An Overlapping Domain Approach for Hemodynamics and FSI
: Partition of Unity, Benchmarks and Beyond

Student thesis: Doctoral ThesisDoctor of Philosophy


The primary objective of this thesis was to develop a new method for solving fluid flow and fluid structure interaction (FSI) in the context of large deformations. For FSI problems that incorporate large deforming solids (e.g. cardiac valves), balancing solution accuracy and computational accuracy remains a challenging aspect which continues to drive interest into the development of new methods. In this context, overlapping do-main (domain decomposition) fluid problem solvers having been proposed as a way to combine the advantages of conforming fluid-solid interfaces, found in interface-tracking methods, and the lack or reduction of element distortion, seen in interface-capturing techniques. However, the stability of these approaches is typically conditioned by the choice of problem-specific stabilization parameters.
Driven to eliminate the need for these stabilization terms, the objective of this thesis is the development of a novel technique for solving incompressible flow, based on the partition of unity finite element method (PUFEM). The crux of this approach is to define the solution spaces as weighted sums of two local fields, background and embedded, each supported by different, unstructured and non-conforming grids. The method is initially implemented in a high-level language in the case of 2D flow problems, using inf-sup stable elements. The method is shown to be accurate and robust through extensive testing, ranging from quasi-static Stokes and Navier-Stokes problems to transient arbitrary Lagrangian-Eulerian (ALE) and FSI applications.
To perform a convergence analysis in the FSI setup, a novel class of benchmark problems was created. The new class of analytical solutions was defined for pulsatile flow inside a channel or tube with elastic walls to verify accurate implementation as well as analyse spatiotemporal convergence of the FSI solver. The class, consisting of 16 different cases and based on different permutations of 2D and 3D domains, quasi-static and transient behaviour, linear and non-linear constitutive laws, displays a wide range of solution complexity and allows for progressive testing of the code. A benchmark problem was also developed to verify the vWERP method within a moving domain, used to estimate pressure differences within the ventricles from 4D flow images. The CFD benchmark problem was based on a simple spherical geometry with analytical domain deformations.
In order to extend PUFEM to clinical applications, the method was implemented into CHeart, an in-house multi-physics FEM solver with a parallel software framework. The implementation allows for simulation of 3D problems as well as the use of a stabilized form of the Navier-Stokes equation in order to simulate flow with high Reynolds numbers, as seen in the heart and cardiovascular system. A number of implementation strategies were explored to improve solver efficiency.
Date of Award1 Sept 2021
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorDavid Nordsletten (Supervisor) & Reza Razavi (Supervisor)

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