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A novel boundary integral formulation for dynamic implicit interfaces

一种新颖的动态隐式接口边界积分公式

基本信息

批准号：
1318975
负责人：
Yen-Hsi Tsai
金额：
$ 20.99万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2013
资助国家：
美国
起止时间：
2013-09-01 至 2017-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1318975&HistoricalAwards=false
关键词：
novel boundary integral formulation dynamic

项目摘要

This project involves a novel formulation for constructing boundary integral methods to solve boundary value problems involving linear partial differential equations on domains with oriented, piecewise smooth boundaries defined implicitly by the corresponding signed distance functions. The proposed framework will facilitate computations in a wide class of computational problems that involve, e.g., the solutions of Poisson's equation or Helmholtz equation defined on time dependent domains with irregular boundaries. Such computational problems are found in multiphase viscous fluid flows, inverse scattering, shape optimization problems, and gradient flows of surface energies modeling e.g. solidification process of a fluid. The aim of our new formulation is to lift the overhead of remeshing that is needed in finite element methods, and to avoid delicate and possibly complicated formulas found in finite difference based methods that depend on how surfaces "cut" through the underlying grids. The formulation is based on averaging a one parameter family of parameterizations of an integral equation defined on the boundary of the domain. By application of the coarea formula, a novel boundary integral equation without any explicit parameterization of the boundaries is derived. The resulting numerical algorithm is simple and is applicable to a variety of meshing or grids. The proposed research program consists of a systematic study of such new type of boundary integral formulation, encompassing (a) numerical integration methods (quadratures) for singular integrals, (b) analytical and numerical treatment of corners and higher co-dimensional manifolds, (c) applications to nonlinear interface dynamics and shape optimizations. The proposed research will contribute directly to a crucial mathematical and computational part of many important applications in science and engineering, from multiphase fluids, seismic imaging in petroleum engineering, inverse scattering in wave propagation, high order nonlinear interface evolution found in the study of solidification of fluids, to bio-mechanical applications. The training of students and post-doctoral researchers provided by the proposed research program will allow them to conduct research in highly inter-disciplinary projects and bring state-of-the-art numerical analysis and computational algorithms to the related areas.

该项目涉及一种新的配方，用于构建边界积分方法，以解决边界值问题，涉及线性偏微分方程的域上的定向，分段光滑的边界由相应的符号距离函数隐式定义。所提出的框架将有助于计算在广泛的一类计算问题，涉及，例如，定义在非规则边界的时间依赖区域上的Poisson方程或Helmholtz方程的解。这样的计算问题在多相粘性流体流、逆散射、形状优化问题和表面能的梯度流建模（例如流体的固化过程）中被发现。我们的新配方的目的是解除的开销重新网格划分，需要在有限元方法，并避免微妙的和可能复杂的公式中发现基于有限差分的方法，取决于如何表面“切割”通过底层网格。该配方是基于平均的一个参数家庭的参数化定义的域的边界上的积分方程。应用共面积公式，导出了一个新的边界积分方程，该方程不需要对边界进行任何显式参数化。所得的数值算法简单，适用于各种网格或网格。拟议的研究计划包括这种新型的边界积分公式的系统研究，包括（a）奇异积分的数值积分方法（求积），（B）角点和更高的余维流形的分析和数值处理，（c）应用于非线性界面动力学和形状优化。拟议的研究将直接有助于在科学和工程中的许多重要应用，从多相流体，石油工程中的地震成像，波传播中的逆散射，高阶非线性界面演化中发现的固化的流体的研究，生物力学应用的一个关键的数学和计算的一部分。拟议的研究计划提供的学生和博士后研究人员的培训将使他们能够在高度跨学科的项目中进行研究，并将最先进的数值分析和计算算法带到相关领域。