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

研究人员开发和应用有效的边界积分方法的运动的接口在三维流动。 该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的数值计算中的一个重大困难。这种力引入高阶（即，高导数）项的演化方程，这导致严重的稳定性约束或'刚度'显式时间积分方法。 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。研究者的方法依赖于使用表面的第一和第二基本系数作为动力学变量，并采用一种特殊的界面参数化结合小尺度下控制方程的分析。这使得隐式时间积分方法在三维流动中的有效应用成为可能。研究人员在无粘流体的典型界面问题中实现了该方法，包括Kelvin-Helmholtz，Rayleigh-Taylor和水波问题，并研究了无粘流中不可伸展弹性片材和3D粘性流中囊泡的动力学。最重要的是，他们开发了一个版本的数值方法，使用区域分解或重叠坐标补丁来描述界面。这有一个额外的好处，提供了一个框架，实现光谱精确和空间自适应的方法。移动边界问题发生在许多不同的领域，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，跳动的心脏和移动的细胞和有机体。边界积分法是模拟运动界面的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，边界积分法的计算效率往往很低。 发展快速、精确的三维界面流边界积分方法，对于理解已有的应用和进一步发展技术将是非常有益的。

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.