Showing papers on "Finite difference method published in 1994"
TL;DR: In this paper, a method for constructing boundary conditions (numerical and physical) of the required accuracy for compact (Pade-like) high-order finite-difference schemes for hyperbolic systems is presented.
728 citations
TL;DR: In this article, the authors presented a multi-parameter family of difference operators when τ⩾3, where τ is the dimension of the difference operator and λ is the number of points in the difference matrix.
703 citations
TL;DR: In this paper, the Navier-Stokes equations for constant viscosity were solved using the SPH method and the expected parabolic and paraboloid velocity profiles were obtained.
Abstract: present a new SPH method that can be used to solve the Navier-Stokes equations for constant viscosity. The method is applied to two-dimensional Poiseuille flow, three-dimensional Hagen Poiseuille flow and two-dimensional isothermal flows around a cylinder. In the former two cases, the temperature of fluid is assumed to be linearly dependent on a coordinate variable x along the flow direction. The numerical results agree well with analytic solutions, and we obtain nearly uniform density distributions and the expected parabolic and paraboloid velocity profiles. The density and ·velocity field in the latter case are compared with the results obtained using a finite difference method. Both methods give similar results for Reynolds number Re=6, 10, 20, 30 and 55, and the differences in the total drag coefficients are about 2~4%. Our numerical simulations indicate that SPH is also an effective numerical method for calculation of viscous flows.
369 citations
TL;DR: In this article, a robust and efficient finite difference algorithm for computing the magnetotelluric response of general three-dimensional (3D) models using the minimum residual relaxation method was developed.
Abstract: We have developed a robust and efficient finite difference algorithm for computing the magnetotelluric response of general three-dimensional (3-D) models using the minimum residual relaxation method. The difference equations that we solve are second order in H and are derived from the integral forms of Maxwell's equations on a staggered grid. The boundary H field values are obtained from two-dimensional transverse magnetic mode calculations for the vertical planes in the 3-D model. An incomplete Cholesky decomposition of the diagonal subblocks of the coefficient matrix is used as a preconditioner, and corrections are made to the H fields every few iterations to ensure there are no H divergences in the solution. For a plane wave source field, this algorithm reduces the errors in the H field for simple 3-D models to around the 0.01% level compared to their fully converged values in a modest number of iterations, taking only a few minutes of computation time on our desktop workstation. The E fields can then be determined from discretized versions of the curl of H equations.
311 citations
TL;DR: In this article, an approximate expression for the history force on a spherical bubble is proposed for finite Reynolds number, Re. Satisfactory agreement is observed between the presently proposed history force and the numerical solution.
Abstract: An approximate expression for the history force on a spherical bubble is proposed for finite Reynolds number, Re. At small time, the history‐force kernel is a constant, which decreases with increasing Re, and the kernel decays as t−2 for large time. For an impulsively started flow over a bubble, accurate finite difference results show that the history force on the bubble decays as t−2 at large time. Satisfactory agreement is observed between the presently proposed history force and the numerical solution.
278 citations
TL;DR: In this article, a method for implementing the general Floquet boundary condition in the finite-difference time-domain algorithm (FDTD) is presented, where the Floquet type of phase shift boundary condition is incorporated into the time domain analysis by illuminating the structure with a combination of sine and cosine excitations to generate a phasor representation at each time step.
Abstract: A method for implementing the general Floquet boundary condition in the finite-difference time-domain algorithm (FDTD) is presented. The Floquet type of phase shift boundary condition is incorporated into the time-domain analysis by illuminating the structure with a combination of sine and cosine excitations to generate a phasor representation of the solution at each time step. With this approach, the characteristics of periodic structures comprised of arbitrarily shaped inhomogeneous geometries can be computed for an arbitrary angle of incidence. Theoretical results are compared for various planar frequency selective surfaces (FSS) and for one with a three-dimensional element, e.g., a thick, double, concentric square loop. >
234 citations
TL;DR: In this article, a new concept of field computation is presented based on the postulate of the existence of linear equations of the discretized nodal values of the fields, different from the conventional equations, but leading to the same solutions.
