Predictor-corrector Technique Research Articles

New propagators for quantum-classical molecular dynamics simulations

The propagation of quantum/classical molecular dynamics equations is investigated from two different points of view. One is to take into account the time-dependence of effective Hamiltonians of quantum and classical subsystems by the combination of Zhu and Zhao’s time-independentlike split operator method [J. Chem. Phys. 105, 9536 (1996)] and a predictor-corrector technique. The other one is to introduce a time-independent quasi-Hamiltonian operator for the total system based on which new split-operator propagators are constructed. High-order quantum wave-packet propagators and the multiple time scale technique can be easily employed in our algorithms. Four new quantum/classical propagators are proposed and tested in a simple model system. It is found that combinations of our algorithms with high-order quantum propagators or the multiple time scale technique can give good results even with very large time steps. These new schemes will be particularly powerful in the case when the propagation of the classical subsystem is most time-consuming.

Implicit Bidiagonal Scheme for Depth-Averaged Free-Surface Flow Equations

A general fast implicit bidiagonal numerical scheme, based on the MacCormack's predictor-corrector technique requiring the inversion of only block bidiagonal matrices, has been developed and subsequently applied for subcritical and supercritical free-surface flow calculations. The model has been applied to depth-averaged steady flows. There are two main advantages of the proposed method: the technique has fast convergence and utilizes a body fitted nonorthogonal local coordinate system to simulate irregular geometry flows. The model is used to analyze a wide variety of hydraulic engineering problems including flows in a converging-diverging subcritical flume, supercritical expansions at various Froude numbers, and supercritical converging chutes. For each of these test cases, the calculated results are compared with experimental data. The comparisons with measurements as well as with other numerical solutions show that the proposed method is comparatively accurate, fast, and reliable.

Proposals for the propagation of quantum time-dependent self-consistent field equations

Three proposals for the propagation of the quantum time-dependent self-consistent field equations are presented. One is obtained by combining Zhu and Zhao's time-independent-like split-operator method [W.S. Zhu, X.S. Zhao, J. Chem. Phys. 105 (1996) 9536] and a predictor–corrector technique. By defining a time-independent quasi-Hamiltonian operator for the total system, two new split-operator propagators are proposed. Comparisons with the simple split-operator and second-order difference methods in a two-dimensional model system, the Henon–Helies potential, show that the three new methods all have better accuracy and higher stability.

Linearized factorization techniques for multidimensional reaction—diffusion equations

An iterative predictor—corrector technique for the elimination of the approximate factorization errors which result from the factorization of linearized θ-methods in multidimensional reaction—diffusion equations is proposed, and its convergence and linear stability are analyzed. Four approximate factorization techniques which do not account for the approximate factorization errors are developed. The first technique uses the full Jacobian matrix of the reaction terms, requires the inversion of, in general, dense matrices, and its approximate factorization errors are second-order accurate in time. The second and third methods approximate the Jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are, however, first-order accurate in time. The fourth approximately factorized method has approximate factorization errors which are second-order accurate in time and requires the inversion of lower and upper triangular matrices. The techniques are applied to a nonlinear, two-species, two-dimensional system of reaction—diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the Jacobian matrix as functions of the allocation of the reaction terms, space and time.

Analysis of Contact Mechanics for a Circular Filamentary Brush/Workpart System

This paper addresses the contact problem associated with the filament/workpart interaction that arises during brushing processes. A discretized model of a filament within the brushing tool is developed by employing Lagrange’s equations in conjunction with special constraint equations that are appropriate for the impact and impending large displacement of a flexible fiber whose tip traverses a flat, rigid surface. This formulation leads to the identification of five nondimensional parameters which fully characterize the filament/workpart contact problem. A damping mechanism is also included which can be used for modeling complex filament interactions that arise during the actual brushing operation. Special consideration is given to examining the initial filament/workpart impact and the subsequent forces that are generated along the contact region. Initial velocity of the filament is determined by employing an inelastic impact mechanics analysis. Time-varying transient response of the filament is then obtained by employing a predictor-corrector technique in conjunction with a finite difference method. Overall brush force is computed by a superposition of filament contact forces exerted onto the workpart surface. Numerical results are reported and compared with experimentally obtained data for an actual brush/workpart system.

Implicit, compact, linearized θ-methods with factorization for multidimensional reaction-diffusion equations

An iterative predictor–corrector technique for the elimination of the approximate factorization errors which result from the factorization of implicit, three-point compact, linearized θ-methods in multidimensional reaction-diffusion equations is proposed, and its convergence and linear stability are analyzed. Four compact, approximate factorization techniques which do not account for the approximate factorization errors and which involve three-point stencils for each one-dimensional operator are developed. The first technique uses the full Jacobian matrix of the reaction terms, requires the inversion of, in general, dense matrices, and its approximate factorization errors are second-order accurate in time. The second and third methods approximate the Jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are, however, first-order accurate in time. The fourth approximately factorized, compact, implicit method has approximate factorization errors which are second-order accurate in time and requires the inversion of lower and upper triangular matrices. The techniques are applied to a nonlinear, two-species, two-dimensional system of reaction-diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the Jacobian matrix as functions of the allocation of the reaction terms, space and time.

Stochastic tracking in nonlinear dynamical systems.

In a previous paper [Phys. Rev. A 46, 7439 (1992)] we have introduced an alternative continuation method which does not require an analytical model, but only an experimental time series. Using a predictor-corrector technique, the method tracks a given unstable orbit through different bifurcation regimes by varying an accessible system parameter. In this method, the continuation parameter was varied deterministically. That is, the location of the parameter is chosen by the experimenter. In this paper we introduce a similar algorithm, but now the parameter is varied randomly. A correction procedure is introduced so that control of an unstable orbit is not lost as the parameter changes. Moreover, we show that the small-amplitude feedback-control technique used for correction allows large-amplitude bursts in the parameter. These features are useful to experimentalists for canceling drift in experiments, which is inevitable at some level.

Analytical modelling of laminated composites

A significant number of research papers has been published on the analytical modelling of composite laminates over the past 20 years. The drive for more accurate analysis has led us to techniques which have become computationally more and more burdensome, while the engineering world continues to use simple, first-older shear deformable plate theory as its primary tool. This paper presents a unique approach to the analysis of thick laminated composites by presenting two simple finite element methods. The first uses the Predictor Corrector technique to extend the simple Mindlin type element to achieve greater accuracy, and the second develops a new Least Squares element which can approximate a C 1 continuous element. The Least Squares element has the capability to incorporate a simplified higher order basis into a piecewise continuous displacement field creating an accurate, yet computationally simple, element. These two methods have the potential to upgrade analysis methods significantly with little additional computational cost. It is hoped that this work can instigate further research into efficient modelling of composite laminates.

Three-dimensional simulation of electrothermal deicing systems

This paper examines three-dimensional transient heat transfer in a multilayered body that is ice covered. The physical application studied is the process of melting and removal of ice from aircraft components by use of electrothermal heaters. To model the ice-phase change, a predictor-corrector technique is used which assumes a phase for each ice gridpoint. This allows the use of the method-of-Douglas three-dimensio nal alternating direction numerical solver to iteratively converge on the correct phase of each ice node for each time step. Three-dimensional results are presented and the usefulness of the code as a design tool is illustrated. Comparisons between experimental deicer test results and numerical simulations are discussed. Nomenclature Cp = specific heat capacity, Btu/lb-°F H = enthalpy, Btu/lb h = heat transfer coefficient, Btu/h-ft2-°F k = thermal conductivity, Btu/h-ft-°F L = latent heat of fusion of ice, Btu/lb Q = rate of heat generation per unit volume, Btu/h-ft3 Rx — jc-direction Fourier number, dimensionless Ry = y-direction Fourier number, dimensionless Rz = z-direction Fourier number, dimensionless T = temperature, °F

Finite element simulation of pulsatile flow through arterial stenosis

The problem of blood flow through a stenosis is solved using the incompressible Navier-Stokes equations in a rigid circular tube presenting a partial occlusion. Calculations are based on a Galerkin finite element method. The time marching scheme employs a predictor-corrector technique using a variable time step. Results are obtained for steady and physiological pulsatile flows. Computational experiments analyse the effect of varying the degree of stenosis, the stricture length, the Reynolds number and Womersley number. The method gives results which agree well with previous computations for steady flows and experimental findings for steady and pulsatile flows.

An improvement of fractional step methods for the incompressible Navier-Stokes equations

Abstract A numerical method for computing three-dimensional, unsteady incompressible flows is presented. The method is a predictor-corrector technique combined with a fractional step method. Each time step is advanced in three sub-steps. The novel feature of the present scheme is that the Poisson equation for the pressure is solved only at the final sub-step resulting in substantial savings in computing time. It is shown that the method allows a larger CFL number and reduces the computing cost without loss of accuracy by satisfying the continuity equation only at the last sub-step. Numerical solutions for the decaying vortices and flow over a backward-facing step are obtained and compared with analytical and other numerical results.

Numerical model of unsteady subsonic aeroelastic behavior

A method for predicting unsteady, subsonic aeroelastic responses was developed. The technique accounts for aerodynamic nonlinearities associated with angles of attack, vortex-dominated flow, static deformations, and unsteady behavior. The fluid and the wing together are treated as a single dynamical system, and the equations of motion for the structure and flow field are integrated simultaneously and interactively in the time domain. The method employs an iterative scheme based on a predictor-corrector technique. The aerodynamic loads are computed by the general unsteady vortex-lattice method and are determined simultaneously with the motion of the wing. Because the unsteady vortex-lattice method predicts the wake as part of the solution, the history of the motion is taken into account; hysteresis is predicted. Two models are used to demonstrate the technique: a rigid wing on an elastic support experiencing plunge and pitch about the elastic axis, and an elastic wing rigidly supported at the root chord experiencing spanwise bending and twisting. The method can be readily extended to account for structural nonlinearities and/or substitute aerodynamic load models. The time domain solution coupled with the unsteady vortex-lattice method provides the capability of graphically depicting wing and wake motion.

