Hybrid Finite Element Methods for Geometric Partial Differential Equations

几何偏微分方程的混合有限元方法

基本信息

批准号：
1913272
负责人：
Ari Stern
金额：
$ 21.26万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1913272&HistoricalAwards=false
关键词：
Hybrid Finite Element Methods Geometric

项目摘要

Finite element methods are among the most important, powerful tools in scientific computing. Their applications include several areas of Federal strategic interest: materials and manufacturing, biomedical engineering and biotechnology, structural engineering and civil infrastructure, environmental engineering, and more. They are widely used by scientists and engineers in academia, industry, and national laboratories to simulate large, complex physical systems. These physical systems often obey "conservation laws," which state that some quantity -- like mass, or energy, or electric charge -- can move around in space, but cannot appear or disappear spontaneously. It is desirable that simulations of these systems also obey these conservation laws, because they are so fundamental; otherwise, the simulated results may not be physically meaningful or trustworthy. However, this is not always the case with current methods. In this project, the PI will develop and analyze finite element methods that obey these conservation laws and preserve related physical properties. The success of this project would lead to new computational methods and improved understanding of current methods for a wide variety of scientific applications. Because the specific applications addressed by the proposed research are of high scientific value, this could have important ramifications for computational physics and engineering.The PI proposes to investigate structure-preserving hybrid finite element methods for partial differential equations (PDEs) containing local symmetries, invariants, and conservation laws. In applications, these locally-invariant geometric structures often have important physical meaning (e.g., conservation of charge in electromagnetics), so it is desirable to devise conservative numerical methods that preserve these structures exactly rather than approximately. Hybrid methods provide a natural framework for this, since one may examine local invariants, element-by-element, in terms of numerical traces and fluxes on their boundaries. The proposed research consists of two main components. (1) Hamiltonian PDEs, which are ubiquitous in physical applications, satisfy the multisymplectic conservation law, which is closely tied to physically-important reciprocity phenomena, traveling waves, dispersion relations, and bifurcations. The PI will extend his recent joint work on multisymplectic hybridizable discontinuous Galerkin (HDG) methods for spatial discretization to time-evolution problems, using spatial HDG semidiscretization and spacetime HDG methods. (2) While the time evolution of Maxwell's equations automatically preserves a divergence constraint associated with charge conservation, this is generally not true for finite element discretizations. Preliminary results show that this can be resolved using a class of hybrid methods, where charge conservation holds in the sense of the numerical electric flux. The PI proposes to extend this by analyzing nonconforming hybrid methods for the Maxwell eigenvalue problem, as well as hybrid methods for Yang-Mills theory using finite element exterior calculus.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

有限元方法是科学计算中最重要，最有力的工具之一。他们的应用包括联邦战略兴趣的几个领域：材料和制造业，生物医学工程和生物技术，结构工程和民用基础设施，环境工程等。它们被学术界，工业和国家实验室的科学家和工程师广泛使用，以模拟大型，复杂的物理系统。这些物理系统经常遵守“保护法”，该法律表明某种数量（例如质量，能量或电荷）可以在太空中四处移动，但不能自发出现或消失。希望这些系统的模拟也遵守这些保护法，因为它们是如此的基本。否则，模拟的结果可能不会在身体上有意义或值得信赖。但是，当前方法并非总是如此。在该项目中，PI将开发和分析遵守这些保护法律并保留相关物理特性的有限元方法。该项目的成功将导致新的计算方法，并提高人们对各种科学应用的当前方法的理解。因为拟议的研究涉及的特定应用具有很高的科学价值，所以这可能对计算物理和工程有重要影响。和保护法。在应用中，这些局部不变的几何结构通常具有重要的物理含义（例如电磁中电荷的保护），因此希望设计保守的数值方法，以确切地保留这些结构而不是大致保存这些结构。混合方法为此提供了一个自然框架，因为人们可以根据数值痕迹和边界上的数值痕迹检查局部不变性。拟议的研究由两个主要组成部分组成。（1）在物理应用中无处不在的汉密尔顿PDE满足了多透明保护定律，该法与物理上重要的互惠现象紧密相关，流行波，分散关系和分支。 PI将使用空间HDG半差异和时空HDG方法将他最近的联合工作扩展到多核混合性不连续的Galerkin（HDG）方法，以进行空间离散化。（2）虽然麦克斯韦方程的时间演变自动保留与电荷保护相关的差异约束，但对于有限元离散化，这通常不正确。初步结果表明，可以使用一类混合方法来解决这一问题，其中电荷保护在数值电通量的意义上。 PI建议通过分析麦克斯韦特征值问题的不合格混合方法，以及使用有限元外观计算的Yang-Mills理论的混合方法，这反映了NSF的法定任务，并被认为是通过使用评估来通过评估的支持，基金会的智力优点和更广泛的影响评论标准。