving the nonlinear problem (1.14) are given.

Firstly we observe that Hencky's law and the requirement that a c e (L) are equivalent to the nonlinear equation:

(3.1) e* P~ (A-' e (u))= L,

which is satisfied if and only if u minimizes in V the following functional

(3.2) ~b (v) =(ix)l (A-I e (v))--L (v).

This minimization problem is just the dual problem considered by Mercier, whose article we refer to for existence and uniqueness questions as well as for approximation with Pi-La~an~e finite elements.

From the numerical point of view, the ill-conditioning of the operator e and the fact that (jx)~ is not a quadratic functional produce slow convergence of classical iterative algorithms as gradient-like methods.

In order to overcome this difficulty, appearinR in many other variational problems, some algorithms have been introduced (Glowinski-Marrocco [6], Gabay-Mercier [5J, Mercier [8], Marrocco [7], Bermudez-Moreno [2]).

These algorithms may be considered as variations of the so-called << penalty- duality ~ i. e. Uzawa's algorithm for the << augmented lagrangian >~ which is given below for the elastoplasticity problem.

By using (2.4), Hencky's law:

(3.4)

because we have

(3.5)

340 A. BERMUDEZ - l- M. VIAI~O: Some numerical

(3.3) a--PK (A -1 e (u)),

may be written in the equivalent form:

0"= (FK)I+,t (A -1 8 (U) -1-,~o'),

(AD.=Az+~,

if A is a maximal monotone operator (see BrSzis [3]). Consequently, the problem (3.1) is equivalent to the two following coupled

equations

1 1 2 ~r~= L (3.6) e* P~ (~-~ . A- e (u) + 1 - ~ )

/ 1 l 2 ..

(3.7) ~ P K

The previous formulation leads to the penalty-duality algorithm:

[ 1 A_ , . 2. ) (3.8) e* V X \ l + 2 . . e (u )-k-~f~ ~" = L

(3.9) ~+1__ pK (1_~2 ~ A-I . 2~ (u) + i-+-L ~

Assuming that the displacement problem has a solution, it is not difficult to show that the sequence {a"} given by (3.9) is strongly convergent to the stress field. But it is obvious that for each step a nonlinear problem similar to (3.1) must be solved.

This last difficulty does not exist in ~the following variation proposed by Mercier [8] :

(3.10)

(3.11)

Observe that the problem (3.11) is linear. Actually, problem.

a~ and uleV are arbitrarily given

~=PK(1 ~ A-'~ ~)+~-~ (u an-' ]

e* A -1 e (un+l)=e*A -1 ~ (un)+2 (L--e* (2~- - a"-l)).

it is an elasticity


We consider another linear iterative algorithm for solving the equation (3.1). It has been introduced by Bermudez-Moreno [2] in an abstract framework, but the convergence results given there require some assumptions which are not satisfied in the present case.

Let u be a solution of the ~ displacement problem >>, and q the element of H defined by

(3.12) q=P~: (A -~ e (u) )--o9 A -1 e (u),

where co is a positive real number. From (3.1) we deduce:

(3.13) a) e* A - l e (u)+e* q = L .

On the other hand, applying (2.4) to (3.12) we obtain

(3.14) q-- (PK)~ ~ (A -1 e (u)+ 2 q),

for all 2 such that 0 < 2 < l , and then (3.14) implies: go

(3.15) q=i -z-~ P~ 1-,t +2

1-2ro+22 ) o) + q 1_2co (A-1 ~ (u)+2 q).

The formulation (3.13) (3.15) leads to consider the following algorithm: q~ arbitrarily given

we* A -1 e (u")+e* qn= L,

(3.16) qn+m= 1_@~ P~C(l_21+2(A- 'e (u~)+2 q " ) ) ) -

(.o (A- ' e (u n) + 2 q"), 1 --2~o

(3.17) q,,+l=p n q,,+m+ (1 --On) q'.

PROVOSZTION 3.1: Assume that the displacement problem has a solution.

Then, if 2 = -1- and p~,e[8, 1] for some ~7>0, we have: 2c.o

i) { qn} is bounded

ii) The sequence { a"} defined by

342 A. BERMUDEZ - I . M. VIANO: Some numerical

(3.18) a, , - -Pr(-1_21+2 (A~-' e (un)+,a, q") ),

is strongly convergent to u, the solution of the stress problem (1.14).

PROOF: Firstly we prove the following

LEMMA 3.1: Let A be a maximal monotone operator strongly non expansive,

PROOF: From (2.1) we deduce

tll x - y l t 2 - - - 2 2 IIA: x - -A: ' yll2+ 2,~ ( A : x +col: x - - A : ~ y--co 1~ ~ y,

(5.21) (l:' x - l z ~ (I--22a))[j l :x-l : 'y[[ ~.

On the other hand, we have (see Pazy [9]):

~A: x +oJ 1:' x e A (1~ ~ x), (3.22)

{A:' y+co 12 '~ y e A (Ix ~~ y).

