1,420 research outputs found
Stable phase field approximations of anisotropic solidification
We introduce unconditionally stable finite element approximations for a phase
field model for solidification, which take highly anisotropic surface energy and kinetic
effects into account. We hence approximate Stefan problems with anisotropic
Gibbs{Thomson law with kinetic undercooling, and quasi-static variants thereof.
The phase field model is given by
#wt + � %(') 't = r: (b(')rw) ;
c
a
� %(')w = " �
� �(r') '
A structure preserving front tracking finite element method for the Mullins--Sekerka problem
We introduce and analyse a fully discrete approximation for a mathematical
model for the solidification and liquidation of materials of negligible
specific heat. The model is a two-sided Mullins--Sekerka problem. The
discretization uses finite elements in space and an independent
parameterization of the moving free boundary. We prove unconditional stability
and exact volume conservation for the introduced scheme. Several numerical
simulations, including for nearly crystalline surface energies, demonstrate the
practicality and accuracy of the presented numerical method.Comment: 24 pages, 9 figure
Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
We present variational approximations of boundary value problems for
curvature flow (curve shortening flow) and elastic flow (curve straightening
flow) in two-dimensional Riemannian manifolds that are conformally flat. For
the evolving open curves we propose natural boundary conditions that respect
the appropriate gradient flow structure. Based on suitable weak formulations we
introduce finite element approximations using piecewise linear elements. For
some of the schemes a stability result can be shown. The derived schemes can be
employed in very different contexts. For example, we apply the schemes to the
Angenent metric in order to numerically compute rotationally symmetric
self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes
to compute geodesics that are relevant for optimal interface profiles in
multi-component phase field models.Comment: 42 pages, 21 figure
Discrete hyperbolic curvature flow in the plane
Hyperbolic curvature flow is a geometric evolution equation that in the plane
can be viewed as the natural hyperbolic analogue of curve shortening flow. It
was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave
phenomena in solid-liquid interfaces. We introduce a semidiscrete finite
difference method for the approximation of hyperbolic curvature flow and prove
error bounds for natural discrete norms. We also present numerical simulations,
including the onset of singularities starting from smooth strictly convex
initial data.Comment: 23 pages, 10 figure
An unconditionally stable finite element scheme for anisotropic curve shortening flow
summary:Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method
Unfitted finite element methods for axisymmetric two-phase flow
We propose and analyze unfitted finite element approximations for the
two-phase incompressible Navier--Stokes flow in an axisymmetric setting. The
discretized schemes are based on an Eulerian weak formulation for the
Navier--Stokes equation in the 2d-meridian halfplane, together with a
parametric formulation for the generating curve of the evolving interface. We
use the lowest order Taylor--Hood and piecewise linear elements for
discretizing the Navier--Stokes formulation in the bulk and the moving
interface, respectively. We discuss a variety of schemes, amongst which is a
linear scheme that enjoys an equidistribution property on the discrete
interface and good volume conservation. An alternative scheme can be shown to
be unconditionally stable and to conserve the volume of the two phases exactly.
Numerical results are presented to show the robustness and accuracy of the
introduced methods for simulating both rising bubble and oscillating droplet
experiments
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