Numerical Methods for Fluid-Structure Interaction Problems with Large Displacements

流体-结构相互作用（FSI）问题出现在许多应用中，例如地质力学、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，数学模型必须捕获血液与血管壁、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的“移动域”流固耦合问题，是具有挑战性的数值求解和分析。快速有效的FSI求解器对于生物工程应用是有价值的，因为数值算法与实验和临床测量的结合提供了一种创新的方法来理解心血管系统的许多组成部分的基本功能及其相互作用。PI将为大位移的非线性流固耦合问题开发一类数值方法和基本理论。研究的目的是为此类问题的算法开发和数值分析做出基础性贡献。拟议的研究将推动我们的能力边界模拟FSI问题的血液动力学，包括骨折在软组织中的传播。 本计画的目标是发展一类数值方法与基础理论，以解决大位移的非线性流固耦合问题。所提出的方法将专门设计用于解决血液动力学问题。我们将考虑弹性和多孔弹性结构，其中固体力学由超弹性本构模型描述。将开发分区和单片方法。将特别注意所提出的方法的数值分析。研究目标将通过以下具体目标来实现：目标1：使用二阶向后微分公式时间离散化和Crank-Nicolson蛙跳时间离散化，开发具有多孔超弹性结构的FSI问题的非迭代区域分解方法;目标2：开发具有厚超弹性结构的FSI问题的非迭代区域分解方法;目标3：开发和分析具有超弹性结构的FSI问题的整体相场方法。该奖项反映了NSF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而被认为值得支持。

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.