SOME NEW FINITE ELEMENT METHODS FOR THE ANALYSIS OF SHOCK AND ACCELERATION WAVES IN NONLINEAR MATERIALS. J. T. ODEN The University of Texas at Austin

Transcription

1 ~ SOME NEW FINITE ELEMENT METHODS FOR THE ANALYSIS OF SHOCK AND ACCELERATION WAVES IN NONLINEAR MATERIALS J. T. ODEN The University of Teas at Austin L. C. WELLFORD, Jr. University of Southern California ABSTRACT In this note, we give a concise summary of a number of new discontinuous finite element schemes that we have developed for the analysis of shock and acceleration waves in nonlinearly elastic solids.. INTRODUCTION Recent theoretical and numerical results indicate that many of the traditional shortcomings of Gaerkin methods of shock wave phenomena can be overcome by using finite elements with built-in discontinuities. The theoretical foundations of discontinuous Gaerkin methods for shock waves is developed in [lj, [2J and the development of numerical schemes to implement this theory is described in [3J. The development of an independent theory for discontinuous finite element methods for acceleration waves is described in [4J. In the present paper, we give a summary account of some aspects of this theory, we describe some new techniques for both shock and acceleration waves, and we present some new numerical results. 2. VARIATIONAL PRINCIPLES FOR WAVES We consider a class of one-dimensional problems in dynamic finite elasticity. We use the following notations: X = material particle E. I, I being an open set of particles whose motion is to be studied. u(x,t) = displacement of particle X at time t, t > o. cr(,t) = first Pioa-Kirchhoff stress at X, t p = mass density of a material body B, I c B, in its reference

2 configuration, t = o. ~(I) p Cm(I) o wmo) p = Soboev space of functions with generalized partial derivatives of order ~ m in Lp(I) = space of m-times continuously differentiable functions on I with compact support on I. = closure of Coo(I) in the Soboev norm o "U II p = I \ I aku Ip dx (2.) wn(i) ~ ~ P I k<m [] Y = I(,t+)- I(X,t-)= jump of a function at particle Y. Iy = }(I(X,t+) + I(X,t-)= average value of a function at a jump. f(x,t) = Body force at X,t. u,u,u X = velocity, acceleration, and displacement gradient (at X,t), respectively. It is well-known that a shock wave in a material body B, the displacement gradient u and the velocity u eperience simple jumps. If Y denotes the particle atxwhich these jumps occur, then the following kinematical compatibility condition must hold [u] + V[u ] = 0 y X y (2.2) Here V is the intrinsic velocity of the shock: dy V = dt (2.3) From momentum balances, it can be shown that v2 = [cr]y p[ux]y (2.4) In addition, an acceleration wave represents the propagation of discontinuities in the acceleration and the second gradients, ux~. If an acceleration wave front eists at particle Z, kinematical compatiblity demands that [U]Z - W 2 [U XX ]Z = 0 (2.5) where W is the intrinsic velocity of the acceleration wave, dz W = dt (2.6) and, from momentum considerations, (2.7)

3 Now we assume that at any time t > 0 there may eist in I, N shock waves defined at the collection of surfaces (particles) {y.}n and M acceleration i=l waves at particles {Z.}~ ; of course, it is possible that one or more of the = Zi and Y. coincide. These physical surfaces define natural partitions of I. J into N+ shockless domaind J. and M+ domains devoid of acceleration waves, K. : I/{Yi} N+ =U J i i=l M+l I/{Z.} = U K. i=l (2.8) With these preliminaries, we may state the following variational principles for waves:. Theorem. Let energy be conserved in a locally integrable sense on I. Then linear momentum is weakly conserved in a body with N shock waves Y i if [f [(pu - pfj_ + (Y X N+ N i=l J. i=l ]dx + [ {pvj[u]y; _ IL + p[cr] v y } = {crv} Y. i 0 for everl velocity field v satisfying the kinematical boundary conditions, wherein I = [O,LJ. Moreover, if there are instead M fronts {Z.} at which acceleration waves occur, momentum is conserved if ' [J i= K i M.. \ 3 [(pu - pf)v + crvx]d + ~ {pwi[uxx]z.. = (2.9) (2.0) for every kinematically admissible velocity v Equation (2.9) was derived in [3] and (2.0) was proved in [4J. They suggest that in the presence of both shock and acceleration waves we should have, for every kinematically admissible velocity v, M+N+2 L +L N [(pi.i- pf)v + crv]d {pvi[u]y i k=l J K i= k k M + \ {pw 3 [u ] + [0-] }(v X ) L Zi Z.. =

4 3. FINITE ELEMENT APPROXIMATIONS We now use (2.9) - (2.l) as a basis for discontinuous finite element approimations. We iqtroduce another partition P of I: 0 = X O.< Xl ~ 000 < X G = L, with hi = X' - i- and h = m~ hi. The interval [ -, Xl] = Ii is a finite element. Net, we construct the Ifdiscrete lf spaces, < j < N: < i < G-; Q.. = I. () J. } lj J (3.) < i < M; < j < G-l; 0.. = I. n K.} lj J (3.2) ~,k(i), (~,k(z» = (V(X): V(X) E ~,k(i) (~,k(i», V(Q) = V(L) = O} (3.3) Pk(O) = {p: p is a polynomial of degree ~ 2 on Q} (3.4) These spaces also have finite-element interpolation properties: (3.5) etc.; see, for eample [5J. The semi discrete finite-element method for shock waves consists of seeking UEH~,k(I}nc (I) such that - cry [Z] } i Y. = {cr(o,t}z(o)}i L o (3.6) whereas the semi discrete finite element method for acceleration waves consists of seeking U E ~,k(i)n C (I) such that M+l f;;~. [(plj - pf)v + crvx]dx +L {PW~[UXX]Zi = M i=l [cr(v)]z. }(VX)Z. = {cr(o,t)v(o)}i L o (3.7)

5 Note that in either case the space of trial functions differs from the space of test functions. 4. APPLICATIONS We summarize briefly the results of some recent appication-s. For more details on some of these results, see [lj - [4J; some of these numerical results we describe here are new. We proceed as follows:. We use simple central and forward differences for the temporal approimations in (3.6) and (3.7); e.g. lj(n~t) = JL (U n + l _ 2U n + Un-) ~t U(n~t) = it (Un - U n - ); un = U(X,n~t) 2. For shock wave problems, we take as the constitutive equation aw aw ai a = au- = ar- au a, a =,2,3 X a X W = W(I l, 2, 3 ) = strain energy per unit initial I = (I, II, III) = principal invariants of deformation tensor. a For incompressible materials, we use aw ai am cr=-o~+pala au X du Ci =,2 where p is the hydrostatic pressure. Generally, we assume p can be eliminated via boundary conditions, such as in the case in which B is a thin cylindrical bar; I is its ais, and the stress normal to I is zero. The eamples to be cited below involve Mooney materials, for which wherein C l, C 2 = material constants. W = C l (I-3) + C 2 (II - 3) 3. For acceleration waves, we limit the present discussion to linear Hookean materials for which a = Eu X. 4. We come to the critical question of the construction of the spaces H~,k(I) and ~,k(i). We give two basic eamples: ~.l(i) Finite-Element Space. First consider a piecewise linear local approimation characterized by four parameters: U(X) = J a + bx c + dx < y > y The parameters are determined in terms of the values U, and U 2 of U at the nodal points at each end of the element and in terms of the jump S in U at the shock front Y. We find that

6 2 U(X) = \ U '4J L aa (X) + sa(i<> + Sy</>(X) a=l (4.) wherein the (X) are the usual Lagrange finite element interpolation functions Ci I(X) = - I h ' (4.2) and a(x} and </>(X) are discontinuous shape functions, l, X < Y a(x) = ; 0, X > y X > < y y (4.3) The entire collection is illustrated in Fig.. It is easily verified that these functions satisfy the kinematical compatibility conditions (2.2). '4J.5 a.5 X 2.5 ~ ~ X </>.5 i':j I: y. h ~ X. Figure. Discontinuous linear trial functions An V~,l(I) Finite-Element Space. For acceleration waves, a similar process can be used if we employ piecewise quadratic approimations of the type in Fig. 2. Here we use

