Dissertationenhttps://kobra.uni-kassel.de:443/handle/123456789/20100617333542024-03-19T09:57:10Z2024-03-19T09:57:10ZA Unifying Theory for Runge-Kutta-like Time Integrators: Convergence and StabilityIzgin, Thomashttps://kobra.uni-kassel.de:443/handle/123456789/154742024-02-21T15:24:47Z2024-01-01T00:00:00ZThe work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are allowed to depend on the solution and the step size. As a result of this, we also refer to them as non-standard additive RK (NSARK) methods. We motivate and introduce modified Patankar (MP) schemes as a subclass of NSARK methods and emphasize their importance. The first major part of this thesis is then dedicated to providing a tool for deriving order conditions for NSARK methods. The proposed approach may yield implicit order conditions, which can be rewritten in explicit form using the NB-series of the stages. The obtained explicit order conditions can be further reduced using Gröbner bases computations. With the presented approach, it was possible for the first time to obtain conditions for the construction of 3rd and 4th order Geometric Conservative (GeCo) as well as 4th order MPRK schemes. Moreover, a new fourth order MPRK method is constructed using our theory and the order of convergence is validated numerically. The second major part is concerned with the stability of nonlinear time integrators preserving at least one linear invariant. We discus how the given approach generalizes the notion of A-stability. We are able to prove that investigating the Jacobian of the generating map is sufficient to understand the stability of the nonlinear method in a neighborhood of the steady state. This approach allows for the first time the investigation of several modified Patankar schemes such as MPRK, SSPMPRK, MPDeC, GeCo and gBBKS methods. During that analysis we additionally demonstrate that GeCo2 and gBBKS methods are not in C^2 and prove asymptotic stability for GeCo1 schemes and, for some PDRS, also for MPRK methods. In the particular case of MPRK schemes, we compute a general stability function in a way that can be easily adapted to the case of PDRS. Finally, our findings support the numerically observed robustness of MP methods while we are able to provide sharp bounds on the time step in the case of conditional stability. Last but not least, the approach from the theory of dynamic systems is used to derive a necessary condition for avoiding unrealistic oscillations of the numerical approximation.
2024-01-01T00:00:00ZIzgin, ThomasThe work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are allowed to depend on the solution and the step size. As a result of this, we also refer to them as non-standard additive RK (NSARK) methods. We motivate and introduce modified Patankar (MP) schemes as a subclass of NSARK methods and emphasize their importance. The first major part of this thesis is then dedicated to providing a tool for deriving order conditions for NSARK methods. The proposed approach may yield implicit order conditions, which can be rewritten in explicit form using the NB-series of the stages. The obtained explicit order conditions can be further reduced using Gröbner bases computations. With the presented approach, it was possible for the first time to obtain conditions for the construction of 3rd and 4th order Geometric Conservative (GeCo) as well as 4th order MPRK schemes. Moreover, a new fourth order MPRK method is constructed using our theory and the order of convergence is validated numerically. The second major part is concerned with the stability of nonlinear time integrators preserving at least one linear invariant. We discus how the given approach generalizes the notion of A-stability. We are able to prove that investigating the Jacobian of the generating map is sufficient to understand the stability of the nonlinear method in a neighborhood of the steady state. This approach allows for the first time the investigation of several modified Patankar schemes such as MPRK, SSPMPRK, MPDeC, GeCo and gBBKS methods. During that analysis we additionally demonstrate that GeCo2 and gBBKS methods are not in C^2 and prove asymptotic stability for GeCo1 schemes and, for some PDRS, also for MPRK methods. In the particular case of MPRK schemes, we compute a general stability function in a way that can be easily adapted to the case of PDRS. Finally, our findings support the numerically observed robustness of MP methods while we are able to provide sharp bounds on the time step in the case of conditional stability. Last but not least, the approach from the theory of dynamic systems is used to derive a necessary condition for avoiding unrealistic oscillations of the numerical approximation.Finite Element Simulations for the Design of Therapeutic Approaches for Retinal DiseasesDörsam, Simonhttps://kobra.uni-kassel.de:443/handle/123456789/148492023-06-27T07:10:42Z2022-01-01T00:00:00ZThe retinal disease age-related macular degeneration is the most common cause of vision loss in industrialized countries. In this thesis, motivated by the drug (antibody) treatment of this disease, we designed long-term three dimensional Finite Element simulations of the drug distribution in the healthy human eye. The underlying model consists of a time-dependent convection-diffusion equation coupled to a stationary Darcy equation describing the flow of the aqueous humor through the vitreous medium. We replaced the in general used Dirichlet boundary condition for the pressure with an inhomogeneous Neumann boundary condition for the velocity to obtain a more realistic description of the flow. The influence of collagen fibers inside the vitreous on drug distribution is accounted for by anisotropic diffusion and the gravity via an additional transport term. The resulting coupled model was solved in a decoupled way: first the Darcy equation by using mixed finite elements, then the convection-diffusion equation by using trilinear Lagrange elements. Krylov subspace methods are used to solve the resulting algebraic system. To cope with large time steps we applied the strong A-stable fractional step theta scheme. With this strategy we achieved second order of convergence in time and space. The developed simulations were used for process optimization for which specific output functionals were evaluated. We found out that an unfavorable injection angle can result in 38% less drug reaching the macula. This thesis provides the first steps toward virtual exploration and improvement of therapy for age-related macular degeneration and for other retinal diseases.; In den Industrieländern ist die altersbedingte Makuladegeneration die häufigste Ursache für Neuerblindung. Die Behandlung mit Medikamenten als Anti-VEGF Blocker gilt als eine der wichtigsten Therapien der altersbedingten Makuladegeneration und anderer Netzhauterkrankungen. In dieser Arbeit haben wir dreidimensionale Langzeitsimulationen der Medikamentenverteilung im menschlichen Auge erstellt mit der Finite-Elemente Methode. Das zugehörige Modell besteht aus einer zeitabhängigen Konvektions-Diffusions-Gleichung, die mit einer stationären Darcy-Gleichung gekoppelt ist, was die Kammerwasserströmung im Glaskörper beschreibt. Wir haben die im allgemeinen benutzte konstante Dirichlet Druckrandbedingung ersetzt durch eine inhomogene Neumann Randbedingung für das Geschwindigkeitseinströmungsprofil, um eine realistischere Beschreibung der Strömung zu erhalten. Die Einflüsse der Kollagenfasern des Glaskörpers auf die Medikamentenausbreitung wurde durch anisotrope Diffusion modelliert und die Wirkung der Schwerkraft auf das Medikament durch einen zusätzlichen Transportterm dargestellt. Dieses mathematische Modell wurde entkoppelt gelöst. Zuerst berechneten wir die numerische Lösung der zeitunabhängigen Darcy-Gleichung mit gemischten Finiten-Elementen, dann die von der Konvektion-Diffusions-Gleichung mit einer Ortsdiskretisierung von trilinearen Lagrange-Elementen. In der Zeitdiskretisierung wurde das stark A-stabile Teilschritt-Theta Verfahren benutzt, welches auch noch für große Zeitschritte geeignet ist. Die Krylov-Unterraum-Verfahren sind dann verwendet worden, für die zu lösenden linearen Gleichungssysteme in jedem Zeitschritt. Unsere Simulationen wurden für die Optimierung von Prozessen in der Anwendung genutzt, wofür auch spezifische Ausgabefunktionale entwickelt worden sind. Wir haben herausgefunden, dass ein ungünstiger Injektionswinkel dazu führen kann, dass 38 Prozent weniger am Medikament die Makula erreicht. Diese Arbeit liefert die ersten Schritte, um die Therapie der altersbedingten Makuladegeneration oder anderer Netzhauterkrankungen virtuell zu erforschen und zu verbessern.
2022-01-01T00:00:00ZDörsam, SimonThe retinal disease age-related macular degeneration is the most common cause of vision loss in industrialized countries. In this thesis, motivated by the drug (antibody) treatment of this disease, we designed long-term three dimensional Finite Element simulations of the drug distribution in the healthy human eye. The underlying model consists of a time-dependent convection-diffusion equation coupled to a stationary Darcy equation describing the flow of the aqueous humor through the vitreous medium. We replaced the in general used Dirichlet boundary condition for the pressure with an inhomogeneous Neumann boundary condition for the velocity to obtain a more realistic description of the flow. The influence of collagen fibers inside the vitreous on drug distribution is accounted for by anisotropic diffusion and the gravity via an additional transport term. The resulting coupled model was solved in a decoupled way: first the Darcy equation by using mixed finite elements, then the convection-diffusion equation by using trilinear Lagrange elements. Krylov subspace methods are used to solve the resulting algebraic system. To cope with large time steps we applied the strong A-stable fractional step theta scheme. With this strategy we achieved second order of convergence in time and space. The developed simulations were used for process optimization for which specific output functionals were evaluated. We found out that an unfavorable injection angle can result in 38% less drug reaching the macula. This thesis provides the first steps toward virtual exploration and improvement of therapy for age-related macular degeneration and for other retinal diseases.
In den Industrieländern ist die altersbedingte Makuladegeneration die häufigste Ursache für Neuerblindung. Die Behandlung mit Medikamenten als Anti-VEGF Blocker gilt als eine der wichtigsten Therapien der altersbedingten Makuladegeneration und anderer Netzhauterkrankungen. In dieser Arbeit haben wir dreidimensionale Langzeitsimulationen der Medikamentenverteilung im menschlichen Auge erstellt mit der Finite-Elemente Methode. Das zugehörige Modell besteht aus einer zeitabhängigen Konvektions-Diffusions-Gleichung, die mit einer stationären Darcy-Gleichung gekoppelt ist, was die Kammerwasserströmung im Glaskörper beschreibt. Wir haben die im allgemeinen benutzte konstante Dirichlet Druckrandbedingung ersetzt durch eine inhomogene Neumann Randbedingung für das Geschwindigkeitseinströmungsprofil, um eine realistischere Beschreibung der Strömung zu erhalten. Die Einflüsse der Kollagenfasern des Glaskörpers auf die Medikamentenausbreitung wurde durch anisotrope Diffusion modelliert und die Wirkung der Schwerkraft auf das Medikament durch einen zusätzlichen Transportterm dargestellt. Dieses mathematische Modell wurde entkoppelt gelöst. Zuerst berechneten wir die numerische Lösung der zeitunabhängigen Darcy-Gleichung mit gemischten Finiten-Elementen, dann die von der Konvektion-Diffusions-Gleichung mit einer Ortsdiskretisierung von trilinearen Lagrange-Elementen. In der Zeitdiskretisierung wurde das stark A-stabile Teilschritt-Theta Verfahren benutzt, welches auch noch für große Zeitschritte geeignet ist. Die Krylov-Unterraum-Verfahren sind dann verwendet worden, für die zu lösenden linearen Gleichungssysteme in jedem Zeitschritt. Unsere Simulationen wurden für die Optimierung von Prozessen in der Anwendung genutzt, wofür auch spezifische Ausgabefunktionale entwickelt worden sind. Wir haben herausgefunden, dass ein ungünstiger Injektionswinkel dazu führen kann, dass 38 Prozent weniger am Medikament die Makula erreicht. Diese Arbeit liefert die ersten Schritte, um die Therapie der altersbedingten Makuladegeneration oder anderer Netzhauterkrankungen virtuell zu erforschen und zu verbessern.Analysis of a Coupled Fluid-Elastic Interaction ProblemLuckas, Michelle Melaniehttps://kobra.uni-kassel.de:443/handle/123456789/143882023-03-15T12:57:26Z2023-01-01T00:00:00ZIn this thesis, a non-linear system of partial differential equations is studied, describing the motions of an elastic structure which is immersed into an incompressible viscous fluid. The displacement of the elastic structure is modelled by a Lamé system and the fluid velocity as well as the fluid pressure are described by the Navier-Stokes equations. The structure and the fluid are coupled via two boundary conditions at the interface which correspond to continuity of velocities and forces. As the elasticity is linearized in the Lamé system, the idea of geometric linearization is used to state the equations on constant domains.
In Chapter 2, the existence of local strong solutions to this system is proven. First, the system is decoupled such that the Lamé system can be solved by using a hidden regularity result and a maximal regularity result for the Stokes system can be applied to solve the fluid part. Then an approximation and a fixed point argument are used to construct a local solution to the coupled linearized system. Finally, the solvability of the non-linear system is concluded by applying another fixed point argument. Provided that the initial data are suffciently small, the existence of global solutions is proven in Chapter 3. The proof is based on energy and higher order estimates which can be derived for the system. As a Corollary, it is shown that the fluid velocity converges to zero for large times.
In Chapter 4, the main result of this thesis is stated. It proves that also the elastic displacement converges for large times and that the limit depends on the geometry of the underlying domain. For a large class of good domains, only a stationary part and an artificial rigid velocity remain for large times. If the structure domain is not good (e.g. if it is a sphere), also a pressure wave with a periodic structure can remain. A new method is derived to prove this convergence result: As the elastic displacement is not necessarily globally bounded in L2, convergence of the time derivative of the elastic displacement is shown first, using the structure of the Lamé system as well as the energy estimate of the coupled system. Then the convergence is lifted to the level of the elastic displacement. The main result is first proven for the simpler case of a good domain, before it is extended to any possible geometry.
2023-01-01T00:00:00ZLuckas, Michelle MelanieIn this thesis, a non-linear system of partial differential equations is studied, describing the motions of an elastic structure which is immersed into an incompressible viscous fluid. The displacement of the elastic structure is modelled by a Lamé system and the fluid velocity as well as the fluid pressure are described by the Navier-Stokes equations. The structure and the fluid are coupled via two boundary conditions at the interface which correspond to continuity of velocities and forces. As the elasticity is linearized in the Lamé system, the idea of geometric linearization is used to state the equations on constant domains.
In Chapter 2, the existence of local strong solutions to this system is proven. First, the system is decoupled such that the Lamé system can be solved by using a hidden regularity result and a maximal regularity result for the Stokes system can be applied to solve the fluid part. Then an approximation and a fixed point argument are used to construct a local solution to the coupled linearized system. Finally, the solvability of the non-linear system is concluded by applying another fixed point argument. Provided that the initial data are suffciently small, the existence of global solutions is proven in Chapter 3. The proof is based on energy and higher order estimates which can be derived for the system. As a Corollary, it is shown that the fluid velocity converges to zero for large times.
In Chapter 4, the main result of this thesis is stated. It proves that also the elastic displacement converges for large times and that the limit depends on the geometry of the underlying domain. For a large class of good domains, only a stationary part and an artificial rigid velocity remain for large times. If the structure domain is not good (e.g. if it is a sphere), also a pressure wave with a periodic structure can remain. A new method is derived to prove this convergence result: As the elastic displacement is not necessarily globally bounded in L2, convergence of the time derivative of the elastic displacement is shown first, using the structure of the Lamé system as well as the energy estimate of the coupled system. Then the convergence is lifted to the level of the elastic displacement. The main result is first proven for the simpler case of a good domain, before it is extended to any possible geometry.Mathematical Modelling and Adaptive Finite Element Simulation of Viscoelastic Fluid-Structure Interaction Systems and Chemical Processes with Applications to OphthalmologyDrobny, Alexanderhttps://kobra.uni-kassel.de:443/handle/123456789/143672023-01-13T14:37:12Z2023-01-01T00:00:00ZThe aim of this thesis is the numerical analysis of nonlinear coupled partial differential equations and their application to ophthalmology. Firstly, we consider fluid-structure interaction problems where the fluid is either Newtonian or viscoelastic. The structure is modelled as a hyperelastic material. The application to ophthalmology lies in the interaction of the vitreous with its surrounding elastic structures like the sclera and the lens. The underlying flow in the vitreous is modelled by a viscoelastic Burgers model or by the Newtonian Navier-Stokes equations depending on the pathology of the eye. Secondly, we consider chemical processes in the context of the treatment of age