Finite difference method

About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.

A finite-difference, time-domain solution for three-dimensional electromagnetic modeling

Abstract: We have developed a finite-difference solution for three-dimensional (3-D) transient electromagnetic problems. The solution steps Maxwell's equations in time using a staggered-grid technique. The time-stepping uses a modified version of the Du Fort-Frankel method which is explicit and always stable. Both conductivity and magnetic permeability can be functions of space, and the model geometry can be arbitrarily complicated. The solution provides both electric and magnetic field responses throughout the earth. Because it solves the coupled, first-order Maxwell's equations, the solution avoids approximating spatial derivatives of physical properties, and thus overcomes many related numerical difficulties. Moreover, since the divergence-free condition for the magnetic field is incorporated explicitly, the solution provides accurate results for the magnetic field at late times.An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homogeneous Dirichlet condition is employed along the subsurface boundaries. Numerical dispersion is alleviated by using an adaptive algorithm that uses a fourth-order difference method at early times and a second-order method at other times. Numerical checks against analytical, integral-equation, and spectral differential-difference solutions show that the solution provides accurate results.Execution time for a typical model is about 3.5 hours on an IBM 3090/600S computer for computing the field to 10 ms. That model contains 100 X 100 X 50 grid points representing about three million unknowns and possesses one vertical plane of symmetry, with the smallest grid spacing at 10 m and the highest resistivity at 100 Omega . m. The execution time indicates that the solution is computer intensive, but it is valuable in providing much-needed insight about TEM responses in complicated 3-D situations.

The finite-difference time-domain method for modeling of seismic wave propagation

Abstract: We present a review of the recent development in finite-difference time-domain modeling of seismic wave propagation and earthquake motion. The finite-difference method is a robust numerical method applicable to structurally complex media. Due to its relative accuracy and computational efficiency it is the dominant method in modeling earthquake motion and it also is becoming increasingly more important in the seismic industry and for structural modeling. We first introduce basic formulations and properties of the finite-difference schemes including promising recent advances. Then we address important topics as material discontinuities, realistic attenuation, anisotropy, the planar free surface boundary condition, free-surface topography, wavefield excitation (including earthquake source dynamics), non-reflecting boundaries, and memory optimization and parallelization.

A relaxation method for solving elliptic difference equations

Abstract: WHEN solving elliptic equations by the method of fmite differences we have to deal with systems of linear algebraic equations, often of a very high order. Given a sufficiently high order of the system, the familiar iterative methods of solution of such systems are very slowly convergent. Numerous works have been devoted to methods of speeding up the convergence of the iterations. These speedingup methods can be split provisionally into two groups. The first group includes methods which use the spectrum of the iterative operators; they are described in detail in text-books [l] and [2]. The second, rather indefinite group includes the so-called “relaxation” methods, that are based essentially on “intuition”, the “computer’s experience”; they are regarded as applicable for nonmechanical computation by a sufficiently experienced group of workers, but as little suited to being carried out on digital computers; it is usually suggested that the relaxation method can be extremely effective [3]. The present article describes a method that we have developed for iterative solution of elliptic difference equations, using an idea not unlike the relaxation method. The present method was put into practice on a digital computer and gave good results. Le us take Poisson’s equation in a rectangular domain:

Two new approximate methods for analyzing free vibration of structural components

Abstract: Two approximate methods, which have not previously been used for structural dynamics problems, are applied to the free vibration analysis of various structural components. The first method is a new version of the complementary energy method. It is shown to be considerably more accurate than the conventional Rayleigh and Rayleigh-Schmidt methods when applied to spatially one-dimensional free vibration problems: prismatic and tapered bars, prismatic beams, and axisymmetric motion of circular membranes. The second method is the differential quadrature method introduced by Bellman and his associates. It is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates.