16 research outputs found
Sharp Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility
In this work, the sharp interface limit of the degenerate Cahn-Hilliard
equation (in two space dimensions) with a polynomial double well free energy
and a quadratic mobility is derived via a matched asymptotic analysis involving
exponentially large and small terms and multiple inner layers. In contrast to
some results found in the literature, our analysis reveals that the interface
motion is driven by a combination of surface diffusion flux proportional to the
surface Laplacian of the interface curvature and an additional contribution
from nonlinear, porous-medium type bulk diffusion, For higher degenerate
mobilities, bulk diffusion is subdominant. The sharp interface models are
corroborated by comparing relaxation rates of perturbations to a radially
symmetric stationary state with those obtained by the phase field model.Comment: 27 pages, 2 figure
Hyperuniform monocrystalline structures by spinodal solid-state dewetting
Materials featuring anomalous suppression of density fluctuations over large
length scales are emerging systems known as disordered hyperuniform. The
underlying hidden order renders them appealing for several applications, such
as light management and topologically protected electronic states. These
applications require scalable fabrication, which is hard to achieve with
available top-down approaches. Theoretically, it is known that spinodal
decomposition can lead to disordered hyperuniform architectures. Spontaneous
formation of stable patterns could thus be a viable path for the bottom-up
fabrication of these materials. Here we show that mono-crystalline
semiconductor-based structures, in particular SiGe layers
deposited on silicon-on-insulator substrates, can undergo spinodal solid-state
dewetting featuring correlated disorder with an effective hyperuniform
character. Nano- to micro-metric sized structures targeting specific
morphologies and hyperuniform character can be obtained, proving the generality
of the approach and paving the way for technological applications of disordered
hyperuniform metamaterials. Phase-field simulations explain the underlying
non-linear dynamics and the physical origin of the emerging patterns.Comment: 6 pages, 3 figures, supplementary information (7 pages) enclose
Hybrid parallelization of an adaptive finite element code
summary:We present a hybrid OpenMP/MPI parallelization of the finite element method that is suitable to make use of modern high performance computers. These are usually built from a large bulk of multi-core systems connected by a fast network. Our parallelization method is based firstly on domain decomposition to divide the large problem into small chunks. Each of them is then solved on a multi-core system using parallel assembling, solution and error estimation. To make domain decomposition for both, the large problem and the smaller sub-problems, sufficiently fast we make use of a hierarchical mesh structure. The partitioning is done on a coarser mesh level, resulting in a very fast method that shows good computational balancing results. Numerical experiments show that both parallelization methods achieve good scalability in computing solution of nonlinear, time dependent, higher order PDEs on large domains. The parallelization is realized in the adaptive finite element software AMDiS
Doubly Degenerate Diffuse Interface Models of Anisotropic Surface Diffusion
We extend the doubly degenerate Cahn-Hilliard (DDCH) models for isotropic
surface diffusion, which yield more accurate approximations than classical
degenerate Cahn-Hilliard (DCH) models, to the anisotropic case. We consider
both weak and strong anisotropies and demonstrate the capabilities of the
approach for these cases numerically. The proposed model provides a variational
and energy dissipative approach for anisotropic surface diffusion, enabling
large scale simulations with material-specific parameters.Comment: 15 pages; 6 figure
A new diffuse-interface model for step flow in epitaxial growth
In this work, we consider epitaxial growth of thin crystalline films. Thereby, we propose
a new diffuse-interface approximation of a semi-continuous model resolving atomic distances in the
growth direction but being coarse-grained in the lateral directions. Mathematically, this leads to a free
boundary problem proposed by Burton, Cabrera and Frank for steps separating terraces of different
atomic heights. The evolution of the steps is coupled to a diffusion equation for the adatom (adsorbed
atom) concentration fulfilling Robin-type boundary conditions at the steps. Our approach allows to
incorporate an Ehrlich-Schwoebel barrier as well as diffusion along step edges into a diffuse-interface
model.
This model results in a Cahn-Hilliard equation with a degenerate mobility coupled to diffusion
equations on the terraces with a diffuse-interface description of the boundary conditions at the steps.
We provide a justification by matched asymptotic expansions formally showing the convergence of the
diffuse-interface model towards the sharp-interface model as the interface width shrinks to zero. The
results of the asymptotic analysis are numerically reproduced by a finite element discretisation
Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization
Short time existence for a surface diffusion evolution equation with
curvature regularization is proved in the context of epitaxially strained
three-dimensional films. This is achieved by implementing a minimizing movement
scheme, which is hinged on the -gradient flow structure underpinning
the evolution law. Long-time behavior and Liapunov stability in the case of
initial data close to a flat configuration are also addressed.Comment: 44 page