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面向等几何分析的超弹性材料模型形状优化

Shape Optimization of Hyperelastic Materials Model for Isogeometric Analysis

  • 摘要: 等几何分析作为一种收敛速度快、 数值解高阶连续性好的数值方法, 广受工程设计领域的关注. 基于其优良的性质, 提出一种针对超弹性材料模型的等几何形状优化方法, 具体研究了面向等几何分析框架的超弹性材料计算以及相适应的形状优化方法. 首先, 在等几何框架下推导了离散矩阵装配流程与非线性系统的求解流程; 其次定义了超弹性材料下以柔度最小化和位移最小化为 2 个目标的数学模型, 继而通过对偶方法详细推导了解析的灵敏度,有效提高优化精度; 最终结合等几何分析与移动渐近线算法构建了高效的优化框架. 对悬臂梁、 带孔平板和开口扳手三组基准算例的形状优化实验表明, 所提方法的有效性和可靠性有一定保证, 实验中, 在几何约束下可在指定点处最多降低 90%位移和 7%的柔度, 且与预期形状相符.

     

    Abstract: Isogeometric analysis, as a numerical method with fast convergence and high-order numerical solutions, has attracted significant attention in the field of engineering design. Based on its excellent properties, a shape optimization method for hyperelastic material models is proposed, focusing on the computation of hyperelastic materials within the isogeometric analysis framework, as well as the corresponding shape optimization method. Firstly, the discrete matrix assembly procedure and the solution process for nonlinear systems are derived within the isogeometric analysis framework. Secondly, mathematical models are defined for hyperelastic materials with the objectives of minimizing compliance and displacement. Furthermore, analytical sensitivities are derived by the dual method to effectively improve optimization accuracy. Finally, an efficient optimization framework is constructed by combining IGA with the method of moving asymptotes. Shape optimization experiments on three benchmark cases of the cantilever beam, plate with a hole, and wrench demonstrate the effectiveness and reliability of the proposed method. In the experiments, the displacement and compliance at specified points can be reduced by 90% and 7% with geometric constraints respectively, which are the same as the expected shapes.

     

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