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Discretization Schemes (discretization + scheme)
Kinds of Discretization Schemes Selected AbstractsA mixed FEM approach to stress-constrained topology optimizationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2008M. Bruggi Abstract We present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion. The main idea is to adopt a mixed finite-element discretization scheme wherein not only displacements (as usual) but also stresses are the variables entering the formulation. By doing so, any stress constraint may be handled within the optimization procedure without resorting to post-processing operation typical of displacement-based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed. Two dual variational principles of Hellinger,Reissner type are presented in continuous and discrete form that, which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes (Int. J. Numer. Meth. Engng. 1984; 24(3):359,373). Extensive numerical simulations are performed and ongoing extensions outlined, including the optimization of elastoplastic and incompressible media. Copyright © 2007 John Wiley & Sons, Ltd. [source] Implicit symmetrized streamfunction formulations of magnetohydrodynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2008K. 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 cascadic conjugate gradient algorithm for mass conservative, semi-implicit discretization of the shallow water equations on locally refined structured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1-2 2002Luca Bonaventura Abstract A semi-implicit, mass conservative discretization scheme is applied to the two-dimensional shallow water equations on a hierarchy of structured, locally refined Cartesian grids. Different resolution grids are fully interacting and the discrete Helmholtz equation obtained from the semi-implicit discretization is solved by the cascadic conjugate gradient method. A flux correction is applied at the interface between the coarser and finer discretization grids, so as to ensure discrete mass conservation, along with symmetry and diagonal dominance of the resulting matrix. Two-dimensional idealized simulations are presented, showing the accuracy and the efficiency of the resulting method. Copyright © 2002 John Wiley & Sons, Ltd. [source] A time-independent approach for computing wave functions of the Schrödinger,Poisson systemNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2008C.-S. Chien Abstract We describe a two-grid finite element discretization scheme for computing wave functions of the Schrödinger,Poisson (SP) system. To begin with, we compute the first k eigenpairs of the Schrödinger,Poisson eigenvalue (ESP) problem on the coarse grid using a continuation algorithm, where the nonlinear Poisson equation is solved iteratively. We use the k eigenpairs obtained on the coarse grid as initial guesses for computing their counterparts of the ESP on the fine grid. The wave functions of the SP system can be easily obtained using the formula of separation of variables. The proposed algorithm has the following advantages. (i) The initial approximate eigenpairs used in the fine grid can be obtained with low computational cost. (ii) It is unnecessary to discretize the partial derivative of the wave function with respect to the time variable in the SP system. (iii) The major computational difficulties such as closely clustered eigenvalues that occur in the SP system can be effectively computed. Numerical results on the ESP and the SP system are reported. In particular, the rate of convergence of the proposed algorithm is O(h4). Copyright © 2007 John Wiley & Sons, Ltd. [source] The effect of overall discretization scheme on Jacobian structure, convergence rate, and solution accuracy within the local rectangular refinement methodNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 8 2001Beth Anne V. Bennett Abstract The local rectangular refinement (LRR) solution-adaptive gridding method automatically produces orthogonal unstructured adaptive grids and incorporates multiple-scale finite differences to discretize systems of elliptic governing partial differential equations (PDEs). The coupled non-linear discretized equations are solved simultaneously via Newton's method with a Bi-CGSTAB linear system solver. The grids' unstructured nature produces a nonstandard sparsity pattern within the Jacobian. The effects of two discretization schemes (LRR multiple-scale stencils and traditional single-scale stencils) on Jacobian bandwidth, convergence speed, and solution accuracy are studied. With various point orderings, for two simple problems with analytical solutions, the LRR multiple-scale stencils are seen to: (1) produce Jacobians of smaller bandwidths than those resulting from the traditional single-scale stencils; (2) lead to significantly faster Newton's method convergence than the single-scale stencils; and (3) produce more accurate solutions than the single-scale stencils. The LRR method, including the LRR multiple-scale stencils, is finally applied to an engineering problem governed by strongly coupled, highly non-linear PDEs: a steady-state lean Bunsen flame with complex chemistry, multicomponent transport, and radiation modeling. Very good agreement is observed between the computed flame height and previously published experimental data. Copyright © 2001 John Wiley & Sons, Ltd. [source] Performance