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Harald Garcke

Publications and source records attributed to Harald Garcke.

At least 127 records · Page 7Linked to original sources
arXiv2014

Numerical approximation of phase field based shape and topology optimization for fluids

We consider the problem of finding optimal shapes of fluid domains. The fluid obeys the Navier--Stokes equations. Inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow. We formulate a corresponding optimization problem where flow outside the fluid domain is penalized. The resulting formulation of the shape optimization problem is shown to be well-posed, hence there exists a minimizer, and first order optimality conditions are derived. For the numerical realization we introduce a mass conserving gradient flow and obtain a Cahn--Hilliard type system, which is integrated numerically using the finite element method. An adaptive concept using reliable, residual based error estimation is exploited for the resolution of the spatial mesh. The overall concept is numerically investigated and comparison values are provided.

math.OC↗
arXiv2014

Stable Numerical Approximation of Two-Phase Flow with a Boussinesq--Scriven Surface Fluid

We consider two-phase Navier--Stokes flow with a Boussinesq--Scriven surface fluid. In such a fluid the rheological behaviour at the interface includes surface viscosity effects, in addition to the classical surface tension effects. We introduce and analyze parametric finite element approximations, and show, in particular, stability results for semi-discrete versions of the methods, by demonstrating that a free energy inequality also holds on the discrete level. We perform several numerical simulations for various scenarios in two and three dimensions, which illustrate the effects of the surface viscosity.

math.NA↗
arXiv2014

Local Well-Posedness for Volume-Preserving Mean Curvature and Willmore Flows with Line Tension

We show the short-time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by volume-preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non-local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a Hanzawa transformation to write the flows as graphs over a fixed reference hypersurface.

math.AP↗
arXiv2014

A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow

A new time discretization scheme for the numerical simulation of two-phase flow governed by a thermodynamically consistent diffuse interface model is presented. The scheme is consistent in the sense that it allows for a discrete in time energy inequality. An adaptive spatial discretization is proposed that conserves the energy inequality in the fully discrete setting by applying a suitable post processing step to the adaptive cycle. For the fully discrete scheme a quasi-reliable error estimator is derived which estimates the error both of the flow velocity, and of the phase field. The validity of the energy inequality in the fully discrete setting is numerically investigated.

math.AP↗
arXiv2013

Multi-material phase field approach to structural topology optimization

Multi-material structural topology and shape optimization problems are formulated within a phase field approach. First-order conditions are stated and the relation of the necessary conditions to classical shape derivatives are discussed. An efficient numerical method based on an $H^1$-gradient projection method is introduced and finally several numerical results demonstrate the applicability of the approach.

math.OC↗
arXiv2013

On the Stable Numerical Approximation of Two-Phase Flow with Insoluble Surfactant

We present a parametric finite element approximation of two-phase flow with insoluble surfactant. This free boundary problem is given by the Navier--Stokes equations for the two-phase flow in the bulk, which are coupled to the transport equation for the insoluble surfactant on the interface that separates the two phases. We combine the evolving surface finite element method with an approach previously introduced by the authors for two-phase Navier--Stokes flow, which maintains good mesh properties. The derived finite element approximation of two-phase flow with insoluble surfactant can be shown to be stable. Several numerical simulations demonstrate the practicality of our numerical method.

math.NA↗
arXiv2013

A Stable Parametric Finite Element Discretization of Two-Phase Navier--Stokes Flow

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Navier--Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. The parametric finite element approximation of the evolving interface is then coupled to a standard finite element approximation of the two-phase Navier--Stokes equations in the bulk. Here enriching the pressure approximation space with the help of an XFEM function ensures good volume conservation properties for the two phase regions. In addition, the mesh quality of the parametric approximation of the interface in general does not deteriorate over time, and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, our finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results in two and three space dimensions.

math.NA↗
arXiv2013

Eliminating Spurious Velocities with a Stable Approximation of Viscous Incompressible Two-Phase Stokes Flow

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. In addition, the mesh quality of the parametric approximation of the interface does not deteriorate, in general, over time; and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, on using a simple XFEM pressure space enrichment, we obtain exact volume conservation for the two phase regions. Furthermore, our fully discrete finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results which, in particular, demonstrate that spurious velocities can be avoided in the classical test cases.

math.NA↗
arXiv2013

Fast image segmentation and restoration using parametric curve evolution with junctions and topology changes

