论文标题
离散的最佳设计问题的形状和拓扑灵敏度分析的统一方法
A unified approach to shape and topological sensitivity analysis of discretized optimal design problems
论文作者
论文摘要
我们引入了一个统一的灵敏度概念，以实现形状和拓扑扰动，并对两个空间维度中离散的PDE受限设计优化问题进行灵敏度分析。我们假设该设计由固定有限元网格上的分段线性和全球连续级别设置函数表示，并将级别集合函数的扰动与相应设计的形状或拓扑的扰动相关联。我们说明了一个受反应扩散方程约束的问题的敏感性分析，并在我们的离散敏感性与形状和拓扑衍生物的连续概念之间建立了联系。最后，我们验证了我们的敏感性，并在基于级别的设计优化算法中说明了它们的应用，在这种算法中，必须在形状和拓扑更新中进行区分。
We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is represented by a piecewise linear and globally continuous level set function on a fixed finite element mesh and relate perturbations of the level set function to perturbations of the shape or topology of the corresponding design. We illustrate the sensitivity analysis for a problem that is constrained by a reaction-diffusion equation and draw connections between our discrete sensitivities and the well-established continuous concepts of shape and topological derivatives. Finally, we verify our sensitivities and illustrate their application in a level-set-based design optimization algorithm where no distinction between shape and topological updates has to be made.
