Optimal Control of Static Contact in Finite Strain Elasticity

有限应变弹性静力接触问题是一类重要的非线性、非光滑力学问题。如果考虑的材料是软的，例如橡胶或生物软组织，则会产生有限应变。他的项目的目的是开发和分析这些问题的最优控制算法。我们构造和分析了一种正则化和同伦方法，其中放宽了接触问题的非穿透条件和弹性的局部可注性条件。研究了正则化问题的性质和同伦的收敛问题。由此得到的正则化最优控制问题仍然是非线性和非凸的，并将用面向函数空间的复合步长方法来求解。这种方法将利用有限应变弹性问题的结构，特别是弹性力学的变分结构和变形的群结构。为了最终逼近原问题的解，我们将开发和分析一种为这类问题量身定做的仿射不变路径跟踪方法。

Static contact problems in the regime of finite strain elasticity are an important class of mechanical problems with nonlinear, non-smooth behaviour. Finite strains occur if the considered materials are soft, for example for rubber or biological soft tissue. Aim of his project is the development and analysis of algorithms for the optimal control of these problems.We construct and analyse a regularization and homotopy approach, where the non-penetration condition of the contact problem and the local injectivity condition of elasticity are relaxed. Properties of the regularized problems and convergence of the homotopy are studied. The resulting regularized optimal control problems will still be nonlinear and non-convex and will be solved by a function space oriented composite step method. This method will exploit problem structure of finite strain elasticity, in particular the variational structure of elasticity and the group structure of deformations. To finally approximate solutions of the original problem, we will develop and analyse an affine invariant path-following method that is tailored for this class of problems.