This review presents developments in structural shape optimization post 2000 from perspective of numerical analysis techniques. Traditional shape optimization with FEM has undergone considerable transformation as developments in CAD, numerical analysis techniques and optimization algorithms have contributed significantly in improving it. Mesh dependency and inconsistent description of the geometry for design and analysis models remained major challenges in traditional FEM based shape optimization. To improve mesh based shape optimization, in-plane regularization and out-of plane filtering, vertex morphing, traction method in node based shape optimization, mesh morphing and adaptive mesh refinement techniques are discussed. Alternative numerical techniques have also been discussed briefly from shape optimization perspectives which includes modified versions of FEM like eXtended FEM (XFEM) with level sets, Fixed Grid (FG) FEM/Eulerian approach, interface enriched generalized FEM (IGFEM) and finite cell method (FCM), isogeometric analysis (IGA) and its variants like eXtended IGA (XIGA) and IGABEM and meshless methods (MMs) like element free Galerkin (EFG) and reproducing Kernal particle method (RKPM). These numerical techniques have different mathematical background and hence possess different capabilities and limitations. The present work highlights these differences and compares them in context of shape optimization. Critical observations and future research recommendations are presented before concluding remarks.

这篇综述从数值分析技术的角度介绍了2000年后结构形状优化的发展。随着CAD，数值分析技术和优化算法的发展，使用FEM进行的传统形状优化已经历了相当大的转变，为改进它做出了巨大贡献。在传统的基于FEM的形状优化中，网格相关性以及用于设计和分析模型的几何图形描述不一致仍然是主要挑战。为了改进基于网格的形状优化，讨论了平面内正则化和平面外滤波，顶点变形，基于节点的形状优化中的牵引方法，网格变形和自适应网格细化技术。还已经从形状优化的角度简要讨论了替代数值技术，其中包括FEM的修改版本，例如具有水平集的扩展FEM（XFEM），固定网格（FG）FEM /欧拉方法，界面丰富的广义FEM（IGFEM）和有限元方法（ FCM），等几何分析（IGA）及其变体，例如扩展的IGA（XIGA）和IGABEM，以及无网格方法（MM），例如无元素Galerkin（EFG）和再生核粒子法（RKPM）。这些数值技术具有不同的数学背景，因此具有不同的功能和局限性。本工作重点介绍了这些差异，并在形状优化的上下文中进行了比较。在结束发言之前，将提出批评性的意见和未来的研究建议。