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Variational approach to relaxed topological optimization: closed form solutions for structural problems in a sequential pseudo-time framework
arXiv - CS - Computational Engineering, Finance, and Science Pub Date : 2021-08-05 , DOI: arxiv-2108.02535
J. Oliver, D. Yago, J. Cante, O. Lloberas-Valls

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 nonsmoothed 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 closedform solution method this possibly being one of its advantages.

中文翻译:

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

这项工作基于以下元素探索了结构计算优化的特定场景:(a) 考虑了 ersatz (bi-material) 近似的宽松优化设置,(b) 基于非平滑特征函数场作为拓扑设计变量的处理, (c) 一致推导松弛拓扑导数,其确定简单、通用且有效,(d) 将整体增加的成本函数拓扑灵敏度公式化为合适的最优性标准,以及 (e) 考虑伪时间框架为问题解,受问题约束演化。在此设置中,表明可以在变分框架中解析求解优化问题,从而导致特征函数的非线性、封闭形式的代数解,然后在每个时间步长中,通过基于伪能量切割算法的定点方法与约束的精确满足相结合,在非线性算法的每次迭代中,通过二分法求解。拓扑解决方案的不适定性(网格依赖)问题,然后通过该伪能量的拉普拉斯平滑很容易解决。在上述上下文中,解决了许多(3D)拓扑结构优化基准,并分析了使用探索的封闭形式解法获得的解,并通过替代水平集方法与它们的解进行了比较。尽管在成本函数和拓扑设计方面获得的结果在两种方法中都非常相似,
更新日期:2021-08-07
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