Robust topology optimization for heat conduction with polynomial chaos expansion
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(2020) 42:284
TECHNICAL PAPER
Robust topology optimization for heat conduction with polynomial chaos expansion André Jacomel Torii1 · Diogo Pereira da Silva Santos2 · Eduardo Morais de Medeiros1 Received: 12 September 2019 / Accepted: 20 April 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract In this work, we present a non-intrusive polynomial chaos expansion (PCE) approach for topology optimization in the context of stationary heat conduction. The robust topology optimization problem and its sensitivity in both its variational and approximate forms are discussed. Sensitivity analysis of the statistical moments and PCE are described in detail. The variational boundary value problems were solved using the finite element method. The material distribution approach with the SIMP model was employed to represent the design. The numerical examples presented show applications addressing heat generation with uncertain magnitude, heat generation at uncertain location and damage with uncertain location. These examples prove that uncertainty-based optimization is able to obtain more robust designs than deterministic approaches. Keywords Topology optimization · Polynomial chaos expansion · Heat conduction · Sensitivity analysis
1 Introduction The main goal of topology optimization is to distribute material inside a given design domain in order to obtain an optimum design [8, 56]. Due to the complexity of the resulting optimization problem, several mathematical structures have been developed for this task (see Sigmund and Maute [56] for a complete review), such as material distribution [8], homogenization [31, 58], shape optimization [1] and topological sensitivity [2, 48]. Topology optimization has also been applied to problems arising from several different phenomena, such as solid mechanics [8, 24, 49], fluid mechanics [11], heat transfer [12, 17, 27, 41], acoustics [68] and optics [34]. An important question that always concerned researchers working on topology optimization was whether or not the optimum designs obtained were robust enough to be Technical Editor: José Roberto de França Arruda. * André Jacomel Torii [email protected] 1
Programa de Pós‑Graduação em Engenharia Civil e Ambiental (PPGECAM), Universidade Federal da Paraíba (UFPB), João Pessoa, PB 58059‑900, Brazil
Laboratório Nacional de Computação Científica (LNCC), Petrópolis, RJ 25651‑075, Brazil
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employed in practice. Several authors argued that designs obtained with topology optimization are frequently nonrobust, in the sense that small perturbations on the imposed conditions of the physical problem may lead to a very poor performance of the design. In fact, in the numerical examples of this work we observe situations where this is indeed true when non-robust optimization is employed. In the last decades, several works showed the importance of modeling uncertainties in optimization problems [5, 9, 54]. It has been observed that designs obtained in the deterministic context often present very poor performance
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