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

非线性材料非等温热化学力学中的变分公式：理论模型开发和并行求解器的协同设计 - 第 2 阶段

• 批准号: 441509557
• 负责人: Professor Björn Kiefer, Ph.D.
• 金额: --
• 依托单位: Institut für Numerische Mathematik und Optimierung
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/441509557?language=en
• 关键词: Variational; Formulations; Non; Isothermal; Thermo

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.

在这个项目中，模拟了复杂的材料行为，其中相互作用的热化学机械过程起着主导作用。这样的问题出现在各种工程领域。我们的方法集中在共同设计的建模，变分原理的基础上，所产生的离散化非线性和线性化问题的数值求解方法，采用最先进的并行迭代求解器从区域分解。协同设计在多领域的情况下是特别重要的，因为许多更可能的组合变分设置，有限元公式和求解器类型和配置是可以想象的。此外，对于复杂的热化学机械问题，解决方案可能只会成为可获得的，如果有效的和并行的求解器，仔细适应特定的变分模型配方，是可用的。在第一阶段，一个共同的软件环境被定义和实现作为一个跨部门的活动之间的群体。它由开源有限元库deal.II（包含用户定义的材料模型和单元公式）、p4 est（用于分解）和快速鲁棒重叠施瓦茨（FROSch）求解器框架组成。FROSch一直在积极开发这个项目。它是开源Trilinos软件库中ShyLU-Frosch-package的一部分。基于这种实现，它是可能的，调查的数值和并行可扩展性的单片FROSch求解器的问题，力学和扩散之间的强，非线性耦合。此外，三级FROSch求解器进行了研究，这将需要在第二阶段，以实现并行可扩展性的挑战性的问题正在调查中。对于第二个供资阶段，化学力学的最小化原则为基础的配方的解决方案战略应伴随着鞍点配方，并在一个闭环与适应FROSch求解器开发。此外，一个主要的重点放在相场（扩散界面）的建模方面的问题。特别是，我们感兴趣的是模拟耗散，非平衡过程，其中多种化学成分可能会发生反应，并跨越多个相的边界扩散。这里，控制方程表现为Allen-Cahn，Cahn-Hilliard和Fickian扩散方程的耦合集。再次，我们将研究的优点和缺点，不同的配方的物理可解释性的边界条件，所需的anterior功能，以及鲁棒性和可扩展性的迭代求解器通过代表性的数值例子。理论方面的其他方面包括制定变分一致的均匀化计划及其扩展到非等温化学机械过程。在求解器方面，我们将开发单片三级方法以及单片非线性方法量身定制的问题。