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Spatial Discretization (spatial + discretization)
Terms modified by Spatial Discretization Selected AbstractsA cut-cell non-conforming Cartesian mesh method for compressible and incompressible flowINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2007J. Pattinson Abstract This paper details a multigrid-accelerated cut-cell non-conforming Cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow. This is done via a single equation set that describes sub-, trans-, and supersonic flows. Cut-cell technology is developed to furnish body-fitted meshes with an overlapping mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. Spatial discretization is effected via an edge-based vertex-centred finite volume method. An alternative dual-mesh construction strategy, similar to the cell-centred method, is developed. Incompressibility is dealt with via an artificial compressibility algorithm, and stabilization achieved with artificial dissipation. In compressible flow, shocks are captured via pressure switch-activated upwinding. The solution process is accelerated with full approximation storage (FAS) multigrid where coarse meshes are generated automatically via a volume agglomeration methodology. This is the first time that the proposed discretization and solution methods are employed to solve a single compressible,incompressible equation set on cut-cell Cartesian meshes. The developed technology is validated by numerical experiments. The standard discretization and alternative methods were found equivalent in accuracy and computational cost. The multigrid implementation achieved decreases in CPU time of up to one order of magnitude. Copyright © 2007 John Wiley & Sons, Ltd. [source] Convergence of MPFA on triangulations and for Richards' equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2008R. A. Klausen Abstract Spatial discretization of transport and transformation processes in porous media requires techniques that handle general geometry, discontinuous coefficients and are locally mass conservative. Multi-point flux approximation (MPFA) methods are such techniques, and we will here discuss some formulations on triangular grids with further application to the nonlinear Richards equation. The MPFA methods will be rewritten to mixed form to derive stability conditions and error estimates. Several MPFA versions will be shown, and the versions will be discussed with respect to convergence, symmetry and robustness when the grids are rough. It will be shown that the behavior may be quite different for challenging cases of skewness and roughness of the simulation grids. Further, we apply the MPFA discretization approach for the Richards equation and derive new error estimates without extra regularity requirements. The analysis will be accompanied by numerical results for grids that are relevant for practical simulation. Copyright © 2008 John Wiley & Sons, Ltd. [source] Optimal operation of GaN thin film epitaxy employing control vector parametrizationAICHE JOURNAL, Issue 4 2006Amit Varshney Abstract An approach that links nonlinear model reduction techniques with control vector parametrization-based schemes is presented, to efficiently solve dynamic constraint optimization problems arising in the context of spatially-distributed processes governed by highly-dissipative nonlinear partial-differential equations (PDEs), utilizing standard nonlinear programming techniques. The method of weighted residuals with empirical eigenfunctions (obtained via Karhunen-Loève expansion) as basis functions is employed for spatial discretization together with control vector parametrization formulation for temporal discretization. The stimulus for the earlier approach is provided by the presence of low order dominant dynamics in the case of highly dissipative parabolic PDEs. Spatial discretization based on these few dominant modes (which are elegantly captured by empirical eigenfunctions) takes into account the actual spatiotemporal behavior of the PDE which cannot be captured using finite difference or finite element techniques with a small number of discretization points/elements. The proposed approach is used to compute the optimal operating profile of a metallorganic vapor-phase epitaxy process for the production of GaN thin films, with the objective to minimize the spatial nonuniformity of the deposited film across the substrate surface by adequately manipulating the spatiotemporal concentration profiles of Ga and N precursors at the reactor inlet. It is demonstrated that the reduced order optimization problem thus formulated using the proposed approach for nonlinear order reduction results in considerable savings of computational resources and is simultaneously accurate. It is demonstrated that by optimally changing the precursor concentration across the reactor inlet it is possible to reduce the thickness nonuniformity of the deposited film from a nominal 33% to 3.1%. © 2005 American Institute of Chemical Engineers AIChE J, 2006 [source] Stability and convergence of optimum spectral non-linear Galerkin methodsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2001He Yinnian Abstract Our objective in this article is to present some numerical schemes for the approximation of the 2-D Navier,Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non-linear Galerkin method; time discretization is done by the Euler scheme and a two-step scheme. Our results show that under the same convergence rate the optimum spectral non-linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd. [source] An Adaptive Strategy for the Local Discontinuous Galerkin Method Applied to Porous Media ProblemsCOMPUTER-AIDED CIVIL AND INFRASTRUCTURE ENGINEERING, Issue 4 2008Esov S. Velázquez DG methods may be viewed as high-order extensions of the classical finite volume method and, since no interelement continuity is imposed, they can be defined on very general meshes, including nonconforming meshes, making these methods suitable for h-adaptivity. The technique starts with an initial conformal spatial discretization of the domain and, in each step, the error of the solution is estimated. The mesh is locally modified according to the error estimate by performing two local operations: refinement and agglomeration. This procedure is repeated until the solution reaches a desired accuracy. The performance of this technique is examined through several numerical experiments and results are compared with globally refined meshes in examples with known analytic solutions. [source] Evaluation of model complexity and space,time resolution on the prediction of long-term soil salinity dynamics, western San Joaquin Valley, CaliforniaHYDROLOGICAL PROCESSES, Issue 13 2006G. Schoups Abstract The numerical simulation of long-term large-scale (field to regional) variably saturated subsurface flow and transport remains a computational challenge, even with today's computing power. Therefore, it is appropriate to develop and use simplified models that focus on the main processes operating at the pertinent time and space scales, as long as the error introduced by the simpler model is small relative to the uncertainties associated with the spatial and temporal variation of boundary conditions and parameter values. This study investigates the effects of various model simplifications on the prediction of long-term soil salinity and salt transport in irrigated soils. Average root-zone salinity and cumulative annual drainage salt load were predicted for a 10-year period using a one-dimensional numerical flow and transport model (i.e. UNSATCHEM) that accounts for solute advection, dispersion and diffusion, and complex salt chemistry. The model uses daily values for rainfall, irrigation, and potential evapotranspiration rates. Model simulations consist of benchmark scenarios for different hypothetical cases that include shallow and deep water tables, different leaching fractions and soil gypsum content, and shallow groundwater salinity, with and without soil chemical reactions. These hypothetical benchmark simulations are compared with the results of various model simplifications that considered (i) annual average boundary conditions, (ii) coarser spatial discretization, and (iii) reducing the complexity of the salt-soil reaction system. Based on the 10-year simulation results, we conclude that salt transport modelling does not require daily boundary conditions, a fine spatial resolution, or complex salt chemistry. Instead, if the focus is on long-term salinity, then a simplified modelling approach can be used, using annually averaged boundary conditions, a coarse spatial discretization, and inclusion of soil chemistry that only accounts for cation exchange and gypsum dissolution,precipitation. We also demonstrate that prediction errors due to these model simplifications may be small, when compared with effects of parameter uncertainty on model predictions. The proposed model simplifications lead to larger time steps and reduced computer simulation times by a factor of 1000. Copyright © 2006 John Wiley & Sons, Ltd. [source] Kalman filter finite element method applied to dynamic ground motionINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 9 2009Yusuke Kato Abstract The purpose of this paper is to investigate the estimation of dynamic elastic behavior of the ground using the Kalman filter finite element method. In the present paper, as the state equation, the balance of stress equation, the strain,displacement equation and the stress,strain equation are used. For temporal discretization, the Newmark ¼ method is employed, and for the spatial discretization the Galerkin method is applied. The Kalman filter finite element method is a combination of the Kalman filter and the finite element method. The present method is adaptable to estimations not only in time but also in space, as we have confirmed by its application to the Futatsuishi quarry site. The input data are the measured velocity, acceleration, etc., which may include mechanical noise. It has been shown in numerical studies that the estimated velocity, acceleration, etc., at any other spatial and temporal point can be obtained by removing the noise included in the observation. Copyright © 2008 John Wiley & Sons, Ltd. [source] Thermal reservoir modeling in petroleum geomechanicsINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 4 2009Shunde