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search.filters.applied.f.access: openAccess × End date: 2019 × Start date: 2019 × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: finite element method ×

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Now showing 1 - 9 of 9
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    Challenges for drift-diffusion simulations of semiconductors: A comparative study of different discretization philosophies
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Farrell, Patricio; Peschka, Dirk
    We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as Voronoi finite-volume discretization schemes for the drift-diffusion equations describing charge transport in bulk semiconductor devices. Three common challenges, that can corrupt the precision of numerical solutions, will be discussed: boundary layers at Ohmic contacts, discontinuties in the doping profile, and corner singularities in L-shaped domains. The influence on the order of convergence is assessed for each computational challenge and the different discretization schemes. Additionally, we provide an analysis of the inner boundary layer asymptotics near Ohmic contacts to support our observations.
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    Functional a posteriori error estimation for stationary reaction-convection-diffusion problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Eigel, Martin; Samrowski, Tatiana
    A functional type a posteriori error estimator for the finite element discretisation of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimisation problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings.
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    Optimal and robust a posteriori error estimates in L∞(L2) for the approximation of Allen-Cahn equations past singularities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Bartels, Sören; Müller, Rüdiger
    Optimal a posteriori error estimates in L∞(L2) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.
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    Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Bartels, Sören; Müller, Rüdiger
    A fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.
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    A reduced-order modeling for efficient design study of artificial valve in enlarged ventricular outflow tracts
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Caiazzo, Alfonso; Guibert, Romain; Vignon-Clementel, Irene E.
    A computational approach is proposed for efficient design study of a reducer stent to be percutaneously implanted in enlarged right ventricular outflow tracts (RVOT). The need for such a device is driven by the absence of bovine or artificial valves which could be implanted in these RVOT to replace the absent or incompetent native valve, as is often the case over time after Tetralogy of Fallot repair. Hemodynamics are simulated in the stented RVOT via a reduce order model based on proper orthogonal decomposition (POD), while the artificial valve is modeled as a thin resistive surface. The reduced order model is obtained from the numerical solution on a reference device configuration, then varying the geometrical parameters (diameter) for design purposes. To validate the approach, forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters, and compared with the results of full CFD simulations. Such an approach could also be useful for uncertainty quantification. Device design, percutaneous pulmonary valve replacement, proper orthogonal decomposition (POD), finite element method, blood flow CFD, repaired Tetralogy of Fallot.
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    A Stokes-consistent backflow stabilization for physiological flows
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Bertoglio, Cristobal; Caiazzo, Alfonso
    In computational fluid dynamics incoming flow at open boundaries, or emphbackflow, often yields to unphysical instabilities for high Reynolds numbers. It is widely accepted that this is due to the incoming energy arising from the convection term, which cannot be empha priori controlled when the velocity field is unknown at the boundary. In order to improve the robustness of the numerical simulations, we propose a stabilized formulation based on a penalization of the residual of a weak Stokes problem on the open boundary, whose viscous part controls the incoming convective energy, while the inertial term contributes to the kinetic energy. We also present different strategies for the approximation of the boundary pressure gradient, which is needed for defining the stabilization term. The method has the advantage that it does not require neither artificial modifications or extensions of the computational domain. Moreover, it is consistent with the Womersley solution. We illustrate our approach on numerical examples
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    A tangential regularization method for backflow stabilization in hemodynamics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bertoglio, Cristóbal; Caiazzo, Alfonso
    In computational simulations of fluid flows, instabilities at the Neumann boundaries may appear during backflow regime. It is widely accepted that this is due to the incoming energy at the boundary, coming from the convection term, which cannot be controlled when the velocity field is unknown. We propose a stabilized formulation based on a local regularization of the fluid velocity along the tangential directions on the Neumann boundaries. The stabilization term is proportional to the amount of backflow, and does not require any further assumption on the velocity profile. The perfomance of the method is assessed on a twoand three-dimensional Womersley flows, as well as considering a hemodynamic physiological regime in a patient-specific aortic geometry.
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    Simulation of composite materials by a Network FEM with error control
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Eigel, Martin; Peterseim, Daniel
    A novel Finite Element Method (FEM) for the computational simulation in particle reinforced composite materials with many inclusions is presented. It is based on a specially designed mesh consisting of triangles and channel-like connections between inclusions which form a network structure. The total number of elements and, hence, the number of degrees of freedom are proportional to the number of inclusions. The error of the method is independent of the possibly tiny distances of neighbouring inclusions. We present algorithmic details for the generation of the problem adapted mesh and derive an efficient residual a posteriori error estimator which enables to compute reliable upper and lower error bounds. Several numerical examples illustrate the performance of the method and the error estimator. In particular, it is demonstrated that the (common) assumption of a lattice structure of inclusions can easily lead to incorrect predictions about material properties.
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    Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Carstensen, Carsten; Eigel, Martin
    A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to π/2 . The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allows the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.