Average Acceleration Method Research Articles

Accurate Representation of External Force in Time History Analysis

It is usually thought that the integration time step should be small enough to represent properly the variation of the dynamic loading with respect to time. However, there is no evaluation criterion that can be used to determine whether the external force is accurately represented. In this paper, criteria for accurate representation of external force are proposed based on analytical results. It is found that amplitude distortion both in the transient response and the steady-state response for each time step is closely related to the step discretization error of external force. In fact, for a negligible period distortion, an amplitude distortion will be less than 5% if the relative step discretization error is constrained to be less than 5% at each time step for the Newmark explicit method, Fox-Goodwin method, and linear acceleration method while for the constant average acceleration method it must be less than 2.5%. This criterion leads to the need of using of eight or more integration time steps to accurately represent a complete cycle of a harmonic loading for the Newmark explicit method, Fox-Goodwin method, and linear acceleration method while for the constant average acceleration method 12 or more integration time steps are required.

Error Propagation in Implicit Pseudodynamic Testing of Nonlinear Systems

Error propagation analysis of an implicit pseudodynamic algorithm has been developed for linear elastic systems. However, these error propagation results might not be applicable to nonlinear systems. Since the pseudodynamic testing method aims at investigating the nonlinear behavior of a seismically loaded structure it is important to perform the error propagation analysis of the implicit pseudodynamic algorithm for nonlinear systems. A technique to conduct the nonlinear error propagation analysis of an implicit pseudodynamic algorithm is constructed and is illustrated by analyzing the constant average acceleration method. Theoretical results reveal that the numerical and error propagation properties for nonlinear systems are generally inherited from those of linear elastic systems although some properties are affected by the step degree of nonlinearity. It is verified that the most important property of unconditional stability is preserved for nonlinear systems for a complete pseudodynamic test procedure even if the convergence error is present. It is concluded that error propagation for the improved implementation is superior to that of the direct implementation and is consistent with that developed for linear elastic systems.

Higher-Order Implicit Dynamic Time Integration Method

A general procedure is presented for the use of higher-order approximations of a system’s dynamic response using new time integration schemes. An implicit method is developed and presented that makes use of higher-order polynomial acceleration variations for multi-degree-of-freedom dynamic structural analyses. A variable θ is used to increase the stability and accuracy of the method. The relative merits and stability aspects of the method are described. The higher-order method is compared with two traditional implicit methods, the Wilson-θ Method and the Constant Average Acceleration Method of the Newmark family. The peak displacement percent difference and percent time difference to peak displacement are used to measure the accuracy of the methods when applied to a multi-degree-of-freedom frame problem.

Second-order inelastic dynamic analysis of 3-D steel frames

This paper presents a reliable numerical procedure for nonlinear time-history analysis of three-dimensional steel frames subjected to dynamic loads. Geometric nonlinearities of member (P- δ) and frame (P-Δ) are taken into account by the use of stability functions in framed stiffness matrix formulation. The gradual yielding along the member length and over the cross-section is included by using a tangent modulus concept and a softening plastic hinge model based on a modified version of Orbison yield surface. A computer program utilizing the average acceleration method for the integration scheme is developed to numerically solve the equation of motion of framed structure formulated in an incremental form. The results of several numerical examples are compared with those derived from using beam element model of ABAQUS program to illustrate the accuracy and the computational efficiency of the proposed procedure.

AN UNCONDITIONALLY STABLE EXPLICIT METHOD FOR STRUCTURAL DYNAMICS

An explicit integration method with unconditional stability was proposed and presented in this paper. Numerical characteristics of this explicit method for linear elastic sys-tems are almost the same as those of the constant average acceleration method while for a nonlinear system it is more efficient in computing than for the constant average acceleration method. This explicit method integrates the most promising advantages possessed by the explicit and implicit methods. No limitation on the time step in satisfying the stability limit, which is the most important property for an implicit method, is theoretically proved for this explicit method. Furthermore, the avoidance of solving any implicit system or using any iterative procedure, which usually brings considerable simplification in practical applications for explicit methods, leads to the very low cost of one explicit step. Thus, the computational effort can be significantly reduced when compared to implicit methods in each time step. Consequently, this explicit method can be used to solve dynamic problems efficiently due to the unconditional stability and the very low cost of explicit steps. Rough guidelines with regards to the selection of a time step in achieving accurate solutions for the constant average acceleration method are also appropriate for the proposed explicit method.

