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Advanced development of time-domain BEM for two-dimensional scalar wave propagation

Advanced development of time-domain BEM for two-dimensional scalar wave propagation
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  INTERNATIONAL OURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 29, 1003-1020 (1990) ADVANCED DEVELOPMENT OF TIME-DOMAIN BEM FOR TWO-DIMENSIONAL SCALAR WAVE PROPAGATION A. S. M. ISRAIL. AND P. K. BANERJEE' Department of Civil Engineering, Stale UniDersity of New York at Bufalo, Bufalo, New York 14260, U.S.A. SUMMARY An improved formulation for solving 2D transient scalar wave propagation problems by the Boundary Element Method (BEM) is presented. The kernels presented are simpler and better behaved than those that have appeared in the published literature. An appropriate set of temporal shape functions for linear variation is used. For spatial variations isoparametric quadratic elements are used. All of these represent significant improvements over the present level of sophistication in the analysis of 2D transient scalar problems. The algorithm is implemented in a general purpose boundary element code known as GPBEST. INTRODUCTION Scalar wave propagation problems are commonly encountered in various engineering disciplines. The propagation of acoustic waves, water waves, electromagnetic waves, etc. is all governed by the scalar wave equation. Kirchoff was the first to formulate the integral equation for scalar wave problems in terms of unknown potentials. However, the solution of such problems started only in the early 1960's with the pioneering work of Friedman and Shaw.2 They solved 20 transient acoustic wave scattering problems by considering them as 3D cylindrical ones with an arbitrary axis length using a time-marching scheme. Later, Banaugh and Goldsmith3 extended it to time-harmonic prob- lems. Groenenboom4 recently reviewed some of these efforts and presented a general bounday element formulation for the solution of transient potential fluid-flow problems in three dimen- sions. It has been extended by Groenenboom et aL5 to incorporate coupling with the Finite Difference technique. Niwa et ~1 ~ nd Fukui' derived the two-dimensional boundary integrodif- ferential equations directly from the three-dimensional form of Love's integral identity and presented displacements due to the diffraction of an incident wave by a cylindrical cavity. Cole et af.8 mplemented a transient time-stepping boundary element algorithm for scalar problems using a constant boundary element, which they found to be unstable, leading to a build up of errors. Nevertheless, this was a significant contribution towards a general 2D ime-domain formulation using a 2D time-dependent fundamental solution. Later Mansur' used a similar time-stepping scheme and incorporated both constant and linear elements in space together with constant and linear temporal variations. However, his transient kernel corresponding to the derivative of potential is obtained through complex manipulation of the Heaviside function appearing in the Graduate Student Professor 0029-5981/90/051003-18 09.00 990 by John Wiley Sons, Ltd. Received 29 April 1989 Revised 28 July 1989  1004 A. S. M. ISRAIL AND P. K. BANERJEE potential kernel. Also the resulting kernel, after convolution, is unable to produce the steady-state kernel in a single large time step, although this reduction to the steady-state value is a very important characteristic of the transient kernel. Moreover, he defined the temporal linear shape functions as a triangular function over two successive time steps, which is not at all justified. In the present paper, the potential-derivative kernel is derived by satisfying the causality condition and by taking the required derivatives in a consistent manner. The resulting kernel is explicit, simpler and, once the convolution integral is evaluated, it reduces to the steady-state counterpart for both constant and linear time-variations. The linear temporal shape functions are appropriately defined within one time step. The resulting formulation leads to very accurate results, as can be seen in the examples presented. Moreover, the algorithm presented incorporates isoparametric quadratic elements and has the capability of tackling problems with multiple regions by satisfying the continuity of potential and fluxes across common interfaces. The problem of so-called 'shadow zones' as discussed by Groenenboom4 and Antes and von EstorlT'O is also investigated in the context of 2D scalar propagation problems. It has been shown that, contrary to their observation, the correct solution is reproduced even for concave surfaces since the formulation automatically obeys the causality and integrations are carried out to at least four digit precision. It is to be mentioned here that apparently the observation of 'shadow zones' has been recently* corrected by those researchers.' '- TIME-DOMAIN BEM FORMULATION I. Governing equation isotropic, homogeneous body is given by The differential equation governing transient scalar wave propagation through an elastic, 1 azp c2 at2 V2p(x, ) - -(x, t) + b(x, t) = 0 where p is the potential, x the position vector, t the time, b the body source, V2 the Laplacian operator and c the speed of wave propagation given by @, ith E the elastic modulus and p the mass density. For a well-posed problem, equation 1) is accompanied by appropriate initial and boundary conditions. 