Showing papers on "Finite difference published in 2007"

3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study

Abstract: We present a finite-difference frequency-domain method for 3D visco-acoustic wave propagation modeling. In the frequency domain, the underlying numerical problem is the resolution of a large sparse system of linear equations whose right-hand side term is the source. This system is solved with a massively parallel direct solver. We first present an optimal 3D finite-difference stencil for frequency-domain modeling. The method is based on a parsimonious staggered-grid method. Differential operators are discretized with second-order accurate staggered-grid stencils on different rotated coordinate systems to mitigate numerical anisotropy. An antilumped mass strategy is implemented to minimize numerical dispersion. The stencil incorporates 27 grid points and spans two grid intervals. Dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3D domain. To assess the feasibility of the method for frequency-domain full-waveform inversion, we computed simulations in the 3D SEG/EAGE overthrust model for frequencies 5, 7, and 10 Hz. Results confirm the huge memory requirement of the factorization (several hundred Figabytes) but also the CPU efficiency of the resolution phase (few seconds per shot). Heuristic scalability analysis suggests that the memory complexity of the factorization is O(35N(4)) for a N-3 grid. Our method may provide a suitable tool to perform frequency-domain full-waveform inversion using a large distributed-memory platform. Further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time- and frequency-domain modeling of wave propagation to perform 3D full-waveform inversion.

A finite volume numerical approach for coastal ocean circulation studies: Comparisons with finite difference models

Abstract: [1] An unstructured grid, finite volume, three-dimensional (3-D) primitive equation coastal ocean model (FVCOM) has been developed for the study of coastal ocean and estuarine circulation by Chen et al. (2003a). The finite volume method used in this model combines the advantage of finite element methods for geometric flexibility and finite difference methods for simple discrete computation. Currents, temperature, and salinity are computed using an integral form of the equations, which provides a better representation of the conservative laws for mass, momentum, and heat. Detailed comparisons are presented here of FVCOM simulations with analytical solutions and numerical simulations made with two popular finite difference models (the Princeton Ocean Model and Estuarine and Coastal Ocean Model (ECOM-si)) for the following idealized cases: wind-induced long-surface gravity waves in a circular lake, tidal resonance in rectangular and sector channels, freshwater discharge onto the continental shelf with curved and straight coastlines, and the thermal bottom boundary layer over the slope with steep bottom topography. With a better fit to the curvature of the coastline using unstructured nonoverlapping triangle grid cells, FVCOM provides improved numerical accuracy and correctly captures the physics of tide-, wind-, and buoyancy-induced waves and flows in the coastal ocean. This model is suitable for applications to estuaries, continental shelves, and regional basins that feature complex coastlines and bathymetry.

Multiscale and Stabilized Methods

Abstract: : This article presents an introduction to multiscale and stabilized methods, which represent unified approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of finite difference, finite volume, and spectral methods, in addition to finite element methods.) The analytical ideas are first illustrated for time-harmonic wave-propagation problems in unbounded fluid domains governed by the Helmholtz equation. This leads to the well-known Dirichlet-to-Neumann formulation. A general treatment of the variational multiscale method in the context of an abstract Dirichlet problem is then presented which is applicable to advective-diffusive processes and other processes of physical interest. It is shown how the exact theory represents a paradigm for subgrid-scale models and posteriori error estimation. Hierarchical p-methods and bubble function methods are examined in order to understand and, ultimately, approximate the "fine-scale Green's function" which appears in the theory. Relationships among so-called residual-free bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of non-symmetric, linear evolution operators formulated in space-time. The variational multiscale method also provides guidelines and inspiration for the development of stabilized methods which have attracted considerable interest and have been extensively utilized in engineering and the physical sciences. An overview of stabilized methods for advective-diffusive equations is presented. A variational multiscale treatment of incompressible viscous flows, including turbulence is also described. This represents an alternative formulation of Large Eddy Simulation which provides simplified theoretical framework of LES with potential for improved modeling.

The Application of Finite Element Analysis to Body Wave Propagation Problems

Abstract: Summary The finite element method is shown to be a powerful tool for the numerical modelling of seismic body wave propagation problems. Applications extend to both problems on a scale of interest to engineers and also to large-scale seismological problems. Solutions are sought in the time domain. Efficient programs have been written to accomplish this. The scope of numerical solutions has been greatly enhanced by the use of a previously reported scheme for exactly cancelling reflections at the boundaries of the model. The finite difference results of Boore and the analytical results of Trifunac for the amplification due to a mountain and an alluvial valley respectively are compared with new finite element results. The new results agree well, although there are some difficulties with resonance in the alluvial valley problem. Boore's SH results have been extended to vertical P and SV incidence. A deep earthquake zone has been modelled realistically in two dimensions and earthquakes simulated at depth. It is suggested that the variation in observed amplitude across the top of the zone, due to refraction away from the slab, may be used to provide an estimate of the thickness of the slab from long-period observations of local earthquakes.

