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.

部分微分方程（PDE）提供了经历连续的空间和/或临时变化的物理现象的原理数学模型，在广泛的科学和工程应用中，其解决方案的近似值至关重要。用于求解PDE的有限元方法（fem）在科学界受到青睐，因为它们在代表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，以及​​它们可以解决解决方案的局部特征的能力，例如急剧过渡（例如，冲击，层）和奇异性（无结合的衍生物）。 FEM通过将感兴趣区域划分为由较小的计算单元组成的网格，并根据这些单元上定义的“简单”功能近似PDE的解决方案。该项目旨在通过允许更多的通用细胞形状和功能类型来提高FEM的灵活性，以更有效地对复杂的材料进行建模。将生产出公共可用的软件，并支持数学理论和数值实验，以说明其在经历挑战和现实特征的问题上的实用性。PI和合作者将开发用于计算的理论和实用算法，并使用有限的元素定义在由相当普遍的曲线性多角形组成的网格上定义的有限元素。在将要解决的问题中包括：（i）所得有限元空间的插值/近似理论，除局部多项式外，该空间还包含特殊的本地函数； （ii）组装有限元线性系统的基础选择以及有效，准确的正交规则； （iii）有效且可靠的后误差估计器以及基于它们的自适应改进技术； （iv）开发在公共存储库上托管的免费软件，其中包括示例问题。融合这项工作的问题类型是涉及复杂材料模型的PDE模型，这些模型可能在材料类型之间具有多个复杂接口。文章中提供的数值示例将突出提出的方法在此类问题上的表现，从而与可行的竞争方法进行了相关的比较。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛的影响审查标准通过评估来获得的支持。