A level‐set‐based shape optimization method for thermoelectric materials

Abstract

AbstractA thermoelectric device is a useful device which directly converts thermal energy to electric energy and vice versa. Although this device has such energy conversion functions, the demands in the industries are limited because of the relatively low conversion performance. In order to enhance this performance, the thermal conductivity of thermoelectric material should be low. According to dramatic improvements of nano‐processing accuracy in recent decades, many thermoelectric materials have been developed with low thermal conductivities by utilizing a unique phenomenon of a nanoscale heat conduction which is called “temperature jump.” However, few structural optimization methods have been proposed for nanoscale thermoelectric materials since a conventional analysis method cannot be applied to a heat conduction problem in a nanostructure. On behalf of the Poisson's equation, we need to introduce a Boltzmann transport equation for describing a nanoscale heat conduction problem with a discontinuity effect on a material interface. The level set method has a potential for dealing with such a boundary condition in structural optimization. That is, material interfaces can be expressed by the iso‐surface of the level set function, and the boundary effects are imposed in the boundaries during the optimization procedures. In this paper, we propose a level‐set‐based shape optimization method for thermoelectric materials. First, an analysis method for a nanoscale heat conduction problem based on the Boltzmann transport equation is introduced. An objective functional is defined for obtaining a low thermal conductivity material. Then, based on an adjoint method, we derive the shape sensitivity, which enables us to consider the effects of the temperature jumps on the material interfaces. Next, a level‐set‐based shape optimization is applied to our optimization problem. Finally, as a validation of our proposed method, several numerical examples are computed.