Discontinuous Immersed Finite Element Methods for Interface Problems

界面问题的不连续浸入式有限元方法

• 批准号: 0713763
• 负责人: Tao Lin
• 金额: $ 15.3万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-15 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0713763&HistoricalAwards=false
• 关键词: Discontinuous; Immersed; Finite; Element; Methods

Immersed finite element (IFE) methods have been developed to solve problems with discontinuous coefficients and they are very competitive because of two essential features: (1) they allow structured meshes and avoid mesh distortion associated with Lagrangian and interface-tracking methods; (2) their basis functions incorporate the interface jump conditions imposed by physics. The objectives of this project are to combine IFE methods with the versatile discontinuous Galerkin (DG) finite element methods to efficiently and accurately capture solution discontinuities, simplify h and h-p adaptivity, and yield efficient parallel methods for interface problems arising from modeling multi-scale and multi-physics procedures in engineering and sciences. The PIs of this project plan to construct and analyze DG methods with IFE spaces that attain optimal or near optimal convergence rates under h and h-p refinement on meshes not necessarily aligned with interfaces. The PIs plan to focus on three representative types of boundary value problems with discontinuous coefficients in two and three dimensions: (i) Second-order elliptic problems, (ii) Linear elasticity systems and fourth-order equations for beams and plates, (iii) Maxwell's equations. The PIs will investigate problems with linear and curved interfaces on meshes consisting of triangles and quadrilaterals in two dimensions, tetrahedrons and hexahedrons in three dimensions. The PIs will perform error analysis and investigate the convergence of the IFE solution to the true solution under h, p and h-p refinement. The PIs will also investigate superconvergence properties of discontinuous IFE solutions and use these results to construct efficient and reliable a-posteriori IFE error estimators for assessing solution quality and guiding adaptivity.Simulating a multi-scale/multi-physics phenomenon often involves a domain consisting of different materials and leads to an interface problem consisting of partial differential equations with discontinuous coefficients, boundary (and initial) conditions, and jump conditions required by pertinent physics across the material interfaces. It is well known that efficiently solving interface problems is critical for numerical simulations in many applications of engineering and sciences, including flow problems, electromagnetic problems, shape/topology optimization problems, to name just a few. IFE methods are competitive methods for solving interface problems and the methods developed in the proposed research projects will have direct impacts on numerical simulations in electric propulsion of plasma engine design/research, optimal packaging of electronic devices, efficient and better image reconstruction in computer tomography, polymer matrix carbon fiber reinforced composites and related material technologies in aerospace and space structures, non-destructive/non-invasive detection of suspicious materials in security check, design of optimal shapes for lighter and stronger structures, and many other application areas of great federal interests.

已经开发了浸没的有限元（IFE）方法来解决不连续系数的问题，并且由于两个基本特征，它们非常有竞争力：（1）它们允许结构化的网格并避免与Lagrangian和接口跟踪方法相关的网格失真； （2）它们的基础功能结合了物理施加的接口跳跃条件。该项目的目的是将IFE方法与多功能不连续的Galerkin（DG）有限元方法相结合，以有效，准确地捕获解决方案不连续性，简化H和H-P适应性，并产生有效的并行方法，用于导致通过建模多层尺度和多型物体工程学的工具和SCIENESS引起的界面问题。该项目计划的PI旨在构建和分析DG方法的IFE空间，这些空间在H和H-P改进下达到最佳或接近最佳的收敛速率，这不一定与接口相符。 PIS计划将两种代表性的边界价值问题类型与两个和三个维度的不连续系数相关：（i）二阶椭圆问题，（ii）光束和板的线性弹性系统和第四阶方程，（III）Maxwell方程。 PI将研究由三角形和四个维度的四角形和四面化，四面体和十六世纪三维的三角形和四边形组成的线性和弯曲界面的问题。 PI将执行误差分析，并研究IFE解决方案与H，P和H-P细化下的真实溶液的收敛性。 PI还将调查不连续的IFE解决方案的超授权特性，并使用这些结果来构建高效且可靠的A-tosterii IFE IFE误差估计器，以评估解决方案质量和指导适应性。构图多尺度/多型物理现象通常会导致材料和互动的界面与界面相差不同，并涉及与界面相差的范围，并涉及界面的范围，并涉及互联方的界面。条件和相关物理所需的跳跃条件在整个材料界面上。众所周知，有效解决界面问题对于工程和科学的许多应用中的数值模拟至关重要，包括流量问题，电磁问题，形状/拓扑优化问题，仅举几例。 IFE方法是解决界面问题的竞争方法，拟议的研究项目中开发的方法将直接影响等离子体发动机设计/研究的电推进的数值模拟，电子设备的最佳包装，电子设备的最佳包装，有效，更好的图像重建，计算机层析成像，聚合物纤维纤维纤维纤维组合和相关的构造和相关的构造和相关的构造中，不及时的空间构造，不可能构成型号的空间，以构造的构造和无关的空间构造，而无效的构造/范围内的空间范围内的空间范围内，无效的空间范围内，无效。在安全检查中检测可疑材料，设计最佳形状以较轻，更强的结构以及许多其他具有巨大联邦利益的应用领域。