Kees Vuik

Delft University of Technology, Delft, South Holland, Netherlands

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Publications (39)28.89 Total impact

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    ABSTRACT: In this paper, we introduce a new position-dependent smoothness-increasing accuracy-conserving (SIAC) filter that retains the benefits of position dependence as proposed in van Slingerland et al. (SIAM J Sci Comput 33:802–825, 2011) while ameliorating some of its shortcomings. As in the previous position-dependent filter, our new filter can be applied near domain boundaries, near a discontinuity in the solution, or at the interface of different mesh sizes; and as before, in general, it numerically enhances the accuracy and increases the smoothness of approximations obtained using the discontinuous Galerkin (dG) method. However, the previously proposed position-dependent one-sided filter had two significant disadvantages: (1) increased computational cost (in terms of function evaluations), brought about by the use of \(4k+1\) central B-splines near a boundary (leading to increased kernel support) and (2) increased numerical conditioning issues that necessitated the use of quadruple precision for polynomial degrees of \(k\ge 3\) for the reported accuracy benefits to be realizable numerically. Our new filter addresses both of these issues—maintaining the same support size and with similar function evaluation characteristics as the symmetric filter in a way that has better numerical conditioning—making it, unlike its predecessor, amenable for GPU computing. Our new filter was conceived by revisiting the original error analysis for superconvergence of SIAC filters and by examining the role of the B-splines and their weights in the SIAC filtering kernel. We demonstrate, in the uniform mesh case, that our new filter is globally superconvergent for \(k=1\) and superconvergent in the interior (e.g., region excluding the boundary) for \(k\ge 2\). Furthermore, we present the first theoretical proof of superconvergence for postprocessing over smoothly varying meshes, and explain the accuracy-order conserving nature of this new filter when applied to certain non-uniform meshes cases. We provide numerical examples supporting our theoretical results and demonstrating that our new filter, in general, enhances the smoothness and accuracy of the solution. Numerical results are presented for solutions of both linear and nonlinear equations solved on both uniform and non-uniform one- and two-dimensional meshes.
    No preview · Article · Nov 2014 · Journal of Scientific Computing
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    ABSTRACT: Superconvergence of discontinuous Galerkin methods is an area of increasing interest due to the ease with which higher order information can be extracted from the approximation. Cockburn, Luskin, Shu, and Suli showed that by applying a B-spline filter to the approximation at the final time, the order of accuracy can be improved from O(hk+1) to O(h2k+1) in the L2-norm for linear hyperbolic equations with periodic boundary conditions (where k is the polynomial degree and h is the mesh element diameter) [Math. Comp. (2003)]. The applicability of this filter for linear hyperbolic problems with non-periodic boundary conditions was computationally extended and renamed a position-dependent smoothness-increasing accuracy-conserving (SIAC) filter by van Slingerland, Ryan, Vuik [SISC (2011)]. However, error estimates in the L2-norm for this new position-dependent SIAC filter were never given. Furthermore, error estimates in the L∞-norm have not been established for the original kernel nor the position-dependent kernel. In this paper, for the first time we establish that it is possible to obtain O(hmin{2k+1,2k+2-d/2}) accuracy in the L∞-norm for the position-dependent SIAC filter, where d is the dimension. Furthermore, we extend the error estimates given by Cockburn et al. so that they are applicable to the entire domain when implementing the position-dependent SIAC filter. We also computationally demonstrate the applicability of this filter for visualization of streamlines.
    Full-text · Article · Sep 2014 · Mathematics of Computation
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    Full-text · Dataset · Apr 2014
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    ABSTRACT: A simple node enrichment strategy using a gradient based error estimator is presented for the Collocated Discrete Least Squares (CDLS) meshless method. Also, a procedure is defined to distribute collocation points according to the field nodes position. Here, shape functions are constructed using the Radial Point Interpolation Method (RPIM). As temporal discretization, a first-order accurate scheme, named the semi-incremental fractional step method, is used. One of the advantages of this scheme is its capability of using large time step sizes for the solution of governing equations on steady state problems. The capability of the presented strategy is shown by investigating the Carreau-Yasuda uid flow model in solving lid-driven cavity flow problems with different curve fitting indices values.
    Full-text · Article · Feb 2014 · Scientia Iranica
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    ABSTRACT: We propose and analyze a generic multi-class kinematic wave traffic flow model: Fastlane. The model takes into account heterogeneity among driver-vehicle units with respect to speed and space occupancy: long vehicles with large headways (e.g. trucks) take more space than short vehicles with short headways (e.g. passenger cars). Moreover, and this is what makes the model unique, this effect is larger when the traffic volume is higher. This state dependent space occupancy is reflected in dynamic passenger car equivalent values. The resulting model is shown to satisfy important requirements such as providing a unique solution and being anisotropic. Simulations are applied to compare Fastlane to other multiclass models. Furthermore, we show that the characteristic velocity depends on the truck share, which is one of the main consequences of our modeling approach.
    Full-text · Article · Jan 2014 · Transportation Research Record Journal of the Transportation Research Board
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    ABSTRACT: An historical overview of the development of traffic flow models is proposed in the form of a model tree. The model tree shows the genealogy of four families: the fundamental relation, microscopic, mesoscopic and macroscopic models. We discuss the families, branches and models. By describing the historical developments of traffic flow modeling, we take one step further back than conventional literature reviews that focus on the current state-of-the-art. This allows us to identify the main trends in traffic flow modeling: (1) convergence of many branches to generalized models, (2) adaptations and extensions of the LWR model to deal with real phenomena, (3) multi-class versions of many models and, (4) the development of hybrid models combining the advantages of different types of models.
    Full-text · Article · Jan 2014
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    ABSTRACT: In this article we demonstrate how network components can be modeled using the kinematic wave model in the Lagrangian formulation. This includes modeling nodes (or discontinuities) such as inflow and outflow boundaries, merges and bifurcations (e.g. ramps) and nonhomogeneous roads. Nodes are usually fixed in space. This makes their implementation in Lagrangian coordinates where the coordinates move with the vehicle more complex than in Eulerian coordinates where the coordinates are fixed in space. To this end we derive an analytical node model. The article then discusses how to implement such sink and source terms in a discretized version of the kinematic wave model in Lagrangian coordinates. In this implementation several choices have to be made. Test results show that even with the most simple choices (discretization based on full vehicle groups and discrete time steps) accurate and plausible results are obtained. We conclude that the Lagrangian formulation can successfully be applied for simulation of networks of nonhomogeneous roads. {\copyright} 2011 Elsevier Ltd. All rights reserved.
    Full-text · Article · Sep 2013 · Transportation Research Part C Emerging Technologies
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    ABSTRACT: The present work explores the massively parallel capabilities of the most advanced architecture of graphics processing units (GPUs) code named "Fermi", on a two-dimensional unstructured cell-centred finite volume code. We use the SIMPLE algorithm to solve the continuity and momentum equations that was fully ported to the GPU. The benefits of this implementation are compared with a serial implementation that traditionally runs on the central processing unit (CPU). The developed codes were assessed with the bench-mark problems of Poiseuille flow, for Newtonian and generalized Newtonian fluids, as well as by the lid-driven cavity and the sudden expansion flows for Newtonian fluids. The parallel (GPU) code accelerated the resolution of those three problems by factors of 19, 10 and 11, respectively, in comparison with the corresponding CPU single core counterpart. The results are a clear indication that GPUs are and will be useful in the field of computational fluid dynamics (CFD) for rheologically simple and complex fluids.
    No preview · Article · Jan 2013 · AIP Conference Proceedings
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    ABSTRACT: Traffic flow models and simulation tools are often used for traffic state estimation and prediction. Recently several multi-class models based on the kinematic wave traffic flow model have been introduced. These multi-class models take into account the heterogeneity of both vehicles and drivers. We analyse two important properties of these models: hyperbolicity and anisotropy. Both properties relate to the propagation speed of disturbances, as can be observed in real traffic. We discuss the importance of traffic flow models to be hyperbolic and anisotropic. Moreover, we develop a framework to analyse whether traffic flow models have these properties. Therefore, we derive a generic formulation of multi-class kinematic wave traffic flow models, rewrite it in the Lagrangian formulation and apply eigenvalue analysis to the resulting system of equations. Our analysis shows that most multi-class kinematic wave traffic flow models are indeed hyperbolic and anisotropic under certain modelling conditions.
