Some results of studies of convergence and accuracy of finite element approximations of certain nonlinear problems encountered in finite elasticity are presented. A general technique for obtaining error bounds is also described together with an existence theorem. Numerical results obtained by solving a representative problem are also included

Oden, J. T.

English

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SOME CONVERGENCE PROPERTIES OF FINITE ELEMENT APPROXIMATIONS OF 
PROBLEMS IN NONLINEAR ELASTICITY WITH MULTI-VALUED SOLUTIONS* 
J. T. Oden 
Texas I n s t i t u t e  f o r  Computational Mechanics 
The Univers i ty  of Texas 
SUMMARY 
Some r e s u l t s  of s t u d i e s  of convergence and accuracy of f i n i t e  element ap- 
proximations of  certain non l inea r  problems encountered i n  f i n i t e  e l a s t i c i t y  
are presented.  A gene ra l  technique f o r  ob ta in ing  e r r o r  bounds is a l s o  de- 
s c r i b e d  toge the r  w i t h  an  e x i s t e n c e  theorem. Numerical r e s u l t s  ob ta ined  by 
so lv ing  a r e p r e s e n t a t i v e  problem are a l s o  included.  
INTRODUCTION 
I n  t h i s  n o t e  I summarize some r e c e n t  r e s u l t s  ob ta ined  on f i n i t e  element 
approximations of c e r t a i n  non l inea r  e l l ipt ic-boundary-value problems i n  f i n i t e  
e l a s t i c i t y .  The r e s u l t s  I quo te  h e r e  are given i n  a more e l a b o r a t e  form else- 
where. I n  r e f e r e n c e  1, Ricardo Nicolau and I repor t ed  some r e s u l t s  on a class 
of problems i n  which b i f u r c a t i o n s  occur .  There we cons ider  cases i n  which, 
f o r  a g iven  set  of e x t e r n a l  f o r c e s ,  no t  only can m u l t i p l e  s o l u t i o n s  occur ,  
b u t  a l o s s  of r e g u l a r i t y  can appa ren t ly  r e s u l t  on c e r t a i n  s o l u t i o n  pa ths .  A 
complete account of t h e s e  r e s u l t s  i s  t o  be publ ished i n  a l e n g t h i e r  ar t ic le .  
The p r i n c i p a l  f e a t u r e s  of t h i s  work are (1) a p r i o r i  e r r o r  estimates and 
proofs  of convergence of f i n i t e  element approximations of h igh ly  nonl inear  
e l a s t i c i t y  problems ( t h e s e  estimates are opt imal)  , (2) e r r o r  estimates f o r  
m u l t i p l e  s o l u t i o n s  of a non l inea r  e l l i p t i c  problem ( t h e s e  estimates are a l s o  
opt imal ,  b u t  t h e  p red ica t ed  bounds are d i f f e r e n t  f o r  d i f f e r e n t  s o l u t i o n  p a t h s ) ,  
(3) a d i scuss ion  of s p e c i f i c  numerical  r e s u l t s  and c e r t a i n  s p e c i a l  problems 
connected wi th  t h e  numerical  a n a l y s i s  of  t h i s  class of problems. 
NOTATION AND PRELIMINARIES 
W e  shall  employ the fo l lowing  n o t a t i o n s  and conventions: 
w = (u,v,w) = displacement  v e c t o r  i n  a material body 13, u,  v, and w be ing  che 
Car t e s i an  components of displacement i n  the material d i r e c t i o n s  
x, Y, z. 
N 
* This  work w a s  supported by t h e  Nat iona l  Science Foundation under Grant ENG- 
75-07846. 
555 
https://ntrs.nasa.gov/search.jsp?R=19770003340 2020-03-22T12:10:45+00:00Z
Vw = grad ien t  of w -- - 
W = s t r a i n  energy p e r  u n i t  volume of t h e  body i n  a r e fe rence  conf igu ra t ion ,  W 
be ing  a n  a p p r o p r i a t e l y  i n v a r i a n t  twice-continuously d i f f e r e n t i a b l e  func- 
t i o n  of Vw. 
Y 
V = V(!,p) = p o t e n t i a l  of t h e  e x t e r n a l  f o r c e s  p e r  u n i t  r e f e r e n c e  volume, p 
