Further insights into an implicit time integration scheme for structural dynamics

Summary
A comprehensive study of the two sub-steps composite implicit time integration scheme for the structural dynamics is presented in this paper. A framework is proposed for the convergence accuracy analysis of the generalized composite scheme. The local truncation errors of the acceleration, velocity, and displacement are evaluated in a rigorous procedure. The presented and proved accuracy condition enables the displacement, velocity, and acceleration achieving second-order accuracy simultaneously, which avoids the drawback that the acceleration accuracy may not reach second order. The different influences of numerical frequencies and time step on the accuracy of displacement, velocity, and acceleration are clarified. The numerical dissipation and dispersion and the initial magnitude errors are investigated physically, which measure the errors from the algorithmic amplification matrix's eigenvalues and eigenvectors, respectively. The load and physically undamped/damped cases are naturally accounted. An optimal algorithm-Bathe composite method (Bathe and Baig, 2005; Bathe, 2007; Bathe and Noh, 2012) is revealed with unconditional stability, no overshooting in displacement, velocity, and acceleration, and excellent performance compared with many other algorithms. The proposed framework also can be used for accuracy analysis and design of other multi-sub-steps composite schemes and single-step methods under physical damping and/or loading. Copyright © 2016 John Wiley & Sons, Ltd.

Accuracy of a composite implicit time integration scheme for structural dynamics

In this paper, a new family of controllably dissipative composite algorithms is developed to obtain reliable numerical response of structural dynamic problems. The proposed algorithm is a self-starting, unconditionally stable and second-order accurate three sub-step composite algorithm. The new method includes two optimal sub-families of algorithms, both of which can control numerical dissipations in the high-frequency range by an intuitive way, and their numerical dissipations can range from the non-dissipative case to the asymptotic annihilating case. Besides, they actually involve only one free parameter and always share the identical effective stiffness matrices inside three sub-step to save the computational cost, which does not hold in some existing sub-step algorithms. Some numerical examples are given to show the superiority of the new algorithm with respect to controllable numerical dissipations and the ability of capturing the free-play nonlinearity.

A novel family of controllably dissipative composite integration algorithms for structural dynamic analysis

A concept of hybrid-stiffness stiffened shell is proposed based on sub-panel elements to achieve a simultaneous buckling pattern, which can provide enhanced design flexibility to fully explore the load-carrying capacity of structures. Then, a novel hybrid model is established to improve the computational efficiency of post-buckling analysis for stiffened shells, where the Numerical Implementation of Asymptotic Homogenization Method is utilized to smear out the stiffeners. On this basis, an integrated optimization framework of sub-panel configurations and weld lands for stiffened shells is presented. Illustrative examples with single and multiple cutouts demonstrate the effectiveness of the proposed framework based on the concept of hybrid-stiffness stiffened shell.

Integrated optimization of hybrid-stiffness stiffened shells based on sub-panel elements

Optimization of a class of composite method for structural dynamics

Nonlinear elastic buckling and postbuckling analysis of cylindrical panels

A robust time integration scheme for snap-through buckling of shallow arches is proposed. The algorithm is a composite method that consists of three sub-steps. Numerical damping is introduced to the system by employing an algorithm similar to the backward differentiation formulas method in the last sub-step. Optimal algorithmic parameters are established based on stability criteria and minimization of numerical damping. The proposed method is accurate, numerically stable, and efficient as demonstrated through several examples involving loss of stability, large deformation, large displacements and large rotations.

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A robust composite time integration scheme for snap-through problems

This paper presents an analytical method to study the non-linear stability and remote unconnected equilibria of shallow arches with non-symmetric geometric imperfections. The exact solutions of the equilibria and critical loads are obtained. Unlike many previous studies, these solutions can be applied to arbitrary shallow arches with arbitrary geometric imperfections. It is found that slightly imperfect arches have multiple remote unconnected equilibria that cannot be obtained in experiments or using finite element simulations if a proper perturbation is not performed. The formulas to directly calculate the critical loads, including those of the remote unconnected equilibria, are also derived. The effect of asymmetric geometric imperfections on the equilibria and critical loads is revealed by applying the derived formulas to half-sine arches with different geometric imperfections.

Non-linear stability and remote unconnected equilibria of shallow arches with asymmetric geometric imperfections

Modeling and vibratory characteristics of a mass-carrying cable system with multiple pulley supports in span range

This paper presents an analytical method to investigate the effects of symmetric and asymmetric elastic supports on the nonlinear equilibria and buckling responses of shallow arches. It is found that arches with symmetric elastic supports can bifurcate into secondary paths with high-order symmetric modes. When a small asymmetry exists in the elastic supports, the equilibria of the arch may abruptly split and lead to the occurrence of remote unconnected equilibria. Such unconnected equilibria can be obtained experimentally or numerically using typical path following controls only with prior knowledge of location of these paths. A small asymmetry in the elastic supports may also make a secondary branch shrink into points connecting surrounding equilibria, resulting in the appearance of more limit points. The analytical solutions are also derived to directly calculate critical loads. We find that the magnitude of the stiffness of symmetric elastic supports has no influence on limits loads and bifurcation loads at branching into secondary paths with symmetric configurations, but greatly affect the bifurcation loads of secondary paths with asymmetric configurations. All critical loads are very sensitive to the degree of asymmetry in the elastic supports. The asymmetry in the supports reduces the top values of all pairs of critical loads compared to the case of symmetric elastic supports. The results obtained from the analytical derivations are confirmed using finite element analysis (FEA).

Yang Zhou

Papers

Nonlinear elastic buckling and postbuckling analysis of cylindrical panels

A robust composite time integration scheme for snap-through problems

Non-linear stability and remote unconnected equilibria of shallow arches with asymmetric geometric imperfections

Modeling and vibratory characteristics of a mass-carrying cable system with multiple pulley supports in span range

Nonlinear Buckling and Postbuckling of Shallow Arches With Vertical Elastic Supports