Abstract: Numerical computations of frequency domain field problems or elliptical partial differential equations may be based on differential equations or integral equations. The new concept of field computation presented in this paper is based on the postulate of the existence of linear equations of the discretized nodal values of the fields, different from the conventional equations, but leading to the same solutions. The postulated equations are local and invariant to excitation. It is shown how the equations can be determined by a sequence of "measures". The measured equations are particularly useful at the mesh boundary, where the finite difference methods fail. The measured equations do not assume the physical condition of absorption, so they are also applicable to concave boundaries. Using the measured equations, one can terminate the finite difference mesh very close to the physical boundary and still obtain robust solutions. It will definitely make a great impact on the way one applies finite difference and finite element methods in many problems. Computational results using the measured equations of invariance in two and three dimensions are presented. >
185 citations
TL;DR: In this paper, a simple and practical numerical method for the liquefaction analysis is formulated using au-p (displacement of the solid phase-pore water pressure) formulation, and the accuracy of the proposed numerical method is addressed through a comparison of the numerical results and the analytical solutions for the transient response of saturated porous solids.
Abstract: The phenomenon of liquefaction is one of the most important subjects in Earthquake Engineering and Coastal Engineering. In the present study, the governing equations of such coupling problems as soil skeleton and pore water are obtained through application of the two-phase mixture theory. Using au-p (displacement of the solid phase-pore water pressure) formulation, a simple and practical numerical method for the liquefaction analysis is formulated. The finite difference method (FDM) is used for the spatial discretization of the continuity equation to define the pore water pressure at the center of the element, while the finite element method (FEM) is used for the spatial discretization of the equilibrium equation. FEM-FDM coupled analysis succeeds in reducing the degrees of freedom in the descretized equations. The accuracy of the proposed numerical method is addressed through a comparison of the numerical results and the analytical solutions for the transient response of saturated porous solids. An elasto-plastic constitutive model based on the non-linear kinematic hardening rule is formulated to describe the stress-strain behavior of granular materials under cyclic loading. Finally, the applicability of the proposed numerical method is examined. The following two numerical examples are analyzed in this study: (1) the behavior of seabed deposits under wave action, and (2) a numerical simulation of shaking table test of coal fly ash deposit.
171 citations
TL;DR: In this article, a numerical finite difference scheme was proposed to simulate the motion of a damped, stiff string interacting with a nonlinear hammer, from which the time histories of string displacement and velocity for each point of the string were computed in the time domain.
Abstract: The first attempt to generate musical sounds by solving the equations of vibrating strings by means of finite difference methods (FDM) was made by Hiller and Ruiz [J. Audio Eng. Soc. 19, 462–472 (1971)]. It is shown here how this numerical approach and the underlying physical model can be improved in order to simulate the motion of the piano string with a high degree of realism. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, as well as the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by a few examples of simulated string waveforms. A brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also included.
170 citations
TL;DR: This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis as well as seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy.
Abstract: SUMMARY This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis. Firstly, drawbacks existing in the method of differential quadrature (DQ) are evaluated and discussed. Then, an improved and simpler generalized differential quadrature method (GDQ) is introduced to overcome the existing drawback and to simplify the procedure for determining the weighting coefficients. Subsequently, the generalized differential quadrature is systematically employed to solve problems in structural analysis. Numerical examples have shown the superb accuracy, efficiency, convenience and the great potential of this method. Numerical approximation methods for solving partial differential equations have been widely used in various engineering fields. Classical techniques such as finite element and finite difference methods are well developed and well known. These methods can provide very accurate results by using a large number of grid points. However, in a large number of cases, reasonably approximate solutions are desired at only a few specific points in the physical domain. In order to get results even at or around a point of interest with acceptable accuracy, conventional finite element and finite difference methods still require the use of a large number of grid points. Consequently, the requirement for computer capacity is often unnecessarily large in such cases. In seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy, the method of differential quadrature (DQ) was introduced by Bellman et a!.'. The method of DQ is a global approximate method. This method is based on the ideas that the derivative of a function with respect to a co-ordinate direction can be expressed as a weighted linear sum of all the function values at all mesh points along that direction and that a continuous function can be approximated by a higher-order polynomial in the overall domain. The DQ method differs from the finite element method (FEM) in two aspects. Firstly, the FEM uses lower-order polynomials to approximate a function on a local element level, while the DQ method approximates a function on the global area using higher-order polynomials. Secondly, the DQ method directly approximates the derivatives of a function at a point, while the FEM approximates a function over a local element and the derivatives can then be derived from the approximate function. In this aspect, the DQ method is more similar to the finite difference method (FDM). However, the FDM is also a local approximation method based on lower-order polynomial approximation. In fact, it can be shown that the FDM is just a special case of the DQ method where it is applied locally on the range [.xi- xi + l]. Owing to the higher-order
166 citations
TL;DR: FDTD equations that allow us to use a nonuniform grid are derived and this grid gives a better accuracy to CPU–resource ratio in a number of circumstances, and tilted and curved boundaries can be described more easily.
Abstract: The finite‐difference time‐domain (FDTD) approximation can be used to solve acoustical field problems numerically. Mainly because it is a time‐domain method, it has some specific advantages. The basic formulation of the FDTD method uses an analytical grid for the discretization of an unknown field. This is a major disadvantage. In this paper, FDTD equations that allow us to use a nonuniform grid are derived. With this grid, tilted and curved boundaries can be described more easily. This gives a better accuracy to CPU–resource ratio in a number of circumstances. The paper focuses on the new formulation and its accuracy. The problem of automatically generating the mesh in a general situation is not addressed. Simulations using quasi‐Cartesian grids are compared to Cartesian grid results.
TL;DR: In this paper, a finite-difference approach was proposed to obtain a unique set of equations for dielectric optical waveguides with stepped refractive index profiles, where the boundary conditions were formulated and included in the finite difference scheme.
Abstract: An important class of dielectric optical waveguides has stepped refractive index profiles. For these waveguides we present a new finite-difference approach free of spurious modes. The coupled difference equations are formulated in terms of the transverse magnetic field components H/sub x/ and H/sub y/. We show how the boundary conditions can be formulated and included in the finite difference scheme to obtain a unique set of equations. For a step-index fiber a comparison of the numerical results with the analytical solution shows that the relative error in the propagation constant is as low as 4.4/spl times/10/sup /spl minus/7/ for an index difference of 7.3%. For a rib waveguide, we compare our results with previously published data based on other methods. Field plots of the dominant and the weak transverse field components of the magnetic field for the fundamental mode of a buried rib waveguide are also given. >
01 Oct 1994
TL;DR: In this paper, a finite difference calculation of the full-vectorial modes of optical waveguides is presented, and the leakage phenomenon of the higher modes of a rib waveguide is predicted by this method.
Abstract: Finite difference calculations of full-vectorial modes of optical waveguides are presented. This method has overcome the limitations of the semivectorial approximation and is able to calculate full-vectorial modes of arbitrary order for a given structure with an arbitrary refractive index profile. Numerical results show that the method is accurate. In addition, the leakage phenomenon of the higher modes of a rib waveguide is predicted by this method.
TL;DR: In this paper, the Yee scheme is shown to be second-order convergent on a non-uniform mesh, despite the fact that the local truncation error is only of first order.
Abstract: The Yee scheme is the principal finite difference method used in computing time domain solutions of Maxwell's equations. On a uniform grid the method is easily seen to be second- order convergent in space. This paper shows that the Yee scheme is also second-order convergent on a nonuniform mesh despite the fact that the local truncation error is (nodally) only of first order.
TL;DR: In this paper, the dispersion equation was shown to be very well behaved and superior to that obtained with the Arakawa grids A-E. The dispersion equations were used for the case of an unstaggered grid for vorticity, divergence and mass.
Abstract: Numerical simulation of geostrophic adjustment in shallow water is discussed for the case of an unstaggered grid for vorticity, divergence, and mass. The dispersion equation is shown to be very well behaved and superior to that obtained with the Arakawa grids A–E.
TL;DR: In this article, the second-order terms associated with geometric nonlinearity were introduced into the basic equation of generalized beam theory to give rise to simple explicit equations for the load to cause buckling in individual modes under either axial load or uniform bending moment.
TL;DR: In this paper, a method of incorporating lumped terminal conditions into a finite-difference, time-domain (FDTD) analysis of multiconductor transmission lines is given, which provides an exact solution of the transmission-line equations via the FDTD technique when the line discretization, /spl Delta/t, and the time discretisation,/spl Delta /t, are chosen such that /spl delta/t=/spl Delta z/v where v is the phase velocity of propagation on the line.
Abstract: A method of incorporating lumped terminal conditions into a finite-difference, time-domain (FDTD) analysis of multiconductor transmission lines is given. The method provides an exact solution of the transmission-line equations via the FDTD technique when the line discretization, /spl Delta/t, and the time discretization, /spl Delta/t, are chosen such that /spl Delta/t=/spl Delta/z/v where v is the phase velocity of propagation on the line. Examples are given to show that in the case of a multiconductor line in an inhomogeneous medium where the mode velocities are not identical, the method gives accurate results with a minimum of computational effort. >
20 Jun 1994
TL;DR: In this paper, the finite difference time-domain algorithm (FDTD) was used to analyze complex periodic structures with a combination of sine and cosine excitations to generate a phasor representation at each time step.
Abstract: Periodic structures, e.g., frequency selective surfaces (FSSs), are used in applications such as the design of bandpass radomes for missiles, subreflectors for dual frequency reflector antenna systems and filters for optics and infrared. In this work, complex periodic structures are analyzed using the finite difference time-domain algorithm (FDTD) in combination with the Floquet boundary condition. The Floquet type of phase shift boundary condition is incorporated in the time-domain analysis by illuminating the structure with a combination of sine and cosine excitations to generate a phasor representation of the solution at each time step. With this approach, the FDTD method can be applied to a frequency selective surface (FSS) geometry of arbitrary shape illuminated by a plane wave at an arbitrary angle of incidence, without the need to store large amounts of data to model the time advance or delay between the periodic cells. The application of the technique will be demonstrated on a thick, doubly-concentric square loop FSS.
TL;DR: In this article, the Von Neumann stability analysis on the finite-difference equations of a pipe shows that the equations are unconditionally stable and an iterative convergence method is applied to the calculation of node pressure at junctions in networks.
TL;DR: In this paper, a full finite difference time domain methodology is developed for electromagnetic wave propagation in a plasma, consistent with central difference approximation of the curl, divergence and gradient operators that appear in the joint equations of Euler and Maxwell, and the coupling effects between the fluid velocity and the electric field.
Abstract: A full finite difference time domain methodology is developed for electromagnetic wave propagation in a plasma The finite difference grid is consistent with central difference approximation of the curl, divergence and gradient operators that appear in the joint equations of Euler and Maxwell, and the coupling effects between the fluid velocity and the electric field To accomplish the time advancement, the central difference approximation is invoked for the time derivatives and leapfrog concepts are employed The resulting difference equations converge to the exact equations, provided that the developed stability requirement is satisfied Finally, numerical results are provided and compared with the inverse fast Fourier transform results of closed-form, frequency domain solutions for the half space problem; the agreement between solutions is shown to be excellent
TL;DR: In this paper, a finite difference approximation to an inverse problem of determining an unknown source parameter p(t) which is a coefficient of the solution u in a linear parabolic equation subject to the specification of u at an internal point along with the usual initial boundary conditions is considered.
Abstract: We consider a finite difference approximation to an inverse problem of determining an unknown source parameter p(t) which is a coefficient of the solution u in a linear parabolic equation subject to the specification of the solution u at an internal point along with the usual initial boundary conditions. The backward Euler scheme is studied and its convergence is proved via an application of the discrete maximum principle for a transformed problem. Error estimates For u and p involve numerical differentiation of the approximation to the transformed problem. Some experimental numerical results using the newly proposed numerical procedure are discussed.
01 Jan 1994
TL;DR: In this article, a method of finite differences is used to solve the forward problem for a given piston configuration; some nontrivial issues arise in determining boundary conditions, and the finite difference equations are then rearranged into a linear system of equations which formulates the inverse problem.
Abstract: In elasticity imaging, a surface deformation is applied to an object using small pistons, and the resulting induced strains in the interior of the object are measured using ultrasonic imaging. Two important problems are considered: (1) the forward problem of determining the strains induced by a known deformation of an object with known elasticity; and (2) the inverse problem of reconstructing elasticity from measured strains and the equations of equilibrium. The method of finite differences is used to solve the forward problem for a given piston configuration; some nontrivial issues arise in determining boundary conditions. The finite difference equations are then rearranged into a linear system of equations which formulates the inverse problem; this system can be solved for the unknown elasticities. This formulation of the inverse problem is completely consistent with the forward problem; this is useful for iterative methods in which the deformation is adaptively changed. A comparison between simulated and actual measured results demonstrate the feasibility of the proposed procedure. >
TL;DR: In this article, a finite difference method was applied to the analysis of the temporal and spatial stability of the three-dimensional boundary layer flow on a swept wing, and the results showed that the algorithm can be reduced significantly by exploiting the special structure of two matrices.
Abstract: The present investigation is concerned with a fourth order accurate finite difference method and its application to the study of the temporal and spatial stability of the three-dimensional compressible boundary layer flow on a swept wing. This method belongs to the class of compact two-point difference schemes discussed by White (1974) and Keller (1974). The method was apparently first used for solving the two-dimensional boundary layer equations. Attention is given to the governing equations, the solution technique, and the search for eigenvalues. A general purpose subroutine is employed for solving a block tridiagonal system of equations. The computer time can be reduced significantly by exploiting the special structure of two matrices.
TL;DR: In this article, Chen et al. compared the results of the spatially horizontally averaged Richards equation (SHARE) model and the averaged Green-Ampt model with the results from a three-dimensional finite difference model of unsaturated flow which is perceived as the reference solution.
Abstract: Two models for horizontally averaged unsaturated flow have been developed from two different approaches in the first (Chen et al., this issue) of these companion papers. In this paper the results from both the spatially horizontally averaged Richards equation (SHARE) model and the averaged Green-Ampt model are compared with the results from a three-dimensional finite difference model of unsaturated flow which is perceived as the reference solution. The results of the averaged Green-Ampt model show very good agreement with the averaged results from the three-dimensional model, while SHARE model results are applicable only when fluctuations in soil parameters are small with respect to their mean values. It is also shown that methods of simple parameter averaging (arithmetic or geometric averages) with the local Richards equation does not yield meaningful results in heterogeneous soils. This study suggests that spatially horizontally averaged simplified models (such as the averaged Green-Ampt model) are attractive alternatives to perturbation models (such as the SHARE model) in heterogeneous fields. Due to their simplicity in formulation, accuracy in predicting average behaviors, and minimal requirement of computer effort, the spatially horizontally averaged simplified models can be easily implemented in large-scale models, such as atmospheric mesoscale models. boundary and initial conditions. The average saturation at each depth was obtained by areal integration of the local scale water saturation over the specified area under the assumption of independent vertical soil columns at field scales. Relative occurrence frequency curves of the spatially varying parameters were required for the solution of the average quantities. The second model developed was the spatially horizontally averaged Richards equation (SHARE) model. This model was developed by using spatial averaging and regular perturbation techniques under the assumption of no source or sink in the study area. The SHARE model was expressed as a system of two coupled one-dimensional partial differential equations in terms of the mean saturation and the cross-covariance of saturation and saturated hydrau-
TL;DR: In this paper, transitions using through-substrate vias are considered and analyzed both in the time and frequency domains using the Finite Difference Time Domain (FDTD) technique and Finite Element Method (FEM) respectively.
