Closed-form integration of singular terms for constant , linear and quadratic boundary elements . Part 2 . SVP wave propagation

Part I of these two related papers considered the analyticalevaluation of singular integrals for anti-plane boundary elements, the results of which were then applied to the evaluation of scattering problems involving SH waves. This second part provides an extension of these results to the more complicated case of in-plane boundary elements, and presents their application to scattering problems involving SV-P waves. First, the singular integrals for constant, linear and quadratic boundary elements are evaluated a in closed form. Thereafter, the formulation is used to model cylindrical inclusions in a two-dimensional elastic medium illuminated by dynamic in-plane (plane-strain) line sources. The boundary element method (BEM) results are then compared with the known analytical solutions for these problems. q 1999 Elsevier Science Ltd. All rights reserved. Keywords:Wave propagation; Shear waves; Elastic inclusions; Scattering; Boundary element method; Singular integrals 1. Introduction Part I of these two related papers [1] considered the analytical evaluation of singular integrals for anti-plane boundary elements and their application to scattering problems with SH waves. This second part extends those results to the more complicated case of in-plane (plane strain) boundary elements, and applies them to scattering problems involving SV-P waves. First, the singular integrals for constant, linear and quadratic boundary elements are evaluated analytically. These exact integrals are then compared with those obtained with Gaussian quadrature, which both demonstrates the validity of the analytical expressions presented and also allows us to assess the accuracy of the numerical method. Thereafter, the formulation is used to model cylindrical inclusions in a two-dimensional elastic medium insonified (or illuminated) by dynamic in-plane (i.e. plane-strain) line sources. Finally, the boundary element method (BEM) results are compared with the known closed-form solutions for these scattering problems. 2. Boundary element formulation The boundary element formulation for in-plane wave motion involves the integrals [2–4] H ij ˆ Z Cl fHij …xk; xl ;nl† dCl …i; j ˆ 1;2† …1† G ij ˆ Z Cl fGij …xk; xl† dCl …i; j ˆ 1;2† …2† in which Gij …xk; xl† and Hij …xk; xl ;nl† are, respectively, the components of the Green’s tensor for displacement and traction components at the observation point xk in direction i caused by a concentrated load acting at the source point xl in direction j. Also, nl is the unit outward normal for the lth boundary segment Cl, andf contains the interpolation functions. The requisite Green’s function for this problem is [5] Gij …x; x0† ˆ i=…4m†{ 2 dij H0…kbr†1 dij =…kbr†bH1…kbr† 2 jH1…kar†c} 2 i=…4m† 2r 2xi 2r 2xj bH2…kbr†2 jH2…kar†c ( )