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Finite Element Discretization (finite + element_discretization)

Distribution by Scientific Domains

Engineering	67%
Mathematics and Statistics	32%

Selected Abstracts

Parallel eigenanalysis of multiaquifer systems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2005
L. Bergamaschi
Abstract Finite element discretizations of flow problems involving multiaquifer systems deliver large, sparse, unstructured matrices, whose partial eigenanalysis is important for both solving the flow problem and analysing its main characteristics. We studied and implemented an effective preconditioning of the Jacobi,Davidson algorithm by FSAI-type preconditioners. We developed efficient parallelization strategies in order to solve very large problems, which could not fit into the storage available to a single processor. We report our results about the solution of multiaquifer flow problems on an SP4 machine and a Linux Cluster. We analyse the sequential and parallel efficiency of our algorithm, also compared with standard packages. Questions regarding the parallel solution of finite element eigenproblems are addressed and discussed. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Optimal use of high-resolution topographic data in flood inundation models

HYDROLOGICAL PROCESSES, Issue 3 2003
P. D. Bates
Abstract In this paper we explore the optimum assimilation of high-resolution data into numerical models using the example of topographic data provision for flood inundation simulation. First, we explore problems with current assimilation methods in which numerical grids are generated independent of topography. These include possible loss of significant length scales of topographic information, poor representation of the original surface and data redundancy. These are resolved through the development of a processing chain consisting of: (i) assessment of significant length scales of variation in the input data sets; (ii) determination of significant points within the data set; (iii) translation of these into a conforming model discretization that preserves solution quality for a given numerical solver; and (iv) incorporation of otherwise redundant sub-grid data into the model in a computationally efficient manner. This processing chain is used to develop an optimal finite element discretization for a 12 km reach of the River Stour in Dorset, UK, for which a high-resolution topographic data set derived from airborne laser altimetry (LiDAR) was available. For this reach, three simulations of a 1 in 4 year flood event were conducted: a control simulation with a mesh developed independent of topography, a simulation with a topographically optimum mesh, and a further simulation with the topographically optimum mesh incorporating the sub-grid topographic data within a correction algorithm for dynamic wetting and drying in fixed grid models. The topographically optimum model is shown to represent better the ,raw' topographic data set and that differences between this surface and the control are hydraulically significant. Incorporation of sub-grid topographic data has a less marked impact than getting the explicit hydraulic calculation correct, but still leads to important differences in model behaviour. The paper highlights the need for better validation data capable of discriminating between these competing approaches and begins to indicate what the characteristics of such a data set should be. More generally, the techniques developed here should prove useful for any data set where the resolution exceeds that of the model in which it is to be used. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Bifurcation modeling in geomaterials: From the second-order work criterion to spectral analyses

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 9 2009
F. Prunier
Abstract The present paper investigates bifurcation analysis based on the second-order work criterion, in the framework of rate-independent constitutive models and rate-independent boundary-value problems. The approach applies mainly to nonassociated materials such as soils, rocks, and concretes. The bifurcation analysis usually performed at the material point level is extended to quasi-static boundary-value problems, by considering the stiffness matrix arising from finite element discretization. Lyapunov's definition of stability (Annales de la faculté des sciences de Toulouse 1907; 9:203,274), as well as definitions of bifurcation criteria (Rice's localization criterion (Theoretical and Applied Mechanics. Fourteenth IUTAM Congress, Amsterdam, 1976; 207,220) and the plasticity limit criterion are revived in order to clarify the application field of the second-order work criterion and to contrast these criteria. The first part of this paper analyses the second-order work criterion at the material point level. The bifurcation domain is presented in the 3D stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. The relevance of this criterion, when the nonlinear constitutive model is expressed in the classical form (d, = Md,) or in the dual form (d, = Nd,), is discussed. In the second part, the analysis is extended to the boundary-value problems in quasi-static conditions. Nonlinear finite element computations are performed and the global tangent stiffness matrix is analyzed. For several examples, the eigenvector associated with the first vanishing eigenvalue of the symmetrical part of the stiffness matrix gives an accurate estimation of the failure mode in the homogeneous and nonhomogeneous boundary-value problem. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A node-based agglomeration AMG solver for linear elasticity in thin bodies

