Three-Dimensional Optimal Shape Design in Heat Transfer Based on Body-fitted Grid Generation

Abstract

This paper is concerned with an optimal shape design (shape optimization) problem in heat transfer. As an inverse steady-state heat transfer problem, given a body locally heated by a specified heat flux and exposed to convective heat transfer on parts of its boundary, the aim is to find the optimal shape of this body such that the temperature is constant on a desired subset of its boundary. The numerical method to achieve this aim consists of a three-dimensional elliptic grid generation technique to generate a mesh over the body and solve for a heat conduction equation. This paper describes a novel sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of grid node positions and the conjugate gradient method (CGM) is used as an optimization algorithm to minimize the difference between the computed temperature on the boundary and desired temperature. The elliptic grid generation technique allows us to map the physical domain (body) onto a fixed computational domain and to discretize the heat conduction equation using the finite difference method (FDM).