Free material optimization of thermal conductivity tensors with asymmetric components

TL;DR

This paper extends Free Material Optimization to include asymmetric thermal conductivity tensors induced by the thermal Hall effect, enabling the design of materials with novel thermal properties.

Contribution

It introduces a new FMO framework for asymmetric tensors, accounting for physical phenomena like the thermal Hall effect, which breaks tensor symmetry.

Findings

Validated the method through numerical experiments

Demonstrated the design of asymmetric thermal conductivity tensors

Showed potential for novel thermal material applications

Abstract

Free Material Optimization (FMO), a branch of topology optimization, in which the design variables are the full constitutive tensors, can provide the most general form of the design problems. Considering the microstructure composed of isotropic materials, the constitutive tensors are yet positive definite and symmetric. On the other hand, it has been reported that the symmetry of this constitutive tensor can be broken in appearance by considering other physical phenomena. In the present study, we focus on the thermal Hall effect, which is explained as the phenomena that induces the temperature gradient orthogonal to a given temperature gradient across a solid when a magnetic field is applied to the solid. This effect makes the thermal conductivity tensor asymmetric and justifies extending the space of the constitutive tensors to be an asymmetric domain. We propose the FMO for asymmetric constitutive tensors, parameterizing the design space so that the physically available property could be naturally satisfied. Several numerical experiments are provided to show the validity and the utility of the proposed method.