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Title: Optimisation of the PFC functional
Author: Bignold, Simon M.
ISNI:       0000 0004 5921 9402
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2016
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In this thesis we develop and analyse gradient-fl ow type algorithms for minimising the Phase Field Crystal (PFC) functional. The PFC model was introduced by Elder et al [EKHG02] as a simple method for crystal simulation over long time-scales. The PFC model has been used to simulate many physical phenomena including liquid-solid transitions, grain boundaries, dislocations and stacking faults and is an area of active physics and numerical analysis research. We consider three continuous gradient fl ows for the PFC functional, the L2-, H-1- and H2-gradient fl ows. The H-1-gradient flow, known as the PFC equation, is the typical flow used for the PFC model. The L2-gradient flow is known as the Swift-Hohenberg equation. The H2-gradient ow appears to be a novel feature of this thesis and will motivate our development of a line search algorithm. We analyse two methods of time discretisation for our gradient fl ows. Firstly, we develop a steepest descent algorithm based on the H2-gradient fl ow. We further develop a convex-concave splitting of the PFC functional, recently proposed by Elsey and Wirth [EW13], to discretise the L2- and H-1-gradient flows. We are able to prove energy stability of both our steepest descent algorithm and the convex-concave splitting scheme of [EW13]. We then use the Lojasiewicz gradient inequality (first developed in [ Loj62]) to prove that all three schemes converge to equilibrium. For numerical simulations we undertake spatial discretisation of our schemes using Fourier spectral methods. We consider a number of implementation issues for our fully discrete algorithms including a striking issue that occurs when the number of spatial grid points is low. We then perform several numerical tests which indicate that our new steepest descent algorithm performs well compared with the schemes of [EW13] and even compared with a Newton type scheme (the trust region method).
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics