Predictor Corrector Process Research Articles

A reduced-order solution for critical buckling temperature of thin-walled structures

This work presents a reduced-order solution for the thermoelastic geometrically nonlinear response of simply-supported thin-walled structures subjected to the purely thermal load. Previously, the reduced-order method was only applicable to the buckling problem with a fixed temperature value. Now, the method is reformulated to achieve the critical buckling temperature from the thermoelastic geometrically nonlinear buckling analysis. The thermal buckling problem is represented using the thermal expansion model and its equivalent mechanical model, respectively. For the equivalent mechanical model, the simply-supported constraints need to be partially released and the temperature field is converted to be a thermal load. The reduced-order model is constructed based on the improve Koiter theory for thermal buckling analysis of the equivalent mechanical model. The thermoelastic geometrically nonlinear response is traced using the reduced-order solution together with a predictor-corrector process. The reduced-order solution of the equivalent mechanical model is validated by fully nonlinear solution of the thermal expansion model. Various plates and shells are selected to demonstrate the accuracy and highly efficiency of the proposed method for thermal buckling analysis.

Imperfection sensitivity study of the thermal–mechanical buckling of laminated composite cylinders using a novel reduced-order modeling method

Laminated composite cylinders subjected to axial compression exhibit the “snap-back” response, the buckling load of which is very sensitive to initial geometric imperfections. An operating environment with high temperatures influences both the buckling behavior and imperfection sensitivity of thin-walled cylinders. This paper proposes a novel reduced-order modeling method for nonlinear buckling and imperfection sensitivity analyses, considering the effects of the initial geometric imperfections and initial temperature field. The Koiter asymptotic theory is reformulated to construct a thermal–mechanical reduced-order model with initial geometric imperfections. The initial temperature field is converted to an additional degree of freedom during the construction of the reduced-order model. An a posteriori account of geometric imperfections is achieved for the thermomechanical reduced system of the perfect structure. A predictor–corrector process is developed to trace the thermoelastic geometrically nonlinear response of an axially compressed cylinder with geometric imperfections. Various numerical examples are designed to validate the high efficiency and satisfactory accuracy of the proposed method for imperfection sensitivity analyses. The optimization process based on the proposed method is applied to determine the most imperfection-sensitive laminate and the “worst” imperfection shape.

A 2D SWE meshless model with fictitious water level at dry nodes

In this study, a two-dimensional shallow-water-equation numerical model is developed for simulating free surface water flows with internal moving wet/dry front boundaries. It is based on the Eulerian aspect and meshless approach. The predictor-corrector process is used for the time marching. The spatial derivatives of physical quantities are computed using the local polynomial approximation with the weighted-least-squares approach. A formula for specifying the fictitious water level, which does not exist physically, at dry nodes is proposed. This is the main novelty of the present study and it significantly reduces computation time compared with the authors’ previous work. Four dam break test cases are employed to validate the capability of the present model.

PHƯƠNG PHÁP LẶP SONG SONG RUNGE-KUTTA HAI BƯỚC

In this paper, we introduce the Parallel iteration of two-step Runge-Kutta methods for solving non-stiff initial-value problems for systems of first-order differential equations (ODEs): y′(t) = f(t, y(t)), for use on parallel computers. Starting with an s−stage implicit two-step Runge-Kutta (TSRK) method of order p, we apply the highly parallel predictor-corrector iteration process in P (EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta method that has order p for all m, and that requires s(m+1) right-hand side evaluations per step of which each s evaluation can be computed parallelly. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of L2. We then solve this system by a predictor–corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows.

Solving Fractional Dynamical System with Freeplay by Combining Memory-Free Approach and Precise Integration Method

The Yuan-Agrawal (YA) memory-free approach is employed to study fractional dynamical systems with freeplay nonlinearities subjected to a harmonic excitation, by combining it with the precise integration method (PIM). By the YA method, the original equations are transformed into a set of first-order piecewise-linear ordinary differential equations (ODEs). These ODEs are further separated as three linear inhomogeneous subsystems, which are solved by PIM together with a predictor-corrector process. Numerical examples show that the results by the presented method agree well with the solutions obtained by the Runge-Kutta method and a modified fractional predictor-corrector algorithm. More importantly, the presented method has higher computational efficiency.

