Computation of Nearly Singular Integrals with Applications to Fluid Dynamics

0102356 BealeThe建议的工作是关于改进的方法计算奇异或近似奇异积分的发展，并与应用程序的数值方法模拟流体流动与移动边界。 在许多科学问题中，感兴趣的量可以用积分来表示，例如边界上的双层势。当这样的积分是在边界附近计算时，通常的求积规则是不准确的，因为被积函数中的导数很大。期望以简单且有效的方式计算这样的值，而不需要对正交点进行特殊选择。本方法使用正则化的奇异性和一个标准的正交规则，其次是来自渐近分析的局部校正。 这种技术将被用来设计一个替代版本的浸入边界法的C。Peskin是一种用于模拟各种生物过程的计算方法。在这种方法中，粘性不可压缩流体受到膜或界面的影响，该膜或界面对流体施加力。 在两个维度上，力可以用所提出的方法可以应用的近似奇异积分来表示。 希望这种修改将扩大实用性的浸入边界法在实际应用中。 还可以应用于计算具有分离不同流体的移动边界的流动的方法。 近奇异积分的技术将被扩展到三维空间的曲面积分，推广已经做的工作，表面上的标记点。 相关的工作将涉及无粘势流中的界面，使用边界积分的改进正则化。许多科学过程涉及移动边界，例如，一滴流体通过另一滴流体的边界，或活组织中的膜。 这种过程的数值模型遇到特殊的困难，因为它是最简单的计算量在一组点，这是有规律地间隔和不变。 目前有几种数值方法，避免从根本上改变该地区的方式近似在每个新的时间。在某些情况下，重要的量可以写成边界上的积分。 这些积分是微分方程的解，通常是奇异的或近似奇异的;也就是说，出现很大的值，代表附近点的强相互作用，就像引力的平方反比定律一样。 需要特殊的技术来精确和有效地计算这样的积分。 本建议涉及这类方法的发展和应用。 使用积分表示来修改计算具有移动边界的流体运动的现有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.