Galerkin

Distribution by Scientific Domains

Kinds of Galerkin

  • discontinuous galerkin
  • element-free galerkin

  • Terms modified by Galerkin

  • galerkin approach
  • galerkin approximation
  • galerkin boundary element method
  • galerkin finite element method
  • galerkin formulation
  • galerkin method
  • galerkin methods
  • galerkin projection
  • galerkin solution
  • galerkin weak form

  • Selected Abstracts


    Some numerical issues using element-free Galerkin mesh-less method for coupled hydro-mechanical problems

    INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 7 2009
    Mohammad Norouz Oliaei
    Abstract A new formulation of the element-free Galerkin (EFG) method is developed for solving coupled hydro-mechanical problems. The numerical approach is based on solving the two governing partial differential equations of equilibrium and continuity of pore water simultaneously. Spatial variables in the weak form, i.e. displacement increment and pore water pressure increment, are discretized using the same EFG shape functions. An incremental constrained Galerkin weak form is used to create the discrete system equations and a fully implicit scheme is used for discretization in the time domain. Implementation of essential boundary conditions is based on a penalty method. Numerical stability of the developed formulation is examined in order to achieve appropriate accuracy of the EFG solution for coupled hydro-mechanical problems. Examples are studied and compared with closed-form or finite element method solutions to demonstrate the validity of the developed model and its capabilities. The results indicate that the EFG method is capable of handling coupled problems in saturated porous media and can predict well both the soil deformation and variation of pore water pressure over time. Some guidelines are proposed to guarantee the accuracy of the EFG solution for coupled hydro-mechanical problems. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    A discontinuous Galerkin method for elliptic interface problems with application to electroporation

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2009
    Grégory Guyomarc'h
    Abstract We solve elliptic interface problems using a discontinuous Galerkin (DG) method, for which discontinuities in the solution and in its normal derivatives are prescribed on an interface inside the domain. Standard ways to solve interface problems with finite element methods consist in enforcing the prescribed discontinuity of the solution in the finite element space. Here, we show that the DG method provides a natural framework to enforce both discontinuities weakly in the DG formulation, provided the triangulation of the domain is fitted to the interface. The resulting discretization leads to a symmetric system that can be efficiently solved with standard algorithms. The method is shown to be optimally convergent in the L2 -norm. We apply our method to the numerical study of electroporation, a widely used medical technique with applications to gene therapy and cancer treatment. Mathematical models of electroporation involve elliptic problems with dynamic interface conditions. We discretize such problems into a sequence of elliptic interface problems that can be solved by our method. We obtain numerical results that agree with known exact solutions. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    Analysis of thick functionally graded plates by local integral equation method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2007
    J. Sladek
    Abstract Analysis of functionally graded plates under static and dynamic loads is presented by the meshless local Petrov,Galerkin (MLPG) method. Plate bending problem is described by Reissner,Mindlin theory. Both isotropic and orthotropic material properties are considered in the analysis. A weak formulation for the set of governing equations in the Reissner,Mindlin theory with a unit test function is transformed into local integral equations considered on local subdomains in the mean surface of the plate. Nodal points are randomly spread on this surface and each node is surrounded by a circular subdomain, rendering integrals which can be simply evaluated. The meshless approximation based on the moving least-squares (MLS) method is employed in the numerical implementation. Numerical results for simply supported and clamped plates are presented. Copyright © 2006 John Wiley & Sons, Ltd. [source]


    Coupling of mesh-free methods with finite elements: basic concepts and test results

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2006
    T. Rabczuk
    Abstract This paper reviews several novel and older methods for coupling mesh-free particle methods, particularly the element-free Galerkin (EFG) method and the smooth particle hydrodynamics (SPH), with finite elements (FEs). We study master,slave couplings where particles are fixed across the FE boundary, coupling via interface shape functions such that consistency conditions are satisfied, bridging domain coupling, compatibility coupling with Lagrange multipliers and hybrid coupling methods where forces from the particles are applied via their shape functions on the FE nodes and vice versa. The hybrid coupling methods are well suited for large deformations and adaptivity and the coupling procedure is independent of the particle distance and nodal arrangement. We will study the methods for several static and dynamic applications, compare the results to analytical and experimental data and show advantages and drawbacks of the methods. Copyright © 2006 John Wiley & Sons, Ltd. [source]


