Grid Finite Difference Method Research Articles

A rotated staggered grid finite-difference with the absorbing boundary condition of a perfectly matched layer

A rotated staggered grid finite-difference (FD) method with a perfectly matched layer (PML) method is proposed for numerically solving elastic wave equations in inhomogeneous elastic and poroelastic media. Compared with a standard staggeredgrid FD, the former has the advantage over the latter in that its physical variables need only to be defined at two locations. In the rotated staggered grid, stress and strain components (or particle velocity and displacement components) are defined at elementary cell centers, and the velocity or displacement components (or the stress and strain components) are defined at vertexes. In this way, no elastic moduli need to be interpolated or averaged. Numerical results from the proposed method have been compared with the standard staggered FD method. The results are in good agreement with each other. Our numerical results show that the proposed algorithm can handle much stronger impedance contrast. This is especially true when simulating fractured medium filled with fluids such as water or gas without giving special treatment. On the other hand, the implemented PML absorbing boundary condition works well in efficiently reducing reflected waves from the artificial interfaces. It generates almost no reflection at artificial interfaces with a boundary of PML thickness of half a wavelength. Our theoretical analysis and numerical tests proved that the PML absorbing algorithm in the rotated staggered grid is almost identical to those in the standard staggered grid. In this paper, we also presented all of the formulations of the PML implementation and modeling examples in elastic, poroelastic, and anisotropic media.

Improvement in the Fault Boundary Conditions for a Staggered Grid Finite-difference Method

Improvement in the Fault Boundary Conditions for a Staggered Grid Finite-difference Method

Comparison between staggered grid finite difference method and stochastic method in simulating strong ground motions

Strong ground motion of an earthquake is simulated by using both staggered grid finite difference method (FDM) and stochastic method, respectively. The acceleration time histories obtained from the both ways and their response spectra are compared. The result demonstrates that the former is adequate to simulate the low-frequency seismic wave; the latter is adequate to simulate the high-frequency seismic wave. Moreover, the result obtained from FDM can better reflect basin effects.

速度・応力型差分法での固体・流体境界の扱いについて

We must consider the seismo-acoustic problem at the irregular fluid-solid boundary in various acoustical and seismological subjects: e. g., the ocean-acoustics problem, the analysis of the data from ocean-bottom seismometers, the analysis of the scattered waves from the irregular core-mantle boundary, and the strong motion prediction at sites close to the ocean. In this paper, we study why and how the fluid-solid or the acoustic-seismic boundary conditions are satisfied in the velocity-stress staggered grid finite-difference method (FDM) which has been widely used for seismo-acoustics problems. A simple analysis of the 2D finite-difference equations together with the comparisons between finite-difference seismograms and the discrete wavenumber method (DWM) seismograms show that the conditions are satisfied provided that (1) the boundary is placed through the shear stress (τxz) grid points (i. e., the rigidity of the shear stress grid points on the boundary must be set to zero), (2) the 2nd-order centered equation with averaged density is applied to the normal velocity component of the grid points right on the boundary, and (3) inadequate higher order finite-difference equations (e. g. the standard 4th-order centered equation) is not applied to grid points neighboring the boundary so that the values with discontinuity at the boundary are not included in the finitedifference equations for grid points off from the boundary. The accuracy of the spatial finitedifference equation at the boundary is O(h) with h being the space increment despite its apparent form of “2nd-order” equation. Violations to these conditions deteriorate the results of computation, especially for the interface wave such as Stoneley wave. In FDM inclined interface is approximated by stairsteps made from the grid. We find that these stairsteps can generate scattered waves with amplitudes up to 20% of the Stoneley wave. We need about 60 grid points per wavelength of the Stoneley wave at the dominant frequency in order to suppress such scattered wave. In the case of basin-like fluid layer with free surface and with “shoreline” we need large number of grid points (about 120 grid points) per wavelength at the dominant frequency in order to model the secondary Rayleigh wave generated (scattered) at the basin edges.

