A novel boundary integral formulation for dynamic implicit interfaces

这个项目涉及到一种新的构造边界积分方法的公式，该方法用于求解区域上的线性偏微分方程组的边值问题，该区域具有由相应的符号距离函数隐式定义的定向分段光滑边界。所提出的框架将有助于计算一大类计算问题，例如，定义在具有不规则边界的依赖于时间的区域上的泊松方程或亥姆霍兹方程的解。这样的计算问题在多相粘性流体流动、逆散射、形状优化问题和表面能梯度流模拟中都存在，例如流体的凝固过程。我们的新公式的目的是消除有限元方法中所需的重新划分网格的开销，并避免在基于有限差分的方法中发现微妙且可能复杂的公式，这些公式依赖于曲面如何通过底层网格进行切割。该公式基于定义在区域边界上的积分方程单参数化族的平均。应用余面积公式，导出了一种新的边界积分方程解，而无需对边界进行任何显式的参数化处理。得到的数值算法简单，适用于各种网格划分或网格划分。提出的研究计划包括对这种新型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)角点和高次共维流形的分析和数值处理，(C)在非线性界面动力学和形状优化中的应用。从多相流体、石油工程中的地震成像、波传播中的逆散射、流体凝固研究中发现的高阶非线性界面演化，到生物力学应用，所提出的研究将直接为科学和工程中的许多重要应用提供重要的数学和计算部分。拟议的研究计划提供的对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。

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.