Design sensitivity analysis and optimization of interface shape for zoned-inhomogeneous thermal conduction problems using boundary integral formulation

A generalized formulation of the shape design sensitivity analysis for two-dimensional steady-state thermal conduction problem as applied to zoned-inhomogeneous solids is presented using the boundary integral and the adjoint variable method. Shape variation of the external and zone-interface boundary is considered. Through an analytical example, it is proved that the derived sensitivity formula coincides with the analytic solution. In numerical implementation, the primal and adjoint problems are solved by the boundary element method. Shape sensitivity is numerically analyzed for a compound cylinder, a thermal diffuser and a cooling fin problem, and its accuracy is compared with that by numerical differentiation. The sensitivity formula derived is incorporated to a nonlinear programming algorithm and optimum shapes are found for the thermal diffuser and the cooling fin problem.