Geometrically nonlinear isogeometric topology optimization via extended finite element method

Abstract

Geometric nonlinearity can accurately simulate and predict the large deformation behavior of structures, which is crucial for ensuring the safety and reliability of engineering designs. To address the limitations of existing linear isogeometric topology optimization methods in accurately describing the nonlinear behavior of boundary elements, this paper proposes a geometric nonlinear isogeometric topology optimization method based on the extended finite element method. Firstly, to accurately capture the nonlinear behavior of boundary elements, Gaussian integration points and weights of boundary elements are regenerated during the optimization process using the extended finite element method. Secondly, an isogeometric topology optimization model is developed based on the principle of minimum complementary work and the bi-directional evolutionary structural optimization method. A geometric nonlinear isogeometric analysis in the total Lagrangian formulation is proposed, and the incremental form of the Newton-Raphson method is employed to solve the nonlinear equilibrium equations, improving the accuracy and stability of structural response analysis. Lastly, the effectiveness and necessity of isogeometric topology optimization considering geometric nonlinearity are demonstrated through 2D and 3D examples. The results show that structures experience significant deformation under full load conditions, leading to the failure of linearly optimized structures, while geometrically nonlinear optimized structures exhibit superior performance.