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2009
Prasad S. Sumant
Abstract This paper describes the development of an efficient and accurate algebraic multigrid finite element solver for analysis of linear elasticity problems in two-dimensional thin body elasticity. Such problems are commonly encountered during the analysis of thin film devices in micro-electro-mechanical systems. An algebraic multigrid based on element interpolation is adopted and streamlined for the development of the proposed solver. A new node-based agglomeration scheme is proposed for computationally efficient, aggressive and yet effective generation of coarse grids. It is demonstrated that the use of appropriate finite element discretization along with the proposed algebraic multigrid process preserves the rigid body modes that are essential for good convergence of the multigrid solution. Several case studies are taken up to validate the approach. The proposed node-based agglomeration scheme is shown to lead to development of sparse and efficient intergrid transfer operators making the overall multigrid solution process very efficient. The proposed solver is found to work very well even for Poisson's ratio >0.4. Finally, an application of the proposed solver is demonstrated through a simulation of a micro-electro-mechanical switch. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A discontinuous enrichment method for the efficient solution of plate vibration problems in the medium-frequency regime

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010
Paolo Massimi
Abstract A discontinuous enrichment method (DEM) is presented for the efficient discretization of plate vibration problems in the medium-frequency regime. This method enriches the polynomial shape functions of the classical finite element discretization with free-space solutions of the biharmonic operator governing the elastic vibrations of an infinite Kirchhoff plate. These free-space solutions, which represent flexural waves and decaying modes, are discontinuous across the element interfaces. For this reason, two different and carefully constructed Lagrange multiplier approximations are introduced along the element edges to enforce a weak continuity of the transversal displacement and its normal derivative, and discrete Lagrange multipliers are introduced at the element corners to enforce there a weak continuity of the transversal displacement. The proposed DEM is illustrated with the solution of sample plate vibration problems with different types of harmonic loading in the medium-frequency regime, away from and close to resonance. In all cases, its performance is found to be significantly superior to that of the classical higher-order finite element method. Copyright © 2010 John Wiley & Sons, Ltd. [source]

Energy,momentum consistent finite element discretization of dynamic finite viscoelasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
M. Groß
Abstract This paper is concerned with energy,momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy,momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three-field formulation, in which the deformation field, the velocity field and a strain-like viscous internal variable field are treated as independent quantities. The new non-linear viscous evolution equation satisfies a non-negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy,momentum consistency is based on the collocation property and an enhanced second Piola,Kirchhoff stress tensor. The obtained coupled non-linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time-integration algorithm in comparison with the ordinary midpoint rule. Both the quasi-rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignment

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2007
C. Miehe
Abstract The paper considers a variational formulation of brittle fracture in elastic solids and proposes a numerical implementation by a finite element method. On the theoretical side, we outline a consistent thermodynamic framework for crack propagation in an elastic solid. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius,Planck inequality in the sense of Coleman's method. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip and minimizes the incremental energy release. On the numerical side, we exploit this variational structure in terms of crack-driving configurational forces. First, a standard finite element discretization in space yields a discrete formulation of the global dissipation in terms configurational nodal forces. As a consequence, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface segments of the mesh. Critical for the success of this procedure is its embedding into an r-adaptive crack-segment reorientation procedure with configurational-force-based directional indicator. Here, successive crack releases appear in discrete steps associated with the given space discretization. These are performed by a staggered loading,release algorithm of energy minimization at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive-definite discrete subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. We demonstrate the performance of the formulation by means of representative numerical simulations. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Stress-adapted numerical form finding of pre-stressed surfaces by the updated reference strategy

