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Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

机译:松弛拓扑优化的变分方法:顺序伪时间框架中结构问题的封闭形式解决方案

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The work explores a specific scenario for structural computational optimization based on the following elements: (a) a relaxed optimization setting considering the ersatz (bi-material) approximation, (b) a treatment based on a non-smoothed characteristic function field as a topological design variable, (c) the consistent derivation of a relaxed topological derivative whose determination is simple, general and efficient, (d) formulation of the overall increasing cost function topological sensitivity as a suitable optimality criterion, and (e) consideration of a pseudo-time framework for the problem solution, ruled by the problem constraint evolution.In this setting, it is shown that the optimization problem can be analytically solved in a variational framework, leading to, nonlinear, closed-form algebraic solutions for the characteristic function, which are then solved, in every time-step, via fixed point methods based on a pseudo-energy cutting algorithm combined with the exact fulfillment of the constraint, at every iteration of the non-linear algorithm, via a bisection method. The issue of the ill-posedness (mesh dependency) of the topological solution, is then easily solved via a Laplacian smoothing of that pseudo-energy.In the aforementioned context, a number of (3D) topological structural optimization benchmarks are solved, and the solutions obtained with the explored closed-form solution method, are analyzed, and compared, with their solution through an alternative level set method. Although the obtained results, in terms of the cost function and topology designs, are very similar in both methods, the associated computational cost is about five times smaller in the closed-form solution method this possibly being one of its advantages. Some comments, about the possible application of the method to other topological optimization problems, as well as envisaged modifications of the explored method to improve its performance close the work. (C) 2019 Elsevier B.V. All rights reserved.
机译:这项工作基于以下要素探索了一种结构计算优化的特定方案:(a)考虑ersatz(双材料)近似的轻松优化设置,(b)基于非平滑特征函数场作为拓扑的处理设计变量;(c)确定简单,通用且有效的松弛拓扑导数的一致推导;(d)将总体成本函数拓扑敏感性确定为合适的最优准则,以及(e)考虑伪伪在这种情况下,表明可以在变分框架中解析求解优化问题,从而得到特征函数的非线性封闭形式的代数解,从而确定了问题求解的时间框架。然后在每个时间步骤中,通过基于伪能量削减算法并结合精确实现的定点方法求解通过二等分方法,在非线性算法的每次迭代中约束条件的变化。然后可以通过该伪能量的拉普拉斯平滑轻松解决拓扑解决方案的不适定性(网格依赖性)问题。在上述情况下,解决了许多(3D)拓扑结构优化基准,并且通过探索的封闭形式求解方法获得的求解,将通过替代级别集方法进行分析,并与之进行比较。尽管就成本函数和拓扑设计而言,所获得的结果在两种方法中都非常相似,但是在闭式求解方法中,相关的计算成本大约小五倍,这可能是其优势之一。一些评论,关于该方法在其他拓扑优化问题中的可能应用,以及对所探索方法的改进以改善其性能的预期工作结束。 (C)2019 Elsevier B.V.保留所有权利。

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