Amplification Matrices Research Articles

On designing and developing single‐step second‐order implicit methods with dissipation control and zero‐order overshoots via subsidiary variables

AbstractHilber and Hughes gave several competitive demands on the implicit methods in 1978. However, there are no such integration methods in the traditional algorithm design. Via imposing subsidiary variables, this article proposes two novel implicit methods to splendidly achieve these competitive demands without increasing computational costs. The two novel methods are self‐starting, unconditionally stable, single‐solve, identically second‐order accurate, controllably dissipative, and non‐overshooting. The novel methods achieve the maximal dissipation of auxiliary variables in the high‐frequency limit, although such the design is not a must. The first novel method uses auxiliary velocity and acceleration variables, whereas the second one employs two auxiliary acceleration variables. After embedding desired numerical characteristics, the novel methods employ four eigenvalues of amplification matrices in the high‐frequency limit as user‐specified parameters to flexibly control numerical high‐frequency dissipation. To reduce the number of use‐specified parameters, this article gives some recommended sub‐families of algorithms, such as optimal low‐frequency dissipation schemes. The article further refines the error analysis to analytically calculate amplitude and phase errors for some implicit methods. The error analysis technique can be applied to almost all direct time integration methods. Comparisons of amplitude and phase errors highlight the superiority of the novel methods over the published algorithms. In addition, some classical measures, such as numerical damping ratios and relative period errors, are analyzed and compared. Numerical examples are solved to show the novel methods' superiority.

Linear and brute force stability of orthogonal moment multiple-relaxation-time lattice Boltzmann methods applied to homogeneous isotropic turbulence.

Multiple-relaxation-time (MRT) lattice Boltzmann methods (LBM) based on orthogonal moments exhibit lattice Mach number dependent instabilities in diffusive scaling. The present work renders an explicit formulation of stability sets for orthogonal moment MRT LBM. The stability sets are defined via the spectral radius of linearized amplification matrices of the MRT collision operator with variable relaxation frequencies. Numerical investigations are carried out for the three-dimensional Taylor-Green vortex benchmark at Reynolds number 1600. Extensive brute force computations of specific relaxation frequency ranges for the full test case are opposed to the von Neumann stability set prediction. Based on that, we prove numerically that a scan over the full wave space, including scaled mean flow variations, is required to draw conclusions on the overall stability of LBM in turbulent flow simulations. Furthermore, the von Neumann results show that a grid dependence is hardly possible to include in the notion of linear stability for LBM. Lastly, via brute force stability investigations based on empirical data from a total number of 22 696 simulations, the existence of a deterministic influence of the grid resolution is deduced. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Highly precise time integration method for linear structural dynamic analysis

SummaryTo achieve high accuracy, conventional time integration methods require small time steps, especially when applied to large‐scale finite element systems. The use of small time steps calls for a large number of computations. In this paper, a strategy is proposed to construct an efficient and highly precise time integration method for linear time‐invariant systems. Adopting the multi‐substep notion, the highly precise time integration method creates an integrated amplification matrix prior to recursive calculations by multiplying the amplification matrices of N equal substeps, where a 2m algorithm and the method of storing incremental matrix are respectively employed to reduce computational cost and rounding error. On the basis of the Newmark method, two highly precise symplectic schemes and one highly precise dissipative scheme are developed. In addition, the consistent stability of the used Newmark schemes is presented. The free vibration of a three‐dimensional truss, the impact of a particle against a rod, and the forced vibrations of a rectangular thin plate and a stiff system are investigated to validate the performance of the proposed method.

Von Neumann stability analysis of first-order accurate discretization schemes for one-dimensional (1D) and two-dimensional (2D) fluid flow equations

In literature, the von Neumann stability analysis of simplified model equations, such as the wave equation, is typically used to determine stability conditions for the non-linear partial differential fluid flow equations (Navier–Stokes and Euler). However, practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations. The goal of this paper is to determine stability conditions for the full system of fluid flow equations – the Euler equations are examined, as any conditions derived for the Euler equations will apply to the Navier–Stokes (NS) equations in the limit of convection-dominated flows. A von Neumann stability analysis is conducted for the one-dimensional (1D) and two-dimensional (2D) Euler equations. The system of equations is discretized on a staggered grid using finite-difference discretization techniques; the use of a staggered grid allows equivalence to finite-volume discretization. By combining the different discretization techniques, ten solution schemes are formulated – eight solution schemes are considered for the 1D Euler equations, and two schemes for the 2D Euler equations. For each scheme, error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum-allowable CFL (Courant–Friedrichs–Lewy) number as a function of Mach number. The predictions are verified for selected schemes using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, which offer a viable alternative to complicated mathematical expressions often reported in published literature, and should benefit everyday CFD (Computational Fluid Dynamics) users. The stability regions are used to discuss the effect of time integration (explicit vs. implicit), density bias in continuity equation and momentum convection term linearization on stability. A comparison of the predicted stability limits for 1D and 2D Euler equations with commonly-used stability conditions arising from the wave equation shows that the stability thresholds for the Euler equations lie well below those predicted by the wave equation analysis; in addition, the 2D Euler stability limits are more restrictive as compared to 1D Euler limits. Since the present analysis accounts for the full system of fluid flow (Euler) equations, the derived stability conditions can be used by CFD practitioners to estimate a timestep or CFL number to guide the stability of their computations.

Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

This paper deals with a generalized multi time step integration used for structural dynamic analysis. The proposed method presents three kinds of implicit schemes in which the accelerations and the velocities of the previous steps are utilized to integrate the equations of motion. This procedure employs three groups of weighted factors calculated by minimizing the numerical errors of displacement and velocity in Taylor series expansion. Moreover, a comprehensive study on mathematical stability of the proposed technique which is performed based on the amplification matrices proves that the new method is more stable than existing schemes such as IHOA. For numerical verification, wide range of dynamic systems including linear and nonlinear, single and multi degrees of freedom, damped and undamped, forced and free vibrations from finite element and finite difference methods are analyzed. These numerical studies demonstrate that efficiency and accuracy of the proposed method are higher than other techniques.

Unconditionally Stable Divergence-Free Vector-Based Meshless Method for Transient Electromagnetic Analysis

A vector-based meshless method or vector meshless method, which is proven to be divergence-free, was recently proposed for transient electromagnetic analysis. In this method, an explicit finite-difference scheme is used for approximating the time derivatives in Maxwell’s equations. Therefore, the stability criterion constraints the maximum time-step. This paper presents the alternating-direction implicit (ADI) formulation of the vector meshless method based on a different form of the ADI algorithm named the leapfrog ADI scheme. The computational cost of the leapfrog ADI scheme is less than the conventional ADI algorithm and algebraically equivalent. In the proposed vector meshless method, Courant–Friedrichs–Lewy limit does not constrain the time-step due to its implicit formulation. Moreover, the stability condition of the conventional vector meshless method and the proposed ADI vector meshless method are investigated through their amplification matrices in this paper. The results show that the proposed method has two features: unconditional stability and divergence-free property. The first one is obtained due to the incorporation of the leapfrog ADI algorithm, and the second one is achieved by the use of vector basis functions. The divergence-free property of the proposed method can lead to obtaining more accurate solutions than the conventional meshless method. In addition, some numerical examples are presented to verify the properties and the efficiency of the proposed method.

On the stability analysis of the PISO algorithm on collocated grids

Abstract A Fourier stability analysis is performed on the standard PISO algorithm for collocated grids. The amplification matrices of the sequence are obtained for one-dimensional and two-dimensional incompressible flow problems and the error amplification is studied for different numerical conditions (i.e. mesh-Reynolds number, Courant number, number of PISO corrections). Stability features are also analyzed for steady state conditions and the performance is compared against the SIMPLE-family algorithms. The effect of the mesh orientation and aspect ratio on two-dimensional flow problems are also investigated. It is found that a grid with a higher refinement in one direction can produce an unstable behavior for small mesh-Reynolds numbers, while stability is ensured for the same conditions with a uniformly refined grid in both directions. The first situation can be corrected by enforcing the coupling between pressure and velocity through more PISO corrections.

