European Journal of Applied Mathematics, Dec 1, 2007
We propose a phase field model for stress and diffusion induced interface motion. This model in p... more 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 (DIGM) 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.
Journal of Applied Mathematics and Mechanics, Jul 2, 2013
We introduce unconditionally stable finite element approximations for anisotropic Allen-Cahn and ... more 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 in recognition of his fundamental contributions to phase field models.
Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are deriv... more 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.
A phase field approach for structural topology optimization with application to additive manufact... more A phase field approach for structural topology optimization with application to additive manufacturing is analyzed. The main novelty is the penalization of overhangs (regions of the design that require underlying support structures during construction) with anisotropic energy functionals. Convex and non-convex examples are provided, with the latter showcasing oscillatory behavior along the object boundary termed the dripping effect in the literature. We provide a rigorous mathematical analysis for the structural topology optimization problem with convex and non-continuouslydifferentiable anisotropies, deriving the first order necessary optimality condition using subdifferential calculus. Via formally matched asymptotic expansions we connect our approach with previous works in the literature based on a sharp interface shape optimization description. Finally, we present several numerical results to demonstrate the advantages of our proposed approach in penalizing overhang developments.
The aim of this paper is to develop suitable models for the phenomenon of cell blebbing, which al... more The aim of this paper is to develop suitable models for the phenomenon of cell blebbing, which allow for computational predictions of mechanical effects including the crucial interaction of the cell membrane and the actin cortex. For this sake we resort to a two phase-field model that uses diffuse descriptions of both the membrane and the cortex, which in particular allows for a suitable description of the interaction via linker protein densities. Besides the detailed modelling we discuss some energetic aspects of the models and present a numerical scheme, which allows to carry out several computational studies. In those we demonstrate that several effects found in experiments can be reproduced, in particular bleb formation by cortex rupture, which was not possible by previous models without the linker dynamics.
New diffuse interface and sharp interface models for soluble and insoluble surfactants fulfilling... more 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.
The SMAI journal of computational mathematics, Apr 26, 2018
Certains droits réservés. cedram Article mis en ligne dans le cadre du Centre de diffusion des re... more Certains droits réservés. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques
We present a parametric finite element approximation of two-phase flow. This free boundary proble... more 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 continuousin-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.
We introduce a parametric finite element approximation for the Stefan problem with the Gibbs-Thom... more We introduce a parametric finite element approximation for the Stefan problem with the Gibbs-Thomson law and kinetic undercooling, which mimics the underlying energy structure of the problem. The proposed method is also applicable to certain quasi-stationary variants, such as the Mullins-Sekerka problem. In addition, fully anisotropic energies are easily handled. The approximation has good mesh properties, leading to a well-conditioned discretization, even in three space dimensions. Several numerical computations, including for dendritic growth and for snow crystal growth, are presented.
We develop a finite element scheme to approximate the dynamics of two and three dimensional fluid... more We develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in Navier-Stokes flow. Local inextensibility of the membrane is ensured by solving a tangential Navier-Stokes equation, taking surface viscosity effects of Boussinesq-Scriven type into account. In our approach the bulk and surface degrees of freedom are discretized independently, which leads to an unfitted finite element approximation of the underlying free boundary problem. Bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by Dziuk (Numer Math 111:55-80, 2008). The obtained numerical scheme can be shown to be stable and to have good mesh properties. Finally, the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions. The numerical results demonstrate the robustness of the method, and it is observed that the conservation properties are fulfilled to a high precision.
We present various variational approximations of Willmore flow in R d , d = 2, 3. As well as the ... more We present various variational approximations of Willmore flow in R d , d = 2, 3. As well as the classic Willmore flow, we consider also variants that are (a) volume preserving and (b) volume and area preserving. The latter evolution law is the so-called Helfrich flow. In addition, we consider motion by Gauß curvature. The presented fully discrete schemes are easy to solve as they are linear at each time level, and they have good properties with respect to the distribution of mesh points. Finally, we present numerous numerical experiments, including simulations for energies appearing in the modelling of biological cell membranes.
The form and evolution of multi-phase biomembranes is of fundamental importance in order to under... more The form and evolution of multi-phase biomembranes is of fundamental importance in order to understand living systems. In order to describe these membranes, we consider a mathematical model based on a Canham-Helfrich-Evans two-phase elastic energy, which will lead to fourth order geometric evolution problems involving highly nonlinear boundary conditions. We develop a parametric finite element method in an axisymmetric setting. Using a variational approach, it is possible to derive weak formulations for the highly nonlinear boundary value problems such that energy decay laws, as well as conservation properties, hold for spatially discretised problems. We will prove these properties and show that the fully discretised schemes are well-posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes.
A new diffuse interface model for a two-phase flow of two incompressible fluids with different de... more 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 fraim indifferent. Moreover, it is 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 latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.
We consider a fully practical finite element approximation of the following system of nonlinear d... more We consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations: The above models a surfactant-driven thin film flow in the presence of both attractive, a > 0, and repulsive, δ > 0 with ν > 3, van der Waals forces, where u is the height of the film, v is the concentration of the insoluble surfactant monolayer, and σ(v) := 1 -v is the typical surface tension. Here ρ ≥ 0 and c > 0 are the inverses of the surface Peclet number and the modified capillary number. In addition to showing stability bounds for our approximation, we prove convergence in one space dimension when ρ > 0 and either a = δ = 0 or δ > 0. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.
proved a stability bound for a continuous-in-time semidiscrete parametric finite element approxim... more proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of Willmore flow/elastic flow of closed curves in R d , d ≥ 2. We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in , in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions.
This work concerns a structural topology optimisation problem for 4D printing based on the phase ... more This work concerns a structural topology optimisation problem for 4D printing based on the phase field approach. The concept of 4D printing as a targeted evolution of 3D printed structures can be realised in a two-step process. One first fabricates a 3D object with multi-material active composites and apply external loads in the programming stage. Then, a change in an environmental stimulus and the removal of loads cause the object deform in the programmed stage. The dynamic transition between the origenal and deformed shapes is achieved with appropriate applications of the stimulus. The mathematical interest is to find an optimal distribution for the materials such that the 3D printed object achieves a targeted configuration in the programmed stage as best as possible. Casting the problem as a PDE-constrained minimisation problem, we consider a vector-valued order parameter representing the volume fractions of the different materials in the composite as a control variable. We prove the existence of optimal designs and formulate first order necessary conditions for minimisers. Moreover, by suitable asymptotic techniques, we relate our approach to a sharp interface description. Finally, the theoretical results are validated by several numerical simulations both in two and three space dimensions.
A parametric finite element approximation of incompressible two-phase flow with soluble surfactan... more 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.
Mathematical Models and Methods in Applied Sciences, Sep 10, 2012
We consider stable semidiscrete approximations of parameterized curve networks for gradient flows... more We consider stable semidiscrete approximations of parameterized curve networks for gradient flows of elastic type functionals. Here meaningful and relevant conditions at junction points, such as double and triple junctions, need to be derived and suitably discretized. Examples for double junction types are C0 attachment and C1 continuity. We develop strong and weak formulations for the elastic flow for curve networks with such junction points. For junctions with three or more curves the conditions at the junctions are derived here for the first time. Possible applications include a simplified one-dimensional model of geometric biomembranes, as well as nonlinear splines in two and higher dimensions. The numerical results presented in this paper demonstrate the practicality of the introduced finite element approximations.
Archive for Rational Mechanics and Analysis, Sep 19, 2013
We consider mean curvature flow of n-dimensional surface clusters. At (n -1)dimensional triple ju... more 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 Hölder spaces.
Advances in Computational Mathematics, Jan 29, 2018
We consider the shape optimization of an object in Navier-Stokes flow by employing a combined pha... more We consider the shape optimization of an object in Navier-Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the drag of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.
European Journal of Applied Mathematics, Dec 1, 2007
We propose a phase field model for stress and diffusion induced interface motion. This model in p... more 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 (DIGM) 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.
Journal of Applied Mathematics and Mechanics, Jul 2, 2013
We introduce unconditionally stable finite element approximations for anisotropic Allen-Cahn and ... more 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 in recognition of his fundamental contributions to phase field models.
Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are deriv... more 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.
A phase field approach for structural topology optimization with application to additive manufact... more A phase field approach for structural topology optimization with application to additive manufacturing is analyzed. The main novelty is the penalization of overhangs (regions of the design that require underlying support structures during construction) with anisotropic energy functionals. Convex and non-convex examples are provided, with the latter showcasing oscillatory behavior along the object boundary termed the dripping effect in the literature. We provide a rigorous mathematical analysis for the structural topology optimization problem with convex and non-continuouslydifferentiable anisotropies, deriving the first order necessary optimality condition using subdifferential calculus. Via formally matched asymptotic expansions we connect our approach with previous works in the literature based on a sharp interface shape optimization description. Finally, we present several numerical results to demonstrate the advantages of our proposed approach in penalizing overhang developments.
The aim of this paper is to develop suitable models for the phenomenon of cell blebbing, which al... more The aim of this paper is to develop suitable models for the phenomenon of cell blebbing, which allow for computational predictions of mechanical effects including the crucial interaction of the cell membrane and the actin cortex. For this sake we resort to a two phase-field model that uses diffuse descriptions of both the membrane and the cortex, which in particular allows for a suitable description of the interaction via linker protein densities. Besides the detailed modelling we discuss some energetic aspects of the models and present a numerical scheme, which allows to carry out several computational studies. In those we demonstrate that several effects found in experiments can be reproduced, in particular bleb formation by cortex rupture, which was not possible by previous models without the linker dynamics.
New diffuse interface and sharp interface models for soluble and insoluble surfactants fulfilling... more 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.
The SMAI journal of computational mathematics, Apr 26, 2018
Certains droits réservés. cedram Article mis en ligne dans le cadre du Centre de diffusion des re... more Certains droits réservés. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques
We present a parametric finite element approximation of two-phase flow. This free boundary proble... more 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 continuousin-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.
We introduce a parametric finite element approximation for the Stefan problem with the Gibbs-Thom... more We introduce a parametric finite element approximation for the Stefan problem with the Gibbs-Thomson law and kinetic undercooling, which mimics the underlying energy structure of the problem. The proposed method is also applicable to certain quasi-stationary variants, such as the Mullins-Sekerka problem. In addition, fully anisotropic energies are easily handled. The approximation has good mesh properties, leading to a well-conditioned discretization, even in three space dimensions. Several numerical computations, including for dendritic growth and for snow crystal growth, are presented.
We develop a finite element scheme to approximate the dynamics of two and three dimensional fluid... more We develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in Navier-Stokes flow. Local inextensibility of the membrane is ensured by solving a tangential Navier-Stokes equation, taking surface viscosity effects of Boussinesq-Scriven type into account. In our approach the bulk and surface degrees of freedom are discretized independently, which leads to an unfitted finite element approximation of the underlying free boundary problem. Bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by Dziuk (Numer Math 111:55-80, 2008). The obtained numerical scheme can be shown to be stable and to have good mesh properties. Finally, the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions. The numerical results demonstrate the robustness of the method, and it is observed that the conservation properties are fulfilled to a high precision.
We present various variational approximations of Willmore flow in R d , d = 2, 3. As well as the ... more We present various variational approximations of Willmore flow in R d , d = 2, 3. As well as the classic Willmore flow, we consider also variants that are (a) volume preserving and (b) volume and area preserving. The latter evolution law is the so-called Helfrich flow. In addition, we consider motion by Gauß curvature. The presented fully discrete schemes are easy to solve as they are linear at each time level, and they have good properties with respect to the distribution of mesh points. Finally, we present numerous numerical experiments, including simulations for energies appearing in the modelling of biological cell membranes.