The inequality (3.20) follows from (3.21) and (3.22) taking (3.19) into account.

In order to prove the proposition 3.1 we apply the lemma 3.1 to

A=Px, x = A - * e ( u ) + 2 q and y=A-*e(u" )+2 q".

From (3.20), we obtain:

'2 • q) q .+ ln_c . ~ --~--[lq+a~ ]a (A-I e (tt) q--.], -- la~(A-le(u.)+2q.)]12+

(3.2S) + IIq-qn+'/2112<- Iiq-qnll2+~llA-' e (u)--A-~ e (u~)l12+

2 1 + --~--(A- �9 (u)--A -1 e (u~), q_qn).

co ]:' x--A~ ~ y--~o ]a~ yll 2

io e .

(3.19) (y~--y2, xl--x,_)> llyl--y2[[ 2, for y~eAx, i= 1, 2

then, the following inequality holds:

(3.2o)

I + IIA':'x-Aa'~ lix-yl[ 2.


Moreover, from (3.13) and (3.16) we obtain the equality:

(A-~ e (u)--A -1 e (u"), q_qn)= --co [iA-l e (u)--A -~ e (un)ll 2.

So that (3.23) implies:

I ~-~--Ilq+eo ]z~ (A -~ e (u) q-2 q)--qn+In--~o ]~ (A -x e (un) +) . q-)ll2+ (3.24)

J ~ + [Pq-qn+~/2[l'<- llq-q"l[ ~.

On the other hand we have

tllq - q" '112-p: trq- q"+~/~112 + 2pn (1 -p~) (q_qn+l/2, q _ q , )

(3.25) (q- ( l"Pn) ~ Ilq-q"lle<a~ II q-q"+'/~li~'-~ (1-0~)][q_q.[]2.

Multiplying (3.24) by p. and using (3.25) we obtain:

t~--~Ilqq-co ]z~ (A -x E (u) q-2 q)--qn+l/2--o) ] z '~ (A -~ e (u ~) +2 qn)llz (3.26)

I ! + [Iq-qn+~ll'_< I[q-q~l[ *,

from which the convergence of the sequence { ]]q-q~ll ) follows and then:

(3.27) {q~+Ve+o2lz~(A-le(un)+)~qn)}---rq+eolz~(A-le(u)+2~q)

as n--> oo, strongly in H.

The proposition is now a consequence of (3.27) and the following equalities:

q+co 1 : (A -1 e (u )+2 q ) = q + ~ A -1 ~ (u), (3.28)

and

1 _~ (un) + ) " ) . (3.29) qn+l/2+o2]a~(A-le(un)+2q")=PX l__2co+2(A e q")

4. Numer ica l results.

We take the following example from Mercier [8]: consider a beam which is clamped on one of its ends and loaded with a force of density F on the other extreme.

544 A. BERMUDEZ - J. M. VIA~O: Some numerical

We neglect the weight of the beam and suppose it is a plane stress problem. We choose:

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

~2=]0, 20[ X ]--2, 2[,

s [--2,2],

FF--{20}X[--2,2] U [0,20] X{--2,2},

t=0,

~0 on [0,20]X{--2,2} , F ( x ) =

[(--s, 0) on {20}X[- -2,2] ,

and as plasticity convex Von Mises' convex subset, i. e.:

(4.6) B = { z e R ~ :[r~ ___k~} (k positive constant),

where 1. ] denotes the euclidean norm and ~ the deviator of 1: given by:

1 (4.7) ~D=. :_ N- tr (r) ~.

Thus, the projection on B is

(r if iv~I <--kV2

(4.8, PB (~') = i]_~. T 1 k}r2+ -~-tr(~)~ if I• I >k::2-.

We take the following values for the physical constants:

(4.9) Young's Modulus: E=15.10 s

(4.10) Poisson's coefficient: v = 0 3 3

(4.11) k -- E I/X. lty" 2 ( l + v )

Finally, the parameter s in (4.5) is taken to be successively s = l S 0 and s=160 .

The discretization of (3.1) is obtained by using linear piecewise finite

methods in elastoplasticity 3 4 5

elements. The elasticity problem at each step has been solved with MODULEF library.

The first triangulation is shown in fig. 1. There are 120 triangles, 77 nodes and 140 degrees of freedom.

The chosen numbering of the nodes yields stiffness matrix whose half- bandwidth is 9.

The plastic triangles for s = 1 6 0 and s = 1 5 0 are shown in figures 2 and 3 respectively. Tables 1 and 2 (resp. 3 and 4) give the number of iterations required for convergence when algorithm (3.16) (3.17) (resp. (3.10) (3.11)) is used. The following test is taken:

(4.12) z 1 11n+I- 111 + I 1= n+l - + T

with ~= 1. The second triangulation is shown in fig. 4. It has 576 triangles, 333 nodes

and 648 degrees of freedom. The half-bandwidth is 11. Figure 5 shows the plastic triangles obtained for s = 150.

