Boundary-element shape sensitivity analysis for thermal problems with nonlinear boundary conditions

Implicit differentiation of the discretized boundary integral equations governing the conduction of heat in solid objects subjected to nonlinear boundary conditions is shown to generate an accurate and economical approach for the computation of shape sensitivities for this class of problems. This approach involves the employment of analytical derivatives of boundary-element kernel functions with respect to shape design variables. A formulation is presented that can consistently account for both temperature-dependent convection and radiation boundary conditions. Several iterative strategies are presented for the solution of the resulting sets of nonlinear equations and the computational performances examined in detail. Multizone analysis and zone condensation strategies are demonstrated to provide substantive computational economies in this process for models with either localized nonlinear boundary conditions or regions of geometric insensitivity to design variables. A series of nonlinear example problems are presented that have closed-form solutions.