A boundary formulation for the dead weight integral in 2-D elastodynamic time domain BEM

MECHANICS RESEARCH COMMUNICATIONS Vol. 19(4),273-278,1992. Printed in the USA. 0093-6413/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press Ltd.

A BOUNDARY FORMULATION FOR THE DEAD WEIGHT INTEGRAL IN 2-D ELASTODYNAMIC TIME DOMAIN BEM

H. Antes

Institute of Applied Mechanics, Technische Universit~t Abt-Jerusalem-Str. 7, D-3300 Braunschweig, Germany (Received 3 June 1991; accepted for print 23 July 1991)

Introduction

The subject of dynamic interaction has been recognized as a very important phenomenon in the dynamic analysis of structures. When infinite or semi-infinite domains are envolved, the most appropriate method to study relevant interaction effects seems to be the Boundary Element Method (BEM) because it correctly accounts for the radiation damping to infinity. Moreover, the method is extremely convenient when the loading on the structure under analysis is limited to the surface, since then the BEM requires only the discretization of the boundary rather than the domain. However, this feature is generally lost when body forces are present due to the need to compute integrals over all the domain [1]. Unfortunately, numerically evaluating a domain integral requires the domain of the problem to be divided into integration cells. This greatly increases the amount of data preparation required and causes the BEM to loose much of its advantage over domain type methods. However, as was shown first by Cruse et al. [2] in elastostatic problems with axisymmetric geometry, for certain types of commonly encountered body forces the domain integral may be transformed to boundary integrals, using the so-called Galerkin vector to achieve the required transformation. Later, the same key was applied by Danson [3] to general two- and three-dimensional geometry. Again, m the direct boundary element formulation of elastostatic problems, Gipson and Camp [4] use random numbers, i.e., a coupling of Monte Carlo quadrature with the boundary element method, to achieve an extremely user - friendly device to handle most general body force integrals. Brebbia, Nardini, and Wrobel [5, 6, 7] developed a special technique for boundary element formulations for time dependent problems, the so-called Dual Reciprocity Method (DRM) which uses special approximations in order to transfer domain integrals to the boundary. Accurate results have been reported for several cases. In

this

paper,

an

analytic

transformation 273

of

the

two-dimensional,

elasto-

274

H. ANTES

dynamic, time-dependent, dead weight domain integral into a boundary integral is presented. The technique is similar to that using the Galerkin vector [2,

3].

This new boundary formulation for the dead weight integral of the boundary integral equations which describe elastodynamic problems has already been successfully used to explore the dynamic behavior of flexible massive structures in unilateral contact with elastic soil [8, 9]. There, e.g., in the case of incident waves, the interface tractions due to the dead weight significantly influence the reaction, i.e., the uplifting or sliding of the heavy structure.

The Basic Time Domain Integral Equations

In the present two-dimensional study of transient wave propagation problems, the material is assumed to be linearly elastic, isotropic, and homogeneous, and the motions are limited to small amplitudes. The gravity effect, i.e., the dead weight is taken into account. The equations of motion, together with the actual initial and boundary conditions, i.e., the actual initial-boundary value problem, can also be described by integral equations. These equations can be deduced by employing Graffi's elastodynamic reciprocal theorem [10] and a subsequent integral transformation [1]. When the initial conditions [1, 11] are zero, one obtains (t'=t-r; the overdot """ indicates time differentiation) t

d0(¢)uJ(¢'t) = f { 0

~F [ ~ll)(x'~;t')Tj(x,r) +

P:i)(x,¢;t')uj(x,r)

(1) -Qli)(x,¢;t')~.(x,r)]dFx, J

+ f

*i

ul)(x,¢;t' ) bj(x,r)dOx } dr

D where d . = 0,5 ~.. whenever ~ lies on a smooth boundary F, while for interior 1j 1l points (~ED) d..=,j ~..1j holds. The actual body force distribution bj(x,r) due to a constant gravitational field acting in direction (-x2) is given by (bl= 0, b2-- -pg), where g is the gravity constant. The appropriate singular solution ~(1), J that is the response at a point x and time t in an unbounded domain without any imposed initial conditions to a unit

