Beyond the Method of Regularized Stokeslets

正则斯托克小波法是一种有效计算不可压缩斯托克流中浸入弹性细丝或边界运动的数值方法。该方法是基于对施加在表面、曲线甚至分散点上的力所产生的流体速度表达式的非具体化。主要思想是遵循Stokes方程基本解的推导但用光滑近似代替函数。发展正则化Stokeslet的动机是双重的：(1)当力在三维表面上分布时，奇异Stokeslet是可积的，但由于积分具有非常尖的核（称为近奇异），数值方法通常难以准确计算非常接近表面的流体点的流动。正则化核提供了一种稳定的计算技术；(2)通过对Stokeslet （Stokeslet偶极子、Stokes偶极子、应力小波、罗小波等）微分，Stokes方程有更多的奇异解。它们较高的奇异性使得曲面上的积分发散，但正则化方法提供了在计算中使用这些元素的方法。提出的工作在几个方面扩展了流行的正则化stokeslet方法。它提出了正则化方法的理论，广义的定义包括高阶元素的系统推导，用于计算微生物群的高效建模。通过误差分析和对速度场的修正，解决了该方法在固体在流体中运动情况下的收敛性问题，提高了收敛速度。该理论还包括为了在计算中获得理论精度而必须满足的blob的期望性质，并提供满足这些性质的blob族。正则化Stokeslets理论扩展到其他流体模型，如多孔介质的Brinkman方程和具有透水壁的通道模型。将周期边界条件在一个、两个或三个坐标方向上的应用方法进行了扩展。该理论的进一步发展为模拟由力驱动的小尺度流动、微生物运动、靠近表面的颗粒周围流动等提供了更丰富、更完整的工具。该项目结合了新的数学理论和计算方法的发展，设计出更准确、更有效的方法来模拟微生物和纤毛周围的流体运动。该项目将改进和扩展一种流行的计算技术，称为正则化Stokeslets方法，该方法在生物流体流动领域的许多应用中非常有用。它的用途包括去除不需要的生物膜，微过滤作为去除颗粒物质的方法，了解细菌鞭毛的运动和行为，了解自我推进的微游泳者所需的力，分析精子运动和其他与人类生殖有关的问题。该方法的流行是由于它在各种各样的应用中都很有用，并且与其他方法相比，它相对容易实现。然而，正则化Stokeslets的方法可以在许多方面得到改进，比如增加新的组件来扩展其适用性，通过执行数学分析来阐明实现模型的新方法，以及通过评估新的计算方法来获得更准确或更快的结果，或者两者兼而有之。

The Method of Regularized Stokeslets is a numerical technique for the efficient computation of the motion of immersed elastic filaments or boundaries in incompressible Stokes flow. The method is based on desingularizing the expression for the fluid velocity resulting from forces applied on surfaces, curves or even scattered points. The main idea is to follow the derivation of the fundamental solution of the Stokes equations but replacing the delta function with a smooth approximation. The motivation for developing regularized Stokeslets is twofold: (1) when forces are distributed over a surface in three dimensions, the singular Stokeslet is integrable but numerical methods often have difficulty computing accurately the flow at fluid points very near the surface since the integrals have very spiky kernels (known as nearly-singular). Regularizing the kernels provides a stable computational technique; (2) the Stokes equations have more singular solutions derived from differentiating the Stokeslet (Stokes doublets, dipoles, stresslets, rotlets, etc.). Their higher singularity makes the integration over surfaces divergent but regularization methods provide a way to use these elements in computations. The proposed work extends the popular method of regularized Stokeslets in several ways. It advances the theory of regularization methods, broadly defined to include the systematic derivation of higher-order elements for the computationally efficient modeling of swarms of microorganisms. The convergence of the method in the case of solid bodies moving in the fluid is addressed based on error analysis and corrections to the velocity field to improve the convergence rate. The theory also includes desired properties that the blobs must satisfy in order to attain the theoretical accuracy in computations, and to provide families of blobs that satisfy these properties. The theory of regularized Stokeslets is expanded to apply to other fluid models, such as the Brinkman equations for porous media and models of channels with permeable walls. The work extends the methods for applications with periodic boundary conditions in one, two or three coordinate directions. The further advancement of the theory provides a richer and more complete set of tools for modeling small-scale flows driven by forces, microorganism motility, flows around particles near surfaces and more. This project combines the development of new mathematical theory and computational methods to devise more accurate and more efficient ways of simulating the fluid motion around microorganisms and cilia. The project will improve and extend a popular computational technique, called the method of regularized Stokeslets, which has been very useful in many applications in areas of biological fluid flows. Its uses include the removal of unwanted biofilms, microfiltration as a method for removing particulate matter, understanding the motion and behavior of bacterial flagella, understanding the forces required by self-propelled microswimmers, analyzing sperm motility and other issues related to human reproduction. The popularity of the method is due to its usefulness in a wide variety of applications and to the relative ease of implementation compared to other methods. However, the method of regularized Stokeslets can be improved in many ways by adding new components that will expand its applicability, by performing mathematical analyses that will shed light on new ways of implementing the model, and by assessing new computational methods that reach the results either more accurately or faster or both.