Minimizing the quadratic mean temperature gradient for the heat-conduction problem using the level-set method

Abstract

This paper presents a numerical algorithm for minimizing the quadratic mean temperature gradient for the heat-conduction problem on the basis of the shape derivative for an elliptical system and the level-set method for a propagating surface. The level-set method as an implicit boundary model is employed to represent the optimal boundaries of heat transfer material. The objective function of the optimization problem is the quadratic mean temperature gradient. The shape of physical domain is treated as the design variable. The material derivative theory of the continuum mechanics and the adjoint method are used to implement the shape sensitivity analysis of the objective function. Since the level-set approach itself cannot generate new holes in the material region, as a remedy, the topological derivative of the elliptic equations that generates new holes to suppress the topological dependence of initialization is introduced. Numerical examples demonstrate that the proposed method is an effective technique for the optimal design of the heat-conduction problem.