Adaptive mesh refinement in fluid-structure interaction in three-dimensions

利兹[1，2]最近的研究开发了一种基于虚拟区域方法的求解流固耦合(FSI)问题的新算法。这已经在2-D和3-D的模型问题上得到了实现和测试：前者在四边形网格上使用局部网格加密，而后者仅基于均匀八面体网格。本项目将从两个重要方面扩展当前的工作：(1)通过使用(A)网格自适应性和(B)预条件迭代求解器来提高3-D算法的效率；(2)将该方法应用于比目前所考虑的模型问题更实际的FSI应用1。3-D实现目前运行非常耗时，因此我们将从两个基本方面寻求提高效率：a.通过实现自适应网格方法。这将涉及将基础求解器(基于八面体上的Taylor-Hood元素)替换为基于等效四面体单元的求解器，然后考虑在四面体和六面体网格上使用自适应软件。最初，自适应性将由固体-流体界面的位置驱动，但随后将考虑其他方法，如高解决方案梯度或基于目标的误差估计。还可以考虑使用机器学习来指导网格的自适应性。通过研究在每个时间步迭代求解线性代数方程的有效预条件算子。这将基于利用我们对线性系统结构的知识的尝试，而不是简单地使用“黑箱”预处理器(如目前所做的那样)。它将建立在大量工作的基础上，这些工作已经开发出了用协调有限元模拟的不可压缩流动的预条件2。到目前为止，文献[1，2]中只考虑了简单的模型问题。我们将与生物医学工程研究所的主管(Marlene Mengoni博士)合作，将开发的技术应用于关节生物力学中出现的一系列具有挑战性的3-D FSI应用。这些将涉及复杂的3-D几何形状、各种固体和材料属性以及一系列外力。除了提供具有挑战性的问题来测试所开发的方法，这一合作还将支持数值技术的验证和潜在的非学术影响的重要途径。参考：[1]Wang Y；Jimack PK；Walkley MA(2017)流体-结构相互作用的单场整体虚域方法。应用力学与工程中的计算机方法，317,1146-1168页。[2]王勇；吉马克PK；Walkley MA(2019)流体-结构相互作用的单场虚拟域法的能量分析。《应用数值数学》，140,165-182页。

Recent research at Leeds [1, 2] has developed a novel algorithm for solving fluid-structure interaction (FSI) problems based upon a fictitious domain approach. This has been implemented and tested on model problems in 2-D and 3-D: the former with the use of local mesh refinement on a quadrilateral mesh and the latter based upon a uniform octahedral mesh only.This project will expand upon the current work in two significant ways: (1) enhancing the efficiency of the 3-D algorithm through the use of (a) mesh adaptivity and (b) preconditioned iterative solvers; (2) applying the method to more practical FSI applications than the model problems so far considered.1. The 3-D implementation is currently highly time-consuming to run and so we will seek to improve the efficiency in two fundamental ways:a. By implementing an adaptive mesh approach. This will involve replacing the underlying solver (based upon Taylor-Hood elements on octahedral) by one that is based on equivalent tetrahedral elements and then considering the use of adaptive software on both tetrahedral and hexahedral meshes. Initially the adaptivity will be driven by where the solid-fluid interface is situated but other approaches will then be considered, such as high solution gradients or goal-based error estimation. The use of machine learning to guide the mesh adaptivity may also be considered.b. By investigating efficient preconditioners for the iterative solution of the linear algebraic equations at each time step. These will be based upon attempts to exploit our knowledge of the structure of the linear systems rather than simply using a "black box" preconditioner (as currently done). It will build upon a substantial body of work that has developed preconditioners for incompressible flows simulated with conforming finite elements.2. So far only simple model problems have been considered in [1, 2]. We will work with a supervisor from the Institute of Biomedical Engineering (Dr Marlene Mengoni) to apply the techniques developed to a series of challenging 3-D FSI applications arising in the bio-mechanics of joints. These will involve complex 3-d geometries, a variety of solid and material properties and a range of applied forces. As well as providing challenging problems on which to test the methods developed, this collaboration will also support validation of the numerical techniques and an important pathway to potential non-academic impact.References:[1] Wang Y; Jimack PK; Walkley MA (2017) A One-Field Monolithic Fictitious Domain Method for Fluid-Structure Interactions. Computer Methods in Applied Mechanics and Engineering, 317, pp. 1146-1168.[2] Wang Y; Jimack PK; Walkley MA (2019) Energy Analysis for the One-Field Fictitious Domain Method for Fluid-Structure Interactions. Applied Numerical Mathematics, 140, pp. 165-182.