Linear Elastic Systems Research Articles

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.

Explicit Pseudodynamic Algorithm with Improved Stability Properties

An explicit pseudodynamic algorithm with an improved stability property is proposed herein. This algorithm is shown to be unconditionally stable for any linear elastic systems and any instantaneous stiffness softening systems. The most attracting stability property is that it can have unconditional stability for the instantaneous hardening systems with the instantaneous degree of nonlinearity less than or equal to 2. This property has never been found among the currently available explicit algorithms. Hence, it may be applied to perform a general pseudodynamic test without considering the stability problem since it is rare for a civil engineering structure whose instantaneous degree of nonlinearity is greater than 2. This explicit algorithm can be implemented as a common explicit pseudodynamic algorithm, such as the use of the Newmark explicit method, since it does not involve any iteration procedure. In addition, it possesses comparable accuracy as that of a general second-order accurate integration method such as the Newmark explicit method. Both numerical and error propagation properties are analytically studied and numerical experiments are used to confirm these properties. Actual pseudodynamic tests attested to the feasibility of this proposed explicit pseudodynamic algorithm.

Nonlinear evaluations of unconditionally stable explicit algorithms

Two explicit integration algorithms with unconditional stability for linear elastic systems have been successfully developed for pseudodynamic testing. Their numerical properties in the solution of a linear elastic system have been well explored and their applications to the pseudodynamic testing of a nonlinear system have been shown to be feasible. However, their numerical properties in the solution of a nonlinear system are not apparent. Therefore, the performance of both algorithms for use in the solution of a nonlinear system has been analytically evaluated after introducing an instantaneous degree of nonlinearity. The two algorithms have roughly the same accuracy for a small value of the product of the natural frequency and step size. Meanwhile, the first algorithm is unconditionally stable when the instantaneous degree of nonlinearity is less than or equal to 1, and it becomes conditionally stable when it is greater than 1. The second algorithm is conditionally stable as the instantaneous degree of nonlinearity is less than 1/9, and becomes unstable when it is greater than 1. It can have unconditional stability for the range between 1/9 and 1. Based on these evaluations, it was concluded that the first algorithm is superior to the second one. Also, both algorithms were found to require commensurate computational efforts, which are much less than needed for the Newmark explicit method in general structural dynamic problems.

Accurate Integration of Nonlinear Systems Using Newmark Explicit Method

AbstractIn the step-by-step solution of a linear elastic system, an appropriate time step can be selected based on analytical evaluation resultsHowever, there is no way to select an appropriate time step for accurate integration of a nonlinear system. In this study, numerical properties of the Newmark explicit method are analytically evaluated after introducing the instantaneous degree of nonlinearity. It is found that the upper stability limit is equal to 2 only for a linear elastic system. In general, it reduces for instantaneous stiffness hardening and it is enlarged for instantaneous stiffness softening. Furthermore, the absolute relative period error increases with the increase of instantaneous degree of nonlinearity for a given product of the natural frequency and the time step. The rough guidelines for accurate integration of a nonlinear system are also proposed in this paper based on the analytical evaluation results. Analytical evaluation results and the feasibility of the rough guidelines proposed for accurate integration of a nonlinear system are confirmed with numerical examples.

A historical perspective of Menabrea’s theorem in elasticity

This paper presents the theorem proposed by Luigi Federico Menabrea to study linear elastic redundant systems. Some of Menabrea’s papers on the subject are examined, as well as the criticism and the corrections brought to his first proof. We consider Menabrea’s work in the frame of the studies of his contemporaries; we try to provide a historical and epistemological background for Menabrea’s theorem and for its consequences in modern mechanics.

Numerical dissipation for pseudodynamic testing of nonlinear systems

It is well known that for linear elastic systems favorable numerical dissipation can suppress the spurious growth of high frequency modes while low frequency modes are almost unaffected. Thus, it is interesting to analytically explore the performance of numerical dissipation in the solution of a nonlinear system. In this paper, a novel technique is introduced to evaluate the numerical and error propagation properties of the γ‐function explicit method for nonlinear systems. It is analytically and numerically verified that this method has favorable numerical dissipation not only for linear elastic but also nonlinear systems. In fact, the suppression of high frequency modes for nonlinear systems is analytically proved. The importance of the spectral radius for a nonlinear system is also addressed. The rate of suppression of the spurious growth of high frequency responses will be accelerated due to stiffness hardening while it is decelerated due to stiffness softening.

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.

Domain decomposition for linear elasticity with DG jumps and mortars

In this article, we present a domain decomposition method for solving a linear elasticity system. Data are transmitted by jumps, as in a discontinuous Galerkin method, and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains. A decoupling algorithm condenses the unknowns on the interface. The matrix of the system is constructed by parallel computation. Numerical experiments confirming the theory are presented.

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.

Energy transfer in the hybrid system dynamics (energy transfer in the axially moving double belt system)

First, as an introduction, using the author’s published references, a short survey of an analytical study of the energy transfer between two coupled subsystems, as well as between a linear and nonlinear oscillators of a hybrid system, in the free and forced vibrations of a different type of inter connections between subsystems is presented. Second, as author’s new research result, an analytical study of the energy transfer between two coupled like-string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system, in the free vibrations is presented. On the basis of the obtained analytical expressions for the kinetic and potential energy of the belts and potential energy of the of light pure elastic distributed layer numerous conclusions are derived. In the pure linear elastic double belt system no transfer energy between different eigen modes of transversal vibrations of the axially moving double belt system, but in every from of the set of the infinite numbers eigen modes, there are transfer energy between belts. Each of the eigen modes of the free transversal vibrations are like two-frequency. The change of the potential energy of the booth belts is four frequency, and interaction part of the potential energy is one frequency in the each eigen mode. Changes of the kinetic energy of the both belts of the sandwich double axially moving bet system is two frequency like oscillatory regimes with two time multiplicities of the eineg frequencies of the corresponding eigen amplitude mode.

Well-Posedness of Systems of Linear Elasticity with Dirichlet Boundary Control and Observation

An open-loop system of linear elasticity with Dirichlet boundary control and collocated observation is considered. The main result we obtained states that this system is well-posed in the sense of Salamon. The result deduces the exponential stability of the closed-loop system under proportional output feedback. Hence it answers positively an open question proposed by Liu and Krstić [IMA J. Appl. Math., 65 (2000), pp. 109–121]. Moreover, the well-posedness, together with the regularity of the open-loop system in the sense of Weiss that was obtained in our separate paper [S. G. Chai and B. Z. Guo, Feedthrough Operator for Linear Elasticity System with Boundary Control and Observation, Preprint, School of Computational and Applied Mathematics, University of the Witwatersrand, South Africa, 2008], makes this infinite-dimensional system parallel in many ways to a linear finite-dimensional system in the framework of well-posed and regular linear infinite-dimensional systems.

Decay estimates for the solutions of the system of crystal acoustics for cubic crystals

The aim of this paper is to study decay estimates for global solutions of the system of linear crystal elasticity in three space dimensions for cubic crystals in the generic nearly isotropic case. Our main result is that for large time, the solutions to the system decay with a decay rate of order t −1/2−1/κ for some κ ∈ N which is strictly positive (cf. Remark 1.2). We should observe that in the isotropic case, the optimal decay rate is t −1 and that the decay rate for the wave equation in n space variables is t −(n−1)/2 . On the other hand, the decay rate for solutions of the Maxwell system for optically biaxial crystals is of order t −1/2 .( See O. Liess,Asymptot. Anal. 4 (1991), 61-95.)

Variational approaches to solving initial-boundary-value problems in the dynamics of linear elastic systems

Problems of the controlled motion of an elastic body are considered in the linear theory. Using the method of integrodifferential relations, a family of quadratic functionals is introduced, which define the state of the elastic body, and variational formulations of the initial-boundary-value problem of dynamics are given. Euler's equations and boundary and terminal relations corresponding to them are obtained from the condition for the functionals to be stationary. It is shown that there is a relation between the proposed formulations and the Hamilton variational principle in the case of boundary-value and time-periodic problems of dynamics. A numerical algorithm is developed for finding the motions of an elastic body, based on piecewise-polynomial approximations and a criterion is proposed for estimating the quality of the approximate solutions. An example of the calculation and analysis of the forced transverse motions of a rectilinear beam with a square cross section is given for the three-dimensional model.

Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity. We study and implement discretizations based on such mixed finite element methods for the smooth domain formulations to the unilateral crack problems. We obtain convergence rates and optimal error estimates and we present some numerical experiments in agreement with the theoretical results.

