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- ItemA model of an electrochemical flow cell with porous layer(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Ehrhardt, Matthias; Fuhrmann, Jürgen; Linke, AlexanderIn this paper we discuss three different mathematical models for fluid-porous interfaces in a simple channel geometry that appears e.g. in thin-layer channel flow cells. Here the difficulties arise from the possibly different orders of the corresponding differential operators in the different domains. A finite volume discretization of this model allows to calculate the limiting current of the H_2 oxidation in a porous electrode with platinum catalyst particles.
- ItemA gradient-robust well-balanced scheme for the compressible isothermal Stokes problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Akbas, Mine; Gallouët, Thierry; Gaßmann, Almut; Linke, Alexander; Merdon, ChristianA novel notion for constructing a well-balanced scheme --- a gradient-robust scheme --- is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient --- if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM-FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straight-forward extension to barotropic situations with nonlinear equations of state is feasible.
- ItemAssessment of reduced order Kalman filter for parameter identification in one-dimensional blood flow models using experimental data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Caiazzo, Alfonso; Caforio, Federica; Montecinos, Gino; Müller, Lucas O.; Blanco, Pablo J.; Toro, Eleutero F.This work presents a detailed investigation of a parameter estimation approach based on the reduced order unscented Kalman filter (ROUKF) in the context of one-dimensional blood flow models. In particular, the main aims of this study are (i) to investigate the effect of using real measurements vs. synthetic data (i.e., numerical results of the same in silico model, perturbed with white noise) for the estimation and (ii) to identify potential difficulties and limitations of the approach in clinically realistic applications in order to assess the applicability of the filter to such setups. For these purposes, our numerical study is based on the in vitro model of the arterial network described by [Alastruey et al. 2011, J. Biomech. 44], for which experimental flow and pressure measurements are available at few selected locations. In order to mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Youngs modulus and thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis based on the generalized sensitivity function, comparing then the results obtained with the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.
- ItemHighly accurate quadrature-based Scharfetter-Gummel schemes for charge transport in degenerate semiconductors(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Patriarca, Matteo; Farrell, Patricio; Fuhrmann, Jürgen; Koprucki, ThomasWe introduce a family of two point flux expressions for charge carrier transport described by drift-diffusion problems in degenerate semiconductors with non-Boltzmann statistics which can be used in Voronoi finite volume discretizations. In the case of Boltzmann statistics, Scharfetter and Gummel derived such fluxes by solving a linear two point boundary value problem yielding a closed form expression for the flux. Instead, a generalization of this approach to the nonlinear case yields a flux value given implicitly as the solution of a nonlinear integral equation. We examine the solution of this integral equation numerically via quadrature rules to approximate the integral as well as Newtons method to solve the resulting approximate integral equation. This approach results into a family of quadrature-based Scharfetter-Gummel flux approximations. We focus on four quadrature rules and compare the resulting schemes with respect to execution time and accuracy. A convergence study reveals that the solution of the approximate integral equation converges exponentially in terms of the number of quadrature points. With very few integration nodes they are already more accurate than a state-of-the-art reference flux, especially in the challenging physical scenario of high nonlinear diffusion. Finally, we show that thermodynamic consistency is practically guaranteed.
- ItemTransient numerical simulation of sublimation growth of SiC bulk single crystals : modeling, finite volume method, results(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2003) Philip, PeterThis work treats transient numerical simulation of growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (also called the modified Lely method). A transient mathematical model of the growth process is presented. Subsequently, the finite volume method for the discretization of evolution equations, which constitutes the basis for the numerical simulations presented in this work, is studied mathematically, proving the existence of discrete solutions. All material data used for numerical simulations in this work are collected in the appendix. Starting with a description of the physical growth procedure, problems arising during the growth process are discussed as well as techniques that are used for process control. It is explained why numerical simulation is an important tool for control, and the advantages of a transient approach are considered. Within the presented transient model, continuous mixture theory is used to obtain balance equations for energy, mass, and momentum inside the gas phase. In particular, reaction-diffusion equations are deduced. Heat conduction is treated inside solid materials. Heat transport by radiation is modeled via the net radiation method for diffuse-gray radiation to allow for radiative heat transfer between the surfaces of cavities. The model includes the semi-transparency of the single crystal via a band approximation. Induction heating is modeled by an axisymmetric complex-valued magnetic scalar potential that is determined as the solution of an elliptic problem. The resulting heat source distribution is calculated from the magnetic potential. The heat sources are updated continuously during the solution of the transient problem for the temperature evolution to allow for changes in the electrical conductivity depending on temperature and for changes due to a moving induction coil. The finite volume method is treated in a rigorous mathematical framework. It allows the discretization of parabolic, hyperbolic, and elliptic partial differential equations, as they arise from the mathematical model of the growth process, including nonlocal contributions due to radiative heat transfer. The general abstract setting consists of a system of nonlinear evolution equations in arbitrary finite space dimension, each evolution equation living on a different polytope domain. In general, each evolution equation has diffusive and convective contributions as well as source and sink terms. Each contribution is permitted to depend on the solution. Discontinuities of the solution are allowed at domain interfaces. Interface conditions in terms of the solution and its flux are considered. Moreover, nonlocal interface conditions are considered. Outer boundary conditions include Dirichlet conditions, flux conditions, emission conditions, and nonlocal conditions. Time discretization is performed by an implicit Euler scheme, where an explicit discretization is allowed in certain dependencies such that the temperature-dependent emissivities can be taken from the previous time step. As usual, the space discretization is performed by integrating the evolution equations over control volumes and then using quadrature formulas. As an axisymmetric setting and cylindrical coordinates are used in the simulations, a treatment of change of variables is included in the abstract considerations. For the case that the evolution equations constitute nonlinear heat equations, still allowing nonlinear diffusion, convection, and source and sink terms, as well as nonlocal interface and boundary conditions as they arise from modeling radiative heat transfer, discrete L∞ - L1 a priori estimates are established for the system resulting from the finite volume discretization. A fixed point argument is then used to prove the existence and uniqueness of discrete solutions. The presented numerical simulations are conducted in an axisymmetric setting. They constitute transient investigations of control parameters affecting the temperature evolution during the heating of the growth apparatus. A cylindrically symmetric finite volume scheme provides the discretization for both the transient nonlinear heat problem and the stationary magnetic potential problem. For different heating powers and different vertical coil positions, the temperature evolution is monitored at the surface of the crystal and at the surface of the source powder as well as at the top and at the bottom of the growth apparatus. It is studied how the temperature difference between source and seed, which is highly relevant to the growth process, is related to the measurable temperature difference between bottom and top. Results concerning the time lack between the heating of the surface of the source powder and the heating of its interior are considered. Finally, the global evolution of temperature and heat sources is investigated.
- ItemAssessing the quality of the excess chemical potential flux scheme for degenerate semiconductor device simulation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Abdel, Dilara; Farrell, Patricio; Fuhrmann, JürgenThe van Roosbroeck system models current flows in (non-)degenerate semiconductor devices. Focusing on the stationary model, we compare the excess chemical potential discretization scheme, a flux approximation which is based on a modification of the drift term in the current densities, with another state-of-the-art Scharfetter-Gummel scheme, namely the diffusion-enhanced scheme. Physically, the diffusion-enhanced scheme can be interpreted as a flux approximation which modifies the thermal voltage. As a reference solution we consider an implicitly defined integral flux, using Blakemore statistics. The integral flux refers to the exact solution of a local two point boundary value problem for the continuous current density and can be interpreted as a generalized Scharfetter-Gummel scheme. All numerical discretization schemes can be used within a Voronoi finite volume method to simulate charge transport in (non-)degenerate semiconductor devices. The investigation includes the analysis of Taylor expansions, a derivation of error estimates and a visualization of errors in local flux approximations to extend previous discussions. Additionally, drift-diffusion simulations of a p-i-n device are performed.
- ItemMAC schemes on triangular Delaunay meshes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Eymard, Robert; Fuhrmann, Jürgen; Linke, AlexanderWe study two classical generalized MAC schemes on unstructured triangular Delaunay meshes for the incompressible Stokes and Navier-Stokes equations and prove their convergence for the first time. These generalizations use the duality between Voronoi and triangles of Delaunay meshes, in order to construct two staggered discretization schemes. Both schemes are especially interesting, since compatible finite volume discretizations for coupled convection-diffusion equations can be constructed which preserve discrete maximum principles. In the first scheme, called tangential velocity scheme, the pressures are defined at the vertices of the mesh, and the discrete velocities are tangential to the edges of the triangles. In the second scheme, called normal velocity scheme, the pressures are defined in the triangles, and the discrete velocities are normal to the edges of the triangles. For both schemes, we prove the convergence in $L^2$ for the velocities and the discrete rotations of the velocities for the Stokes and the Navier-Stokes problem. Further, for the normal velocity scheme, we also prove the strong convergence of the pressure in $L^2$. Linear and nonlinear numerical examples illustrate the theoretical predictions.
- ItemConvergence of an implicit Voronoi finite volume method for reaction-diffusion problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Fiebach, André; Glitzky, Annegret; Linke, AlexanderWe investigate the convergence of an implicit Voronoi finite volume method for reaction- diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. The numerical scheme uses boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh-independent global upper and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. In order to illustrate the preservation of qualitative properties by the numerical scheme, we present a long-term simulation of the Michaelis-Menten-Henri system. Especially, we investigate the decay properties of the relative free energy and the evolution of the dissipation rate over several magnitudes of time, and obtain experimental orders of convergence for these quantities.
- ItemMulti-dimensional modeling and simulation of semiconductor nanophotonic devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Kantner, Markus; Höhne, Theresa; Koprucki, Thomas; Burger, Sven; Wünsche, Hans-Jürgen; Schmidt, Frank; Mielke, Alexander; Bandelow, UweSelf-consistent modeling and multi-dimensional simulation of semiconductor nanophotonic devices is an important tool in the development of future integrated light sources and quantum devices. Simulations can guide important technological decisions by revealing performance bottlenecks in new device concepts, contribute to their understanding and help to theoretically explore their optimization potential. The efficient implementation of multi-dimensional numerical simulations for computer-aided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in device-scale modeling of quantum dot based single-photon sources and laser diodes by self-consistently coupling the optical Maxwell equations with semiclassical carrier transport models using semi-classical and fully quantum mechanical descriptions of the optically active region, respectively. For the simulation of realistic devices with complex, multi-dimensional geometries, we have developed a novel hp-adaptive finite element approach for the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties of the photonic structures. For electrically driven devices, we introduced novel discretization and parameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semiconductors at cryogenic temperature. Our methodical advances are demonstrated on various applications, including vertical-cavity surface-emitting lasers, grating couplers and single-photon sources.
- ItemInverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Fuhrmann, Jürgen; Linke, Alexander; Merdon, Christian; Neumann, Felix; Streckenbach, Timo; Baltruschat, Helmut; Khodayari, MehdiThin layer flow cells are used in electrochemical research as experimental devices which allow to perform investigations of electrocatalytic surface reactions under controlled conditions using reasonably small electrolyte volumes. The paper introduces a general approach to simulate the complete cell using accurate numerical simulation of the coupled flow, transport and reaction processes in a flow cell. The approach is based on a mass conservative coupling of a divergence-free finite element method for fluid flow and a stable finite volume method for mass transport. It allows to perform stable and efficient forward simulations that comply with the physical bounds namely mass conservation and maximum principles for the involved species. In this context, several recent approaches to obtain divergence-free velocities from finite element simulations are discussed. In order to perform parameter identification, the forward simulation method is coupled to standard optimization tools. After an assessment of the inverse modeling approach using known real-istic data, first results of the identification of solubility and transport data for O2 dissolved in organic electrolytes are presented. A plausibility study for a more complex situation with surface reactions concludes the paper and shows possible extensions of the scope of the presented numerical tools.