    Imposition of essential boundary conditions by displacement constraint equations in meshless methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2001
    Xiong Zhang
    Abstract One of major difficulties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary ,u, and one consisting of the remaining unknowns. A simplified displacement constraint equations method (SDCEM) is also proposed, which results in a efficient scheme with sufficient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded stiffness matrix. Numerical results show that the accuracy of the present method is higher than that of the modified variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin-based meshless method, such as element-free Galerkin methods, reproducing kernel particle method, meshless local Petrov,Galerkin method. Copyright © 2001 John Wiley & Sons, Ltd. [source]


    On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment,

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
    I. Kalashnikova
    Abstract A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L2 inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well-posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well-posed and stable far-field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty-like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd. [source]


    An optimally convergent discontinuous Galerkin-based extended finite element method for fracture mechanics

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2010
    Yongxing Shen
    Abstract The extended finite element method (XFEM) enables the representation of cracks in arbitrary locations of a mesh. We introduce here a variant of the XFEM rendering an optimally convergent scheme. Its distinguishing features are as follows: (a) the introduction of singular asymptotic crack tip fields with support on only a small region around the crack tip (the enrichment region), (b) only one and two enrichment functions are added for anti-plane shear and planar problems, respectively and (c) the relaxation of the continuity between the enrichment region and the rest of the domain, and the adoption of a discontinuous Galerkin (DG) method therein. The method is provably stable for any positive value of a stabilization parameter, and by weakly enforcing the continuity between the two regions it eliminates ,blending elements' partly responsible for the suboptimal convergence of some early XFEMs. Moreover, the particular choice of enrichment functions results in a surprisingly sparse stiffness matrix that remains reasonably conditioned as the mesh is refined. More importantly, the stress intensity factors can be extracted with a satisfactory accuracy as primary unknowns. Quadrature strategies required for the optimal convergence are also discussed. Finally, the DG method was modified to retain stability based on an inf-sup condition. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    An adaptive spacetime discontinuous Galerkin method for cohesive models of elastodynamic fracture

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
    Reza Abedi
    Abstract This paper describes an adaptive numerical framework for cohesive fracture models based on a spacetime discontinuous Galerkin (SDG) method for elastodynamics with elementwise momentum balance. Discontinuous basis functions and jump conditions written with respect to target traction values simplify the implementation of cohesive traction,separation laws in the SDG framework; no special cohesive elements or other algorithmic devices are required. We use unstructured spacetime grids in a h -adaptive implementation to adjust simultaneously the spatial and temporal resolutions. Two independent error indicators drive the adaptive refinement. One is a dissipation-based indicator that controls the accuracy of the solution in the bulk material; the second ensures the accuracy of the discrete rendering of the cohesive law. Applications of the SDG cohesive model to elastodynamic fracture demonstrate the effectiveness of the proposed method and reveal a new solution feature: an unexpected quasi-singular structure in the velocity response. Numerical examples demonstrate the use of adaptive analysis methods in resolving this structure, as well as its importance in reliable predictions of fracture kinetics. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    A hybridizable discontinuous Galerkin method for linear elasticity

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2009
    S.-C. Soon
    Abstract This paper describes the application of the so-called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block-wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of k+1 for any k,0. The third feature is that, by using an element-by-element post-processing, a new approximate displacement can be obtained that converges at the order of k+2, whenever k,2. Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6-7 2009
    F. Nobile
    Abstract We consider the problem of numerically approximating statistical moments of the solution of a time-dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen,Loève expansions driven by a finite number of uncorrelated random variables. After approximating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Lower-bound limit analysis by using the EFG method and non-linear programming

