Robust Solvers for Coupled Problems with Applications to Electromagnetism and Poromechanics

用于电磁学和孔隙力学应用耦合问题的鲁棒求解器

• 批准号: 1620063
• 负责人: Xiaozhe Hu
• 金额: $ 14.42万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-15 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620063&HistoricalAwards=false
• 关键词: Robust; Solvers; Coupled; Problems; Applications

The goal of this project is to develop, implement, and study robust and efficient computational (Partial Differential Equation) solvers for large-scale systems of equations that describe coupled physical problems. In particular, we aim to investigate applications in the computation of electromagnetic phenomena around obstacles, as well as poromechanic applications, such as the study of groundwater flow in porous media. Designing these solvers represents an important class of challenging problems in computational mathematics, because those coupled systems usually describe complex multiphysics phenomena across different time and spatial scales. Currently, most efficient and robust solvers are developed for single-physics problems, whereas each tool we develop will strongly consider the importance of the inherent coupling. The research will provide new computational paradigms for electromagnetics and poroelasticity, both of which have crucial applications in physics and engineering, such as fusion energy applications, shale gas recovery, carbon dioxide consolidation, and cardiac muscle behavior, to name a few. Finally, the project supports one graduate student. Through training and collaboration with investigators and other experts in the field, they will become involved in the broader research communities of scientific computing and engineering. Each of the applications described above corresponds to a discretized coupled system of partial differential equations (PDEs). Due to the complexity of the multi-physics and multi-scale phenomena described by such models, an essential component is efficient and robust nonlinear and linear solvers, due to the fact that the computational time needed to simulate complex physics is in many cases dominated by solving the large-scale linear systems of equations representing the discretized PDEs. Therefore, this research focuses on developing, analyzing, and implementing efficient iterative methods and preconditioners for coupled PDE systems. More precisely, this is achieved by two possible approaches. When structure-preserving discretizations are applied, properties of the PDE models that are carried over to the discrete model are used to derive an exact block factorization and we will design novel preconditioners without approximating the Schur complements. For general numerical schemes, monolithic multigrid methods will be developed by generalizing the multigrid theoretical framework for indefinite coupled systems. Finally, by applying these new iterative solvers and studying their interplay with accurate discretization schemes, the investigators will create robust and efficient numerical simulations for systems such as Maxwell's equations and those describing poromechanics. More generally, the new solvers developed here will provide insight on how to use analytic tools in the design of algebraic solvers and novel theoretical foundations for the design of robust solvers for general coupled PDEs will be considered.

该项目的目标是开发，实施和研究描述耦合物理问题的大规模方程系统的鲁棒和高效的计算（偏微分方程）求解器。 特别是，我们的目标是研究应用在计算周围的障碍物的电磁现象，以及孔隙力学的应用，如研究地下水在多孔介质中的流动。设计这些求解器代表了计算数学中一类重要的挑战性问题，因为这些耦合系统通常描述跨不同时间和空间尺度的复杂多物理场现象。 目前，大多数高效和强大的求解器是针对单物理问题开发的，而我们开发的每个工具都将强烈考虑固有耦合的重要性。 该研究将为电磁学和孔隙弹性提供新的计算范例，这两者在物理学和工程学中都有重要的应用，例如聚变能应用，页岩气回收，二氧化碳固结和心肌行为，仅举几例。最后，该项目支持一名研究生。通过培训和与该领域的调查人员和其他专家合作，他们将参与更广泛的科学计算和工程研究社区。上述每个应用程序对应于一个离散化的耦合偏微分方程（PDE）系统。 由于这种模型所描述的多物理场和多尺度现象的复杂性，一个重要的组成部分是高效和鲁棒的非线性和线性求解器，这是由于模拟复杂物理场所需的计算时间在许多情况下由求解表示离散化PDE的大规模线性方程组所支配。 因此，本研究的重点是发展，分析和实施有效的迭代方法和耦合PDE系统的预条件。更确切地说，这是通过两种可能的方法来实现的。当结构保持离散化应用，性质的PDE模型进行到离散模型被用来推导出一个确切的块分解，我们将设计新的预条件，而不近似的舒尔补。对于一般的数值格式，整体多重网格方法将通过推广不确定耦合系统的多重网格理论框架来发展。最后，通过应用这些新的迭代求解器并研究它们与精确离散化方案的相互作用，研究人员将为麦克斯韦方程组和描述孔隙力学的方程组等系统创建鲁棒且高效的数值模拟。更一般地说，这里开发的新的求解器将提供洞察力如何使用分析工具的代数求解器的设计和新的理论基础，一般耦合偏微分方程的鲁棒求解器的设计将被考虑。

项目成果

An Adaptive Multigrid Method Based on Path Cover

• DOI: 10.1137/18m1194493
• 发表时间: 2018-06
• 期刊: SIAM J. Sci. Comput.
• 作者: Xiaozhe Hu;Junyuan Lin;L. Zikatanov