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.     

Questioning numerical integration methods for microsphere (and microplane) constitutive equations

In the last few years, more and more complex microsphere models have been proposed to predict the mechanical response of various polymers. Similarly than for microplane models, they consist in deriving a one-dimensional force vs. stretch equation and to integrate it over the unit sphere to obtain a three-dimensional constitutive equation. In this context, the focus of authors is laid on the physics of the one-dimensional relationship, but in most of the case the influence of the integration method on the prediction is not investigated.Here we compare three numerical integration schemes: a classical Gaussian scheme, a method based on a regular geometric meshing of the sphere, and an approach based on spherical harmonics. Depending on the method, the number of integration points may vary from 4 to 983,040! Considering simple quantities, i.e. principal (large) strain invariants, it is shown that the integration method must be carefully chosen. Depending on the quantities retained to described the one-dimensional equation and the required error, the performances of the three methods are discussed. Consequences on stress–strain prediction are illustrated with a directional version of the classical Mooney–Rivlin hyperelastic model. Finally, the paper closes with some advices for the development of new microsphere constitutive equations.     

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.     

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.     

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.     

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.     

Asymptotic–numerical methods and Pade approximants for non-linear elastic structures

In this paper, we apply asymptotic–numerical methods for computing non-linear equilibrium paths of elastic beam, plate and shell structures. The non-linear branches are sought in the form of asymptotic expansions, and they are determined by solving numerically (FEM) several linear problems with a single stiffness matrix. A large number of terms of the series can be easily computed by using recurrence formulas. In comparison with a more classical step-by-step procedure, the method is rapid and automatic. We show, with some examples, that the choice of the expansion's parameter and the use of Padé approximants play an important role in the determination of the size of the domain of convergence.     

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.     

Explicit Runge–Kutta methods for the integration of rate-type constitutive equations

Modern constitutive models have the potential to improve the quality of engineering calculations involving non-linear anisotropic materials. The adoption of complex models in practice, however, depends on the availability of reliable and accurate solution methods for the stress point integration problem. This paper presents a modular implementation of explicit Runge–Kutta methods with error control, that is suitable for use, without change, with any rate-type constitutive model. The paper also shows how the complications caused by the algebraic constraint of conventional plasticity are resolved through a simple subloading modification. With this modification any rate-independent model can be implemented without difficulty, using the integration module as an accurate and robust standard procedure. The effectiveness and efficiency of the method are demonstrated through a comparative evaluation of second and fifth-order formulas, applied to a complex constitutive model for natural clay, full details of which are given. This work was undertaken with the financial support of the UK Engineering and Physical Sciences Research Council: Grant no. GR/S84897/01.     

A-stable two-step time integration methods with controllable numerical dissipation for structural dynamics

A comprehensive study of A-stable linear two-step time integration methods for structural dynamics analysis is presented in this paper. An optimal A-stable linear two-step (OALTS) time integration method is revealed with desirable performance on low-frequency accuracy and high-frequency numerical dissipation properties. The OALTS time integration method is implemented in a direct integration manner for the second-order equations of structural dynamics; is implicit, A-stable, and second-order accurate in displacement, velocity, and acceleration, simultaneously; is easily started; and is numerical dissipation controllable. The OALTS time integration method shows desirable performance on spectral radius distribution, dissipation and dispersion errors, and overshooting behavior, where the results of some typical algorithms in the literature are also compared. Benchmark examples with/without physical damping are performed to validate the accuracy, stability, and efficiency of the OALTS time integration method.     

复阻尼结构动力方程的增维精细积分法

为了避开求解复阻尼结构强迫激励动力学方程的积分运算,引入增维精细积分方法。根据复阻尼系统复化对偶原则,将动力学方程和激励波对偶复化为实部和虚部的形式,推导出增维矩阵的精细积分求解过程。结果表明,由于不用求解迭代矩阵H的逆矩阵,避免了矩阵奇异带来的计算解的不稳定性。在计算矩阵仅增加一维的情况下,化积分运算为代数运算,扩大了精细积分法的应用范围。通过对比增维精细积分法和频域法计算结果,二者结果保持较高的一致性。     

A family of noniterative integration methods with desired numerical dissipation

A new family of unconditionally stable integration methods for structural dynamics has been developed, which possesses the favorable numerical dissipation properties that can be continuously controlled. In particular, it can have zero damping. This numerical damping is helpful to suppress or even eliminate the spurious participation of high frequency modes, whereas the low frequency modes are almost unaffected. The most important improvement of this family method is that it involves no nonlinear iterations for each time step, and thus it is very computationally efficient when compared with a general second‐order accurate integration method, such as the constant average acceleration method. Copyright © 2014 John Wiley & Sons, Ltd.