Showing papers on "Finite difference method published in 1995"
Book•
11 Sep 1995
1,541 citations
TL;DR: This work systematically analyze the performance of implicit-explicit IMEX schemes, propose improved new schemes, and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.
Abstract: Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized partial differential equations (PDEs) of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes, and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.For the prototype linear advection-diffusion equation, a stability analysis for first-, second-, third-, and fourth-order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay...
1,155 citations
TL;DR: In this article, a von Neumann stability analysis of the equations of smoothed particle hydrodynamics (SPH) along with a critical discussion of various parts of the algorithm is presented.
600 citations
TL;DR: It is shown that the eigenmodes for electromagnetic waves in an inhomogeneous dielectric medium can be obtained with an algorithm that scales linearly with the size of the system, using discretization of the Maxwell equations in both the spatial and the time domain.
Abstract: We show that the eigenmodes for electromagnetic waves in an inhomogeneous dielectric medium can be obtained with an algorithm that scales linearly with the size of the system. The method employs discretization of the Maxwell equations in both the spatial and the time domain and the integration of the Maxwell equations in the time domain. The spectral intensity can then be obtained by a Fourier transform. We applied the method to a few problems of current interest, including the photonic band structure of a periodic dielectric structure, the effective dielectric constants of some three-dimensional and two-dimensional systems, and the defect states of a periodic dielectric structure with structural defects.
317 citations
TL;DR: In this article, the effects of the viscosity/temperature parameter G r the thermal-diffusion parameter Sr (Soret number) and the diffusion-thermo parameter Df (Dufour number) have been examined on the flow field of a hydrogen-air mixture as a non-chemical reacting fluid pair.
293 citations
TL;DR: In this paper, Harten et al. presented high-resolution finite difference schemes for solving the nonlinear model Boltzmann equations for the computations of rarefied gas flows.
262 citations
TL;DR: In this article, the Navier-Stokes equations were approximated to fourth-order accuracy with stencils extending only over a 3 x 3 square of points, and the key advantage of the new compact 4-order scheme is that it allows direct iteration for low-to-mediwn Reynolds numbers.
Abstract: SUMMARY We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity fonn, can be approximated to fourth--order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low~to-mediwn Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re $1500 point-SOR iteration is used and the convergence is fast.
253 citations
Book•
1 Jan 1995
TL;DR: The current state of simulation of Injection Molding is summarized in this article, where the authors present an overview of the state-of-the-art state of the art.
Abstract: Part 1 The Current State of Simulation: Introduction, Stress and Strain in Fluid Mechanics, Material Properties of Polymers, Governing Equations, Approximations for Injection Molding, Numerical Methods for Solution Part 2 Improving Molding Simulation: Improved Fiber Orientation Modeling, Improved Mechanical Property Modeling, Long Fiber-Filled Materials, Crystallization, Effects of Crystallizations on Rheology and Thermal Properties, Colorant Effects, Prediction of Post-Molding Shrinkage and Warpage, Additional Issues of Injection-Molding Simulation, Epilogue Appendices: History of Injection-Molding Simulation, Tensor Notation, Derivation of Fiber Evolution Equations, Dimensional Analysis of Governing Equations, The Finite Difference Method, The Finite Element Method, Numerical Methods for the 2.5D Approximation, Three-Dimensional FEM for Mold Filling Analysis, Level Set Method, Full Form of Mori-Tanaka Model.
244 citations
TL;DR: In this paper, a particle velocity-stress, finite-difference method is developed for the simulation of wave propagation in 2-D heterogeneous poroelastic media, instead of the prevailing second-order differential equations, they consider a first-order hyperbolic system that is equivalent to Biot's equations.
