Multistep Discretizations Research Articles

A quick multi-step discretization and parallelization wear simulation model for crown gear coupling with misalignment angle

Compared with other gear transmissions travelling along meshing line, the motion of crown gear coupling is much more complex, which makes more difficult for its tooth surface wear simulation. To reduce the computational costs associated with the wear simulation of crown gear coupling with misalignment angle, a quick multi-step discretization and parallelization wear simulation model is proposed. The model combines the finite element method based on self-compiled high-precision finite element mesh, kinematic analysis and Archard's formula to predict the tooth surface wear. The characteristics of the relative sliding distance are also discussed, while the effects of the misalignment angle and load torque on the wear depth are investigated. The numerical results show that the maximum relative sliding distance increases with an increase in the misalignment angle, and the relative sliding distance between the teeth near the pure tilted area is larger than that near the pure pivoted area. In addition, when crown gear coupling is aligned, the tooth surface doesn't produce wear, while the tooth surface wear pattern is symmetrical about the middle of the tooth width.

Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations

<p style='text-indent:20px;'>The stability and convergence of the Fourier pseudo-spectral method are analyzed for the three dimensional incompressible Navier-Stokes equation, coupled with a variety of time-stepping methods, of up to fourth order temporal accuracy. An aliasing error control technique is applied in the error estimate for the nonlinear convection term, while an a-priori assumption for the numerical solution at the previous time steps will also play an important role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical stability is preserved. These semi-implicit numerical schemes use a combination of explicit Adams-Bashforth extrapolation for the nonlinear convection term, as well as the pressure gradient term, and implicit Adams-Moulton interpolation for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in details. It is proved that, the Fourier pseudo-spectral method coupled with the carefully designed time-discretization is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Some numerical results are also presented to verify the established convergence rates of the proposed schemes.</p>

Método Espectro Nodal de Grades Compostas com Aproximação Exponencial na Formulação de Ordenadas Discretas em Meios Homogêneos Não-multiplicativos

Neutron transport problems in non-multiplying media appear in several applications in science and engineering, with emphasis on the design of shielding structures.  In this work, we present the composite spatial grids spectral nodal method with an exponential approximation for neutron transport problems in homogeneous non-multiplying media.  We use as a mathematical model the discrete ordinates formulation in two-dimensional Cartesian geometry with one-speed approximation and isotropic scattering source. The discretization process is done in two steps like is as composite spatial grid methods. An exponential function is used to approximate the transverse leakage terms in the region. The multi-step discretization mechanism transforms the two-dimensional problem into two coupled one-dimensional problems, that are solved using the alternative directions iterative scheme. Numerical results are offered for a model problem, showing that the proposed method generates accurate results with a low computational cost.

Parameter range reduction from partial data in systems of differential algebraic equations

This paper presents an algorithm for parameter range reduction from partial data in systems of differential algebraic equations. Interval arithmetic and linear multistep discretizations are used to prune regions of parameter space that are inconsistent with available time series data. Methods of splitting parameter space to ensure further progress are discussed. The algorithm outputs a reduced range and an approximation for each parameter value. These outputs are shown to have beneficial effect as input to traditional parameter estimation schemes. The range reduction algorithm is tested on three systems with partial data sets.

Linear Multistep Discretization Methods With Variable Step-Size in Nonlinear Wave Digital Structures for Virtual Analog Modeling

There is a growing interest in Virtual Analog modeling algorithms for musical audio processing designed in the Wave Digital WD domain. Such algorithms typically employ a discretization strategy based on the trapezoidal rule with fixed sampling step, though this is not the only option. In fact, alternative discretization strategies possibly with an adaptive sampling step can be quite advantageous, particularly when dealing with nonlinear systems characterized by stiff equations. In this paper, we propose a unified approach for modeling capacitors and inductors in the WD domain using generic linear multi-step discretization methods with variable time-step size, and provide generalized adaptation conditions. We also show that the proposed approach for implementing dynamic energy-storing elements in the WD domain is particularly suitable to be combined with a recently developed technique for efficiently solving a class of circuits with multiple one-port nonlinearities, called Scattering Iterative Method. Finally, as examples of application, we develop WD models for a Van Der Pol oscillator and a dynamic diode-based ring modulator, which use different discretization methods.

