Mini Symposium on Computational Methods for Interfaces and Surfaces
Organiser: Charlie Elliott
Friday 3 July 2009

Speakers

• Lubomir Banas (Heriot-Watt) Finite element approximation of a phase field model for multi-fluid incompressible magnetohydrodynamics
  

We propose an implicit finite element method for a phase field approximation of time dependent multi-fluid incompressible magnetohydrodynamics. The proposed discretization satisfies a discrete energy law. We discuss computational aspects of the method and present some numerical experiments.

• Martin Rumpf (Bonn) Geodesics in Shape Space via Variational Time Discretization

A variational approach to defining geodesics in the space of implicitly described shapes is introduced in this paper. The proposed framework is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1-1 property of the induced flow in shape space. For decreasing time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the shape space. Considering shapes as boundary contours, the proposed shape metric is identical to a physical dissipation in a viscous fluid model of optimal transportation. If the pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, for decreasing time step size one obtains an additional optical flow term controlling the transport of the shape by the underlying motion field. The implementation of the proposed approach is based on a level set representation of shapes, which allows topological transitions along the geodesic path. For the spatial discretization a finite element approximation is employed both for the pairwise deformations and for the level set representation. The numerical relaxation of the energy is performed via an efficient multi--scale procedure in space and time. Examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach.

• Bjorn Stinner (Warwick) On a Surface Finite Element Method for Biomembranes with Lipid Decomposition
  

Bilayers consisting of lipids molecules are the basic component of cell membranes. Vesicles formed from such biomembranes show a variety of interesting shapes that can be explained by its elastic bending energy. Due to inhomogeneities the lipids may separate and form different phases on the membrane which results in an energy contribution from the phase interfaces. We have been numerically studying equilibrium shapes, i.e., local energy minima by relaxing suitable initial shapes. A suitable (kind of) gradient flow dynamics has been defined for this purpose where the inter-membrane domains are described using the phase field methodology. The governing equations consist of pde on the membrane surface describing the phase separation coupled to a geometric evolution law for the membrane. The discretisation is based on representing the membrane by a triangulated surface on which quadratic parametric FEs are defined. The convergence as grid parameter and diffuse interface thickness tend to zero has been numerically investigated. Further issues are the sharp interface limit of the phase-field approach and adaptive mesh refinement.

• V Styles (Sussex) Primal-dual active set methods for Allen-Cahn variational inequalities

We propose a primal-dual active set method for local and non-local Allen-Cahn variational inequalities. Properties of the method are discussed and numerical simulations are presented that demonstrate its efficiency.

See also:
Mathematics Research Centre
Mathematical Interdisciplinary Research at Warwick (MIR@W)
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