Abstract: A particle velocity-stress, finite-difference method is developed for the simulation of wave propagation in 2-D heterogeneous poroelastic media. Instead of the prevailing second-order differential equations, we consider a first-order hyperbolic system that is equivalent to Biot's equations. The vector of unknowns in this system consists of the solid and fluid particle velocity components, the solid stress components, and the fluid pressure. A MacCormack finite-difference scheme that is fourth-order accurate in space and second-order accurate in time forms the basis of the numerical solutions for Biot's hyperbolic system. An original analytic solution for a P-wave line source in a uniform poroelastic medium is derived for the purposes of source implementation and algorithm testing. In simulations with a two-layer model, additional «slow» compressional incident, transmitted, and reflected phases are recorded when the damping coefficient is small. This «slow» compressional wave is highly attenuated in porous media saturated by a viscous fluid. From the simulation we also verified that the attenuation mechanism introduced in Biot's theory is of secondary importance for «fast» compressional and rotational waves. The existence of seismically observable differences caused by the presence of pores has been examined through synthetic experiments that indicate that amplitude variation with offset may be observed on receivers and could be diagnostic of the matrix and fluid parameters. This method was applied in simulating seismic wave propagation over an expanded steam-heated zone in Cold Lake, alberta in an area of enhanced oil recovery (EOR) processing. The results indicate that a seismic surface survey can be used to monitor thermal fronts
231 citations
TL;DR: In this paper, the authors investigated the three-dimensional development of controlled transition in a flat-plate boundary layer using direct numerical simulation (DNS) using the complete Navier-Stokes equations.
Abstract: The three-dimensional development of controlled transition in a flat-plate boundary layer is investigated by direct numerical simulation (DNS) using the complete Navier-Stokes equations. The numerical investigations are based on the so-called spatial model, thus allowing realistic simulations of spatially developing transition phenomena as observed in laboratory experiments. For solving the Navier-Stokes equations, an efficient and accurate numerical method was developed employing fourth-order finite differences in the downstream and wall-normal directions and treating the spanwise direction pseudo-spectrally. The present paper focuses on direct simulations of the wind-tunnel experiments by Kachanov et al. (1984, 1985) of fundamental breakdown in controlled transition. The numerical results agreed very well with the experimental measurements up to the second spike stage, in spite of relatively coarse spanwise resolution. Detailed analysis of the numerical data allowed identification of the essential breakdown mechanisms. In particular, from our numerical data, we could identify the dominant shear layers and vortical structures that are associated with this breakdown process.
229 citations
TL;DR: In this article, the behavior of a Godunov-projection method for the incompressible Navier-Stokes equations as a function of the resolution of the computational mesh is presented.
TL;DR: In this article, the analytical partial derivatives of the plane wave EXAFS function can be calcalated using cubic spline, which decreases the CPU time needed for a refinement by over 60% for a three shell system compared to a refinement with partial derivaties calculated with the finite difference method.
Abstract: Parameterization of phase and backscattering amplitude with cubic splines is described. Using the cubic spline, the analytical partial derivatives of the plane wave EXAFS function can be calcalated. The use of analytical partial derivatives decreases the CPU time needed for a refinement by over 60% for a three shell system compared to a refinement with partial derivaties calculated with the finite difference method.
TL;DR: In this article, a two-dimensional airfoil with either a bilinear or cubic structural nonlinearity in pitch, and subject to incompressible flow has been analyzed using Wagner's function.
Book•
9 Nov 1995
TL;DR: In this article, the authors present a FORTRANet program for simulating groundwater flow and quality in a Porous Media environment, using finite difference methods and finite element methods for solving hydrodynamic dispersion equations.
Abstract: Contents: Introduction.- Hydrodynamic Dispersion in Porous Media.- Analytical Solutions of Hydrodynamic Dispersion Equations.- Finite Difference Methods and the Method of Characteristics for Hydrodynamic Dispersion Equations.- Finite Element Methods for Solving Hydrodynamic Dispersion Equations.- Numerical Solutions of Advection-Dominated Problems.- Mathematical Models of Groundwater Quality.- Applications of Groundwater Quality Models.- Conclusions.- Appendix A: The Related Parameters in the Modeling of Mass Transport in Porous Media.- Appendix B: A FORTRAN Program for Simultaneously Simulating Groundwater Flow and Quality.- References.
