Level set-based shape optimization for temperature minimization problems in nanostructures

Abstract

Heat conduction in nanostructures is ballistic process, whereas heat conduction at macroscale is diffusive. When a system has two types of materials, temperature discontinuities occur on the material interfaces. Thermal designs for electric devices in nanostructures utilize these unique phenomena, especially temperature jumps on material interfaces and they let us obtain innovative material properties. However, almost all conventional designs for nanoscale heat conduction problems have been dependent on heuristic and experimental approaches. In order to develop high-performance electric devices, we propose a level set-based shape optimization method for temperature minimization problems in nanostructures. First, we define an optimization problem for a nanoscale heat conduction problem based on the Boltzmann transport equation, then obtain a shape sensitivity through an adjoint method, which can precisely consider the temperature discontinuities on material interfaces. Next, we expand a level set-based shape optimization method to the design problem. Finally, we confirm the validity of our proposed method through a numerical example.