Yin Abstract Thermal oil recovery processes involve high pressures and temperatures, leading to large volume changes and induced stresses. These cannot be handled by traditional reservoir simulation because it does not consider coupled geomechanics effects. In this paper we present a fully coupled, thermal half-space model using a hybrid DDFEM method. A finite element method (FEM) solution is adopted for the reservoir and the surrounding thermally affected zone, and a displacement discontinuity method is used for the surrounding elastic, non-thermal zone. This approach analyzes stress, pressure, temperature and volume change in the reservoir; it also provides stresses and displacements around the reservoir (including transient ground surface movements) in a natural manner without introducing extra spatial discretization outside the FEM zone. To overcome spurious spatial temperature oscillations in the convection-dominated thermal advection,diffusion problem, we place the transient problem into an advection,diffusion,reaction problem framework, which is then efficiently addressed by a stabilized finite element approach, the subgrid-scale/gradient subgrid-scale method. Copyright © 2008 John Wiley & Sons, Ltd. [source] Numerical stability and error analysis for the incompressible Navier,Stokes equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2002S. Prudhomme Abstract This paper describes a strategy to control errors in finite element approximations of the time-dependent incompressible Navier,Stokes equations. The approach involves estimating the errors due to the discretization in space, using information from the residuals in the momentum and continuity equations. Following a numerical stability analysis of channel flows past a cylinder, it is concluded that the errors due to the residual in the continuity equation should be carefully controlled since it appears to be the source of unphysical perturbations artificially created by the spatial discretization. The performance of the adaptive strategy is then tested for lid-driven oblique cavity flows. Copyright © 2002 John Wiley & Sons, Ltd. [source] Adaptive ICT procedure for non-linear seepage flows with free surface in porous mediaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2002N. H. Sharif Abstract This paper focuses on adaptive finite element (FE)-methods for computation of the motion of viscous fluid interfaces fundamentally encountered in multiphase flow problems in porous media. An interface capturing technique (ICT)-procedure is formulated with a stabilized finite element scheme in a Eulerian framework to solve the two-dimensional (2D) and three-dimensional (3D) Navier,Stokes equation in porous media. Global mesh refinements of the discretized domain and local mesh refinements in the vicinity of the interface are used for the spatial discretization. The ICT is embedded into the finite element scheme by adding an extra advection equation and an additional unbounded degree of freedom to the number of the unknowns. Problems of non-linear free surface seepage flow in earth-fill dams are simulated in order to validate the performance of the FE-ICT. Computations for steady non-linear seepage flows in 2D and 3D are obtained for homogenous, isotropic and isothermal porous media. Copyright © 2002 John Wiley & Sons, Ltd. [source] Polygonal finite elements for topology optimization: A unifying paradigmINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2010Cameron Talischi Abstract In topology optimization literature, the parameterization of design is commonly carried out on uniform grids consisting of Lagrangian-type finite elements (e.g. linear quads). These formulations, however, suffer from numerical anomalies such as checkerboard patterns and one-node connections, which has prompted extensive research on these topics. A problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members, leading to mesh-dependent sub-optimal designs. Thus, to address the geometric features of the spatial discretization, we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions. More specifically, we consider polygonal meshes constructed from Voronoi tessellations, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities. Copyright © 2009 John Wiley & Sons, Ltd. [source] A hypersingular time-domain BEM for 2D dynamic crack analysis in anisotropic solidsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2009M. Wünsche Abstract A hypersingular time-domain boundary element method (BEM) for transient elastodynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack-faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the crack-opening-displacements (CODs). Special crack-tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time-domain BEM. Copyright © 2008 John Wiley & Sons, Ltd. [source] State-space time integration with energy control and fourth-order accuracy for linear dynamic systemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2006Steen Krenk Abstract A fourth-order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented. It is derived by integrating the phase-space representation and evaluating the resulting displacement and velocity integrals via integration by parts, substituting