Multiscale Methods for the Rapid Solution of Boundary Integral Equations in Geometrically Complicated Domains

几何复杂域中边界积分方程快速求解的多尺度方法

• 批准号: 0074553
• 负责人: Johannes Tausch
• 金额: $ 7.3万
• 依托单位: Southern Methodist University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074553&HistoricalAwards=false
• 关键词: Multiscale; Methods; Rapid; Solution; Boundary

The proposed research is concerned with multiscale discretizations of boundary integral operators on geometrically complicated surfaces. The standard approach, generating multiscale discretizations in parameter spaces and lifting them on the surface, is not efficient when a large number of parameter patches is required. This project is concerned with an alternative approach to multiscale discretizations. The basis is generated in a hierarchical decomposition of the three-space and subsequently restricted on the boundary surface. This construction leads to a sparse representation of the integral operator even for complicated geometries. It is planned to apply this approach to boundary integral formulations of general elliptic as well as time dependent parabolic problems.The range of important engineering applications which the proposed research could impact is quite diverse. Examples include the analysis of integrated circuit interconnect and micromechanical systems. In these areas finding computationally efficient numerical methods for complex three dimensional structures is important in order to generate prototypes interactively on a computer. In the past the major computational tool has been the Fast Multipole Method (FMM), because of its relative simplicity and its flexibility to geometry and integral operator. We construct a multiscale basis by using a hierarchical decomposition of the three-space. The same hierarchy of cubes is also used by the FMM, and therefore the proposed approach is able to combine the flexibility of the FMM with the strengths of multiscale methods. Thus the planned research can lead to faster and more robust algorithms as well as to a better understanding of theoretical and practical issues of both approaches.

所提出的研究涉及几何复杂曲面上边界积分算子的多尺度离散。 标准的方法，在参数空间中生成多尺度离散化，并将它们提升到曲面上，当需要大量的参数补丁时，效率不高。 这个项目关注的是多尺度离散化的另一种方法。 该基础是在三维空间的分层分解中生成的，随后被限制在边界表面上。 这种结构导致了稀疏表示的积分算子，即使是复杂的几何形状。 计划将这种方法应用于一般椭圆和时间依赖抛物问题的边界积分公式，所提出的研究可能影响的重要工程应用的范围是相当多样化的。 例子包括集成电路互连和微机械系统的分析。 在这些领域中，找到计算效率高的数值方法，复杂的三维结构是很重要的，以便在计算机上交互式生成原型。 在过去，主要的计算工具一直是快速多极子方法（FMM），因为它相对简单，其灵活的几何和积分算子。 我们构造了一个多尺度的基础，通过使用三个空间的层次分解。 FMM也使用相同层次的立方体，因此所提出的方法能够将FMM的灵活性与多尺度方法的优点联合收割机结合起来。 因此，计划的研究可以导致更快，更强大的算法，以及更好地理解这两种方法的理论和实践问题。