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Extrapolation Scheme (extrapolation + scheme)
Selected AbstractsA 2-D spectral-element method for computing spherical-earth seismograms,II.GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2008Waves in solid, fluid media SUMMARY We portray a dedicated spectral-element method to solve the elastodynamic wave equation upon spherically symmetric earth models at the expense of a 2-D domain. Using this method, 3-D wavefields of arbitrary resolution may be computed to obtain Fréchet sensitivity kernels, especially for diffracted arrivals. The meshing process is presented for varying frequencies in terms of its efficiency as measured by the total number of elements, their spacing variations and stability criteria. We assess the mesh quantitatively by defining these numerical parameters in a general non-dimensionalized form such that comparisons to other grid-based methods are straightforward. Efficient-mesh generation for the PREM example and a minimum-messaging domain decomposition and parallelization strategy lay foundations for waveforms up to frequencies of 1 Hz on moderate PC clusters. The discretization of fluid, solid and respective boundary regions is similar to previous spectral-element implementations, save for a fluid potential formulation that incorporates the density, thereby yielding identical boundary terms on fluid and solid sides. We compare the second-order Newmark time extrapolation scheme with a newly implemented fourth-order symplectic scheme and argue in favour of the latter in cases of propagation over many wavelengths due to drastic accuracy improvements. Various validation examples such as full moment-tensor seismograms, wavefield snapshots, and energy conservation illustrate the favourable behaviour and potential of the method. [source] Numerical aspects of a real-time sub-structuring technique in structural dynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2007R. Sajeeb Abstract A time domain coupling technique, involving combined computational and experimental modelling, for vibration analysis of structures built-up of linear/non-linear substructures is developed. The study permits, in principle, one or more of the substructures to be modelled experimentally with measurements being made only on the interfacial degrees of freedom. The numerical and experimental substructures are allowed to communicate in real time within the present framework. The proposed strategy involves a two-stage scheme: the first is iterative in nature and is implemented at the initial stages of the solution in a non-real-time format; the second is non-iterative, employs an extrapolation scheme and proceeds in real time. Issues on time delays during communications between different substructures are discussed. An explicit integration procedure is shown to lead to solutions with high accuracy while retaining path sensitivity to initial conditions. The stability of the integration scheme is also discussed and a method for numerically dissipating the temporal growth of high-frequency errors is presented. For systems with non-linear substructures, the integration procedure is based on a multi-step transversal linearization method; and, to account for time delays, we employ a multi-step extrapolation scheme based on the reproducing kernel particle method. Numerical illustrations on a few low-dimensional vibrating structures are presented and these examples are fashioned after problems of seismic qualification testing of engineering structures using real-time substructure testing techniques. Copyright © 2007 John Wiley & Sons, Ltd. [source] Efficient preconditioning techniques for finite-element quadratic discretization arising from linearized incompressible Navier,Stokes equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2010A. El Maliki Abstract We develop an efficient preconditioning techniques for the solution of large linearized stationary and non-stationary incompressible Navier,Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non-stationary case. The time discretization procedure uses the Gear scheme and the second-order Taylor,Hood element P2,P1 is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r,(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P1,P2 hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented. Copyright © 2009 John Wiley & Sons, Ltd. [source] Complete basis set extrapolations of dispersion, exchange, and coupled-clusters contributions to the interaction energy: a helium dimer study,INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 12 2008gorzata Jeziorska Abstract Effectiveness of various extrapolation schemes in predicting complete basis set (CBS) values of interaction energies has been investigated for the helium dimer as a function of interatomic separation R. The investigations were performed separately for the leading dispersion and exchange contributions to the interaction energy and for the interaction energy computed using the coupled cluster method with single and double excitations (CCSD). For all these contributions, practically exact reference values were obtained from Gaussian-type geminal calculations. Sequences of orbital basis sets augmented with diffuse and bond functions or augmented with two sets of diffuse functions have been employed, with the cardinal numbers up to X = 7. The functional form EX = ECBS + A(X , k),, was applied for the extrapolations, where EX is the contribution to the interaction energy computed with a basis set of cardinal number X. The main conclusion of this work is that CBS extrapolations of an appropriate functional form generally improve the accuracy of the interaction energies at a very small additional computational cost (of the order of 10%) and should be recommended in calculations of interatomic and intermolecular potentials. The effectiveness of the extrapolations significantly depends, however, on the interatomic separation R and on the composition of the basis set. Basis sets with midbond functions, well known to provide at a given size much more accurate nonextrapolated results than bases lacking such functions, have been found to perform best also in extrapolations. The X,1 extrapolations of dispersion energies computed with midbond function turned out to be very efficient (except at large R), reducing the errors by an order of magnitude for small X and a factor of two for large X (where the errors of nonextrapolated results are already very small). If midbond functions are not used, the X,3 formula is most appropriate for the dispersion energies. For the exchange component of the interaction energy, the best results are obtained,in both types of basis sets,with the X,4 extrapolation, which leads (in both cases) to almost an order of magnitude reduction of the error. The X,3 and (X , 1),3 extrapolations work also well, but give smaller improvements. The correlation component of the CCSD interaction energy extrapolates best with , between 2 and 3 for bases with midbond functions and between 3 and 4 for bases without such functions. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008 [source] Molecular dynamics simulations of fluid methane properties using ab initio intermolecular interaction potentialsJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 12 2009Shih-Wei Chao Abstract Intermolecular interaction energy data for the methane dimer have been calculated at a spectroscopic accuracy and employed to construct an ab initio potential energy surface (PES) for molecular dynamics (MD) simulations of fluid methane properties. The full potential curves of the methane dimer at 12 symmetric conformations were calculated by the supermolecule counterpoise-corrected second-order Mřller-Plesset (MP2) perturbation theory. Single-point coupled cluster with single and double and perturbative triple excitations [CCSD(T)] calculations were also carried out to calibrate the MP2 potentials. We employed Pople's medium size basis sets [up to 6-311++G(3df, 3pd)] and Dunning's correlation consistent basis sets (cc-pVXZ and aug-cc-pVXZ, X = D, T, Q). For each conformer, the intermolecular carbon,carbon separation was sampled in a step 0.1 Ĺ for a range of 3,9 Ĺ, resulting in a total of 732 configuration points calculated. The MP2 binding curves display significant anisotropy with respect to the relative orientations of the dimer. The potential curves at the complete basis set (CBS) limit were estimated using well-established analytical extrapolation schemes. A 4-site potential model with sites located at the hydrogen atoms was used to fit the ab initio potential data. This model stems from a hydrogen,hydrogen repulsion mechanism to explain the stability of the dimer structure. MD simulations using the ab initio PES show quantitative agreements on both the atom-wise radial distribution functions and the self-diffusion coefficients over a wide range of experimental conditions. © 2008 Wiley Periodicals, Inc. J Comput Chem 2009 [source] | |||||||||||||