Abstract: Ordinary Differential Equations.- The Analytical Behaviour of Solutions.- Numerical Methods for Second-Order Boundary Value Problems.- Parabolic Initial-Boundary Value Problems in One Space Dimension.- Analytical Behaviour of Solutions.- Finite Difference Methods.- Finite Element Methods.- Two Adaptive Methods.- Elliptic and Parabolic Problems in Several Space Dimensions.- Analytical Behaviour of Solutions.- Finite Difference Methods.- Finite Element Methods.- Time-Dependent Problems.- The Incompressible Navier-Stokes Equations.- Existence and Uniqueness Results.- Upwind Finite Element Method.- Higher-Order Methods of Streamline Diffusion Type.- Local Projection Stabilization for Equal-Order Interpolation.- Local Projection Method for Inf-Sup Stable Elements.- Mass Conservation for Coupled Flow-Transport Problems.- Adaptive Error Control.

Abstract: A new methodology to build discrete models of boundary-value problems is presented. The h-pcloud method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. This new method uses radial basis functions of varying size of supports and with polynomialreproducing properties of arbitrary order. The approximating properties of the h-p cloud functions are investigated in this article and a several theorems concerning these properties are presented. Moving least squares interpolants are used to build a partition of unity on the domain of interest. These functions are then used to construct, at a very low cost, trial and test functions for Galerkin approximations. The method exhibits a very high rate of convergence and has a greater -exibility than traditional h-p finite element methods. Several numerical experiments in I-D and 2-D are also presented. @ 1996 John Wiley & Sons, Inc. In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational effort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion and separation to model fracture, fragmentation, free surfaces, etc. Moreover, in most computer-aided design work, the generation of an appropriate mesh constitutes, by far, the costliest portion of the computer-aided analysis of products and processes. These are among the reasons that interest in so-called meshless methods has grown rapidly in recent years. Most meshless methods require a scattered set of nodal points in the domain of interest. In these methods, there may be no fixed connectivities between the nodes, unlike the finite element or finite difference methods. This feature has significant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis. The analysis of problems such as crack propagation, penetration, and large deformations, can, in principle, be greatly simplified by the use of meshless methods. A growing crack, for example, can be modeled by simply extending the free surfaces that correspond to the crack [ 11. The analysis of large deformation problems by, e.g., finite element methods, may require the continuous remeshing of the domain to avoid the breakdown of the calculation due to

Abstract: Direct numerical simulations and computational aeroacoustics require an accurate finite difference scheme that has a high order of truncation and high-resolution characteristics in the evaluation of spatial derivatives. Compact finite difference schemes are optimized to obtain maximum resolution characteristics in space for various spatial truncation orders. An analytic method with a systematic procedure to achieve maximum resolution characteristics is devised for multidiagonal schemes, based on the idea of the minimization of dispersive (phase) errors in the wave number domain, and these are applied to the analytic optimization of multidiagonal compact schemes. Actual performances of the optimized compact schemes with a variety of truncation orders are compared by means of numerical simulations of simple wave convections, and in this way the most effective compact schemes are found for tridiagonal and pentadiagonal cases, respectively. From these comparisons, the usefulness of an optimized high-order tridiagonal compact scheme that is more efficient than a pentadiagonal scheme is discussed. For the optimized high-order spatial schemes, the feasibility of using classical high-order Runge-Kutta time advancing methods is investigated.

Abstract: We study the numerical approximation of an integro-differential equation which is intermediate between the heat and wave equations. The proposed discretization uses convolution quadrature based on the first- and second-order backward difference methods in time, and piecewise linear finite elements in space. Optimal-order error bounds in terms of the initial data and the inhomogeneity are shown for positive times, without assumptions of spatial regularity of the data.

Abstract: A new mathematical formulation is presented for the systematic development of perfectly matched layers from Maxwell's equations in properly constructed anisotropic media. The proposed formulation has an important advantage over the original Berenger's perfectly matched layer in that it can be implemented in the time domain without any splitting of the fields. The details of the numerical implementation of the proposed perfectly matched absorbers in the context of the finite-difference time-domain approximation of Maxwell's equations are given. Results from three-dimension (3-D) simulations are used to illustrate the effectiveness of the media constructed using the proposed approach as absorbers for numerical grid truncation.

Abstract: In this paper, an efficient finite-difference time-domain algorithm (FDTD) is presented for solving Maxwell's equations with rotationally symmetric geometries. The azimuthal symmetry enables us to employ a two-dimensional (2-D) difference lattice by projecting the three-dimensional (3-D) Yee-cell in cylindrical coordinates (r, /spl phi/, z) onto the r-z plane. Extensive numerical results have been derived for various cavity structures and these results have been compared with those available in the literature. Excellent agreement has been observed for all of the cases investigated.

Abstract: An innovative method of analysis was developed to simulate the non-linear seismic finite-amplitude liquid sloshing in two-dimensional containers. In view of the irregular and time-varying liquid surface, the method employed a curvilinear mesh system to transform the non-linear sloshing problem from the physical domain with an irregular free-surface boundary into a computational domain in which rectangular grids can be analysed by the finite difference method. Non-linearities associated with both the unknown location of the free surface and the high-order differential terms were considered. The Crank-Nicolson time marching scheme was employed and the resulting finite difference algorithm is unconditionally stable and very lightly damped with respect to the temporal co-ordinate. In order to minimize numerical instability caused by the computational dispersion in spatially discretized surface wave, a second-order dissipation term was added to the system to filter out the spurious high-frequency waves. Sloshing effects and structural response were measured in terms of sloshing amplitude, base shear and overturning moment generated by the hydrodynamic pressure of the liquid exerted on the container walls. Simulation results of liquid sloshing induced by earthquake and harmonic base excitations were compared with those of the linear wave theory and the limitations of the latter in assessing the response of seismically excited liquids were addressed.

Abstract: A hybrid method is presented for incorporating general terminations into the solution of lossy multiconductor transmission lines (MTLs). The terminations are characterized by a state-variable formulation which allows a general characterization of dynamic as well as nonlinear elements in the termination networks. The method combines the second-order accuracy of the finite difference-time domain (FDTD) algorithm for the MTL with the absolutely stable, backward Euler discretization of the state-variable representations of the termination networks. A compact matrix formulation of the recursion relations at the interface between the MTL and the termination networks allows a straightforward coding of the algorithm. Skin effect losses of the line conductors as well as the effect of an incident field are easily incorporated into the algorithm. Several numerical examples are given which contain dynamic and nonlinear elements in the terminations. These examples demonstrate the validity of the method and show that the temporal and spatial step sizes can be maximized, thereby minimizing the computational burden.

Abstract: The perfectly matched layer (PML) boundary condition for the Helmoltz equation is developed and applied to the finite-difference beam propagation method. Its effectiveness is verified by way of examples.

Abstract: In this paper, a finite difference method is developed to analyze the guided-wave properties of a class of two-dimensional photonic crystals (irregular dielectric rods). An efficient numerical scheme is developed to deal with the deterministic equations resulting from a set of finite difference equations for inhomogeneous periodic structures. Photonic band structures within an irreducible Brillouin zone are investigated for both in-plane and out-of-plane propagation. For out-of-plane propagation, the guided waves are hybrid modes; while for in-plane propagation, the guided waves are either TE or TM modes, and there exist photonic bandgaps within which wave propagation is prohibited. Photonic bandgap maps for squares, veins, and crosses are investigated to determine the effects of the filling factor, the dielectric contrast, and lattice constants, on the band-gap width and location. Possible applications of photonic bandgap materials are discussed.

Abstract: A new formulation is presented for the systematic development of perfectly matched layers (PML) from Maxwell's equations in properly constructed anisotropic media. The proposed formulation has an important advantage over the original Berenger's PML in that it can be implemented in the time domain without any splitting of the fields. Results from 3D simulations illustrate the effectiveness of the proposed method.

Abstract: An active and efficient method of including frequency-dependent conductor losses into the time-domain solution of the multiconductor transmission line equations is presented. It is shown that the usual A+B/spl radic/s representation of these frequency-dependent losses is not valid for some practical geometries. The reason for this the representation of the internal inductance the at lower frequencies. A computationally efficient method for improving this representation in the finite-difference time-domain (FDTD) solution method is given and is verified using the conventional time-domain to frequency-domain (TDFD) solution technique.

Abstract: Two high-accuracy finite-difference schemes for simulating long-range linear wave propagation are presented: a maximum-order scheme and an optimized scheme. The schemes combine a seven-point spatial operator and an explicit six-stage low-storage time-march method of Runge--Kutta type. The maximum-order scheme can accurately simulate the propagation of waves over distances greater than five hundred wavelengths with a grid resolution of less than twenty points per wavelength. The optimized scheme is found by minimizing the maximum phase and amplitude errors for waves which are resolved with at least ten points per wavelength, based on Fourier error analysis. It is intended for simulations in which waves travel under three hundred wavelengths. For such cases, good accuracy is obtained with roughly ten points per wavelength.

Abstract: The perfectly matched layer (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. Recently, it has been pointed out that this absorbing boundary condition is the same as coordinate stretching in the complex space. In this paper, the corresponding coordinate stretching is analyzed in the discretized space of Maxwell's equations as described by the Yee algorithm. The corresponding dispersion relationship is derived for a PML medium and then the problem of reflection from a single interface is solved. A perfectly matched interface is shown not to exist in the discretized space, even though it exists in the continuum space. Numerical simulations both using finite difference method and finite element method confirm that such discretization error exists. A numerical scheme using the finite element method is then developed to optimize the PML with respect to its parameters. Examples are given to demonstrate the performance of the optim...

Abstract: A three‐dimensional finite‐difference (FD) method is used to simulate sonic wave propagation in a borehole with an inhomogeneous solid formation. The second‐order FD scheme solves the first‐order elastic wave equations with central differencing in both space and time via staggered grids. Liao’s boundary condition is used to reduce artificial reflections from the finite computational domain. In the staggered grids, sources have to be implemented at the discrete center in order to maintain the appropriate symmetry in an axisymmetric borehole environment. The FD scheme is validated for multipole sources in three special media: (i) a homogeneous medium; (ii) a homogeneous formation with a fluid‐filled borehole; and (iii) a horizontally layered formation. The staircase approximation of a circular borehole introduces little error in dipole wave fields, although it causes a noticeable phase velocity error in the monopole Stoneley wave. This error has been drastically reduced by using a material averaging scheme ...

Abstract: In this article, we construct and analyse a family of finite difference schemes for the acoustic wave equation with variable coefficients. These schemes are fourth-order accurate in space and time in the case of smooth media and are designed to remain stable and "optimal" for reflection-transmission phenomena in the case of discontinuous coefficients. Together with a detailed mathematical study, various numerical results are presented.

Abstract: A numerical simulation of butterfly valve flows is a useful technique to investigate the physical phenomena of the flow field. A three-dimensional numerical analysis was carried out on incompressible fluid flows in a butterfly valve by using FLUENT, which solves difference equations. Characteristics of the butterfly valve flows at different valve disk angles with a uniform incoming velocity were investigated. Comparisons of FLUENT results with other results, i.e., experimental results, were made to determine the accuracy of the employed method. Results of the three-dimensional analysis may be useful in the valve design.

Abstract: The lattice Boltzmann method (LBM) is used to simulate flow in an infinite periodic array of octagonal cylinders. Results are compared with those obtained by a finite difference (FD) simulation solved in terms of streamfunction and vorticity using an alternating direction implicit scheme. Computed velocity profiles are compared along lines common to both the lattice Boltzmann and finite difference grids. Along all such slices, both streamwise and transverse velocity predictions agree to within 0ċ5% of the average streamwise velocity. The local shear on the surface of the cylinders also compares well, with the only deviations occurring in the vicinity of the corners of the cylinders, where the slope of the shear is discontinuous. When a constant dimensionless relaxation time is maintained, LBM exhibits the same convergence behaviour as the FD algorithm, with the time step increasing as the square of the grid size. By adjusting the relaxation time such that a constant Mach number is achieved, the time step of LBM varies linearly with the grid size. The efficiency of LBM on the CM-5 parallel computer at the National Center for Supercomputing Applications (NCSA) is evaluated by examining each part of the algorithm. Overall, a speed of 13ċ9 GFLOPS is obtained using 512 processors for a domain size of 2176×2176.

Abstract: Two basic refinements of the finite-difference method for 3-D dc resistivity modeling are presented. The first is a more accurate formula for the source singularity removal. The second is the analytic computation of the source terms that arise from the decomposition of the potential into the primary potential because of the source current and the secondary potential caused by changes in the electrical conductivity. Three examples are presented: a simple two-layered model, a vertical contact, and a buried sphere. Both accurate and approximate Dirichlet boundary conditions are used to compute the secondary potential. Numerical results show that for all three models, the average percentage error of the apparent resistivity obtained by the modified finite-difference method with accurate boundary conditions is less than 0.5%. For the vertical contact and the buried sphere models, the error caused by the approximate boundary condition is less than 0.01%.