A Cartesian grid finite-difference method for 2D incompressible viscous flows in irregular geometries

A method for generating a non-uniform Cartesian grid for irregular two-dimensional (2D) geometries such that all the boundary points are regular mesh points is given. The resulting non-uniform grid is used to discretize the Navier–Stokes equations for 2D incompressible viscous flows using finite-difference approximations. To that end, finite-difference approximations of the derivatives on a non-uniform mesh are given. We test the method with two different examples: the shallow water flow on a lake with irregular contour and the pressure driven flow through an irregular array of circular cylinders.

Heat and mass transfer in par-baked bread during freezing

To understand the phenomena governing the heat and mass transfers during the freezing of par-baked bread, simulation of freezing in an infinite two-layer cylinder was performed. A fixed grid finite difference method was used in the solution of simultaneous heat and moisture transfer (SHMT) equations according to Lees' three-level scheme. The model accommodates the effects of temperature dependent variables such as apparent specific heat, enthalpy, thermal conductivity, and water activity, and predicts the temperature profiles and weight losses. The model was verified by freezing of par-backed breads with cylindrical shape within an experimental freezer. The RMSE between experimental results and model predictions is 0.505 °C for surface temperature, and 1.8 °C for center temperature. The precision on the loss water is 10%. These results indicate that the method could be successfully applied to SHMT operations in foods freezing.

Heat and mass transfer in par-baked bread during freezing

To understand the phenomena governing the heat and mass transfers during the freezing of par-baked bread, simulation of freezing in an infinite two-layer cylinder was performed. A fixed grid finite difference method was used in the solution of simultaneous heat and moisture transfer (SHMT) equations according to Lees' three-level scheme. The model accommodates the effects of temperature dependent variables such as apparent specific heat, enthalpy, thermal conductivity, and water activity, and predicts the temperature profiles and weight losses. The model was verified by freezing of par-backed breads with cylindrical shape within an experimental freezer. The RMSE between experimental results and model predictions is 0.505 °C for surface temperature, and 1.8 °C for center temperature. The precision on the loss water is 10%. These results indicate that the method could be successfully applied to SHMT operations in foods freezing.

Application of the Staggered Grid Finite Difference Method to the Computation of 3D Induction Logging Response

AbstractThe finite difference method based on staggered grids is applied to simulate the induction logging response in 3D media. The invariance property of the Krylov subspace is used to solve the resulting large sparse complex symmetric linear system. To improve iteration convergence the pseudo inverse of the coefficient matrix is employed when constructing the Krylov subspace instead of the coefficient matrix itself. In each iteration four 3D Poisson equations are computed to get a new Lanczos vector with the incomplete Cholesky decomposition conjugate method. Generally the desirable solution is achieved with the iteration less than 20 times. Also, a new material averaging formula is proposed to get a reasonable average of the conductivity, which guarantees the conservation of the electric current.

Three-Dimensional Electromagnetic Inversion Combining a Finite Difference Forward Solver With Integral Equations Jacobians

In an attempt to overcome long computer run times in direct calculation of resistivity parameter Jacobians in 3-D EM inversion, we combine forward calculations for the staggered grid finite difference (FD) method with Jacobians estimated from the forward results using the integral equations (IE) method. The second-order electric field formulation in the FD method yields the electric field everywhere in the earth efficiently. The IE formulation for the Jacobians utilizes the electric field just within the parameter of interest to define a Born term for the Jacobian, plus a depolarization term reflecting the incremental change in the electric field in all parameters. The amplitude of the depolarization decreases rapidly with distance from the parameter allowing a lumped cell approach to setting up the IE expression to be solved for the Jacobian. The lumped cells may grow geometrically from the parameter in accord with EM scaling, requiring only 100-200 cells to be considered for each parameter. Although this constitutes a sizeable pre-multiplier, the expression for computational cost of the total Jacobian set is only linearly proportional to the number of parameters.

