Topology optimization with geometrical feature constraints based on the partial differential equation system for geometrical features (Overhang constraints considering geometrical singularities in additive manufacturing)
Topology optimization with geometrical feature constraints based on the partial differential equation system for geometrical features (Overhang constraints considering geometrical singularities in additive manufacturing)
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DOI：
10.1299/transjsme.19-00129
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
T. Yamada;J. Masamune;H. Teramoto;Takahiro Hasebe;H. Kuroda
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文献类型：
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作者：
T. Yamada;J. Masamune;H. Teramoto;Takahiro Hasebe;H. Kuroda
本文旨在开发一种基于几何形状特征的偏微分方程（PDE）的无支撑结构增材制造（AM）拓扑优化中的几何特征约束方案。首先，简要介绍了拓扑优化的基本概念和基于水平集的拓扑优化方法。其次，制定了几何形状特征的偏微分方程系统。这里，使用偏微分方程的解析解讨论状态变量分布的各个方面。基于讨论，制定了指示包括几何奇点的扩展法向矢量的函数。第三，在没有支撑结构的增材制造中产品形状的几何要求（即所谓的悬垂约束）在二维上得到了阐明。还阐明了将所提出的概念扩展到三维问题的方法。此外，还讨论了悬伸约束中的几何奇点。基于偏微分方程系统和明确的几何要求，制定了包括几何奇点的悬伸约束。考虑悬垂约束，制定了线弹性问题的拓扑优化问题。构建了基于水平集的拓扑优化算法，利用有限元法求解线弹性问题和偏微分方程的控制方程，并更新水平集函数。最后，提供了二维数值例子来验证所提出方法的有效性和实用性。
This paper aims to develop a scheme for geometrical feature constraints in topology optimization for Additive Manufacturing (AM) without support structures based on the Partial Differential Equation (PDE) of geometrical shape features. To begin with, the basic concept of topology optimization and a level set-based topology optimization method are briefly described. Second, the PDE system for geometrical shape features is formulated. Here, aspects of the distribution of state variables are discussed using an analytical solution of the PDE. Based on the discussion, a function indicating the extended normal vector including geometrical singularity points is formulated. Third, geometrical requirements of product shape in AM without support structures – the so-called overhang constraint – are clarified in two-dimensions. A way of extending of the proposed concept to three-dimensional problems is also clarified. Additionally, geometrical singularities in the overhang constraint are discussed. Based on the PDE system and the clarified geometrical requirements, the overhang constraint including geometrical singularities is formulated. A topology optimization problem of the linear elastic problem is formulated considering the overhang constraint. A level set-based topology optimization algorithm is constructed where the Finite Element Method (FEM) is used to solve the governing equation of the linear elastic problem and the PDE, and to update the level set function. Finally, two-dimensional numerical examples are provided to confirm the validity and utility of the proposed method.