Von Neumann Stability Analysis of a Segregated Pressure-Based Solution Scheme for One-Dimensional and Two-Dimensional Flow Equations

The goal of this paper is to derive the von Neumann stability conditions for the pressure-based solution scheme, semi-implicit method for pressure-linked equations (SIMPLE). The SIMPLE scheme lies at the heart of a class of computational fluid dynamics (CFD) algorithms built into several commercial and open-source CFD software packages. To the best of the authors' knowledge, no readily usable stability guidelines appear to be available for this popularly employed scheme. The Euler equations are examined, as the inclusion of viscosity in the Navier–Stokes (NS) equation serves to only soften the stability limits. First, the one-dimensional (1D) Euler equations are studied, and their stability properties are delineated. Next, a rigorous stability analysis is carried out for the two-dimensional (2D) Euler equations; the analysis of the 2D equations is considerably more challenging as compared to analysis of the 1D form of equations. The Euler equations are discretized using finite differences on a staggered grid, which is used to achieve equivalence to finite-volume discretization. Error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum allowable Courant–Friedrichs–Lewy (CFL) number as a function of Mach number. The predictions are verified using the Riemann problem, and very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, as compared to complicated mathematical expressions often reported in published literature. Since our analysis accounts for the solution scheme along with the full system of flow equations, the conditions reported in this paper offer practical value over the conditions that arise from analysis of simplified 1D model equations.

Analysis of velocity-components decoupled projection method for the incompressible Navier–Stokes equations

We study the temporal accuracy and stability of the velocity-components decoupled projection method (VDPM) for fully discrete incompressible Navier–Stokes equations. In particular, we investigate the effect of three formulations of the nonlinear convection term, which include the advective, skew-symmetric, and divergence forms, on the temporal accuracy and stability. Second-order temporal accuracy of the VDPM for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l2-norm. Considering the energy evolution, we demonstrate that the VDPM is stable when the time step is less than or equal to a constant. Stability diagrams, which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices, are obtained using von Neumann analysis. These diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term. Numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability, and the effects of the formulations of the nonlinear convection term are analyzed. Overall, our results indicate that the VDPM along with an advective discrete convection operator is almost unconditionally stable, second-order accurate in time, and computationally efficient because of the non-iterative solution procedure in solving the decoupled momentum equations.

Fourier analysis of the SIMPLE serials

ABSTRACTIn this article, error amplification matrices have been developed for SIMPLE and five revised SIMPLE variants by Fourier decomposition. Comparisons among six algorithms are made to check the effect of pressure–velocity coupling enhancements with respect to SIMPLE in terms of error damping. The Fourier analysis indicates that the SIMPLE variants developed on SIMPLER show better performance on high-frequency error damping, and can be used as better multigrid smoothers. The analysis also demonstrates that some pressure–velocity coupling enhancements can accelerate overall damping rate and promote convergence without weakening the convergence conditions.

New implicit higher order time integration for dynamic analysis

In this paper new implicit time integration called N-IHOA is presented for dynamic analysis of high damping systems. Here, current displacement and velocity are assumed to be functions of the velocities and accelerations of several previous time steps, respectively. This definition causes that only one set of weighted factors is calculated from the Taylor series expansion which leads to a simple approach and reduce the computational efforts. Moreover a comprehensive study on stability of the proposed method i.e., NIHOA compared with IHOA integration which is performed based on amplification matrices proves the ability of the N-IHOA in high damping vibrations such as control systems. Also, wide range of numerical examples which contains single/multi degrees of freedom, damped/un-damped, free/forced vibrations from finite element/finite difference demonstrate that the accuracy and efficiency of the proposed time integration is more than the common approaches such as the IHOA, the Wilson-O and the Newmark-B.

The value of numerical amplification matrices in time integration methods

Several frequently overlooked concepts in linear elastodynamics are reviewed. First that the amplification matrix may be obtained numerically and if obtained this way, it may give some extra information on the programmed algorithm. Second that the adequate dimension of that matrix depends on the algorithm at hand. Such dimension equals the number of independent initial conditions that must be prescribed. Third that those initial conditions should be consistent with the problem at hand and with the algorithm. Overshooting phenomena present in some time integration algorithms may be a consequence of overlooking such issues.

Fourier Analysis of the Effect of Momentum Interpolation on the SIMPLE Series Formulated on a Collocated Grid

The error amplification matrices of three SIMPLE series formulated on a staggered grid and a collocated grid using three momentum interpolation methods are derived. The effects of grid layout and momentum interpolation methods on convergence properties of SIMPLE series are checked. The results indicate that the convergence conditions are available by damping out the highest and lowest errors. The SIMPLE series show almost identical convergence rates under a wide range of useful velocity relaxation factors, regardless of grid layout and interpolation method. However, the high-frequency reduction shows significant variation with grid layout and interpolation method, which maybe referred to in multigrid application.

Linear coherent distributed estimation with cluster-based sensor networks

The authors consider distributed estimation using sensor network with coherent multiple access channel model and LMMSE fusion rule. The sensors in the network are divided into a number of clusters. Sensors within the same cluster are allowed to collaborate through an amplification matrix to form a message this then transmitted. They formulate the problem of choosing the amplification matrices as an optimal power allocation problem under a total power constraint. The solution gives the optimal amplification matrices as scaled outer products of the observation gain and the channel gain vectors. The authors show that collaboration improves performance and, in simulations, demonstrate that the amount of improvement is closely related to the amount of collaboration.

Tensor-Based Channel Estimation and Iterative Refinements for Two-Way Relaying With Multiple Antennas and Spatial Reuse

Relaying is one of the key technologies to satisfy the demands of future mobile communication systems. In particular, two-way relaying is known to exploit the radio resources in a very efficient manner. In this contribution, we consider two-way relaying with amplify-and-forward (AF) MIMO relays. Since AF relays do not decode the signals, the separation of the data streams has to be performed by the terminals themselves. For this task both nodes require reliable channel knowledge of all relevant channel parameters. Therefore, we examine channel estimation schemes for two-way relaying with AF MIMO relays. We investigate a simple Least Squares (LS) based scheme for the estimation of the compound channels as well as a tensor-based channel estimation (TENCE) scheme which takes advantage of the special structure in the compound channel matrices to further improve the estimation accuracy. Note that TENCE is purely algebraic (i.e., it does not require any iterative procedures) and applicable to arbitrary antenna configurations. Then we demonstrate that the solution obtained by TENCE can be improved by an iterative refinement which is based on the structured least squares (SLS) technique. In this application, between one and four iterations are sufficient and consequently the increase in computational complexity is moderate. The iterative refinement is optional and targeted for cases where the channel estimation accuracy is critical. Moreover, we propose design rules for the training symbols as well as the relay amplification matrices during the training phase to facilitate the estimation procedures. Finally, we evaluate the achievable channel estimation accuracy of the LS-based compound channel estimation scheme as well as the tensor-based approach and its iterative refinement via numerical computer simulations.

