Cleberson Dors

Explicit time-domain approaches based on numerical Green's functions computed by finite differences - The ExGA family

The present paper describes a new family of time stepping methods to integrate dynamic equations of motion. The scalar wave equation is considered here; however, the method can be applied to time-domain analyses of other hyperbolic (e.g., elastodynamics) or parabolic (e.g., transient diffusion) problems. The algorithms presented require the knowledge of the Greens function of mechanical systems in nodal coordinates. The finite difference method is used here to compute numerically the problem Greens function; however, any other numerical method can be employed, e.g., finite elements, finite volumes, etc. The Greens matrix and its time derivative are computed explicitly through the range [0,@Dt] with either the fourth-order Runge-Kutta algorithm or the central difference scheme. In order to improve the stability of the algorithm based on central differences, an additional matrix called step response is also calculated. The new methods become more stable and accurate when a sub-stepping procedure is adopted to obtain the Greens and step response matrices and their time derivatives at the end of the time step. Three numerical examples are presented to illustrate the high precision of the present approach.

Theory of equivalent staggered-grid schemes: application to rotated and standard grids in anisotropic media

The previous finite-difference numerical schemes designed for direct application to second-order elastic wave equations in terms of displacement components are strongly dependent on Poisson’s ratio. This fact makes theses schemes useless for modelling in offshore regions or even in onshore regions where there is a high Poisson’s ratio material. As is well known, the use of staggered-grid formulations solves this drawback. The most common staggered-grid algorithms apply central-difference operators to the first-order velocity–stress wave equations. They have been one of the most successfully applied numerical algorithms for seismic modelling, although these schemes require more computational memory than those mentioned based on second-order wave equations. The goal of the present paper is to develop a general theory that enables one to formulate equivalent staggered-grid schemes for direct application to hyperbolic second-order wave equations. All the theory necessary to formulate these schemes is presented in detail, including issues regarding source application, providing a general method to construct staggered-grid formulations to a wide range of cases. Afterwards, the equivalent staggered-grid theory is applied to anisotropic elastic wave equations in terms of only velocity components (or similar displacements) for two important cases: general anisotropic media and vertical transverse isotropy media using, respectively, the rotated and the standard staggered-grid configurations. For sake of simplicity, we present the schemes in terms of velocities in the second- and fourth-order spatial approximations, with second-order approximation in time for 2D media. However, the theory developed is general and can be applied to any set of second-order equations (in terms of only displacement, velocity, or even stress components), using any staggered-grid configuration with any spatial approximation order in 2D or 3D cases. Some of these equivalent staggered-grid schemes require less computer memory than the corresponding standard staggered-grid formulation, although the programming is more evolved. As will be shown in theory and practice, with numerical examples, the equivalent staggered-grid schemes produce results equivalent to corresponding standard staggered-grid schemes with computational advantages. Finally, it is important to emphasize that the equivalent staggered-grid theory is general and can be applied to other modelling contexts, e.g., in electrodynamical and poroelastic wave propagation problems in a systematic and simple way.

A new finite difference scheme for modeling acoustic wave propagation

SUMMARY This abstract report a new simple scheme based on finite difference method (FDM) for acoustic wave propagation. This scheme is named here as Equivalent Staggered Grid scheme (ESG), because it is constructed to give responses numerically equivalent to the standard acoustic staggered grid schemes (SSG), derived from the Virieux type formulations. The relations used in ESG are obtained by double application of SSG simple approximations which produces a new central difference formulas. These formulas are then applied to the second order acoustic wave equation in terms of pressure, in the same way traditionally applied to the first order system of equations in terms of pressure and velocity in order to generate the ESG scheme. The main characteristics of the new scheme are that its properties of accuracy and stability are equal to SSG, but with memory requirements like non staggered grid schemes. The demonstration of numerical equivalence between ESG and SSG schemes are verified through the comparison of their responses in some examples.

New FD Operators to Solve Second Order Elastic Wave Equation

The finite difference method (FDM) is one of the most applied and studied numerical methods for seismic wave propagation due to its computational efficiency. There are several available algorithms based on FDM, each one presenting advantages and disadvantages. Concerning offshore elastic wave simulations, it is well known that numerical schemes based on staggered grids are the most applied and successful schemes, but they spend a great amount of computational memory. It is because of the need to store both velocity and stress components. On the other hand, simple grid FDM based schemes based on second-order elastic wave equations in terms of only displacement components present unacceptable responses when modelling high Poisson ration material layers (specially water). This paper presents a new FDM staggered-grid scheme (ESG) based only in displacements – where new stable DF approximations are applied directly to the second-order wave equation – able to model offshore regions as well as staggered grid schemes. The ESG has the same stability and accuracy characteristics as staggered grid schemes but spends as memory as simple grid schemes. In other words, it is presented a staggered grid scheme that takes advantage of the two better features of staggered and simple grid schemes.