标题： 基于非匹配有限元的水平集拓扑优化与自动形状微分 显示英文标题

标题： Level-set topology optimisation with unfitted finite elements and automatic shape differentiation

摘要： 在本文中，我们开发了用于非拟合离散化的自动形状微分技术，并将这些技术与非拟合方法的最新形状微积分进展联系起来。我们将现有的解析形状微积分结果扩展到域边界与背景域边界相交的情况。我们进一步表明，使用开发的自动形状微分技术，无论网格大小如何，都可以以机器精度恢复这些解析导数，大大减轻了这些量的解析推导负担。此外，我们还表明，我们还可以恢复对称的形状Hessian矩阵。我们在Julia包GridapTopOpt和更广泛的Gridap生态系统中为串行和分布式计算框架实现了这些技术。作为此实现的一部分，我们提出了一种新的基于图的方法用于孤立体积检测。通过考虑线性弹性轮的三维最小柔度拓扑优化以及在Stokes流中的流体-结构相互作用问题中的线性弹性结构，我们展示了非拟合自动形状微分框架和我们实现的适用性。该实现具有通用性，使GridapTopOpt能够在不进行形状导数的解析计算的情况下，在非结构化网格上解决更广泛的问题，并避免在域边界处平滑材料属性时出现的问题。该软件是开源的，可在https://github.com/zjwegert/GridapTopOpt.jl获取。 显示英文摘要

摘要： In this paper we develop automatic shape differentiation techniques for unfitted discretisations and link these to recent advances in shape calculus for unfitted methods. We extend existing analytic shape calculus results to the case where the domain boundary intersects with the boundary of the background domain. We further show that we can recover these analytic derivatives to machine precision regardless of the mesh size using the developed automatic shape differentiation techniques, drastically reducing the burden associated with the analytic derivation of these quantities. In addition, we show that we can also recover the symmetric shape Hessian. We implement these techniques for both serial and distributed computing frameworks in the Julia package GridapTopOpt and the wider Gridap ecosystem. As part of this implementation we propose a novel graph-based approach for isolated volume detection. We demonstrate the applicability of the unfitted automatic shape differentiation framework and our implementation by considering the three-dimensional minimum compliance topology optimisation of a linear elastic wheel and of a linear elastic structure in a fluid-structure interaction problem with Stokes flow. The implementation is general and allows GridapTopOpt to solve a wider range of problems on unstructured meshes without analytic calculation of shape derivatives and avoiding issues that arise when material properties are smoothed at the domain boundary. The software is open source and available at https://github.com/zjwegert/GridapTopOpt.jl.