BOUNDARY

FORMULATION

F O R DEAD W E I G H T

INTEGRALS

275

impulse at time r in the direction x and located at point g, is given by [1, 1 12] ~l,i)(x,,;t,) 1 (H(clt'-r) [ r2 ] = (2R l + ) r r - R16ij j ~ c r2 1~1 ,i ,j 2 H(c2t'-r) [ r2 ] c r2 (2R2 + ~r~) r ,i r ,j - ( R + )6j 2 1/2 ~(c a 2-72 Ra t - r 2) ; a = 1,2 ; r = Ix - gl ,

(2)

where H is the Heaviside step-function, c I and c 2 are the dilatational shear wave velocities, and /9 denotes the mass density of the medium. inferior commas indicate space differentiation.

and The

The

kernels

~(i) and Qj * (i) are obtained from the corresponding singular J tractions ~!i) after carrying out integration by part with respect to time and J can be found in [11].

Transformation to Boundary Integrals

In order to transform the domain integral in Equ. (1) to boundary integrals, a generalized form of Gauss' theorem will be used, which may be expressed as follows [3]: If Ajkl...is a general Cartesian tensor field, then

f ~a . A jkl.., d~r~g D

=

~

f Ajkl...ni(x)dFx

(3)

F

holds, w h e r e ni(x ) is the unit outward normal at x. The actual body force domain integral contains two different types of singularities:

_'g-r

(a)

r2

r ,~.r ,j.

(b)

1

r

r

tensor

functions

gi/r)

with

and

r2

6ij

(4a)

and

1

6...

(4b)

The higher order tensor fields Gijk(r) can be found analytically so that

276

H,

0

ANTES

Gijk(r ) = gij(r)

(5)

k

Substituting this special tensor field into Gauss' theorem (3) gives (6)

= f Gijk,kdr2x = f G~jk nk(X)d/"x • ~2 F

f g~j d~2x f2

For the case of a body with a constant mass density p in a constant gravitational field, i.e. for a constant body force bj(x,z) = bjo, one obtains explicitly (~ E /-'): *i

f u! )(x,~;t')b.(x,z)df2v = } J ~9 I"

1 (i)

2 (i)

1

= bjo j" [H(c,t'-r)Rjk(X,~;t') - H(c2t'-r)Rjk(X,~;t')Jnk(X)dr x

(7)

F+ where

1(i) Rjk(X'~;t')

=

~

l

r,k [Rt(r,ir,j"~ii)

~

"

c t' + R clt,(2r "r,l. J_)] " ~ij)In( l r

(8) 2<~) ) Rjk(X,¢;t,

1 r ,k[ c 2t' + R 2)] T ~ R r ,1r ,j - c t'(2r,ir d- ~ij)ln( r "

=

and /"+ = {r [ r < cut' } is the boundary of that part of the domain ~2 which the p- and s-waves, respectively, starting at point ~ have already traversed (see Figure 1). When the wave fronts have reached all points x in the domain f2, F + coincides with F. .~.._../:

F

I"

i

.\

r'

.= r

f i -+~~;'s ,

/

{x I ~ t =

r)

C~

~ = (~ I~ = ~)

¢ \

FIG. 1 Definition of the boundary parts F + , Fr, and Fe enclosing the domain covered by a wave started at point ~

BOUNDARY FORMULATION

FOR DEAD WEIGHT

INTEGRALS

277

On the boundary part Fr, i.e., on the wave fronts, the integrands are identically zero; therefore, the boundary integral along f' r can be cancelled. On the boundary part Fe, i.e., in a circular distance r = e from the singularity point g, the following special relations hold (if A~ = (a1-¢o2> 0): (i) r,t = cos ~ and r,2 = sin ¢~, such that r,knk = - 1, (ii) d/' x = -edna, where ~ varies from ¢~1 to ~2" Then, one obtains

~2 t' Cl f [ .... ] d/"x = " 2-~ f [ d~ij + (2r,ir,j" d}ij)ln¢2 ] d, _

t '

(9)

wbith eo s ( ~ + ~2 )

t2j(,t,¢o:) =

l

sin(¢~l+¢2 )] . (10)

Ls i n(wI + Ca2) -cos(ca +¢a2)j On a smooth boundary, we have A¢~ = n which yields ....

