Topology Optimization Incorporating Level Set Boundary Expressions Using a Particle Method
Topology Optimization Incorporating Level Set Boundary Expressions Using a Particle Method
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DOI：
10.1299/kikaia.77.2054
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
M. Manabe;T. Yamada;K. Izui;S. Nishiwaki
中科院分区：
文献类型：
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作者：
M. Manabe;T. Yamada;K. Izui;S. Nishiwaki
结构的拓扑优化已被应用于非线性结构问题，然而传统的几何非线性结构拓扑优化方法由于使用网格，在使用FEM （Finite Element Method）进行非线性分析时遇到困难。在这项研究中，我们提出了一种新的基于水平集的拓扑优化方法，考虑几何非线性，使用无网格/粒子技术来优化大变形的弹性结构。在该方法中，由于不依赖网格进行几何非线性分析，因此采用粒子法MPS （Moving Particle semi - implicit）进行响应分析。本文首先提出了一个基于水平集方法的拓扑优化问题，并给出了一种利用Tikhonov正则化方法对优化问题进行正则化的方法。推导了更新水平集函数的反应扩散方程，构造了利用有限元法求解平衡方程和更新水平集函数时的反应扩散方程的优化算法。其次，展示了粒子相互作用模型和MPS方法中几何非线性的处理，并解释了基于水平集的拓扑优化与MPS方法相结合的实现。最后，给出了几个数值算例，验证了所提出的拓扑优化方法对几何非线性问题的有效性。
Topology optimization for structures has been applied to nonlinear structural problems, however conventional topology optimization methods for structures with geometrical nonlinearity encounter difficulties during nonlinear analysis using the FEM (Finite Element Method), due to the use of a mesh. In this study, we propose a new level set-based topology optimization method considering geometrical nonlinearity using a mesh-free/particle technique, for optimizing elastic structures that undergo large deformation. In the proposed method, the MPS (Moving Particle Semiimplicit) method, a particle method, is used for the response analysis, since it does not rely on a mesh for geometrically nonlinear analysis. In this paper, first, a topology optimization problem is formulated based on the level set method and a method for regularizing the optimization problem by the Tikhonov regularization method is explained. The reactiondiffusion equation that updates the level set function is then derived and an optimization algorithm, which uses the FEM to solve the equilibrium equations and the reaction-diffusion equation when updating the level set function, is constructed. Next, the particle interaction model and the treatment of geometrical nonlinearity in the MPS method are shown, and the implementation of combining the level set-based topology optimization and the MPS methods is explained. Finally, several numerical examples are provided to demonstrate the effectiveness of the proposed method of topology optimization for geometrically nonlinear problems.