Nonlinear impact behaviour of laminated composite shells in hygrothermal environments

The nonlinear transient response of laminated composite shell panels subjected to low velocity impact in hygrothermal environments is investigated using finite element method. The present formulation considers doubly curved thick shells and includes large deformations with Green- Lagrange strains. The analysis is carried out using quadratic eight—noded isoparametric element. A modified Hertzian contact law is incorporated into the finite element program to evaluate the impact force. The nonlinear equation is solved using the Newmark average acceleration method in conjunction with an incremental modified Newton-Raphson scheme. The validity of the present analysis is demonstrated by comparing the present results with the solutions available in the literature. A parametric study is carried out to investigate the effects of the curvature and side to thickness ratios of simply supported composite cylindrical and spherical shell panels.

ERRORS IN NUMERICAL SOLUTION OF EQUATION OF MOTION OF LIGHTLY DAMPED SDOF SYSTEM NEAR RESONANCE

This study investigates the error which occurs when numerically integrating the equation of motion of a single degree of freedom system excited by a harmonic force near resonance. The Constant Average Acceleration method was considered in particular as it features in many finite element software packages. It was found that a considerable error in the calculated responses occurs in systems with low damping due to the well known phenomenon of period elongation. However, the error is reduced for systems with higher damping and/or when smaller time step is used. With regard to this, recommendations are given as to the time steps required to obtain solutions with a pre-defined level of accuracy.

Nonlinear transient analysis of laminated composite shells in hygrothermal environments

The nonlinear transient response of composite shell panels subjected to mechanical load in hygrothermal environments is investigated using finite element method. The present formulation considers doubly curved thick shells and includes large deformations with Green–Lagrange strains. The analysis is carried out using quadratic eight-noded isoparametric element. The nonlinear time dependent equation is solved by using the Newmark average acceleration method in conjunction with an incremental modified Newton–Raphson scheme. The validity of the model is demonstrated by comparing the present results with the solutions available in the literature. A parametric study is carried out varying the curvature ratios and side to thickness ratios of composite cylindrical, spherical and hyperbolic paraboloid shell panels with simply supported boundary conditions.

Design spaces, measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design—illustration to structural dynamics

For the first time, for time discretized operators, we describe and articulate the importance and notion of design spaces and algorithmic measures that not only can provide new avenues for improved algorithms by design, but also can distinguish in general, the quality of computational algorithms for time-dependent problems; the particular emphasis is on structural dynamics applications for the purpose of illustration and demonstration of the basic concepts (the underlying concepts can be extended to other disciplines as well). For further developments in time discretized operators and/or for evaluating existing methods, from the established measures for computational algorithms, the conclusion that the most effective (in the sense of convergence, namely, the stability and accuracy, and complexity, namely, the algorithmic formulation and algorithmic structure) computational algorithm should appear in a certain algorithmic structure of the design space amongst comparable algorithms is drawn. With this conclusion, and also with the notion of providing new avenues leading to improved algorithms by design, as an illustration, a novel computational algorithm which departs from the traditional paradigm (in the sense of LMS methods with which we are mostly familiar with and widely used in commercial software) is particularly designed into the perspective design space representation of comparable algorithms, and is termed here as the forward displacement non-linearly explicit L-stable (FDEL) algorithm which is unconditionally consistent and does not require non-linear iterations within each time step. From the established measures for comparable algorithms, simply for illustration purposes, the resulting design of the FDEL formulation is then compared with the commonly advocated explicit central difference method and the implicit Newmark average acceleration method (alternately, the same conclusion holds true against controllable numerically dissipative algorithms) which pertain to the class of linear multi-step (LMS) methods for assessing both linear and non-linear dynamic cases. The conclusions that the proposed new design of the FDEL algorithm which is a direct consequence of the present notion of design spaces and measures, is the most effective algorithm to-date to our knowledge in comparison to the class of second-order accurate algorithms pertaining to LMS methods for routine and general non-linear dynamic situations is finally drawn through rigorous numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.

