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## Preconditioner
Kinds of Preconditioner
## Selected Abstracts## Simple preconditioners for the conjugate gradient method: experience with test day models JOURNAL OF ANIMAL BREEDING AND GENETICS, Issue 3 2002I. STRANDÉNPreconditioned conjugate gradient method can be used to solve large mixed model equations quickly. Convergence of the method depends on the quality of the preconditioner. Here, the effect of simple preconditioners on the number of iterations until convergence was studied by solving breeding values for several test day models. The test day records were from a field data set, and several simulated data sets with low and high correlations among regression coefficients. The preconditioner matrices had diagonal or block diagonal parts. Transformation of the mixed model equations by diagonalization of the genetic covariance matrix was studied as well. Preconditioner having the whole block of the fixed effects was found to be advantageous. A block diagonal preconditioner for the animal effects reduced the number of iterations the higher the correlations among animal effects, but increased memory usage of the preconditioner. Diagonalization of the animal genetic covariance matrix often reduced the number of iterations considerably without increased memory usage. Einfache Preconditioners für die `Conjugate Gradient Method': Erfahrungen mit Testtagsmodellen Die `Preconditioned Conjugate Gradient Methode' kann benutzt werden um große `Mixed Model' Gleichungssysteme schnell zu lösen. In diesem Beitrag wurde der Einfluss von einfachen Preconditioners auf die Anzahl an Iterationen bis zur Konvergenz bei der Schätzung von Zuchtwerten bei verschiedenen Testtagsmodellen untersucht. Die Testtagsdaten stammen aus einem Felddatensatz und mehreren simulierten Datensätzen mit unterschiedlichen Korrelationen zwischen den Regressionskoeffizienten. Die Preconditioner Matrix bestand aus Diagonalen oder Blockdiagonalen Teilen. Eine Transformation der Mixed Modell Gleichungen durch Diagonalisierung der genetischen Kovarianzmatrix wurde ebenfalls untersucht. Preconditioners mit dem Block der fixen Effekte zeigten sich immer überlegen. Ein Blockdiagonaler Preconditioner für den Tiereffekt reduzierte die Anzahl an Iterationen mit höher werden Korrelationen zwischen den Tiereffekten, aber erhöhte den Speicherbedarf. Eine Diagonalisierung der genetischen Kovarianzmatrix reduzierte sehr oft die Anzahl an Iterationen erheblich ohne den Speicherbedarf zu erhöhen. [source] ## Fast iterative solution of large undrained soil-structure interaction problems INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2003Kok-Kwang PhoonAbstract In view of rapid developments in iterative solvers, it is timely to re-examine the merits of using mixed formulation for incompressible problems. This paper presents extensive numerical studies to compare the accuracy of undrained solutions resulting from the standard displacement formulation with a penalty term and the two-field mixed formulation. The standard displacement and two-field mixed formulations are solved using both direct and iterative approaches to assess if it is cost-effective to achieve more accurate solutions. Numerical studies of a simple footing problem show that the mixed formulation is able to solve the incompressible problem ,exactly', does not create pressure and stress instabilities, and obviate the need for an ad hoc penalty number. In addition, for large-scale problems where it is not possible to perform direct solutions entirely within available random access memory, it turns out that the larger system of equations from mixed formulation also can be solved much more efficiently than the smaller system of equations arising from standard formulation by using the symmetric quasi-minimal residual (SQMR) method with the generalized Jacobi (GJ) preconditioner. Iterative solution by SQMR with GJ preconditioning also is more elegant, faster, and more accurate than the popular Uzawa method. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Performance of Jacobi preconditioning in Krylov subspace solution of finite element equations INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 4 2002F.-H. LeeAbstract This paper examines the performance of the Jacobi preconditioner when used with two Krylov subspace iterative methods. The number of iterations needed for convergence was shown to be different for drained, undrained and consolidation problems, even for similar condition number. The differences were due to differences in the eigenvalue distribution, which cannot be completely described by the condition number alone. For drained problems involving large stiffness ratios between different material zones, ill-conditioning is caused by these large stiffness ratios. Since Jacobi preconditioning operates on degrees-of-freedom, it effectively homogenizes the different spatial sub-domains. The undrained problem, modelled as a nearly incompressible problem, is much more resistant to Jacobi preconditioning, because its ill-conditioning arises from the large stiffness ratios between volumetric and distortional deformational modes, many of which involve the similar spatial domains or sub-domains. The consolidation problem has two sets of degrees-of-freedom, namely displacement and pore pressure. Some of the eigenvalues are displacement dominated whereas others are excess pore pressure dominated. Jacobi preconditioning compresses the displacement-dominated eigenvalues in a similar manner as the drained problem, but pore-pressure-dominated eigenvalues are often over-scaled. Convergence can be accelerated if this over-scaling is recognized and corrected for. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Adaptive preconditioning of linear stochastic algebraic systems of equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2007Y. T. FengAbstract This paper proposes an adaptively preconditioned iterative method for the solution of large-scale linear stochastic algebraic systems of equations with one random variable that arise from the stochastic finite element modelling of linear elastic problems. Firstly, a Rank-one posteriori preconditioner is introduced for a general linear system of equations. This concept is then developed into an effective adaptive preconditioning scheme for the iterative solution of the stochastic equations in the context of a modified Monte Carlo simulation approach. To limit the maximum number of base vectors used in the scheme, a simple selection criterion is proposed to update the base vectors. Finally, numerical experiments are conducted to assess the performance of the proposed adaptive preconditioning strategy, which indicates that the scheme with very few base vectors can improve the convergence of the standard Incomplete Cholesky preconditioning up to 50%. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## A note on least squares methods INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 2 2006G. F. CareyAbstract We examine the relationship of preconditioned L2 residual, Sobolev gradient and H,1 least squares methods. Of particular interest are: (1) a demonstration that the Sobolev gradient approach is simply a form of preconditioning for the standard L2 scheme, and (2) that the Sobolev preconditioner is related to the additional solve step in the H,1 formulation. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Application of the additive Schwarz method to large scale Poisson problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2004K. M. SinghAbstract This paper presents an application of the additive Schwarz method to large scale Poisson problems on parallel computers. Domain decomposition in rectangular blocks with matching grids on a structured rectangular mesh has been used together with a stepwise approximation to approximate sloping sides and complicated geometric features. A seven-point stencil based on central difference scheme has been used for the discretization of the Laplacian for both interior and boundary grid points, and this results in a symmetric linear algebraic system for any type of boundary conditions. The preconditioned conjugate gradient method has been used as an accelerator for the additive Schwarz method, and three different methods have been assessed for the solution of subdomain problems. Numerical experiments have been performed to determine the most suitable set of subdomain solvers and the optimal accuracy of subdomain solutions; to assess the effect of different decompositions of the problem domain; and to evaluate the parallel performance of the additive Schwarz preconditioner. Application to a practical problem involving complicated geometry is presented which establishes the efficiency and robustness of the method. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## A priori pivoting in solving the Navier,Stokes equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2002S. Ø. WilleAbstract Mixed finite element formulations of incompressible Navier,Stokes Equations leads to non-positive definite algebraic systems inappropriate for iterative solution techniques. However, introducing a suitable preconditioner, the mixed finite element equation system becomes positive definite and solvable by iterative techniques. The present work suggests a priori pivoting sequences for parallel and serial implementations of incomplete Gaussian factorization. Tests are performed for the driven cavity problem in two and three dimensions. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 6 2002R. S. TuminaroAbstract Multilevel methods offer the best promise to attain both fast convergence and parallel efficiency in the numerical solution of parabolic and elliptic partial differential equations. Unfortunately, they have not been widely used in part because of implementation difficulties for unstructured mesh solvers. To facilitate use, a multilevel preconditioner software module, ML, has been constructed. Several methods are provided requiring relatively modest programming effort on the part of the application developer. This report discusses the implementation of one method in the module: a two-level Krylov,Schwarz preconditioner. To illustrate the use of these methods in computational fluid dynamics (CFD) engineering applications, we present results for 2D and 3D CFD benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010Clark R. DohrmannAbstract Overlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which are uniformly bounded, which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e. methods based on nonoverlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains. In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered. In addition, in spite of using overlapping subdomains to define the local components of the preconditioner, values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive-definite problems and the standard preconditioned conjugate gradient method. Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## A study on the lumped preconditioner and memory requirements of FETI and related primal domain decomposition methods INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2008Yannis FragakisAbstract In recent years, domain decomposition methods (DDMs) have emerged as advanced solvers in several areas of computational mechanics. In particular, during the last decade, in the area of solid and structural mechanics, they reached a considerable level of advancement and were shown to be more efficient than popular solvers, like advanced sparse direct solvers. The present contribution follows the lines of a series of recent publications on the relationship between primal and dual formulations of DDMs. In some of these papers, the effort to unify primal and dual methods led to a family of DDMs that was shown to be more efficient than the previous methods. The present paper extends this work, presenting a new family of related DDMs, thus enriching the theory of the relations between primal and dual methods, with the primal methods, which correspond to the dual DDM that uses the lumped preconditioner. The paper also compares the numerical performance of the new methods with that of the previous ones and focuses particularly on memory requirement issues related to the use of the lumped preconditioner, suggesting a particularly memory-efficient formulation. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## A preconditioned conjugate gradient approach to structural reanalysis for general layout modifications INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2007Zhengguang LiAbstract This paper presents a preconditioned conjugate gradient approach to structural static reanalysis for general layout modifications. It is suitable for all types of layout modifications, including the general case in which some original members and nodes are deleted and other new members and nodes are added concurrently. The approach is based on the preconditioned conjugate gradient technique. The preconditioner is constructed, and an efficient implementation for applying the preconditioner is presented, which requires the factorization of the stiffness matrix corresponding to the newly added degrees of freedom only. In particular, the approach can adaptively monitor the accuracy of approximate solutions. Numerical examples show that the condition number of the preconditioned matrix is remarkably reduced. Therefore, the fast convergence and accurate results can be achieved by the approach. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Improved implementation and robustness study of the X-FEM for stress analysis around cracks INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2005E. BéchetAbstract Numerical crack propagation schemes were augmented in an elegant manner by the X-FEM method. The use of special tip enrichment functions, as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh, which is of industrial interest, especially when a numerical model for crack propagation is desired. This paper improves the implementation of the X-FEM method for stress analysis around cracks in three ways. First, the enrichment strategy is revisited. The conventional approach uses a ,topological' enrichment (only the elements touching the front are enriched). We suggest a ,geometrical' enrichment in which a given domain size is enriched. The improvements obtained with this enrichment are discussed. Second, the conditioning of the X-FEM both for topological and geometrical enrichments is studied. A preconditioner is introduced so that ,off the shelf' iterative solver packages can be used and perform as well on X-FEM matrices as on standard FEM matrices. The preconditioner uses a local (nodal) Cholesky based decomposition. Third, the numerical integration scheme to build the X-FEM stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme. A 2D benchmark problem is designed to show the improvements and the robustness. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## A comparison of eigensolvers for large-scale 3D modal analysis using AMG-preconditioned iterative methods INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2005Peter ArbenzAbstract The goal of our paper is to compare a number of algorithms for computing a large number of eigenvectors of the generalized symmetric eigenvalue problem arising from a modal analysis of elastic structures. The shift-invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however, a sparse direct factorization is required for the resulting set of linear equations. Instead, our paper considers the use of preconditioned iterative methods. We present a brief review of available preconditioned eigensolvers followed by a numerical comparison on three problems using a scalable algebraic multigrid (AMG) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2005A. P. PeirceAbstract We present a novel multigrid (MG) procedure for the efficient solution of the large non-symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi-layered elastic medium. The governing equations involve a highly non-linear coupled system of integro-partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N3). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green's functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub-sets of the finest grid are used to treat mixed variable problems associated with ,pinch points.' Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss,Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2002K. K. PhoonAbstract Finite element simulations of very large-scale soil,structure interaction problems (e.g. excavations, tunnelling, pile-rafts, etc.) typically involve the solution of a very large, ill-conditioned, and indefinite Biot system of equations. The traditional preconditioned conjugate gradient solver coupled with the standard Jacobi (SJ) preconditioner can be very inefficient for this class of problems. This paper presents a robust generalized Jacobi (GJ) preconditioner that is extremely effective for solving very large-scale Biot's finite element equations using the symmetric quasi-minimal residual method. The GJ preconditioner can be formed, inverted, and implemented within an ,element-by-element' framework as readily as the SJ preconditioner. It was derived as a diagonal approximation to a theoretical form, which can be proven mathematically to possess an attractive eigenvalue clustering property. The