柔度极小化问题拓扑Optimization的非凸性
Revisiting Non-Convexity in Topology Optimization of Compliance Minimization Problems
摘要：
目的：这是一个尝试，以更好地弥合数学和工程/物理方面的主题之间的差距。我们跟踪不同来源的非凸化的背景下，拓扑最优化问题从域离散化，通过惩罚的离散性和过滤方法的影响，并结束与连续方法的说明。设计/方法/途径：从顺应性最小化问题的全局最优值出发，我们采用分析工具来研究中间密度惩罚如何影响问题的凸性，各种滤波技术的潜在惩罚效应，如何使用连续方法来接近全局最优值，以及初始猜测如何在确定最终最优值时具有一定的权重。调查结果：惩罚的非凸化效果的中间密度元素只是盖过任何其他类型的非凸化引入到问题中，主要是由于其严重性和局部性。强烈建议使用延拓方法来克服局部最小值问题，尽管其步骤和收敛标准取决于应用类型而留给用户。创意/价值：在这篇文章中，我们提出了一个基于密度的拓扑最优化问题的非凸性的来源，重点是线弹性柔度最小化的全面处理。我们特别强调了潜在的惩罚样的各种过滤技术的影响，通过详细的数学处理。
英文摘要：
Purpose: This is an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic. We trace the different sources of non-convexification in the context of topology optimization problems starting from domain discretization, passing through penalization for discreteness and effects of filtering methods, and end with a note on continuation methods. Design/Methodology/Approach: Starting from the global optimum of the compliance minimization problem, we employ analytical tools to investigate how intermediate density penalization affects the convexity of the problem, the potential penalization-like effects of various filtering techniques, how continuation methods can be used to approach the global optimum, and how the initial guess has some weight in determining the final optimum. Findings: The non-convexification effects of the penalization of intermediate density elements simply overshadows any other type of non-convexification introduced into the problem, mainly due to its severity and locality. Continuation methods are strongly recommended to overcome the problem of local minima, albeit its step and convergence criteria are left to the user depending on the type of application. Originality/Value: In this article, we present a comprehensive treatment of the sources of non-convexity in density-based topology optimization problems, with a focus on linear elastic compliance minimization. We put special emphasis on the potential penalization-like effects of various filtering techniques through a detailed mathematical treatment.
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