Structure-dependent Integration Method Research Articles

Insight Into Feasibility of Structure-Dependent Methods for Dynamic Analysis

The first family of structure-dependent integration methods have been successfully developed for nonlinear dynamic analysis. Although its numerical properties were evaluated and its performance was numerically corroborated for both linear and nonlinear systems, its feasibility is still under debate due to the lack of a theoretical background. It seems that an eigen-based theory can provide a fundamental basis for the proof of the feasibility of structure-dependent integration methods. This can be manifested from each major stage of the development of structure-dependent integration methods. Therefore, the development of the first family of structure-dependent integration methods will be presented and the correlation between each major stage and an eigen-based theory will be explored and explained. Besides, this developing sequence can lay a typical procedure for developing a general structure-dependent integration method.

An Unusual Damped Stability Property and its Remedy for an Integration Method

An unusual stability property is found for a structure-dependent integration method since it exhibits a different nonlinearity interval of unconditional stability for zero and nonzero damping. Although it is unconditionally stable for the systems of stiffness softening and invariant as well as most systems of stiffness hardening, an unstable solution that is unexpected is obtained as it is applied to solve damped stiffness hardening systems. It is found herein that a nonlinearity interval of unconditional stability for a structure-dependent method may be drastically shrunk for nonzero damping when compared to zero damping. In fact, it will become conditionally stable for any damped stiffness hardening systems. This might significantly restrict its applications. An effective scheme is proposed to surmount this difficulty by introducing a stability factor into the structure-dependent coefficients of the integration method. This factor can effectively amplify the nonlinearity intervals of unconditional stability for structure-dependent methods. A large stability factor will result in a large nonlinearity interval of unconditional stability. However, it also introduces more period distortion. Consequently, a stability factor must be appropriately selected for accurate integration. After choosing a proper stability factor, a structure-dependent method can be widely and easily applied to solve general structural dynamic problems.

One-Parameter Controlled Non-Dissipative Unconditionally Stable Explicit Structure-Dependent Integration Methods with No Overshoot

Although many families of integration methods have been successfully developed with desired numerical properties, such as second order accuracy, unconditional stability and numerical dissipation, they are generally implicit methods. Thus, an iterative procedure is often involved for each time step in conducting time integration. Many computational efforts will be consumed by implicit methods when compared to explicit methods. In general, the structure-dependent integration methods (SDIMs) are very computationally efficient for solving a general structural dynamic problem. A new family of SDIM is proposed. It exhibits the desired numerical properties of second order accuracy, unconditional stability, explicit formulation and no overshoot. The numerical properties are controlled by a single free parameter. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as the WBZ-α method, HHT-α method and generalized-α method. Although this family method has unconditional stability for the linear elastic and stiffness softening systems, it becomes conditionally stable for stiffness hardening systems. This can be controlled by a stability amplification factor and its unconditional stability is successfully extended to stiffness hardening systems. The computational efficiency of the proposed method proves that engineers can do the accurate nonlinear analysis very quickly.

An Eigen-Based Theory For Structure-Dependent Integration Methods for Nonlinear Dynamic Analysis

An eigen-based theory for structure-dependent integration methods is constructed and it can provide the fundamental basis for the successful development of this type of integration methods. It is proved that a structure-dependent integration method can simultaneously combine unconditional stability and explicit formulation since it is proposed to accurately integrate low-frequency modes while no instability is guaranteed for high-frequency modes. In general, it is promising for solving inertial problems, where the total response is dominated by low-frequency modes. In addition, it can be explicitly and implicitly implemented for time integration although an explicit implementation is of practical significance since it can save many computational efforts due to the combination of unconditional stability and explicit formulation. A typical procedure to develop structure-dependent integration methods is constructed. In general, a coupled equation of motion for a multiple degree of freedom system can be decomposed into a set of uncoupled modal equations of motion by means of an eigen-decomposition technique. Next, an eigen-dependent integration method is developed to solve each modal equation of motion. Consequently, all the eigen-dependent integration methods are combined to form a structure-dependent integration method by employing a reverse procedure of the eigen-decomposition technique.

