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

基本信息

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

项目摘要

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.

该项目的目的是为描述耦合物理问题的大规模方程系统开发，实施和研究强大，有效的计算（部分微分方程）求解器。特别是，我们旨在调查在障碍物周围电磁现象的计算中的应用，以及诸如多孔培养基中地下水流量的研究。设计这些求解器代表了计算数学中一系列重要的挑战性问题，因为这些耦合系统通常会在不同的时间和空间尺度上描述复杂的多物理现象。当前，为单物理问题开发了最有效，最强大的求解器，而我们开发的每个工具都会强烈考虑固有耦合的重要性。这项研究将为电磁学和毛弹性提供新的计算范式，这两种都在物理和工程中都有至关重要的应用，例如融合能源应用，页岩气回收，二氧化碳巩固和心肌行为，仅举几个。最后，该项目支持一名研究生。通过与该领域的研究人员和其他专家的培训和合作，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偏微分方程（PDE）的离散耦合系统。由于这种模型描述的多物理学和多尺度现象的复杂性，因此，基本组件是有效且可靠的非线性和线性求解器，因为在许多情况下，模拟复杂物理学所需的计算时间是通过求解代表定位PDES的大型线性系统来支配的。因此，这项研究的重点是开发，分析和实施耦合PDE系统的有效迭代方法和预处理。更确切地说，这是通过两种可能的方法来实现的。当应用结构保存离散化时，将使用转移到离散模型的PDE模型的属性将用于得出确切的块分解，我们将设计新颖的预调节器而不近似Schur补充。对于一般数值方案，将通过概括无限耦合系统的多方理论框架来开发整体式多族方法。最后，通过应用这些新的迭代求解器并通过准确的离散化方案研究它们的相互作用，研究人员将为麦克斯韦方程和描述门学的系统创建强大而有效的数值模拟。更普遍地，此处开发的新求解器将提供有关如何在代数求解器设计中使用分析工具的见解，以及为一般耦合PDE的强大求解器设计的新型理论基础。