.M!t 97

CONTINUOUS AND DISCONTINUOUS FINITE ELEMENT APPROXIMATIONS

OF SHOCK WAVES IN NONLINEAR ELASTIC SOLIDS

J. T. Oden and L. C. Wellford, Jr.

8.1 Introduction. This paper contains a fairly concise summary of someof our recent ~heoretical results on finite-element-Galerkin methods for the analysisof shock and acceleration waves in certain nonlinear elastodynamics problems. Ourfirst encounter with shock phenomena in nonlinear elastodynamics was in the contextof numerical experiments with extended versions of our finite element codes, originallydeveloped some years ago to study finite static deformations of isotropic hyperelas-tic bodies (for a summary of this work, see [1]). To handl e such phenomen, we developedsome finite element/finite difference schemes of the shock smearing type (e.g.--a Lax-Wendroff type scheme) and, on the basis of a wide collection of numerical experiments,these appeared to be satisfactory for certain classes of problems. These schemes,which, together with some of the numerical results, are described in [2-5J, do notproduce a sharp definition of the shock front and involve a large and often unaccept-able degree of artificial viscosity. What is worse, a recent theoretical study [6Jshows that for certain choices of the artificial viscosity,some of these schemes candiverge. This situation led us to the consideration of shock-fitting schemes, butwe still wished to retain the usual advantageous features of the finite element method.The only recourse was to consider discontinuous shape functions and thus to developfinite elements which contain built-in jumps so as to accommodate shocks. We havesince developed a fairly complete theory of such approximations for one-dimensionalmaterial bodies with strain energy functions given as polynomials in certain strainmeasures. We describe some of these results below. We believe that our approach willprovide an attractive new technique for shock calculations in all physical problemsin which shock phenomena naturally occur.

8.2 Motion of a Slab of Hyperelastic Material. We consider the transversemotion of a thick slab of isotropic, compressible, hyperelastic material subjected tobody forces f(x,t) along the x-axis, which is directed transverse to the slab. If ais the slab thickness, then the slab consists of the material particles, {(x,y,z):-00 < y,z < "'; 0 s x s a; x,y,z" R). For given data f and initial conditions, wewish to determine the transverse displacement u(x,t) of particle x for each time t,,[O,TJ. We assume throughout this work that prior to this displacement the body isin a natural unstressed state.

Since the problem is nonlinear, we must anticipate pOSSible surfaces Y(t)of discontinuity of the veJocity u(x,t) = 3u(x,t)/3t and the displacement gradientux(x,t) = 3u(X,t)/3x which may be propagating through the body at any time t. If thefunction Y(t) defines the particle at which such discontinuities occur at time,t then

,IiI

III

II

I.III:

III

150

We

the basic (l

identified

in the sen

for every ~1

At(or tJ (x, t»fonner condo

Any functiol

jargon, cal'II

lowi ng variT

Then the li

ever

(8.1)

(8.2)

(8.7)

(8.8)

(8.9)

(8.6)

(8.5)

(8.4)

(8.3)

dYV = dt

= Strain energy per unit reference volume

= Longitudinal extension ratio

= Average of ~ at V.,

= Jump in ~ at Vi

p(x) = mass density of the body in the reference

configuration

~weAl = W(u )x

~V = 1/2 (~(V:.t) + ~(V:,t))i ' ,

A(x,t) = 1 + ux(x,t)

C1 (x, t) = Piola-Kirchhoff stress

I{Vi}~=O -'L Jk ' t E [O,tJk=o

Of course, at any specific time, there may be several shock waves propagating through

the body. Thus, V t " [O,T], we assume that the interval I = {x: 0 $ X S a)(y,z)

€ R 2) contains N + 1 shock waves {Yk(t))~=o with intrinsic velocities Vk = dYk/dt,

o s k s N. The open sets

are called shockless subdomains whenever 0 = Vo s Vl s ... s YN=a, V t " [O,t]; when

ever Vi_l = Vi' Ji = 0. Thus, and

yet) describes a shock wave, and the intrinsic speed of this wave is

To complete the description of the problem, we introduce the following no-

tations (here ~(x,t) is an arbitrary function of x and t):

(8.15 )

(8.14 )

(8.10)

(8.11 )

(8.12)

(8.l3a)

