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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.
- 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.
- ItemA continuum model for yttria-stabilised zirconia incorporating triple phase boundary, lattice structure and immobile oxide ions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Vágner, Petr; Guhlke, Clemens; Miloš, Vojtěch; Müller, Rüdiger; Fuhrmann, JürgenA continuum model for yttria-stabilised zirconia (YSZ) in the framework of non-equilibrium thermodynamics is developed. Particular attention is given to i) modeling of the YSZ-metal-gas triple phase boundary, ii) incorporation of the lattice structure and immobile oxide ions within the free energy model and iii) surface reactions. A finite volume discretization method based on modified Scharfetter-Gummel fluxes is derived in order to perform numerical simulations. The model is used to study the impact of yttria and immobile oxide ions on the structure of the charged boundary layer and the double layer capacitance. Cyclic voltammograms of an air-half cell are simulated to study the effect of parameter variations on surface reactions, adsorption and anion diffusion.