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
• 负责人: J. Thomas Beale
• 金额: $ 20.56万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312654&HistoricalAwards=false
• 关键词: Development; Analysis; Numerical; Methods; Fluid

This work concerns numerical methods for material interfaces moving in viscous fluid flow. Such models are used especially for biological processes on small scales, in which the interface exerts a force on the fluid. The work to be done will focus on the efficient computation of surface integrals representing viscosity-dominated flow, or Stokes flow, and error analysis of finite difference methods for more general Navier-Stokes flow, such as the immersed interface method, in which the equations are discretized on a regular grid, with modifications near the interface to account for the forces. The methods considered should be at least second-order accurate. In the first project a relatively simple method will be developed for computing singular or nearly singular integrals on smooth surfaces, such as the velocity integrals for three-dimensional Stokes flow, evaluated on or near the surface. This work will improve and generalize earlier work, in which a standard quadrature of a regularized integral is combined with corrections found by analysis near the singularity. This method should be accurate even for grid points near the surface, allowing more flexibility in computing the motion of the surface. It could be used as part of a computation for Navier-Stokes flow. In the second project, error estimates in maximum norm will be derived for finite difference methods such as the immersed interface method. Recent results of the proposer showing a gain in regularity for finite difference versions of Poisson or diffusion equations will be used to clarify the relationship between the accuracy of numerical solutions and the corrections needed near the interface and also the choice of time discretization. The work will include convergence proofs for simplified interface problems with Navier-Stokes flow and maximum norm stability of the approximate projection on divergence-free vector fields. A number of scientific problems involve moving boundaries in fluids, such as a drop of one fluid in another, or the motion of an elastic membrane in living tissue. Numerical study of such problems has special difficulties, since it is desirable to discretize the fluid variables on a fixed grid, while the moving boundary must be represented separately, together with its influence on the fluid motion. It is difficult to discretize the moving surface in a way that is simple and practical. For Stokes flow, dominated by viscosity, integral formulations of the fluid variables are widely used. The proposed method of integration promises to be simpler and more efficient than standard methods and require less effort with the surface. Thus it could contribute to the practicality of three-dimensional simulations as the models become more realistic. The second project emphasizes estimates of maximum errors, since these are likely to be largest near interfaces. They are less well developed than estimates in integral norms. Such estimates will be used for methods such as the immersed interface method, the decomposition of Navier-Stokes flow mentioned above, and approximate projection methods. Error analysis of existing numerical methods should improve understanding of their validity and limitations. It can also show how choices in the methods affect their accuracy and suggest improvements.

本工作涉及粘性流体流动中材料界面运动的数值方法。这种模型特别用于小尺度的生物过程，其中界面对流体施加一个力。要做的工作将集中在表示粘性主导流或斯托克斯流的表面积分的有效计算，以及更一般的纳维-斯托克斯流的有限差分方法的误差分析，例如浸入界面法，其中方程在规则网格上离散，在界面附近进行修改以解释力。所考虑的方法至少应具有二阶精度。在第一个项目中，将开发一种相对简单的方法，用于计算光滑表面上的奇异或近奇异积分，例如在表面或表面附近计算三维斯托克斯流的速度积分。这项工作将改进和推广早期的工作，其中正则积分的标准正交与奇点附近分析发现的修正相结合。这种方法应该是准确的，即使网格点接近表面，允许更灵活地计算表面的运动。它可以作为纳维-斯托克斯流计算的一部分。在第二个项目中，将对有限差分方法（如浸入界面法）的最大范数进行误差估计。作者最近的研究结果表明，泊松方程或扩散方程的有限差分版本在规律性上有所提高，将用于澄清数值解的精度与界面附近所需的修正之间的关系，以及时间离散化的选择。工作将包括简化的Navier-Stokes流界面问题的收敛性证明和无散度向量场上近似投影的最大范数稳定性。许多科学问题涉及流体中的移动边界，例如一种流体在另一种流体中的一滴，或者活组织中弹性膜的运动。这类问题的数值研究具有特殊的困难，因为希望在固定网格上离散流体变量，而运动边界必须单独表示，以及它对流体运动的影响。用一种简单实用的方法对运动曲面进行离散化是很困难的。对于以粘度为主的斯托克斯流，广泛采用流体变量的积分公式。所提出的积分方法比标准方法更简单、更有效，并且对曲面的处理更少。因此，随着模型变得更加真实，它可以为三维模拟的实用性做出贡献。第二个项目强调对最大误差的估计，因为这些估计可能在接口附近最大。它们不如积分范数的估计发展得好。这些估计将用于浸没界面法、上述Navier-Stokes流分解和近似投影法等方法。对现有数值方法的误差分析应提高对其有效性和局限性的认识。它还可以显示方法中的选择如何影响其准确性，并提出改进建议。