发表机构
Queensland University of Technology; Umeå University（昆士兰科技大学; 于默奥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出形状微积分与可扩展自动形状微分框架，用于非拟合多相拓扑优化，经验证高精度，支持大规模并行，并应用于各向异性扩散、线弹性和流固耦合问题。
AI 中文摘要
我们提出了形状微积分技术以及一个可扩展的自动形状微分框架，用于在由多个水平集函数定义的非拟合离散化上进行多相拓扑优化。首先，我们利用凸几何的概念，为多相系统建立了离散情形下一般且精确的形状微积分表达式。为补充这一理论基础，我们引入了一个基于多面体切割的开源多相自动形状微分框架。该计算框架通过有限差分和我们建立的精确表达式进行验证，与后者匹配至接近机器精度。此外，所提出的自动形状微分在分布式计算环境中具有可扩展性，在13,824个计算机核心上展示了接近理想的弱扩展性，最高可处理16.5亿个有限元。我们通过求解各向异性扩散、线弹性和流固耦合的非拟合多相拓扑优化问题来展示我们的实现。这些理论和计算贡献共同为使用非拟合有限元方法推进多相拓扑优化提供了稳健且易于获取的基础。特别是，这些方法能够求解涉及多相和多物理系统且具有非平凡边界条件的拓扑优化问题。开源软件可在该https URL获取。
英文摘要
We present shape calculus techniques and a scalable automatic shape differentiation framework for multi-phase topology optimisation on unfitted discretisations defined by several level-set functions. First we establish general, exact shape calculus expressions in the discrete case for multi-phase systems by leveraging concepts from convex geometry. To complement this theoretical foundation, we introduce an open-source multi-phase automatic shape differentiation framework based on polytopal cutting. This computational framework is validated against both finite differences and our established exact expressions, matching the latter to near machine precision. Furthermore, the proposed automatic shape differentiation is scalable across distributed computing environments, demonstrating near ideal weak scaling up to 1.65 billion finite elements across 13,824 computer cores. We demonstrate our implementation by solving unfitted multi-phase topology optimisation problems for anisotropic diffusion, linear elasticity, and fluid--structure interaction. Together, these theoretical and computational contributions provide a robust and accessible foundation for advancing multi-phase topology optimisation using unfitted finite element methods. In particular, the methods enable the solution of topology optimisation problems involving multi-phase and multi-physics systems with non-trivial boundary conditions. The open-source software is available at https://github.com/zjwegert/GridapTopOpt.jl.