Abstract: MIC and MMIC packages capable of good performance at frequencies as high as 60 GHz need to have small volume, low weight, microstrip and/or coplanar waveguide (CPW) compatibility and exhibit negligible electrical interference with the rest of the circuit. In order to acquire some of these characteristics, special provisions need to be made during circuit layout and design, resulting in high-density packages. The designed circuits have a large number of interconnects which are printed on electrically small surface areas and communicate through the substrate in a direct through-via fashion or electromagnetically through appropriately etched apertures. In a circuit environment of this complexity, parasitic effects such as radiation and cross talk are intensified, thus, making the vertical interconnection problem very critical. In this paper, transitions using through-substrate vias are considered and analyzed both in the time and frequency domains using the Finite Difference Time Domain (FDTD) technique and the Finite Element Method (FEM), respectively. The merits of each method in conjunction with accuracy, computational efficiency and versatility are discussed and results are compared showing excellent agreement. Specifically, a microstrip short-circuit, a microstrip ground pad, a CPW-to-microstrip through-via transition and a channelized CPW-to-microstrip transition are analyzed and their electrical performance is studied. >
TL;DR: In this paper, the concentration polarization effects on solute separation in ultra-filtration have been studied and a comparison between a finite-difference solution and an analytical solution for the mass transport differential equation is presented.
TL;DR: In this paper, the Hilbert uniqueness method was used to solve the exact and approximate boundary controllability problems for the adjoint heat equation using convex duality, and a combination of finite difference methods for the time discretization, finite element methods for space discretisation, and of conjugate gradient and operator splitting methods for iterative solution of discrete control problems.
Abstract: The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate boundary controllability problems for the heat equation. Using convex duality, we reduce the solution of the boundary control problems to the solution of identification problems for the initial data of an adjoint heat equation. To solve these identification problems, we use a combination of finite difference methods for the time discretization, finite element methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods discussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.
TL;DR: In this article, a new family of implicit, single-step time integration methods is presented for solving structural dynamics problems, which are unconditionally stable, second-order accurate and asymptotically annihilating.
TL;DR: In this article, a simple method to construct finite-difference equations and grids entirely inheriting admitted Lie groups of transformations of initial differential models is developed, and the criterion of invariance of difference equations together with difference grid is discussed.
Abstract: The present paper is concerned with continuous groups of transformations in a space of discrete variables. The criterion of invariance of difference equations together with difference grid is discussed. One simple method to construct finite-difference equations and grids entirely inheriting admitted Lie groups of transformations of initial differential models is developed.
TL;DR: This paper describes how to interpolate in a conservative manner when solving systems of conservative laws on overlapping grids and solves a two-dimensional viscous Burger’s equation to demonstrate the conservative nature of the scheme.
Abstract: This paper describes how to interpolate in a conservative manner when solving systems of conservative laws on overlapping grids. Overlapping grids are a flexible and efficient way to create grids for complicated regions. Before now, however, no general method had been developed for conservative interpolation. The basic idea consists of assuming that the interpolation coefficients are free parameters and then deriving constraints to ensure that the interpolation is conservative. A system of equations is then solved to determine the coefficients. A two-dimensional viscous Burger’s equation is solved to demonstrate the conservative nature of the scheme.