effectiveness of the GJ preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three-dimensional footing problem. This paper casts a new perspective on the potentialities of the simple diagonal preconditioner, which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Physics-based preconditioner for iterative algorithms in multi-scatterer and multi-boundary method of moments formulations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002Jürgen v. HagenAbstract An efficient method to solve electromagnetic scattering problems involving several metallic scatterers or bodies composed of dielectric and metallic regions is proposed. So far, the method of moments has successfully been applied to large arrays of identical scatterers when it was combined with preconditioned iterative algorithms to solve for the linear system of equations. Here, the method is generalized to geometries that are composed of several metallic elements of different shapes and sizes, and also to scatterers that are composed of metallic and dielectric regions. The method uses in its core an iterative algorithm, preferably the transpose-free quasi-minimum residual (TFQMR) algorithm, and a block diagonal Jacobi preconditioner. For best performance, the blocks for the preconditioner are chosen according to individual scatterers or groups of scatterers for the array case, and according to the electric and magnetic current basis functions for dielectric/metallic scatterers. The iterative procedure converges quickly for an optimally chosen preconditioner, and is robust even for a non-optimal preconditioner. Reported run times are compared to run times of an efficiently programmed LU factorization, and are shown to be significantly lower. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Efficient preconditioning techniques for finite-element quadratic discretization arising from linearized incompressible Navier,Stokes equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2010A. El MalikiAbstract We develop an efficient preconditioning techniques for the solution of large linearized stationary and non-stationary incompressible Navier,Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non-stationary case. The time discretization procedure uses the Gear scheme and the second-order Taylor,Hood element P2,P1 is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r,(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P1,P2 hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Low-cost implicit schemes for all-speed flows on unstructured meshes INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2008T. KloczkoAbstract Matrix-free implicit treatments are now commonly used for computing compressible flow problems: a reduced cost per iteration and low-memory requirements are their most attractive features. This paper explains how it is possible to preserve these features for all-speed flows, in spite of the use of a low-Mach preconditioning matrix. The proposed approach exploits a particular property of a widely used low-Mach preconditioner proposed by Turkel. Its efficiency is demonstrated on some steady and unsteady applications. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## A comparison of preconditioners for incompressible Navier,Stokes solvers INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2008M. ur RehmanAbstract We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier,Stokes equations. These systems are of the so-called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium-sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## A control volume finite-element method for numerical simulating incompressible fluid flows without pressure correction INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2007Ahmed OmriAbstract This paper presents a numerical model to study the laminar flows induced in confined spaces by natural convection. A control volume finite-element method (CVFEM) with equal-order meshing is employed to discretize the governing equations in the pressure,velocity formulation. In the proposed model, unknown variables are calculated in the same grid system using different specific interpolation functions without pressure correction. To manage memory storage requirements, a data storage format is developed for generated sparse banded matrices. The performance of various Krylov techniques, including Bi-CGSTAB (Bi-Conjugate Gradient STABilized) with an incomplete LU (ILU) factorization preconditioner is verified by applying it to three well-known test problems. The results are compared to those of independent numerical or theoretical solutions in literature. The iterative computer procedure is improved by using a coupled strategy, which consists of solving simultaneously the momentum and the continuity equation transformed in a pressure equation. Results show that the strategy provides useful benefits with respect to both reduction of storage requirements and central processing unit runtime. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## A preconditioned semi-staggered dilation-free finite volume method for the incompressible Navier,Stokes equations on all-hexahedral elements INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2005Mehmet SahinAbstract A new semi-staggered finite volume method is presented for the solution of the incompressible Navier,Stokes equations on all-quadrilateral (2D)/hexahedral (3D) meshes. The velocity components are defined at element node points while the pressure term is defined at element centroids. The continuity equation is satisfied exactly within each elements. The checkerboard pressure oscillations are prevented using a special filtering matrix as a preconditioner for the saddle-point problem resulting from second-order discretization of the incompressible Navier,Stokes equations. The preconditioned saddle-point problem is solved using block preconditioners with GMRES solver. In order to achieve higher performance FORTRAN source code is based on highly efficient PETSc and HYPRE libraries. As test cases the 2D/3D lid-driven cavity flow problem and the 3D flow past array of circular cylinders are solved in order to verify the accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Numerical simulation of cavitating flow in 2D and 3D inducer geometries INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005O. Coutier-DelgoshaAbstract A computational method is proposed to simulate 3D unsteady cavitating flows in spatial turbopump inducers. It is based on the code FineTurbo, adapted to take into account two-phase flow phenomena. The initial model is a time-marching algorithm devoted to compressible flow, associated with a low-speed preconditioner to treat low Mach number flows. The presented work covers the 3D implementation of a physical model developed in LEGI for several years to simulate 2D unsteady cavitating flows. It is based on a barotropic state law that relates the fluid density to the pressure variations. A modification of the preconditioner is proposed to treat efficiently as well highly compressible two-phase flow areas as weakly compressible single-phase flow conditions. The numerical model is applied to time-accurate simulations of cavitating flow in spatial turbopump inducers. The first geometry is a 2D Venturi type section designed to simulate an inducer blade suction side. Results obtained with this simple test case, including the study of its general cavitating behaviour, numerical tests, and precise comparisons with previous experimental measurements inside the cavity, lead to a satisfactory validation of the model. A complete three-dimensional rotating inducer geometry is then considered, and its quasi-static behaviour in cavitating conditions is investigated. Numerical results are compared to experimental measurements and visualizations, and a promising agreement is obtained. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Efficient preconditioning of the discrete adjoint equations for the incompressible Navier,Stokes equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005René SchneiderAbstract Preconditioning of the discrete adjoint equations is closely related to preconditioning the linear systems arising in the Newton linearization of the discretized flow equations. We investigate the use of an optimal preconditioner for both problems on the example of a finite element discretization of the steady state incompressible Navier,Stokes equations. It is demonstrated that complications arising from the use of a zero mean pressure condition in the problem formulation can be overcome by modifying the preconditioner suitably. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## A domain decomposition method for modelling Stokes flow in porous materials INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2002Guangli LiuAbstract An algorithm is presented for solving the Stokes equation in large disordered two-dimensional porous domains. In this work, it is applied to random packings of discs, but the geometry can be essentially arbitrary. The approach includes the subdivision of the domain and a subsequent application of boundary integral equations to the subdomains. This gives a block diagonal matrix with sparse off-block components that arise from shared variables on internal subdomain boundaries. The global problem is solved using a biconjugate gradient routine with preconditioning. Results show that the effectiveness of the preconditioner is strongly affected by the subdomain structure, from which a methodology is proposed for the domain decomposition step. A minimum is observed in the solution time versus subdomain size, which is governed by the time required for preconditioning, the time for vector multiplications in the biconjugate gradient routine, the iterative convergence rate and issues related to memory allocation. The method is demonstrated on various domains including a random 1000-particle domain. The solution can be used for efficient recovery of point velocities, which is discussed in the context of stochastic modelling of solute transport. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Simple preconditioners for the conjugate gradient method: experience with test day models JOURNAL OF ANIMAL BREEDING AND GENETICS, Issue 3 2002I. STRANDÉNPreconditioned conjugate gradient method can be used to solve large mixed model equations quickly. Convergence of the method depends on the quality of the preconditioner. Here, the effect of simple preconditioners on the number of iterations until convergence was studied by solving breeding values for several test day models. The test day records were from a field data set, and several simulated data sets with low and high correlations among regression coefficients. The preconditioner matrices had diagonal or block diagonal parts. Transformation of the mixed model equations by diagonalization of the genetic covariance matrix was studied as well. Preconditioner having the whole block of the fixed effects was found to be advantageous. A block diagonal preconditioner for the animal effects reduced the number of iterations the higher the correlations among animal effects, but increased memory usage of the preconditioner. Diagonalization of the animal genetic covariance matrix often reduced the number of iterations considerably without increased memory usage. Einfache Preconditioners für die `Conjugate Gradient Method': Erfahrungen mit Testtagsmodellen Die `Preconditioned Conjugate Gradient Methode' kann benutzt werden um große `Mixed Model' Gleichungssysteme schnell zu lösen. In diesem Beitrag wurde der Einfluss von einfachen Preconditioners auf die Anzahl