Computation of Nearly Singular Integrals with Applications to Fluid Dynamics
近奇异积分的计算及其在流体动力学中的应用
基本信息
- 批准号:0102356
- 负责人:
- 金额:$ 16.55万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2001
- 资助国家:美国
- 起止时间:2001-08-01 至 2005-01-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
[01:23 . 56]提出的工作是发展计算奇异或近奇异积分的改进方法,并应用于模拟具有移动边界的流体流动的数值方法。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。当这样的积分在边界附近求值时,通常的求积分规则是不准确的,因为被积函数的导数很大。我们希望用一种简单而有效的方法来计算这些值,而不需要特别选择正交点。本方法采用奇异点正则化和标准正交规则,然后由渐近分析得到局部修正。该技术将用于设计C. Peskin浸入边界法的替代版本,这是一种用于模拟各种生物过程的计算方法。在这种方法中,粘性的、不可压缩的流体受到膜或界面的影响,膜或界面对流体施加一个力。在二维空间中,力可以用近似奇异积分表示,所提出的方法可以应用于这种积分。希望这一改进将扩展浸入边界法在实际应用中的适用性。还可以应用于具有分离不同流体的移动边界的流的计算方法。本文将近似奇异积分的方法推广到三维空间的曲面积分中,对曲面上的标记点所做的工作进行推广。相关工作将涉及使用改进的正则化边界积分的无粘势流中的界面。许多科学过程涉及移动的边界,例如,一种液体流过另一种液体的边界,或者活组织中的膜。这些过程的数值模型遇到了特殊的困难,因为在一组规则间隔和不变的点上计算数量是最简单的。目前有几种数值方法可以避免从根本上改变在每个新时间近似区域的方式。在某些情况下,重要的量可以写成边界上的积分。这些积分是微分方程的解,通常是奇异或近似奇异的;也就是说,出现较大的值,表示附近点的强相互作用,就像在引力的平方反比定律中一样。要准确有效地计算这类积分,需要特殊的技术。本建议涉及这类方法的发展和应用。利用积分表示来改进现有计算具有运动边界的流体运动方法的设计,可能会提高这些数值方法做出可靠预测的能力。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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J. Thomas Beale其他文献
The adjoint double layer potential on smooth surfaces in $$\mathbb {R}^3$$ and the Neumann problem
- DOI:
10.1007/s10444-024-10111-0 - 发表时间:
2024-04-19 - 期刊:
- 影响因子:2.100
- 作者:
J. Thomas Beale;Michael Storm;Svetlana Tlupova - 通讯作者:
Svetlana Tlupova
Decay of solutions of the Stokes system arising in free surface flow on an infinite layer
无限层上自由表面流中斯托克斯系统解的衰减
- DOI:
- 发表时间:
2020 - 期刊:
- 影响因子:0
- 作者:
J. Thomas Beale;Takaaki Nishida and Yoshiaki Teramoto - 通讯作者:
Takaaki Nishida and Yoshiaki Teramoto
On the Accuracy of Vortex Methods at Large Times
- DOI:
10.1007/978-1-4612-3882-9_2 - 发表时间:
1988 - 期刊:
- 影响因子:0
- 作者:
J. Thomas Beale - 通讯作者:
J. Thomas Beale
Scattering frequencies of resonators
谐振器的散射频率
- DOI:
10.1002/cpa.3160260408 - 发表时间:
1973 - 期刊:
- 影响因子:3
- 作者:
J. Thomas Beale - 通讯作者:
J. Thomas Beale
Solving Partial Differential Equations on Closed Surfaces with Planar Cartesian Grids
- DOI:
10.1137/19m1272135 - 发表时间:
2019-08 - 期刊:
- 影响因子:0
- 作者:
J. Thomas Beale - 通讯作者:
J. Thomas Beale
J. Thomas Beale的其他文献
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{{ truncateString('J. Thomas Beale', 18)}}的其他基金
Development and Analysis of Numerical Methods for Fluid Interfaces
流体界面数值方法的开发和分析
- 批准号:
1312654 - 财政年份:2013
- 资助金额:
$ 16.55万 - 项目类别:
Standard Grant
Numerical Methods for Moving Interfaces in Fluids
流体中移动界面的数值方法
- 批准号:
0806482 - 财政年份:2008
- 资助金额:
$ 16.55万 - 项目类别:
Continuing Grant
Computational Methods for Singular and Nearly Singular Integrals with Applications to Fluid Dynamics
奇异和近似奇异积分的计算方法及其在流体动力学中的应用
- 批准号:
0404765 - 财政年份:2004
- 资助金额:
$ 16.55万 - 项目类别:
Standard Grant
Mathematical Sciences: Analysis of Fluid Motion
数学科学:流体运动分析
- 批准号:
9403402 - 财政年份:1995
- 资助金额:
$ 16.55万 - 项目类别:
Standard Grant
Mathematical Sciences: Nonlinear Partial Differential Equations
数学科学:非线性偏微分方程
- 批准号:
9102782 - 财政年份:1991
- 资助金额:
$ 16.55万 - 项目类别:
Continuing Grant
Mathematical Sciences: Nonlinear Partial Differential Equations
数学科学:非线性偏微分方程
- 批准号:
8800347 - 财政年份:1988
- 资助金额:
$ 16.55万 - 项目类别:
Continuing Grant
Mathematical Sciences: Vortex Methods for Incompressible Flow
数学科学:不可压缩流的涡旋方法
- 批准号:
8408393 - 财政年份:1984
- 资助金额:
$ 16.55万 - 项目类别:
Standard Grant
Free Surfaces and Numerical Methods in Fluid Mechanics
流体力学中的自由表面和数值方法
- 批准号:
8101639 - 财政年份:1981
- 资助金额:
$ 16.55万 - 项目类别:
Standard Grant
Nonlinear Partial Differential Equations
非线性偏微分方程
- 批准号:
7800908 - 财政年份:1978
- 资助金额:
$ 16.55万 - 项目类别:
Standard Grant
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