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2008
    Shenshen Chen
    Abstract Intended to avoid the complicated computations of elasto-plastic incremental analysis, limit analysis is an appealing direct method for determining the load-carrying capacity of structures. On the basis of the static limit analysis theorem, a solution procedure for lower-bound limit analysis is presented firstly, making use of the element-free Galerkin (EFG) method rather than traditional numerical methods such as the finite element method and boundary element method. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration. The reduced-basis technique is adopted to solve the mathematical programming iteratively in a sequence of reduced self-equilibrium stress subspaces with very low dimensions. The self-equilibrium stress field is expressed by a linear combination of several self-equilibrium stress basis vectors with parameters to be determined. These self-equilibrium stress basis vectors are generated by performing an equilibrium iteration procedure during elasto-plastic incremental analysis. The Complex method is used to solve these non-linear programming sub-problems and determine the maximal load amplifier. Numerical examples show that it is feasible and effective to solve the problems of limit analysis by using the EFG method and non-linear programming. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    An element-wise, locally conservative Galerkin (LCG) method for solving diffusion and convection,diffusion problems

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2008
    C. G. Thomas
    Abstract An element-wise locally conservative Galerkin (LCG) method is employed to solve the conservation equations of diffusion and convection,diffusion. This approach allows the system of simultaneous equations to be solved over each element. Thus, the traditional assembly of elemental contributions into a global matrix system is avoided. This simplifies the calculation procedure over the standard global (continuous) Galerkin method, in addition to explicitly establishing element-wise flux conservation. In the LCG method, elements are treated as sub-domains with weakly imposed Neumann boundary conditions. The LCG method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging. It is also shown in this paper that the proposed LCG method is identical to the standard global Galerkin (GG) method, at both steady and unsteady states, for an inside node. Thus, the method, has all the advantages of the standard GG method while explicitly conserving fluxes over each element. Several problems of diffusion and convection,diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the LCG method. Both linear and quadratic elements are used in the calculations. For convection-dominated problems, Petrov,Galerkin weighting and high-order characteristic-based temporal schemes have been implemented into the LCG formulation. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    Conservation properties of a time FE method.

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2005
    Part IV: Higher order energy, momentum conserving schemes
    Abstract In the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite-dimensional Hamiltonian systems with symmetry is shown. The result of this development is the enhanced Galerkin (eG) finite element method in time. The conservation of the eG method is generally related to its collocation property. Total energy conservation, in particular, is obtained by a new projection technique. The eG method is, moreover, based on objective time discretization of the used strain measure. This paper is concerned with particle dynamics and semi-discrete non-linear elastodynamics. The related numerical examples show good performance in presence of stiffness as well as for calculating large-strain motions. Copyright © 2005 John Wiley & Sons, Ltd. [source]


    On singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems with corners

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004
    A. Dimitrov
    Abstract In this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin,Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P+,Q+,2R)d=0 is obtained, where the saddle-point-type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa-like scheme is used. This technique needs only one direct matrix factorization as well as few matrix,vector products for finding all eigenvalues in the interval ,,(,) , (,0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface-breaking crack in an incompressible elastic material and the three-dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd. [source]


    A posteriori error approximation in EFG method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2003
    L. Gavete
    Abstract Recently, considerable effort has been devoted to the development of the so-called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the ,a posteriori error' should be approximated. An ,a posteriori error' approximation based on the moving least-squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least-squares approximations. Different selection strategies of the moving least-squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error. The performance of the developed approximation of the error is illustrated by analysing different examples for two-dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Time accurate consistently stabilized mesh-free methods for convection dominated problems