Godunov computational fluid dynamics method for extreme flow velocities and any equation of state

An inviscid computational fluid dynamics code that can handle multiple component species, simple chemical reactions, a completely general equation of state, and velocities up to hundreds of kilometers per second is described. Radiation effects are not included. The code uses a sliding grid technique to handle multiple moving zones containing different media. Third-order extrapolations and interpolations are used to determine cell boundary values of the primitive variables. Limiting techniques then are applied to these values to maintain code accuracy and stability. First- and second-order Godunov procedures are used to calculate the cell boundary fluxes. The cell center values are updated in time using an explicit MacCormack predictor-corrector technique that is second-order accurate in time. The code was proofed with benchmark test cases, including Riemann's shock tube problem at pressure ratios up to 104, impacts of ideal-gas and dense media zones at closing velocities up to 220 km/s, detonation waves in stoichiometric hydrogen-oxyge n mixtures, ideal-gas wedge and cone flows, and a wedge flow of iron at a speed of 110 km/s. The results of these proof tests are discussed. The agreement between the CFD results and the exact solutions is found to be excellent.

A predictor-corrector technique for constrained least-squares regularization

The paper describes a numerical strategy for the approximate solution of nonlinear, discretized, inverse problems by regularization. It is assumed that the solution of the associated direct problems and the computation of Frechet derivatives are expensive. In order to minimize the amount of work, a predictor-corrector type algorithm is proposed. From a series of solutions to problems with a coarse discretization one obtains a starting approximation for a problem with a fine discretization.

On the Numerical Inversion of the Schwarz-Christoffel Conformal Transformation

Considering the numerical optimization approach for the inversion of the Schwarz-Christoffel conformal transformation formula, some improvements in integration procedures and some topics in optimization methods are discussed. Predictor-corrector techniques are introduced in order to map internal lines after the polygonal boundary is transformed these are applied to the dielectric interface in inhomogeneous line cross sections, allowing conformal transformation of quasi-TEM structures by purely numerical methods. Some examples of computations are presented, and some results are compared to known analytical calculations.

A fully implicit method for simulation of the one-dimensional solidification of a binary alloy

A finite difference method is developed which overcomes the difficulties of the disparate time scales and small solute diffusion layers inherent to one-dimensional models for the solidification of binary alloys. The boundary immobilization method applied with a coordinate transformation which stretches the solute boundary layer adjacent to the interface is combined with centred difference approximations to generate a set of ordinary differential equations for the solute and temperature fields and the melt/solid interface location. Numerical integration by a variable step-size fully-implicit trapezoid method is compared to an explicit Adams—Bashforth predictor—corrector technique. The implicit method is found to be from two to twelve times more efficient when the Lewis number (the ratio of mass to thermal diffusivities) is small, as is typically the case for metals and semiconductors.

Inverse perturbation method for structural redesign with frequency and mode shape constraints

A procedure is described for the redesign of undamped structural systems to meet natural frequency and/or mode shape objectives. The procedure can be applied to large or small modal changes and is based on a single finite element analysis of the baseline system. Perturbation of the baseline system is used to develop a set of equations that characterize the redesign process. Depending on the number of modal objectives and design variables, the problem is formulated as underconstrained, properly constrained, or overconstrained. AH three problems are solved using an incremental predictor-corrector technique within the feasibility domain defined by the practical constraints imposed on the design variables. The procedure is illustrated by the redesign of a 1254 degree-of-freedom casting for a frequency objective.

A simple algorithm for the nonlinear dynamic analysis of networks

A simple and economical algorithm is presented for the nonlinear dynamic analysis of networks, with lumped masses on the nodes, bars obeying uniaxial stress-strain laws and no damping. The step-by-step algorithm of the trapezoidal rule is used, combined with the predictor-corrector technique, with only two corrections per step. The prediction of an upper-bound for the eigenfrequencies permits the use of a constant steplength throughout the whole algorithm. An example problem and an error analysis of the algorithm are presented.

A three-dimensional model for spark ignition engines: Initial validation

A three-dimensional model for the cold flow motion and/or combustion of a homogeneous charge in Spark Ignition Engines was developed. An explicit predictor-corrector technique was employed. The model was validated in its two-dimensional version by comparing the computed TDC turbulence intensity with a parametric set of experimental data and with the results obtained with a different, previously validated, two-dimensional model. The fully three-dimensional model was then tested to simulate the cold flow motion of a homogeneous charge in a pancake-like chamber. It was concluded that the model can be used for cold flow simulation of Spark Ignition Engine.

One-Dimensional Model for Pulverized Coal Combustion and Gasification

Abstract A one-dimensional model has been developed for pulverized coal combustors and gasifiers. The model describes the response of a coal particle system to its thermal, chemical and physical environment. Moisture vaporization, coal devolatilization, heterogeneous char oxidation, gas particle interchange, radiation, gas phase oxidation, primary and secondary stream mixing, and heat losses are considered. A predictor-corrector solution technique was used to solve the ordinary non-linear differential equations. Several combustor and gasifier predictions are shown. The model predictions are compared with experimental data. The effects of particle size and distribution are shown to be particularly important. Significant rate controlling processes include initial particle heat-up and char surface reaction.