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在第一阶段开发的方法，这些方法是基于大多数静态（拟）变分不等式的对偶公式到时间相关的方法。再一次，与标准原始离散化相比，将双变量作为单独的场添加到应力和相关表面牵引力的近似中，可以获得高度精确的近似。这些方法的守恒性质对时变问题特别有利。将开发基于应力的自适应方法，该方法采用位移重建进行后验误差估计。与适当构造的多网格求解器相结合，这将导致对此类（拟）变分不等式的非常有效的整体处理。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。对于准静态情况，最初采用时间步进方法。这样做的好处是，在SPP的第一阶段已经开发的技术可以用于空间适应性。这包括后验误差估计以及应力空间的单调多电平方法。然而，这个项目的第二个阶段的最终目标确实包括对时间相关（拟）变分不等式的自适应时空离散化的发展。为此，对于标准抛物线问题已经很好理解的技术，将适当地修改用于随时间变化的摩擦接触问题。一阶系统最小二乘泛函被用作时空误差和相关自适应的后验估计的起点。对于动态摩擦接触，需要解决更具挑战性的稳定性问题，这些问题可以使用该主题早期工作的专业知识来解决。最后，将针对出现的时空有限元公式开发时空多重网格方法，从而形成高效的整体求解策略。

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.