being a real loading  parameter.  
C = aw/aVw = stress t e n s o r  E C(w) 
U = , s p a c e  of admiss ib le  displacements  = { w: I (W f V)dXdYdZ < a; w = 0 on 352) - - Y Y  
Y -  n -., 
8 6  
(Here 52 is a bounded open set of p a r t i c l e s  composing t h e  i n t e r i o r  of t h e  
body 13 and 8R is  i t s  boundary) 
To i n d i c a t e  va r ious  dependences, w e  a l s o  use  such n o t a t i o n s  as C(w), 
VV(w,p) , etc. -.,- - 
The p o t e n t i a l  V(w,p) is  assumed t o  b e  of  t h e  form - 
where pf  i s  a body f o r c e  t e r m  and V (w,p) i s  non l inea r  i n  w. 
n o t a t i o n s ,  w e  a l s o  in t roduce  t h e  ope ra to r  
To s impl i fy  - 0 ,  ., 
Then, formal ly ,  A i s  g iven  by 
avo(w,P> - 
A(w,p) = -DivC(w) - - - aw - 
W e  are concerned wi th  non l inea r  boundary-value problems of t h e  fo l lowing  
type: f i n d  w Lf such t h a t  - 
We are p a r t i c u l a r l y  concerned w i t h  Ga le rk in  approximations of ( 3 ) .  W e  
i n t roduce  a real  parameter h ,  O <  h < 1, which, of course ,  corresponds t o  t h e  
mesh parameter i n  f i n i t e  element approximations , and denote  { Uh) 
fami ly  of f in i te -d imens iona l  subspaces of U such t h a t  (J U is dense i n  u .  
= a  
O<hLl 
O<h<l 
556 
The Galerkin approximation of (3)  t hen  amounts t o  r e so lv ing  t h e  fo l lowing  
problem: f i n d  w Uh such t h a t  -h 
Upon s u b t r a c t i n g  ( 4 )  from (3)  evalua ted  on q = q we o b t a i n  t h e  or tho-  - -hy  g o n a l i t y  condi t ion :  
SOME HYPOTEHSES ON THE STRESS AND POTENTIAL OPERATORS 
I n  many problems i n  f i n i t e  e l a s t i c i t y ,  i t  appears  t o  be  j u s t i f i e d  t o  
make hypotheses of t h e  fol lowing type  concerning t h e  ope ra to r  A and t h e  space  
U :  
I. The ope ra to r  A of (1) maps U i n t o  its t o p o l o g i c a l  d u a l  U'; LI is a 
r e f l e x i v e  Banach space w i t h  norm I Iwl I - U' 
11. The displacemgnt f i e l d  i n  t h e  body correspondi%g t o  a given load  p 
is contained i n  a space  U w i t h  s t r o n g e r  topology than  U, U being  densely and 
cont inuously imbedded i n  U .  
111. The ope ra to r  A is weakly cont inuous;  i .e. i f  ( w  1 is any sequence n 
converging weakly t o  w then  A(wn,p) converges weakly t o  A(w ,p ) .  
-0 ., -0 , 
I V .  The ope ra to r  A i s  coe rc ive ;  i.e. 
V. A s u f f i c i e n t  cond i t ion  t h a t  I1 holds  is t h a t  A b e  a p o t e n t i a l  opera- 
t o r  w i th  a Gateaux d i f f e r e n t i a l  DA such t h a t  (DA(w + O(w - w ) )  * q,w - w ) = 0 
as n + U. 
YO -n -0 -n -0 
f o r  any sequence { w 1 converging weakly t o  w , V q -ll -0 .., 
V I .  A s u f f i c i e n t  cond i t ion  f o r  coerc iveness  i s  t h a t  t h e r e  e x i s t s  a 
cons tan t  1-1 > 0 such t h a t  
(A(~J,P) - A(y2,P), yl - W2') 2 Y o /  - -2 1 1 '  u - 1-I (7) 
where y is a p o s i t i v e  cons t an t  and p > 1. 
0 
557 
V I I .  There e x i s t  func t ions  B: X U +lR and C: U X +By B weakly con- 
t inuous,  such t h a t  V w ,w ,w < U, -1 -2 -3 
where y is a p o s i t i v e  constant and p > 0. 
Theorem 1 (Existence). L e t  e i t h e r  of t h e  following hold: 
( i )  Conditions I, 111, and I V  above, o r  
( i i )  Conditions I, I V Y  and V, or 
( i i i )  Conditions I, 111, and V I ,  o r  
( i v )  Conditions I, I V Y  and V I .  
Then t h e r e  e x i s t s  a t  least one vec tor  w U t h a t  s a t i s f i e s  (3) f o r  each - p,f E U’.. 
W e  emphasize t h a t  t h e  operator A is not  necessar i ly  monotone. 
