Abstract: Several issues related to the application of very high-order schemes for the finite difference simulation of the full Navier-Stokes equations are investigated. The schemes utilize an implicit, approximately factored time-integration method coupled with spatial fourth- and sixth-order compact-difference formulations and a filtering strategy of up to tenth order. For this last aspect a consistent optimization approach is developed to treat points near the boundary resulting in minimal degradation of accuracy. The problems investigated exhibit many of the challenging features of practical flows and include several with complications introduced by curvilinear meshes, viscous effects, unsteadiness, and three-dimensionality. The high-order method is observed to be very robust for every problem considered. The algorithm is demonstrated to be highly accurate compared to both second-order and upwind-biased methods. For several cases, particularly very-low-Mach-number flows, filtering is determined to be a superior alternative to scalar damping

Abstract: In this paper, a finite-difference time-domain method that is free of the constraint of the Courant stability condition is presented for solving electromagnetic problems. In it, the alternating direction implicit (ADI) technique is applied in formulating the finite-difference time-domain (FDTD) algorithm. Although the resulting formulations are computationally more complicated than the conventional FDTD, the proposed FDTD is very appealing since the time step used in the simulation is no longer restricted by stability but by accuracy. As a result, computation speed can be improved. It is found that the number of iterations with the proposed FDTD can be at least three times less than that with the conventional FDTD with the same numerical accuracy.

Abstract: This paper presents two dimensional (2-D) finite difference time domain (FDTD) simulations of low-loss right-angle waveguide bends, T-junctions and crossings, based on high index-contrast waveguides. Such structures are essential for the dense integration of optical components. Excellent performance characteristics are obtained by designing the waveguide intersection regions as low-Q resonant cavities with certain symmetries and small radiation loss. A simple analysis, based on coupled mode theory in time, is used to explain the operation principles and agrees qualitatively with the numerical results.

Abstract: This book deals with the simulation of the incompressible Navier-Stokes equations for laminar and turbulent flows. The book is limited to explaining and employing the finite difference method. It furnishes a large number of source codes which permit to play with the Navier-Stokes equations and to understand the complex physics related to fluid mechanics. Numerical simulations are useful tools to understand the complexity of the flows, which often is difficult to derive from laboratory experiments. This book, then, can be very useful to scholars doing laboratory experiments, since they often do not have extra time to study the large variety of numerical methods; furthermore they cannot spend more time in transferring one of the methods into a computer language. By means of numerical simulations, for example, insights into the vorticity field can be obtained which are difficult to obtain by measurements. This book can be used by graduate as well as undergraduate students while reading books on theoretical fluid mechanics; it teaches how to simulate the dynamics of flow fields on personal computers. This will provide a better way of understanding the theory. Two chapters on Large Eddy Simulations have been included, since this is a methodology that in the near future will allow more universal turbulence models for practical applications. The direct simulation of the Navier-Stokes equations (DNS) is simple by finite-differences, that are satisfactory to reproduce the dynamics of turbulent flows. A large part of the book is devoted to the study of homogeneous and wall turbulent flows. In the second chapter the elementary concept of finite difference is given to solve parabolic and elliptical partial differential equations. In successive chapters the 1D, 2D, and 3D Navier-Stokes equations are solved in Cartesian and cylindrical coordinates. Finally, Large Eddy Simulations are performed to check the importance of the subgrid scale models. Results for turbulent and laminar flows are discussed, with particular emphasis on vortex dynamics. This volume will be of interest to graduate students and researchers wanting to compare experiments and numerical simulations, and to workers in the mechanical and aeronautic industries.

Abstract: We discuss the foundations and state-of-the-art of the method of analytical regularization (MAR) (also called the semi-inversion method). This is a collective name for a family of methods based on conversion of a first-kind or strongly-singular second-kind integral equation to a second-kind integral equation with a smoother kernel, to ensure point-wise convergence of the usual discretization schemes. This is done using analytical inversion of a singular part of the original equation; discretization and semi-inversion can be combined in one operation. Numerous problems being solved today with this approach are reviewed, although in some of them, MAR comes in disguise.

Abstract: In this paper, we first extend the Sunde logarithmic approximation for the single-wire line ground impedance to the case of a multiconductor line. The new approximate forms are compared to the general expressions which involve integrals over an infinitely long interval and an excellent agreement is found. The inverse Fourier transform of the ground impedance presents singularities which complicate the numerical solution of the transmission line equations. The order of the singularity is reduced by 1, and a careful numerical treatment is then employed to derive an equivalent and numerically more appropriate form of coupling equations in which there is no longer a singular term. Finally, finite-difference time-domain (FDTD) solutions of the coupling equations are presented and the theory is applied to calculate lightning-induced voltages on a multiconductor line. The lightning-induced voltages are calculated for the case of lossless/lossy, single-conductor/multiconductor lines and the effect of ground losses and the presence of other conductors on the magnitude and shape of induced voltages are illustrated.

