13 results
Search Results
Now showing 1 - 10 of 13
- ItemOn thermodynamic consistency of a Scharfetter-Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Koprucki, Thomas; Rotundo, Nella; Farrell, Patricio; Doan, Duy Hai; Fuhrmann, JürgenDriven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter-Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter-Gummel schemes.
- ItemWhat company does my news article refer to? Tackling multiclass problems with topic modeling(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Lübbering, Max; Kunkel, Julian; Farrell, PatricioWhile it is technically trivial to search for the company name to predict the company a new article refers to, it often leads to incorrect results. In this article, we compare the two approaches bag-of-words with k-nearest neighbors and Latent Dirichlet Allocation with k-nearest neighbor by assessing their applicability for predicting the S&P 500 company which is mentioned in a business news article or press release. Both approaches are evaluated on a corpus of 13k documents containing 84% news articles and 16% press releases. While the bag-of-words approach yields accurate predictions, it is highly inefficient due to its gigantic feature space. The Latent Dirichlet Allocation approach, on the other hand, manages to achieve roughly the same prediction accuracy (0.58 instead of 0.62) but reduces the feature space by a factor of seven.
- ItemMultilevel interpolation of divergence-free vector fields(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Farrell, Patricio; Gillow, Kathryn; Wendland, HolgerWe introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multiscale approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example.
- ItemA novel surface remeshing scheme via higher dimensional embedding and radial basis functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dassi, Franco; Farrell, Patricio; Si, HangMany applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy.
- ItemBlock preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Farrell, Patricio; Pestana, JenniferSymmetric collocation methods with radial basis functions allow approximation of the solution of a partial differential equation, even if the right-hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported radial basis functions and a multiscale technique. But the condition number and sparsity will still deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction (ARASM). Numerical results verify the effectiveness of the preconditioners.
- ItemNumerical methods for drift-diffusion models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Farrell, Patricio; Rotundo, Nella; Doan, Duy Hai; Kantner, Markus; Fuhrmann, Jürgen; Koprucki, ThomasThe van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume disretization scheme and recent efforts to generalize this approach to general statistical distribution functions.
- ItemComparison of thermodynamically consistent charge carrier flux discretizations for Fermi-Dirac and Gauss-Fermi statistics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Farrell, Patricio; Patriarca, Matteo; Fuhrmann, Jürgen; Koprucki, ThomasWe compare three thermodynamically consistent ScharfetterGummel schemes for different distribution functions for the carrier densities, including the FermiDirac integral of order 1/2 and the GaussFermi integral. The most accurate (but unfortunately also most costly) generalized ScharfetterGummel scheme requires the solution of an integral equation. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newtons method. We discuss the quality of this approximation and plot the resulting currents for FermiDirac and GaussFermi statistics. Finally, by comparing two modified (diffusion-enhanced and inverse activity based) ScharfetterGummel schemes with the more accurate generalized scheme, we show that the diffusion-enhanced ansatz leads to considerably lower flux errors, confirming previous results (J. Comp. Phys. 346:497-513, 2017).
- ItemTetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dassi, Franco; Kamenski, Lennard; Farrell, Patricio; Si, HangGiven a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.
- 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.
- ItemAn Anisoptropic Surface Remeshing Strategy Combining Higher Dimensional Embedding with Radial Basis Functions(Amsterdam [u.a.] : Elsevier, 2016) Dassi, Franco; Farrell, Patricio; Si, HangMany applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy.