The Library
Particles and biomembranes : a variational PDE approach.
Tools
Hobbs, Graham (2016) Particles and biomembranes : a variational PDE approach. PhD thesis, University of Warwick.
|
PDF
WRAP_Theses_Hobbs_2016.pdf - Submitted Version - Requires a PDF viewer. Download (2862Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3072653~S15
Abstract
We examine mathematical models for small deformations of membranes. First we review physically well-established models, posed in the Monge gauge, from a mathematical perspective. We produce a variational framework in which well posedness can be studied and finite element methods applied. The methods are used to investigate the effects of point forces, point displacement constraints and point curvature constraints. Such models are suitable for the study of deformations induced by filaments contained in the cell cytoskeleton and by embedded protein inclusions. In particular we study the membrane mediated interactions between filaments and also between inclusions.
We then introduce a new linearised model which describes small deformations of closed surfaces that are minimisers of Helfrich-type energies. The deformed surface is described as a graph over the Helfrich minimising undeformed surface. This is the natural generalisation of the Monge gauge to initially curved surfaces. We focus on a Willmore energy which gives rise to spheres and a family of tori as undeformed surfaces and also introduce surface tension on a sphere. Again we study deformations induced by filaments. A variational formulation is produced which is similar to the Monge gauge case and we formulate a numerical method to study membrane mediated interactions.
Finally we introduce an abstract splitting method which allows a high order PDE to be solved by an equivalent system of lower order equations. We give conditions which ensure well posedness of the system and produce a finite element method whose solution converges to the solution of the full system. The theory is applied to show convergence for the numerical methods used for the surface deformations model. We provide examples which show the theoretical error estimates are achieved.
Item Type: | Thesis (PhD) | ||||
---|---|---|---|---|---|
Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Membranes (Biology) -- Mathematical models, Finite element method, Differential equations, Partial | ||||
Official Date: | September 2016 | ||||
Dates: |
|
||||
Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Elliott, Charles M. | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | viii, 175 leaves : illustrations | ||||
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