Stress-Based Methods for Variational Inequalities in Solid Mechanics: Finite Element Discretization and Solution by Hierarchical Optimization

这个项目的目标是扩展的方法，在第一阶段的基础上的双重制定主要是静态（准）变分不等式的时间依赖的。再次，添加对偶变量作为一个单独的字段导致高度准确的近似的应力和相关的表面牵引力相比，标准的原始离散化。这些方法的守恒性质是特别有利的时间依赖problems.Adaptive应力为基础的方法将开发采用位移重建的后验误差估计。结合适当构造的多重网格求解器，这将导致一个非常有效的整体治疗这种（准）变分不等式。这些方法将首先在准静态问题的背景下进行研究，然后扩展到动态摩擦接触。对于准静态的情况下，一个时间步进的方法最初使用。这具有的优点是，在SPP的第一阶段期间已经开发的技术可以用于空间自适应性。这包括后验误差估计以及单调多级方法的应力spaces.The最终目标的第二个项目阶段，然而，包括时间相关的（准）变分不等式的自适应时空离散的发展。为此，已经很好地理解为标准的抛物线问题的技术将适当地修改为时间相关的摩擦接触问题。一阶系统最小二乘泛函被用作空时误差和相关自适应性的后验估计的起点。对于动态摩擦接触，需要解决更具挑战性的稳定性问题，这可以通过使用该主题早期工作的专业知识来完成。最后，时空多重网格方法将开发所产生的时空有限元制剂导致一个高效的整体解决方案的战略。

The goal of this project is the extension of the methods developed during the first phase based on the dual formulation of mostly static (quasi-)variational inequalities to time-dependent ones. Again, adding the dual variable as a separate field leads to highly accurate approximations for the stresses and for the associated surface traction forces compared to standard primal discretizations. The conservation properties of these methods are particularly advantageous for time-dependent problems.Adaptive stress-based methods will be developed which employ displacement reconstructions in the a posteriori error estimation. In combination with suitably constructed multigrid solvers, this will lead to a very efficient overall treatment of such (quasi-)variational inequalities. These methods will first be studied in the context of quasi-static problems and then extended to dynamic frictional contact. For the quasi-static case, a time-stepping approach is initially used. This has the advantage that the techniques already developed during the first phase of the SPP can be used for the spatial adaptivity. This includes the a posteriori error estimators as well as the monotone multilevel method for the stress spaces.The ultimate goal of this second project phase does, however, consist of the development of adaptive space-time discretizations for time-dependent (quasi-)variational inequalities. To this end, techniques which are already well understood for standard parabolic problems will be appropriately modified for the time-dependent frictional contact problems. A first-order system least squares functional is used as a starting point for a posteriori estimation of the space-time error and associated adaptivity. For dynamical frictional contact, more challenging stability issues need to be addressed which can be done using expertise from earlier work on that topic. Finally, space-time multigrid methods will be developed for the arising space-time finite element formulations leading to a highly efficient overall solution strategy.