Numerical Methods for Fluid-Structure Interaction Problems with Large Displacements

流固相互作用（FSI）问题出现在许多应用中，如地质力学、空气动力学和血流动力学（血液动力学）。在血液动力学应用中，数学模型必须捕捉血液与血管壁、软组织或心肌的弹性结构动力学之间的非线性耦合。这些结构动力学产生了“移动域”FSI问题，这对数值解决和分析具有挑战性。快速高效的FSI求解器对于生物工程应用非常有价值，因为数值算法与实验和临床测量相结合，为理解心血管系统许多组成部分的基本功能及其相互作用提供了一种创新的方法。PI将为具有大位移的非线性FSI问题开发一类数值方法和基础理论。本研究旨在为此类问题的算法和数值分析的发展做出根本性贡献。拟议的研究将推动我们在血流动力学中模拟FSI问题的能力，包括软组织中的骨折扩展。这个项目的目标是发展一类数值方法和基础理论来解决大位移的非线性FSI问题。建议的方法将专门为血流动力学引起的问题而设计。我们将考虑弹性和孔弹性结构，其中固体力学是由超弹性本构模型描述的。将开发分区方法和整体方法。将特别注意所提出方法的数值分析。目标1：利用二阶后向微分公式时间离散化和Crank-Nicolson Leapfrog时间离散化技术，发展具有多孔超弹性结构的FSI问题的非迭代域分解方法；目标2：发展非迭代的区域分解方法，用于厚超弹性结构的FSI问题；目标3：针对超弹性结构的FSI问题开发和分析单片相场方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Fluid-structure interaction (FSI) problems arise in many applications, such as geomechanics, aerodynamics, and blood flow dynamics (hemodynamics). In hemodynamic applications, mathematical models must capture the non-linear coupling between blood and the elastic structural dynamics of vessel walls, soft tissue, or cardiac muscles. These structural dynamics create 'moving domain' FSI problems that are challenging to numerically solve and analyze. Fast and efficient FSI solvers are valuable for bioengineering applications since the combination of numerical algorithms with experimental and clinical measurements provides an innovative approach to understanding the basic function of many components of the cardiovascular system and their mutual interaction. The PI will develop a class of numerical methods and underlying theory for non-linear FSI problems with large displacements. The research aims at making fundamental contributions to development of algorithms and numerical analysis of such problems. The proposed research will push the boundaries of our ability to model FSI problems in hemodynamics, including fracture propagation in soft tissue. The goal of this project is the development of a class of numerical methods and underlying theory for solving non-linear FSI problems with large displacements. Proposed methods will be specially designed for problems arising from hemodynamics. We will consider elastic and poroelastic structures where solid mechanics are described by hyperelastic constitutive models. Both partitioned and monolithic methods will be developed. Special attention will be given to numerical analysis of the proposed methods. The research goals will be achieved through the following specific aims: Aim 1: Development of noniterative, domain decomposition methods for FSI problems with porohyperelastic structures using secondorder Backward Differentiation Formula time discretization and the Crank-Nicolson Leapfrog time discretization; Aim 2: Development of non-iterative, domain decomposition methods for FSI problems with thick, hyperelastic structures; and Aim 3: Development and analysis of a monolithic, phase-field approach for FSI problems with hyperelastic structures.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.