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2005
R. Wüchner
Abstract In this paper we present the updated reference strategy for numerical form finding of pre-stressed membranes, which is based on standard finite element discretization. The singularities of the inverse problem are regularized by a homotopy mapping. A projection scheme is proposed where anisotropic pre-stress is defined with respect to an additional reference plane, which reflects the initially developable surface of membrane strips in the production process. Physically problematic combinations of edge geometry and surface stress are solved by a self-adaptive stress correction scheme. The algorithm is based on a local criterion derived from differential geometry. Several examples illustrate the success of each idea and implementation. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2005
Jean-François Remacle
Abstract An anisotropic adaptive analysis procedure based on a discontinuous Galerkin finite element discretization and local mesh modification of simplex elements is presented. The procedure is applied to transient two- and three-dimensional problems governed by Euler's equation. A smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified. The mesh metric field in smooth portions of the domain is controlled by a Hessian matrix constructed using a variational procedure to calculate the second derivatives. The transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Hierarchic finite element bases on unstructured tetrahedral meshes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2003
Mark Ainsworth
Abstract The problem of constructing hierarchic bases for finite element discretization of the spaces H1, H(curl), H(div) and L2 on tetrahedral elements is addressed. A simple and efficient approach to ensuring conformity of the approximations across element interfaces is described. Hierarchic bases of arbitrary polynomial order are presented. It is shown how these may be used to construct finite element approximations of arbitrary, non-uniform, local order approximation on unstructured meshes of curvilinear tetrahedral elements. Copyright © 2003 John Wiley & Sons, Ltd. [source]

The modeling and numerical analysis of wrinkled membranes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2003
Hongli Ding
Abstract In this paper three fundamental issues regarding modeling and analysis of wrinkled membranes are addressed. First, a new membrane model with viable Young's modulus and Poisson's ratio is proposed, which physically characterizes stress relaxation phenomena in membrane wrinkling, and expresses taut, wrinkled and slack states of a membrane in a systematic manner. Second, a parametric variational principle is developed for the new membrane model. Third, by the variational principle, the original membrane problem is converted to a non-linear complementarity problem in mathematical programming. A parametric finite element discretization and a smoothing Newton method are then used for numerical solution. The proposed membrane model and numerical method are capable of delivering convergent results for membranes with a mixture of wrinkled and slack regions, without iteration of membrane stresses. Three numerical examples are provided. Copyright © 2003 John Wiley & Sons, Ltd. [source]

A general non-linear optimization algorithm for lower bound limit analysis

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2003
Kristian Krabbenhoft
Abstract The non-linear programming problem associated with the discrete lower bound limit analysis problem is treated by means of an algorithm where the need to linearize the yield criteria is avoided. The algorithm is an interior point method and is completely general in the sense that no particular finite element discretization or yield criterion is required. As with interior point methods for linear programming the number of iterations is affected only little by the problem size. Some practical implementation issues are discussed with reference to the special structure of the common lower bound load optimization problem, and finally the efficiency and accuracy of the method is demonstrated by means of examples of plate and slab structures obeying different non-linear yield criteria. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Implicit symmetrized streamfunction formulations of magnetohydrodynamics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2008
K. S. Kang
Abstract We apply the finite element method to the classic tilt instability problem of two-dimensional, incompressible magnetohydrodynamics, using a streamfunction approach to enforce the divergence-free conditions on the magnetic and velocity fields. We compare two formulations of the governing equations, the standard one based on streamfunctions and a hybrid formulation with velocities and magnetic field components. We use a finite element discretization on unstructured meshes and an implicit time discretization scheme. We use the PETSc library with index sets for parallelization. To solve the nonlinear problems on each time step, we compare two nonlinear Gauss-Seidel-type methods and Newton's method with several time-step sizes. We use GMRES in PETSc with multigrid preconditioning to solve the linear subproblems within the nonlinear solvers. We also study the scalability of this simulation on a cluster. 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 2008
M. ur Rehman
Abstract 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]

Efficient preconditioning of the discrete adjoint equations for the incompressible Navier,Stokes equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005
René Schneider
Abstract 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 new stable space,time formulation for two-dimensional and three-dimensional incompressible viscous flow

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2001
Donatien N'dri
Abstract A space,time finite element method for the incompressible Navier,Stokes equations in a bounded domain in ,d (with d=2 or 3) is presented. The method is based on the time-discontinuous Galerkin method with the use of simplex-type meshes together with the requirement that the space,time finite element discretization for the velocity and the pressure satisfy the inf,sup stability condition of Brezzi and Babu,ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two-dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 9 2009
Jing Li
Abstract A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite systems of linear equations of the form (K,,2M)u=f, which arise from solving eigenvalue problems when an inverse shifted method is used and also from the finite element discretization of Helmholtz equations. Here, both K and M are symmetric positive definite. The proposed BDDC method is closely related to the previous dual,primal finite element tearing and interconnecting method (FETI-DP) for solving this type of problems (Appl. Numer. Math. 2005; 54:150,166), where a coarse level problem containing certain free-space solutions of the inherent homogeneous partial differential equation is used in the algorithm to accelerate the convergence. Under the condition that the diameters of the subdomains are small enough, the convergence rate of the proposed algorithm is established, which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with a decrease of the subdomain diameters. These results are supported by numerical experiments of solving a two-dimensional problem. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Preconditioners for the discretized time-harmonic Maxwell equations in mixed form

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2007
Chen Greif
Abstract We introduce a new preconditioning technique for iteratively solving linear systems arising from finite element discretization of the mixed formulation of the time-harmonic Maxwell equations. The preconditioners are motivated by spectral equivalence properties of the discrete operators, but are augmentation free and Schur complement free. We provide a complete spectral analysis, and show that the eigenvalues of the preconditioned saddle point matrix are strongly clustered. The analytical observations are accompanied by numerical results that demonstrate the scalability of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Algebraic multilevel preconditioning of finite element matrices using local Schur complements

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2006
J. K. Kraus
Abstract We consider an algebraic multilevel preconditioning technique for SPD matrices arising from finite element discretization of elliptic PDEs. In particular, we address the case of non-M matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The left upper block of the considered multiplicative two-level preconditioner is approximated using incomplete factorization techniques. The coarse-grid element matrices are simply Schur complements computed from local neighbourhood matrices, i.e. small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation that can be analysed by regarding (local) macro elements. These components, when combined in the framework of an algebraic multilevel iteration, yield a robust and efficient linear solver. The presented numerical experiments include also the Lamé differential equation for the displacements in the two-dimensional plane-stress elasticity problem. Copyright © 2005 John Wiley & Sons, Ltd. [source]

An algebraic multigrid method for finite element discretizations with edge elements

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2002
S. Reitzinger
Abstract This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H0(curl,,). The finite element spaces are generated by Nédélec's edge elements. A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl-operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ,discrete' gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework. Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright © 2002 John Wiley & Sons, Ltd. [source]

A distributed memory parallel implementation of the multigrid method for solving three-dimensional implicit solid mechanics problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2004
A. Namazifard
Abstract We describe the parallel implementation of a multigrid method for unstructured finite element discretizations of solid mechanics problems. We focus on a distributed memory programming model and use the MPI library to perform the required interprocessor communications. We present an algebraic framework for our parallel computations, and describe an object-based programming methodology using Fortran90. The performance of the implementation is measured by solving both fixed- and scaled-size problems on three different parallel computers (an SGI Origin2000, an IBM SP2 and a Cray T3E). The code performs well in terms of speedup, parallel efficiency and scalability. However, the floating point performance is considerably below the peak values attributed to these machines. Lazy processors are documented on the Origin that produce reduced performance statistics. The solution of two problems on an SGI Origin2000, an IBM PowerPC SMP and a Linux cluster demonstrate that the algorithm performs well when applied to the unstructured meshes required for practical engineering analysis. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier,Stokes equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2002
Volker John
Abstract This paper presents a numerical study of the 3D flow around a cylinder which was defined as a benchmark problem for the steady state Navier,Stokes equations within the DFG high-priority research program flow simulation with high-performance computers by Schafer and Turek (Vol. 52, Vieweg: Braunschweig, 1996). The first part of the study is a comparison of several finite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary fitted higher order finite element methods are in general most accurate. Our numerical study improves the hitherto existing reference values for the benchmark parameters considerably. The second part of the study deals with efficient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The flexible GMRES method with a multiple discretization multigrid method proves to be the best solver. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Higher-order finite element discretizations in a benchmark problem for incompressible flows

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2001
Volker John
Abstract We present a numerical study of several finite element discretizations applied to a benchmark problem for the two-dimensional steady state incompressible Navier,Stokes equations defined in Schäfer and Turek (The benchmark problem ,Flow around a cylinder'. In Flow Simulation with High-Performance Computers II. Notes on Numerical Fluid Mechanics, vol. 52, Hirschel EH (ed.). Vieweg: Wiesbaden, 1996; 547,566). The discretizations are compared with respect to the accuracy of the computed benchmark parameters. Higher-order isoparametric finite element discretizations turned out to be by far the most accurate. The discrete systems obtained with higher-order discretizations are solved with a modified coupled multigrid method whose behaviour within the benchmark problem is also studied numerically. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Sparse factorization of finite element matrices using overlapped localizing solution modes