MPI-based parallel finite element approaches for implicit nonlinear dynamic analysis employing sparse PCG solvers

This paper presents three formulations combining domain decomposition based finite element method with linear preconditioned conjugate gradient (LPCG) technique for solving large-scale problems in structural mechanics on parallel processing machines. In the first formulation called the Global Interface Formulation (GIF), the PCG algorithm is applied on the assembled interface stiffness coefficient matrices of all submeshes. The second formulation called Local Submesh Formulation (LSF) operates on the local unassembled submesh matrices and the preconditioner is constructed using the local submesh information. In the third formulation called Local Interface Formulation (LIF), the sparse PCG algorithm is formulated using the unassembled local schur complement matrices of submeshes. Both diagonal and incomplete Cholesky preconditioners have been employed. These domain decomposition based PCG algorithms have been implemented within a finite element code for nonlinear implicit transient dynamic analysis. Time integration is performed using Newmark-β constant average acceleration method. The parallel finite element code uses an MPI-based message passing approach to provide portable parallel execution on shared, distributed and distributed shared memory computers. Numerical experiments have been conducted on PARAM-10000, an Indian parallel supercomputer to evaluate the performance of the implicit parallel nonlinear finite element code employing the three proposed PCG formulations. Numerical studies indicate that the proposed parallel PCG formulations are highly adaptive for parallel computing and superior in performance when compared to the conventional domain decomposition algorithm with parallel direct solver. The LSF formulation, which is amenable for efficient implementation of communications by way of overlapping with computations found to be superior in performance compared to other two PCG formulations.

Studies of Newmark method for solving nonlinear systems: (II) Verification and guideline

Numerical properties of the Newmark method in the solution of nonlinear systems derived in the accompanying paper are thoroughly confirmed with numerical examples herein. It seems that analytical results can reveal the insight of the Newmark method in the step‐by‐step solution of linear and nonlinear systems. Although the constant average acceleration method is unconditionally stable for linear elastic systems these explorations confirm that it might lead to instability for nonlinear systems. In addition, numerical accuracy for period distortion and amplitude change is also shown to be consistent with the analytical predictions. Therefore, the performance of the Newmark method in the step‐by‐step solution of nonlinear systems is well investigated. As a result, a rough guideline to yield accurate solutions for the use of step‐by‐step integration methods to solve nonlinear systems is proposed.

Large free vibrations of a beam carrying a moving mass

The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution. In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.

Integrated equations of motion for direct integration methods

In performing the dynamic analysis, the step size used in a step-by-step integration method might be much smaller than that required by the accuracy consideration in order to capture the rapid chances of dynamic loading or to eliminate the linearization errors. It was first found by Chen and Robinson that these difficulties might be overcome by integrating the equations of motion with respect to time once. A further study of this technique is conducted herein. This include the theoretical evaluation and comparison of the capability to capture the rapid changes of dynamic loading if using the constant average acceleration method and its integral form and the exploration of the superiority of the time integration to reduce the linearization error. In addition, its advantage in the solution of the impact problems or the wave propagation problems is also numerically demonstrated. It seems that this time integration technique can be applicable to all the currently available direct integration methods.

A Comparison of Numerical Methods Applied to a Fractional Model of Damping Materials

The use of fractional derivatives in the constitutive equations of systems with damping materials provides a powerful tool for modeling these systems because the model does not exhibit many of the short comings of those based on integer-order derivatives. The resulting equations of motion possess closed-form solutions only for single-degree-of-freedom systems and only for a small number of loadings. For practical applications, therefore, the equations of motion must be solved using numerical methods. This paper presents two numerical schemes to solve single-degree- and multi-degree-of-freedom systems with fractional damp ing subjected to a number of commonly used loading conditions. The techniques employed are based on the central difference method and the average acceleration method. Whenever possible, the numerical results are compared with the analytical solutions. The results of the two numerical methods are essentially identical, with the exact solutions for zero initial conditions, but differ for nonzero conditions and large damping. For small damping, the average method has the advantage of its simpler formulation, especially with regard to the starting values. For arbitrary damping, however, the central difference method, in view of its robustness, is the preferred method.

From the trapezoidal rule to higher-order accurate and unconditionally stable time-integration method for structural dynamics

Since the trapezoidal rule was originally proposed as the constant average acceleration method by Newmark, it has been modified to various forms. However, most of the modifications were limited to its numerical coefficients and, as a result, little change in the characteristics was obtained like numerical dissipation with the loss of accuracy. In this work, the constant average acceleration, which is a fundamental feature of the trapezoidal rule, is investigated and generalized. Using the generalization of average acceleration concept, a higher-order accurate and unconditionally stable time-integration method is developed. The stability analysis is carried out for a system of single degree of freedom. The analysis proves that the present method is unconditionally stable. To observe the accuracy of the method, several numerical tests are performed and the results are compared with the results from the trapezoidal rule and the exact solution. From the numerical tests, it has been known that the present method has a higher order accuracy.

Time-dependent analysis of anisotropic viscoelastic composite shell structures

A numerical algorithm for analyzing the dynamic responses of anisotropic viscoelastic composite shell structures has been developed in the real time domain. The approach is based on variational principles and direct time integration by the Newmark average acceleration method. Recursion formulas have been obtained in order to reduce computer storage by requiring only the previous time solution to compute the subsequent one. Using the presently developed procedure, the dynamic responses of the viscoelastic graphite/epoxy composite plate and shell structures subjected to various loads are presented. The influence of temperature on the laminated shells is also examined.

Assessment of the accuracy of the newmark method in transient analysis of wave propagation problems

AbstractThe Newmark average acceleration method is non‐dissipative and unconditionally stable, but its accuracy in transient analysis of wave propagation problems depends not only on the spatial discretization but also on the temporal discretization. It has been found that the effects of spatial and temporal discretization when considered separately as commonly done, are far from adequate for most transient analysis. A better criterion manipulating the interdependent relationship between mesh size and time‐step magnitude is imperative to achieve sufficiently accurate results of analysis. In this paper, the accuracy of the Newmark method is investigated by considering the two basic sources of errors, namely, numerical amplitude dissipation and velocity dispersion. The effects of both spatial and temporal discretizations are considered. A new technique to describe the characteristics of various frequency spectra is established. A criterion for mesh design and the selection of time‐step magnitude is also proposed based on the combined effects of the amplitude dissipation and velocity dispersion. The efficiency and effectiveness of the proposed criterion are demonstrated using two one‐dimensional wave propagation problems. A two‐dimensional application shows that this criterion is equally applicable to multidimensional problems.

Effects of time step size on the response of a bilinear system, I: Numerical study

Numerical simulations have been carried out to study the dynamic response of a bilinear structural system subjected to harmonic excitations. By using Newmark's average acceleration method with a very small time step, subharmonic and chaotic responses have been obtained for certain parameter values. Poincaré maps, a divergence analysis with respect to initial conditions, and a bifurcation diagram with respect to frequency are presented. The effects of time step size on the numerical solutions of chaos are discussed. Insufficiently small time steps can lead to spurious existence of subharmonics and incomplete chaotic attractors. With small time steps, the correct shape of the chaotic attractor can be obtained. However, the response time history is sensitive to the time step size.

An unconditionally stable parallel transient algorithm with multi-step independent subdomain computation

A parallel algorithm for computing the transient response of linear structures is presented. In this algorithm, the transient response of subdomains can be integrated independently for multiple time steps at a time. The subdomain transient solutions that are obtained through independent computation are then corrected to obtain the solution of the global problem within each subdomain, using interface portions of independent subdomain solutions from every time step. The algorithm implements the Newmark average acceleration method, so it is unconditionally stable. Operation counts show that this algorithm requires significantly fewer operations per time step than the Newmark algorithm requires when applied to the global problem as a whole. A numerical example is presented.

A generalized least‐squares family of algorithms for transient dynamic analysis

AbstractBy use of the generalized least‐squares procedure, in conjunction with a finite element approximation in time, a simple three‐time‐level family of time integration schemes is derived. This results in fourth‐order accurate unconditionally stable algorithms and stable eighth‐order accurate non‐dissipative algorithms. Numerical examples show the accuracy of the proposed schemes in comparison with the Fox‐Goodwin formula and Newmark's average acceleration method.