Modeling of 3‐D Electromagnetic Responses in Frequency domain by Using the Staggeredgrid Finite Difference Method

AbstractThree dimensional modeling of electromagnetic fields is essential in understanding the responses of the electromagnetic fields in the earth. Although various numerical modeling techniques have been developed, there are still much to do for the practical use. In this study, the staggered‐grid finite difference method is used to simulate the three dimensional electromagnetic responses. It is suitable for the magnetic dipole in any direction. The calcuation results are compared with that obtained from analytic solution, integral equation and other means. It is shown that the staggered grid finite difference method combined with the Bi‐conjugate gradient iterative solver processed with divergence correction and incomplete Cholesky decomposition procedures can efficiently improve the three dimensional electromagnetic calculation with high accuracy and stability.

断層近傍強震動シミュレーションのための滑り速度時間関数の近似式

In order to simulate near-field ground motions accurately, it is necessary to use a physically reasonable time function of the slip velocity on a fault. Although the analytical expression of slip velocity is preferred for the simulation, the slip velocity on a fault is currently estimated numerically for each set of dynamic parameters of source process. In this study, an approximate expression of slip velocity is proposed on the basis of the numerical solution of 2D and 3D crack simulations including a slip-weakening friction law. For the time of slip onset to the time of maximum slip velocity, the slip velocity is approximated by a quadratic function based on a solution of a 2D steady state crack with a slip-weakening friction law. For a latter part of the slip history, the slip velocity is approximated on the basis of results of 3D crack simulation using the staggered grid finite difference method with the slip-weakening friction law. The important parameters for describing the slip velocity are maximum amplitude of the slip velocity, the time of the maximum slip velocity, and the rise time.

Numerical procedure for simulation of multicomponent and multi-layered phase diffusion

A mass balance error-free numerical procedure for simulation of multicomponent and multi-layered phase diffusional reactions has been suggested, based on an explicit fixed grid finite difference method. Basically, a local equilibrium was assumed at each phase interface and one dimensional movement of interfaces was considered. Attention was paid to the treatments of newly formed thin phase layers and different molar volumes among interacting phases, together with the removal of a mass balance error. Specially derived finite difference forms were used to treat phase layers thinner than two inter-grid distances. A new flux balance equation which is independent from the molar volume differences among phases and leaves no mass balance error was developed by a transformation of space variable system and by a systematic analysis of sources of mass balance error, respectively. Through some model simulations, it could be shown that the present numerical procedure cantreat multi-layered phase diffusion including thin layered phases and can reproduce transitions of layer sequences during diffusional reactions successfully, leaving no mass balance error.

VARIABLE GRID FINITE DIFFERENCE METHOD FOR TWO-DIMENSIONAL TWO-PHASE IMMISCIBLE FLOW

An explicit, time-dependent variable grid finite difference method is introduced and analyzed for approximating the solution of a scalar conservation law in two dimension. The scheme is stable, and the numerical solution is proved to converges to the relevant physical solution.

Estimation of wave propagation characteristics around fracture using simulation techniques: T. Kikuchi & M. Abe, Geothermal Science & Technology, 5(1–2), 1995, pp 53–62

Fractures play an important role in a geothermal system because they are not only conduits of hot water but reservoirs themselves. Therefore, it is important to estimate positions and permeability of fractures from the view point of reservoir evaluation. To estimate fracture properties, the method using seismic waves is one of the most effective. In this paper, simulations are carried out to clarify the wave propagation around a fracture, with the aim of developing a method to estimate the apertures and permeability of fractures. The fractures are simulated as two parallel plates up to the present. However, the real fracture is more complicated and has many contact points. Therefore, we have attempted to simulate the real fracture using the staggered grid finite-difference method. The horizontal and vertical displacement components around a fracture with contact points were calculated. Stoneley wave was found to exist under the following conditions, (1) if the aperture of the fracture is 0.1 m, (2) the dominant frequency of the source is about 175Hz, (3) the source is located in the fluid of the fracture or on the surface of the fracture. Stoneley wave reflects at the point of contact under conditions mentioned earlier.

