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.

本文提出了一种计算框架，用于通过结合非线性屈曲约束来优化大变形下稳定结构的设计。提出了一种抑制与低密度单元相关的寄生屈曲模式的新策略。该策略取决于构建一个伪质量矩阵，该矩阵为仅被低密度元素包围的 DOF 分配小的伪质量，并退化为实体区域的单位矩阵。开发了一种新的优化程序，可以处理简单和多个特征值，其中推导出简单特征值的一致灵敏度和多个特征值的方向导数，并将其用于基于梯度的优化算法 - 移动渐近线的方法。自适应线性能量插值方法也被纳入非线性分析中，以处理大变形下的低密度单元变形。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大变形下目标载荷下的结构稳定性。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，可以很好地控制优化结构的安全裕度，即目标载荷与一阶临界载荷之间的差距。还展示了有趣的结构行为，例如模式切换和多个分叉。对于具有低对称性或高对称性的系统，非线性稳定性约束可以确保大变形下目标载荷下的结构稳定性。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，可以很好地控制优化结构的安全裕度，即目标载荷与一阶临界载荷之间的差距。还展示了有趣的结构行为，例如模式切换和多个分叉。对于具有低对称性或高对称性的系统，非线性稳定性约束可以确保大变形下目标载荷下的结构稳定性。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，可以很好地控制优化结构的安全裕度，即目标载荷与一阶临界载荷之间的差距。还展示了有趣的结构行为，例如模式切换和多个分叉。通过选择不同的稳定性约束值可以很好地控制优化结构的稳定性。还展示了有趣的结构行为，例如模式切换和多个分叉。通过选择不同的稳定性约束值可以很好地控制优化结构的稳定性。还展示了有趣的结构行为，例如模式切换和多个分叉。