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

这项工作的目的是发展计算奇异或近奇异积分的有效方法，并将这些方法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一层源而产生的调和势函数。对于震源附近的点，由于导数较大，评估并不是例行的。这项工作是在早期NSF资助的研究中发展起来的，其方法是系统地将积分正则化，在网格点上作为标准积分进行评估，然后添加局部校正项。这些修正是以解析的方式得出的。对于曲面，使用重叠栅格。这项工作将分为几个部分：一类通过表面积分定义的三维势的静态问题将被处理，扩展了前面的工作。为了允许边界不光滑，将推导出一种计算二维曲线上带角点的积分的方法。在粘性、不可压缩的2D流体流动中应用移动边界的计算将作为浸没边界方法或浸没界面方法的一种可能的改进而被测试。计算的对象是无粘性的三维运动边界或两种不同流体之间的界面；水波是一个重要的情况，在其他情况下，将使用正则化来控制物理不稳定性。科学过程通常涉及移动边界，如一种液体滴通过另一种液体，或活组织中弹性膜的运动。对这类现象进行数值模拟涉及特殊困难，目前正在使用几种方法。重要的量，如压强的跳跃，通常可以写成奇异积分，就像所描述的那样。本文所开发的技术可用于将积分计算结合到具有移动边界的粘性流体流动的数值方法中。如果这导致这些方法的改进，它们可能会更广泛地用于预测双流体系统或具有移动膜的生物过程。在无粘性流动中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.