Higher Order Traction Time Approximation InTime Domain Scalar Wave Equation Analysis

Abstract

In the present paper the traditional BEM formulation for domain scalar wave propagation analysis is extended to a new class of problems. A procedure to consider linear interpolation for traction (flux) is worked out and the discontinuities are taken into account by coupling the standard boundary integral equation for displacement (potential) and the boundary integral equation for velocity. Time integration of boundary kernels is performed analytically for linear approximation of displacements and tractions. A numerical example is presented in order to assess the accuracy of the proposed formulation. INTRODUCTION Linear and constant interpolation functions have usually been used in domain boundary element analysis in order to represent, respectively, the dependence of potential (displacement) and flux (traction/' . In the standard BEM formulation*, the use of constant approximation is the only way to consider traction discontinuities. Higher order interpolation, which implicitly requires continuity of tractions, is unable to represent wave reflection when the wave front reaches boundary parts where the potential function (displacement) is prescribed. Errors originated from the continuity requirement for tractions are critical in finite domain analysis. Naturally, a question arises: is it possible to use higher order interpolation functions for tractions in a domain BEM analysis? The answer to this question is yes, although not with the standard BEM formulation. This is accomplished in a more general procedure, developed in this paper, which employs linear interpolation for the tractions. In the procedure discussed here, an additional set of equations, provided by the velocity boundary integral equation applied to the nodes having prescribed essential boundary conditions, is incorporated to the Transactions on Modelling and Simulation vol 18, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X 108 Boundary Elements original set of equations. Matrices are partially assembled in such a way that, at the end of the process, two unknown values of tractions at the same time node are obtained. The boundary integral representations for the potential and its derivative are presented by making use of the concept of finite part of an integral (FPI), due to Hadamard*. In order to obtain the discretized versions of these representations, linear variation is assumed to both the potential and its normal derivative (flux) and, as usual in BEM domain analysis, integration is performed analytically. Compact expressions of the integrated kernels, corresponding to typical positions of the wave front of the fundamental solution, are presented for completeness. Linear boundary elements are used in the boundary discretization. In order to assess the accuracy of the formulation, a numerical example is analysed at the end of the paper. INTEGRAL EQUATION The domain integral representation corresponding to the scalar wave equation can be written as (source density and initial conditions contributions have not been included): u/X,t;%,T)u(X,T)dT dF(X) (1) The fundamental solution u*(X,t£,T) corresponds to the effect of an impulse at t=r located at X=£ and is defined as follows: u*(X,t; W = M^ 2 2 H[c(t-i)-r] = U*(X,t; %,t)H[c(t-T)-r] (2) ' In expression (2) H[c(t-x)-r] stands for the Heaviside function (r is the distance between the field (X) and the source (£) points). The symbol =f on the second term on the right-hand-side of equation (1) stands for the finite part of an integral, as defined by Hadamard*: * U(X,tT T)u(X,T)dT- U*(X,t; ,T)u(X,T)l 'o (3) and Ur(X,t; ,T) is given by: u*(X,t; S,T) = ^ 2 3/2 H[c(t-t)-r] = U*(X,t; [c(t-x)-r] (4) TIME DERIVATIVE BOUNDARY INTEGRAL EQUATION The derivative of equation (1) with respect to can be written as: t = == U, (X,t£,T)p(X,T)dTdr(X) JrJo (5) The fundamental function u,(X,t; 4,1) that appears in the first term on the right-hand-side of equation (5) is defined as follows: u(X,t; W = 2 2 (t-T) -r ] 3/2 H[c(t-T)-r] = U(X,t; ,T)H[c(t-T>-r] (6) The FPI integral and the derivative which appear in the right-handside of equation (5) are defined, respectively, as follows: