Kees Vuik

Technische Universiteit Delft, Delft, South Holland, Netherlands

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Publications (24)14.89 Total impact

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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.
    08/2005: pages 53 - 60; , ISBN: 9783527604838
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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.
    12/2003: pages 103-129;
  • Kees Vuik
    Materials Transactions - MATER TRANS. 01/2003; 44(7):1448-1456.
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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
    Journal of Computational and Applied Mathematics 09/2001; · 0.99 Impact Factor
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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
    08/2001;
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    ABSTRACT: For the solution of practical complex problems in arbitrarily shaped domains, simple
    08/2001;
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    ABSTRACT: For the solution of practical flow problems in arbitrarily shaped domains, simple Schwarz
    08/2001;
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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.
    Computers & Mathematics with Applications 11/1999; 38(s 9–10):245–249. · 2.07 Impact Factor
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    ABSTRACT: Theory An iterative numerical method for solving the wave equation in an inhomogeneous medium with constant density is presented. The method is based on a Krylov iterative method and enhanced by a powerful preconditioner. For the preconditioner, a complex Shifted- Laplace operator is proposed, designed specifically for the wave equation. A multigrid method is used to approximately compute the inverse of the preconditioner. Numerical examples on 2D problems show that the combined method is robust and applicable for a wide range of frequencies. Extension to 3D is straightforward. The time-harmonic wave problem is represented by the Helmholtz equation:
    SEG Technical Program Expanded Abstracts 01/1999; 23(1).
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    ABSTRACT: The transition from homogeneous to localized deformations during the loading of a soil specimen within a ®nite element com-putation is often characterized by a bifurcation point, indicating loss of uniqueness of the solution. The signalling of a bifurcation point is done via the eigenvalues of the structural sti€ness matrix resulting from the ®nite element discretization. Eigenvectors related to negative eigenvalues can be used to perturb a homogeneous state and to obtain a localized deformation mode. This procedure is called branch switching. Several methods are proposed to perform this branch switching.
    Computer Methods in Applied Mechanics and Engineering 07/1998; 190(5). · 2.62 Impact Factor
  • Journal of the Mechanical Behavior of Materials. 06/1998; 9(2):115-120.
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    ABSTRACT: A general numerical model is described for the dissolution kinetics of stoichiometric second phases for one-dimensional cases in ternary systems. The model is applicable to both infinite and finite media and handles both complete and incomplete dissolution. It is shown that the dissolution kinetics of stoichiometric multicomponent second phase particles can differ strongly from that of the mono-element particles. The influence of the soft-impingement, ratio of the diffusion coefficients, stoichiometry, composition and the geometry of the dissolving stoichiometric phase is shown. The model is applied to an AlMgSi-alloy.
    Materials Science and Engineering A 05/1998; 246(s 1–2):93–103. · 2.11 Impact Factor
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    ABSTRACT: this paper, however, we are only interested in the convergence behaviour of the ICCG process for problems with layers with large contrasts in the coefficients. For that reason we simplify the equation considerably and assume that we have to solve the stationary linearized 2D diffusion equation, in a layered region:
    04/1998;
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    ABSTRACT: The dissolution of a disk-likeAl2Cuparticle is considered. A characteristic property is that initially the particle has a nonsmooth boundary. The mathematical model of this dissolution process contains a description of the particle interface, of which the position varies in time. Such a model is called a Stefan problem. It is impossible to obtain an analytical solution for a general two-dimensional Stefan problem, so we use the finite element method to solve this problem numerically. First, we apply a classical moving mesh method. Computations show that after some time steps the predicted particle interface becomes very unrealistic. Therefore, we derive a new method for the displacement of the free boundary based on the balance of atoms. This method leads to good results, also, for nonsmooth boundaries. Some numerical experiments are given for the dissolution of anAl2Cuparticle in anAl–Cualloy.
    Journal of Computational Physics 01/1998; · 2.14 Impact Factor
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    ABSTRACT: A numerical analysis of the homogenisation treatment of aluminium alloys under industrial circumstances is presented. The basis of this study is a mathematical model which is applicable to the dissolution of stoichiometric multicomponent phases in both finite and infinite ternary media. It handles both complete and incomplete particle dissolution as well as the subsequent homogenisation of the matrix. The precipitate volume fraction and matrix homogeneity are followed during the entire homogenisation treatment. First, the influence of the metallurgical parameters, such as particle size distribution, initial matrix concentration profile and particle geometry on the dissolution- and matrix homogeneity kinetics is analysed. Then, the impact of the heating-rate and local temperature on the homogenisation kinetics is investigated. Conclusions for an optimal homogenisation treatment of aluminium alloys may be drawn. The model presented is general but the calculations were performed for the system Al–Mg–Si with an Al-rich matrix and Mg2Si-precipitates.
