We present a worst-case approach to topology optimization (TO) for maximum stiffness under boundary displacement parametrized by a matrix-valued scaling function times an uncertain vector giving its direction. The objective function in the TO problem is the minimum of the potential energy maximized over the set of boundary displacements, which in the absence of prescribed loads means maximizing the reaction loads arising from enforcing the boundary displacement. It is shown that the TO problem can be cast as the minimization of the maximum eigenvalue of a matrix depending on solutions to a small number of (linear elastic) state problems. Numerical solution of this potentially non-smooth problem using algorithms for smooth optimization, a non-linear semi-definite programming reformulation, and a non-smooth bundle method is discussed and tested.

我们提出了一种拓扑优化 (TO) 的最坏情况方法，用于在边界位移下通过矩阵值缩放函数乘以给出其方向的不确定向量参数化的最大刚度。TO 问题中的目标函数是在一组边界位移上最大化的势能的最小值，在没有规定载荷的情况下，这意味着最大化由强制边界位移引起的反应载荷。结果表明，根据少数（线性弹性）状态问题的解决方案，可以将 TO 问题转化为矩阵最大特征值的最小化。讨论并测试了使用平滑优化算法、非线性半定规划重构和非平滑捆绑方法对这个潜在的非平滑问题的数值解。