Parallel multilevel solvers for coupled interface problems

虽然在过去的几十年里，并行有限元模拟领域取得了巨大的进展，但对复杂的、有约束的问题的并行求解，如力学和流体力学，仍然是一个具有挑战性的任务。通常，对于同质问题和结构化网格，可以相对直接地实现良好的可扩展性。使用从Krylov子空间方法到区域分解方法或多层方法的各种求解方法，也成功地进行了涉及非结构网格和自适应网格的并行模拟。然而，约束或异质问题的并行处理仍然远不是微不足道的。这是由于要解决的离散系统的更复杂的数学结构以及用于组装、求解和并行数据交换的更先进的数据结构。如果必须解决与时间相关的界面，例如液滴撞击壁面(液-气流)或裂纹扩展的情况，则会产生额外的、到目前为止几乎不能解决的困难。这些界面不仅影响离散化，而且影响迭代求解方法的稳健性。此外，接口的特殊处理对几何对象在大型机器上的并行分布提出了挑战，有效的解决方法及其可扩展和灵活的实现不仅对底层方法提出了很高的要求，而且对所使用的软件也提出了很高的要求。在这个项目中，我们将开发和实现鞍点问题的并行多级求解器，这些求解器能够以健壮的方式处理与时间相关的接口。为此，我们将采用一种新的方法来构建多层层次结构，该方法基于非标准转移算子和依赖于解的粗网格空间。作为这项工作的基础，该项目的一个中心目标是开发和实施一个独立的高级库来管理分布式几何对象。为了促进这个库的广泛适用性，我们将在两个不同的软件和应用程序环境中使用它，即RWTH的Drops和USI的ObsLib++。尽管两相流的RWTH和计算力学和裂纹扩展的USI的应用领域似乎不同，但这两种应用都导致了大的鞍点问题，并且由于时间相关界面的数值处理而变得复杂。对于这个项目，必须具备处理依赖时间的界面约束、鞍点问题的多层方法和求解器、非标准传输运算符以及用于高效并行数据管理的HPC软件的丰富专业知识。RWTH和USI的申请者共同以互补的方式提供这种专业知识，这推动了这个研究小组的形成和瑞士与德国的合作。

Although during the last decades tremendous progress has been achieved in the area of parallel finite element simulations, the parallel solution of complex and constrained problems in, e. g., mechanics and fluid mechanics, still remains a challenging task. Usually, good scalability can be achieved relatively straightforwardly for homogeneous problems and structured meshes. Parallel simulations involving unstructured and adaptive meshes have also been successfully carried out, using a wide variety of solution methods, ranging from Krylov subspace methods to domain decomposition approaches or multilevel methods. The parallel treatment of constrained or heterogeneous problems, however, is still far from trivial.This is caused by the more complex mathematical structure of the discrete systems to be solved and by the more advanced data structures employed for assembling, solving, and parallel data exchange.An additional and up to now only scarcely addressed difficulty arises if time dependent interfaces have to be resolved, as is the case in, e.g., droplets impacting on a wall (liquid-gas flow) or crack propagation. These interfaces do not only influence the discretization, but also the robustness of iterative solution methods. Moreover, the special treatment of the interface poses a challenge for the parallel distribution of geometric objects across large scale machines.Efficient solution methods for these problems and their scalable and flexible implementations put a high demand not only on underlying methodology but also on the software used. In this project we will develop and implement parallel multilevel solvers for saddle point problems which are able to deal with time dependent interfaces in a robust manner. To this end, we will employ a new approach for the construction of multilevel hierarchies, which is based on non-standard transfer operators and solution dependent coarse grid spaces. As a basis for this work, one central goal of this project is the development and implementation of a stand-alone high level library for the management of distributed geometric objects. In order to foster the broad applicability of this library, we will use it within two different software and application environments, namely DROPS at RWTH and ObsLib++ at USI. Despite the seemingly different application fields, RWTH with two-phase flow and USI with computational mechanics and crack propagation, both applications lead to large saddle point problems, additionally complicated by the numerical treatment of time dependent interfaces. For this project strong expertise in the handling of time dependent interfacial constraints, multilevel methods and solvers for saddle point problems, non-standard transfer operators, and HPC software for efficient parallel data management is mandatory. The applicants at RWTH and USI together provide in a complementary manner this expertise, which motivated the formation of this research team and the Swiss-German cooperation.