A novel boundary integral formulation for dynamic implicit interfaces

这个项目涉及一个新的构造边界积分方法的公式来解决线性偏微分方程的边值问题，这些边值问题涉及由相应的有符号距离函数隐式定义的有向分段光滑边界的域。所提出的框架将有助于在广泛的计算问题中进行计算，例如，泊松方程或亥姆霍兹方程的解在不规则边界的时间依赖域上定义。这类计算问题存在于多相粘性流体流动、逆散射、形状优化问题和梯度流动的表面能建模中，例如流体的凝固过程。我们的新公式的目的是提高在有限元方法中需要的重网格的开销，并避免在基于有限差分的方法中发现的依赖于表面如何“切割”底层网格的微妙和可能复杂的公式。该公式是基于在区域边界上定义的积分方程的单参数参数化族的平均。利用共面积公式，导出了一种不需要显式参数化边界的边界积分方程。所得的数值算法简单，适用于各种网格或网格。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（正交），(b)角和高协维流形的解析和数值处理，(c)非线性界面动力学和形状优化的应用。所提出的研究将直接有助于科学和工程中许多重要应用的关键数学和计算部分，从多相流体，石油工程中的地震成像，波传播中的逆散射，流体凝固研究中的高阶非线性界面演化，到生物力学应用。拟议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，并为相关领域带来最先进的数值分析和计算算法。

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.