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Elements of the theory of partial differential equations that are relevant for the modeling process are discussed. Elliptic, parabolic, hyperbolic equations as well as the Navier-Stokes and the Euler equations are treated. Finally, boundary layer theory is discussed.

We derive a Cahn-Hilliard-Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. Via a coupling of the Cahn-Hilliard-Darcy equations to a system of reaction-diffusion equations a multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, can be included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth is investigated numerically., Comment: 43 pages, 56 figures

Electrical networks and space frames in its stationary state can be described by sets of linear equations of similar structure. This is extendable to alternating current circuits by the use of complex parameters. The set of equations can be written in a form equivalent to a constrained optimization problem, giving rise to a dual form by means of Lagrange multipliers.

Viele Anwendungen in Naturwissenschaften und Technik fuhren auf Problemstellungen, bei denen die Geometrie des Gebietes, auf dem eine Gleichung gelost werden soll, a priori unbestimmt ist. Ist eine partielle Differentialgleichung in einem Gebiet zu losen, von dem ein Teil des Randes unbekannt ist, so spricht man von einem Problem mit freiem Rand. Zusatzlich zu den ublichen Randbedingungen, die gebraucht werden, um die partielle Differentialgleichung zu losen, sind in diesem Fall weitere Bedingungen am freien Rand zu stellen. Probleme mit freiem Rand tauchen unter anderem bei folgenden Fragestellungen auf: Schmelz- und Erstarrungsphanomene (Stefan–Problem), Hindernisprobleme fur elastische Membranen, Kontaktprobleme bei elastischen Verformungen, Wachstum von Tumoren, Stromungen mit freien Oberflachen und Bewertung von Finanzderivaten.

One-dimensional oscillations are described by linear ordinary differential equations with closed form solutions extendable to forced oscillations. Besides the Lagrangian the Hamiltonian form of mechanics is introduced. Further applications are the motion of space frames (from Chapter 3) and two species models from population dynamics. By means of qualitative analysis (using phase portraits) the long time behavior can be analyzed. The principle of linearized stability reduces the stability question to linear systems, where it can be completely studied. Finally also variational problems and optimal control problems are discussed.

Thermodynamics is concerned with the definition and relation of notions like temperature, pressure and volume, with the first law of thermodynamics as a basis. The additional notion of entropy is characterized by the second law of thermodynamics. Temperature can be viewed as an integrating factor. Equilibrium conditions can be formulated by means of thermodynamic potentials. Legendre transforms and differential forms are introduced as auxiliary means. The theory is extended to mixtures to include chemical reactions, i.e., the mass action law in equilibrium and also kinetic reactions.

Diffuse interface models have become an important analytical and numerical method to model two-phase flows. In this contribution we review the subject and discuss in detail a thermodynamically consistent model with a divergence free velocity field for two-phase flows with different densities. The model is derived using basic thermodynamical principles, its sharp interface limits are stated, existence results are given, different numerical approaches are discussed and computations showing features of the model are presented.

New diffuse interface and sharp interface models for soluble and insoluble surfactants fulfilling energy inequalities are introduced. We discuss their relation with the help of asymptotic analysis and present an existence result for a particular diffuse interface model.

Curve evolution schemes are introduced for image segmentation based on a region-based contour model allowing for junctions, vector-valued images, and topology changes. 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.

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., 53 pages, 26 figures

We propose a general control framework for two-phase flows with variable densities in the diffuse interface formulation, where the distribution of the fluid components is described by a phase field. The flow is governed by the diffuse interface model proposed in Abelset al.[M3AS22(2012) 1150013]. On the basis of the stable time discretization proposed in Garckeet al.[Appl. Numer. Math.99(2016) 151] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. We present numerical examples with distributed and boundary controls, and also consider the case, where the initial value of the phase field serves as control variable.

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., Comment: 31 pages, 2 figures

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.

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. The local convergence behaviour of the primal-dual active set algorithm is studied by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency.