1 May 1995
TL;DR: A new finite element method in the time domain based on the Whitney forms gives a direct way to solve Maxwell equations in general unstructured meshes.
Abstract: A new finite element method in the time domain based on the Whitney forms is presented. Using edge elements and face elements for space discretization of the fields and a leap-frog scheme in time, the algorithm gives a direct way to solve Maxwell equations in general unstructured meshes. >
TL;DR: In this article, the authors considered a Cauchy problem for the heat equation in the quarter plane, where data are given at x = 1 and a solution is sought in the interval 0
Abstract: We consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x=1 and a solution is sought in the interval 0
TL;DR: In this paper, the authors correct some misunderstandings that have appeared in the literature concerning the order of accuracy of the QUICK scheme for steady-state convective modeling and other related convection-diffusion schemes are also considered.
TL;DR: In this article, a conservative difference scheme is presented for the initial-boundary value problem for generalized Zakharov equations, which can be implicit or semiexplicit depending on the choice of a parameter.
Abstract: . A conservative difference scheme is presented for the initial-boundary value problem for generalized Zakharov equations. The scheme canbe implicit or semiexplicit depending on the choice of a parameter. On thebasis of a priori estimates and an inequality about norms, convergence of thedifference solution is proved in order 0(h2 + t2) , which is better than previous results. IntroductionThe Zakharov equations [20](1.1) iEt + Exx-NE = 0,(1.2) ^Ntt-{N+\E\2)xx = 0describe the propagation of Langmuir waves in plasmas. Here the complexunknown function E is the slowly varying envelope of the highly oscillatoryelectric field, and the unknown real function N denotes the fluctuation of the ion density about its equilibrium value.The global existence of a weak solution for the Zakharov equations in one dimension is proved in [19], and existence and uniqueness of a smooth solutionfor the equations are obtained provided smooth initial data are prescribed.Numerical methods for the Zakharov equations are studied only in [5, 9, 10,
TL;DR: In this article, the 3D finite difference method is used to simulate borehole wave propagations in an isotropic as well as an anisotropic formation, and the finite difference results agree excellently with the analytic solutions of a point force source in the transversely isropic medium.
Abstract: In this paper the three‐dimensional finite difference method is used to simulate borehole wave propagations in an isotropic as well as an anisotropic formation. The finite difference results agree excellently with the analytic solutions of a point force source in the transversely isotropic medium. The finite difference synthetics are also in very good agreement with the discrete wave‐number solutions for fluid‐filled borehole wave propagation. The finite difference synthetics are compared with ultrasonic lab measurements in a scaled borehole model. The borehole is drilled along the X axis in an orthorhombic phenolite solid. Both monopole and dipole logs agree well. The observations of the shear wave splitting in the dipole logs are confirmed by the finite difference simulations. The 3‐D finite difference method is applied to the fluid‐filled borehole wave propagation in the tilted isotropic formation and in the orthorhombic phenolite formation. In a borehole drilled along the Z axis in a phenolite formati...
TL;DR: The flux-corrected transport (FCT) as discussed by the authors algorithm, adapted from hydrodynamics, reduces the numerical dispersion in finite-difference wavefield continuation by incorporating diffusion into the wave field continuation process.
Abstract: Finite-difference acoustic-wave modeling and reverse-time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well. However, conventional finite-difference methods for solving the acoustic- or elastic-wave equation suffer from numerical dispersion when too few samples per wavelength are used. The flux-corrected transport (FCT) algorithm, adapted from hydrodynamics, reduces the numerical dispersion in finite-difference wavefield continuation. The flux-correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion. Incorporating the flux-correction procedure in conventional finite-difference modeling or reverse-time migration can provide finite-difference solutions with no numerical dispersion even for impulsive sources. The FCT correction, which can be applied to finite-difference approximations of any order in space and time, is an efficient alternative to use for finite-difference approximations of increasing order. Through demonstrations of modeling and migration on both synthetic and field data, we show the benefits of the FCT algorithm, as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse.
