Hybrid Optimization Techniques for Multi-State Optimal Design Problems

This paper addresses optimal design problems governed by multi-state stationary diffusion equations, aiming at the simultaneous optimization of the domain shape and the distribution of two isotropic materials in prescribed proportions. Existence of generalized solutions is established via a hybrid approach combining homogenization-based relaxation in the interior with suitable restrictions on admissible domains. Based on this framework, we propose a numerical method that integrates homogenization and shape optimization. The domain boundary is evolved using a level set method driven by the shape derivative, while the interior material distribution is updated via an optimality criteria algorithm. The approach is demonstrated on a representative example.