and numerical behavior of the second-order scheme of precise time-step integration for transient dynamic analysisNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2007Hang Ma Abstract Spurious high-frequency responses resulting from spatial discretization in time-step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time-step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second-order scheme of the precise integration method (PIM). Taking the Newmark-, method as a reference, the performance and numerical behavior of the second-order PIM for elasto-dynamic impact-response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock-excited rod with rectangular, half-triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine-like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] A conservative scheme for the shallow-water system on a staggered geodesic grid based on a Nambu representationTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 639 2009Matthias Sommer Abstract A conservative spatial discretization scheme is constructed for a shallow-water system on a geodesic grid with C-type staggering. It is derived from the original equations written in Nambu form, which is a generalization of Hamiltonian representation. The term ,conservative scheme' refers to one that preserves the constitutive quantities, here total energy and potential enstrophy. We give a proof for the non-existence of potential enstrophy sources in this semi-discretization. Furthermore, we show numerically that in comparison with traditional discretizations, such schemes can improve stability and the ability to represent conservation and spectral properties of the underlying partial differential equations. Copyright © 2009 Royal Meteorological Society [source] View factor calculation using the Monte Carlo method and numerical sensitivityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2006M. R. Vuji Abstract The geometrical view (configuration) factor plays a crucial role in radiative heat transfer simulations and several methods, such as integration, the Monte Carlo and the hemi-cube method have been introduced to calculate view factors in recent years. In this paper the Monte Carlo method combined with the finite element (FE) technique is investigated. Results describing the relationships between different discretization schemes, number of rays used for the view factor calculation, CPU time and accuracy are presented. The interesting case where reduced accuracy is obtained with increased refinement of FE mesh is discussed. Copyright © 2005 John Wiley & Sons, Ltd. [source] A unified formulation for continuum mechanics applied to fluid,structure interaction in flexible tubesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2005C. J. Greenshields Abstract This paper outlines the development of a new procedure for analysing continuum mechanics problems with a particular focus on fluid,structure interaction in flexible tubes. A review of current methods of fluid,structure coupling highlights common limitations of high computational cost and solution instability. It is proposed that these limitations can be overcome by an alternative approach in which both fluid and solid components are solved within a single discretized continuum domain. A single system of momentum and continuity equations is therefore derived that governs both fluids and solids and which are solved with a single mesh using finite volume discretization schemes. The method is validated first by simulating dynamic oscillation of a clamped elastic beam. It is then applied to study the case of interest,wave propagation in highly flexible tubes,in which a predicted wave speed of 8.58 m/s falls within 2% of an approximate analytical solution. The method shows further good agreement with analytical solutions for tubes of increasing rigidity, covering a range of wave speeds from those found in arteries to that in the undisturbed fluid. Copyright © 2005 John Wiley & Sons, Ltd. [source] Displacement/pressure mixed interpolation in the method of finite spheresINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2001Suvranu De Abstract The displacement-based formulation of the method of finite spheres is observed to exhibit volumetric ,locking' when incompressible or nearly incompressible deformations are encountered. In this paper, we present a displacement/pressure mixed formulation as a solution to this problem. We analyse the stability and optimality of the formulation for several discretization schemes using numerical inf,sup tests. Issues concerning computational efficiency are also discussed. Copyright © 2001 John Wiley & Sons, Ltd. [source] Unstructured finite volume discretization of two-dimensional depth-averaged shallow water equations with porosityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2010L. Cea Abstract This paper deals with the numerical discretization of two-dimensional depth-averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small-scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small-scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two-dimensional shallow water equations with porosity, both of them are high-order schemes. The numerical schemes proposed are well-balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high-order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd. [source] A gradient smoothing method (GSM) for fluid dynamics problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2008G. R. Liu Abstract A novel gradient smoothing method (GSM) based on irregular cells and strong form of governing equations is presented for fluid dynamics problems with arbitrary geometries. Upon the analyses about the compactness and the positivity of coefficients of influence of their stencils for approximating a derivative, four favorable schemes (II, VI, VII and VIII) with second-order accuracy are selected among the total eight proposed discretization schemes. These four schemes are successively verified and carefully examined in solving Poisson's equations, subjected to changes in the number of nodes, the shapes of cells and the irregularity of triangular cells, respectively. Numerical results imply us that all the four schemes give very good results: Schemes VI and VIII produce a slightly better accuracy than the other two schemes on irregular cells, but at a higher cost in computation. Schemes VII and VIII that consistently rely on gradient smoothing operations are more accurate than Schemes II and VI in which directional correction is imposed. It is interestingly found that GSM is insensitive to the irregularity of meshes, indicating the robustness of the presented GSM. Among the four schemes of GSM, Scheme VII outperforms the other three schemes, for its outstanding overall performance in terms of numerical accuracy, stability and efficiency. Finally, GSM solutions with Scheme VII to some benchmarked compressible flows including inviscid flow over NACA0012 airfoil, laminar flow over flat plate and turbulent flow over an RAE2822 airfoil are presented, respectively. Copyright © 2008 John Wiley & Sons, Ltd. [source] Spectral analysis of flux vector splitting finite volume methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2001Tapan K. Sengupta Abstract New results are presented here for finite volume (FV) methods that use flux vector splitting (FVS) along with higher-order reconstruction schemes. Apart from spectral accuracy of the resultant methods, the numerical stability is investigated which restricts the allowable time step or the Courant,Friedrich,Lewy (CFL) number. Also the dispersion relation preservation (DRP) property of various spatial and temporal discretization schemes is investigated. The DRP property simultaneously fixes space and time steps. This aspect of numerical schemes is important for simulation of high-Reynolds number flows, compressible flows with shock(s) and computational aero-acoustics. It is shown here that for direct numerical simulation applications, the DRP property is more restrictive than stability criteria. Copyright © 2001 John Wiley & Sons, Ltd. [source] Volume-of-fluid-based model for multiphase flow in high-pressure trickle-bed reactor: Optimization of numerical parametersAICHE JOURNAL, Issue 11 2009Rodrigo J. G. Lopes Abstract Aiming to understand the effect of various parameters such as liquid velocity, surface tension, and wetting phenomena, a Volume-of-Fluid (VOF) model was developed to simulate the multiphase flow in high-pressure trickle-bed reactor (TBR). As the accuracy of the simulation is largely dependent on mesh density, different mesh sizes were compared for the hydrodynamic validation of the multiphase flow model. Several model solution parameters comprising different time steps, convergence criteria and discretization schemes were examined to establish model parametric independency results. High-order differencing schemes were found to agree better with the experimental data from the literature given that its formulation includes inherently the minimization of artificial numerical dissipation. The optimum values for the numerical solution parameters were then used to evaluate the hydrodynamic predictions at high-pressure demonstrating the significant influence of the gas flow rate mainly on liquid holdup rather than on two-phase pressure drop and exhibiting hysteresis in both hydrodynamic parameters. Afterwards, the VOF model was applied to evaluate successive radial planes of liquid volume fraction at different packed bed cross-sections. © 2009 American Institute of Chemical Engineers AIChE J, 2009 [source] The effect of overall discretization scheme on Jacobian structure, convergence rate, and solution accuracy within the local rectangular refinement methodNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 8 2001Beth Anne V. Bennett Abstract The local rectangular refinement (LRR) solution-adaptive gridding method automatically produces orthogonal unstructured adaptive grids and incorporates multiple-scale finite differences to discretize systems of elliptic governing partial differential equations (PDEs). The coupled non-linear discretized equations are solved simultaneously via Newton's method with a Bi-CGSTAB linear system solver. The grids' unstructured nature produces a nonstandard sparsity pattern within the Jacobian. The effects of two discretization schemes (LRR multiple-scale stencils and traditional single-scale stencils) on Jacobian bandwidth, convergence speed, and solution accuracy are studied. With various point orderings, for two simple problems with analytical solutions, the LRR multiple-scale stencils are seen to: (1) produce Jacobians of smaller bandwidths than those resulting from the traditional single-scale stencils; (2) lead to significantly faster Newton's method convergence than the single-scale stencils; and (3) produce more accurate solutions than the single-scale stencils. The LRR method, including the LRR multiple-scale stencils, is finally applied to an engineering problem governed by strongly coupled, highly non-linear PDEs: a steady-state lean Bunsen flame with complex chemistry, multicomponent transport, and radiation modeling. Very good agreement is observed between the computed flame height and previously published experimental data. Copyright © 2001 John Wiley & Sons, Ltd. [source] Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov-Fokker-Planck systemNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2005M. Asadzadeh Abstract We prove stability estimates and derive optimal convergence rates for the streamline diffusion and discontinuous Galerkin finite element methods for discretization of the multi-dimensional Vlasov-Fokker-Planck system. The focus is on the theoretical aspects, where we deal with construction and convergence analysis of the discretization schemes. Some related special cases are implemented in M. Asadzadeh [Appl Comput Meth 1(2) (2002), 158,175] and M. Asadzadeh and A. Sopasakis [Comput Meth Appl Mech Eng 191(41,42) (2002), 4641,4661]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] Block factorized preconditioners for high-order accurate in time approximation of the Navier-Stokes equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2003Alessandro Veneziani Computationally efficient solution methods for the unsteady Navier-Stokes incompressible equations are mandatory in real applications of fluid dynamics. A typical strategy to reduce the computational cost is to split the original problem into subproblems involving the separate computation of velocity and pressure. The splitting can be carried out either at a differential level, like in the Chorin-Temam scheme, or in an algebraic fashion, like in the algebraic reinterpretation of the Chorin-Temam method, or in the Yosida scheme (see 1 and 19). These fractional step schemes indeed provide effective methods of solution when dealing with first order accurate time discretizations. Their extension to high order time discretization schemes is not trivial. To this end, in the present work we focus our attention on the adoption of inexact algebraic factorizations as preconditioners of the original problem. We investigate their properties and show that some particular choices of the approximate factorization lead to very effective schemes. In particular, we prove that performing a small number of preconditioned iterations is enough to obtain a time accurate solution, irrespective of the dimension of the system at hand. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 487,510, 2003 [source] Eulerian backtracking of atmospheric tracers.THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 615 2006II: Numerical aspects Abstract In Part I of this paper, a mathematical equivalence was established between Eulerian backtracking or retro- transport, on the one hand, and adjoint transport with respect to an air-mass-weighted scalar product, on the other. The time symmetry which lies at the basis of this mathematical equivalence can however be lost through discretization. That question is studied, and conditions are explicitly identified under which discretization schemes possess the property of time symmetry. Particular consideration is given to the case of the LMDZ model. The linear schemes used for turbulent diffusion and subgrid-scale convection are symmetric. For the Van Leer advection scheme used in LMDZ, which is nonlinear, the question of time symmetry does not even make sense. Those facts are illustrated by numerical simulations performed in the conditions of the European Transport EXperiment (ETEX). For a model that is not time-symmetric, the question arises as to whether it is preferable, in practical applications, to use the exact numerical adjoint, or the retro-transport model. Numerical results obtained in the context of one-dimensional advection show that the presence of slope limiters in the Van Leer advection scheme can produce in some circumstances unrealistic (in particular, negative) adjoint sensitivities. The retro-transport equation, on the other hand, generally produces robust and realistic results, and always preserves the positivity of sensitivities. Retro-transport may therefore be preferable in sensitivity computations, even in the context of variational assimilation. Copyright © 2006 Royal Meteorological Society [source] Evaluation of three spatial discretization schemes with the Galewsky et al. testATMOSPHERIC SCIENCE LETTERS, Issue 3 2010Seoleun Shin Abstract We evaluate the Hamiltonian particle methods (HPM) and the Nambu discretization applied to shallow-water equations on the sphere using the test suggested by Galewsky et al. (2004). Both simulations show excellent conservation of energy and are stable in long-term simulation. We repeat the test also using the ICOSWP scheme to compare with the two conservative spatial discretization schemes. The HPM simulation captures the main features of the reference solution, but wave 5 pattern is dominant in the simulations applied on the ICON grid with relatively low spatial resolutions. Nevertheless, agreement in statistics between the three schemes indicates their qualitatively similar behaviors in the long-term integration. Copyright © 2010 Royal Meteorological Society [source] | ||||||||||||||||