标题： 用于二维和三维大规模并行拓扑优化的55行代码 显示英文标题

标题： A 55-line code for large-scale parallel topology optimization in 2D and 3D

摘要： 本文提出了一段55行的Python代码，用于基于开源有限元计算软件(FEniCS)的二维和三维拓扑优化(TO)，该代码配备了各种有限元工具和求解器。PETSc被用作线性代数后端，这使得计算时间显著少于标准Python库。该代码是基于流行的固体各向同性材料惩罚(SIMP)方法设计的。还介绍了多工况、不同边界条件以及被动单元的集成等扩展。因此，该实现是基于SIMP的三维和二维问题拓扑优化中最紧凑的实现。利用欧几里得距离矩阵的概念来对滤波器的权重矩阵进行矢量化计算，我们实现了计算时间的显著减少，并使代码能够处理复杂的基结构配置。我们还介绍了该代码在大规模拓扑优化问题上的扩展，支持在复杂结构配置上的并行计算，这可以帮助学生和研究人员通过密集网格探索拓扑优化问题的新见解。附录-A包含完整的代码，网站:\url{https://github.com/iitrabhi/topo-fenics}也包含完整的代码。 显示英文摘要

摘要： This paper presents a 55-line code written in python for 2D and 3D topology optimization (TO) based on the open-source finite element computing software (FEniCS), equipped with various finite element tools and solvers. PETSc is used as the linear algebra back-end, which results in significantly less computational time than standard python libraries. The code is designed based on the popular solid isotropic material with penalization (SIMP) methodology. Extensions to multiple load cases, different boundary conditions, and incorporation of passive elements are also presented. Thus, this implementation is the most compact implementation of SIMP based topology optimization for 3D as well as 2D problems. Utilizing the concept of Euclidean distance matrix to vectorize the computation of the weight matrix for the filter, we have achieved a substantial reduction in the computational time and have also made it possible for the code to work with complex ground structure configurations. We have also presented the code's extension to large-scale topology optimization problems with support for parallel computations on complex structural configuration, which could help students and researchers explore novel insights into the TO problem with dense meshes. Appendix-A contains the complete code, and the website: \url{https://github.com/iitrabhi/topo-fenics} also contains the complete code.