arXiv subjects

Patrik Knopf

Publications and source records attributed to Patrik Knopf.

At least 37 records · Page 2Linked to original sources

arXiv2023

Phase-Field Methods for Spectral Shape and Topology Optimization

We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to represent the shapes over which we minimize. The idea behind this method is to modify the Laplace operator by introducing phase-field dependent coefficients in order to extend the eigenvalue problem on a fixed design domain containing all admissible shapes. The resulting shape and topology optimization problem can then be formulated as an optimal control problem with PDE constraints in which the phase-field function acts as the control. For this optimal control problem, we establish first-order necessary optimality conditions and we rigorously derive its sharp interface limit. Eventually, we present and discuss several numerical simulations for our optimization problem.

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arXiv2022

A diffuse-interface approach for solid-state dewetting with anisotropic surface energies

We present a diffuse-interface model for the solid-state dewetting problem with anisotropic surface energies in ${\mathbb R}^d$ for $d\in\{2,3\}$. The introduced model consists of the anisotropic Cahn--Hilliard equation, with either a smooth or a double-obstacle potential, together with a degenerate mobility function and appropriate boundary conditions on the wall. Upon regularizing the introduced diffuse-interface model, and with the help of suitable asymptotic expansions, we recover as the sharp-interface limit the anisotropic surface diffusion flow for the interface together with an anisotropic Young's law and a zero-flux condition at the contact line of the interface with a fixed external boundary. Furthermore, we show the existence of weak solutions for the regularized model, for both smooth and obstacle potential. Numerical results based on an appropriate finite element approximation are presented to demonstrate the excellent agreement between the proposed diffuse-interface model and its sharp-interface limit.

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arXiv2022

Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms

We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn-Hilliard type equation to describe the tumor evolution, a velocity equation for the associated volume-averaged velocity field, and a convective reaction-diffusion type equation to describe the density of the chemical species. The velocity equation is either represented by Darcy's law or by the Brinkman equation. We first construct a global weak solution of the multiphase Cahn-Hilliard-Brinkman model. After that, we show that such weak solutions of the system converge to a weak solution of the multiphase Cahn-Hilliard-Darcy system as the viscosities tend to zero in some suitable sense. This means that the existence of a global weak solution to the Cahn-Hilliard-Darcy system is also established.

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arXiv2021

On the two and one-half dimensional Vlasov-Poisson system with an external magnetic field: global well-posedness and stability of confined steady states

The time evolution of a two-component collisionless plasma is modeled by the Vlasov-Poisson system. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that an external magnetic field is present in order to confine the plasma in a given infinitely long cylinder. After discussing global well-posedness of the corresponding Cauchy problem, we construct stationary solutions which indeed have support away from their confinement device. Then, in the main part of this work we investigate the stability of such steady states, both with respect to perturbations in the initial data, where we employ the energy-Casimir method, and also with respect to perturbations in the external magnetic field.

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arXiv2021

On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems. We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliard equation with dynamic boundary conditions.

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arXiv2021

Strong well-posedness, stability and optimal control theory for a mathematical model for magneto-viscoelastic fluids

In this article, we study the strong well-posedness, stability and optimal control of an incompressible magneto-viscoelastic fluid model in two dimensions. The model consists of an incompressible Navier--Stokes equation for the velocity field, an evolution equation for the deformation tensor, and a gradient flow equation for the magnetization vector. First, we prove that the model under consideration posseses a global strong solution in a suitable functional framework. Second, we derive stability estimates with respect to an external magnetic field. Based on the stability estimates we use the external magnetic field as the control to minimize a cost functional of tracking-type. We prove existence of an optimal control and derive first-order necessary optimality conditions. Finally, we consider a second optimal control problem, where the external magnetic field, which represents the control, is generated by a finite number of fixed magnetic field coils.

