Constant Average Acceleration Method Research Articles

Comparison of capability of time integration methods in capturing dynamic loading

Numerical properties of the time integration method proposed by the first author of this paper in 2007 are the same as those of the constant average acceleration method (AAM) for linear elastic systems, except that the capability to capture dynamic loading was not explored. It was found that there were different quadrature equations to predict the next step displacement increment. A modified quadrature equation of this method was derived so that the equation to determine the next step displacement was numerically equivalent to the equation used in the constant AAM. It was verified that the original form of this method, in general, had a better capability to capture dynamic loadings than the constant AAM. This excellent property, in addition to computational efficiency, will help to make this method competitive with general secondorder accurate integration methods.

A new family of explicit time integration methods

It is very promising for an integration method to possess unconditional stability and the explicitness of each time step simultaneously. For this purpose, a novel family of explicit methods, which can have unconditional stability, is developed and presented. For linear elastic systems, the numerical properties of the proposed family method are the same as the Newmark family method since they have the same characteristic equation. A subfamily of this family can have unconditional stability for linear elastic systems. However, the most important properties of this subfamily are unconditional stability for nonlinear systems, and comparable accuracy when compared to second-order accurate methods, such as the Newmark explicit method and constant average acceleration method. Hence, this subfamily might be very useful for general structural dynamic problems, where the response is dominated by low frequency modes only. This is because that the unconditional stability and comparable accuracy allow the use of a large time step, and the explicitness of each time step involves no iterative procedure. As a result, many computational efforts can be saved.

Nonlinear Performance of Explicit Pseudodynamic Algorithms

A previously published explicit method has been proved to have the same numerical properties as those of constant average acceleration method for linear elastic systems. Although its application to the pseudodynamic testing of nonlinear systems with high frequency modes has been conducted and thus unconditional stability for nonlinear systems was indicated, there is still lack of an analytical proof. In order to explore the nonlinear performance of this explicit pseudodynamic algorithm, a new parameter of instantaneous degree of nonlinearity is introduced to monitor the stiffness change between the stiffness at the end of a time step and the initial stiffness. This parameter enables basic analysis and error propagation analysis for a nonlinear system. In addition, it can also be applied to construct the rough guidelines to select an appropriate time step to conduct a pseudodynamic test although it is almost impossible to achieve this goal using the currently available techniques for a nonlinear system. Analytical results reveal that this algorithm can have unconditional stability for instantaneous stiffness softening and linear elastic systems while it has conditional stability for instantaneous stiffness hardening systems. The proposed rough guidelines for selecting a time step to yield a reliable pseudodynamic test are confirmed with numerical examples.

Numerical characteristics of constant average acceleration method in solution of nonlinear systems

The currently available evaluation technique for the characteristics of an integration method is usually constructed from a whole step‐by‐step integration procedure for linear elastic systems. In this paper, a technique based on linearized analysis is proposed to evaluate the numerical properties of a step‐by‐step integration method for each time step for both linear and nonlinear systems. For this purpose, two parameters are introduced for each time step. One is used to describe the degree of nonlinearity (DON) and the other is employed to monitor the degree of convergence (DOC) as an iteration procedure is used. It seems that the proposed evaluation technique can reliably predict the numerical properties of an integration method in the solution of nonlinear systems. Furthermore, a guideline to have accurate solutions of nonlinear systems is proposed and confirmed with numerical examples. Although the evaluation technique is conducted on a time step, it is still indicative for a complete integration procedure since this procedure consists of each time step. It is verified that the constant average acceleration method (CAAM) is spectrally stable for nonlinear systems in a whole integration procedure based on a linearized stability analysis.

An Efficient Computing Explicit Method for Structural Dynamics

An integration algorithm, which integrates the most important advantage of explicit methods of the explicitness of each time step and that of implicit methods of the possibility of unconditional stability, is presented herein. This algorithm is analytically shown to be unconditionally stable for any linear elastic and nonlinear systems except for the instantaneous stiffness hardening systems with the instantaneous degree of nonlinearity larger than 43 based on a linearized stability analysis. Hence, its stability property is better than the previously published algorithm (Chang, 2007, “Improved Explicit Method for Structural Dynamics,” J. Eng. Mech., 133(7), pp. 748–760), which is only conditionally stable for instantaneous stiffness hardening systems although it also possesses unconditional stability for linear elastic and any instantaneous stiffness softening systems. Due to the explicitness of each time step, the possibility of unconditional stability, and comparable accuracy, the proposed algorithm is very promising for a general structural dynamic problem, where only the low frequency responses are of interest since it consumes much less computational efforts when compared with explicit methods, such as the Newmark explicit method, and implicit methods, such as the constant average acceleration method.

