Collaborative Research: Fast High-Order Methods for Vesicle-Fluid and Membrane-Fluid Interaction and Adhesion

研究人员开发了数值算法，用于模拟不可压缩可变形膜与斯托克斯流体、其他膜和刚性壁的相互作用。这项工作的目标是实现对由大量相互作用的小泡组成的系统的高精度快速模拟。所研究的数值方法基于边界积分公式，该公式只使用定义在表面上的场，因此不需要对3D空间进行离散，这大大简化了对可变形边界的模拟。对于由边界积分公式离散化和线性化得到的稠密线性方程组，使用研究者先前发展的核无关快速多极子算法的扩展来求解。工作包括两个主要方向：基于网格和局部谱基上的构造性流形结构的可变形曲面离散格式，以及边界积分和膜-流体相互作用的线性和非线性求解器。研究人员的目标是创造有效的计算工具来解决各种生物和生物医学背景下出现的问题。潜在的应用包括模拟血细胞在血液流动中的行为，在液体中形成膜，以及设计基于囊泡的靶向药物输送机制。针对这类问题的有效工具的开发使测试涉及复杂边界、大量变形细胞或囊泡的场景成为可能。

The investigators develop numerical algorithms for simulation of incompressible deformable membranes interacting with Stokesian fluids, other membranes and rigid walls. The goal of this work is to enable high-accuracy fast simulations of systems consisting of large numbers of interacting vesicles. Investigated numerical methods are based on a boundary integral formulation which only uses fields defined on the surface and therefore eliminates the need for discretization of 3D space, which considerably simplifies simulation of deformable boundaries. The dense linear systems of equations resulting from discretization and linearization of the boundary integral formulation are solved using extensions of kernel-independent fast multipole algorithms previously developed by the investigators. The work includes two main directions: discretization schemes for deformable surfaces, based on constructive manifold structures on meshes and localized spectral bases, and boundary integral and membrane-fluid interaction linear and nonlinear solvers. The investigators aim to create efficient computational tools for solving problems arising in a variety of biological and biomedical contexts. Potential applications include simulation of blood cell behavior in blood flow, formation of membranes in a fluid and design of targeted drug delivery mechanisms based on vesicles. Development of efficient tools for such problems makes it possible to test scenarios involving complex boundaries, large numbers of deforming cells or vesicles.