STABILITY AND ACCURACY OF LATTICE BOLTZMANN SCHEMES FOR ANISOTROPIC ADVECTION-DIFFUSION EQUATIONS

The stability of the numerical schemes for anisotropic advection-diffusion equations derived from the lattice Boltzmann equation with the D2Q4 particle velocity model is analyzed through eigenvalue analysis of the amplification matrices of the scheme. Accuracy of the schemes is investigated by solving benchmark problems, and the LBM scheme is compared with traditional implicit schemes. Numerical experiments demonstrate that the LBM scheme produces stable numerical solutions close to the analytical solutions when the values of the relaxation parameters in x and y directions are greater than 1.9 and the Courant numbers satisfy the stability condition. Furthermore, the numerical solutions produced by the LBM scheme are more accurate than those of the Crank–Nicolson finite difference scheme for the case where the Courant numbers are set to be values close to the upper bound of the stability region of the scheme.

Examination, Clarification, and Simplification of Stability and Dispersion Analysis for ADI-FDTD and CNSS-FDTD Schemes

We describe a rigorous analysis of unconditional stability for the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) and Crank-Nicholson split step (CNSS-FDTD) schemes avoiding use of the von Neumann spectral criterion. The proof is performed in the spectral domain, and uses skew-Hermitivity of matrix terms in the ADI and CNSS total amplification matrices. A bound for the total ADI amplification matrix is provided. While the CN and CNSS-FDTD amplification matrices are unitary, the bound on the ADI-FDTD depends on the Courant number. Importantly, we have found that the ADI-FDTD amplification matrix is not normal, i.e., the unit spectral radius alone cannot be used to prove the ADI-FDTD unconditional stability. The paper also shows that the ADI-FDTD and CNSS-FDTD schemes share the same dispersion equation by a similarity of their total amplification matrices, and the two schemes have identical numerical dispersion in the frame of plane waves.

NUMERICAL SCHEMES OBTAINED FROM LATTICE BOLTZMANN EQUATIONS FOR ADVECTION DIFFUSION EQUATIONS

Stability and accuracy of the numerical schemes obtained from the lattice Boltzmann equation (LBE) used for numerical solutions of two-dimensional advection-diffusion equations are presented. Three kinds of velocity models are used to determine the moving velocity of particles on a squre lattice. A system of explicit finite difference equations are derived from the LBE based on the Bhatnagar, Gross and Krook (BGK) model for individual velocity model. In order to approximate the advecting velocity field, a linear equilibrium distribution function is used for each of the moving directions. The stability regions of the schemes in the special case of the relaxation parameter ω in the LBE being set to ω=1 are found by analytically solving the eigenvalue problems of the amplification matrices corresponding to each scheme. As for the cases of general relaxation parameters, the eigenvalue problems are solved numerically. A benchmark problem is solved in order to investigate the relationship between the accuracy of the numerical schemes and the order of the Peclet number. The numerical experiments result in indicating that for the scheme based on a 9-velocity model we can find the parameters depending on the order of the given Peclet number, which generate accurate solutions in the stability region.

Some Remarks on the Stability of Time-Domain Electromagnetic Simulations

Aspects of the stability of time-domain update schemes for electromagnetic problems are examined based on the consideration of system amplification matrices arising from the discretization of Maxwell's equations in the sequence space. We discuss time-domain stability for a given discretization scale and under three different assumptions of decreasing generality: general updates, general leapfrog updates, and reciprocal leapfrog updates.

A Fourier analysis of the IPSA/PEA algorithms applied to multiphase flows with mass transfer

The error amplification matrices for two variations of a pressure-based Eulerian–Eulerian multiphase algorithm are developed using the method of Fourier decomposition. The algorithms examined here include Spalding's IPSA/PEA, and a revised form proposed by Siebert and Antal appropriate for flows with mass transfer. The error amplification matrices are developed for a single spatial dimension, but for an arbitrary number of phases or fluids. The ramifications arising from these amplification matrices are explored in this article. For two-phase applications the revised form produces a broader theoretical range of convergent behavior for different interphase momentum and mass transfer rates. For four-phase applications both methods appear to be conditionally stable, and produce similar convergence behaviors. Large differences in interphase mass and momentum transfer rates between phasic pairs appear to adversely affect the algorithm's stability range.