(8.13b)V Vi " Q

qX)d:<= 0

151

q(x,t) = x-component of heat flux vector

e(x,t) = Internal energy density

{~(.,t))~ = ~(a,t) - ~(O,t)

We shall say that energy is conserved in a locally integrable sense when-

for every kinematically admissible velocity v.The proof of this theorem is given in [7J and follows from manipulations of

the basic global balance laws of mechanics with jump conditions.8.3. Some Constitutive Properties. Now for the class of materials we have

identified, the form of the Piola-Kirchhoff stress is detennined by the strain energyin the sense that

ever

and

At the boundary surfaces x=o, x=a of the slab, we may prescribe u(x,t)(or tJ (x,t») either as a function of time or the Piola-Kirchhoff stress o(x,t). Theformer conditions are termed kinematic boundary conditions in the physical literature.Any function Vex) satisfying the kinematic conditions is, in keeping with classicaljargon, called kinematically admissible.

With these conventions and notations now established, we are led to the fol-lowing variational principle:

Theorem 8.1. Let the energy be conserved in a locally integrable sense.Then the 1inear momentum isba 1anced ina weak sense if and on ly if

(8.7)

(8. I)

(8.2)

(8.3)

(8.5)

(8.4)

(8.6)

(8.8)

(8.9)

t" [O,t]; when

3gating throughx s aJ(y,z)

!S Vk = dYk/dt.

'ollowing no-

152

where

Wi t

numerous ot

T

(ithat obeys t

where g(.,.)

where {." > d

(i

(i:> 0 such thai

Th(

a nonlinear (

(8.21)

(8.20)

(8.17)

(8.19)

(8.16)

(8.18)

12 = 1 + A2 ,

P

WeAl = L i\A 2kk=O

p

a(u ) = ~ C (1 + u )2k-lx L k xk=l

Ilull 1(I) =Wm

11 = 2 + A2 •

weAl = L Cijk(Il (A) - 3) i(I2(A) - 3)j (I3(A) - 1) ki ,j ,k

Wl(I), for any m ~ 0, is also the completion of Cl(I) in the normm

1 °1The subspace of'll (I) such that u(O) = u(a) = 0 is denoted'll (I), and the dual of01 m_l m'II (1) is denoted'll (I), q = m/(m-l). We shall also use the notation

m q

We shall assume that WeAl corresponds to a compressibile isotropic materialin which there is a polynomial dependence of energy on the principal invariants ofstrain; e.g.

where p is an integer> O. The Piola-Kirchhoff stress is then

Thus, we can find combinations of the material constants C"k that will allow us tolJexpress the strain energy function in the fonn

where Ii(A) are the invariants given, for our particular choice of constraints, bythe simple relations,

where Ck = 2kCk ~ O.Since we are interested in weak solutions, it makes sense at this point to

regard the displacement gradient Ux in a generalized sense. In view of (8.19) and(8.14), integrals of powers of u of order s 2p appear in the weak momentum equation.

x 1Hence, we can regard u as an element of the Sobolev space W2p(I) (1 < 2p < CD):

(8.25)

(8.24)

(8.23)

(8.22)

•

(8.26)

l/j

153

Ilux + vxlltk(2P+l)/(2P_1l(I)

ux

{jo

u* = (1 - e)u + ev ,Os e s 1x x x

2p-l

g(ux'vx) = c Lk=O

(DGo(u~)(uX - vx)'wx> = (o(ux) - o(vx)'wx>

1V U,v,W" W2p(I)

where

The proof of this theorem is lengthy, and is given in [8J together withnumerous other related results:

where g(. ,.) is the nonnegative function.

where ~,.>denotes a duality pairing on Lq(I) and L2p(I), q = 2p/2p-1.

(ii) o(u ) is continuous, in the sense that for any u, v € W12 (I),x p

With these notations in hand, we next cite some fundamental properties.

