Structure-Preserving Hybrid Finite Element Methods

保结构混合有限元方法

• 批准号: 2208551
• 负责人: Ari Stern
• 金额: $ 23.76万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208551&HistoricalAwards=false
• 关键词: Structure; Preserving; Hybrid; Finite; Element

Finite element methods are among the cornerstones of modern scientific computing, combining high performance with strong theoretical guarantees of accuracy and stability. 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. However, in order for the simulations produced by these methods to be physically meaningful and trustworthy, it is desirable that they capture certain fundamental physical laws present in the original systems. Certain systems satisfy "conservation laws," which state that some quantity (like mass, energy, or charge) can move through space but not appear or disappear spontaneously, while other systems satisfy "dissipation laws" that require these quantities to grow or decay at a certain rate (e.g., decay of energy due to friction). Although classical finite element methods make it difficult to express these physical laws, the PI has shown that "hybrid" finite element methods provide a way to do so. In this project, the PI will conduct further investigations into hybrid finite element methods, developing and analyzing computational techniques for preserving these important physical structures. The success of this project would lead to new computational methods and improved understanding of current methods for a wide variety of high-value scientific applications, with important ramifications for computational physics and engineering.Many partial differential equations (PDEs), particularly those encountered in physical applications, contain local symmetries, conservation laws, and related structures. At first glance, some of these local structures appear difficult for a finite element method to capture, and consequently, much of the research in this direction has been restricted to finite difference methods on rectangular grids. However, in recent years, the PI has developed a successful research program showing that hybridization provides a route around the apparent obstacles to local-structure-preserving finite elements. This project will investigate several new questions in this direction, organized around two main themes. (1) Local conservation laws with quadratic densities are common in Hamiltonian PDEs, because they arise from linear point symmetries. However, conventional finite element discretization breaks local symmetry, resulting only in global conservation laws. The PI aims to circumvent this obstacle by using hybrid methods, and to extend this to non-conservative systems where a quadratic quantity is dissipated rather than conserved. (2) The PI and his collaborators have recently succeeded in hybridizing finite element exterior calculus (FEEC), which uses differential forms to unify several families of methods for Laplace-type operators. The PI will investigate the application of hybridized FEEC methods to Hamiltonian PDEs, specifically focusing on multisymplectic structure preservation, and to investigate hybrid FEEC methods with strongly (rather than merely weakly) conservative fluxes, including nonconforming and hybridizable discontinuous Galerkin (HDG) methods.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.

有限元方法是现代科学计算的基石之一，它将高性能与准确性和稳定性的强大理论保证结合起来。它们的应用包括联邦战略利益的几个领域：材料和制造、生物医学工程和生物技术、结构工程和土木基础设施、环境工程等等。它们被学术界、工业界和国家实验室的科学家和工程师广泛使用来模拟大型、复杂的物理系统。然而，为了使这些方法产生的模拟具有物理意义和可信性，它们需要捕获原始系统中存在的某些基本物理定律。某些系统满足“守恒定律”，该定律规定某些量（如质量、能量或电荷）可以在空间中移动，但不会自发出现或消失，而其他系统则满足“耗散定律”，要求这些量在一定的速率（例如，由于摩擦而导致的能量衰减）。尽管经典的有限元方法很难表达这些物理定律，但 PI 表明“混合”有限元方法提供了一种实现这一点的方法。在该项目中，PI 将进一步研究混合有限元方法，开发和分析用于保存这些重要物理结构的计算技术。该项目的成功将带来新的计算方法，并提高对各种高价值科学应用的当前方法的理解，对计算物理和工程学产生重要影响。许多偏微分方程（PDE），特别是在物理领域遇到的偏微分方程（PDE）应用程序，包含局部对称性、守恒定律和相关结构。乍一看，其中一些局部结构似乎很难用有限元方法捕获，因此，这个方向的大部分研究都仅限于矩形网格上的有限差分方法。然而，近年来，PI 开发了一项成功的研究计划，表明混合技术为解决局部结构保留有限元的明显障碍提供了一条途径。该项目将围绕两个主题研究这个方向的几个新问题。 (1) 具有二次密度的局部守恒定律在哈密顿偏微分方程中很常见，因为它们源自线性点对称性。然而，传统的有限元离散化打破了局部对称性，仅产生全局守恒定律。 PI 旨在通过使用混合方法来规避这一障碍，并将其扩展到二次量被耗散而不是守恒的非保守系统。 (2) PI 和他的合作者最近成功地混合了有限元外微积分 (FEEC)，它使用微分形式来统一拉普拉斯型算子的几个方法族。 PI 将研究混合 FEEC 方法在哈密顿偏微分方程中的应用，特别关注多辛结构保存，并研究具有强（而不仅仅是弱）保守通量的混合 FEEC 方法，包括非相容和可混合的不连续伽辽金 (HDG) 方法。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。