the time derivatives from the original differential equations. The resulting algorithm has an exact energy equation, in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix. This implies unconditional stability of the algorithm, and the relative phase error is of fourth-order. An optional high-frequency algorithmic damping is constructed by optimal combination of three different damping matrices, each proportional to either the mass or the stiffness matrix. This leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth-order in the frequency. Thus, the low-frequency response is virtually undamped, and the algorithm remains third-order accurate even when algorithmic damping is included. The accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium, where the algorithmic damping is used to reduce dispersion due to the spatial discretization, leading to a smooth solution with a clearly defined wave front. Copyright © 2005 John Wiley & Sons, Ltd. [source] Theory and numerics of geometrically non-linear open system mechanicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2003E. Kuhl Abstract The present contribution aims at deriving a general theoretical and numerical framework for open system thermodynamics. The balance equations for open systems differ from the classical balance equations by additional terms arising from possible local changes in mass. In contrast to existing formulations, these changes not only originate from additional mass sources or sinks but also from a possible in- or outflux of matter. Constitutive equations for the mass source and the mass flux are discussed for the particular model problem of bone remodelling in hard tissue mechanics. Particular emphasis is dedicated to the spatial discretization of the coupled system of the balance of mass and momentum. To this end we suggest a geometrically non-linear monolithic finite element based solution technique introducing the density and the deformation map as primary unknowns. It is supplemented by the consistent linearization of the governing equations. The resulting algorithm is validated qualitatively for classical examples from structural mechanics as well as for biomechanical applications with particular focus on the functional adaption of bones. It turns out that, owing to the additional incorporation of the mass flux, the proposed model is able to simulate size effects typically encountered in microstructural materials such as open-pored cellular solids, e.g. bones. Copyright © 2003 John Wiley & Sons, Ltd. [source] Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2002Christian Miehe Abstract The paper investigates computational procedures for the treatment of a homogenized macro-continuum with locally attached micro-structures of inelastic constituents undergoing small strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a setting of smooth single-surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity. Focusing on macro-strain-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed linear displacements, periodic fluctuations and constant stresses on the boundary of the micro-structure and discuss their numerical solutions based on a spatial discretization of the fine-scale displacement fluctuation field. The performance of the proposed methods is demonstrated for the model problem of von Mises-type elasto-visco-plasticity of the constituents and applied to a comparative study of micro-to-macro transitions of inelastic composites. Copyright © 2002 John Wiley & Sons, Ltd. [source] Examination for adjoint boundary conditions in initial water elevation estimation problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2010T. KurahashiArticle first published online: 23 JUL 200 Abstract I present here a method of generating a distribution of initial water elevation by employing the adjoint equation and finite element methods. A shallow-water equation is employed to simulate flow behavior. The adjoint equation method is utilized to obtain a distribution of initial water elevation for the observed water elevation. The finite element method, using the stabilized bubble function element, is used for spatial discretization, and the Crank,Nicolson method is used for temporal discretizations. In addition to a method for optimally assimilating water elevation, a method is presented for determining adjoint boundary conditions. An examination using the observation data including noise data is also carried out. Copyright © 2009 John Wiley & Sons, Ltd. [source] A further work on multi-phase two-fluid approach for compressible multi-phase flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2008Yang-Yao Niu Abstract This paper is to continue our previous work Niu (Int. J. Numer. Meth. Fluids 2001; 36:351,371) on solving a two-fluid model for compressible liquid,gas flows using the AUSMDV scheme. We first propose a pressure,velocity-based diffusion term originally derived from AUSMDV scheme Wada and Liou (SIAM J. Sci. Comput. 