Theorem 8.2. Let the Piola-Kirchhoff stress, as given by (8.19), represent1 1a nonlinear operator from L2 (I) intoL (IJ'-2 + - = 1. Thenp q p q

(i) o(u ) is strongly monotone on '1121 (I); i.e., there exists a constant yx p

> 0 such that

(iii) o(ux) has a linear Gateaux derivative DGo on L(L2p(I), Lq(I))W2~/(2P-l)that obeys the weak Lagrange formula

(8.18)

(8.16)

(8.19)

(8.17 )

(8.21)

(8.20)

allow us to

traints, by

:ropic materialIvariants of

1s point to8.19) andum equation.< CD):

dual of

The semi-discrete, finite-element-Ga1erkin (FEG) approximation of the varia-tional problem (8.14) is essentially this: find the function U • ~,k(l,Q,P)n CO(I)

such ~hat

8.4. Discontinuous Finite Element Approximations. We now describe a directGa1erkin approximation of (8.14) obtaineo using finite elements with interior discon-tinuities in the shape functions. We begin, as is standard in finite element approx-imations, by partitioning the domain I into subdomains Ik via the partion P: 0 = xO

1 G-1 k- 1 k .< x <. • . < x = a, Ik = {x: X s X s x , 1 $ k s G- 1} . Then I = Uk Ik and 1 f

. k-1 k maxhk = dla Ik = x - x , then we use a mesh parameter, h = k hk. We thus have twopartitions of I; the partition Q defined by the N shock surfaces and the finite ele-ment partion P. If L is any subinterval of I, we denote by Pk(L) the space of poly-nomials of degree s k on L.

A general class of dsicontinuous finite element approximations can be de-fined as elements of the finite-dimensional space,

picted in ffigure, i.l

In this ca!

where S(t)

i.e, thesecients Sf. istrength of

F

where fleisconventiona"thods and Utinuous shal

Th

a special fipolynomial fj

shocl<less dodescribing a

It

(8.27)

(8.30)

(8.28)

(8.29)

<; j ,;G-1)flPk(Q..) ; 1 sis N,lJ

v 'II € ~,k(I,Q,P)

i=l

Q .. = J. n I.lJ 1 J

N

L (pf,W)J. + {o(.,t)W(·)}~; = 1 1

[(plJ,lnJ. + (o(Ux),~/x)J.J1 1

N-l

+~

i=1

N

L

~,k(I,Q,P)={V(x)

Here Qij is the shockless portion of element Ij:

154

where V is deSigned to satisfy the kinematical constaints, and wherein

(U,W)J. = 1 UWdX1 ~i

I

155

(8.35)

(8.34)

{

X11' x ~ Y

1 - *, x > y (8.33)

Ne k

U(x) = L lf~a(x) + L l\~(x)a=l t1=1

x , I (8.31)e

{

X, x ~ YD1 (x) = 0, x > y

In this case, the shock strength parameters are

where Set) is the shock strength

The situation is now clear. Our FEG approximation amounts to constructing~ special finite element nlesh over a given domain in which conventional piecewisepolynomial finite element approximations are used in collections of elements depictingshock1ess domains, while special elements containing jump terms are used in a submeshdescribing a boundary-layer on each side of the shock front.

It follows that each element U(x) in ~,k(I,Q,p)n CO(I) is of the form

where Ne is the number of nodes of element Ie(in our case, Ne = 2). Here Wa(x) areconventional, piecewise polynomial, local bases functions for the finite element me-thods and ur is the value of U(x) at node I~. The basis functions D6(X) are discon-tinuous shape functions with the property that for element Ie

1)6(x k) = 0, k = e, e - 1 )(8.32)

D6(x) = 0, Yi tIe' 1 s i ! N - 1

i.e, these functions vanish whenever no shock is present in element I. The coeffi-ecients S6 in (8.31) represent time-dependent parameters defining,for each t, thestrength of the shock.

For example, a piecewise linear discontinuous approximation of the typed~-

picted in Fig. 8.1 is obtained if we use the local basis functions shown in thatfigure, i.e.

,I. (x) = l-!. ~ (x) = !.'1'1 h' 2 h

(8.29)

(8.28)

(8.30)

(8.27)

the varia-In C°(I)

1 be de-

be a director discon-nt approx-: 0 = xO

Ik and ifhave two

1ite e1e-of poly-

1 ~ y ~ I' X

h ~

~

whi ch is prelWe

~l (X)

element is 01

~

X

ljIl(X) 1

X

TIl (X) V1I'" y .1 X

where

PrEto note that

ThEthe kinemati<

s

X

156

U(X)

Figure 8.1. Discontinuous finite element shape functions.

