A mathematical game for topology optimization of cooling systems

We propose a topology optimization-based method for optimal design of cooling systems in the form of a mathematical game between two players trying to reach a compromise between limiting the amount of a cooling medium used and obtaining low temperatures in the design domain. The flow of the cooling medium is governed by a Stokes-Brinkman flow model with penalty, while the temperature is governed by a stationary convection-diffusion problem whose solution is approximated using a finite element method with consistent stabilization. Existence of solution for the continuum problems and finite element convergence are shown. The idea and performance of the proposed design method are illustrated by numerical examples based on a problem-setting inspired by an industrial design problem for a gas turbine part. The method exhibits good convergence and is able to generate meaningful design concepts representing various levels of compromise between limited use of cooling medium and low temperatures.