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arXiv2021

Phase-field dynamics with transfer of materials: The Cahn--Hilliard equation with reaction rate dependent dynamic boundary conditions

The Cahn--Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn--Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein, Miranville and Schimperna (Physica D, 2011) and the model by Liu and Wu (Arch.~Ration.~Mech.~Anal., 2019). Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.

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arXiv2021

Convergence of a Robin boundary approximation for a Cahn--Hilliard system with dynamic boundary conditions

We prove the existence of unique weak solutions to an extension of a Cahn--Hilliard model proposed recently by C.~Liu and H.~Wu (2019), in which the new dynamic boundary condition is further generalised with an affine linear relation between the surface and bulk order parameters. As a first approach to tackle more general and nonlinear relations, we investigate the existence of unique weak solutions to a regularisation by a Robin boundary condition. Included in our analysis is the case where there is no diffusion for the surface order parameter, which causes new difficulties for the analysis of the Robin system. Furthermore, for the case of affine linear relations, we show the weak convergence of solutions as the regularisation parameter tends to zero, and derive an error estimate between the two models. This is supported by numerical experiments which also demonstrate some non-trivial dynamics for the extended Liu--Wu model that is not present in the original model.

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arXiv2021

On the nonlocal Cahn--Hilliard equation with nonlocal dynamic boundary condition and boundary penalization

The Cahn--Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. In recent times, various dynamic boundary conditions have been introduced to model interactions of the materials with the boundary more precisely. To take long-range interactions of the materials into account, we propose a new model consisting of a nonlocal Cahn--Hilliard equation subject to a nonlocal dynamic boundary condition that is also of Cahn--Hilliard type and contains an additional boundary penalization term. We rigorously derive our model as the gradient flow of a nonlocal total free energy with respect to a suitable inner product of order $H^{-1}$ which contains both bulk and surface contributions. The total free energy is considered as nonlocal since it comprises convolutions in the bulk and on the surface of the phase-field variables with certain interaction kernels. The main difficulties arise from defining a suitable kernel on the surface and from handling the resulting boundary convolution. In the main model, the chemical potentials in the bulk and on the surface are coupled by a Robin type boundary condition depending on a specific relaxation parameter related to the rate of chemical reactions. We prove weak and strong well-posedness of this system, and we investigate the singular limits attained when the relaxation parameter tends to zero or infinity. By this approach, we also obtain weak and strong well-posedness of the corresponding limit systems.

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arXiv2020

Weak solutions of the Cahn-Hilliard system with dynamic boundary conditions: A gradient flow approach

The Cahn-Hilliard equation is the most common model to describe phase separation processes of a mixture of two components. For a better description of short-range interactions of the material with the solid wall, various dynamic boundary conditions have been considered in recent times. New models with dynamic boundary conditions have been proposed recently by C. Liu and H. Wu [arXiv:1710.08318]. We prove the existence of weak solutions to these new models by interpreting the problem as a suitable gradient flow of a total free energy which contains volume as well as surface contributions. The formulation involves an inner product which couples bulk and surface quantities in an appropriate way. We use an implicit time discretization and show that the obtained approximate solutions converge to a weak solution of the Cahn-Hilliard system. This allows us to substantially improve earlier results which needed strong assumptions on the geometry of the domain. Furthermore, we prove that this weak solution is unique.

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arXiv2020

Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth

In this paper, we study a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn-Hilliard type equation for the phase field variable coupled to a reaction diffusion equation for the nutrient and a Brinkman type equation for the velocity. The system is equipped with homogeneous Neumann boundary conditions for the tumor variable, the chemical potential and the nutrient as well as a "no-friction" boundary condition for the velocity. The control acts as a medication by cytotoxic drugs and enters the phase field equation. The cost functional is of standard tracking type and is designed to track the phase field variable during the evolution and at some fixed final time. We prove that the model satisfies the basics for calculus of variations and we establish first-order and second-order conditions for local optimality. Moreover, we present a globality condition for critical controls and we show that the optimal control is unique on small time intervals.