, I i

II; I< ~ I' .1;1 I

I: I. ~till I

II ~, ,, .I

I !~

(8.36)

(8.37)

(8.38)

of motion of a finite

k =B

= - Y[ UX ] V

= py

i=1

= -SV

N

Mae = Li=l

N

rlle = 2i=l

N

2

i=l

157

L ijaqay + LSllpt.y + dxy(UX)

a IIN-l

+~

Theorem 8.3. The discontinuous FEG approxinmtion (8.31) and (8.33) satisfiesthe kinematical compatibility condition for waves.

Proof. While this property should be obvious, it is nevertheless interestingto note that

where

which is precisely the compatibility condition. •We remark that the following form of the equations

element is obtained when (8.31) is introduced into (8.29):

-1 s

158

i -rlla = [N/l]V .~a.

1 1

N

fa = L (pf, 1/Ia)J.i= 1 1

r = {o~ )qa a 0

N

qay = L (p~a,NY)J·i=l 1

N

Plly = L (pNll,Ny)J.i= 1 1

N

dy = L (o(Ux), NY'X)Ji

i=l

Mi = [N.]y Nlly U i Yi

N

py = L (pf ,NY)J.i= 1 1

In (8.37) we recognizemaaand fa as the local mass matrix and consistent

generalized f~rce vector for a typical element. KB(~) is a nonlinear stiffness term

and rllB and rla represent amendments to the mass and stiffness properties of the ele-

ment due to jumps. We remark that in (8.37) and (8.38) the descrete analogues of the

jump conditions must be added, as indicated, to the descrete equations of motion.

This is characteristic of variational methods of approximation in that, typically, the

governing equation together with boundary-, initia1-, and jump-conditions are set

forth in a single expression. When (8.37) and (8.38) are expanded, we discover that

a finite element analogue of the differential equations governing shock growth and

decay precipitates from these general equations.

To solve numerically the wave problem, we must, of course, also discretize

(8.37) and (8.38) in the temporal variable. Toward this end, we may introduce any

of several explicit finite difference schemes, but our experience is that it does

not pay to be too sophisticated. For example, we may use simple central difference

approximations,

etc.

consider i'

approximat

the error

such that

Notice tha

typica 1 fi

mspaces Hh'

H~·k(I .Q. F

if

and, at tl

(8.39)

(8.41)

(8.40)

159

N 11-1

L {(pU,W)J. + (a(Ux),WX)J)- L [aW]V.i=l' , i=l '

N-l

L+'{(PVi[U ]Y. + [o(Ux) ]yi)Wy.)i= 1 ' ,

II

L (pf,l~) , V'll, H~,k(IJQ,P)

i=l

and, at the shock surfaces,

if

62tvcr = ~(Uu 1 _'2Uu + vcr 1) % UU(n6t)6t n+ n n-

Notice that when (8.31) is substituted into (8.39) we obtain (8.37) locally, for a

typical finite element.We must first establish some preliminary properties of our approximating

spaces Hmh,k(I,Q,P). Whenever yeo) = yea) = 0, V V " M c H~,k(I,Q,P) we denote M =om kHh' (I. Q. P) . 0

1. We identify an element Z(x,t)~H~,k(I,Q,P) as a local energy projection

such that

etc.

8.5. Accuracy and Convergence of Semidiscrete Approximations. We shallconsider in this article the convergence theory surrounding the semidiscrete FEGapproximation of the weak equations of motion (8.14); that is, we wish to determinethe error in the solution of the Galerkin approximation problem: find

discretizeoduce any

j t does:Iifference

consistenttiffness term; of the ele-logues of the'motion.:ypica 11y, the

are setscover thatrowth and

where

C,

nical, and I

its basic stracting thlthe projectled to the

Ibfinite-elemeLet the intrthat Vi (ux)that the fir1owi ng ineqL

Notice thatimplying tha

In

4.mation shalltive cons tan(8.42)

(8.44)

(8.45)

(8.46)

(8.47)

(8.43)

v

l!,;;jsk+l

= 1,2, ... ,N

O~jsk+1

E = u - Z

e=E+E

e = u - U

E = Z - U

Ilell ~ IIEli + IIEII

IIVllwj(J.)P 1

160

and

ClearlY

where

3. We denote bye, E, and E the actual semidescrete approximation error,the projection error of U and Z, and the interpolation error between the actual

solution u and Z, respectively. That is,

2. We assume that the space H~,k(I,Q,P)is endowed with the followingfinite element interpolation properties:

and an inverse assumption,

161

(8.49)