an Iterationen bis zur Konvergenz bei der Schätzung von Zuchtwerten bei verschiedenen Testtagsmodellen untersucht. Die Testtagsdaten stammen aus einem Felddatensatz und mehreren simulierten Datensätzen mit unterschiedlichen Korrelationen zwischen den Regressionskoeffizienten. Die Preconditioner Matrix bestand aus Diagonalen oder Blockdiagonalen Teilen. Eine Transformation der Mixed Modell Gleichungen durch Diagonalisierung der genetischen Kovarianzmatrix wurde ebenfalls untersucht. Preconditioners mit dem Block der fixen Effekte zeigten sich immer überlegen. Ein Blockdiagonaler Preconditioner für den Tiereffekt reduzierte die Anzahl an Iterationen mit höher werden Korrelationen zwischen den Tiereffekten, aber erhöhte den Speicherbedarf. Eine Diagonalisierung der genetischen Kovarianzmatrix reduzierte sehr oft die Anzahl an Iterationen erheblich ohne den Speicherbedarf zu erhöhen. [source] ## Shifted SSOR preconditioning technique for electromagnetic wave scattering problems MICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 4 2009J. Q. ChenAbstract To efficiently solve large dense complex linear system arising from electric field integral equations (EFIE) formulation of electromagnetic scattering problems, the multilevel fast multipole method (MLFMM) is used to accelerate the matrix-vector product operations. The symmetric successive over-relaxation (SSOR) preconditioner is constructed based on the near-field matrix of the EFIE and employed to speed up the convergence rate of iterative methods. This technique can be greatly improved by shifting the near-field matrix of the EFIE with the principle value term of the magnetic field integral equation (MFIE) operator. Numerical results demonstrate that this method can reduce both the number of iterations and the computational time significantly with low cost for construction and implementation of preconditioners. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 1035,1039, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24254 [source] ## Application of the preconditioned GMRES to the Crank-Nicolson finite-difference time-domain algorithm for 3D full-wave analysis of planar circuits MICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 6 2008Y. YangAbstract The increase of the time step size significantly deteriorates the property of the coefficient matrix generated from the Crank-Nicolson finite-difference time-domain (CN-FDTD) method. As a result, the convergence of classical iterative methods, such as generalized minimal residual method (GMRES) would be substantially slowed down. To address this issue, this article mainly concerns efficient computation of this large sparse linear equations using preconditioned generalized minimal residual (PGMRES) method. Some typical preconditioning techniques, such as the Jacobi preconditioner, the sparse approximate inverse (SAI) preconditioner, and the symmetric successive over-relaxation (SSOR) preconditioner, are introduced to accelerate the convergence of the GMRES iterative method. Numerical simulation shows that the SSOR preconditioned GMRES method can reach convergence five times faster than GMRES for some typical structures. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1458,1463, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23396 [source] ## A new multilevel algebraic preconditioner for the diffusion equation in heterogeneous media NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2010Yu KuznetsovAbstract We develop and analyze a new multilevel preconditioner for algebraic systems arising from the finite volume discretization of 3D diffusion,reaction problems in highly heterogeneous media. The system matrices are assumed to be symmetric M -matrices. The preconditioner is based on a special coarsening algorithm and the inner Chebyshev iterative procedure. The condition number of the preconditioned matrix does not depend on the coefficients in the diffusion operator. Numerical experiments confirm theoretical results and reveal the competitiveness of the new preconditioner with respect to a well-known algebraic multigrid preconditioner. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## Fast solvers with block-diagonal preconditioners for linear FEM,BEM coupling NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2009Stefan A. FunkenAbstract The purpose of this paper is to present optimal preconditioned iterative methods to solve indefinite linear systems of equations arising from symmetric coupling of finite elements and boundary elements. This is a block-diagonal preconditioner together with a conjugate residual method and a preconditioned inner,outer iteration. We prove the efficiency of these methods by showing that the number of iterations to preserve a given accuracy is bounded independent of the number of unknowns. Numerical examples underline the efficiency of these methods. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Preconditioners for ill-posed Toeplitz matrices with differentiable generating functions NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2009C. EstaticoAbstract Both theoretical analysis and numerical experiments in the literature have shown that the Tyrtyshnikov circulant superoptimal preconditioner for Toeplitz systems can speed up the convergence of iterative methods without amplifying the noise of the data. Here we study a family of Tyrtyshnikov-based preconditioners for discrete ill-posed Toeplitz systems with differentiable generating functions. In particular, we show that the distribution of the eigenvalues of these preconditioners has good regularization features, since the smallest eigenvalues stay well separated from zero. Some numerical results confirm the regularization effectiveness of this family of preconditioners. Copyright © 2009 John Wiley & Sons, Ltd. [source] |