A simple explicit arc-length method using the dynamic relaxation method with kinetic damping

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Abstract

The explicit arc-length method is simulated to trace the post-buckling equilibrium path of structures by using dynamic relaxation method with kinetic damping. This method based on the cylindrical arc-length method does not require the computation and formulation of any tangent stiffness matrix to search the snap-through or snap-back problems. The convergence to the solution is achieved by using only vector equation with kinetic damping technique. Two approaches for cylindrical arc-length control are formulated with incremental and total displacement constraint. The merits of the explicit arc-length method, in tracing the post-buckling behavior of structures, are demonstrated by analyzing the numerical examples.

Introduction

In nonlinear finite element method (FEM) formulations, the Newton Raphson (NR) method is accepted as the effective and general numerical method to calculate the structural response by using implicit algorithm with a quadratic convergence rate. For the explicit algorithm, the dynamic relaxation method (DRM), which was proposed by Day [1], may be the successful one for the static nonlinear problems. In DRM process, the structural responses are quasi-statically damped and converged into the static equilibrium state consequently. The viscous or kinetic damping technique [2] can be used.

The DRM has been applied to the highly nonlinear problems in various fields, such as shape-finding and load analysis of cable-membrane tension structures [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], post-buckling analysis [15], [16], [17], [18], [19], [20], [21], [22], [23] and hydro dynamic FEM codes [24], [25], [26]. The effective multi-core parallel solution algorithm can be applied to the DRM procedures [7], [27], [28].

The characteristic feature of the DRM is that it does not need to formulate the stiffness matrix, but vector equations to obtain the unknown displacements. Therefore the numerical procedures are remarkably reduced and simplified. The derivation, assemble and linear equation routine for stiffness matrix are omitted consequently. The efficiency of the DRM is mainly dependant on the applied problems. The occurrence of singularity of stiffness matrix in highly nonlinear problems is the main reason to use the explicit DRM. However, the ineffectiveness of low convergence rate is the main drawback of its non-use of a matrix operation.

To trace the nonlinear equilibrium path of the problems, the arc-length method (ALM) was developed by the many pioneering researchers [29], [30], [31], [32], [33], [34], [35], [36]. The ALM has been accepted as the most effective numerical method for the post-buckling problems with NR type algorithms. And this type of algorithm has the difficulties to solve the singularity of the stiffness matrix near the critical point. Therefore explicit DRM can be applied to the highly nonlinear post-buckling problems because of its non-using matrix operation.

Han and Lee [14] have demonstrated the stabilizing process of unstable structures. Papadrakakis [15] has solved the post-buckling problems of space structures by using explicit algorithm with simple displacement incremental procedures. Ramesh and Krishnamoorthy [16] have proposed the explicit ALM to obtain the post-buckling nonlinear equilibrium path. In their formulation, the cylindrical arc-length constraint equation, proposed by Crisfield [33], has been used with the norm of total displacement and viscous damping [17], [18], [19], [20], [21] for the DRM procedures. However Pasqualino and Estefen [21] used the norm of incremental displacement for explicit ALM differently.

In this paper, the simple explicit cylindrical ALM has been used to trace the post-buckling equilibrium path of the structures by combining DRM with kinetic damping technique [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. The convergence to the solution is achieved by using only vector quantities. And the stiffness matrix and viscous damping terms are not required in its overall procedure. By using the kinetic damping technique without viscous damping term, the DRM procedures are simplified remarkably. The merits of the using explicit ALM, in tracing the post-buckling behavior of structures, are demonstrated by analyzing the numerical verification examples.

Section snippets

The formulation of DRM

The DRM proposed by Day [1] is an iterative nonlinear algorithm to find static equilibrium state of structures by minimizing the total potential energy of the structures. The fundamental feature of DRM is that the static equilibrium state of the structures is determined by the damped motion of themselves during the pseudo quasi-static time integration. A brief review of the DRM procedures is as follows [3], [10], [11], [14], [15].

The dynamic equilibrium equation can be written as follows:pt=Mat+

DRM with kinetic damping

The comprehensive detailed descriptions of the DRM with kinetic damping are given in the literatures [2], [3], [4], [7], [8], [9], [10], [11], [12], [13], [14]. We briefly introduce the simple and effective kinetic damping technique that applied in this study.

By considering the DRM without viscous damping, the analysis becomes simple, because the number of parameters can be reduced by the kinetic damping technique. In other words, the numerical analysis in the DRM can be controlled only with

Numerical method

In this section, by using the DRM with kinetic damping, the explicit nonlinear numerical algorithms are formulated. The explicit ALM can be expressed by adapting the DRM with the cylindrical arc-length constraint that proposed by Crisfield [33], [34], [35]. The explicit ALM, proposed by Ramesh and Krishnamoorthy [16] firstly, has been adopted the cylindrical arc-length constraint equation that proposed by Crisfield. The total displacement [16] or incremental displacement [21] can be used for

Verification examples

In an effort to verify the computational merits and limitations of the proposed explicit ALM by using the DRM with kinetic damping, various numerical examples were solved with the proposed approach, and comparisons were made with previously published results.

Conclusion

In this study, two types of explicit ALM are presented to trace the post-buckling equilibrium path of structures by using DRM with kinetic damping. The kinetic damping technique is used for the quasi-static damped motion of the structures to enable the DRM process simple an efficient for explicit ALM algorithm. By using the proposed explicit ALM, we obtained the successful results for complex snap-back problems that have not yet reported by explicit ALM previously. The proposed explicit ALM

Acknowledgements

This research was supported by WCU (World Class University) program through the National Research. Foundation of Korea funded by the Ministry of Education, Science and Technology (R32-2008-000-20042-0).

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