Biomembranes and vesicles consisting of multiple phases can attain a multitude of shapes, undergoing complex shape transitions. We study a Cahn--Hilliard model on an evolving hypersurface coupled to Navier--Stokes equations on the surface and in the surrounding medium to model these phenomena. The evolution is driven by a curvature energy, modelling the elasticity of the membrane, and by a Cahn--Hilliard type energy, modelling line energy effects. A stable semidiscrete finite element approximation is introduced and, with the help of a fully discrete method, several phenomena occurring for two-phase membranes are computed., 61 pages, 17 figures

In this paper, we introduce a novel approach for active contours with free endpoints. A scheme for image segmentation is presented based on a discrete version of the Mumford–Shah functional where the contours can be both closed and open curves. Additional to a flow of the curves in normal direction, evolution laws for the tangential flow of the endpoints are derived. Using a parametric approach to describe the evolving contours together with an edge-preserving denoising, we obtain a fast method for image segmentation and restoration. The analytical and numerical schemes are presented followed by numerical experiments with artificial test images and with a real medical image.

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 version

We consider an optimal control problem for a diffuse interface model of tumor growth. The state equations couples a Cahn-Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions for both the cytotoxic concentration and the treatment time., Comment: 39 pages

We consider shape and topology optimization of an object in fluid flow governed by the Navier-Stokes equations. Shapes are modelled with the help of a phase field approach and the solid body is relaxed to be a porous medium. The phase field method uses a Ginzburg-Landau functional in order to approximate a perimeter penalization. We focus on surface functionals and carefully introduce a new modelling variant, show existence of minimizers and derive first order necessary conditions. These conditions are related to classical shape derivatives by identifying the sharp interface limit with the help of formally matched asymptotic expansions. Finally, we present numerical computations based on a Cahn-Hilliard type gradient descent which demonstrate that the method can be used to solve shape optimization problems for fluids with the help of the new approach.

We propose and analyze a primal-dual active set method for discretized versions of the local and nonlocal Allen–Cahn variational inequalities. An existence result for the nonlocal variational inequality is shown in a formulation involving Lagrange multipliers for local and nonlocal constraints. Local convergence of the discrete method is shown by interpreting the approach as a semismooth Newton method. Properties of the method are discussed and several numerical simulations demonstrate its efficiency. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

In this paper we derive thermodynamically consistent higher order phase field models for the dynamics of biomembranes in incompressible viscous fluids. We start with basic conservation laws and an appropriate version of the second law of thermodynamics and obtain generalizations of models introduced by Du, Li and Liu [3] and Jamet and Misbah [11]. In particular we derive a stress tensor involving higher order derivatives of the phase field and generalize the classical Korteweg capillarity tensor.

On the basis of our previous work, we introduce novel fully discrete, fully practical parametric finite element approximations for geometric evolution equations of curves in the plane. The fully implicit approximations are unconditionally stable and intrinsically equidistribute the vertices at each time level. We present iterative solution methods for the systems of nonlinear equations arising at each time level and present several numerical results. The ideas easily generalize to the evolution of curve networks and to anisotropic surface energies. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

We study the coupled surface and grain boundary motion in bi- and tricrystals in three-space dimensions, building on previous work by the authors on the simplified two-dimensional case. The motion of the interfaces, which in this paper are presented by two-dimensional hypersurfaces, is described by two types of normal velocities: motion by mean curvature and motion by surface diffusion. Three hypersurfaces meet at triple-junction lines, where junction conditions need to hold. Similarly, boundary conditions are prescribed where an interface meets an external boundary, and these conditions naturally give rise to contact angles. We present a variational formulation of the flows, which leads to a fully practical finite-element approximation that exhibits excellent mesh properties, with no mesh smoothing or remeshing required in practice. For the introduced parametric finite-element approximation we show well posedness and, in general, unconditional stability, i.e. there is no restriction on the chosen time-step size. Moreover, the induced discrete equations are linear and easy to solve. A generalisation to anisotropic surface energies is straightforward. Several numerical results in two- and three-space dimensions are presented, including simulations for thermal grooving and sintering. Three-dimensional simulations featuring quadruple junction points, non-standard boundary contact angles and fully anisotropic surface energies are also presented.