N

yIIE(T)112p + LW~p(I) i=l

Comments on the proof. The proof of this theorem is lengthy and very tech-nical, and we shall not reproduce it here. However, it is fitting to discuss brieflyits basic structure. We begin by evaluating (8.14) at v = W E H~,m(I ,Q,P), aM sub-tracting the resulting equations from (18.39). Then setting W = Et = ~i,making use ofthe projection properties (8.40) and (8.41) and the Lagrange formula (8.26), we areled to the ordinary differential equation,

IItf(t)II LCD(L2

(I))+ cllleIIL<»[,p(I)S

C2{11~~(0)IIL (I) +g(Zx(C),Ux(0))1/2I1E(O)lIw1

(1)2 2p

4. We require that an auxiliary condition be satisfied; i.e., the approxi-mation shall be said to satisfy the amplitude condition whenever there exists a posi-tive constant µ such that

where

Theorem 8.4. Let the conditions of Theorem 8.2 hold, and let the space offinite-element approximations H~,k(I,Q,P) be such that (8.40), (8,41) and (8.48) hold.Let the intrinsic shock wave velocities Vi be preserved in H~,k(I,Q,P) in the sensethat Vi(ux) = Vi(Ux) = Vi(Zx)' Then there exist positive constants C1 and C2 suchthat the finite-element-Galerkin approximation error e of (8.45) satisfies the fol-lowing inequality:

TJ Vif(o+(Zx) - a+(Ux»)E~

o

-(a-(Zx) - O-(Uxl)E+ dt} ~µIIE(T)1I2~ (8.48)x W

2p(I)

Notice that when we linearize the problem the classical Betti reciprocal relationsimplying that this condition is always satisfied.

In [7J we prove the following theorem.

..

(8.43)

(8.42)

(8.44)

(8.45)

(8.46)

(8.47)

'0 llowi n9

on error,ctua1

8.

of our theorto the fullycriteria forspatial (matof the mostin [9].

01at t = nf.t:

where

Thushockless dOlTits first dereither zero (semidescrete

(8.50)

(8.51)

i=1

162

N

2: (pEtt,Et)L2(Ji)i=l

N-1

+L

supsis; N

1 d 1/2'2 dt lip EtIlL2(I)

+ ii 1I.1/"ttlli,IJj) + ,'n lIall2"lIi,IJ,)}N

+ ~ 2 v~ IIltllli2(y.; (J., J.+1))h 1 1 1i=1

,t ~ ~IOGa(u:)"IIL_:Ji)

This part is fairly standard. Next, we introduce the projections of (8.40) and (8.41)use the Sobo1ev imbedding theorems, and a number of elementary inequalities to trans-form this differential equation into an alternate one,

1 d {1/2 2 ( >}"2 dt lip EtIlL2(I) + o(Zx) - o(U) .Ex .

N-l }

+ ~ ~ {(o+(Zx) - ''+(Ux»)E~ - (o-(Zx) - o-(Ux))E:1=1

Integrating this equation form 0 to t, applying the Gronwall inequality,introducing the amplitude condition (8.48), taking the supremum over all t € [O,T].and then introducing the triangle inequality (8.47) gives the desired result (8.49).

It is now a simple matter to obtain a priori error estimates.

Theorem 8.5. Let the conditions of Theorem 8.4 hold and let the spacesH~,k(I ~;) possess the interpolation properties (8.43) and (8.44). Moreover, let

. k+l k+1 )the Solutlon of (8.14) be such that u, ut E LCD

(W2p (I») and Utt " L2

(W2p (I) . Thenthe approximation error e = u - U in the semidescrete FEG scheme (8.39) is such that

I

IIiI

(8.54)

•

(8.53)

(8.52)

~(2p-1)

2kp 4k 2(2p_l)2 ' ~ '

en = e(n t) = En + En

}En = E(n6t) = un _ Zn (8.55)

En = E(n6t) = Zn _ Un

= min

21- (2p-1)'1lutli (k+l(J.»)