A dissipative family of eigen-based integration methods for nonlinear dynamic analysis

A novel family of controllable, dissipative structure-dependent integration methods is derived from an eigen-based theory, where the concept of the eigenmode can give a solid theoretical basis for the feasibility of this type of integration methods. In fact, the concepts of eigen-decomposition and modal superposition are involved in solving a multiple degree of freedom system. The total solution of a coupled equation of motion consists of each modal solution of the uncoupled equation of motion. Hence, an eigen-dependent integration method is proposed to solve each modal equation of motion and an approximate solution can be yielded via modal superposition with only the first few modes of interest for inertial problems. All the eigen-dependent integration methods combine to form a structure-dependent integration method. Some key assumptions and new techniques are combined to successfully develop this family of integration methods. In addition, this family of integration methods can be either explicitly or implicitly implemented. Except for stability property, both explicit and implicit implementations have almost the same numerical properties. An explicit implementation is more computationally efficient than for an implicit implementation since it can combine unconditional stability and explicit formulation simultaneously. As a result, an explicit implementation is preferred over an implicit implementation. This family of integration methods can have the same numerical properties as those of the WBZ-α method for linear elastic systems. Besides, its stability and accuracy performance for solving nonlinear systems is also almost the same as those of the WBZ-α method. It is evident from numerical experiments that an explicit implementation of this family of integration methods can save many computational efforts when compared to conventional implicit methods, such as the WBZ-α method.

Non-iterative methods for dynamic analysis of nonlinear velocity-dependent problems

A class of nonlinear velocity-dependent problems must be solved iteratively for conventional integration methods since there exists no completely explicit integration method among them. A completely explicit structure-dependent integration method is developed in this work, and it can be applied to solve such problems non-iteratively. A completely explicit method for time integration is characterized by the adoption of explicit difference formulas for both the displacement and velocity increment. It can be derived from an eigen-based theory, and this theory can provide a fundamental basis for the feasibility of time integration. The new method generally has no serious disadvantages of weak instability and unusual overshoot in high-frequency steady-state responses that have been found in the current completely explicit structure-dependent integration methods. It is analytically and numerically verified that it can be used to solve nonlinear velocity-dependent problems without nonlinear iterations and can have a comparable accuracy in contrast to the solutions obtained from conventional integration methods with an iteration procedure. Its computational efficiency due to no nonlinear iterations is also affirmed by numerical examples.

A dual family of dissipative structure-dependent integration methods for structural nonlinear dynamics

A dual family of dissipative structure-dependent integration methods is proposed for structural nonlinear dynamics. It not only can be a family of two-step integration methods but also can be a family of one-step integration methods correspondingly. This family of methods is derived from an ingenious arrangement of the displacement difference equation, where the previous step data are applied to replace the previous two-step data by means of the asymptotic equation of motion. It has desirable properties, such as unconditional stability, explicitness of each time step, second-order accuracy and high-frequency numerical damping. In addition, it has no adverse properties that have been found in some structure-dependent integration methods, such as weak instability, conditional stability for stiffness hardening systems, high-frequency overshooting in steady-state responses and poor capability of seizing high nonlinearity. This family of methods contains most current semi-explicit, structure-dependent integration methods as it is a family of two-step integration methods although it also covers many brand-new members of two-step integration methods, and it is a brand-new family of one-step integration methods. It has the same properties as those of the generalized- $$\alpha $$ method for linear elastic systems. However, it saves many computational efforts for solving inertial problems due to no involvement of nonlinear iterations per time step.

A remedy for a family of dissipative, non-iterative structure-dependent integration methods

A family of the structure-dependent methods seems very promising for time integration since it can simultaneously have desired numerical properties, such as unconditional stability, second-order accuracy, explicit formulation and numerical dissipation. However, an unusual overshoot, which is essentially different from that found by Goudreau and Taylor in the transient response, has been experienced in the steady-state response of a high frequency mode. The root cause of this unusual overshoot is analytically explored and then a remedy is successfully developed to eliminate it. As a result, an improved formulation of this family method can be achieved.

Elimination of Overshoot in Forced Vibration Responses for Chang Explicit Family Methods

AbstractAn unusual overshooting behavior might be experienced in the forced response of a high-frequency mode for a structure-dependent integration method. This unusual overshooting is different fr...

An unusual amplitude growth property and its remedy for structure-dependent integration methods

An unusual amplitude growth in the steady-state response of a high frequency mode for a structure-dependent integration method is numerically and analytically identified. This is a brand new type of amplitude growth and it has never been found in the conventional integration methods. The root cause of this unusual amplitude growth can be revealed by examining the local truncation error constructed from a forced vibration error, where the dominant error term for high frequency modes plays the key role for this unusual amplitude growth. An effective remedy is proposed by introducing a load-dependent term into the difference equation for displacement and/or velocity increment to remove the dominant error term. As a result, this adverse amplitude growth can be automatically removed. Consequently, the inclusion of the load-dependent term in the formulation of the difference equation for displacement increment and/or velocity increment is inevitable for a general structure-dependent integration method.