We present a variational formulation of motion by minus the Laplacian of curvature and mean curvature flow, as well as related second and fourth order flows of a closed hypersurface in R 3 . On introducing a parametric finite element approximation, we prove stability bounds and compare our scheme with existing approaches. The presented scheme has very good properties with respect to the distribution of mesh points and, if applicable, volume conservation.

In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network., Comment: submitted

The focus of this workshop was the modeling, analysis and nu- merical simulation of biological membranes. The fundamental models which characterize equilibria are based on bending energies such as the Helfrich energy and suitable generalizations. Talks in the workshop focused on equi- librium shapes of membranes and vesicles, the corresponding relaxation dy- namics and the coupling with outer or inner flows.

We propose a phase field model for stress and diffusion-induced interface motion. This model, in particular, can be used to describe diffusion-induced grain boundary motion and generalizes a model of Cahn, Fife and Penrose as it more accurately incorporates stress effects. In this paper we will demonstrate that the model can also be used to describe other stress-driven interface motion. As an example, interface motion resulting from interactions of interfaces with dislocations is studied.

We present a variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow, as well as related flows. The proposed scheme covers both the closed-curve case and the case of curves that are connected via triple junction points. On introducing a parametric finite-element approximation, we prove stability bounds and report on numerical experiments, including regularized crystalline mean curvature flow and regularized crystalline surface diffusion. The presented scheme has very good properties with respect to the distribution of mesh points and, if applicable, area conservation.

We present a variational formulation of combined motion by minus the Laplacian of curvature and mean curvature flow, as well as related flows. The proposed scheme covers both the closed curve case and the case of curves that are connected via triple or quadruple junction points or intersect the external boundary. On introducing a parametric finite element approximation, we prove stability bounds and compare our scheme with existing approaches. The presented scheme has very good properties with respect to the equidistribution of mesh points and, if applicable, area conservation.

Vesicles and many biological membranes are made of two monolayers of lipid molecules and form closed lipid bilayers. The dynamical behaviour of vesicles is very complex and a variety of forms and shapes appear. Lipid bilayers can be considered as a surface fluid and hence the governing equations for the evolution include the surface (Navier--)Stokes equations, which in particular take the membrane viscosity into account. The evolution is driven by forces stemming from the curvature elasticity of the membrane. In addition, the surface fluid equations are coupled to bulk (Navier--)Stokes equations. We introduce a parametric finite element method to solve this complex free boundary problem, and present the first three dimensional numerical computations based on the full (Navier--)Stokes system for several different scenarios. For example, the effects of the membrane viscosity, spontaneous curvature and area difference elasticity (ADE) are studied. In particular, it turns out, that even in the case of no viscosity contrast between the bulk fluids, the tank treading to tumbling transition can be obtained by increasing the membrane viscosity. Besides the classical tank treading and tumbling motions, another mode (called the transition mode in this paper, but originally called the vacillating-breathing mode and subsequently also called trembling, transition and swinging mode) separating these classical modes appears and is studied by us numerically. We also study how features of equilibrium shapes in the ADE and spontaneous curvature models, like budding behaviour or starfish forms, behave in a shear flow., Minor corrections; 15 pages, 26 figures

A parametric finite element approximation of incompressible two-phase flow with soluble surfactants is presented. The Navier-Stokes equations are coupled to bulk and surfaces PDEs for the surfactant concentrations. At the interface adsorption, desorption and stress balances involving curvature effects and Marangoni forces have to be considered. A parametric finite element approximation for the advection of the interface, which maintains good mesh properties, is coupled to the evolving surface finite element method, which is used to discretize the surface PDE for the interface surfactant concentration. The resulting system is solved together with standard finite element approximations of the Navier-Stokes equations and of the bulk parabolic PDE for the surfactant concentration. Semidiscrete and fully discrete approximations are analyzed with respect to stability, conservation and existence/uniqueness issues. The approach is validated for simple test cases and for complex scenarios, including colliding drops in a shear flow, which are computed in two and three space dimensions.

