Topology optimization of thermal problems in a nonsmooth variational setting: closed-form optimality criteria

TL;DR

This paper extends the Relaxed Variational Approach to thermal topology optimization, providing closed-form optimality criteria and demonstrating its effectiveness through 3D examples and comparisons with level set methods.

Contribution

The paper introduces a novel application of the RVA to thermal problems, deriving closed-form optimality criteria and integrating finite element discretization for improved optimization.

Findings

RVA-based solutions outperform level set methods in certain metrics

The method effectively handles 3D thermal topology optimization problems

Computational cost is comparable or lower than existing approaches

Abstract

This paper extends the nonsmooth Relaxed Variational Approach (RVA) to topology optimization, proposed by the authors in a preceding work, to the solution of thermal optimization problems. First, the RVA topology optimization method is briefly discussed and, then, it is applied to a set of representative problems in which the thermal compliance, the deviation of the heat flux from a given field and the average temperature are minimized. For each optimization problem, the relaxed topological derivative and the corresponding adjoint equations are presented. This set of expressions are then discretized in the context of the finite element method and used in the optimization algorithm to update the characteristic function. Finally, some representative (3D) thermal topology optimization examples are presented to asses the performance of the proposed method and the Relaxed Variational Approach solutions are compared with the ones obtained with the level set method in terms of the cost function, the topology design and the computational cost.