Boundary formulations for shape sensitivity of temperature dependent conductivity problems

Abstract

AbstractUsed in concert with the Kirchhoff transformation, implicit differentiation of the discretized boundary integral equations governing the conduction of heat in solids with temperature dependent thermal conductivity is shown to generate an accurate and economical approach for computation of shape sensitivities. For problems with specified temperature and heat flux boundary conditions, a linear problem results for both the analysis and sensitivity analysis. In problems with either convection or radiation boundary conditions, a non‐linear problem is generated. Several iterative strategies are presented for the solution of the resulting sets of non‐linear equations and the computational performances examined in detail. Multi‐zone analysis and zone condensation strategies are demonstrated to provide substantive computational economies in this process for models with either localized non‐iinear boundary conditions or regions of geometric insensitivity to design variables. A series of non‐linear example problems is presented that have closed form solutions. Exact anaytical expressions tor the shape sensitivities associated with these problems are developed and these are compared with the sensitivities computed using the boundary element formulation.