Curve evolution schemes for image segmentation based on a region based contour model allowing for junctions, vector-valued images and topology changes are introduced. Together with an a posteriori denoising in the segmented homogeneous regions this leads to a fast and efficient method for image segmentation and restoration. An uneven spread of mesh points is avoided by using the tangential degrees of freedom. Several numerical simulations on artificial test problems and on real images illustrate the performance of the method.

cs.CV↗
arXiv2013

Diffuse interface modelling of soluble surfactants in two-phase flow

Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are derived from balance equations and an energy inequality so that thermodynamic consistency is guaranteed. Via a formal asymptotic analysis, they are related to sharp interface models. Both cases of dynamic as well as instantaneous adsorption are covered. Flexibility with respect to the choice of bulk and surface free energies allows to realise various isotherms and relations of state between surface tension and surfactant. Some numerical simulations display the effectiveness of the presented approach.

physics.flu-dyn↗
arXiv2013

Finite-Element Approximation of One-Sided Stefan Problems with Anisotropic, Approximately Crystalline, Gibbs--Thomson Law

We present a finite-element approximation for the one-sided Stefan problem and the one-sided Mullins--Sekerka problem, respectively. The problems feature a fully anisotropic Gibbs--Thomson law, as well as kinetic undercooling. Our approximation, which couples a parametric approximation of the moving boundary with a finite element approximation of the bulk quantities, can be shown to satisfy a stability bound, and it enjoys very good mesh properties which means that no mesh smoothing is necessary in practice. In our numerical computations we concentrate on the simulation of snow crystal growth. On choosing realistic physical parameters, we are able to produce several distinctive types of snow crystal morphologies. In particular, facet breaking in approximately crystalline evolutions can be observed.

physics.comp-ph↗
arXiv2012

Phase Field Models versus Parametric Front Tracking Methods: Are they accurate and computationally efficient?

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification. Particular emphasis is put on Stefan problems, and their quasi-static variants, with applications to crystal growth. New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.

physics.comp-ph↗
arXiv2012

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 {align*} \vartheta\,w_t + λ\,\varrho(φ)\,φ_t & = \nabla \,.\, (b(φ)\,\nabla\, w) \,, \cPsi\,\tfrac{a}α\,\varrho(φ)\,w & = ε\,\tfracρα\,μ(\nabla\,φ)\,φ_t -ε\,\nabla \,.\, A'(\nabla\, φ) + ε^{-1}\,Ψ'(φ) {align*} subject to initial and boundary conditions for the phase variable $φ$ and the temperature approximation $w$. Here $ε> 0$ is the interfacial parameter, $Ψ$ is a double well potential, $\cPsi = \int_{-1}^1 \sqrt{2\,Ψ(s)}\;{\rm d}s$, $\varrho$ is a shape function and $A(\nabla\,φ) = \tfrac12\,|γ(\nabla\,φ)|^2$, where $γ$ is the anisotropic density function. Moreover, $\vartheta \geq 0$, $λ> 0$, $a > 0$, $α> 0$ and $ρ\geq 0$ are physical parameters from the Stefan problem, while $b$ and $μ$ are coefficient functions which also relate to the sharp interface problem. On introducing the novel fully practical finite element approximations for the anisotropic phase field model, we prove their stability and demonstrate their applicability with some numerical results.

math.NA↗
arXiv2012

On an Incompressible Navier-Stokes/Cahn-Hilliard System with Degenerate Mobility

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain by allowing for a degenerate mobility. The model has been developed by Abels, Garcke and Grün for fluids with different densities and leads to a solenoidal velocity field. It is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system, such that an energy estimate is fulfilled which follows from the fact that the model is thermodynamically consistent.

math.AP↗
arXiv2012

Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels, Garcke, and Grün for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system. The density of the mixture depends on an order parameter.

math.AP↗
arXiv2012

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.

math.NA↗
arXiv2012

Mean curvature flow with triple junctions in higher space dimensions

We consider mean curvature flow of n-dimensional surface clusters. At (n-1)-dimensional triple junctions an angle condition is required which in the symmetric case reduces to the well-known 120 degree angle condition. Using a novel parametrization of evolving surface clusters and a new existence and regularity approach for parabolic equations on surface clusters we show local well-posedness by a contraction argument in parabolic Hoelder spaces.

math.AP↗