The Library
Optimisation of the PFC functional
Tools
Bignold, Simon M. (2016) Optimisation of the PFC functional. PhD thesis, University of Warwick.
|
PDF
WRAP_Theses_Bignold_2016.pdf - Submitted Version - Requires a PDF viewer. Download (15Mb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b2876074~S1
Abstract
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).
Item Type: | Thesis (PhD) | ||||
---|---|---|---|---|---|
Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Crystals -- Simulation methods | ||||
Official Date: | June 2016 | ||||
Dates: |
|
||||
Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Ortner, Christoph ; Elliott, Charlie | ||||
Extent: | ix, 178 leaves ; illustrations (colour), charts | ||||
Language: | eng |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year