A Topology Optimization Method for Geometrically Nonlinear Problems Incorporating Level Set Boundary Expressions and a Particle Method

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Abstract

Structural topology optimization has been applied to nonlinear structural problems, but conventional methods considering geometrical nonlinearity encounter difficulties during nonlinear analysis using the FEM (Finite Element Method). In this study, we propose a new level set-based topology optimization method considering geometrical nonlinearity, using a meshfree particle technique, for optimizing elastic structures that undergo finite displacement. In the proposed method, the MPS (Moving Particle Semi-implicit) method is used for solving the state equation, since it does not use 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 using the Tikhonov regularization method is explained. The reaction-diffusion equation that updates the level set function is then derived and an optimization algorithm, which uses the FEM to solve 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 described and the implementation that combines the level set-based topology optimization with the MPS method is explained. Finally, several numerical examples are provided to demonstrate the effectiveness of the proposed topology optimization method for geometrically nonlinear problems.

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