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2003
    Antonio Huerta
    Abstract The behaviour of high-order time stepping methods combined with mesh-free methods is studied for the transient convection,diffusion equation. Particle methods, such as the element-free Galerkin (EFG) method, allow to easily increase the order of consistency and, thus, to formulate high-order schemes in space and time. Moreover, second derivatives of the EFG shape functions can be constructed with a low extra cost and are well defined, even for linear interpolation. Thus, consistent stabilization schemes can be considered without loss in the convergence rates. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Adaptive crack propagation analysis with the element-free Galerkin method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2003
    Gye-Hee Lee
    Abstract In this paper, an adaptive analysis of crack propagation based on the error estimation by the element-free Galerkin (EFG) method is presented. The adaptivity analysis in quasi-static crack propagation is achieved by adding and/or removing the nodes along the background integration cells, those are refined or recovered according to the estimated errors. These errors are obtained basically by calculating the difference between the values of the projected stresses and original EFG stresses. To evaluate the performance of the proposed adaptive procedure, the crack propagation behaviour is investigated for several examples. The results of these examples show the efficiency and accuracy of the proposed scheme in crack propagation analysis. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    Moving kriging interpolation and element-free Galerkin method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003
    Lei Gu
    Abstract A new formulation of the element-free Galerkin (EFG) method is presented in this paper. EFG has been extensively popularized in the literature in recent years due to its flexibility and high convergence rate in solving boundary value problems. However, accurate imposition of essential boundary conditions in the EFG method often presents difficulties because the Kronecker delta property, which is satisfied by finite element shape functions, does not necessarily hold for the EFG shape function. The proposed new formulation of EFG eliminates this shortcoming through the moving kriging (MK) interpolation. Two major properties of the MK interpolation: the Kronecker delta property (,I(sJ)=,IJ) and the consistency property (,In,I(x)=1 and ,In,I(x)xIi=xi) are proved. Some preliminary numerical results are given. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    Discrete singular convolution and its application to the analysis of plates with internal supports.

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2002
    Part 1: Theory, algorithm
    Abstract This paper presents a novel computational approach, the discrete singular convolution (DSC) algorithm, for analysing plate structures. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied. Approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels. Unified features of the DSC algorithm for solving differential equations are explored. It is demonstrated that different methods of implementation for the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. The use of the algorithm for the vibration analysis of plates with internal supports is discussed. Detailed formulation is given to the treatment of different plate boundary conditions, including simply supported, elastically supported and clamped edges. This work paves the way for applying the DSC approach in the following paper to plates with complex support conditions, which have not been fully addressed in the literature yet. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    A control volume capacitance method for solidification modelling with mass transport

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002
    K. Davey
    Abstract Capacitance methods are popular methods used for solidification modelling. Unfortunately, they suffer from a major drawback in that energy is not correctly transported through elements and so provides a source of inaccuracy. This paper is concerned with the development and application of a control volume capacitance method (CVCM) to problems where mass transport and solidification are combined. The approach adopted is founded on theory that describes energy transfer through a control volume (CV) moving relative to the transporting mass. An equivalent governing partial differential equation is established, which is designed to be transformable into a finite element system that is commonly used to model transient heat-conduction problems. This approach circumvents the need to use the methods of Bubnov,Galerkin and Petrov,Galerkin and thus eliminates many of the stability problems associated with these approaches. An integration scheme is described that accurately caters for enthalpy fluxes generated by mass transport. Shrinkage effects are neglected in this paper as all the problems considered involve magnitudes of velocity that make this assumption reasonable. The CV approach is tested against known analytical solutions and is shown to be accurate, stable and computationally competitive. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    Essential boundary condition enforcement in meshless methods: boundary flux collocation method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002
    Cheng-Kong C. Wu
    Abstract Element-free Galerkin (EFG) methods are based on a moving least-squares (MLS) approximation, which has the property that shape functions do not satisfy the Kronecker delta function at nodal locations, and for this reason imposition of essential boundary conditions is difficult. In this paper, the relationship between corrected collocation and Lagrange multiplier method is revealed, and a new strategy that is accurate and very simple for enforcement of essential boundary conditions is presented. The accuracy and implementation of this new technique is illustrated for one-dimensional elasticity and two-dimensional potential field problems. Copyright © 2001 John Wiley & Sons, Ltd. [source]


