Analysis and Numerical Techniques for Optimal Control Problems Involving Variational Inequalities Arising in Elastoplasticity

在外加载荷的影响下，固体会偏离其静止形状。在施加的载荷或应力足够小的情况下，许多固体表现出线性弹性和可逆行为。然而，如果外加载荷引起的应力超过某一阈值(屈服应力)，材料的行为将从弹性区域切换到所谓的塑性区域。在这种状态下，整个加载过程不再是可逆的，即使在取消加载后仍保持永久变形。在数学上，这导致了一种涉及变分不等式的描述。例如，希望将塑性变形作为金属零件的工业成形技术。寻找影响期望最终变形的适当的时变载荷的任务导致了弹塑性系统的最优控制问题。这个项目的主要目标是研究这些优化问题，量化离散化带来的误差，并开发快速算法来解决这些问题。

Solid bodies depart from their rest shape under the influence of applied loads. In case the applied loads or stresses are sufficiently small, many solids exhibit a linearly elastic and reversible behavior. If, however, the stress induced by the applied loads exceeds a certain threshold (the yield stress), the material behavior switches from the elastic to the so-called plastic regime. In this state, the overall loading process is no longer reversible and permanent deformations remain even after the loads are withdrawn. Mathematically, this leads to a description involving variational inequalities.Plastic deformation is desired for instance as an industrial shaping technique of metal workpieces. The task of finding appropriate time-dependent loads which effect a desired final deformation leads to optimal control problems for elastoplasticity systems. The main goals of this project are to investigate these optimization problems, to quantify the error due to discretization, and to develop fast algorithms for their solution.