101 research outputs found
On the stable discretization of strongly anisotropic phase field models with applications to crystal growth
We introduce unconditionally stable finite element approximations for
anisotropic Allen--Cahn and Cahn--Hilliard equations. These equations
frequently feature in phase field models that appear in materials science. On
introducing the novel fully practical finite element approximations we prove
their stability and demonstrate their applicability with some numerical
results.
We dedicate this article to the memory of our colleague and friend Christof
Eck (1968--2011) in recognition of his fundamental contributions to phase field
models.Comment: 20 pages, 8 figure
A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
We develop a robust solver for a second order mixed finite element splitting
scheme for the Cahn-Hilliard equation. This work is an extension of our
previous work in which we developed a robust solver for a first order mixed
finite element splitting scheme for the Cahn-Hilliard equaion. The key
ingredient of the solver is a preconditioned minimal residual algorithm (with a
multigrid preconditioner) whose performance is independent of the spacial mesh
size and the time step size for a given interfacial width parameter. The
dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text
overlap with arXiv:1709.0400
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 robust solver for a second order mixed finite element method for the Cahn–Hilliard equation
We develop a robust solver for a second order mixed finite element splitting scheme for the Cahn–Hilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the Cahn–Hilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild
Phase Separation Dynamics in Isotropic Ion-Intercalation Particles
Lithium-ion batteries exhibit complex nonlinear dynamics, resulting from
diffusion and phase transformations coupled to ion intercalation reactions.
Using the recently developed Cahn-Hilliard reaction (CHR) theory, we
investigate a simple mathematical model of ion intercalation in a spherical
solid nanoparticle, which predicts transitions from solid-solution radial
diffusion to two-phase shrinking-core dynamics. This general approach extends
previous Li-ion battery models, which either neglect phase separation or
postulate a spherical shrinking-core phase boundary, by predicting phase
separation only under appropriate circumstances. The effect of the applied
current is captured by generalized Butler-Volmer kinetics, formulated in terms
of diffusional chemical potentials, and the model consistently links the
evolving concentration profile to the battery voltage. We examine sources of
charge/discharge asymmetry, such as asymmetric charge transfer and surface
"wetting" by ions within the solid, which can lead to three distinct phase
regions. In order to solve the fourth-order nonlinear CHR
initial-boundary-value problem, a control-volume discretization is developed in
spherical coordinates. The basic physics are illustrated by simulating many
representative cases, including a simple model of the popular cathode material,
lithium iron phosphate (neglecting crystal anisotropy and coherency strain).
Analytical approximations are also derived for the voltage plateau as a
function of the applied current
Existence of solutions to an anisotropic degenerate Cahn-Hilliard-type equation
We prove existence of solutions to an anisotropic Cahn-Hilliard-type
equation with degenerate diffusional mobility. In particular, the mobility
vanishes at the pure phases, which is typically used to model motion by
surface diffusion. The main difficulty of the present existence result is the
strong non-linearity given by the fourth-order anisotropic operator. Imposing
particular assumptions on the domain and assuming that the strength of the
anisotropy is sufficiently small enables to establish appropriate auxiliary
results which play an essential part in the present existence proof. In
addition to the existence we show that the absolute value of the
corresponding solutions is bounded by 1
Linearly Preconditioned Nonlinear Solvers for Phase Field Equations Involving p-Laplacian Terms
Phase field models are usually constructed to model certain interfacial dynamics. Numerical simulations of phase-field models require long time accuracy, stability and therefore it is necessary to develop efficient and highly accurate numerical methods. In particular, the unconditionally energy stable , unconditionally solvable, and accurate schemes and fast solvers are desirable.
In this thesis, We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. The results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods.
Based on the PSD framework, we also proposed two efficient and practical Preconditioned Nonlinear Conjugate Gradient (PNCG) solvers. The main idea of the preconditioned solvers is to use a linearized version of the nonlinear operator as a metric for choosing the initial search direction. And the hybrid conjugate directions as the following search direction. In order to make the proposed solvers and scheme much more practical, we also investigate an adaptive time stepping strategy for time dependent problems.
Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection, the square phase field crystal model and functionalized Cahn-Hilliard equation – are carried out to verify the efficiency of the schemes and solvers
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