related macular degeneration which is one of the most common reasons for blindness. We derive a two-compartment model consisting of the vitreous and the retina. The pharmacokinetic and the pharmacodynamic is modelled by a system of convection-diffusion-reaction equations in the vitreous and by a system of diffusion-reaction equations in the retina. Both topics lead to nonlinear partial differential equations in each subdomain with an additional coupling on the common interfaces. To cope with this in the context of viscoelastic fluid-structure interaction we employ the arbitrary Lagrangian Eulerian transformation. The numerical methods are based on the derivation of monolithic variational formulations and the linearization using Newton’s method. The temporal discretization is realized using single-step methods like the Crank-Nicolson scheme. For the spatial discretization we use the finite element method. The numerical methods are analysed on various benchmark problems. Furthermore we apply the dual-weighted residual method to the considered problems and examine its performance when used for adaptive mesh refinement. The results show the superiority of the dual-weighted residual method over global mesh refinement for all considered problems. Additionally the benchmark simulations on viscoelastic fluid-structure interaction show that the viscoelasticity of the fluid significantly influences the drag, the lift and the displacement of the beam and as a results also influences the fluid flow itself through the two-way coupling of the structure and the fluid. For viscoelastic fluid-structure interaction in the human eye the numerical simulations show that the maximal stress in the viscoelastic vitreous is about 6:5 times higher than in the vitrectomized vitreous. Regarding the drug therapy we compare two commonly used drugs, namely aflibercept and ranibizumab concerning their efficacy and analyse different dosages for the drug. We show among others that only about 20% of the drug reaches the retina through the inner-limiting membrane and that 50% of the concentration of the disease has been rebuilt in the retina after 38:19 days for a single drug injection.
2023-01-01T00:00:00ZDrobny, AlexanderThe aim of this thesis is the numerical analysis of nonlinear coupled partial differential equations and their application to ophthalmology. Firstly, we consider fluid-structure interaction problems where the fluid is either Newtonian or viscoelastic. The structure is modelled as a hyperelastic material. The application to ophthalmology lies in the interaction of the vitreous with its surrounding elastic structures like the sclera and the lens. The underlying flow in the vitreous is modelled by a viscoelastic Burgers model or by the Newtonian Navier-Stokes equations depending on the pathology of the eye. Secondly, we consider chemical processes in the context of the treatment of age related macular degeneration which is one of the most common reasons for blindness. We derive a two-compartment model consisting of the vitreous and the retina. The pharmacokinetic and the pharmacodynamic is modelled by a system of convection-diffusion-reaction equations in the vitreous and by a system of diffusion-reaction equations in the retina. Both topics lead to nonlinear partial differential equations in each subdomain with an additional coupling on the common interfaces. To cope with this in the context of viscoelastic fluid-structure interaction we employ the arbitrary Lagrangian Eulerian transformation. The numerical methods are based on the derivation of monolithic variational formulations and the linearization using Newton’s method. The temporal discretization is realized using single-step methods like the Crank-Nicolson scheme. For the spatial discretization we use the finite element method. The numerical methods are analysed on various benchmark problems. Furthermore we apply the dual-weighted residual method to the considered problems and examine its performance when used for adaptive mesh refinement. The results show the superiority of the dual-weighted residual method over global mesh refinement for all considered problems. Additionally the benchmark simulations on viscoelastic fluid-structure interaction show that the viscoelasticity of the fluid significantly influences the drag, the lift and the displacement of the beam and as a results also influences the fluid flow itself through the two-way coupling of the structure and the fluid. For viscoelastic fluid-structure interaction in the human eye the numerical simulations show that the maximal stress in the viscoelastic vitreous is about 6:5 times higher than in the vitrectomized vitreous. Regarding the drug therapy we compare two commonly used drugs, namely aflibercept and ranibizumab concerning their efficacy and analyse different dosages for the drug. We show among others that only about 20% of the drug reaches the retina through the inner-limiting membrane and that 50% of the concentration of the disease has been rebuilt in the retina after 38:19 days for a single drug injection.