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.

Using basic thermodynamic principles we derive a Cahn--Hilliard--Darcy model for tumour growth including nutrient diffusion, chemotaxis, active transport, adhesion, apoptosis and proliferation. The model generalises earlier models and in particular includes active transport mechanisms which ensure thermodynamic consistency. We perform a formally matched asymptotic expansion and develop several sharp interface models. Some of them are classical and some are new which for example include a jump in the nutrient density at the interface. A linear stability analysis for a growing nucleus is performed and in particular the role of the new active transport term is analysed. Numerical computations are performed to study the influence of the active transport term for specific growth scenarios., Comment: 47 pages, 51 figures, minor typos corrected, new figures added, accepted version

A new formulation for shape and topology optimization in a Stokes flow is introduced. The investigated problem minimizes the total potential power of the flow. By combining a porous medium and a phase field approach we obtain a well-posed problem in a diffuse interface setting that can be reformulated into a problem without state equations. We can derive a sharp interface problem with zero permeability outside the fluid region as a \(\Gamma \)-limit of this porous medium—phase field problem.

In this paper, we introduce a novel parametric finite element 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.

We propose and investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple a bulk-diffusion to an evolution equation on the membrane. The latter describes the relaxation dynamics for an energy which takes lipid–phase separation and lipid–cholesterol interaction energy into account. It takes the form of an (extended) Cahn–Hilliard equation. Different laws for the exchange term represent equilibrium and nonequilibrium models. We present a thermodynamic justification, analyze the respective qualitative behavior and derive asymptotic reductions of the model. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively. Finally we perform numerical simulations and investigate the long-time behavior of the model and its parameter dependence. Both the mathematical analysis and the numerical simulations show the emergence of raft-like structures in the nonequilibrium case whereas in the equilibrium case only macrodomains survive in the long-time evolution.

Spinodal decomposition, i.e., the separation of a homogeneous mixture into different phases, can be modeled by the Cahn-Hilliard equation - a fourth order semilinear parabolic equation. If elastic stresses due to a lattice misfit become important, the Cahn-Hilliard equation has to be coupled to an elasticity system to take this into account. Here, we present a discretization based on finite elements and an implicit Euler scheme. We first show solvability and uniqueness of solutions. Based on an energy decay property we then prove convergence of the scheme. Finally we present numerical experiments showing the impact of elasticity on the morphology of the microstructure.

Existence of solutions to a Cahn–Hilliard system taking elastic stresses into account is shown. The analysis is hindered by a logarithmic singularity and a quadratic term in the displacement gradient. This makes careful a priori estimates for both terms necessary. In particular, a global version of L p -estimates for gradients of nonlinear elliptic systems is shown in order to control the logarithmic singularity.

The linearizedstability of stationary solutions to the surface diffusion flow with angle conditions and no-flux conditions as boundary conditions is studied. We perform a linearized stability analysis in which the H-1 -gradient flow structure plays a key role. As a byproduct our analysis also gives a criterion for the stability of critical points of the length functional of curves which come into contact with the outer boundary. Finally, we study the linearized stability of several examples.

We propose a multi-order parameter phase field system and a sharp interface model to describe bidirectional diffusion induced grain boundary motion in the presence of triple junctions. Numerical approximations of the models are presented together with some computational results.

We propose a general control framework for two-phase flows with variable densities in the diffuse interface formulation, where the distribution of the fluid components is described by a phase field. The flow is governed by the diffuse interface model proposed in [Abels, Garcke, Grun, M3AS 22(3):1150013(40), 2012]. On the basis of the stable time discretization proposed in [Garcke, Hinze, Kahle, APPL NUMER MATH, 99:151-171, 2016] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. We present numerical examples with distributed and boundary controls, and also consider the case, where the initial value of the phase field serves as control variable.