Existence for an energetic problem in contact mechanics with dry friction

AbstractA simplified contact problem with dry friction is considered for a non–linear elastic system with two degrees of freedom. The simplification consits in neglecting kinetic energies or equivalently inertial forces. By proving existence it is shown under which conditions such simplifications are justifiable. The main focus is on the influence of a curved obstacle surface on the question of existence. The friction is modelled according to the Coulomb law and the coefficient of friction may vary along the obstacle surface. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Linearized Stability and Accuracy for Step-by-Step Solutions of Certain Nonlinear Systems

In this work, stability and accuracy of the Newmark method for nonlinear systems are obtained from a linearized analysis. This analysis reveals that an unconditionally stable integration method for linear elastic systems is unconditionally stable for nonlinear systems and a conditionally stable integration method for linear elastic systems remains conditionally stable for nonlinear systems except that its upper stability limit might vary with the step degree of nonlinearity and step degree of convergence. A sufficient condition to have a stable computation for nonlinear systems in a whole step-by-step integration procedure is also developed in this study. Furthermore, it is also found that numerical accuracy in the solution of nonlinear systems is closely related to the step degree of nonlinearity and step degree of convergence although its characteristics are similar to those of the preceding works for linear elastic systems. Since these results are obtained from a linearized analysis, they can be applicable to the nonlinear systems that satisfied the simplifications for the analysis but may not be applicable to general nonlinear systems.

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.

PERFORMANCE RELIABILITY BASED OPTIMIZATION CRITERION FOR ELASTIC STRUCTURES SUBJECT TO RANDOM LOADS

A stochastic optimization which makes use of a reliability constraint, expressing a performance requirement, is proposed for linear elastic systems subject to random dynamic loads. The criterion selected for the optimum design is the minimization of a deterministic measure, whereas the constraint condition requires that system failure probability must be smaller than a given allowable level, where failure is associated to the first exceeding of system response from a safe domain. The proposed method is then developed for the optimum design of a Tuned Mass Damper, utilized for reducing undesirable vibrations originated in structures by random loads.Unlike conventional TMD designs based on minimizing the mean-square response, a reliability-based performance measure is then considered in the proposed design, which properly accounts for uncertainties in loads. A sensitivity analysis is carried out and results are shown in a useful manner for TMD design decision support.

Semiactive control strategy for a phase II smart base isolated benchmark building

This paper presents a semiactive control strategy for the seismic protection of the phase II smart base isolated benchmark building subjected to near-fault earthquakes. Magnetorheological (MR) fluid dampers controlled by an optimal direct output feedback control algorithm with a voltage generator are used as semiactive control devices. The benchmark building is an eight-story base isolated building, and the superstructure is considered to be a linear elastic system with lateral–torsional behavior. To reduce the energy transmitted to the structure from the ground, a combination of linear and nonlinear bearings is installed at the base of the structure. It is difficult to design the controller for this benchmark problem because of the nonlinear behavior of the isolation members. Linear control theory is frequently applied to nonlinear structures because there are no appropriate and well-defined control theories that can consider general types of nonlinearities. Therefore, to improve control performance, the nonlinearities of the isolation members are considered indirectly with the linear control theory during the design process of the controllers for the MR dampers. Numerical simulation results under various earthquake inputs show that the proposed control strategy is effective in reducing the responses of the isolation members as well as the superstructure. Therefore, the proposed control system could be used as an improved control strategy for base isolated buildings subjected to near-fault earthquakes. Copyright © 2008 John Wiley & Sons, Ltd.

PERFORMANCE OF THE HHT-α METHOD FOR THE SOLUTION OF NONLINEAR SYSTEMS

The HHT-α method previously developed shows favorable numerical dissipation, in addition to unconditional stability, in the solution of linear elastic systems. However, its performance in the solution of nonlinear systems has not been studied yet. In this paper, the numerical properties of a subfamily of this integration method with -⅓ ≤ α ≤ 0, β = ¼(1 - α)2and γ = ½-α for the solution of nonlinear systems are analytically explored using a linearized analysis, where the stiffness term in total form is used to determine the restoring force. A theoretical proof of unconditional stability for nonlinear systems is presented. The subfamily of the integration method is verified to possess favorable numerical dissipation for both linear and nonlinear systems. Period distortion affected by the step degree of either nonlinearity or convergence is studied. Although the analysis is conducted for a single-degree-of-freedom nonlinear system, the application to a multiple-degree-of-freedom nonlinear system is also illustrated. It is confirmed that the performance of the HHT-α method for nonlinear systems is generally the same as that for linear elastic systems, except for the high dependence on the step degree of nonlinearity.