Energy-law preserving continuous finite element methods for simulation of liquid crystal and multi-phase flows
Abstract
The liquid crystal (LC) flow model is a coupling between
orientation (director field) of LC molecules and a flow field.
The model may probably be one of simplest complex fluids and
is very similar to a Allen-Cahn phase field model for
multiphase flows if the orientation variable is replaced by a
phase function. There are a few large or small parameters
involved in the model (e.g. the small penalty parameter for
the unit length LC molecule or the small phase-change
parameter, possibly large Reynolds number of the flow field,
etc.). We propose a C^0 finite element formulation in space
and a modified midpoint scheme in time which accurately
preserves the inherent energy law of the model. We use C^0
elements because they are simpler than existing C^1 element
and mixed element methods. We emphasise the energy law
preservation because from the PDE analysis point of view the
energy law is very important to correctly catch the evolution
of singularities in the LC molecule orientation. In addition
we will see numerical examples that the energy law preserving
scheme performs better under some choices of parameters. We
shall apply the same idea to a Cahn-Hilliard phase field model
where the biharmonic operator is decomposed into two Laplacian
operators. But we find that under our scheme non-physical
oscillation near the interface occurs. We figure out the
reason from the viewpoint of differential algebraic equations
and then remove the non-physical oscillation by doing only one
step of a modified backward Euler scheme at the initial time.
A number of numerical examples demonstrate the good
performance of the method. At the end of the talk we will show
how to apply the method to compute a superconductivity model,
especially at the regime of Hc2 or beyond. The talk is based
on a few joint papers with Chun Liu, Qi Wang, Xingbin Pan and
Roland Glowinski, etc.
A new look at Newton's method
Abstract
Current methods for globalizing Newton's Method for solving systems of nonlinear equations fall back on steps biased towards the steepest descent direction (e.g. Levenberg/Marquardt, Trust regions, Cauchy point dog-legs etc.), when there is difficulty in making progress. This can occasionally lead to very slow convergence when short steps are repeatedly taken.
This talk looks at alternative strategies based on searching curved arcs related to Davidenko trajectories. Near to manifolds on which the Jacobian matrix is singular, certain conjugate steps are also interleaved, based on identifying a Pareto optimal solution.
Preliminary limited numerical experiments indicate that this approach is very effective, with rapid and ultimately second order convergence in almost all cases. It is hoped to present more detailed numerical evidence when the talk is given. The new ideas can also be incorporated with more recent ideas such as multifilters or nonmonotonic line searches without difficulty, although it may be that there is no longer much to gain by doing this.
14:00
"Multicellular modelling using computers: biomechanical calibration, active processes, and emergent tissue dynamics"
14:00
Mathematical Modelling of fungal mycelia : a question of scale
16:30
Boundary Value Problems on Measure Chains
Abstract
When modelling a physical or biological system, it has to be decided
what framework best captures the underlying properties of the system
under investigation. Usually, either a continuous or a discrete
approach is adopted and the evolution of the system variables can then
be described by ordinary or partial differential equations or
difference equations, as appropriate. It is sometimes the case,
however, that the model variables evolve in space or time in a way
which involves both discrete and continuous elements. This is best
illustrated by a simple example. Suppose that the life span of a
species of insect is one time unit and at the end of its life span,
the insect mates, lays eggs and then dies. Suppose the eggs lie
dormant for a further 1 time unit before hatching. The `time-scale' on
which the insect population evolves is therefore best represented by a
set of continuous intervals separated by discrete gaps. This concept
of `time-scale' (or measure chain as it is referred to in a slightly
wider context) can be extended to sets consisting of almost arbitrary
combinations of intervals, discrete points and accumulation points,
and `time-scale analysis' defines a calculus, on such sets. The
standard `continuous' and `discrete' calculus then simply form special
cases of this more general time scale calculus.
In this talk, we will outline some of the basic properties of time
scales and time scale calculus before discussing some if the
technical problems that arise in deriving and analysing boundary
value problems on time scales.
14:30
Neurogenesis in the developing spinal cord: making the right number of neurons at the right time
17:00
Chemotactic Cell Movement and its Role in Development
Abstract
In St John's College.
Oxford Life Sciences Modelling Colloquia Series
16:30