Variational Formulations in the Non-Isothermal Thermo-Chemo-Mechanics of Nonlinear Materials: Co-Design of Theoretical Model Development and Parallel Solvers - Phase 2

在这个项目中，复杂材料的行为被模拟，其中相互作用的热-化学-机械过程起主导作用。这样的问题出现在各种工程领域。我们的方法集中在基于变分原理的建模和离散化的非线性和线性化问题的数值求解方法的共同设计上，这些方法使用了区域分解中最先进的并行迭代求解器。联合设计在多领域方案中特别重要，因为可以想象变分设置、有限元公式以及求解器类型和配置的更多可能的组合。此外，对于复杂的热-化学-机械问题，只有在有有效和并行的解算器可用时，才可能获得解，这些解算器仔细地适应特定的变分模型公式。在第一阶段，定义了一个通用的软件环境，并将其作为小组之间的跨部门活动来实施。它由开源有限元库Deal.II(具有用户定义的材料模型和单元公式)、p4est(用于分解)和快速而稳健的重叠Schwarz(Frosch)求解器框架组成。Frosch一直在积极开发这一项目。它是开源Trilinos软件库中ShyLU-Frosch包的一部分。基于这种实现，可以研究单片Frosch求解器对力学和扩散之间具有强非线性耦合的问题的数值和并行可伸缩性。此外，还研究了三级Frosch求解器，在第二阶段将需要它们来实现对所研究的具有挑战性的问题的并行可伸缩性。对于第二个供资阶段，最大限度地减少基于原则的化学力学配方的解决战略应伴随着鞍点配方，并在采用适应的Frosch解算器的闭环系统中开发。此外，在建模方面，主要侧重于相场(扩散界面)问题。特别是，我们对模拟耗散、非平衡过程感兴趣，在这种过程中，多个化学成分可能会反应并扩散到多个相的边界。在这里，控制方程表现为Allen-Cahn、Cahn-Hilliard和Fickian扩散方程的耦合组。同样，我们将通过典型的数值例子来研究不同的公式在边界条件的物理可解释性、所需的ansatz函数以及迭代求解器的稳健性和可伸缩性方面的优缺点。理论方面的其他方面包括变分一致的均匀化方案的形成及其对非等温化学-机械过程的扩展。在求解器方面，我们将开发针对问题量身定做的整体式三层方法以及整体式非线性方法。

In this project, complex material behavior is modeled, in which interacting thermo-chemo-mechanical processes play a dominant role. Such problems arise in a variety of engineering fields. Our approach centers on the co-design of modeling, based on variational principles, and numerical solution methods for the arising discretized nonlinear and linearized problems, that employ state-of-the-art parallel iterative solvers from domain decomposition. The co-design is of particular importance in multi-field scenarios, since many more possible combinations of variational settings, finite element formulations and solver types and configurations are conceivable. Moreover, for complex thermo-chemo-mechanical problems, solutions may only become obtainable, if efficient and parallel solvers, carefully adapted for the particular variational model formulation, are available. During the first phase, a common software environment was defined and implemented as a cross-sectional activity between the groups. It is composed of the open source finite element library deal.II (with user-defined material models and element formulations), p4est (for the decomposition), and the Fast and Robust Overlapping Schwarz (FROSch) solver framework. FROSch has been under active development in this project. It is part of the ShyLU-FROSch-package in the open source Trilinos software library. Based on this implementation it was possible to investigate the numerical and parallel scalability of monolithic FROSch solvers for problems with strong, nonlinear coupling between mechanics and diffusion. Moreover, three-level FROSch solvers have been investigated, which will be needed in the second phase to achieve parallel scalability for the challenging problems under investigation. For the second funding phase, solution strategies for minimization principle-based formulations of chemo-mechanics shall be accompanied by saddle point formulations and developed in a closed loop with adapted FROSch solvers. Moreover, a major emphasis is placed on phase-field (diffuse interface) problems on the modeling side. In particular, we are interested in simulating dissipative, non-equilibrium processes, in which multiple chemical components may react and diffuse across the boundaries of multiple phases. Here, the governing equations appear as coupled sets of Allen-Cahn, Cahn-Hilliard and Fickian diffusion equations. Again, we will study the advantages and disadvantages of different formulations regarding physical interpretability of boundary conditions, required ansatz functions, as well as robustness and scalability of the iterative solvers by means of representative numerical examples. Other aspects on the theoretical side include the formulation of variationally-consistent homogenization schemes and their extension to non-isothermal chemo-mechanical processes. On the solver side we will develop monolithic three-level approaches as well as monolithic nonlinear approaches tailored to the problems.