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

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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.