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High-Order XFEM with Applications to Two-Phase Flows

Saxby, Ben Alexander

[Thesis]. Manchester, UK: The University of Manchester; 2014.

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Abstract

In this thesis we investigate the accuracy of high-order Extended Finite Element Methods (XFEMs) for the solution of discontinuous problems, with a view to computing high-order solutions to a two-phase flow problem.We start by demonstrating optimal exponential rates of convergence for a spectral/hp element method applied to a smooth problem. We then consider an immersed method on a fixed background mesh that uses level sets to capture the location of a discontinuity and the XFEM to characterise this discontinuity on element interiors. We present an improvement to the modified XFEM of [Moes et. al., 2003] and then use it to solve both a Poisson problem and a linear elasticity problem with discontinuities modelled independently of the mesh.Very close to optimal rates of convergence are recovered for the Poisson problem with both straight and quadratically curved interfaces for approximations up to order p=4. These rates are better than those published in the literature for the XFEM with a curved weak discontinuity, and they are also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order p>1. Almost optimal rates of convergence are then also recovered for an elastic problem with a circular discontinuity for approximations up to order p=4.The use of the XFEM for time-dependent problems is investigated, and a novel level set update method that retains the signed distance property without need for reinitialisation is also presented.Finally we apply these methods to the time-dependent simulation of a two-phase flow problem. We validate the method against both an analytic dispersion relation for relaxation under small interface perturbations and an existing implementation for large interface perturbations. We then present a proof-of-concept implementation of a high-order immersed method for an oscillating tank flow problem and demonstrate the ability of our implementation to simulate problems with large amplitude interface deformations.