A boundary element approach for recovery of share sensitivities in three-dimensional elastic solids

Abstract

A boundary element formulation for design sensitivities has been developed for application in three-dimensional shape optimization of elastic solids. Both displacement and stress sensitivities can be determined, at any stage of the design process, by relatively straightforward numerical integration procedures. Further simplification in calculating displacement sensitivities is accomplished by using a special rigid-body integral identity to remove singularities. The issue of accuracy was addressed by analyzing the problem of an infinite elastic solid with a triaxial ellipsoidal cavity. It turns out that this test case is considerably more general than the usual axisymmetric examples employed in prior elastostatic error analyses. Furthermore, the stress concentration can be increased indefinitely by decreasing the two aspect ratios that define the cavity surface. More importantly, the availability of an analytical solution makes it possible to obtain an exact measure of error. For completeness, stresses and stress sensitivities were obtained for three progressively eccentric cavity shapes, using three different singular integration schemes. Generally, the numerical predictions and exact results were in excellent agreement. In the worst case, with a stress concentration of about 4.1, the best stress sensitivity prediction was within 2 percent of the exact value. This is remarkably accurate, given that the corresponding cavity model consisted of only twelve elements per octant.