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.

中文翻译：

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非光滑变分设置中热问题的拓扑优化：封闭形式的最优准则

本文将作者在先前工作中提出的非光滑松弛变分方法 (RVA) 扩展到拓扑优化，以解决热优化问题。首先，简要讨论了 RVA 拓扑优化方法，然后将其应用于一组具有代表性的问题，其中热柔量、给定场的热通量偏差和平均温度最小化。对于每个优化问题，都给出了松弛拓扑导数和相应的伴随方程。然后，这组表达式在有限元方法的上下文中被离散化，并用于优化算法以更新特征函数。最后，