TL;DR: In this paper, a numerical simulation program was developed to predict the transient behavior of fiber orientations together with a mold filling simulation for short-fiber-reinforced thermoplastics in arbitrary three-dimensional injection mold cavities.
Abstract: The present study develops a numerical simulation program to predict the transient behavior of fiber orientations together with a mold filling simulation for short-fiber-reinforced thermoplastics in arbitrary three-dimensional injection mold cavities. The Dinh-Armstrong model including an additional stress due to the existence of fibers is incorporated into the Hele-Shaw equation to result in a new pressure equation governing the filling process. The mold filling simulation is performed by solving the new pressure equation and energy equation via a finite element/finite difference method as well as evolution equations for the second-order orientation tensor via the fourth-order Runge-Kutta method. The fiber orientation tensor is determined at every layer of each element across the thickness of molded parts with appropriate tensor transformations for arbitrary three-dimensional cavity space.
Book•
17 May 1995
TL;DR: In this article, the authors present a model for the propagation and migration of a seismic wave field, based on the Kirchhoff migration and the stationary phase of the seismic wavefield.
Abstract: to seismic migration.- Harmonic analysis, delta functions, and all that.- Equations of motion for the earth.- Elastic wave equations.- Ray theory.- Kirchhoff migration.- Kirchhoff migration/inversion.- The method of stationary phase.- Downward continuation of the seismic wavefield.- Plane wave decomposition of seismograms.- Numerical methods for tracing rays.- Finite difference methods for wave propagation and migration.- SU user's manual.- SUB user's guide.
TL;DR: The finite volume method as mentioned in this paper uses the divergence theorem, considers a finite volume around the point and discretizes the surface bounding the volume, and gives an approximate value for the derivative of a field at a given point using the values of the fields at a few locations neighboring the point.
Abstract: Like the finite difference method, the finite volume method gives an approximate value for the derivative of a field at a given point using the values of the field at a few locations neighboring the point. The method uses the divergence theorem, considers a “finite volume” around the point and discretizes the surface bounding the volume. When the finite volumes considered are regular polyhedra, one obtains the expressions corresponding to standard centered finite differences, but the finite volume method is more general than the finite difference method because it may deal directly with irregular grids. It is possible to give a finite volume formulation of the elastodynamic problem, using dual volumes, that correspond, in the regular case, to the staggered grids used in the finite difference method. The scheme thus obtained is more general than the one obtained using finite differences, as the “grids” may be totally unstructured, but at the cost of having, in the general case, only a first-order accuracy. Although the scheme is not consistent, numerical tests suggest that it is stable and convergent. This implementation of a finite volume method does not provide a way for a more general treatment of the boundaries than the conventional finite difference method.
TL;DR: In this article, the authors used a fractional step procedure with high-order spatial discretization to solve the pressure Poisson equation of a flow past a normal flat plate at low Reynolds numbers.
Abstract: Well-resolved two-dimensional numerical simulations of the unsteady separated flow past a normal flat plate at low Reynolds numbers have been performed using a fractional step procedure with high-order spatial discretization. A fifth-order upwind-biased scheme is used for the convective terms and the diffusive terms are represented by a fourth-order central difference scheme. The pressure Poisson equation is solved using a direct method based on eigenvalue decomposition of the coefficient matrix. A systematic study of the flow has been conducted with high temporal and spatial resolutions for a series of Reynolds numbers. The interactions of the vortices shed form the shear layers in the near-and far-wake regions are studied. For Reynolds numbers less than 250 the vortices are observed to convect parallel to the freestream. However, at higher Reynolds numbers (500 and 1000), complex interactions including vortex pairing, tearing and deformations are seen to occur in the far-wake region. Values of the drag coefficient and the wake closure length are presented and compared with previous experimental and numerical studies.
