Computational Methods for Singular and Nearly Singular Integrals with Applications to Fluid Dynamics

本文的目的是发展计算奇异积分或近似奇异积分的有效方法，并将其应用于具有移动边界的流体流动的数值模拟。 科学问题的数学表述通常涉及奇异积分，例如由于曲线或曲面上的源层而产生的调和势函数。 对于靠近源的点，由于导数较大，因此评估不是常规的。 这项工作的方法，在早期的NSF资助的研究，是正规化的积分在一个系统的方式，评估在网格点作为一个标准的积分，然后添加本地校正项。 修正是通过解析推导出来的。 对于曲面，使用重叠栅格。 这项工作将在几个部分：一类静态问题的三维潜力定义通过表面积分将被处理，扩展早期的工作。 为了允许边界不平滑，将推导出一种用于计算2D中具有角的曲线上的积分的方法。 应用于粘性不可压缩二维流体流动中的移动边界的计算将作为浸没边界方法或浸没界面方法的可能改进进行测试。 在没有粘性的情况下，计算将针对两种不同流体之间的移动边界或界面进行;水波是一种重要的情况，在其他情况下，正则化将用于控制物理不稳定性。科学过程通常涉及移动边界，例如一滴流体移动通过另一种流体，或者活组织中弹性膜的运动。 这种现象的数值模拟涉及特殊的困难，并在使用几种方法。 重要的量，如压力的跳跃，通常可以写成像上面描述的那样的奇异积分。 在这项工作中开发的技术可用于将积分计算到数值方法的粘性流体流动与移动边界。 如果这导致这些方法的改进，它们可以更广泛地用于预测双流体系统或具有移动膜的生物过程。 无粘性流动的应用可以提高对完全非线性水波和不稳定流体层中混合开始的理解。

The purpose of this work is to develop efficient methods for computing singular or nearly singular integrals and to apply the methods to the numerical simulation of fluid flow with moving boundaries. The mathematical formulation of scientific problems often involves singular integrals, such as a harmonic potential function due to a layer of sources on a curve or surface. For points near the source, evaluation is not routine because of large derivatives. The approach of this work, developed in earlier NSF-funded research, is to regularize the integral in a systematic way, evaluate at grid points as for a standard integral, and then add local correction terms. The corrections are derived analytically. For surfaces, overlapping grids are used. The work will be in several parts: A class of static problems for 3D potentials defined through surface integrals will be treated, extending earlier work. In order to allow for boundaries that are not smooth, a method will be derived for computing integrals on curves with corners in 2D. Application to the computation of moving boundaries in viscous, incompressible 2D fluid flow will be tested as a possible improvement in the immersed boundary method or immersed interface method. Computations will be done for a moving boundary or interface between two different fluids in 3D without viscosity; water waves are one important case, and in other cases regularization will be used to control physical instabilities.Scientific processes often involve moving boundaries, such as a drop of one fluid moving through another, or the motion of an elastic membrane in living tissue. Numerical modeling of such phenomena involves special difficulties, and several approaches are in use. Important quantities, such as a jump in pressure, can often be written as singular integrals like the ones described. The techniques developed in this work could be used to incorporate integral calculations into numerical methods for viscous fluid flow with moving boundaries. If this leads to improvement in these methods, they could be more widely useful for predictions of two-fluid systems or biological processes with moving membranes. The application to flow without viscosity could improve understanding of fully nonlinear water waves and the onset of mixing in an unstable fluid layer.