681 research outputs found
Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow
We apply a phase field approach for a general shape optimization problem of a
stationary Navier-Stokes flow. To be precise we add a multiple of the
Ginzburg--Landau energy as a regularization to the objective functional and
relax the non-permeability of the medium outside the fluid region. The
resulting diffuse interface problem can be shown to be well-posed and
optimality conditions are derived. We state suitable assumptions on the problem
in order to derive a sharp interface limit for the minimizers and the
optimality conditions. Additionally, we can derive a necessary optimality
system for the sharp interface problem by geometric variations without stating
additional regularity assumptions on the minimizing set
Segmentation and Restoration of Images on Surfaces by Parametric Active Contours with Topology Changes
In this article, a new method for segmentation and restoration of images on
two-dimensional surfaces is given. Active contour models for image segmentation
are extended to images on surfaces. The evolving curves on the surfaces are
mathematically described using a parametric approach. For image restoration, a
diffusion equation with Neumann boundary conditions is solved in a
postprocessing step in the individual regions. Numerical schemes are presented
which allow to efficiently compute segmentations and denoised versions of
images on surfaces. Also topology changes of the evolving curves are detected
and performed using a fast sub-routine. Finally, several experiments are
presented where the developed methods are applied on different artificial and
real images defined on different surfaces
Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis
We consider a diffuse interface model for tumor growth consisting of a
Cahn--Hilliard equation with source terms coupled to a reaction-diffusion
equation, which models a tumor growing in the presence of a nutrient species
and surrounded by healthy tissue. The well-posedness of the system equipped
with Neumann boundary conditions was found to require regular potentials with
quadratic growth. In this work, Dirichlet boundary conditions are considered,
and we establish the well-posedness of the system for regular potentials with
higher polynomial growth and also for singular potentials. New difficulties are
encountered due to the higher polynomial growth, but for regular potentials, we
retain the continuous dependence on initial and boundary data for the chemical
potential and for the order parameter in strong norms as established in the
previous work. Furthermore, we deduce the well-posedness of a variant of the
model with quasi-static nutrient by rigorously passing to the limit where the
ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is
small.Comment: 33 pages, minor typos corrected, accepted versio
Segmentation of Three-dimensional Images with Parametric Active Surfaces and Topology Changes
In this paper, we introduce a novel parametric method for segmentation of
three-dimensional images. We consider a piecewise constant version of the
Mumford-Shah and the Chan-Vese functionals and perform a region-based
segmentation of 3D image data. An evolution law is derived from energy
minimization problems which push the surfaces to the boundaries of 3D objects
in the image. We propose a parametric scheme which describes the evolution of
parametric surfaces. An efficient finite element scheme is proposed for a
numerical approximation of the evolution equations. Since standard parametric
methods cannot handle topology changes automatically, an efficient method is
presented to detect, identify and perform changes in the topology of the
surfaces. One main focus of this paper are the algorithmic details to handle
topology changes like splitting and merging of surfaces and change of the genus
of a surface. Different artificial images are studied to demonstrate the
ability to detect the different types of topology changes. Finally, the
parametric method is applied to segmentation of medical 3D images
Sharp interface limit for a phase field model in structural optimization
We formulate a general shape and topology optimization problem in structural
optimization by using a phase field approach. This problem is considered in
view of well-posedness and we derive optimality conditions. We relate the
diffuse interface problem to a perimeter penalized sharp interface shape
optimization problem in the sense of -convergence of the reduced
objective functional. Additionally, convergence of the equations of the first
variation can be shown. The limit equations can also be derived directly from
the problem in the sharp interface setting. Numerical computations demonstrate
that the approach can be applied for complex structural optimization problems
Multiscale Problems in Solidification Processes
Our objective is to describe solidification phenomena in alloy systems. In the classical approach, balance equations in the phases are coupled to conditions on the phase boundaries which are modelled as moving hypersurfaces. The Gibbs-Thomson condition ensures that the evolution is consistent with thermodynamics. We present a derivation of that condition by defining the motion via a localized gradient flow of the entropy.
Another general framework for modelling solidification of alloys with multiple phases and components is based on the phase field approach. The phase boundary motion is then given by a system of Allen-Cahn type equations for order parameters. In the sharp interface limit, i.e., if the smallest length scale ± related to the thickness of the diffuse phase boundaries converges to zero, a model with moving boundaries is recovered. In the case of two phases
it can even be shown that the approximation of the sharp interface model by the phase field model is of second order in ±. Nowadays it is not possible to simulate the microstructure evolution in a whole workpiece. We present a two-scale model derived by homogenization methods including a mathematical justification by an estimate of the model error
Thermodynamically Consistent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities
A new diffuse interface model for a two-phase flow of two incompressible
fluids with different densities is introduced using methods from rational
continuum mechanics. The model fulfills local and global dissipation
inequalities and is also generalized to situations with a soluble species.
Using the method of matched asymptotic expansions we derive various sharp
interface models in the limit when the interfacial thickness tends to zero.
Depending on the scaling of the mobility in the diffusion equation we either
derive classical sharp interface models or models where bulk or surface
diffusion is possible in the limit. In the two latter cases the classical
Gibbs-Thomson equation has to be modified to include kinetic terms. Finally, we
show that all sharp interface models fulfill natural energy inequalities.Comment: 34 page
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