A boundary contour formulation for design sensitivity analysis in two-dimensional linear elasticity

Abstract

A formulation for computing first-order shape design sensitivities in two-dimensional (2-D) linear elastostatics by the boundary contour method (BCM), along with a numerical implementation using quadratic boundary elements, is presented in this paper. Here, the direct differentiation approach is analytically applied to the appropriate boundary contour equations in order to derive the sensitivities of all the physical quantities (displacements, tractions and stresses) on the boundary as well as those for displacements and stresses inside the body under consideration. The nonsingular formulation of the BCM is used for computing the boundary displacements, and boundary stresses at “off contour” regular points. A regular boundary point is a point on the boundary where it is locally smooth; an off contour point lies inside a boundary element. Their corresponding sensitivities are obtained in a straightforward manner from the resulting regular sensitivity formulation. Also, the stress sensitivities at the boundary nodes can be recovered easily from the global displacement shape functions described in a Cartesian coordinate system. Finally, through three numerical examples for which analytical solutions exist, it is shown that the BCM can provide remarkably accurate numerical results for shape sensitivities.