FINITE ELEMENT APPROXIMATIONS AND ERROR BOUNDS 
The subspaces U i n  ( 4 )  are assumed t o  be constructed using f i n i t e  ele- 
h 
ment methods. Thus, the  so lu t ion  domain R is pa r t i t i oned  i n t o  E subdomains 
R over which i s  approximated by piecewise polynomials of degree - < k. If 
w - -h 
U 
e . .  
U n U and w is  i t s  p ro jec t ion  i n t o  U h ,  i t  i s  w e l l  known tha t  t he  subspace 
can be designed s o  t h a t  t h e  following hold: 
h 
h being the  mesh parameter and cf a p o s i t i v e  number. 
( i i )  
558 
I n  (10) and ( l l ) ,  C and C1 are cons tan ts  independent of h. 
0 
W e  proceed t o  determine e r r o r  bounds as follows: 
1. The approximation e r r o r  is e = w - w - - - 
2. 
3. For s u f f i c i e n t l y  s m a l l  h, w e  assume t h a t  
owing t o  t h e  cont inui ty  of B(  ,* ) Thus 
1.1’0 
by v i r t u e  of (10) , wherein B(w) = B(w‘,w). 
4 .  
v Y -  
Combining t h e  r e s u l t  1 wi th  (13 ) ,  w e  see t h a t  as h -f 0, a pos i t i ve  
constant C e x i s t s  such t h a t  2 
559 
Thus, f o r  s u f f i c i e n t l y  smooth w, 
gence f o r  t h e  non l inea r  problem so l o  
h e  opt imal  -- rate of conver- 
Theorem 2. 
t o  t h e  non l inea r  
approximation of 
L e t  (8),  (9 ) ,  and (13) ho ld  and l e t  t h e r e  e x i s t  s o l u t i o n s  
boundary-value problem ( 3 ) .  L e t  w U b e  a f i n i t e  element 
w i n  a subspace Uh i n  possess ing  p r o p e r t i e s  (10) and (11). h h  -- 
Then t h e  approximation e r r o r  e = w - w s a t i s f i e s  t h e  bound (14) as h -t 0. 
Moreover, i f  V > 0 and w is s u f f i c i e n t l y  smooth, t h e  opt imal  rate of conver- 
gence is  obtain<d f o r  t h e  non l inea r  problem. 
- - -h 
AN EXAMPLE AND NUMERICAL EXPERIMENTS 
The fol lowing example i s  descr ibed  i n  [ l ] :  
where h = 1 + u' (u = u(x ) ,  v = v ( x ) ) ,  E 0 , * * - , E 4 ,  K 
In t h i s  case, 
are cons tan t s ,  and p - > 0, 
0 
"1 
-4 W (I) = Ref lex ive  Sobolev space = { (u ,v)  : (1.' I + Iv' I 4)dx < 03, 
u(0)  = u(L) = v(0 )  = v(L)  = 01. 
The func t ions  B(y,y) and C(z,y) are complicated func t ions  of t h e  c 
nen t s  u and v and are g iven  i n  [l]. In  t h i s  case, t h e  ope ra to r  A is  n o t  mono- 
tone. 
T e s t  problems were so lved  us ing  piecewise linear f i n i t e  element (k = 1). 
The problem does n o t  ve unique s o l u t i o n s  f o r  p p cr ' 
computed s o l u t i o n s  f a r i o u s  va lues  of p f o r  t h e  case 
E2 = 0.8, E3 = 0.5, E 3 
is reached a t  p = 0.5.  
= -0.1, E4 = -0.2, KO = 1.0. Obse 
F igure  2 shows t h e  rate of convergence a c t u a l l y  ob ta  
computed by comparing t h e  s o l u t i o n  f o r  coa r se  meshes wi th  t h a t  obtained f o r  
100 elements.  A s  p red ic t ed ,  t h e  rate of convergence is 
O(h' + h'+') = O(h + h 5'2) = O(h) 
REFERENCES 
$. Oden, J. T. and Nicolau,  R., "Analysis of F i n i t e  Element Approximations 
of a Boundary-Value Problem i n  F i n i t e  E l a s t i c i t y , "  Formulations and Compu- 
t a t i o n a l  Algorithms i n  F i n i t e  Element Analys is ,  Edi ted by J. Bathe, J. T. 
Oden, and W. Wunderlich, M.L.T. P r e s s ,  Cambridge, Mass., 1976. 
56 1 
LOAD 
PARAMETER 
P , 8  
. t  
.6 
.5 
.4 
. 3  
.2 
.1 
MAXIMUM DISPLACEMENT COMPONENTS 
F igure  1.- Computed equi l ibr ium paths .  
Log h 
Figure 2.- Computed rates of convergence. 
562 


Some convergence properties of finite element approximations of problems in nonlinear elasticity with multi-valued solutions

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