Collaborative Research: Efficient surface-based numerical methods for 3D interfacial flow with surface tension

研究人员开发并应用了三维流动中界面运动的有效边界积分方法。这些方法解决了三维流动中具有表面张力或弹性力的流体界面数值计算中的一个重大困难。这种力在演化方程中引入了高阶（即高导数）项，这导致了显式时间积分方法的严重稳定性约束或“刚度”。此外，高阶项出现在非线性和非局部算子中，使得稳定隐式方法的有效应用变得困难。研究人员的方法依赖于使用表面的第一和第二基本系数作为动力变量，并采用了一种特殊的界面参数化方法，并结合了对小尺度控制方程的分析。这使得隐式时间积分方法在三维流中的有效应用成为可能。研究人员将该方法应用于无粘流体的典型界面问题，包括Kelvin-Helmholtz、Rayleigh-Taylor和水波问题，并研究了无粘流动中不可扩展弹性片和三维粘性流动中的囊泡的动力学。最重要的是，他们开发了一种使用区域分解或重叠坐标块来描述界面的数值方法。这样做的额外好处是为实现光谱精确和空间自适应方法提供了一个框架。移动边界问题出现在许多不同的领域，例如流体动力学、材料科学和生物学。具体的例子包括移动的海浪、生长的癌症肿瘤、跳动的心脏以及移动的细胞和有机体。研究人员开发了精确和有效的“边界积分”数值方法模拟移动边界的应用。边界积分法是模拟运动界面最精确的数值方法之一，但当界面受到表面张力或弹性力的作用时，边界积分法往往效率低下。发展具有表面张力或弹性力的三维界面流动的快速、准确的边界积分方法将对理解现有的应用和进一步发展技术有很大的好处。

The investigators develop and apply efficient boundary integral methods for the motion of interfaces in 3D flow. The methods address a significant difficulty in the numerical computation of fluid interfaces with surface tension or elastic forces in 3D flow. Such forces introduce high order (i.e., high derivative) terms into the evolution equations, which lead to severe stability constraints or `stiffness' for explicit time-integration methods. Furthermore, the high order terms appear in nonlinear and nonlocal operators, making the efficient application of stable implicit methods difficult.The investigators' method relies on using the first and second fundamental coefficients of the surface as dynamical variables, and employs a special parameterization of the interface combined with an analysis of the governing equations at small scales. This enables the efficient application of implicit time-integration methods for 3D flow. The investigators implement the method in canonical interface problems for inviscid fluids, including the Kelvin-Helmholtz, Rayleigh-Taylor, and water wave problems, and study the dynamics of inextensible elastic sheets in inviscid flow and vesicles in 3D viscous flow. Most importantly, they develop a version of the numerical method which uses domain decomposition or overlapping coordinate patches to describe the interface. This has the added benefit of providing a framework for the implementation of spectrally accurate and spatially adaptive methods.Moving boundary problems occur in many diverse areas in, for example, fluid dynamics, materials science, and biology. Specific examples include traveling ocean waves, growing cancer tumors, beating hearts and moving cells and organisms. The investigators develop accurate and efficient `boundary integral'numerical methods for the simulation of moving boundaries in applications.Boundary integral methods are among the most accurate numerical methods for the simulation of moving interfaces, but are often inefficient when the interface is acted on by surface tension or elastic forces. The development of fast and accurate boundary integral methods for 3D interfacial flow with surface tension or elastic forces will be of great benefit in understanding existing applications and developing technology further.