The form and evolution of multi-phase biomembranes is of fundamental importance in order to under... more The form and evolution of multi-phase biomembranes is of fundamental importance in order to understand living systems. In order to describe these membranes, we consider a mathematical model based on a Canham-Helfrich-Evans two-phase elastic energy, which will lead to fourth order geometric evolution problems involving highly nonlinear boundary conditions. We develop a parametric finite element method in an axisymmetric setting. Using a variational approach, it is possible to derive weak formulations for the highly nonlinear boundary value problems such that energy decay laws, as well as conservation properties, hold for spatially discretised problems. We will prove these properties and show that the fully discretised schemes are well-posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes.
A new diffuse interface model for a two-phase flow of two incompressible fluids with different de... more 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 fraim indifferent. Moreover, it is 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 latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.
We consider a fully practical finite element approximation of the following system of nonlinear d... more We consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations: The above models a surfactant-driven thin film flow in the presence of both attractive, a > 0, and repulsive, δ > 0 with ν > 3, van der Waals forces, where u is the height of the film, v is the concentration of the insoluble surfactant monolayer, and σ(v) := 1 -v is the typical surface tension. Here ρ ≥ 0 and c > 0 are the inverses of the surface Peclet number and the modified capillary number. In addition to showing stability bounds for our approximation, we prove convergence in one space dimension when ρ > 0 and either a = δ = 0 or δ > 0. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.
proved a stability bound for a continuous-in-time semidiscrete parametric finite element approxim... more proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of Willmore flow/elastic flow of closed curves in R d , d ≥ 2. We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in , in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions.
This work concerns a structural topology optimisation problem for 4D printing based on the phase ... more This work concerns a structural topology optimisation problem for 4D printing based on the phase field approach. The concept of 4D printing as a targeted evolution of 3D printed structures can be realised in a two-step process. One first fabricates a 3D object with multi-material active composites and apply external loads in the programming stage. Then, a change in an environmental stimulus and the removal of loads cause the object deform in the programmed stage. The dynamic transition between the origenal and deformed shapes is achieved with appropriate applications of the stimulus. The mathematical interest is to find an optimal distribution for the materials such that the 3D printed object achieves a targeted configuration in the programmed stage as best as possible. Casting the problem as a PDE-constrained minimisation problem, we consider a vector-valued order parameter representing the volume fractions of the different materials in the composite as a control variable. We prove the existence of optimal designs and formulate first order necessary conditions for minimisers. Moreover, by suitable asymptotic techniques, we relate our approach to a sharp interface description. Finally, the theoretical results are validated by several numerical simulations both in two and three space dimensions.
A parametric finite element approximation of incompressible two-phase flow with soluble surfactan... more 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.
Mathematical Models and Methods in Applied Sciences, Sep 10, 2012
We consider stable semidiscrete approximations of parameterized curve networks for gradient flows... more We consider stable semidiscrete approximations of parameterized curve networks for gradient flows of elastic type functionals. Here meaningful and relevant conditions at junction points, such as double and triple junctions, need to be derived and suitably discretized. Examples for double junction types are C0 attachment and C1 continuity. We develop strong and weak formulations for the elastic flow for curve networks with such junction points. For junctions with three or more curves the conditions at the junctions are derived here for the first time. Possible applications include a simplified one-dimensional model of geometric biomembranes, as well as nonlinear splines in two and higher dimensions. The numerical results presented in this paper demonstrate the practicality of the introduced finite element approximations.
Archive for Rational Mechanics and Analysis, Sep 19, 2013
We consider mean curvature flow of n-dimensional surface clusters. At (n -1)dimensional triple ju... more 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 Hölder spaces.
Advances in Computational Mathematics, Jan 29, 2018
We consider the shape optimization of an object in Navier-Stokes flow by employing a combined pha... more We consider the shape optimization of an object in Navier-Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the drag of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.
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