    Efficient computation of order and mode of corner singularities in 3D-elasticity

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2001
    A. Dimitrov
    Abstract A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non-smooth domains in three-dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin,Petrov finite element method. A quadratic eigenvalue problem (P+,Q+,2R) u=0 is obtained, with explicitly analytically defined matrices P,Q,R. Moreover, the three matrices are found to have optimal structure, so that P,R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ,e(,),(,0.5,1.0) (no eigenpairs can be ,lost') as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill-in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd. [source]


    Numerical simulations of viscous flows using a meshless method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2008
    Changfu You
    Abstract This paper uses the element-free Galerkin (EFG) method to simulate 2D, viscous, incompressible flows. The control equations are discretized with the standard Galerkin method in space and a fractional step finite element scheme in time. Regular background cells are used for the quadrature. Several classical fluid mechanics problems were analyzed including flow in a pipe, flow past a step and flow in a driven cavity. The flow field computed with the EFG method compared well with those calculated using the finite element method (FEM) and finite difference method. The simulations show that although EFG is more expensive computationally than FEM, it is capable of dealing with cases where the nodes are poorly distributed or even overlap with each other; hence, it may be used to resolve remeshing problems in direct numerical simulations. Flows around a cylinder for different Reynolds numbers are also simulated to study the flow patterns for various conditions and the drag and lift forces exerted by the fluid on the cylinder. These forces are calculated by integrating the pressure and shear forces over the cylinder surface. The results show how the drag and lift forces oscillate for high Reynolds numbers. The calculated Strouhal number agrees well with previous results. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    One-dimensional simulation of supercritical flow at a confluence by means of a nonlinear junction model applied with the RKDG2 method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2008
    G. Kesserwani
    Abstract We investigate the one-dimensional computation of supercritical open-channel flows at a combining junction. In such situations, the network system is composed of channel segments arranged in a branching configuration, with individual channel segments connected at a junction. Therefore, two important issues have to be addressed: (a) the numerical solution in branches, and (b) the internal boundary conditions treatment at the junction. Going from the advantageous literature supports of RKDG methods to a particular investigation for a supercritical benchmark, the second-order Runge,Kutta discontinuous Galerkin (RKDG2) scheme is selected to compute the water flow in branches. For the internal boundary handling, we propose a new approach by incorporating the nonlinear model derived from the conservation of the momentum through the junction. The nonlinear junction model was evaluated against available experiments and then applied to compute the junction internal boundary treatment for steady and unsteady flow applications. Finally, a combining flow problem is defined and simulated by the proposed framework and results are illustrated for many choices of junction angles. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    A finite element strategy for the solution of interface tracking problems

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2005
    C. Devals
    Abstract A finite element-based numerical strategy for interface tracking is developed for the simulation of two-phase flows. The method is based on the solution of an advection equation for the so-called ,pseudo-concentration' of one of the phases. To obtain an accurate description of the interface, a streamline upwind Petrov,Galerkin (SUPG) scheme is combined with an automatic mesh refinement procedure and a filtering technique, making it possible to generate an oscillation-free pseudo-concentration field. The performance of the proposed approach is successfully tested on four classical two-dimensional benchmark problems: the advection skew to the mesh, the transport of a square shape in a constant velocity flow field, the transport of a cut-out cylinder in a rotating flow field and the transport of a disc in a shear flow. Copyright © 2005 John Wiley & Sons, Ltd. [source]