L", W2p

1 3 ]

1/(2p-l)2kp/(2p-1)3 Iluttil (Wk+1(J.l)_

+ h L2 2p 1

IletIIL)L2(I)) + cllleIlE)L2P(I)) S C2{llet(0)IIL",(L2(I»)

N

+g(u(o), U(0»)1/2 Ile(0)IIP1 + 2: [h2kP/(2P-1)2

W2p(I) i=l-

163

where

Thus, if we use polynomials of order k to construct approximations over theshockless domains, and if the global FEG approximation is continuous, with jumps inits first derivatives at the shock surfaces Yi, and if initial error in energy iseither zero or the same order as the error for t > 0, then the order of error in thesemidescrete FEG approximations is

8.6. Some Additional Remarks. The most interesting and practical featuresof our theory have to do with the extensions of the results of the previous sectionto the fully discrete case. This is not a trivial extension, and involves constructingcriteria for numerical stability as well as a priori error estimates containing thespatial (material) mesh parameter h and the temporal mesh size 6t. We summarize someof the most striking results in the following theorems, the proofs of which are givenin [9].

Once again we decompose the errors as in (8.45), but now we evaluate themat t = n6t:

(8.50)

(8.51)

~uality," CO,T],t (8.49).

40) and (8.41).ies to trans-

spaceser, 1et)). Thensuch that

}

scheme is

Here k is ts ity and Ct,

is partitic

Thl

numeri ca llyis so only iLet us intro

(8.57)

(8.56)

(8.58)

v~ IIIExlilL (V.(J1., J

iii»)

1 '" 1

un+1/2 = -}<un+l + un)

6 u - ~I n+l nt n+1/2 - ~u - u )

164

0t (x) = lilt(x(n+1 )lIt)- x(n t))h+1/2

+ ~ sup1 sis N -

We also define

In addition, we introduce a discrete amplitude condition such that there exist con-stants y and a such that

We can then prove the following theorem:

Theorem 8.6. Let the intrinsic wave speed v~, 1 sis N be duplicated by1

the fully descrete FEG approximation. Let (8.57) hold, and let Utttt

L2

(L2(J

i»),

1 ~ i s N. Then there exist positive constants Cl

and C2

such that

where

, I

I'

(8.63)

(8.61)

(8.62)

•(8.60)

(8.59)

sup IE;~IsisN 1

appearing in (8.58)

-1C =£....6 413p

-1C _U8-4

r-lC4 = 1--2p

~=iY~1

ck2(lVn~-

165

sup IIun II xo s n s v

sups i ,;N

E,~ = sup n1 x E Ji

-ny =

IluliL (X) = SUp IllSt ullx'" 0 s n s V n+1/2

rr-111Cs = L-...4

-1C7 = L2p

Cl,C2 = the positive constants

k2r--lC =~

3 411P

With the notation (8.61) and (8.62), we find that the fully descrete FEGscheme is stable when positive constants A and B exist such that

/ = sup n IDGo(Unll ,0 ~ n s r - 11 X < J. x

1

C = Ck~ +9 p

C = Ck2~10 -----;r-

Here k is the constant appearing in the inverse hypothesis (8.44), p is the mass den-sity and a, 13,C and 11are positive numbers. We assume that the time interval [O,T]is partitioned into r time step 6t. Then we denote

The proof of the previous theorem assumes implicitly that the scheme isnumerically stable. However, a closer look at details of the proof reveals that thisis so only if certain functions of the mesh parameter remain bounded as 6t, h -> o.

Let us introduce the following constants:

(8.56)

(8.57)

(8.58)

exist con-

icated by2(Ji»),

o < v <

Theorm-Utt ( L2(W2p (Jhold, where v aThen the fullyare s ta b 1e in t

such that

Equations (8.63be complex or n,sufficient cond

(8.64)

(8.68)

(8.67)

(8.66)

(8.65)

}

r) 2

1 IRate!B Cl Cl

sha 11 not reprlCn + 4(Cn)2 - 2C" '9 9 9

I temp ora 1 step ,are off set by

166

B > 1 + C66t + C76t2

B > C2 + C86

B > ~ + CnA2 + C A2y 9 1

1B > 2y + C1A

n 2B > C11 + C~t + C10A

Let there be a specific real number v such that

We can always adjust C5 so that this choice of A is less than y - C26tl/2v - C56t.For this choice of 6t. C2 + C86t can always be less that some number B. as can 1 +

6tC6 + 6t2C7. Thus (8.63) and (8.64) can always be satisfied for a given 6t. IfC~ > 0, the first member of (8.65) is satisfied whenever the second member is; other-wise, if C~ < 0, then (8.65)2 is satisfied whenever (8.65) is. It remains to be choosev so that (8.65)3 is also satisfied.