Numerical Treatment of the Moving Interface in Diffusional Reactions

A numerical procedure for the simulation of diffusional growth in binary and multicomponent alloys has been suggested. The sources of mass balance error in both the representative movable grid and the fixed grid finite difference method were carefully analyzed. Paying attention to the removal or reduction of mass balance error, a new flux balance equation was developed together with a correction of the interface position, based on an explicit fixed grid finite difference method. In the simulation of multicomponent diffusion, a local equilibrium was assumed at the moving interface. The interface compositions were calculated by solving the nonlinear equations which come from a supposition that each solute component should give the same interface moving velocity by the newly developed flux balance equation. The reliability of the present method was demonstrated by comparing the results with the analytical solution or those from phase equilibrium calculation.

Variable grid finite difference methods for study of longitudinally varying planar waveguides

A new discretisation procedure based on a finite difference approach is presented for the analysis of arbitrary two-dimensional optical dielectric waveguides. It is based on a longitudinally variant discretisation grid that allows an accurate treatment of waveguide changes of geometry to be obtained. Comparisons with the well known beam propagation method show a strong improvement in accuracy and computational time saving.

Dynamics of Disconnected Risers Under Rigid and Compliant Hang-Off

Abstract An efficient solution scheme to simulate the nonlinear motions of hanging risers based on an adaptive nonuniform grid finite difference method and an implicit time integration scheme is presented. Dynamic buckling-type response of hanging risers under rigid hang-off due to heave acceleration of the support platform in extreme excitation conditions is studied, and the important parameters affecting the response are identified. Significant reduction of motions and resulting stresses is obtained by employing compliant hang-off.

High accuracy ion optics computing

Computer simulation of focused ion beams for surface analysis of materials by SIMS, or for microfabrication by ion beam lithography plays an important role in the design of low energy ion beam transport and optical systems. Many computer packages currently available, are limited in their applications, being inaccurate or inappropriate for a number of practical purposes. This work describes an efficient and accurate computer programme which has been developed and tested for use on medium sized machines. The programme is written in Algol 68 and models the behaviour of a beam of charged particles through an electrostatic system. A variable grid finite difference method is used with a unique data structure, to calculate the eletric potential in an axially symmetric region, for arbitrary shaped boundaries. Emphasis has been placed upon finding an economic method of solving the resulting set of sparse linear equations in the calculation of the electric field and several of these are described. Applications include individual ion lenses, extraction optics for ions in surface analytical instruments and the design of columns for ion beam lithography. Computational results have been compared with analytical calculations and with some data obtained from individual einzel lenses by Harting and Read ( Electrostatic Lenses, Elsevier, Amsterdam ( 1976)).

The stream function coordinate system for solution of one-dimensional unsteady compressible flow problems

A one-dimensional unsteady compressible isentropic flow problem is solved using a floating grid finite difference method. GENMIX, a computer program for solution of two-dimensional parabolic flow problems, has been adopted for this purpose. The advantages of utilizing the staggered floating grid method are demonstrated through solution of flow in a shocktube. The present method is able to precisely locate the discontinuities in temperature and density profiles.

An adaptive grid finite difference method for conservation laws

Adaptive grid finite difference methods for computing time-accurate solutions of nonlinear hyperbolic conservation laws in one space dimension are studied. The basic approach is to decouple the determination of the moving grid from the solution of the differential equation on the moving grid. The grid is determined by an elliptic grid generation technique suitably modified for time-accurate computations. Two methods, Godunov's scheme and an artificial viscosity method, are used to discretize the differential equation. The technique is applied to problems involving the Buckley-Leverett equation modelling the flow of two immiscible fluids in a porous medium. Substantial improvement in computational efficiency is obtained by using the adaptive grid technique. Extensions of this approach to several space dimensions and systems of equations are discussed.