Finally, tables 5 and 6 give the number of iterations for convergence of the algorithms (3.16) (3.17) and (3.10) (3.11), respectively. In (4.12) we have taken e = 102, the stress being of order 104.

Computing time on MULTICS 4B 68 DPS2 computer are also given.

pn = p 1. 0.9 0.8 0.7 0.6 0.5 0.4 O) \ \

1. * 27 25 28 33 39

0.9 * 29 21 24 27 33

0.8 * 29 15 16 18 21

0.7 * 30 16 16 20 25

0.6 * 31 24 20 24 29

0.5 * 31 20 24 28 27

0.4 * 30 24 27

40

24

31

TABLE 1. Iterations: algorithm (3.16), (3.17) s = 150. Triangulation 1. Computing time (15 iterations): 115 s.

346 A. BERMUaEZ - 1. M. VIA~O: Some numerical

P" = P 1. 0.9 0.8 0.7 0.6 0.5 0.4 c o \

1. * 31 33 38 44

0.9 * 29 29 53 38

0.8 * 29 23 27 31 37

0.7 * 30 18 21 24 28 34

0.6 * 30 20 23 26 26 53

0.5 * 51 24 22 27 33 41

0.4 * 28 27 32 48

TABLE 2. Iterations: algorithm (3.16), (3.17), s = 160. Triangulat ion 1.

(*) For 500 iterations the error is greater than 102.

,~, I terations

1. 42

1.5 28

2. 23

2.5 29

3. 35

TABLE 5. Iterations: algorithm (3.10), (3.11), s = 150. Triangulation 1. Comput ing time (23 iterations): 209 s.

2,

2.5

3.

TAaL~ 4. Iterations: Algorithm

~, Iterations

1. 55

1.5 30

28

34

40

(3.10), (3.11), s = 160. Triangulat ion 1,

methods in elastoplasticlty 347

pn = p 1. 0.9 0.8 0.7 0.6 0.5 0.4

1. * 90 99

0.9 * 101 112

0.8 * 78 86

0.7 * 65 72

0.6 * 51 56

0.5 * 32 35

0.4 * 37 41

111 125

125 141 163

96 109 127

81

TABLE 5. Iterations: Algorithm (3.16), (3.17), s = 150. Triangulation 2. Computing time (32 iterations) 1064 s.

(*) For 200 iterations the error is greater than 103.

,~ Iterations

1.5 142

2. 110

2.5 88

3. 70

3.5 53

4. 55

4.5 56

TABLE 6. Iterations: Algorithm (3.10), (3.11), s = 150. Triangulation 2. Computing time (53) iterations): 1848s.

348 A . BERMUDEZ - J. M . VIA.rio: Some numerical

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u- e-

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methods in eiastopiasticlty 349

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350 A. B.~,Mtm~z - I. M. VIA~O: Some numerical

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methods in elastoplasticity 35I

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352 A. BERMUDEZ - J. M, VIA~rO: Some numerical

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REFERENCES

[1] A. BERMUDEZ, C. MOP, NO, Application o/ pursuit method to optimal controF problems. To be published.

[2] A. BEmVIUDEZ, C. MORENO, Duality methods for solving-variational inequalities. Computer and Mathematics with Applications, 7, (1981) 43-58.

[5] H. BREZIS, Opdrateurs maximaux monotones et semi-groups de contraction dans les espaces de HiIbert (1975). North Holland~

[4] G. DUVAUT, ~. L. LIONS, Les Indquations en Mdcanique et en Physique (1972). Dunod.

[5] D. GABAY, B. MERCIER, A dual algorithm /or the solution of non-linear variational problems via finite element approximation. Computers and Mathematics with Applications, 2, (1976), 17-40.

[6] R. GLOWlNSKI, A. MARROCCO, Sur l'approximation par dldments finis d'ordre 1 et la rdsolution par pdnalisation dualit~ d'une classe de probl~mes de Dirichlet non- lin~aires, R. A. I. R. O. (1975) 41-76.

[7] A. MARROCCO, Expdriences num~riques sur des problkmes non-lindaires rdsolus par dldments finis et Lagrangien AugmentO. Rapport de Recherche n o 309 Ida Laboria.

[8] B. MERCIER, Sur la thdorie et l'anaIyse numdrique de problkmes de pIasticitd. ThOse d'Etat. Universit6 Paris VI (1977).

[9] A. P,~Y, Semi-groups of nonlinear contractions in Hilbert space. Problems of nonlinear Analysis C. I. M. E. (1971), Ed. Cremonese.

[10] E. H. ZARANTONELLO, Projections on convex sets in Hilbert space and spectral theory. Contributions to nonlinear Functional Analysis (1971). Academic Press.

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Some numerical methods in elastoplasticity