    Buffeting in transonic flow prediction using time-dependent turbulence model

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005
    A. Kourta
    Abstract In transonic flow conditions, the shock wave/turbulent boundary layer interaction and flow separations on wing upper surface induce flow instabilities, ,buffet', and then the buffeting (structure vibrations). This phenomenon can greatly influence the aerodynamic performance. These flow excitations are self-sustained and lead to a surface effort due to pressure fluctuations. They can produce enough energy to excite the structure. The objective of the present work is to predict this unsteady phenomenon correctly by using unsteady Navier,Stokes-averaged equations with a time-dependent turbulence model based on the suitable (k,,) turbulent eddy viscosity model. The model used is based on the turbulent viscosity concept where the turbulent viscosity coefficient (C,) is related to local deformation and rotation rates. To validate this model, flow over a flat plate at Mach number of 0.6 is first computed, then the flow around a NACA0012 airfoil. The comparison with the analytical and experimental results shows a good agreement. The ONERA OAT15A transonic airfoil was chosen to describe buffeting phenomena. Numerical simulations are done by using a Navier,Stokes SUPG (streamline upwind Petrov,Galerkin) finite-element solver. Computational results show the ability of the present model to predict physical phenomena of the flow oscillations. The unsteady shock wave/boundary layer interaction is described. Copyright © 2005 John Wiley & Sons, Ltd. [source]


    Superconvergence and H(div) projection for discontinuous Galerkin methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2003
    Peter Bastian
    Abstract We introduce and analyse a projection of the discontinuous Galerkin (DG) velocity approximations that preserve the local mass conservation property. The projected velocities have the additional property of continuous normal component. Both theoretical and numerical convergence rates are obtained which show that the accuracy of the DG velocity field is maintained. Superconvergence properties of the DG methods are shown. Finally, numerical simulations of complicated flow and transport problem illustrate the benefits of the projection. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Positivity-preserving, flux-limited finite-difference and finite-element methods for reactive transport

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2003
    Robert J. MacKinnon
    Abstract A new class of positivity-preserving, flux-limited finite-difference and Petrov,Galerkin (PG) finite-element methods are devised for reactive transport problems. The methods are similar to classical TVD flux-limited schemes with the main difference being that the flux-limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite-element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity-preserving property. Analysis of the latter scheme shows that positivity-preserving solutions of the resulting difference equations can only be guaranteed if the flux-limited scheme is both implicit and satisfies an additional lower-bound condition on time-step size. We show that this condition also applies to standard Galerkin linear finite-element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time-step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Numerical investigation of the first instabilities in the differentially heated 8:1 cavity

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2002
    F. Auteri
    Abstract We present a new Galerkin,Legendre spectral projection solver for the simulation of natural convection in a differentially heated cavity. The projection method is applied to the study of the first non-stationary instabilities of the flow in a 8:1 cavity. Statistics of the periodic solution are reported for a Rayleigh number of 3.4×105. Moreover, we investigate the location and properties of the first Hopf bifurcation and of the three successive bifurcations. The results confirm the previous finding in the range of Rayleigh numbers investigated that the flow instabilities originate in the boundary layer on the vertical walls. A peculiar phenomenon of symmetry breaking and symmetry restoring is observed portraying the first steps of the transition to chaos for this flow. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    LS-DYNA and the 8:1 differentially heated cavity

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2002
    Mark A. Christon
    Abstract This paper presents results computed using LS-DYNA's new incompressible flow solver for a differentially heated cavity with an 8:1 aspect ratio at a slightly super-critical Rayleigh number. Three Galerkin-based solution methods are applied to the 8:1 thermal cavity on a sequence of four grids. The solution methods include an explicit time-integration algorithm and two second-order projection methods,one semi-implicit and the other fully implicit. A series of ad hoc modifications to the basic Galerkin finite element method are shown to result in degraded solution quality with the most serious effects introduced by row-sum lumping the mass matrix. The inferior accuracy of a lumped mass matrix relative to a consistent mass matrix is demonstrated with the explicit algorithm which fails to obtain a transient solution on the coarsest grid and exhibits a general trend to under-predict oscillation amplitudes. The best results are obtained with semi-implicit and fully implicit second-order projection methods where the fully implicit method is used in conjunction with a ,smart' time integrator. Copyright © 2002 John Wiley & Sons, Ltd. [source]