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A finite element based level set method for structural topology optimization.

机译：基于有限元的水平集方法，用于结构拓扑优化。

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摘要

A finite element (FE) based level set method is proposed for structural topology optimization problems in this thesis. The level set method has become a popular tool for structural topology optimization in recent years because of its ability to describe smooth structure boundaries and handle topological changes. There are commonly two stages in the optimization process: the stress analysis stage and the boundary evolution stage. The first stage is usually performed with the finite element method (FEM) while the second is often realized by solving the level set equation with the finite difference method (FDM). The first motivation for developing the proposed method is the desire to unify the techniques of both stages within a uniform framework. In addition, there are many problems involving irregular design domains in practice, the FEM is more powerful than the FDM in dealing with these problems. This is the second motivation for this study.;Solving the level set equation with the standard Galerkin FEM might produce unstable results because of the hyperbolic characteristic of this equation. Therefore, the streamline diffusion finite element method (SDFEM), a stabilized method, is employed to solve the level set equation. In addition to the advantage of simplicity, this method generates a system of equations with a constant, symmetric, and positive defined coefficient matrix. Furthermore, this matrix can be diagonalized by virtue of the lumping technique in structural dynamics. This makes the cost in solving and storing quite low. It is more important that the lumped coefficient matrix may help to improve the stability under some circumstance.;The reinitialization equation is also solved with the SDFEM and an extra diffusion term is added to improve the stability near the boundary. We propose a criterion to select the factor of the diffusion term. Due to numerical errors and the diffusion term, boundary will drift during the process of reinitialization. To constrain the boundary from moving, a Dirichlet boundary condition is enforced. Within the framework of FEM, this enforcement can be conveniently preformed with the Lagrangian multiplier method or the penalty method.;Velocity extension is discussed in this thesis. A natural extension method and a partial differential equation (PDE)-based extension method are introduced. Some related topics, such as the "ersatz" material approach and the recovery of stresses, are discussed as well.;The accuracy of the finite element based level set method (FELSM) is compared with that of the finite difference based level set method (FDLSM). The FELSM is a first-order accurate algorithm but we prove that its accuracy is enough for structural optimization problems considered in this study. Even higher-order accurate FDLSM schemes are used, the numerical results are still the same as those obtained by FELSM. It is also shown that if the Courant-Friedreichs-Lewy (CFL) number is large, the FELSM is more robust and accurate than FDLSM.;Numerical examples are involved in this thesis to illustrate the reliability of the proposed method. Problems on both regular and irregular design domains are considered and different meshes are tested and compared.
机译：本文针对结构拓扑优化问题提出了一种基于有限元的水平集方法。水平集方法由于能够描述平滑的结构边界和处理拓扑变化，因此近年来已成为结构拓扑优化的流行工具。优化过程通常有两个阶段：应力分析阶段和边界演化阶段。第一阶段通常使用有限元方法（FEM）进行，而第二阶段通常通过使用有限差分法（FDM）求解水平集方程来实现。开发提出的方法的第一个动机是希望在一个统一的框架内统一两个阶段的技术。另外，在实践中存在很多涉及不规则设计域的问题，在处理这些问题方面，有限元法比FDM更为强大。这是本研究的第二个动机。；由于该方程的双曲特性，用标准的Galerkin FEM解决水平集方程可能会产生不稳定的结果。因此，采用流线扩散有限元方法（SDFEM）（一种稳定方法）来求解水平集方程。除了简单性的优点之外，该方法还生成具有常数，对称和正定义系数矩阵的方程组。此外，借助于结构动力学中的集总技术，可以将该矩阵对角线化。这使得解决和存储的成本非常低。集总系数矩阵在一定情况下有助于提高稳定性。（SDFEM）还解决了重新初始化方程，并增加了扩散项以提高边界附近的稳定性。我们提出了一个选择扩散项因子的准则。由于数值误差和扩散项，边界在重新初始化过程中会漂移。为了限制边界移动，必须执行Dirichlet边界条件。在有限元框架内，可以通过拉格朗日乘数法或罚分法方便地执行这种强制执行。介绍了自然扩展方法和基于偏微分方程（PDE）的扩展方法。还讨论了一些相关主题，例如“ ersatz”材料方法和应力的恢复。；比较了基于有限元的水平集方法（FELSM）和基于有限差分的水平集方法的精度（ FDLSM）。 FELSM是一阶精确算法，但我们证明其准确性足以解决本研究中考虑的结构优化问题。即使使用了更高阶的精确FDLSM方案，数值结果仍然与FELSM获得的结果相同。还表明，如果Courant-Friedreichs-Lewy（CFL）数较大，则FELSM比FDLSM更为健壮和准确。本文通过算例说明了该方法的可靠性。考虑规则和不规则设计领域的问题，并测试和比较不同的网格。

著录项

  • 作者

    Xing, Xianghua.;

  • 作者单位

    The Chinese University of Hong Kong (Hong Kong).;

  • 授予单位 The Chinese University of Hong Kong (Hong Kong).;
  • 学科 Applied Mechanics.;Engineering Mechanical.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 113 p.
  • 总页数 113
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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