Evolution phenomena on time dependent domains arise in various physical and biological applications. An abstract framework using time dependent Hilbert and Banach spaces has been developed that enables to study well-posedness of such problems [15]. Some applications involving moving hypersurfaces are discussed in [18].
Time dependence may enter abstract spaces also in ways other than via an evolving domain. Open questions concern problems that can be formulated in weighted Sobolev spaces where the weights are time dependent.
One PhD student has been working on this topic, funded by the EPSRC via the MASDOC CDT.
In [12], a general computational approach to cell motility is presented. It is based on a geometric evolution law for the cell boundary coupled to a system of reaction-advection-diffusion equations on the cell boundary modelling the involved chemistry. The lower computational cost is the main motivation for the dimensional reduction to a purely surface based model.
In a cooperation with the Warwick Medical School and the Department of SystemsBiology we investigated the quantification of this reduced approach [17]. Developing ways to compare with experimental measurements and solving the related optimisation (fitting) problems lies at the heart of the project.
One post-doc worked on this project who was internally funded by the University of Warwick in terms of the Warwick Impact Fund. Also, one PhD student is working on this topic, funded by the EPSRC via the MASDOC CDT.
One focus lies on discontinuous Galerkin methods which may prove advantageous
in related projects (cell motility, surface active agents) since equations
there may be advection dominated.
A first publication on linearly approximated stationary surfaces [13] was
followed by a second one on higher order approximations [16].
Related to geometric flows (as featuring in the cell motility project), mixed finite element methods on surfaces are studied. Regarding results for systems coupling geometric equations with surface PDEs, a convergence result has been obtained recently. Another interesting topic are self-repulsive energies that serve to avoid self-intersection of moving curves or surfaces.
Surface active agents (surfactants) at interfaces between two fluid phases influence the surface tension and, hence, the motion of the surface. Two-phase flow can be modelled with a Navier-Stokes-Cahn-Hilliard system where double 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) have been derived and analysed [14].
Current investigations concern the extensions to more than two phases where the focus lies on the conditions around the triple junctions.
One PhD student has finished work on analytical and modelling questions. Another one has started to develop and investigate efficient numerical schemes. The work has been funded by the Engineering and Physical Sciences Research Council (EPSRC) via the MASDOC CDT. Previously, the project was also supported by the German Research Foundation (DFG).
Subject of this project have been advection-diffusion processes of conserved surface quantities on moving surfaces. The goal has been the modelling and numerical approximation if the interface is represented by a thin layer. Such a situation occurs, for instance, if the phase field methodology is used to tackle the free boundary problem describing the surface. Most importantly one needs to ensure that the equations in the limit as the interface thickness tends to zero are correctly recovered. But also other useful properties such as thermodynamic consistency or an advantageous structure should be preserved when defining the governing equations in the layer.
Early work has been on an approach to surface PDEs on curves [7] for which also a numerical schemes has been developed that can deal with the degeneracy [8]. More recently, a coupled bulk-surface PDE system could be tackled with significant extensions of the previous ideas in the stationary case (to appear). Ongoing work concerns the parabolic case including evolving surfaces and domains.
The project was supported by the German Research Foundation (DFG).
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 is to couple a geometric evolution equation for the membrane to a pde on the membrane for the phase separation in such a way that the energy decays in time [10].
Numerically, isoparametric surface finite elements on triangulated surfaces are the method of choice. The discretisation of the geometric evolution equation involves an equation for the vertex positions which yields an evolution of the triangulated surface [9,11].
The project was financed by the Engineering and Physical Sciences Research Council (EPSRC) and was previously by the German Research Foundation (DFG).
Using the phase field methodology, clusters of bubbles with possibly anisotropic surface energies and tilings may be computd by minimisation of interfacial energy. This work was based on te preceeding project on alloy solidification. Instead of equations for the alloy components, constraints on the volumes of the phases are considered.
The use of obstacle potentials leading to a variational inequality made the development and analysis of a numerical method to realise the constraints interesting. Using ideas from optimisation a semi-smooth Newton method has been investigated [5,6].
The structures on the scale of some microns in castings of metallic alloys are of great importance for the mechanical properties as tensile and shear strength but also for electrical properties. As the microstructure is a result of the production process our aim has been to develop, analyse, and numerically approximate models describing the solidification of alloys.
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 [1,2]. For the case of two phases the approximation property could even be improved. Taking higher order correction terms in the phase field model into account the approximation could be shown to be of second order in the small length parameter with which the interface thickness scales [3].
Thermodynamically motivated choices for the free energies of the possible phases lead to nonlinearities with growth properties that are analytically challenging to handle. Under suitable assumptions on diffusivities and other parameters entering the system of nonlinear parabolic differential equations existence and uniqueness of weak solution could be shown [4].
The project analysis, modelling and simulation of multi-scale, multi-phase solidification in alloy systems was funded by the German Research Foundation (DFG).