A Fitted Finite Element Method for the Modeling of Complex Materials

复杂材料建模的拟合有限元方法

• 批准号: 2012285
• 负责人: Jeffrey Ovall
• 金额: $ 20万
• 依托单位: Portland State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012285&HistoricalAwards=false
• 关键词: Fitted; Finite; Element; Method; Modeling

Partial differential equations (PDEs) provide the principle mathematical models of physical phenomena that undergo continuous spatial and/or temporal variation, and the approximation of their solutions is of fundamental importance in a broad spectrum of scientific and engineering applications. Finite element methods (FEM) for solving PDEs are favored in the scientific community because of their flexibility in representing materials with complex geometries and spatially-varying material properties, and their ability to resolve local features of the solution, such as sharp transitions (e.g. shocks, layers) and singularities (unbounded derivatives). FEM works by partitioning the region of interest into a mesh consisting of smaller computational cells, and approximating the solution of the PDE in terms of “simple” functions defined on these cells. This project aims to increase the flexibility of FEM by allowing for significantly more general cell shapes and function types, to more efficiently model complex materials. Publicly-available software will be produced, together with supporting mathematical theory and numerical experiments illustrating its practical performance on problems exhibiting challenging and realistic features.The PI and collaborators will develop theory and practical algorithms for computing with finite elements defined on meshes consisting of rather general curvilinear polygons. Among the issues that will be addressed are: (i) interpolation/approximation theory for the resulting finite element spaces, which contain special locally-harmonic functions in addition to local polynomials; (ii) basis selection and efficient and accurate quadrature rules for assembling the finite element linear system; (iii) efficient and reliable a posteriori error estimators, and self-adaptive refinement techniques based on them; (iv) development of freely-available software, hosted on a public repository, that includes example problems. The types of problems that motivate this work are those involving PDE models of complex materials that may have multiple complicated interfaces between material types. Numerical examples supplied in the articles will highlight the performance of the proposed method on such problems, making relevant comparisons with competing approaches where feasible.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.

偏微分方程（PDE）提供了经历连续空间和/或时间变化的物理现象的基本数学模型，并且其解的近似在广泛的科学和工程应用中具有根本重要性。用于求解偏微分方程的有限元方法（FEM）在科学界受到青睐，因为它们在表示具有复杂几何形状和空间变化的材料特性的材料方面具有灵活性，并且能够解决方案的局部特征，例如急剧转变（例如冲击，层）和奇点（无界导数）。FEM的工作原理是将感兴趣的区域划分为由较小的计算单元组成的网格，并根据定义在这些单元上的“简单”函数来近似PDE的解。该项目旨在通过允许更通用的单元形状和功能类型来增加FEM的灵活性，以更有效地模拟复杂材料。将制作公开可用的软件，以及支持的数学理论和数值实验，说明其在具有挑战性和现实特征的问题上的实际性能。PI和合作者将开发理论和实用算法，用于在由相当一般的曲线多边形组成的网格上定义的有限元计算。其中将解决的问题是：（一）插值/逼近理论的有限元空间，其中包含特殊的局部调和函数，除了局部多项式;（二）基的选择和有效和准确的积分规则，组装有限元线性系统;（三）有效和可靠的后验误差估计，和自适应细化技术的基础上; ㈣开发免费提供的软件，存放在公共储存库中，其中包括例题。激发这项工作的问题类型是那些涉及复杂材料的PDE模型，这些材料类型之间可能有多个复杂的界面。文章中提供的数值例子将突出所提出的方法在这些问题上的性能，在可行的情况下与竞争方法进行相关比较。该奖项反映了NSF的法定使命，并被认为是值得支持的，通过使用基金会的知识价值和更广泛的影响审查标准进行评估。