Numerical Analysis and Methods for Fluid Deformable Surfaces and Their Interaction with the Bulk

变形的流体表面在细胞和组织生物学以及乳液和泡沫的建模中是普遍存在的。 计算机建模在更好地理解涉及这些表面以及界面现象的过程中起着越来越重要的作用。本项目的目的是开发精确和可靠的数值方法，用于模拟流体可变形表面及其与主体的相互作用。这种方法的发展将有助于理解脂质双层，肌动蛋白皮质，上皮细胞片的功能，以及其他薄结构表现出面内粘度和横向流动性的特性。该项目为一名研究生提供研究训练。该项目将考虑表面上的流体模型。对于这些，基于连续性的建模导致在时间依赖流形上提出的偏微分方程系统。例如，脂质膜是流体薄层，其可以被建模为具有弯曲弹性的二维粘性表面流体。该项目将为可变形表面上的流体系统开发和分析几何非拟合有限元方法，如表面斯托克斯和表面纳维尔-斯托克斯方程、与平面内弹性耦合的切向流体方程和控制平面外（几何）运动的方程，以及界面-体积耦合流体系统。该项目的重点是分裂计划产生的耦合系统。该方法建立在制定的控制方程的切向微分，并采用时间无关的不适合的背景网格。该方法将允许隐含给定的复杂形状，可能会经历拓扑transitions.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Fluid surfaces that deform are ubiquitous in cell and tissue biology as well as in modeling of emulsions and foams. Computer modeling plays an increasingly important role in better understanding of processes involving such surfaces as well as interfacial phenomena. The present project aims to develop accurate and reliable numerical methods for the simulation of fluidic deformable surfaces and their interaction with the bulk. Development of such methods will facilitate understanding the functionality of lipid bilayers, the actin cortex, epithelial cell sheets, and the properties of other thin structures exhibiting in-plane viscosity and lateral mobility. The project provides research training for a graduate student.The project will consider models of fluids on surfaces. For these, continuum-based modeling leads to systems of partial differential equations posed on time-dependent manifolds. For example, lipid membranes are fluidic thin layers that can be modeled as two-dimensional viscous surface fluids with bending elasticity. The project will develop and analyze a geometrically unfitted finite element method for fluid systems posed on deformable surfaces such as surface Stokes and surface Navier-Stokes equations, tangential fluid equations coupled with in-plane elasticity and equations governing out-of-plane (geometrical) motions, as well as interface-bulk coupled fluid systems. The project focus is on splitting schemes for the resulting coupled systems. The methods build on formulation of governing equations in terms of tangential differential calculus and employ time-independent unfitted background meshes. The approach will allow for implicitly given complex shapes that may undergo topological transitions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.