AbstractThe 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 Award||1 Sept 2021|
|Supervisor||David Nordsletten (Supervisor) & Reza Razavi (Supervisor)|