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 4 2008
J.-S. Choi
Abstract Local-global solution (LOGOS) modes provide a computationally efficient framework for developing fast, direct solution methods for electromagnetic simulations. In this article, we demonstrate that the LOGOS framework yields fast direct solutions for finite element discretizations of the wave equation in two dimensions. For fixed-frequency applications, numerical examples demonstrate that the memory and CPU complexities of the proposed solver are nearly linear. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1050,1054, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23298 [source]

Efficient algorithms for multiscale modeling in porous media

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2010
Mary F. Wheeler
Abstract We describe multiscale mortar mixed finite element discretizations for second-order elliptic and nonlinear parabolic equations modeling Darcy flow in porous media. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. We discuss the construction of multiscale mortar basis and extend this concept to nonlinear interface operators. We present a multiscale preconditioning strategy to minimize the computational cost associated with construction of the multiscale mortar basis. We also discuss the use of appropriate quadrature rules and approximation spaces to reduce the saddle point system to a cell-centered pressure scheme. In particular, we focus on multiscale mortar multipoint flux approximation method for general hexahedral grids and full tensor permeabilities. Numerical results are presented to verify the accuracy and efficiency of these approaches. Copyright © 2010 John Wiley & Sons, Ltd. [source]

On some versions of the element agglomeration AMGe method

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2008
Ilya Lashuk
Abstract The present paper deals with element-based algebraic multigrid (AMGe) methods that target linear systems of equations coming from finite element discretizations of elliptic partial differential equations. The individual element information (element matrices and element topology) is the main input to construct the AMG hierarchy. We study a number of variants of the spectral agglomerate AMGe method. The core of the algorithms relies on element agglomeration utilizing the element topology (built recursively from fine to coarse levels). The actual selection of the coarse degrees of freedom is based on solving a large number of local eigenvalue problems. Additionally, we investigate strategies for adaptive AMG as well as multigrid cycles that are more expensive than the V -cycle utilizing simple interpolation matrices and nested conjugate gradient (CG)-based recursive calls between the levels. The presented algorithms are illustrated with an extensive set of experiments based on a matlab implementation of the methods. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2008
J. K. Kraus
Abstract We construct optimal order multilevel preconditioners for interior-penalty discontinuous Galerkin (DG) finite element discretizations of three-dimensional (3D) anisotropic elliptic boundary-value problems. In this paper, we extend the analysis of our approach, introduced earlier for 2D problems (SIAM J. Sci. Comput., accepted), to cover 3D problems. A specific assembling process is proposed, which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. A new bound for the constant , in the strengthened Cauchy,Bunyakowski,Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approach

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 9 2007
J. Haslinger
Abstract This paper deals with a fast method for solving large-scale algebraic saddle-point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside the original domain. This approach has a significantly higher convergence rate; however, the algebraic systems resulting from finite element discretizations are typically non-symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle-point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved using a projected Krylov subspace method for non-symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non-projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy. Copyright © 2007 John Wiley & Sons, Ltd. [source]

On algebraic multigrid methods derived from partition of unity nodal prolongators

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2006
Tim Boonen
Abstract This paper is concerned with algebraic multigrid for finite element discretizations of the divgrad, curlcurl and graddiv equations on tetrahedral meshes with piecewise linear shape functions. First, an edge, face and volume prolongator are derived from an arbitrary partition of unity nodal prolongator for a tetrahedral fine mesh, using the formulas for edge, face and volume elements. This procedure can be repeated recursively. The implied coarse topology and the normalization of the prolongators are analysed. It is proved that the range spaces of the nodal prolongator and of the derived edge, face and volume prolongators form a discrete de Rham complex if these prolongators have full rank. It is shown that on simplicial meshes, the constructed edge prolongator is a generalization of the Reitzinger,Schöberl prolongator. The derived edge and face prolongators are applied in an algebraic multigrid method for the curlcurl and graddiv equations, and numerical results are presented. Copyright © 2006 John Wiley & Sons, Ltd. [source]