    Full-text · Article · Jan 2013 · Transportmetrica A: Transport Science
  • No preview · Article · Jan 2011
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    Conference Paper: Fast Newton load flow
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    ABSTRACT: The Newton-Raphson method is widely used to solve load flow problems. Traditionally a direct solver is used to solve the linear systems within this method. In this paper we explore the use of an iterative method to solve the linear systems, leading to an inexact Newton-Krylov method. The main parameters of this method are the preconditioner and the forcing terms. Several candidate choices for these parameters are discussed and tested. With the proper preconditioner, and forcing terms, the inexact Newton-Krylov method is shown to greatly improve on using a direct solver.
    Full-text · Conference Paper · May 2010
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    ABSTRACT: The present paper consists of the formulation of a model for particle dissolution in a multi-component alloy taking into account cross-diffusion effects. The model consists of a Stefan condition to compute the velocity of the interface separating the particle and the solvent phase. The influence of the cross-diffusion terms on the particle dissolution rate is shown and it is concluded that its impact can be significant.
    No preview · Chapter · Aug 2005
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    Fred Vermolen · Kees Vuik · Guus Segal
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    ABSTRACT: Large linear systems are solved for modeling many scientific and engineering applications. Often these systems result from a discretization of model equations using Finite Elements, Finite Volumes or Finite Differences. The systems tend to become very large for three dimensional problems. Some models involve both time and space as independent parameters and therefore it is necessary to solve such a linear system efficiently at all time-steps.
    Full-text · Chapter · Dec 2003
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    ABSTRACT: During the homogenisation process of Al-Mg-Si extrusion alloys, plate-like β-Al5FeSi particles transform to multiple rounded α-Al12(FeMn)3Si particles. The rate of this β to α transformation determines the time which is required to homogenise the aluminium sufficiently for extrusion. In this paper, a finite element approach is presented which model the development of fraction transformed with time, in the beginning of the transformation, as a function of homogenisation temperature, as-cast microstructure and concentration of alloying elements. We treat the β to α transformation mathematically as a Stefan problem, where the concentration and the position of the moving boundaries of the α and β particles are determined. For the boundary conditions of the model thermodynamic calculations are used (Thermo-Calc). The influence of several process parameters on the modeled transformed fraction, such as the temperature and initial thickness of the β plates, are investigated. Finally the model is validated with experimental data.
    Full-text · Article · Jul 2003 · MATERIALS TRANSACTIONS
  • Kees Vuik

    No preview · Article · Jan 2003
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    Fred Vermolen · Kees Vuik
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    ABSTRACT: Dissolution of stoichiometric multi-component particles in ternary alloys is an important process occurring during the heat treatment of as-cast aluminium alloys prior to hotextrusion. A mathematical model is proposed to describe such a process. In this model an equation is given to determine the position of the particle interface in time, using two diffusion equations which are coupled by nonlinear boundary conditions at the interface. Some results concerning existence, uniqueness, and monotonicity are given. Furthermore, for an unbounded domain an analytical approximation is derived. The main part of this work is the development of a numerical solution method. Finite differences are used on a grid which changes in time. The discretization of the boundary conditions is important to obtain an accurate solution. The resulting nonlinear algebraic system is solved by the Newton-Raphson method. Numerical experiments illustrate the accuracy of the numerical method. The numerical solution is compared with the analytical approximation. Keywords: Stefan problem, moving grid method, stoichiometric particle dissolution, ternary alloy homogenisation AMS Subject Classification: 35R35, 65M06, 80A22 1
    Full-text · Article · Sep 2001 · Journal of Computational and Applied Mathematics
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    Guus Segal · Kees Vuik
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    ABSTRACT: In this paper we consider the solution of the systems of non-linear equations
    Preview · Article · Aug 2001
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    Erik Brakkee · Kees Vuik · Piet Wesseling
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    ABSTRACT: For the solution of practical complex problems in arbitrarily shaped domains, simple
    Preview · Article · Aug 2001
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    Erik Brakkee · Kees Vuik · Piet Wesseling
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    ABSTRACT: For the solution of practical flow problems in arbitrarily shaped domains, simple Schwarz
    Preview · Article · Aug 2001
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    ABSTRACT: In a previous article [1], the eigenvalues of the elasto-plastic material matrix of a Drucker-Prager nonassociated soil model were analyzed with special attention to the occurrence of complex eigenvalues. The link between this analysis on material level to stress states which arise in a numerical computation is made in this article. © 1999 Elsevier Science Ltd. All rights reserved.
    Preview · Article · Nov 1999 · Computers & Mathematics with Applications