Model Hierarchy for the Shape Optimization of a Microchannel Cooling System

TL;DR

This paper develops a hierarchical modeling approach for optimizing the shape of microchannel cooling systems, using shape calculus and reduced models to achieve efficient and accurate design improvements.

Contribution

It introduces three models, including two reduced models derived via homogenization and transformation, and demonstrates their effectiveness in shape optimization with less computational effort.

Findings

Reduced models perform similarly to the full model in optimization tasks.

Optimization results show significant computational savings with reduced models.

Shape derivatives and adjoint systems are effectively derived using a material derivative free approach.

Abstract

We model a microchannel cooling system and consider the optimization of its shape by means of shape calculus. A three-dimensional model covering all relevant physical effects and three reduced models are introduced. The latter are derived via a homogenization of the geometry in 3D and a transformation of the three-dimensional models to two dimensions. A shape optimization problem based on the tracking of heat absorption by the cooler and the uniform distribution of the flow through the microchannels is formulated and adapted to all models. We present the corresponding shape derivatives and adjoint systems, which we derived with a material derivative free adjoint approach. To demonstrate the feasibility of the reduced models, the optimization problems are solved numerically with a gradient descent method. A comparison of the results shows that the reduced models perform similarly to the original one while using significantly less computational resources.