Abstract: In this study, we use a time domain numerical model based on the fully nonlinear extended Boussinesq equations [Wei et al., 1995] to investigate surface wave transformation and breaking-induced nearshore circulation. The energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum equations, with the viscosity strongly localized on the front face of the breaking waves. Wave run-up on the beach is simulated using a moving shoreline technique. We employ quasi fourth-order finite difference schemes to solve the governing equations. Satisfactory agreement is found between the numerical results and the laboratory measurements of Haller et al. [1997], including wave height, mean water level, and longshore and cross-shore velocity components. The model results reveal the temporal and spatial variability of the wave-induced nearshore circulation, and the instability of the rip current in agreement with the physical experiment. Insights into the vorticity associated with the rip current and wave diffraction by underlying vortices are obtained.

Abstract: An equivalent new expression of the triphasic mechano-electrochemical theory [9] is presented and a mixed finite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement us, modified electrochemical/chemical potentials ϵw, ϵ+and ϵ− (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modified Newton–Raphson iterative procedure is employed to handle the non-linear terms. The resulting first-order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One-dimensional (1-D) linear isoparametric element is developed. The final algebraic equations form a non-symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satisfied automatically. The finite element formulation has been used to investigate a 1-D triphasic stress relaxation problem in the confined compression configuration and a 1-D triphasic free swelling problem. The formulation accuracy and convergence for 1-D cases are examined with independent finite difference methods. The FEM results are in excellent agreement with those obtained from the other methods. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract: In this paper, the development of a fourth- (respectively third-) order compact scheme for the approximation of first (respectively second) derivatives on non-uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi- and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed.

Abstract: 1 Introduction: 1.1 Numerical solution process. 2 Computer aided design in magnetics: 2.1 Finite element based CAD systems 2.2 Design strategies. 3 Electromagnetic fields: 3.1 Quasi stationary fields 3.2 Boundary value problem 3.3 Field equations in partial differential form. 4 Potentials and formulations: 4.1 Magnetic vector potential 4.2 Electric vector potential for conducting current 4.3 Electro-static scalar potential 4.4 Magnetic scalar potential 4.5 A? -formulation 4.6 AV-formulation 4.7 In-plane formulation 4.8 AV-formulation with v?B motion term 4.9 Gauge conditions 4.10 Subsequent treatment of the Maxwell equations. 5 Field computation and numerical techniques: 5.1 Magnetic equivalent circuit 5.2 Point mirroring method 5.3 The numerical solution of partial differential equations 5.4 Finite difference method 5.5 Finite element method 5.6 Material modelling 5.7 Numerical implementation of the FEM 5.8 Adaptive refinement for 2D triangular meshes 5.9 Coupling of field and circuit equations 5.10 Post-processing. 6 Coupled field problems: 6.1 Coupled fields 6.2 Strong and weak coupling 6.3 Coupled problems 6.4 Classification of coupled field problems. 7 Numerical optimisation: 7.1 Electromagnetic optimisation problems 7.2 Optimisation problem definition 7.3 Methods. 8 Linear system equation solvers: 8.1 Methods 8.2 Computational costs. 9 Modelling of electrostatic and magnetic devices: 9.1 Modelling with respect to the time 9.2 Geometry modelling 9.3 Boundary conditions 9.4 Transformations. 10 Examples of computed models: 10.1 Electromagnetic and electrostatic devices 10.2 Coupled thermo-electromagnetic problems 10.3 Numerical optimisation

Abstract: A finite difference scheme using a modified marker-and-cell (MAC) method is applied to investigate the characteristics of non-linear wave motions and their interactions with a stationary three-dimensional body inside a numerical wave tank (NWT). The Navier-Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co-ordinate system. The viscous stresses and surface tension are neglected in the dynamic free-surface condition, and the fully non-linear kinematic free-surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS-MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential-based fully non-linear NWT, and the second-order diffraction computation

Abstract: Finite‐difference (FD) modeling of complicated structures requires simple algorithms. This paper presents a new elastic FD method for spatially irregular grids that is simple and, at the same time, saves considerable memory and computing time. Features like faults, low‐velocity layers, cavities, and/or nonplanar surfaces are treated on a fine grid, while the remaining parts of the model are, with equal accuracy, represented on a coarse grid. No interpolation is needed between the fine and coarse parts due to the rectangular grid cells. Relatively abrupt transitions between the small and large grid steps produce no numerical artifacts in the present method. Planar or nonplanar free surfaces, including underground cavities, are treated in a way similar to internal grid points but with consideration of the zero‐valued elastic parameters and density outside the free surface (vacuum formalism). A theoretical proof that vacuum formalism fullfills the free‐surface conditions is given. Numerical validation is per...

