SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1510-1541, January 2021.
Shape gradients have been widely used in numerical shape gradient descent algorithms for shape optimization. The two types of shape gradients, i.e., the distributed one and the boundary type, are equivalent at the continuous level but exhibit different numerical behaviors after finite element discretization. To be more specific, the boundary type shape gradient is more popular in practice due to its concise formulation and convenience in combining with shape optimization algorithms but has lower numerical accuracy. In this paper we provide a simple yet useful boundary correction for the normal derivatives of the state and adjoint equations, motivated by their continuous variational forms, to increase the accuracy and possible effectiveness of the boundary shape gradient in PDE-constrained shape optimization. We consider particularly the state equation with Dirichlet boundary conditions and provide a preliminary error estimate for the correction. Numerical results show that the corrected boundary type shape gradient has comparable accuracy to that of the distributed one. Moreover, we give a theoretical explanation for the comparable numerical accuracy of the boundary type shape gradient with that of the distributed shape gradient for Neumann boundary value problems.

中文翻译：

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用于偏微分方程约束形状优化的边界型离散形状梯度

SIAM 数值分析杂志，第 59 卷，第 3 期，第 1510-1541 页，2021 年 1 月。
形状梯度已广泛用于形状优化的数值形状梯度下降算法。分布型和边界型两种形状梯度在连续水平上是等价的，但在有限元离散化后表现出不同的数值行为。更具体地说，边界型形状梯度由于其表述简洁、便于与形状优化算法结合而在实践中更受欢迎，但其数值精度较低。在本文中，我们为状态和伴随方程的法向导数提供了一种简单而有用的边界校正，受其连续变分形式的启发，以提高边界形状梯度在 PDE 约束形状优化中的准确性和可能的​​有效性。我们特别考虑了具有 Dirichlet 边界条件的状态方程，并为校正提供了初步的误差估计。数值结果表明，修正后的边界型形状梯度与分布式的具有相当的精度。此外，我们对 Neumann 边界值问题的边界型形状梯度与分布形状梯度的数值精度的可比性给出了理论解释。