Determination of transfer capability region using boundary tracing method

AbstractIn electricity markets, transfer capability is a parameter showing the potential of a considered source–sink pair to transfer power. This paper enhances transfer capability monitoring by proposing a method to determine this parameter with respect to a region referred to as the ‘transfer capability region’ (TCR). The boundary of a TCR is traced by using the modified predictor–corrector process on a plane of real and reactive powers. The TCR shows a set of feasible loading points at the sink area. Moreover, the shape of TCR always changes according to the system parameters. This paper also defines the outermost boundary of the TCR when the considered parameters are set to be free. We apply the proposed method to test systems, and then compare to the maximum loading points obtained by the conventional transfer capability calculation. © 2012 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

A New Class of Highly Accurate Solvers for Ordinary Differential Equations

We introduce a new class of predictor-corrector schemes for the numerical solution of the Cauchy problem for non-stiff ordinary differential equations (ODEs), obtained via the decomposition of the solutions into combinations of appropriately chosen exponentials; historically, such techniques have been known as exponentially fitted methods. The proposed algorithms differ from the classical ones both in the selection of exponentials and in the design of the quadrature formulae used by the predictor-corrector process. The resulting schemes have the advantage of significantly faster convergence, given fixed lengths of predictor and corrector vectors. The performance of the approach is illustrated via a number of numerical examples.

A Predictor-Corrector Process with Refinement for First-Year Calculus Transition Support

In this article we discuss an approach to the critical problem of student retention and the high school-college transition. The context of the discussion is centered on science and engineering students for whom calculus in the first year in college is an essential requirement for on-time graduation. This transition/retention problem has become increasingly critical in light of the much publicized perception that proportionally fewer students arrive in college ready to undertake a serious college calculus course. The article describes the overall approach to this problem adopted at Clarkson University, and then concentrates on one particular aspect: our pre-first year diagnostic testing and summer programs. While these do not represent a complete answer, the statistics support the belief that they have significant positive impact, especially among those students identified as having the greatest need of such reinforcement.

Error analysis for the predictor–corrector process relating to ill-conditioned linear system of equations

In [Xinyuan Wu, An effective iterative procedure process for large scale linear system of equations, Appl. Math. Comput. 180 (1) (2006) 160–166; Xinyuan Wu, Note on predictor–corrector process for ill-conditioned linear system of equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.05.075], the author proposed predictor–corrector processes for large scale linear system of algebraic equations with ill-conditioned coefficient matrix, however, the error analysis for the predictor–corrector processes relating to ill-conditioned linear system of equations has not been given. This paper presents the results on error analysis for the predictor–corrector process proposed in Wu (in press).

Note on predictor–corrector process for ill-conditioned linear system of equations

In [Xinyuan Wu, An effective iterative procedure process for large scale linear system of equations, Appl. Math. Comput., in press], the author proposed a predictor–corrector process for large scale linear system of algebraic equations with ill-conditioned coefficient matrix. Basing on the same principle, this article introduces another iterative process which simplifies and reduces the computational procedure. Numerical experiments are included to show the efficiency of the process.

An effective predictor–corrector process for large scale linear system of equations

This short note introduces a predictor–corrector process for large scale system with ill-conditioned matrix. Numerical experiments are carried out to show the process robust performance for ill-conditioned system of equations.

PH—Postharvest Technology: Numerical Simulation Static-Bed Drying of Barley

The last decade has seen enormous progress in mathematical modelling and numerical simulation in areas related with fluid dynamics and heat and mass transfer. This is because of the increasingly widespread use of powerful, low-cost digital computers. Future efforts should be directed towards developing computer codes that can be implemented by users who may not be well versed in computational skills. Here a computer program has been developed, for simulating a non-equilibrium mathematical model for static-deep-bed drying of barley. The model consists of a set of four non-linear partial differential equations as a result of mass and energy balances and heat transfer equations, together with appropriate diffusion equation. The set of equations is solved by means of a finite-difference method, using a second-order iterative predictor–corrector process to solve the first two equations in the time domain, and a first-order iterative process for the second two, in the space domain. Published experimental results are used to validate the model. The numerical results are in good agreement with the experimental data relative to heated-air drying system, and the prediction accuracy is enhanced with respect to equilibrium models.

A class of explicit parallel two-step Runge-Kutta methods

The aim of the present paper is to construct a class of two-step Runge-Kutta methods of arbitrarily high order for application to parallel computers. Starting with ans-stage implicit two-step Runge-Kutta method of orderp withk=p/2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P(EC) m E mode. In this way, we obtain an explicit two-step Runge-Kutta method that has orderp for allm and that requiresk(m+1) right-hand side evaluations per step of which eachk evaluation can be computed in parallel. By a number of numerical experiments we show the superiority of the parallel predictor-corrector methods proposed here over parallel method available in the literature.

Tidal and long wave propagation—a finite element approach

The governing equations of tidal flow are solved for a two dimensional domain using the finite element method. The appropriate vertically integrated differential equations are replaced by an equivalent set of integral equations which result in a coupled matrix relationship. Three time stepping schemes are investigated. These are the Adams-Moulton multi-step predictor corrector process, the trapezoidal finite difference integration scheme, and the finite element in time. The algorithm is used to predict long wave forms, in rectangular channels, arising from tidal variations and constant wind forces. The tidal propagation in the southern North Sea is also presented.