Abstract: We report on recent advances in the modeling of magnetic losses in steel laminations used in electromagnetic devices. The integrated-lamination moving dynamic Preisach model, used to evaluate the dynamic magnetization loops under distorted unidirectional flux patterns, is described. The main goal is the comparison of two numerical procedures, using the finite element-finite difference technique and the finite element-fixed point technique, respectively, each properly taking into account the hysteresis characteristics by the Preisach theory. Moreover, attention is paid to the identification of the material parameters entering the moving dynamic Preisach model. Finally, the two techniques are validated by the comparison of numerical experiments and measurements on two different materials. Here, global as well as local quantities in the lamination structure are evaluated.

Abstract: Summary I have undertaken 3-D finite difference (FD) modelling of seismic scattering fromfree-surface topography. Exact free-surface boundary conditions for arbitrary 3-D topographies have been derived for the particle velocities. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank–Nicholson method in time. I simulate scattering from teleseismic P waves by using plane incident wave fronts and real topography from a 60 × 60 km area that includes the NORESS array of seismic receiver stations in southeastern Norway. Synthetic snapshots and seismograms of the wavefield show clear conversion from P to Rg (short-period fundamental-mode Rayleigh) waves in areas of rough topography, which is consistent with numerous observations. By parallelization on fast supercomputers, it is possible to model higher frequencies and/or larger areas than before.

Abstract: Heat transport at the microscale is of vital importance in microtechnology applications. In this study, we develop a finite difference scheme of the Crank-Nicholson type by introducing an intermediate function for the heat transport equation at the microscale. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 697–708, 1999

Abstract: Electrical capacitance tomography (ECT) with circular sensors has previously been investigated. For some industrial applications such as circulating fluidised beds, square sensors are required. Research into this specific area has been carried out for the first time. To generate sensitivity maps, the Laplace equation is solved using a finite difference method. Both the linear back-projection algorithm and an iterative algorithm have been implemented for image reconstruction. Experimental results are promising.

Abstract: Multidimensional positive definite advection transport algorithm (MPDATA) is an iterative process for approximating the advection equation, which uses a donor cell approximation to compensate for the truncation error of the originally specified donor cell scheme. This step may be repeated an arbitrary number of times, leading to successively more accurate solutions to the advection equation. In this paper, we show how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes. The analysis is first done in one dimension to illustrate the method and then is repeated in two dimensions. The existence of cross terms in the truncation analysis of the two-dimensional equations introduces an extra complication into the calculation. We discuss the implementation of our new antidiffusive velocities and provide some examples of applications, including a third-order accurate scheme.

Abstract: This paper is concerned with numerical solutions of a general class of coupled nonlinear parabolic equations by the finite difference method. Three monotone iteration processes for the finite difference system are presented, and the sequences of iterations are shown to converge monotonically to a unique solution of the system, including an existence-uniqueness-comparison theorem. A theoretical comparison result for the various monotone sequences and an error analysis of the three monotone iterative schemes are given. Also given is the convergence of the finite difference solution to the continuous solution of the parabolic boundary-value problem. An application to a reaction-diffusion model in chemical engineering and combustion theory is given.

Abstract: An analysis for compressible fluid spiral groove thrust bearings (SGTBs) and face seals (SGFSs) is presented. Zeroth- and first-order equations rendering the static and dynamic performance of SGFSs, respectively, are solved using the finite element method with a successive approximation scheme. Comparison of the present isothermal compressible fluid model for static and dynamic SGTB and SGFS performance validates previous narrow groove theory, finite difference, and finite element analyses. A discussion follows to indicate the importance of using a small number of grooves to prevent instabilities from negative damping in SGTBs and SGFSs when pressurization is lost. Force coefficients are shown to reach asymptotic limits as the axial excitation frequency increases.

Abstract: The finite difference time domain method is used to calculate the specific absorption rate (SAR) due to a butterfly surface coil in a realistic tissue model of the leg. The resulting temperature distribution and temperature changes are found using a finite difference solution to the bioheat transfer equation. Reasonable agreement is found between predicted temperature changes and those measured in vivo provided that the resulting hyperthermia does not induce noticeable changes in perfusion. The method is applicable to radiofrequency dosimetry problems associated with high B o field magnetic resonance systems and where knowledge of spatial variation in SAR is important in assessing the safety of new magnetic resonance procedures.