Assessments of Structure-Dependent Integration Methods with Explicit Displacement and Velocity Difference Equations

AbstractA family of structure-dependent integration methods has been proposed by Guiet al. for time integration. Although it has desirable numerical properties, such as unconditional stability, explicit formulation and second-order accuracy, it has some adverse properties, such as a poor capability to capture structural nonlinearity, an overshoot in a high frequency steady- state response and a weak instability in the high frequency response of nonzero initial conditions. The causes of these adverse properties are explored. A poor capability to capture structural nonlinearity may originate from the convergence rate of 1 in velocity error. This family method has an overshoot in a high frequency steady-state response and this overshoot can be eliminated by adding a load-dependent term into the displacement difference equation. It is also analytically verified that the family method generally has no weak instability. However, the special member withλ= 4, i.e., CR explicit method, is shown to have a weak instability. Thus, it must be prohibited from practical applications although many applications of this method were found in the literature.

Assessments of dissipative structure-dependent integration methods

Two Chang-a dissipative family methods and two KR-a family methods were developed for time integration recently. Although the four family methods are in the category of the dissipative structure-dependent integration methods, their performances may be drastically different due to the detrimental property of weak instability or overshoot for the two KR-a family methods. This weak instability or overshoot will result in an adverse overshooting behavior or even numerical instability. In general, the four family methods can possess very similar numerical properties, such as unconditional stability, second-order accuracy, explicit formulation and controllable numerical damping. However, the two KR-a family methods are found to possess a weak instability property or overshoot in the high frequency responses to any nonzero initial conditions and thus this property will hinder them from practical applications. Whereas, the two Chang-a dissipative family methods have no such an adverse property. As a result, the performances of the two Chang-a dissipative family methods are much better than for the two KR-a family methods. Analytical assessments of all the four family methods are conducted in this work and numerical examples are used to confirm the analytical predictions.

Improved formulation for a structure-dependent integration method

Structure-dependent integration methods seem promising for structural dynamics applications since they can integrate unconditional stability and explicit formulation together, which can enable the integration methods to save many computational efforts when compared to an implicit method. A newly developed structure-dependent integration method can inherit such numerical properties. However, an unusual overshooting behavior might be experienced as it is used to compute a forced vibration response. The root cause of this inaccuracy is thoroughly explored herein. In addition, a scheme is proposed to modify this family method to overcome this unusual overshooting behavior. In fact, two improved formulations are proposed by adjusting the difference equations. As a result, it is verified that the two improved formulations of the integration methods can effectively overcome the difficulty arising from the inaccurate integration of the steady-state response of a high frequency mode.

Unusual Overshooting in Steady-state Response for Structure-dependent Integration Methods

It is numerically illustrated that an unusual overshooting phenomenon might occur in the solution of a forced vibration response if a structure-dependent integration method is applied to conduct the analysis, such as Chang explicit method [2002] and CR explicit method [2008]. This overshooting may occur in the steady-state response of a high frequency mode and it becomes significant as the natural frequency of the mode increases. This unusual overshooting behavior can be successfully detected by using a local truncation error derived from a forced vibration response rather than a free vibration response. A missing loading term in the difference equation for displacement increment is responsible for this unusual overshoot. It is analytically and numerically verified that the unusual overshooting behavior will automatically disappear after introducing the missing loading term into the difference equation for displacement increment.

A Loading Correction Scheme for a Structure-Dependent Integration Method

A structure-dependent integration method may experience an unusual overshooting behavior in the steady-state response of a high frequency mode. In order to explore this unusual overshooting behavior, a local truncation error is established from a forced vibration response rather than a free vibration response. As a result, this local truncation error can reveal the root cause of the inaccurate integration of the steady-state response of a high frequency mode. In addition, it generates a loading correction scheme to overcome this unusual overshooting behavior by means of the adjustment the difference equation for displacement. Apparently, these analytical results are applicable to a general structure-dependent integration method.

A family of dissipative structure-dependent integration methods

new family of structure-dependent integration methods is developed to enhance with desired numerical damping. This family method preserves the most important advantage of the structure-dependent integration method, which can integrate unconditional stability and explicit formulation together, and thus it is very computationally efficient. In addition, its numerical damping can be continuously controlled with a parameter. Consequently, it is best suited to solving an inertia-type problem, where the unimportant high frequency responses can be suppressed or even eliminated by the favorable numerical damping while the low frequency modes can be very accurately integrated.

A general technique to improve stability property for a structure-dependent integration method

Summary A general technique is proposed to improve the stability property of a structure-dependent integration method, which is very computationally efficient in solving inertia-type problems when compared with conventional integration methods, by introducing a stability amplification factor into the coefficient matrices of difference equations. As a result, an improved structure-dependent integration method with a better stability property can be achieved. The concept, derivation, and validation of this technique are intensively studied and are presented in this work. It is evident that the technique can be applied to any structure-dependent integration method to enhance its stability property. Copyright © 2014 John Wiley & Sons, Ltd.