A Starting Method for the Three-Point Adams Predictor-Corrector Method

Certain higher order iterative procedures for the numerical solution of ordinary differential equations require a separate procedure for determining starting values. Such is the case in the Adams and Milne methods. The work involved in programming the computation of starting values may be of the same magnitude as that required in programming the method itself. This paper shows that the three-point Adams formula leads to a method which may be considered “self-starting” in the sense that very minor modifications of the method provide starting values of sufficient accuracy for continuing the solution.Let the equation under solution be y′ = ƒ(y, t), and let the values y′(t0) = y0′, y′(t0 + h) = y1′, y(t0) = y0, y(t0 + h) = y1, be given. The general three-point predictor-corrector process consists of estimating (in some unspecified way) a value y2′ of y2′ computing a first estimate y2 by means of a closed three-point integration formula; obtaining the (presumably) better value y2′ = ƒ(y2, t0 + 2h); and then repeating the process until some convergence criterion is satisfied, either for y2′ or for y2.The process could have started by first estimating y2, rather than y2′. It is important to note that as long as the step size h and the first estimate y2′ of y2′ result in a converging process, the values to which there is convergence are independent of the method of prediction.The Adams three-point formula is yn+1 = yn + h/12[5ƒ(tn+1, yn+1) + 8ƒ(tn, yn) - ƒ(tn-1, yn-1)]. (1) Tn, the single step truncation error, is given by Tn = - h4/24ƒ′′′(x), tn-1 l x l tn+1. The quantities yn, y′n+1, yn′, y′n-1 are assumed exact. Let the initial conditions y0′, y0 be specified for y′ = ƒ(y, t).As a first approximation, let y(0)-1′ = y0′, y(0)+1′ = y0′, y(0)-1 = y0, y(0)+1 = y0. The superscript zero in parentheses indicates that these are the initial estimates of y, y′. The starting procedure consists of performing corrective iterations in the forward (+1) and backward (-1) direction as follows: y(i)+1 = y0 + h/12[5ƒ(t+1, y(i-1)+1) + 8ƒ(t0, y0) - ƒ(t-1, y(j)-1)], y(j)-1 = y0 - h/12[5ƒ(t-1, y(j-1)-1) + 8ƒ(t0, y0) - ƒ(t+1, y(i)+1)].(2) The superscript notation employed does not imply any order to the forward and backward iterations. In the following analysis, however, it will be assumed that the process starts with a forward iteration, and alternates thereafter.It is necessary to show the conditions under which the process converges and the conditions under which the resulting values are accurate enough to warrant continuing the solution by the Adams method.Let y+1 be the value of y at t+1 to which the iterative process converges (if at all). The error at the ith forward iteration is defined as a(i), such that y(i)+1 = y+1 + a(i).(3) The error in the backward direction is b(j), such that y(j)-1 = y-1 + b(j). (3′) Substituting the error definition into equation (2) yields a(i) = 5h/12{ƒ[t+1, y+1 + a(i-1)] - ƒ[t+1, y+1]} - h/12{ƒ[t-1, y-1 + b(i)] - ƒ[t-1, y-1]}, (4) b(j) = - 5h/12{ƒ[t-1, y-1 + b(i-1)] - ƒ[t-1, y-1]} + h/12{ƒ[t+1, y+1 + a(i)] - ƒ[t+1, y+1]}.Let g+1 = ∂ƒ[t+1, y(t0 + x1)]/∂y, 0 l x1 l h, and (5) g-1 = ∂ƒ[t-1, y(t0 + x2)]/∂y, -h l x2 l 0. Let it be assumed that g+1 and g-1 are insensitive to small changes in x1 and x2; then, by the law of the mean, equations (4) can be written as a(i) = 5h/12 g+1a(i-1) - h/12 g-1b(j), b(j) = 5h/12 g-1a(j-1) + h/12 g+1a(i).(6) The order in which the iterations are performed may now be specified by letting i = 2k + 3, j = 2k + 2, k = -1, 0, 1, ···, ∞. Equations (6) can be reduced to a single equation in either a or b. Choosing b results in b(2k+4) + 5h/12 (g-1 - g+1)b(2k+2) - 24h2/12 g+1g-1b(2k) = 0. (7) The condition for convergence of the starting process is that b → 0 as k → ∞; i.e., that the roots of equations (7), when considered as a polynomial in b2, be less than one in magnitude. The conditions for convergence are then V- 5h/12 (g+1 - g-1) ± h/12 √[5(g+1 - g-1]2 - 4g+1g-1V