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arXiv2020

Shape and Topology Optimization involving the eigenvalues of an elastic structure: A multi-phase-field approach

A cost functional involving the eigenvalues of an elastic structure, that is described by a multi-phase-field equation, is optimized. This allows us to handle topology changes and multiple materials. We prove continuity and differentiability of the eigenvalues and we establish the existence of a global minimizer to our optimization problem. We further derive first-order necessary optimality conditions for local minimizers. Moreover, an optimization problem combining eigenvalue and compliance optimization is also discussed.

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arXiv2019

Stratified periodic water waves with singular density gradients

We consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis.

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arXiv2019

Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation

In this paper, we study a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn-Hilliard type equation for the phase field variable coupled to a reaction diffusion equation for the nutrient and a Brinkman type equation for the velocity. The system is equipped with homogeneous Neumann boundary conditions for the tumor variable and the chemical potential, Robin boundary conditions for the nutrient and a "no-friction" boundary condition for the velocity. The control acts as a medication by cytotoxic drugs and enters the phase field equation. The cost functional is of standard tracking type and is designed to track the variables of the state equation during the evolution and the distribution of tumor cells at some fixed final time. We prove that the model satisfies the basics for calculus of variations and we establish first-order necessary optimality conditions for the optimal control problem.

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arXiv2019

Confined steady states of a Vlasov-Poisson plasma in an infinitely long cylinder

We consider the two-dimensional Vlasov-Poisson system to model a two-component plasma whose distribution function is constant with respect to the third space dimension. First, we show how this two-dimensional Vlasov-Poisson system can be derived from the full three-dimensional system. The existence of compactly supported steady states with vanishing electric potential in a three-dimensional setting has already been investigated by A. L. Skubachevskii [15]. We show that his approach can easily be adapted to the two-dimensional system. However, our main result is to prove the existence of compactly supported steady states even with a nontrivial self-consistent electric potential.

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arXiv2018

Optimal control of a Vlasov-Poisson plasma by fixed magnetic field coils

We consider the Vlasov-Poisson system that is equipped with an external magnetic field to describe the time evolution of the distribution function of a plasma. An optimal control problem where the external magnetic field is the control itself has already been investigated in [arXiv:1708.02464] and [arXiv:1708.02753]. However, in real technical applications it will not be possible to choose the control field in such a general fashion as it will be induced by fixed field coils. In this paper we will use the fundamentals that were established in [arXiv:1708.02464] and [arXiv:1708.02753] to analyze an optimal control problem where the magnetic field is a superposition of the fields that are generated by N fixed magnetic field coils. Thereby, the aim is to control the plasma in such a way that its distribution function matches a desired distribution function at some certain final time T as closely as possible.

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arXiv2018

Optimal control of a Vlasov-Poisson plasma by an external magnetic field

The aim of various technical applications (for example fusion research) is to control a plasma by magnetic fields in a desired fashion. In our model the plasma is described by the Vlasov-Poisson system that is equipped with an external magnetic field. We will prove that this model satisfies some basic properties that are necessary for calculus of variations. After that, we will analyze an optimal control problem with a tracking type cost functional with respect to the following topics: Necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality, uniqueness of the optimal control under certain conditions.

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arXiv2017

Optimal control of a Vlasov-Poisson plasma by an external magnetic field - Analysis of a tracking type optimal control problem

In the paper "Optimal control of a Vlasov-Poisson plasma by an external magnetic field - The basics for variational calculus" [arXiv:1708.02464] we have already introduced a set of admissible magnetic fields and we have proved that each of those fields induces a unique strong solution of the Vlasov-Poisson system. We have also established that the field-state operator that maps any admissible field onto its corresponding solution is continuous and weakly compact. In this paper we will show that this operator is also Fréchet differentiable and we will continue to analyze the optimal control problem that was introduced in [arXiv:1708.02464]. More precisely, we will establish necessary and sufficient conditions for local optimality and we will show that an optimal solution is unique under certain conditions.

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