Stable Numerical Solution for a Class of Structured Differential-Algebraic Equations by Linear Multistep Methods

It is known that when we apply a linear multistep method to differential-algebraic equations (DAEs), usually the strict stability of the second characteristic polynomial is required for the zero stability. In this paper, we revisit the use of linear multistep discretizations for a class of structured strangeness-free DAEs. Both explicit and implicit linear multistep schemes can be used as underlying methods. When being applied to an appropriately reformulated form of the DAEs, the methods have the same convergent order and the same stability property as applied to ordinary differential equations (ODEs). In addition, the strict stability of the second characteristic polynomial is no longer required. In particular, for a class of semi-linear DAEs, if the underlying linear multistep method is explicit, then the computational cost may be significantly reduced. Numerical experiments are given to confirm the advantages of the new discretization schemes.

English

The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed.

Small-Signal Stability Analysis of Large Power Systems With Inclusion of Multiple Delays

The paper focuses on the small-signal stability analysis of large power systems with inclusion of multiple delayed signals. The following four techniques are compared: 1) a Chebyshev discretization scheme of an equivalent partial differential equations that resembles the original delay differential-algebraic equations (DDAEs); 2) an approximation of the time integration operator; 3) a linear multistep discretization of the DDAEs based on an high-order implicit time-integration scheme; and 4) the well-known Pade approximants. These techniques are compared using a GPU-based parallel implementation of the Shur method and QR factorization and tested through a real-world transmission system.

A semi-analytical load distribution model for side-fit involute splines

A semi-analytical load distribution model for side-fit involute splines has been proposed by extending the formulations of elastic bodies in contact. The spline compliances considered in the model consist of tooth bending and shear, tooth base flexibility, tooth contact, and torsional deformations. This model allows input of arbitrary loading components and arbitrary initial separations. With this, load distribution characteristics of splines under various combined loading conditions can be determined. In addition, the effects of design variations and manufacturing errors can also be quantified. Aside from the versatility of the model in terms of its capabilities, a new multi-step discretization solution scheme is devised to reduce computational effort thus qualifying the semi-analytical model as a design tool. Numerical results for an example spline under various loading conditions including: (i) pure torsion, (ii) combined torsion and radial loads, and (iii) combined torsion, radial loads and tilting moments, are presented and compared to those predicted by a deformable-body model for the verification of the results.

Numerical integration of positive linear differential-algebraic systems

In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent non-negative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge---Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We analyze these conditions for some common Runge---Kutta and multistep methods and observe that for index-1 systems and stiffly accurate Runge---Kutta methods, positivity is conditionally preserved under similar conditions as for ODEs. For higher index problems, however, none of the common methods is suitable.

Parallel multistep methods for linear evolution problems

Time-stepping procedures for the solution of evolution equations can be performed on parallel architecture by parallelizing the space computation at each time step. This, however, requires heavy communication between processors and becomes inefficient when many time-steps are to be computed and many processors are available. In such cases parallelization in time is advantageous. In this paper we present a method for parallelization in time of linear multistep discretizations of linear evolution problems; we consider a model parabolic and a model hyperbolic problem, and their, respectively, A(theta)-stable and A-stable linear multistep discretizations. The method consists of a discrete decoupling procedure, whereby N+1 decoupled Helmholtz problems with complex frequencies are obtained; N being the number of time steps computed in parallel. The usefulness of the method rests on our ability to solve these Helmholtz problems efficiently. We discuss the theory and give numerical examples for multigrid preconditioned iterative solvers of relevant complex frequency Helmholtz problems. The parallel approach can easily be combined with a time-stepping procedure, thereby obtaining a block time-stepping method where each block of steps is computed in parallel. In this way we are able to optimize the algorithm with respect to the number of processors available, the difficulty of solving the Helmholtz problems, and the possibility of both time and space adaptivity. Extensions to other linear evolution problems and to Runge-Kutta time discretization are briefly mentioned.