An explicit method with improved stability property

AbstractA new explicit method is presented. This method has an enhanced stability property when compared with the previously published explicit method (J. Eng. Mech. 2007; 133(7):748–760), which is unconditionally stable for linear elastic and any instantaneous stiffness softening systems based on linearized analysis whereas it is only conditionally stable for any instantaneous stiffness hardening system. In contrast to the previously published explicit method, the most important improvement of the new explicit method is that it can have unconditional stability for general instantaneous stiffness hardening systems in addition to linear elastic and instantaneous stiffness softening systems. Since these stability results are obtained from a linearized stability analysis they are applicable to the non‐linear systems that satisfied the simplifications used for the analysis. This explicit method is also shown to be second‐order accurate. It is computationally efficient in the solution of a general structural dynamic problem, where the total response is dominated by low‐frequency modes, when compared with explicit methods, such as the Newmark explicit method and the family of explicit methods developed by Tamma et al. (Comput. Methods Appl. Mech. Eng. 2003; 192:257–290), and implicit methods, such as the constant average acceleration method. Some numerical examples are used to confirm the enhanced stability property and the efficiency in computing. Copyright © 2008 John Wiley & Sons, Ltd.

Explicit finite element method for calculation and analysis to the elasto-plastic dynamic response of fluid-saturated porous media

In order to describe the elasto-plastic dynamic response of fluid-saturated porous media, the incremental elasto-plastic wave propagation equations of fluid-saturated porous media are developed by the fundamental theory of continuum mechanics and appointing to the characteristic of fluid-saturated porous media. Then, the space discretization of these equations is performed to get their Galerkin formula. At last, the time discretization of this formula is carried out with the integral method which consists of central difference method and Newmark constant average acceleration method to get the explicit time integral formula for solving the wave propagation equations of porous media. On the basis of the integral formula mentioned above, the time-domain explicit finite element method is developed for calculation and analysis of the elasto-plastic dynamic response of fluid-saturated porous media. In this method, the decoupling technique is adopted and it does not need to solve simultaneous linear equations in each time step, so the computational effort and memory requirement can be reduced considerably by using this method.

Improved Explicit Method for Structural Dynamics

An explicit method, which simultaneously has the most promising advantages of the explicit and implicit methods, is presented. It is shown that numerical properties of the proposed explicit method are exactly the same as those of the constant average acceleration method for linear elastic systems. However, for nonlinear systems, it has unconditional stability for an instantaneous stiffness softening system and conditional stability for an instantaneous stiffness hardening system. This conditional stability property is much better than that of the Newmark explicit method. Hence, the proposed explicit method is possible to have the most important property of unconditional stability for an implicit method. On the other hand, this method can be implemented as simply as an explicit method, and hence, possesses the most important property of explicit implementation for an explicit method. Apparently, the integration of these two most important properties of explicit and implicit methods will allow the proposed explicit method to be competitive with other integration methods for structural dynamics.

Slosh dynamics of liquid-filled containers with submerged components using pressure-based finite element method

Seismic response of liquid storage tank can be strongly influenced by the presence of submerged components. This modification of the dynamic characteristics of the liquid tank systems with internal components can be very useful for improving their seismic behavior. In this paper, a pressure-based finite element technique has been developed to analyze the slosh dynamics of a partially filled rigid container with bottom-mounted submerged components. The fluid is assumed homogeneous, isotropic, inviscid, and to exhibit only limited compressibility. The problem is linearized assuming the frequency of the exciting oscillation not in the immediate neighborhood of the natural slosh frequency, so that the slope of the free surface is small. The linearized problem is spatially discretized using the Galerkin weighted residual method. Earthquake excitations are used as the prescribed boundary condition. The solution is advanced in time using Newmark's constant average acceleration method. The developed code has been used to investigate the effect of a bottom-mounted rectangular component on the slosh dynamics of a liquid-filled rigid container. Numerical results obtained are compared with the existing solutions to validate the code. The parametric study of the tank–fluid system shows the importance of height, width and location of the submerged structural components.

Nonlinear error propagation analysis of implicit pseudodynamic algorithm

An implementation of the HHT‐α method for pseudodynamic testing has been proposed by Chang and Mahin (1993). This implicit algorithm has been shown to be unconditionally stable and to possess less error propagation than any other implicit pseudodynamic algorithm. However, these conclusions are true only for linear elastic systems but might not be applicable to nonlinear systems. A technique is proposed to analyze the nonlinear error propagation of the implicit pseudodynamic algorithm, where only the member of the constant average acceleration method is investigated for simplicity. It is analytically and numerically verified that the constant average acceleration method might lead to numerical instability in the pseudodynamic testing of a nonlinear system although it is unconditionally stable for a linear elastic system. Furthermore, analytical results of the nonlinear error propagation analysis and numerical experiments conclude that the error propagation of the improved implementation is much better than that of the direct implementation. This is consistent with results from use on linear elastic systems.

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.

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.

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.

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.

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.

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.

Scheme to Improve Numerical Analysis of Hysteretic Dynamic Systems

This note presents a procedure that basically consists of the following: (1) The detection, as the step-by-step integration advances, of changes in load-deformation behavior; (2) the determination of the precise times at which such changes occur; (3) the reduction of the time step to calculate the response of the system at those times; and (4) the continuation of the step-by-step integration with adjusted load-deformation properties and the regular time step selected for the analysis. Presented also are the equations for the implementation of the proposed procedure in the particular case of Newmark's constant average acceleration method.