On the stability of explicit methods for the numerical integration of the equations of motion in finite element methods

This paper deals with the stability of the numerical solutions of a dynamic finite element analysis. The solutions are obtained through a stepwise integration of the equations of motion. Upper bounds on the steplength of the integration are obtained from a stability analysis of using a simple finite difference approximation for the equations of motion, and are shown to depend strongly on the particular element is use and on how the mass of the element is distributed at its nodes. As an example, the two-dimensional wave propagation in a semi-infinite plate subjected to a suddenly applied moment along its edge is studies. Through the example, we show that the bound on the steplength, obtained from the simple analysis, can provide a useful guide on choosing the steplength in other higher order integration methods. In particular, we show that, for stability considerations, the upper bound on the steplength should also hold for a fourth order explicit method. In order to achieve an acceptable accuracy of the solution, we show that the steplength should be approximately one half of the bound for the higher order explicit method as well as a higher order implicit method. Solution of the example has been compared with that of the Timoshenko theory.     

Computational methods for some non-linear wave equations

Computational methods based on a linearized implicit scheme and a predictor-corrector method are proposed for the solution of the Kadomtsev–Petviashvili (KP) equation and its generalized from (GKP). The methods developed for the KP equation are applied with minor modifications to the generalized case. An inportant advantage to be gained from the use of the linearized implicit method over the predictor-corrector method which is conditionally stable, is the ability to vary the mesh length, and thereby reducing the computational time. The methods are analysed with respect to stability criteria. Numerical results portraying a single line-soliton solution and the interaction of two-line solitons are reported for the KP equation. Moreover, a lump-like soliton (a solitary wave which decays to zero in all space dimensions) and the interaction of two lump solitons are reported for the KP equation.     

Three-dimensional differential equations dynamic analysis for non-linear structures

A new dynamic model is investigated for the solution of the three-dimensional structural analysis problem of a non-linear structure subjected under seismic forces. Such problem is reduced to the solution of a system of ordinary differential equations of the second kind and this system is numerically evaluated by using a kind of finite elements and by solving the corresponding eigenvalues-eigenvectors problem. An application of structural analysis is given to the determination of the eigenvalues and eigenvectors of a 10-floor building consisting of reinforced concrete and subjected to an horizontal seismic vibration.     

Explicit numerical integration algorithm for a class of non-linear kinematic hardening model

 An explicit updating algorithm has been developed for the Armstrong–Frederick family of non-linear kinematic hardening model, based on the trapezoidal and the backward Euler integration method. The algorithm provides a computationally efficient method for implementing the non-linear kinematic hardening model in finite element codes. It is shown that the trapezoidal method performs better with the original Armstrong–Frederick rule, while the backward Euler rule provides an improved accuracy to the modified multiple back-stress model that incorporates a weight function for dynamic recovery. Numerical examples are presented to illustrate the performance of the algorithm developed, and a comparison with the experimental observation shows that the modified constitutive model indeed provides a more accurate prediction to the long term mean stress relaxation.     

Efficient path integration methods for nonlinear dynamic systems

New approaches for numerical implementation of the path integration (PI) method are described. In essence the PI method is a stepwise calculation of the joint probability density function (PDF) of a set of state space variables describing a white noise excited nonlinear dynamic system. The basic idea behind the proposed procedure is to apply a splines interpolation method to the logarithm of the calculated PDF to obtain an accurate representation of the PDF over the whole domain and not only at the chosen grid points. This exploits the fact that the logarithm of the PDF shows a more polynomial behaviour than the PDF itself, and therefore is better adapted to a splines representation. It is demonstrated that the proposed techniques may lead to significantly improved performance in calculating the response statistics of large classes of nonlinear oscillators excited by white or coloured noise when compared to other available implementations of the PI method. An advantage of the new approaches is that they allow time-variant dynamic systems to be analysed without significant increase in computer time. Numerical results for both 2D and 3D problems are presented.     

A one-step method for direct integration of structural dynamic equations

The choice on an efficient direct integration procedure for linear structural dynamic equations of motion is discussed. It is suggested that as accuracy parameter the truncation error on the exponential terms contained in the modal contributions of the exact solution be assumed. This error does not always coincide with the local truncation error. These considerations were used to design an unconditionally stable one-step method whose accuracy is 0(h⁴). Numerical comparisons with some well-known integration schemes showed the efficiency of the proposed method.     

A simple spatial integration scheme for solving Cauchy problems of non-linear evolution equations

In this study, we address a new and simple non-iterative method to solve Cauchy problems of non-linear evolution equations without initial data. To start with, these ill-posed problems are analysed by utilizing a semi-discretization numerical scheme. Then, the resulting ordinary differential equations at the discretized times are numerically integrated towards the spatial direction by the group-preserving scheme (GPS). After that, we apply a two-stage GPS to integrate the semi-discretized equations. We reveal that the accuracy and stability of the new approach is very good from several numerical experiments even under a large random noisy effect and a very large time span.     

Method of approximating series for numerical integration of systems of differential equations

A method of numerical integration of systems of differential equations is proposed that can be used for equations that describe processes occurring in every field of physics, namely, fluid mechanics, nuclear physics, solid-state physics, etc. The APPROX program package, which implements the method of approximating series, makes it possible to write programs for computations in no more than 2–3 hours and reduces the calculation time by 1–2 orders in comparison with finite-difference methods.Kharkov Polytechnic Institute. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 2, pp. 238–244, February, 1994.     

