Topology optimization of nonlinear heat conduction problems using adjoint design sensitivity analysis method

We develop an adjoint design sensitivity analysis (DSA) method applicable to nonlinear heat conduction problems in steady state. Design sensitivity expressions with respect to thermal conductivity are derived. Using the already factorized system matrix, we compute the sensitivity with trivial costs. The DSA method is further utilized to topology optimization problems, where design variables are parameterized into normalized bulk material densities. Objective function and constraint are a thermal compliance and an allowable material volume of the system, respectively. We face a convergence difficulty in this thermal topology optimization problem just like in the geometrically nonlinear problems of solid mechanics. We can avoid the difficulty by enforcing the smooth variation of design variables. Through numerical examples, the developed DSA method is verified, yielding very accurate sensitivity and requiring trivial CPU time compared with finite difference method. Also, we notice that after the optimization, it yields physically meaningful topology results and overall temperature in the domain is significantly decreased.