Shape Sensitivity Formulation for Axisymmetric Thermal Conducting Solids

Abstract

A direct differentiation method is presented for the shape design sensitivity analysis of axisymmetric thermal conducting solids. Based purely on the standard boundary integral equation (BIE) formulation, a new BIE is derived using the material derivative concept. Design derivatives in terms of shape change are directly calculated by solving the derived BIE. The present direct method has a computational advantage over the adjoint variable method, in the sense that it avoids the problem of solving for the adjoint system with the singular boundary condition. Numerical accuracy of the method is studied through three examples. The sensitivities by the present method are compared with analytic sensitivities for two problems of a hollow cylinder and a hollow sphere, and are then compared with those by finite differences for a thermal diffuser problem. As a practical application to numerical optimization, an optimal shape of the thermal diffuser to minimize the weight under a prescribed constraint is found by use of an optimization routine.