The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion

Abstract: Numerical modeling of seismic wave propagation and earthquake motion is an irreplaceable tool in investigation of the Earth’s structure, processes in the Earth, and particularly earthquake phenomena. Among various numerical methods, the finite-difference method is the dominant method in the modeling of earthquake motion. Moreover, it is becoming more important in the seismic exploration and structural modeling. At the same time we are convinced that the best time of the finite-difference method in seismology is in the future. This monograph provides tutorial and detailed introduction to the application of the finitedifference (FD), finite-element (FE), and hybrid FD-FE methods to the modeling of seismic wave propagation and earthquake motion. The text does not cover all topics and aspects of the methods. We focus on those to which we have contributed. We present alternative formulations of equation of motion for a smooth elastic continuum. We then develop alternative formulations for a canonical problem with a welded material interface and free surface. We continue with a model of an earthquake source. We complete the general theoretical introduction by a chapter on the constitutive laws for elastic and viscoelastic media, and brief review of strong formulations of the equation of motion. What follows is a block of chapters on the finite-difference and finite-element methods. We develop FD targets for the free surface and welded material interface. We then present various FD schemes for a smooth continuum, free surface, and welded interface. We focus on the staggered-grid and mainly optimally-accurate FD schemes. We also present alternative formulations of the FE method. We include the FD and FE implementations of the traction-at-split-nodes method for simulation of dynamic rupture propagation. The FD modeling is applied to the model of the deep sedimentary Grenoble basin, France. The FD and FE methods are combined in the hybrid FD-FE method. The hybrid method is then applied to two earthquake scenarios for the Grenoble basin. Except chapters 1, 3, 5, and 12, all chapters include new, previously unpublished material and results.

Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

Abstract: . We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.

Numerical simulation of flows around two circular cylinders by mesh‐free least square‐based finite difference methods

Abstract: In this paper, the mesh-free least square-based finite difference (MLSFD) method is applied to numerically study the flow field around two circular cylinders arranged in side-by-side and tandem configurations. For each configuration, various geometrical arrangements are considered, in order to reveal the different flow regimes characterized by the gap between the two cylinders. In this work, the flow simulations are carried out in the low Reynolds number range, that is, Re = 100 and 200. Instantaneous vorticity contours and streamlines around the two cylinders are used as the visualization aids. Some flow parameters such as Strouhal number, drag and lift coefficients calculated from the solution are provided and quantitatively compared with those provided by other researchers.

Staggered-grid split-node method for spontaneous rupture simulation

Abstract: [1] We adapt the traction-at-split-node method for spontaneous rupture simulations to the velocity-stress staggered-grid finite difference scheme. The staggered-grid implementation introduces both velocity and stress discontinuities via split nodes. The staggered traction components on the fault plane are interpolated to form the traction vector at split nodes, facilitating alignment of the vectors of sliding friction and slip velocity. To simplify the split-node partitioning of the equations of motion, spatial differencing is reduced from fourth to second order along the fault plane, but in the remainder of the grid the spatial differencing scheme remains identical to conventional spatially fourth-order three-dimensional staggered-grid schemes. The resulting staggered-grid split node (SGSN) method has convergence rates relative to rupture-time, final-slip, and peak-slip-velocity metrics that are very similar to the corresponding rates for both a partly staggered split-node code (DFM) and the boundary integral method. The SGSN method gives very accurate solutions (in the sense that errors are comparable to the uncertainties in the reference solution) when the median resolution of the cohesive zone is 4.4 grid points. Combined with previous results for other grid types and other fault-discontinuity approximations, the SGSN results demonstrate that accuracy in finite difference solutions to the spontaneous rupture problem is controlled principally by the scheme used to represent the fault discontinuity, and is relatively insensitive to the grid geometry used to represent the continuum. The method provides an efficient and accurate means of adding spontaneous rupture capability to velocity-stress staggered-grid finite difference codes, while retaining the computational advantages of those codes for problems of wave propagation in complex media.

Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation

Abstract: We consider the three-dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress-free boundary condition. Building on our recently developed theory for difference methods for second order hyperbolic systems [H.-O. Kreiss, N. A. Petersson, J. Ystrom, SIAM J. Numer. Anal., 40 (2002), pp. 1940-1967], we develop an explicit, second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities. The spatial discretization is self-adjoint, and the stability is obtained through an energy estimate. Seismic events are often modeled using singular source terms, and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source. Several numerical examples are given.