Estimation methods using dynamic phasors for numerical distance protection

Several new methods for faulted transmission line parameters estimation in phasor and time domain are proposed, that eventually improve the overall performance of numerical distance relays. The concept of dynamic phasors is introduced to accommodate the time-variant nature of the current and voltage signals during transients and faults. Based on dynamic phasor transmission line models, direct and indirect estimation methods are derived. For the proposed indirect estimation method, stability of prediction error dynamics is assured by using the Lyapunov direct method. Presented estimation techniques are compared with a conventional stationary phasor solution as well as with a recursive least-square estimator derived in the discrete time domain. In the evaluation, more realistic assumptions are considered with regards to distortion of the input voltage and current signals along with the variable fault resistance because of arcing faults. Simulation results and actual field measurements are included for performance evaluation of the proposed estimators.     

A new family of explicit time integration methods for linear and non-linear structural dynamics

A new family of explicit single-step time integration methods with controllable high-frequency dissipation is presented for linear and non-linear structural dynamic analyses. The proposed methods are second-order accurate and completely explicit with a diagonal mass matrix, even when the damping matrix is not diagonal in the linear structural dynamics or the internal force vector is a function of velocities in the non-linear structural dynamics. Stability and accuracy of the new explicit methods are analysed for the linear undamped/damped cases. Furthermore, the new methods are compared with other explicit methods.     

Two high-order numerical methods for solving boundary-value nonlinear ordinary differential equations

Two numerical methods for solving two-point boundary-value problems associated with systems of first-order nonlinear ordinary differential equations are described. The first method, which is based on Lobatto quadrature, requires four internal function evaluations for each subinterval. It does not need derivatives and is of order h⁷, where h is the space chop. The second method, which is similar to the first but is based on Lobatto–Hermite quadrature, makes the additional use of derivatives to achieve O(h⁹) accuracy. Results of computational experiments comparing these methods with other known methods are given.     

The direct numerical integration of linear matrix differential equations using pade approximations

A class of approximations to the matrix linear differential equation is presented. The approximations range, in accuracy, from the simplest forward difference scheme to the exact solution. The infinite series defining the exponential matrix is used to ascertain the accuracy of the various approximations. A clear distinction is made between approximations to the system equations and the forcing function, with the forcing term being represented by a piecewise linear function. Special application is given to the equations arising in structural dynamics of the form For these structural dynamic equations, the measure of the energy of the system is used to analyse the stability of the numerical approximations.     

A survey of parallel numerical methods for initial value problems for ordinary differential equations

The parallel solution of initial value problems for ordinary differential equations has become an active area of research. Recent developments in this area are surveyed with particular emphasis on traditional forward-step methods that offer the potential for effective small-scale parallelism on existing machines.<>     

Matrix differential equation and higher-order numerical methods for problems of non-linear creep,viscoelasticity and elasto-plasticity

The constitutive equation is assumed in a very general form which includes as special cases non-linear creep, incremental elasto-plasticity as well as viscoelasticity represented by a chain of n standard solid models. Subdividing the structure into N finite elements, the problem of structural analysis is formulated with a system of 6N(n + 1) ordinary non-linear first-order differential equations in terms of the components of stresses and strains in the elements. This formulation enables one to apply Runge–Kutta methods or the predictor–corrector methods.     

A comparison of boundary methods for the numerical solution of hyperbolic systems of equations

Summary This paper compares the numerical solution of a linear system of hyperbolic partial differential equations in one and two space dimensions with the analytic solution. A two step Lax-Wendroff difference scheme is used in the interior region and various methods are used at the boundaries. The accuracy of the overall solution is tabulated for each of the boundary methods. Of particular interest here is the accuracy of the various boundary methods which are used.     

A numerical method for homogenization in non-linear elasticity

In this work, homogenization of heterogeneous materials in the context of elasticity is addressed, where the effective constitutive behavior of a heterogeneous material is sought. Both linear and non-linear elastic regimes are considered. Central to the homogenization process is the identification of a statistically representative volume element (RVE) for the heterogeneous material. In the linear regime, aspects of this identification is investigated and a numerical scheme is introduced to determine the RVE size. The approach followed in the linear regime is extended to the non-linear regime by introducing stress–strain state characterization parameters. Next, the concept of a material map, where one identifies the constitutive behavior of a material in a discrete sense, is discussed together with its implementation in the finite element method. The homogenization of the non-linearly elastic heterogeneous material is then realized through the computation of its effective material map using a numerically identified RVE. It is shown that the use of material maps for the macroscopic analysis of heterogeneous structures leads to significant reductions in computation time.     

High-frequency equations for non-linear vibrations of thermopiezoelectric shells

This paper develops a system of 2D shear deformable equations so as to analyze the non-linear vibrations of shells on the basis of the 3D fundamental equations of thermopiezoelectricity with a second sound effect. First, a differential type of variational principles is presented for the 3D fundamental equations. Next, the system of 2D approximate equations of successively higher orders is deduced from the 3D fundamental equations with the aid of the variational principle and the series expansions of the field variables of thermopiezoelectric shells. The system of 2D equations which is established in invariant differential and variational forms governs all the types of vibrations of thermopiezoelectric shells at both low and high frequencies. All the mechanical, electrical and thermal effects of higher orders are taken into account for the case of large electric fields, infinitesimal temperature variations and large deflections. Lastly, attention is confined to some of special cases involving types of vibrations, geometry and material properties. Besides, the uniqueness is investigated in solutions of the system of fully linearized 2D equations of thermopiezoelectric shells and the conditions sufficient for the uniqueness are enumerated.