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Time-stepping Scheme (time-stepping + scheme)
Selected AbstractsFinite element formulation and algorithms for unsaturated soils.INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 9 2003Part I: Theory Abstract This paper presents a complete finite-element treatment for unsaturated soil problems. A new formulation of general constitutive equations for unsaturated soils is first presented. In the incremental stress,strain equations, the suction or the pore water pressure is treated as a strain variable instead of a stress variable. The global governing equations are derived in terms of displacement and pore water pressure. The discretized governing equations are then solved using an adaptive time-stepping scheme which automatically adjusts the time-step size so that the integration error in the displacements and pore pressures lies close to a specified tolerance. The non-linearity caused by suction-dependent plastic yielding, suction-dependent degree of saturation, and saturation-dependent permeability is treated in a similar way to the elastoplasticity. An explicit stress integration scheme is used to solve the constitutive stress,strain equations at the Gauss point level. The elastoplastic stiffness matrix in the Euler solution is evaluated using the suction as well as the stresses and hardening parameters at the start of the subincrement, while the elastoplastic matrix in the modified Euler solution is evaluated using the suction at the end of the subincrement. In addition, when applying subincrementation, the same rate is applied to all strain components including the suction. Copyright © 2003 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] Two-dimensional transonic aeroservoelastic computations in the time domainINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2001L. Djayapertapa Abstract A computational method to perform transonic aeroelastic and aeroservoelastic calculations in the time domain is presented, and used to predict stability (flutter) boundaries of 2-D wing sections. The aerodynamic model is a cell-centred finite-volume unsteady Euler solver, which uses an efficient implicit time-stepping scheme and structured moving grids. The aerodynamic equations are coupled with the structural equations of motion, which are derived from a typical wing section model. A control law is implemented within the aeroelastic solver to investigate active means of flutter suppression via control surface motion. Comparisons of open- and closed-loop calculations show that the control law can successfully suppress the flutter and results in an increase of up to 19 per cent in the allowable speed index. The effect of structural non-linearity, in the form of hinge axis backlash is also investigated. The effect is found to be strongly destabilizing, but the control law is shown to still alleviate the destabilizing effect. Copyright © 2001 John Wiley & Sons, Ltd. [source] Efficient and accurate time adaptive multigrid simulations of droplet spreadingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2004P. H. Gaskell Abstract An efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non-linear lubrication equations. The algorithm is fully implicit and has embedded within it an adaptive time-stepping scheme that enables the same to be optimized in a controlled manner subject to a specific error tolerance. The method is first validated against a range of analytical and existing numerical predictions commensurate with droplet spreading and then used to simulate a series of new, three-dimensional flows consisting of droplet motion on substrates containing topographic and wetting heterogeneities. The latter are of particular interest and reveal how droplets can be made to spread preferentially on substrates owing to an interplay between different topographic and surface wetting characteristics. Copyright © 2004 John Wiley & Sons, Ltd. [source] A time-marching finite element method for an electromagnetic scattering problemMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2003Tri Van Abstract In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional polyhedral cavity. Finite element methods based on the variational formulation derived in Van and Wood (Adv. Comput. Math., to appear) are considered. Existence and uniqueness of the discrete problem is proved by using Babuska,Brezzi theory. Finite element error estimate and stability of the Newmark scheme are also established. Copyright © 2003 John Wiley & Sons, Ltd. [source] Numerical simulations of type III planetary migration , I. Disc model and convergence testsMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 1 2008A. Pepli ABSTRACT We investigate the fast (type III) migration regime of high-mass protoplanets orbiting in protoplanetary discs. This type of migration is dominated by corotational torques. We study the details of flow structure in the planet's vicinity, the dependence of migration rate on the adopted disc model and the numerical convergence of models (independence of certain numerical parameters such as gravitational softening). We use two-dimensional hydrodynamical simulations with adaptive mesh refinement, based on the flash code with improved time-stepping scheme. We perform global disc simulations with sufficient resolution close to the planet, which is allowed to freely move throughout the grid. We employ a new type of equation of state in which the gas temperature depends on both the distance to the star and planet, and a simplified correction for self-gravity of the circumplanetary gas. We find that the migration rate in the type III migration regime depends strongly on the gas dynamics inside the Hill sphere (Roche lobe of the planet) which, in turn, is sensitive to the aspect ratio of the circumplanetary disc. Furthermore, corrections due to the gas self-gravity are necessary to reduce numerical artefacts that act against rapid planet migration. Reliable numerical studies of type III migration thus require consideration of both the thermal and the self-gravity corrections, as well as a sufficient spatial resolution and the calculation of disc,planet attraction both inside and outside the Hill sphere. With this proviso, we find type III migration to be a robust mode of migration, astrophysically promising because of a speed much faster than in the previously studied modes of migration. [source] A mass-conservative version of the semi-implicit semi-Lagrangian HIRLAMTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 635 2008P. H. Lauritzen Abstract A mass-conservative version of the semi-implicit semi-Lagrangian High-Resolution Limited-Area Model (HIRLAM) is presented. The explicit continuity equation is solved with the so-called cell-integrated semi-Lagrangian (CISL) method. To allow for long time steps, the CISL scheme is coupled with a recently developed semi-implicit time-stepping scheme that involves the same non-complicated elliptic equation as in HIRLAM. Contrarily to the traditional semi-Lagrangian method, the trajectories are backward in the horizontal and forward in the vertical, i.e. cells moving with the flow depart from model layers and arrive in a regular column, and their vertical displacements are computed from continuity of mass and hydrostatic balance in the arrival column. This involves just two-dimensional upstream integrals and allows for a Lagrangian discretization of the energy conversion term in the thermodynamic equation. Preliminary validation of the new model version is performed using an idealized baroclinic wave test case. The accuracy of the new formulation of HIRLAM is comparable to the reference version though it is slightly more diffusive. A main finding is that the new discretization of the energy conversion term leads to more accurate simulations compared to the traditional ,Eulerian' treatment. Copyright © 2008 Royal Meteorological Society [source] Efficient solution techniques for implicit finite element schemes with flux limitersINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2007M. MöllerArticle first published online: 20 MAR 200 Abstract The algebraic flux correction (AFC) paradigm is equipped with efficient solution strategies for implicit time-stepping schemes. It is shown that Newton-like techniques can be applied to the nonlinear systems of equations resulting from the application of high-resolution flux limiting schemes. To this end, the Jacobian matrix is approximated by means of first- or second-order finite differences. The edge-based formulation of AFC schemes can be exploited to devise an efficient assembly procedure for the Jacobian. Each matrix entry is constructed from a differential and an average contribution edge by edge. The perturbation of solution values affects the nodal correction factors at neighbouring vertices so that the stencil for each individual node needs to be extended. Two alternative strategies for constructing the corresponding sparsity pattern of the resulting Jacobian are proposed. For nonlinear governing equations, the contribution to the Newton matrix which is associated with the discrete transport operator is approximated by means of divided differences and assembled edge by edge. Numerical examples for both linear and nonlinear benchmark problems are presented to illustrate the superiority of Newton methods as compared to the standard defect correction approach. Copyright © 2007 John Wiley & Sons, Ltd. [source] Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006R. K. Lin Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] | ||||||||||