A Fitted Finite Element Method for the Modeling of Complex Materials

偏微分方程（PDEs）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，其解的近似在广泛的科学和工程应用中具有重要的基础意义。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材料特性的材料方面具有灵活性，并且它们能够解决解决方案的局部特征，例如急剧过渡（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，并根据在这些单元上定义的“简单”函数近似求解PDE。该项目旨在通过允许更一般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，连同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，用于计算由相当一般的曲线多边形组成的网格上定义的有限元。将解决的问题包括：(i)所得到的有限元空间的插值/近似理论，除了局部多项式外，它还包含特殊的局部调和函数；（ii）有限元线性系统装配的基础选择和高效准确的正交规则；（iii）高效可靠的后验误差估计器，以及基于它们的自适应改进技术；（iv）开发免费软件，托管在公共存储库上，其中包括示例问题。激发这项工作的问题类型是那些涉及复杂材料的PDE模型的问题，这些模型可能在材料类型之间具有多个复杂的界面。文中提供的数值示例将突出所提出的方法在此类问题上的性能，并在可行的情况下与竞争方法进行相关比较。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.