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剂和泡沫的建模中普遍存在。计算机模拟在更好地理解涉及这类表面以及界面现象的过程中发挥着越来越重要的作用。本项目旨在开发准确可靠的数值方法来模拟流体可变形表面及其与块体的相互作用。这些方法的发展将有助于理解脂双层、肌动蛋白皮质、上皮细胞膜的功能，以及其他具有面内粘度和侧向流动性的薄结构的特性。该项目为研究生提供研究培训。该项目将考虑表面上的流体模型。对于这些，基于连续介质的建模导致了在依赖时间的流形上提出的偏微分方程组。例如，脂膜是流体薄层，可以模拟为具有弯曲弹性的二维粘性表面流体。该项目将开发和分析可变形表面上流体系统的几何不匹配有限元方法，如表面Stokes和表面Navier-Stokes方程，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-整体耦合流体系统。该项目的重点是为产生的耦合系统的分裂方案。该方法建立在切向微积分形式的控制方程的基础上，并使用与时间无关的不拟合背景网格。该方法将允许隐含地给定可能经历拓扑转换的复杂形状。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.