1997; 18(3):633,657) to enhance its robustness. The scheme can be applied to gas and liquid fluids universally. We then employ the stratified flow model Chang and Liou (J. Comput. Physics 2007; 225:240,873) for spatial discretization. By defining the fluids in different regions and introducing inter-phasic force on cell boundary, the stratified flow model allows the conservation laws to be applied on each phase, and therefore, it is able to capture fluid discontinuities, such as the fluid interfaces and shock waves, accurately. Several benchmark tests are studied, including the Ransom's Faucet problem, 1D air,water shock tube problems, 2D shock-water column and 2D shock-bubble interaction problems. The results indicate that the incorporation of the new dissipation into AUSM+ -up scheme and the stratified flow model is simple, accurate and robust enough for the compressible multi-phase flows. Copyright © 2008 John Wiley & Sons, Ltd. [source] A collocated, iterative fractional-step method for incompressible large eddy simulationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Giridhar Jothiprasad Abstract Fractional-step methods are commonly used for the time-accurate solution of incompressible Navier,Stokes (NS) equations. In this paper, a popular fractional-step method that uses pressure corrections in the projection step and its iterative variants are investigated using block-matrix analysis and an improved algorithm with reduced computational cost is developed. Since the governing equations for large eddy simulation (LES) using linear eddy-viscosity-based sub-grid models are similar in form to the incompressible NS equations, the improved algorithm is implemented in a parallel LES solver. A collocated grid layout is preferred for ease of extension to curvilinear grids. The analyzed fractional-step methods are viewed as an iterative approximation to a temporally second-order discretization. At each iteration, a linear system that has an easier block-LU decomposition compared with the original system is inverted. In order to improve the numerical efficiency and parallel performance, modified ADI sub-iterations are used in the velocity step of each iteration. Block-matrix analysis is first used to determine the number of iterations required to reduce the iterative error to the discretization error of. Next, the computational cost is reduced through the use of a reduced stencil for the pressure Poisson equation (PPE). Energy-conserving, spatially fourth-order discretizations result in a 7-point stencil in each direction for the PPE. A smaller 5-point stencil is achieved by using a second-order spatial discretization for the pressure gradient operator correcting the volume fluxes. This is shown not to reduce the spatial accuracy of the scheme, and a fourth-order continuity equation is still satisfied to machine precision. The above results are verified in three flow problems including LES of a temporal mixing layer. Copyright © 2008 John Wiley & Sons, Ltd. [source] Acoustic upwinding for sub- and super-sonic turbulent channel flow at low Reynolds numberINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2007H. C. de LangeArticle first published online: 13 FEB 200 Abstract A recently developed asymmetric implicit fifth-order scheme with acoustic upwinding for the spatial discretization for the characteristic waves is applied to the fully compressible, viscous and non-stationary Navier,Stokes equations for sub- and super-sonic, mildly turbulent, channel flow (Re,=360). For a Mach number of 0.1, results are presented for uniform (323, 643 and 1283) and non-uniform (expanding wall-normal, 323 and 643) grids and compared to the (incompressible) reference solution found in (J. Fluid. Mech. 1987; 177:133,166). The results for uniform grids on 1283 and 643 nodes show high resemblance with the reference solution. Expanding grids are applied on 643 - and 323 -node grids. The capability of the proposed technique to solve compressible flow is first demonstrated by increasing the Mach number to 0.3, 0.6 and 0.9 for isentropic flow on the uniform 643 -grid. Next, the flow speed is increased to Ma=2. The results for the isothermal-wall supersonic flows give very good agreement with known literature results. The velocity field, the temperature and their fluctuations are well resolved. This means that in all presented (sub- and super-sonic) cases, the combination of acoustic upwinding and the asymmetric high-order scheme provides sufficient high wave-number damping and low wave-number accuracy to give numerically stable and accurate results. Copyright © 2007 John Wiley & Sons, Ltd. [source] Non-oscillatory relaxation methods for the shallow-water equations in one and two space dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2004Mohammed Seaïd Abstract In this paper, a new family of high-order relaxation methods is constructed. These methods combine general higher-order reconstruction for spatial discretization and higher order implicit-explicit schemes or TVD Runge,Kutta schemes for time integration of relaxing systems. The new methods retain all the attractive features of classical relaxation schemes such as neither Riemann solvers nor characteristic decomposition are needed. Numerical experiments with the shallow-water equations in both one and two space dimensions on flat and non-flat topography demonstrate the high resolution and the ability of our relaxation schemes to better