Novel Finite Element Methods for Three-Dimensional Anisotropic Singular Problems

三维各向异性奇异问题的新颖有限元方法

基本信息

批准号：
1819041
负责人：
Hengguang Li
金额：
$ 19.9万
依托单位：
Wayne State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819041&HistoricalAwards=false
关键词：
Novel Finite Element Methods Three

项目摘要

Elliptic partial differential equations can possess singular solutions in various practical models. The efficacy of numerical simulation highly depends on smoothness and can severely deteriorate in the presence of singularities. The development of effective finite element methods (FEMs) for singular partial differential equations has been a central focus in computational mathematics; however, most of the established methods are for two-dimensional problems. The design of effective three-dimensional (3D) methods is more technically involved and less explored, largely due to the constraints imposed by the 3D geometry and by anisotropic structure in singularities. Existing 3D algorithms are usually tetrahedron-based and sensitive to the domain geometry and to the degree of polynomials, which limits their applications in practical computing. This project aims to develop a novel mesh algorithm that is well-structured, flexible with different 3D elements, and simple in implementation. The new algorithm is expected to significantly improve the effectiveness of 3D numerical simulations in areas where anisotropic solutions frequently occur, including mathematical models in aerospace engineering (e.g., aircraft design), in mechanical engineering (e.g., crack propagation in civil infrastructure), in fluid mechanics, and in electromagnetism.This research project has two main components. (1) Innovative numerical algorithms. The investigator plans to develop a new family of anisotropic meshes that facilitate optimal FEMs approximating 3D anisotropic singular solutions. The mesh construction follows a simple, explicit, and unified approach and applies to four basic 3D elements: the tetrahedron, hexahedron, wedge, and pyramid. With unconventional but implementation-friendly features, these algorithms have shown effectiveness in early numerical tests. (2) Rigorous theoretical investigations and applications. The investigator plans to devise new analytical tools to justify and broaden the applications of the new FEMs, especially when the mesh is anisotropic. This includes (i) optimal error analysis; (ii) new regularity estimates for 3D anisotropic problems; (iii) fast numerical solvers for linear systems from anisotropic meshes; and (iv) efficient implementations in high-performance computing environments.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.

椭圆形的部分微分方程可以在各种实际模型中具有单数解。数值模拟的功效高度取决于平滑度，并且在存在奇异性的情况下可能会严重恶化。开发有效的有限元方法（FEM）用于奇异偏微分方程已成为计算数学的核心重点。但是，大多数已建立的方法用于二维问题。有效的三维（3D）方法的设计在技术上更涉及，并且探索较少，这在很大程度上是由于3D几何形状和各向异性结构在奇异性中施加的限制。现有的3D算法通常基于四面体，对域几何形状和多项式程度敏感，这限制了其在实际计算中的应用。该项目旨在开发一种新颖的网格算法，该算法结构良好，灵活，具有不同的3D元素，并且实现简单。预计新算法将显着提高经常发生各向异性解决方案的3D数值模拟的有效性，包括航空工程中的数学模型（例如飞机设计），机械工程（例如，在流动的机械师和电子机械师中，都有两种构造的项目，在机械工程中（例如，公民基础设施中的裂纹）。（1）创新的数值算法。研究人员计划开发一个新的各向异性网络家族，以促进近似于3D各向异性奇异溶液的最佳FEM。网格结构遵循一种简单，明确且统一的方法，并适用于四个基本的3D元素：四面体，六面体，楔子和金字塔。这些算法具有非常规但实施的特征，在早期的数值测试中显示出有效性。（2）严格的理论研究和应用。研究人员计划设计新的分析工具，以证明和扩大新的FEM的应用，尤其是在网眼处于各向异性的情况下。这包括（i）最佳错误分析；（ii）3D各向异性问题的新规律性估计；（iii）来自各向异性网格的线性系统的快速数值求解器；（iv）在高性能计算环境中的有效实施。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。