Local RBF‐FD solutions for steady convection–diffusion problems

Abstract: This paper describes the application of radial basis function (RBF) based finite difference type scheme (RBF-FD) for solving steady convection–diffusion equations. Numerical studies are made using multiquadric (MQ) RBF. By varying the shape parameter in MQ, the accuracy of the solution is seen to be highly improved for large values of Reynolds' numbers. The developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non-oscillatory. Copyright © 2007 John Wiley & Sons, Ltd.

Research Advances in the Dynamic Stability Behavior of Plates and Shells: 1987–2005—Part I: Conservative Systems

Abstract: This paper reviews most of the recent research done in the field of dynamic stability/ dynamic instability/ parametric excitation /parametric resonance characteristics of structures with special attention to parametric excitation of plate and shell structures. The solution of dynamic stability problems involves derivation of the equation of motion, discretization and determination of dynamic instability regions of the structures. The purpose of this study is to review most of the recent research on dynamic stability in terms of the geometry (plates, cylindrical, spherical and conical shells), type of loading (uniaxial uniform, patch, point loading ….), boundary conditions (SSSS, SCSC, CCCC ….), method of analysis (exact, finite strip, finite difference, finite element, differential quadrature and experimental ….), the method of determination of dynamic instability regions (Lyapunovian, perturbation and Floquet’s methods ), order of theory being applied (thin, thick, 3D, nonlinear….), shell theory used (Sanders’, Love’s and Donnell’s), materials of structures (homogeneous, bimodulus, composite, FGM….) and the various complicating effects such as geometrical discontinuity, elastic support, added mass, fluid structure interactions, non-conservative loading and twisting etc. The important effects on dynamic stability of structures under periodic loading have been identified and influences of various important parameters are discussed. Review on the subject for non-conservative systems in detail will be presented in Part-2.

Frequency response modelling of seismic waves using finite difference time domain with phase sensitive detection (TD–PSD)

Abstract: SUMMARY This paper describes an efficient approach for computing the frequency response of seismic waves propagating in 2- and 3-D earth models within which the magnitude and phase are required at many locations. The approach consists of running an explicit finite difference time domain (TD) code with a time harmonic source out to steady-state. The magnitudes and phases at locations in the model are computed using phase sensitive detection (PSD). PSD does not require storage of time-series (unlike a fast Fourier transform), reducing its memory requirements. Additionally, the response from multiple sources can be obtained from a single finite difference run by encoding each source with a different frequency. For 2-D models with many sources, this time domain phase sensitive detection (TD–PSD) approach has a higher arithmetic complexity than direct solution of the finite difference frequency domain (FD) equations using nested dissection re-ordering (FD–ND). The storage requirements for 2-D finite difference TD–PSD are lower than FD–ND. For 3-D finite difference models, TD–PSD has significantly lower arithmetic complexity and storage requirements than FD–ND, and therefore, may prove useful for computing the frequency response of large 3-D earth models.

Uniform boundary stabilization of the finite difference space discretization of the 1 − d wave equation

Abstract: The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of the energy of the classical semi-discrete system in which the 1−d Laplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing size h. Actually, the decay rate tends to zero as h goes to zero. Then we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. Our method of proof relies essentially on discrete multiplier techniques.

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: Internal resonance activation, reduced-order models and nonlinear normal modes

Abstract: Resonant multi-modal dynamics due to planar 2:1 internal resonances in the non-linear, finite-amplitude, free vibrations of horizontal/inclined cables are parametrically investigated based on the second-order multiple scales solution in Part I [1] (in press). The already validated kinematically non-condensed cable model accounts for the effects of both non-linear dynamic extensibility and system asymmetry due to inclined sagged configurations. Actual activation of 2:1 resonances is discussed, enlightening on a remarkable qualitative difference of horizontal/inclined cables as regards non-linear orthogonality properties of normal modes. Based on the analysis of modal contribution and solution convergence of various resonant cables, hints are obtained on proper reduced-order model selections from the asymptotic solution accounting for higher-order effects of quadratic nonlinearities. The dependence of resonant dynamics on coupled vibration amplitudes, and the significant effects of cable sag, inclination and extensibility on system non-linear behavior are highlighted, along with meaningful contributions of longitudinal dynamics. The spatio-temporal variation of non-linear dynamic configurations and dynamic tensions associated with 2:1 resonant non-linear normal modes is illustrated. Overall, the analytical predictions are validated by finite difference-based numerical investigations of the original partial-differential equations of motion.

Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves

Abstract: A method is proposed for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the preprocessing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing.