Fast Integral Methods for Adaptive Incompressible Flow Simulations

这项研究致力于发展和实现基于快速多极子方法(FMM)的线性方程积分方程法，用于不可压缩流动的自适应有限差分方法。这门研究课程的科学目标是为研究不可压缩流动系统中的现象提供新的方法，而现有方法不容易接近这些现象。对于不可压缩流动系统，散度约束需要求解整体椭圆型方程，这大大增加了模拟的复杂性和计算量。许多流体系统还包含大范围的相关长度尺度，这就需要使用空间和时间自适应方法。在适用的情况下，积分方程法提供了高效、自适应、高阶方法的可能性，这些方法很容易适用于复杂的计算几何，并且高度并行。FMM将发展三个不同的应用，并与自适应流体解算器相结合：求解泊松方程，求解表面拉普拉斯方程，以及脉冲位势的快速求和，每个应用都在特定的物理问题的背景下。Boussinesq对流的数值研究中，强局域锋面的发展，需要求解标准的泊松方程。模拟两个流体系统中准地转热标量波和内波的锋面形成，需要一个等价于表面拉普拉斯算子的拟微分算子求逆。在不可压缩流动中模拟柔性薄膜的一种新方法要求在所有网格点上计算膜力，并求解泊松方程以加强不可压缩性。对于所有这些物理问题，自适应方法提供了研究用均匀网格方法无法接近的问题的机会。拟议的研究可能影响的重要流体流动应用的范围是相当不同的。例如，模拟海洋或气候，预测石油回收或地下污染物流动，模拟燃烧或核反应，以及模拟心脏或肾脏等器官中的血液流动。对于这些类型的应用，线性方程的求解代表了整个计算机模型中计算最密集的部分。使用积分方程法来求解这些模型中的线性方程，与目前的大多数计算机模拟相比，是一个明显的变化。这些方程求解效率的任何提高，都将直接转化为科学家运行更大、更准确模型的能力。使积分方程法成为更标准方法的一种有吸引力的替代方法所需的数学和计算技术直到最近10年才得到充分发展。由于这些方法的复杂性和新颖性，在涉及流体流动的应用中开发它们的力量的工作很少。将作为新方法的测试案例进行的应用代表了当前数值技术不足以回答科学家感兴趣的基本问题的问题。开发的算法也将作为未来适用于更复杂问题的方法的垫脚石。

The proposed research is dedicated to the development and implementation of integral equation methods for linear equations based on the Fast Multipole Method (FMM) for use with adaptive finite difference methods for incompressible flow. The scientific objective of this course of research is to produce new methods for the study of phenomena in incompressible flow systems which are not easily approached with existing methods. For incompressible flow systems, the divergence constraint requires the solution of global elliptic equations which greatly increases the complexity and computational cost of simulation. Many fluid systems also contain a large range of relevant length scales which necessitates the use of spatially and temporally adaptive methods. When applicable, integral equation methods offer the possibility of efficient, adaptive, high-order methods which are readily applicable to complex computational geometries and are also highly parallel. Three distinct applications of FMM will be developed and coupled with adaptive fluid solvers:solution of the Poisson equation, solution of the surface Laplacian,and the fast summation of impulse potentials, each in the context of aspecific physical problem. The numerical study of Boussinesqconvection, in which strong localized fronts develop, requires that astandard Poisson equation be solved. Simulating front formation ofthe quasigeostrophic thermal scalar and internal waves in two fluidsystems requires that a psuedo-differential operator equivalent to asurface Laplacian be inverted. A new approach to modeling thinflexible membranes in incompressible flows requires that membraneforces be evaluated at all grid points as well as a Poisson equationbe solved to enforce incompressibility. For all of these physicalproblems, adaptive methods offer the opportunity to study problemswhich are unapproachable with uniform mesh methods.The range of important fluid flow applications which the proposedresearch could impact is quite diverse. Examples include modeling theocean or climate, predicting oil recovery or contaminant flows in theground, simulating combustion or nuclear reactions, and modeling theflow of blood in organs like the heart or kidneys. For these types ofapplications, the solution of linear equations represents the mostcomputationally intensive part of the overall computer model. The useof integral equation methods to solve the linear equations withinthese models represents a distinct change from the majority of currentcomputer simulations. Any increase in the efficiency in which theseequations can be solved translates directly into the ability forscientists to run larger and more accurate models. The mathematical and computational techniques necessary to make integralequation methods an attractive alternative to more standard approacheshas only been fully developed in the last 10 years. Because of thecomplexity and newness of these methods, little work has been done toexploit their power in applications involving fluid flow. Theapplications that will be pursued as test cases for the new methodsrepresent problems for which current numerical techniques areinadequate for answering fundamental questions of interest toscientists. The algorithms developed will also serve as a steppingstone for future methods applicable to more complicated problems.