TL;DR: A new multilevel methodology is developed in this study to provide a successful numerical simulation for the whole process of flow transition in 3D flat plate boundary layers, including linear growth, secondary instability, breakdown, and transition on a relatively coarse grid with low CPU cost.
TL;DR: The first-order flux difference splitting scheme and its second-order extensions are investigated for their applicability to dam-break problems and a general entropy-satisfying formula is incorporated, which significantly improves the applicability of the Roe scheme.
Abstract: The first-order flux difference splitting scheme and its second-order extensions are investigated for their applicability to dam-break problems. Roe's first-order explicit scheme is first formulate...
TL;DR: A variationally consistent approach for elastoplastic finite element analysis is presented in this article, where a weak formulation of the finite-step boundary value problem is obtained by enforcing the stationarity of a mixed functional of Hu-Washizu type.
TL;DR: In this paper, the flow of a sediment layer that forms on an inclined plate as a consequence of the steady sedimentation of spherical particles was investigated theoretically as well as experimentally.
Abstract: The flow of a sediment layer that forms on an inclined plate as a consequence of the steady sedimentation of spherical particles was investigated theoretically as well as experimentally. The theoretical analysis was based on the model proposed by Nir & Acrivos (1990), modified to include shear-induced diffusion due to gradients in the shear stress as well as a slip velocity along the wall due to the finite size of the particles. The resulting set of partial differential equations, which is amenable to a similarity-type solution both near the leading edge as well as far downstream, was solved numerically using a finite difference scheme thereby yielding theoretical predictions for the particle concentration and velocity profiles, plus the local sediment layer thickness, all along the plate. In addition, a new experimental technique based on laser Doppler anemometry was developed and was used to measure the particle velocity profiles in the highly concentrated sediment layer as well as the corresponding slip coefficient which relates the slip velocity to the velocity gradient adjacent to a wall. The thickness profile of the sediment layer was also measured experimentally by means of video imaging. It was found that the experimental results thus obtained for the particle velocity profile and for the local sediment layer thickness were in very good agreement with the corresponding theoretical predictions especially considering that the latter did not make use of any adjustable parameters.
TL;DR: The parallel electromagnetic simulation program TEMPEST has been generalized to analyze three-dimensional problems in photolithography as discussed by the authors, which combines together techniques for rigorous simulation of electromagnetic scattering and diffraction, a novel and efficient absorbing boundary condition, and synthesis of partially coherent images including optical system aberrations.
Abstract: The parallel electromagnetic simulation program TEMPEST has been generalized to analyze three-dimensional problems in photolithography. TEMPEST, which has been made available on the National Center for Supercomputing Applications, combines together techniques for rigorous simulation of electromagnetic scattering and diffraction, a novel and efficient absorbing boundary condition, and synthesis of partially coherent images including the effects of optical system aberrations. The electromagnetic solution is based on the time-domain finite difference method, but exploits the power of parallel computer architectures. Equations suitable for parallel implementation are given. Simulation time is fifteen to twenty minutes with 64 M simulation nodes on a CM-5 with 512 nodes and 16 GBytes of memory. The usefulness and effectiveness of the program for photolithographic applications are demonstrated by considering problems in projection printing of polarization and transmission effects in contact holes, and reflective notching which causes undesired exposure in photoresists. >
TL;DR: By invoking the slow-wave approximation, the wave equation resumes the form of the Fresnel equation as mentioned in this paper, and codes developed previously for the paraxial beam propagation can be extended to simulate the backward reflection and diffraction at any angle.
Abstract: By invoking the slow-wave approximation, the wave equation resumes the form of the Fresnel equation. Codes developed previously for the paraxial beam propagation can be extended to simulate the backward reflection and diffraction at any angle. Results of planar waveguide gratings and a beveled corner bend are presented. >