,11>

F Moreover, in any case, the time integration can be performed analytically. Finally, the time-depondent dead weight domain integral will result in the following expression, where only the integral along the boundary part F + remains to be evaluated numerically:

t

*(i

j'bjo j" ui ' ( x , ~ ; t - r ) d F x dr o t2 = bjo

tt(clt - r) rj +

(x,~;t) - tt(c2t - r ) r

(x,~;t) r,ndF x

t2 cI } + 2- (A0 ~lj + sin/t O I2ij(~1,~2 ) I n ~ )

(12)

where

l(i) r J (x,~;t)

= ~I

~-

EtRl(r, ir,j

+ In

ct l

) +R

r

I {~(~l)2Jij" tZ(r, ri ,j

" ~¢~ij)} 1

278

H. ANTES

(13) 2(iy rj (x,{;t) = ~ 1

~1 [ t2R2(r, i r , j ct "ln2

- ~aj)

+R 2 r 2{~ (~)~..ij + t2 ( r , i r , j

-~ij)} ]

When r tends to zero, only pseudo-singularities appear in the boundary integral which vanish when the contributions from similar terms (refering to the p-wave and the s-wave, i.e., to e 1 and c2) are calculated simultaneously. Thus, this time-dependent boundary representation of the dead weight integral can be integrated without difficulty.

Acknowledgement The financial support by the Deutsche Forschungsgemeinsehaft through project SFB 151/C1 is gratefully acknowledged.

References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

H. Antes, Finite elements anal. design 1, 313 (1985) T. A. Cruse, D. W. Snow and R. B. Wilson, Computers and Structures 16, 445 (1977). D. J. Danson, Boundary Element Methods, Proc. III. Int. Seminar, Irvine (Ed.: C. A. Brebbia), 105, Springer, Berlin (1981) G. S. Gipson and C. V. Camp, Boundary Elements VII, Proc. VII. Int. Confer., Como (Eds.: C. A. Brebbia, G. Maier), 2, 13-17, Springer , Berlin (1985). D. Nardini and C. A. Brebbia, Topics in Boundary Element Research (Ed.: C. A. Brebbia) Vol. 2: Time-dependent and Vibration Problems, 191, Springer , Berlin, New York (1985). C. A. Brebbia and D. Nardini, Comp. and Maths. with Appls. 12B, 1061 (1986). I. C. Wrobel, C. A. Brebbia, and D. Nardini, Finite Elements in Water Resources VI (Eds.: A. Sa da Costa et al.), Springer, Berlin (1986) H. Antes and B. Steinfeld, Boundary Elements X (Ed.: C. A. Brebbia) Vol. 4: Geomechanics, Wave Propagation and Vibrations, 45, Springer, Berlin, New York (1988) H. Antes and B. Steinfeld, Proc. 2nd. Nat. Congr. on Mechanics, Athens, (Ed.: A.N. Kounadis) 1, 165, HSTAM, Athens (1989) D. Graffi, Mere. Accad. Sci. Bologn. 18, 103 (1947). H. Antes and O. yon Estorff, Proc. 3rd. Conf. Soil Dynamics, Princeton (Ed.: Cakmak, A.S.) 291, Developm. Geotechn. Engng. 44, Elsevier, New York (1987) W. J. Mansur, Ph. D. Thesis, Univ. of Southampton (1983)