resolve the solution in the presence of shocks and dry areas without using either Riemann solvers or front tracking techniques. Copyright © 2004 John Wiley & Sons, Ltd. [source] Reduced order state-space models from the pulse responses of a linearized CFD schemeINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2003Ann L. Gaitonde This paper describes a method for obtaining a time continuous reduced order model (ROM) from a system of time continuous linear differential equations. These equations are first put into a time discrete form using a finite difference approximation. The unit sample responses of the discrete system are calculated for each system input and these provide the Markov parameters of the system. An eigenvalue realization algorithm (ERA) is used to construct a discrete ROM. This ROM is then used to obtain a continuous ROM of the original continuous system. The focus of this paper is on the application of this method to the calculation of unsteady flows using the linearized Euler equations on moving meshes for aerofoils undergoing heave or linearized pitch motions. Applying a standard cell-centre spatial discretization and taking account of mesh movement a continuous system of differential equations is obtained which are continuous in time. These are put into discrete time form using an implicit finite difference approximation. Results are presented demonstrating the efficiency of the system reduction method for this system. Copyright © 2003 John Wiley & Sons, Ltd. [source] Studies on some singular potentials in quantum mechanicsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 6 2005Amlan K. RoyArticle first published online: 10 MAY 200 Abstract A simple methodology is suggested for the efficient calculation of certain central potentials having singularities. The generalized pseudospectral method used in this work facilitates nonuniform and optimal spatial discretization. Applications have been made to calculate the energies, densities, and expectation values for two singular potentials of physical interest, viz., (i) the harmonic potential plus inverse quartic and sextic perturbation and (ii) the Coulomb potential with a linear and quadratic term for a broad range of parameters. The first 10 states belonging to a maximum of ,, = 8 and 5 for (i) and (ii) have been computed with good accuracy and compared with the most accurate available literature data. The calculated results are in excellent agreement, especially in light of the difficulties encountered in these potentials. Some new states are reported here for the first time. This offers a general and efficient scheme for calculating these and other similar potentials of physical and mathematical interest in quantum mechanics accurately. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [source] Optimal operation of GaN thin film epitaxy employing control vector parametrizationAICHE JOURNAL, Issue 4 2006Amit Varshney Abstract An approach that links nonlinear model reduction techniques with control vector parametrization-based schemes is presented, to efficiently solve dynamic constraint optimization problems arising in the context of spatially-distributed processes governed by highly-dissipative nonlinear partial-differential equations (PDEs), utilizing standard nonlinear programming techniques. The method of weighted residuals with empirical eigenfunctions (obtained via Karhunen-Loève expansion) as basis functions is employed for spatial discretization together with control vector parametrization formulation for temporal discretization. The stimulus for the earlier approach is provided by the presence of low order dominant dynamics in the case of highly dissipative parabolic PDEs. Spatial discretization based on these few dominant modes (which are elegantly captured by empirical eigenfunctions) takes into account the actual spatiotemporal behavior of the PDE which cannot be captured using finite difference or finite element techniques with a small number of discretization points/elements. The proposed approach is used to compute the optimal operating profile of a metallorganic vapor-phase epitaxy process for the production of GaN thin films, with the objective to minimize the spatial nonuniformity of the deposited film across the substrate surface by adequately manipulating the spatiotemporal concentration profiles of Ga and N precursors at the reactor inlet. It is demonstrated that the reduced order optimization problem thus formulated using the proposed approach for nonlinear order reduction results in considerable savings of computational resources and is simultaneously accurate. It is demonstrated that by optimally changing the precursor concentration across the reactor inlet it is possible to reduce the thickness nonuniformity of the deposited film from a nominal 33% to 3.1%. © 2005 American Institute of Chemical Engineers AIChE J, 2006 [source] Anisotropic curve shortening flow in higher codimensionMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2007Paola Pozzi Abstract We consider the evolution of parametric curves by anisotropic mean curvature flow in ,n for an arbitrary n,2. After the introduction of a spatial discretization, we prove convergence