In summary, we use the following procedure:1. Pick 6t

2. Calculate the right hand sides of (8.64) and choose a specific small Bthat just satisfies (8.64).

3. Then choose v to satisfy (8.65); i.e.

Then choose a 6t and select A so that

where

where

, "

III

(8.71)

•

(8.70)

(8.69)

167

18-28Cl

+ #, At' lI"ttttlll,(l,{J; J)}

o < v < mi n

1 2+ IIEttllL (L (1)) + h ~up Vi IIIExlllL(V.(J. J, »)

'" 2 1 ~ 1 S N - 1 '" 1 l' 1+1

Rates of convergence can now be established by Simply using (8.43). Weshall not reproduce them here. We remark that a fairly strong restriction on thetemporal step size may be represented by (8.63) for numerical stability, but theseare off set by benefits of a very accurate scheme.

Equations (8.63) can now be satisfied by picking an acceptable A. If v is found tobe complex or negative, a new 6t must be selected and the procedure is repeated. Asufficient condition for stability is then to choose h so that

Theorem 8.7. Let the conditions Theorem 8.6 hold, u, ut " L (W2k+l(J.)),

k+1 '" p 1Utt € L2(W2p (Ji»), and Utttt € L2(L2(Ji») for 1 sis N. In addition, let (8.70)hold, where v and 6t satisfy (8.69) and are determined using the above procedure.Then the fully des crete FEG scheme is stable in the L",(L2(I») sense, the velocitiesare stable in the L~(L2p(I)) sense, and then there exist positive constants w and ~such that

(8.65)

(8.64)

(8.66)

(8.67)

(8.68)

<ic sma 11 8

v - C56t.scan 1 +

6t. Ifr is; other-s to be choose

168

Acknow1edgfflent. Early phases of our work were supported by the U.S. Na-tional Science Foundation under Grant Gk-3907l. In recent months, we have been ableto continue this work due to support of the U.S. Army Research Office in Durham underContract

REFERENCES

[1] J. T. Oden, Finite Elements of Nonlinear Continua, McGraw-Hill, N.V.1972.

[2] J. T. Oden, Approximations and Numerical Analysis of Finite Deforma-tions of Elastic Solids, Nonlinear Elasticity, Academic Press, N.Y., 1973, pp. 175-228.

[3] J. T. Oden, Formulation and Application of Certain Primal and MixedFinite Element Models of Finite Deformations of Elastic Bodies.

[4] R. B. Frost, J. T. Oden, and L. C. Wellford, Jr., A Finite ElementAnalysis of Shock and Finite-Amplitude Waves in One-Dimensional Hyperelastic Bodiesat Finite Strain, Int'l J. of Solids and Structures, (to appear).

[5] J. T. Oden, J. E. Key, and R. B. Fost, A Note on the Analysis ofNonlinear Dynamics of Elastic Membranes by the Finite Element Method, Computers andStructures, Vol. 4, No.2, 1974, pp. 445-452.

[6] L. C. Wellford, Jr., and J. T. Oden, On Some Finite Element Methodsfor Certain Nonlinear Second Order Hyperbolic Equations, TICQM Report, 74-7, Austin,1974.

[7] L. C. Wellford, Jr., and J. T. Oden, A Theory of Discontinuous Finite-Element Ga1erkin Approximations of Shock Waves in Nonlinear Elastic Solids, TICOMReport, 74- 6, Austin, 1974.

[8J J. T. Oden, and L. C. Wellford, Jr., Finite Element Approximationsof a Class of Nonlinear Two-Point Boundary Value Problems in Elasticity, TICOMRepor~, 74-5, Austin, 1974. -----

[9] L. C. Wellford, Jr., and J. T. Oden, A Theory of Discontinuous Finite-Element Galerkin Approximations of Nonlinear Wave Phenomena, TICOM Report, 74-6,Austin, 1974.

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