2. Boundary integral representation The solution of equation (1) for a unit body source is known as the fundamental solution and is denoted by G(x, t; t, r). It represents the potential at a point x at time t due to a unit source applied at t at a preceding time T. The fundamental solution for two-dimensional problems is well known.I3 The corresponding kernel for the potential derivative is obtained (as discussed later) by F(x, t; t, r) = (aG/dn)(x, ; t, T). The fundamental solution state, mentioned above, can be com- bined with the actual state through the use of the dynamic reciprocal identity, to yield the following integral eq~ation,'~ c(t)p(tt t) = I I CW ; t, Wx, ) - W ; t, ~P(x, 1 dr dW (2) so Suggested by one of the reviewers  BEM FOR 2-D SCALAR WAVE PROPAGATION 1005 where and c(() is known as the jump term and has the following values. (i) 1.0 for within the volume, V (ii) 0.5 for on the smooth boundary, S and (iii) 0 for outside V. In equation (2), the body is assumed to be initially at rest and no sources are present within the body. Solution of equation (2) is the essence of the Boundary Element Method. 3. Numerical implementation and solution procedure Equation (2) is an exact representation of the transient problem since no approximations have been introduced yet. However, for the solution of practical problems, suitable approximations are needed for both.spatia1 and temporal variations of the field variables. As will be seen later, the time-functions involved are explicit and simple enough to perform the time-integration analyti- cally. However, numerical techniques are warranted for the spatial-integration. For temporal discretization, the time interval from 0 to t is divided into N equal increments of duration At, i.e. t = NAt. Within each time step, the variables can be assumed to remain constant or vary linearly. Depending on such variations, the solution algorithm will be elaborated. (i) Constant time variation. Since the field-variables remain constant during a time step, they can be taken out of the time-integral, so that only the kernels have to be integrated. The time-integrated kernel is denoted by GN+'- = 11:1 (x, NAt; r r)d7 (3) Combining this and a similar expression for the F-kernel in equation (2), one obtains N c(()pN(() = 1 [G '- @(x) - FN'l- pn(~)]dS(~) (4) n= 1 where pN ) stands for the potential at time t = NAt. field-variables and they are given, in terms of the nodal quantities, by For the spatial discretization, quadratic variation is assumed over both the geometry and where i = 1,2 (for 2D), a = 1,2,3 3 nodes). NJq) are the shape functions and q the intrinsic co-ordinate of the elements. The Jacobian of transformation is  1006 A. S. M. ISRAIL AND P. K. BANERJEE After the spatial discretization, explained above, equation (4) takes the form where M represents the total number of boundary elements. F-kernel, the singular integral of the F-kernel is evaluated in the following way: Since the transient F-kernel has the same type of singularity as the corresponding steady-state where F a S nd F are the transient and steady-state F-kernels respectively. The first integral on the right side of equation (8) involving the steady-state kernel is singular and its evaluation using the uniform potential and the resulting zero flux solution is well known. However, for this process the body must have a closed boundary. Thus, for problems with an open boundary, the region of interest must be enclosed with special kinds of elements known as 'enclosing elements'. The steady-state kernel is integrated only during the first time step. More- over, these elements do not have to be as fine as those used for modelling the problem. Thus, the additional computational effort warranted by these elements is not significant. Details of this can be found in papers by Ahmad and Banerjee and Henry and Banerjee.16 The second integral is non-singular and its numerical treatment does not pose any special difficulty. In general the kernel functions are not continuous over the integration intervals and intelligent subsegmenta- tion needs to be done so that these are evaluated to at least four digit precision. Moreover, if the wave has reached over part of an element the integration is carried out only over that portion of the element. Equation (7) is written for every boundary node and after the appropriate integration, can be cast into the following set of equations: where { p } an { q } are vectors of boundary nodal quantities, with the superscript referring to the time step index. At time t, only half of the boundary values are unknown. The remaining are known; so are the values at all previous times. Thus, putting the unknowns on the left, equation (9) can be rearranged as N- 1 [A']{XN} = [S']{ Y ) - 1 [GN'l-n](@} - FN -n](p }) (10) n=l which is put into the following simplified form: [A']{XN} = [B']{ Y } + {RN} in which { X } and { Y }, respectively, are the vectors of the unknown and known variables and {RN} epresent the effects of past dynamic history on the current time node, N. Solution of equation ll), which involves only real quantities, can be carried out with any standard technique.  BEM FOR 2.D SCALAR WAVE PROPAGATION 1007 where All ?) and MJT) are the temporal interpolation functions corresponding to the local time nodes 1 and 2 (Figure l), and are expressed by ? - n-l M1(T) = At (13) ?,,-I < T < tn n - At M2('T) = Here, the time-integration involves the kernel and the temporal shape function products. After the analytical time-integration, two sets of kernels are obtained, one corresponding to each time node. This is in contrast to one set of kernels for the constant variation case. The time-integrated linear kernels are Expressions for the F-kernel are similar. The subscripts refer to the local time nodes. Following the procedure, similar to constant formulation, one obtains N-1 [A:]{XN) = [B:]{ Y } - 1 [GY+'- + Gy-r]{qr) r=l - F;+I- + Fy-n]{p*}) which can be simplified to CA:I{XN} = CB:1{YN} + {RN} Figure 1. Local time nodes at a time step
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