estimates for the proposed finite-element model. Numerical tests and simulations based on a fully discrete semi-implicit stable algorithm are presented. Copyright © 2007 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] On the long-time stability of the Crank,Nicolson scheme for the 2D Navier,Stokes equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007Florentina Tone Abstract In this article we study the stability for all positive time of the Crank,Nicolson scheme for the two-dimensional Navier,Stokes equations. More precisely, we consider the Crank,Nicolson time discretization together with a general spatial discretization, and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme is stable, provided a CFL-type condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] Method of lines with boundary elements for 1-D transient diffusion-reaction problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2006P.A. Ramachandran Abstract Time-dependent differential equations can be solved using the concept of method of lines (MOL) together with the boundary element (BE) representation for the spatial linear part of the equation. The BE method alleviates the need for spatial discretization and casts the problem in an integral format. Hence errors associated with the numerical approximation of the spatial derivatives are totally eliminated. An element level local cubic approximation is used for the variable at each time step to facilitate the time marching and the nonlinear terms are represented in a semi-implicit manner by a local linearization at each time step. The accuracy of the method has been illustrated on a number of test problems of engineering significance. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006 [source] Fixed-point type iterations in numerical simulations for static and dynamic elastoplasticityPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003Yury Vetyukov Mag. Dynamics of the elastoplastic media is stated in terms of plastic eigenstrains acting upon a background elastic problem with fixed (initial) stiffness. The plastic flow is described by specifically introduced plastic multipliers, which minimizes the number of unknown variables to be determined within the time step. A fixed-point type iteration strategy is suggested for the computation of the plastic multipliers, which has been implemented for the finite element spatial discretization. Numerical experiments show the correctness and the effectiveness of the developed algorithm. [source] Adaptive finite element procedures for elastoplastic problems at finite strainsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003A. Koch Dipl.-Ing. A major difficulty in the context of adaptive analysis of geometrically nonlinear problems is to provide a robust remeshing procedure that accounts both for the error caused by the spatial discretization and for the error due to the time discretization. For stability problems, such as strain localization and necking, it is essential to provide a step,size control in order to get a robust algorithm for the solution of the boundary value problem. For this purpose we developed an easy to implement step,size control algorithm. In addition we will consider possible a posteriori error indicators for the spatial error distribution of elastoplastic problems at finite strains. This indicator is adopted for a density,function,based adaptive remeshing procedure. Both error indicators are combined for the adaptive analysis in time and space. The performance of the proposed method is documented by means of representative numerical examples. [source] On the practical importance of the SSP property for Runge,Kutta time integrators for some common Godunov-type schemesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2005David I. Ketcheson Abstract We investigate through analysis and computational experiment explicit second and third-order strong-stability preserving (SSP) Runge,Kutta time discretization methods in order to gain perspective on the practical necessity of the SSP property. We consider general theoretical SSP limits for these schemes and present a new optimal third-order low-storage SSP method that is SSP at a CFL number of 0.838. We compare results of practical preservation of the TVD property using SSP and non-SSP time integrators to integrate a class of semi-discrete Godunov-type spatial discretizations. Our examples involve numerical solutions to Burgers' equation and the Euler equations. We observe that ,well-designed' non-SSP and non-optimal SSP schemes with SSP coefficients less than one provide comparable stability when used with time steps below the standard CFL limit. Results using a third-order non-TVD CWENO scheme are also presented. We verify that the documented SSP methods with the number of stages greater than the order provide a useful enhanced stability region. We show by analysis and by numerical experiment that the non-oscillatory third-order reconstructions used in (Liu and Tadmor Numer. Math. 1998; 79:397,425, Kurganov and Petrova Numer. Math. 2001; 88:683,729) are in general only second- and first-order accurate, respectively. Copyright © 2005 John Wiley & Sons, Ltd. [source] | ||||||||||||||||