    Materials Science and Engineering A 01/1998; · 2.11 Impact Factor
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    J. Frank, A. Segal, K. Vuik
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    ABSTRACT: In this paper we outline a parallel implementation of Krylov-accelerated Schwarz domain decomposition in which subdomain problems are solved to low precision. By so doing, computational time is focused on the convergence of the global iteration rather than wasted on ineffective subdomain iterations. We consider the GCR method using classical GramSchmidt and Householder orthogonalization methods. Our goal is to apply this approach to the incompressible Navier-Stokes equations. For the parallel implementation, we assume a distributed memory system with message passing. 1 Schwarz Domain Decomposition Domain decomposition is a natural approach to solving partial differential equations on complicated domains [6, 15]. It also almost begs to be parallelized. In this paper we develop a parallel implementation of domain decomposition for the incompressible Navier-Stokes equations within the framework of the ISNaS program. The ISNaS program solves the incompressible continuity and momentum equa...
    09/1996;
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    ABSTRACT: In theory the bi-Lanczos algorithm is an attractive possibility for solving the unsymmetric eigenproblem. In practice, however, it turns out to be unstable due to loss of orthogonality of the iteration vectors. In this paper we discuss the possibilities of reorthogonalizing the bi-Lanczos iteration vectors. To save part of the advantage of the bi-Lanczos method over the other most commonly used Krylov subspace based method of Arnoldi (bi-Lanczos lacks the growth of the number of vector operations per iteration), we propose partial reorthogonalization. In that case reorthogonalization takes place if and only if too much orthogonality is lost, so that the results are accurate enough, whereas the algorithm is still less time consuming than Arnoldi. The theory presented here is an extension of the theory available for partial reorthogonalization of the symmetric Lanczos algorithm.
    Computers & Structures 01/1995; · 1.51 Impact Factor
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    Guus Segal, Kees Vuik, Kees Kassels
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    ABSTRACT: In this paper we consider symmetric and antisymmetric periodic boundary conditions for flows governed by the incompressible Navier-Stokes equations. Classical periodic boundary conditions are studied as well as symmetric and antisymmetric periodic boundary conditions in which there is a pressure difference between inlet and outlet. The implementation of this type of boundary conditions in a finite element code using the penalty function formulation is treated and also the implementation in a finite volume code based on pressure correction. The methods are demonstrated by computation of a flow through a staggered tube bundle.
    International Journal for Numerical Methods in Fluids 06/1994; 18(12):1153 - 1165. · 1.35 Impact Factor
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    ABSTRACT: Recently Eirola and Nevanlinna have proposed an iterative solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest variants of GMRES, in which a preconditioner is constructed at each iteration step by a suitable approximation process, e.g., by GMRES itself. Keywords: GMRES, nonsymmetric linear systems, iterative solver, ENmethod This version is dated June 23, 1992 Introduction The GMRES method, proposed in [13], is a popular method for the iterative solution of sparse linear systems with an unsymmetric nonsingular matrix. In its original form, so-called full GMRES, it is optimal in the sense that it minimizes the residual over the current Krylov subspace. However, it is often too expensive since the required orthogonalization per iteration step grows quadratically with the number of steps. For that reason, one often uses in practice variants of GMRES. The most wellknown variant, already suggested i...
    05/1994;
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    Kees Vuik
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    ABSTRACT: In this paper we compare two recently proposed methods, FGMRES [5] and GMRESR [7], for the iterative solution of sparse linear systems with an unsymmetric nonsingular matrix. Both methods compute minimal residual approximations using preconditioners, which may be different from step to step. The insights resulting from this comparison lead to better variants of both methods. Keywords: FGMRES, GMRESR, non symmetric linear systems, iterative solver. AMS(MOS) subject classification. 65F10 1 Introduction Recently two new iterative methods, FGMRES [5] and GMRESR [7] have been proposed to solve sparse linear systems with an unsymmetric and nonsingular matrix. Both methods are based on the same idea: the use of a preconditioner, which may be different in every iteration. However, the resulting algorithms lead to somewhat different results. In [5] the GMRES method is given for a fixed preconditioner. Thereafter, it is shown that a slightly adapted algorithm: FGMRES can be used in combination...
    04/1994;