The Free Material Design problem for stationary heat equation on low dimensional structures

TL;DR

This paper addresses the optimal design of conductivity tensors for stationary heat equations on low-dimensional structures, providing explicit solutions and a measure-theoretic framework for minimizing thermal compliance.

Contribution

It introduces a novel measure-based approach to the free material design problem for heat conduction, with explicit solutions and examples on low-dimensional structures.

Findings

Explicit solutions for optimal conductivity tensors are derived.

The solution framework involves non-negative tensor-valued measures.

Examples demonstrate the applicability of the theoretical results.

Abstract

For a given balanced distribution of heat sources and sinks, $Q$, we find an optimal conductivity tensor field, $\hat C$, minimizing the thermal compliance. We present $\hat C$ in a rather explicit form in terms of the datum. Our solution is in a cone of non-negative tensor-valued finite Borel measures. We present a series of examples with explicit solutions.49J20, %Singular parabolic equations secondary: 49K20, 80M50