Showing papers on "Finite difference method published in 1999"

A new FDTD algorithm based on alternating-direction implicit method

Abstract: In this paper, a new finite-difference time-domain (FDTD) algorithm is proposed in order to eliminate the Courant-Friedrich-Levy (CFL) condition restraint. The new algorithm is based on an alternating-direction implicit method. It is shown that the new algorithm is quite stable both analytically and numerically even when the CFL condition is not satisfied. Therefore, if the minimum cell size in the computational domain is required to be much smaller than the wavelength, this new algorithm is more efficient than conventional FDTD schemes in terms of computer resources such as central-processing-unit time. Numerical formulations are presented and simulation results are compared to those using the conventional FDTD method.

High-Order-Accurate Methods for Complex Unsteady Subsonic Flows

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

A finite-difference time-domain method without the Courant stability conditions

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.

High-density integrated optics

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.

Fluid Flow Phenomena: A Numerical Toolkit

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.

A semi-implicit finite difference method for non-hydrostatic, free-surface flows

Abstract: In this paper a semi-implicit finite difference model for non-hydrostatic, free-surface flows is analyzed and discussed. It is shown that the present algorithm is generally more accurate than recently developed models for quasi-hydrostatic flows. The governing equations are the free-surface Navier–Stokes equations defined on a general, irregular domain of arbitrary scale. The momentum equations, the incompressibility condition and the equation for the free-surface are integrated by a semi-implicit algorithm in such a fashion that the resulting numerical solution is mass conservative and unconditionally stable with respect to the gravity wave speed, wind stress, vertical viscosity and bottom friction. Copyright © 1999 John Wiley & Sons, Ltd.

Computational Partial Differential Equations: Numerical Methods and Diffpack Programming

Abstract: From the Publisher: The target audience of this book is students and researchers in computational sciences who need to develop computer codes for solving partial differential equations. The exposition is focused on numerics and software related to mathematical models in solid and fluid mechanics. The book teaches finite element methods and basic finite difference methods from a computational point of view. The main emphasis regards development of flexible computer programs, using the numerical library Diffpack. The application of Diffpack is explained in detail for problems including model equations in applied mathematics, heat transfer, elasticity, and viscous fluid flow. Diffpack is a modern software development environment based on C++ and object-oriented programming.

Preconditioning for the Steady-State Navier--Stokes Equations with Low Viscosity

Abstract: We introduce a preconditioner for the linearized Navier--Stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single eigenvalue equal to 1, so that performance is independent of both viscosity and mesh size. For other boundary conditions, we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon. We also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration, with virtually no degradation of performance.

Numerical Methods in Electromagnetism

Abstract: 1. Basic Principles of Electromagnetic Fields 2. Overview of Computational Methods in Electromagnetics 3. The Finite Difference Method 4. Variational and Galerkin Methods 5. Shape Functions 6. The Finite Element Method 7. Integral Equations 8. Open Boundary Problems 9. High- Frequency Problems with Finite Elements 10. Low-Frequency Applications 11. Solution of Equations A. Vector Operators B. Triangle Area in Terms of Vertex Coordinates C. Fourier Transform Mehtod D. Integrals of Area Coordinates E. Integrals of Voluume Coordinates F. Gauss-Legendre Quadrature Formulae, Abscissae and, Weight Coefficients G. Shape Functions for 1D Finite Elements H. Shape Functions for 2D Finite Elements I. Shape Functions for 3D Finite Elements References Index

The method of analytical regularization in wave-scattering and eigenvalue problems: foundations and review of solutions

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.

Transient analysis of multiconductor lines above a lossy ground

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.

Boussinesq modeling of a rip current system

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.

A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues

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.

Positivity-Preserving Numerical Schemes for Lubrication-Type Equations

Abstract: Lubrication equations are fourth order degenerate diffusion equations of the form $h_t + abla \cdot (f(h) abla \Delta h) = 0$, describing thin films or liquid layers driven by surface tension. Recent studies of singularities in which $h\to 0$ at a point, describing rupture of the fluid layer, show that such equations exhibit complex dynamics which can be difficult to simulate accurately. In particular, one must ensure that the numerical approximation of the interface does not show a false premature rupture. Generic finite difference schemes have the potential to manifest such instabilities especially when underresolved. We present new numerical methods, in one and two space dimensions, that preserve positivity of the solution, regardless of the spatial resolution, whenever the PDE has such a property. We also show that the schemes can preserve positivity even when the PDE itself is only known to be nonnegativity preserving. We prove that positivity-preserving finite difference schemes have unique positive solutions at all times. We prove stability and convergence of both positivity-preserving and generic methods, in one and two space dimensions, to positive solutions of the PDE, showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes. We generalize the positivity-preserving property to a finite element framework and show, via concrete examples, how this leads to the design of other positivity-preserving schemes.

Sound generation by shock–vortex interactions

Abstract: Two-dimensional, unsteady, compressible flow fields produced by the interactions between a single vortex or a pair of vortices and a shock wave are simulated numerically. The Navier–Stokes equations are solved by a finite difference method. The sixth-order-accurate compact Pade scheme is used for spatial derivatives, together with the fourth-order-accurate Runge–Kutta scheme for time integration. The detailed mechanics of the flow fields at an early stage of the interactions and the basic nature of the near-field sound generated by the interactions are studied. The results for both a single vortex and a pair of vortices suggest that the generation and the nature of sounds are closely related to the generation of reflected shock waves. The flow field differs significantly when the pair of vortices moves in the same direction as the shock wave than when opposite to it.

Solving Hyperbolic PDEs Using Interpolating Wavelets

Abstract: A method is presented for adaptively solving hyperbolic PDEs. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. The adaptability is performed automatically by thresholding the wavelet coefficients. Operations such as differentiation and multiplication are fast and simple due to the one-to-one correspondence between point values and wavelet coefficients in the interpolating basis. Treatment of boundary conditions is simplified in this sparse point representation (SPR). Numerical examples are presented for one- and two-dimensional problems. It is found that the proposed method outperforms a finite difference method on a uniform grid for certain problems in terms of flops.

Compact finite difference schemes on non-uniform meshes. Application to direct numerical simulations of compressible flows

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.

Numerical modelling and design of electrical machines and devices

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

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

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

Numerical methods for stochastic parabolic PDEs

Abstract: We develop a convergence theory for finite difference approximations of reaction diffusion equations forced by an additive space–time white noise. Special care is taken to develop the estimates in terms of non-smooth initial data and over a long time interval, motivated by an abstract approximation theory of ergodic properties developed by Shardlow& Stuart.

Direct numerical simulations of three-dimensional bubbly flows

Abstract: Direct numerical simulations of the motion of many buoyant bubbles are presented. The Navier–Stokes equation is solved by a front tracking/finite difference method that allows a fully deformable interface. The evolution of 91 nearly spherical bubbles at a void fraction of 6% is followed as the bubbles rise over 100 bubble diameters. While the individual bubble velocities fluctuate, the average motion reaches a statistical steady state with a rise Reynolds number of about 25.