Since 2024

Stochastic Surface PDEs

With Paola Pozzi

Motivated by applications in cell biology we study the numerical approximation of PDEs on evolving surfaces with stochastic sources and reactions. There are conceptual questions related to defining the noise terms on an evolving manifold. Surface finite elements are used for the spatial approximation, whilst a variety of time stepping schemes is investigated. As for stochastic PDEs on flat domains, the regularity of solutions limits the convergence properties.

Publications

A surface finite element scheme for a stochastic PDE on an evolving curve preprint

Since 2023

Health Monitoring of Fistulas

With Yongmann Chung and Risa Romy

Patients on haemodialysis due to chronic kidney disease rely on vascular access via arteriovenous fistulas. Unfortunately, their function deteriorates over time until they fail. The underlying causes are still not fully understood. In this project we monitor fistula health using ultrasound imaging and computational fluid dynamics modelling. Some haemodynamic characteristics of fistulas are known to be related to their detoriation thanks to other imaging techniques but which are expensive. Ultrasound images are much cheaper but, in turn, suffer from significantly more noise and artefacts.
One objective is to filter these for that the blood vessel geometry can be extracted with sufficient accuracy. A second objective is to develop an effective computational pipeline from clinical image data to key features.

Funding

Novel fistula health monitoring for dialysis patients (PhD student) Kidney Research UK

Since 2006

Diffuse Interface and Domain Approximation

With Andrew (Kei Fong) Lam, Luke Benfield, Matt Collins, Andreas Dedner

The subject of this project are coupled systems of balance equations on moving interfaces and adjacent bulk domains. The goal has been the modelling and numerical approximation if the interface is represented in terms of a phase field variable, which itself might not be given but subject to an evolution equation again. An immediate question is whether the equations in the limit as the interface thickness tends to zero are correctly recovered. But also other useful properties such as natural energetic properties and other advantageous structure for numerical methods ideally are preserved.
More recently, the focus has been on automatising the process of formulating a diffuse interface apperoximation and numerically solving the thus smoothed problem with standard adaptive finite element and suitable time stepping schemes. Specifically, the Unified Form Language has been used to transform free boundary problems into fully discrete variational formulations and to feed them into the Distributed Unified Numerics Environment.

Publications

Analysis of the diffuse domain approach for a bulk-surface coupled PDE system doi
Numerical computation of advection and diffusion on evolving diffuse interfaces doi
Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface doi

Funding

Doctoral Training Allowances (PhD student) EPSRC/UKRI
MASDOC CDT (PhD student) EPSRC
Phase field modelling of two-phase flow with surfactants (postdoc) DFG

2015 - 2023

Finite Element Approximation of Geometric Evolutions

With Paola Pozzi

Finite element methods for geometric evolution equations and related problems have been a very active research area since the pioneering work by G Dziuk on mean curvature flow. We have been focusing on curves and studying various extensions, such as the coupling with a balance equation on the curve, triods, and fourth order evolutions. Of particular interest are schemes that augment the geometric evolution in normal direction with a suitable tangential movement such that a good mesh quality is maintained, thus enabling long term computations.

Publications

Convergence of a scheme for elastic flow with tangential mesh movement doi
On motion by curvature of a network with a triple junction> doi
Elastic flow interacting with a lateral diffusion process: The one-dimensional graph case doi
Curve shortening flow coupled to lateral diffusion doi

2011 - 2020

Moving Boundary Problems in Cell Biology

With Chandrasekhar Venkataraman, Charlie Elliott, Andreas Dedner, Adam Nixon, Till Bretschneider

Thanks to changing geometries and a complicated, largely unknown biochemistry cells are a attractive topic for continuum modelling. Typically, equations for the mechanics that describe the movement are coupled with reaction-advection-diffusion equations within the cell, on its boundary, and possibly even outside of it. Dimensionally reduced approaches focuse on the cell boundary and computatinally model it with a moving mesh. Specifically, cell movement and cell blebbing have been investigated.

Publications

Mathematical modelling in cell migration: tackling biochemistry in changing geometrie doi
A finite element method for a fourth order surface equation with application to the onset of cell blebbing doi
Parameter identification problems in the modelling of cell motility doi
Modelling cell motility and chemotaxis with evolving surface finite elements doi

Funding

MASDOC CDT (PhD student) EPSRC
Numerical analysis and computation for partial differential equations on surfaces (Postdoc) EPSRC

2006-2019

Diffuse Interface Models for Surfactants

With Andrew (Kei Fong) Lam, Oliver Dunbar

Surface active agents (surfactants) at interfaces between two fluid phases influence the surface tension and, hence, the motion of the surface. Multi-phase flow can be modelled with a Navier-Stokes-Cahn-Hilliard system where obstacle potentials lead to variational inequalities. Models allowing for general isotherms (relating the surface concentration of the surfactant to the concentration in the adjacent fluids) and equations of state (relating the surface tension to the surfactant concentration) are derived and analysed. In the case of more than two fluid phases suitable conditions around the triple junctions have to be recovered.

Publications

Phase field modelling of surfactants in multiphase flow doi
Diffuse interface modelling of soluble surfactants in two-phase flow doi

Funding

MASDOC CDT (two PhD students) EPSRC
Phase field modelling of two-phase flow with surfactants (postdoc) DFG

2006-2011

Phase Separation on Elastic Biological Membranes

With Charlie Elliott

Lipid bilayers are the basic component of cell boundaries and consist of multiple lipids that may decompose forming two different domains with different viscoelastic properties. In order to contribute to understanding this lateral phase separation we aim for computing closed equilibrium two-phase shapes by means of an appropriate gradient flow dynamics. The membrane energy consists of an elastic bending energy and an excess free energy emerging from the phase interfaces. The idea was to couple a geometric evolution equation for the membrane to a phase separation equation on the moving membrane in such a way that the energy decays in time. Numerically, isoparametric surface finite elements on triangulated surfaces were the method of choice.

Publications

Modeling and computation of two phase geometric biomembranes using surface finite elements doi
A surface phase field model for two-phase biological membranes doi
Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements doi

Funding

Numerical analysis and computation for partial differential equations on surfaces (Postdoc) EPSRC

2001-2007

Phase Field Modelling of Alloy Solidification

With Harald Garcke, Britta Nestler

The microstructures in metallic alloys are of great importance for the mechanical properties such as tensile and shear strength. It is a result of the production process. We have developed a general phase field model allowing for arbitrary numbers of components and phases. By matched asymptotic expansions the model can be related to a classical model with moving phase boundaries as the diffuse interface thickness tends to zero. For the case of two phases the approximation property could even be improved. Choices for the free energies of the possible phases that are motivated by the material properties lead to nonlinearities with growth properties that are analytically challenging to handle. Subject to suitable assumptions on the parameters the system of nonlinear parabolic differential equations has solutions existence, and in some cases even uniqueness could be proved.

Publications

Weak solutions to a multi-phase field system of parabolic equations related to alloy solidification preprint
Second order phase field asymptotics for multi-component systems doi
Multicomponent alloy solidification: Phase-field modeling and simulations doi
A diffuse interface model for alloys with multiple components and phases doi

Funding

Analysis, modelling and simulation of multi-scale, multi-phase solidification in alloy systems (PhD student) DFG