7 U - Ia + bx + cx2 d + ex + fx 2 < Y > Y (4.4) together with the equations, a = U d + eh + fh 2 = U 2 d + 2eh + 4fh 2 = U 3 a + by + cy2 = d + ey + fy2 (4.5) Suppose that a shock also eists at Z which has strength S. Then where 3 U(X) = \ U L aa (X) + R8(X) + Q<t>(X) + An(X) (4.6) a=l 3 X X h 2h2 { X. 8(X) = 0, X < Y X > Y 2 3X + ~ ' X ~ Y - 2h 2h <t>(x) = ( 2 3X+L.,X>Y - "2 h 2h2 (4.7) - -+_ 2 I3(X) - - 2h 2h2 n(x) = X2, X < Y 2 0, X > y The generalized coordinates Rand Q in (4.3) are defined as follows: R = S - AZ Q = SZ - AZ 2 2 (4.8) It is easy to prove that this choice of interpoants satisfies the proper kinematical compatibility conditions at the shock front; observe that [UX]Z + (S - AY)[8 ]z + A[nz]z = S [U] = - Sy Z 2 Thus (2.2) holds. Note also that when S = 0, then R = -AZ and Q = -"2AZ. We remark that the position of a shock at n~t is conveniently obtained via the formula

8 n [a ]y psn (4.9) We also remark that special provisions must be made to handle the reflection of waves. When, for eample, a shock wave impacts a fied particle, momentum and kinematical jump conditions must be satisfied. We know that after a reflection, the particle velocity at a wall must be zero. We have, then, at a fied wall Y, p'j2[u X ] - [a] = 0 and V[U ] + 0- = 0. y y y wherein U X ' a, U are known and U and V must be calculated. In actual calculations, we use the Newton-Raphson method to solve this system of nonlinear equations n y h Figure 2. Discontinuous Quadratic Trial Functions With Discontinuity on Left Side of Internal Node 5. SOME NUMERICAL EXAMPLES. As a first eample, we mention a problem described in [3J which concerns the motion of a cylindrical bar of Mooney material,.36 in. long, an initial area of in 2, C = 24 bs/in 2, C 2 =.5 bs./in. 2, p = 0-4 b.-sec 2 /in 4 subjected to a time-varying end force F(t) at its free end which is lbs at t = 0 and is increased linearly to lbs in 2.0 sec. The other end of the bar is fied. 2h

9 A typical result is shown in Fig. 3, where our method is compared with a shock-smearing scheme. Obviously, the discontinuous finite-element scheme provides for a rather sharp definition of the structure of the shock Stress 40 (P.s. I.) , \ \,,,,, It = sec. Shock Fitting Method Parabolic Regularization Method Figure 3. Shock Structure calculated using discontinuous finite elements Particle Velocity.7 (In./Sec ) t =.25 0 sec. Eact Distance from the Free End of the Bar (In.) Solution Figure 4. Velocity profile for compressive acceleration wave.

10 Acceleration wave calculations are more delicate. In Fig. 4 we see the calculated acceleration amplitude in a illeary elastic bar of modulus 00 lbs/in 2, p = 0-4, V = 000 in./sec., h = in., and ~t = sec. The bar is fied at = 0.5, free at = 0, and is subjected to an applied force p(t) at the free end which varies linearly in time from a magnitude of 0.0 at 5 = 0 to 6.0 psi at t = 0 sec., after which it is held constant. Details of the calculation are given in [4J. Acknowledgement. We are pleased to acknowledge support of this work by the Army Research Office in Durham under Contract No. DAHC04-75-G REFERENCES. Wellford, L. C. Jr. and Oden, J. T., "A Theory of Discontinuous Finite Element Galerkin Approimations of Shock Waves in Nonlinear Elastic Solids I. Variational Methods for Nonlinear Waves," Computer Methods in Applied Mechanics and Engineering, Vol. 7, No. (to appear) -2. Wellford, L. C. Jr. and Oden, J. T., "ATheory of Discontinuous Finite Element Gaerkin Approimations of Shock Waves in Nonlinear Elastic Solids II. Accuracy and Convergence, II Computer Methods in App ied Mechani cs and Engineering, Vol. 7, No.2 to appear). 3. Wellford, L. C. Jr. and Oden, J. T" "Discontinuous Finite Element Approimations for the Analysis of Shock Waves in Nonlinearly Elastic Solids," Journal of Computational Physics ( to appear). 4. Oden, J. T. and Wellford, L. C. Jr., "Discontinuous Finite Element Approimations for the Analysis of Acceleration Waves in Elastic Solids," The Mathematics of Finite Elements with A ications, Brunel University, Ubridge, England, April, 975 to be pu ished by Academic Press, London) 5. Oden, J. T. and Reddy, J. N., Mathematical Theory of Finite Elements, Wiley Interscience, New York (to appear).