基于参数非设计域区域的线弹性切割拓扑优化
Cut Topology Optimization for Linear Elasticity with Coupling to Parametric Nondesign Domain Regions
AI总结：
本文提出了一种基于切割有限元方法的密度型拓扑优化方法，用于线弹性问题，通过允许设计域切割背景网格并加入稳定化项来保证方法稳定性，同时考虑界面条件和非设计域区域的耦合，支持使用参数化网格有限元或同构几何分析进行非设计域区域的独立离散化。
摘要：
我们开发了一种基于切割有限元方法的密度型拓扑优化方法，用于线弹性问题。具体而言，设计域通过切割有限元进行离散化，允许在结构固定背景网格上表示复杂的几何形状。设计域的几何形状可以以任意方式切割背景网格，并在切割边界附近加入某些稳定化项，以保证方法的稳定性。此外，除了标准的Dirichlet和Neumann条件外，我们还考虑了界面条件，使设计域能够与已知部分的结构耦合。这些已知部分的结构，称为非设计域区域，通常代表由设计师提供的几何部分。非设计域区域可以独立于设计域进行离散化，例如使用参数化网格有限元或同构几何分析。界面和Dirichlet条件基于Nitsche的方法，并且对于密度参数的整个范围都是稳定的。特别是，当密度趋于零时，我们获得一个无应力的Neumann条件。
英文摘要：
We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represents parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche's method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero.
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