Computation of Nearly Singular Integrals with Applications to Fluid Dynamics

这项拟议的工作涉及到计算奇异或近奇异积分的改进方法的发展，以及在模拟具有移动边界的流体流动的数值方法中的应用。在许多科学问题中，感兴趣的量可以用积分来表示，例如边界上的双电层势。当这样的积分在边界附近计算时，通常的求积规则是不准确的，因为被积函数中的导数很大。希望以一种简单而有效的方式来计算这样的值，而不需要选择特殊的积分点。本方法使用奇点的正则化和标准的求积规则，然后由渐近分析得到局部校正。这项技术将被用来设计C.Peskin沉没边界方法的替代版本，C.Peskin是一种计算方法，已被用于模拟各种生物过程。在这种方法中，粘性、不可压缩的流体受到施加在流体上的力的膜或界面的影响。在二维中，力可以用近似奇异积分来表示，所提出的方法可以应用于该积分。希望这一修正能扩大浸没边界法在实际应用中的应用。也可以应用于具有分隔不同流体的移动边界的流动的计算方法。将近奇异积分技术推广到三维空间中的曲面积分，推广了曲面上标记点已有的工作。相关工作将涉及无粘、势流中的界面，使用改进的边界积分正则化。许多科学过程涉及移动边界，例如，一滴液体通过另一滴液体的边界，或活组织中的膜。这种过程的数值模型遇到了特殊的困难，因为在一组规则间隔且不变的点上计算量是最简单的。目前有几种数值方法可以避免在每个新的时刻从根本上改变区域的近似方式。在某些情况下，重要的量可以写成边界上的积分。这些积分以微分方程解的形式出现，通常是奇异的，或者几乎是奇异的；也就是说，出现大的值，表示附近的点之间的强烈相互作用，如在重力的平方反定律中。要准确有效地计算这种积分，需要特殊的技术。本提案涉及这类方法的发展和应用。使用积分表示法来修改现有的具有移动边界的流体运动计算方法的设计，可能会提高这些数值方法做出可靠预测的能力。

0102356BealeThe proposed work is concerned with the development of improved methods for computing singular or nearly singular integrals, and with applications to numerical methods for simulating fluid flow with moving boundaries. In many scientific problems, quantities of interest can be written in terms of integrals such as a double layer potential on a boundary. When such an integral is evaluated near the boundary, usual quadrature rules are inaccurate because of large derivatives in the integrand. It is desirable to calculate such values in a way that is simple and efficient, without requiring a special choice of quadrature points. The present approach uses regularization of the singularity and a standard quadrature rule, followed by local corrections derived from asymptotic analysis. This technique will be used to design an alternate version of the immersed boundary method of C. Peskin, a computational method which has been used to model various biological processes. In this method viscous, incompressible fluid is influenced by a membrane or interface which exerts a force on the fluid. In two dimensions, the force can be expressed in terms of nearly singular integrals to which the proposed approach can be applied. It is hoped that this modification will extend the usefulness of the immersed boundary method in realistic applications. Application may also be made to methods for computing flow with moving boundaries separating different fluids. The technique for nearly singular integrals will be extended to surface integrals in three-space, generalizing work already done for marker points on surfaces. Related work will concern interfaces in inviscid, potential flow using improved regularizations of boundary integrals.Many scientific processes involve moving boundaries, for example, the boundary of a drop of one fluid moving through another, or a membrane in living tissue. Numerical models of such processes encounter special difficulties, since it is simplest to compute quantities at a set of points which are regularly spaced and unchanging. There are presently several numerical approaches which avoid changing fundamentally the way the region is approximated at each new time. In some cases important quantities can be written as integrals over boundaries. These integrals arise as solutions of differential equations and are usually singular, or nearly singular; that is, large values appear, representing the strong interaction of nearby points, as in the inverse square law of gravity. Special techniques are needed to calculate such integrals accurately and efficiently. This proposal is concerned with the development and application of methods of this type. The use of integral representations to modify the design of existing methods for computing fluid motion with moving boundaries might improve the ability of these numerical methods to make reliable predictions.