On the Convergence Behaviour of Variable Stepsize Multistep Methods for Singularly Perturbed Problems

In this note, we investigate the convergence behaviour of linear multistep discretizations for singularly perturbed systems, emphasising the features of variable stepsizes. We derive a convergence result for A(ϕ)-stable linear multistep methods and specify a refined error estimate for backward differentiation formulas. Important ingredients in our convergence analysis are stability bounds for non-autonomous linear problems that are obtained by perturbation techniques.

One- and multistep discretizations of index 2 differential algebraic systems and their use in optimization

An approach to solve constrained minimization problems is to integrate a corresponding index 2 differential algebraic equation (DAE). Here, corresponding means that the ω-limit sets of the DAE dynamics are local solutions of the minimization problem. In order to obtain an efficient optimization code, we analyze the behavior of certain Runge–Kutta and linear multistep discretizations applied to these DAEs. It is shown that the discrete dynamics reproduces the geometric properties and the long-time behavior of the continuous system correctly. Finally, we compare the DAE approach with a classical SQP-method.

Stabilized multistep methods for index 2 Euler-Lagrange DAEs

We consider multistep discretizations, stabilized by β-blocking, for Euler-Lagrange DAEs of index 2. Thus we may use “nonstiff” multistep methods with an appropriate stabilizing difference correction applied to the Lagrangian multiplier term. We show that orderp =k + 1 can be achieved for the differential variables with orderp =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low orderk-step Adams-Moulton methods. This approach is related to the recently proposed “half-explicit” Runge-Kutta methods.

Convergence of multistep discretizations of DAEs

Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.

Optimal open multistep discretization formulas for real-time simulation

The performance of digital real-time simulations is considered. A figure of merit is derived that quantifies a simulation's fidelity in terms of the time-domain discrepancy between its output and that of the plant it simulates, assuming that the plant is linearizable and asymptotically stable. This performance index is then used in deriving an easily automated procedure for calculating optimal values for free parameters in plant discretizations based on a generalized form of open linear multistep integration formulas. The theory is demonstrated in simulating the rigid-body dynamics of a fully articulated helicopter rotor blade system.

On resolvent conditions and stability estimates

As many numerical processes for time discretization of evolution equations can be formulated as analytic mappings of the generator, they can be represented in terms of the resolvent. To obtain stability estimates for time discretizations, one therefore would like to carry known estimates on the resolvent back to the time domain. For different types of bounds of the resolvent of a linear operator, bounds for the norm of the powers of the operator and for their sum are given. Under similar bounds for the resolvent of the generator, some new stability bounds for one-step and multistep discretizations of evolution equations are then obtained.

Supplement to Stability with Large Step Sizes for Multistep Discretizations of Stiff Ordinary Differential Equations

Supplement to Stability with Large Step Sizes for Multistep Discretizations of Stiff Ordinary Differential Equations

Stability with Large Step Sizes for Multistep Discretizations of Stiff Ordinary Differential Equations

One-leg and multistep discretizations of variable-coefficient linear systems of ODEs having both slow and fast time scales are investigated analytically. The stability properties of these discretizations are obtained independent of ODE stiffness and compared. The results of numerical computations are presented in tables, and it is shown that for large step sizes the stability of one-leg methods is better than that of the corresponding linear multistep methods.

Stability with large step sizes for multistep discretizations of stiff ordinary differential equations

In this paper we consider a large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales, a slow one and a fast one. A small parameter ε \varepsilon characterizes the stiffness of these systems. We approximate a system of ODE’s in this set by a general class of multistep discretizations which includes both one-leg and linear multistep methods. We determine sufficient conditions under which each solution of a multistep method is uniformly bounded, with a bound which is independent of the stiffness of the system of ODE’s, when the step size resolves the slow time scale but not the fast one. We call this property stability with large step sizes. The theory presented in this paper lets us compare properties of one-leg methods and linear multistep methods when they approximate variable coefficient systems of stiff ODE’s. In particular, we show that one-leg methods have better stability properties with large step sizes than their linear multistep counterparts. This observation is consistent with results obtained by Dahlquist and Lindberg [11], Nevanlinna and Liniger [33] and van Veldhuizen [42]. Our theory also allows us to relate the concept of D-stability (van Veldhuizen [42]) to the usual notions of stability and stability domains and to the propagation of errors for multistep methods which use large step sizes.