This paper proposes a computational framework for the design optimization of stable structures under large deformations by incorporating nonlinear buckling constraints. A novel strategy for suppressing spurious buckling modes related to low-density elements is proposed. The strategy depends on constructing a pseudo-mass matrix that assigns small pseudo masses for DOFs surrounded by only low-density elements and degenerates to an identity matrix for the solid region. A novel optimization procedure is developed that can handle both simple and multiple eigenvalues wherein consistent sensitivities of simple eigenvalues and directional derivatives of multiple eigenvalues are derived and utilized in a gradient-based optimization algorithm - the method of moving asymptotes. An adaptive linear energy interpolation method is also incorporated in nonlinear analyses to handle the low-density elements distortion under large deformations. The numerical results demonstrate that, for systems with either low or high symmetries, the nonlinear stability constraints can ensure structural stability at the target load under large deformations. Post-analysis on the B-spline fitted designs shows that the safety margin, i.e., the gap between the target load and the 1st critical load, of the optimized structures can be well controlled by selecting different stability constraint values. Interesting structural behaviors such as mode switching and multiple bifurcations are also demonstrated.

翻译：本文提出了一种用于设计优化可模拟大变形下稳定结构的计算框架，并结合非线性屈曲约束。该研究提出了抑制与低密度单元相关的虚假屈曲模态的新策略。该策略依赖于构建一个伪质量矩阵，为仅被低密度单元包围的DOFs分配小的伪质量，并且对于固体区域退化为一个单位矩阵。还开发了一种新的优化过程，可以处理简单和多个特征值，其中导出了简单特征值的一致敏感度和多个特征值的方向导数，并在渐进移动算子法中利用。非线性分析中还加入了自适应线性能量插值方法，以处理大变形下低密度元素的扭曲。数值结果表明，对于具有低或高对称性的系统，非线性稳定性约束可以确保在大变形下以目标载荷时结构的稳定性。对B样条拟合设计的后分析表明，通过选择不同的稳定性约束值，优化结构的安全裕度，即目标载荷和第一临界载荷之间的差距，可以得到良好控制。还展示了有趣的结构行为，如模态切换和多次分支。