Robust structures in compliance minimization

确定弹性材料的哪种几何形状在消耗最少材料量的情况下最能支撑载荷是一个经典的工程问题。通常，最优的几何结构会包含无限精细的微观结构和无限多的、无限精细的孔。在数学上，这个问题要么是放松的，要么是正则化的，以承认最优解（能量最小化）的存在。在这个项目中，我们考虑了一种弱形式的正则化，它允许最小值的存在，但仍然表现出非常精细（但有限）的结构。我们的目的是研究这些复杂结构在确定性和随机意义上的稳定性。特别是，我们的目标是处理材料参数的不确定性以及加载场景，以了解这些结构的鲁棒性。

It is a classic engineering problem to identify which geometry of an elastic material best supports a load while consuming a minimum material amount. Typically, the optimal geometry would involve an infinitely fine microstructure with infinitely many, infinitely fine holes. Mathematically, the problem is either relaxed or regularized to admit the existence of the optimal solutions (energy minimizers). In this project we consider a weak form of regularization, which allows for existence of minimizers which nevertheless exhibit very fine (but finite) structures. Our aim is to study stability of these complicated structures, both in the deterministic as well as stochastic sense. In particular, we aim to deal with the uncertainty in the material parameters as well as loading scenarios, to understand robustness of these structures.