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：本文提出一种计算框架，通过引入非线性屈曲约束，实现大变形条件下稳定结构的优化设计。针对低密度单元诱发的伪屈曲模态，提出一种新型抑制策略：通过构建伪质量矩阵，为仅由低密度单元包围的自由度赋予微小伪质量，而实体区域则退化为单位矩阵。开发了一种可处理简单特征值与多重特征值的新型优化程序，其中推导并应用了简单特征值的灵敏度和多重特征值的方向导数，将其整合至梯度优化算法——移动渐近线法中。同时，在大变形非线性分析中引入自适应线性能量插值方法，以处理低密度单元的畸变问题。数值结果表明，无论系统具有低对称性还是高对称性，非线性稳定性约束均能确保大变形下目标载荷的结构稳定性。对B样条拟合设计进行后分析表明，通过选择不同稳定性约束值，可精确控制优化结构的安全裕度（即目标载荷与首阶临界载荷之间的间隙）。研究还揭示了模态转换与多重分岔等有趣的力学行为。