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A novel family of composite sub-step algorithms with controllable numerical dissipations is proposed in this paper to obtain reliable numerical responses in structural dynamics. The new scheme is a self-starting, unconditionally stable and second-order-accurate two-sub-step algorithm with the same computational cost as the Bathe algorithm. The new algorithm can control continuously numerical dissipations in the high-frequency range in an intuitive way, and the ability of numerical dissipations can range from the non-dissipative trapezoidal rule to the asymptotic annihilating Bathe algorithm. Besides, the new algorithm only involves one free parameter and always achieves the identical effective stiffness matrices inside two sub-steps, which is not always achieved in three Bathe-type algorithms, to reduce the computational cost in the analysis of linear systems. Some numerical examples are given to show the superiority of the new algorithm over the Bathe algorithm and the CH- α algorithm. [ABSTRACT FROM AUTHOR]

• Further studies on the Bathe algorithm is given and hence a novel class of the Bathe-like algorithm is presented. • It provides a new family of algorithms with the optimal numerical characteristics. • It gives a novel family of algorithms, which can be regarded as the alternative to the original Bathe algorithm. • This study shows that the Bathe algorithm can reduce to the trapezoidal rule and the backward Euler formula. The Bathe algorithm is superior to the trapezoidal rule in solving nonlinear problems involving large deformations and long-time durations. Generally, the parameter γ = 2 − 2 is highly recommended due to its optimal numerical properties. This paper further studies this implicit composite sub-step algorithm and thus presents a class of the Bathe-like algorithm. It not only gives a novel family of composite algorithms whose numerical properties are the exactly same as the original Bathe algorithm with γ = 2 − 2 , but also provides the generalized alternative to the original Bathe algorithm with any γ. In this study, it has been shown that the Bathe-like algorithm, including the original Bathe algorithm, can reduce to two common single-step algorithms: the trapezoidal rule and the backward Euler formula. Besides, a new parameter called the algorithmic mode truncation factor is firstly defined to describe the numerical property of the Bathe-like algorithm and it can estimate which modes to be damped out. Finally, numerical experiments are provided to show the superiority of the Bathe-like algorithm over some existing methods. For example, the novel Bathe-like algorithms are superior to the original Bathe algorithm when solving the highly nonlinear pendulum. [ABSTRACT FROM AUTHOR]

• The new algorithm is second-order accurate, unconditionally stable (L-stable) and self-starting. • The new algorithm shares the identical effective stiffness matrices inside three sub-step. • There is no overshoot for the proposed algorithm when nonzero initial displacement and/or velocity are used. • The second-order accuracy is obtained in its final form, but it is not required in each sub-step. In this paper, a novel three sub-step composite algorithm with desired numerical properties is developed. The proposed method is a self-starting, unconditionally stable and second-order accurate implicit algorithm without overshoot. Particularly, the second-order accuracy in time is achieved in its final form, but it is not required in each sub-step. Its unique algorithmic parameter is analyzed to achieve the unconditional stability and it shares the identical effective stiffness matrix inside three sub-steps to save the computational cost in linear analyses. The same as the Bathe algorithm, the proposed algorithm is always L-stable, meaning that the spurious high-frequency modes can be effectively eliminated. Three numerical examples are simulated to illustrate the superiority of the proposed algorithm over some existing implicit algorithms. The first numerical simulation, solving a linear single-degree-of-freedom system, shows less period elongation errors and the second-order accuracy of the present scheme. The second one, a clamped-free bar excited by the end load, shows the ability of effectively damping out the unexpected high-frequency modes. The last example solves the nonlinear mass-spring system with variable degree-of-freedoms and illustrates that the composite sub-step algorithm can save more computational cost than the traditional implicit algorithm when the integration step size is selected appropriately. [ABSTRACT FROM AUTHOR]

Highlights • The new algorithm is the second-order accurate, unconditionally stable (L-stable) and self-starting. • The new algorithm shares the identical effective stiffness matrices inside two sub-steps. • The new method does not involve any artificial parameters and additional variable, such as the Lagrange multipliers. • The new scheme achieves the same numerical properties as the Bathe algorithm, but requires less matrix-vector operations. Abstract This paper presents a new composite sub-steps algorithm for solving reliable numerical responses in structural dynamics. The newly developed algorithm is a two sub-steps, second-order accurate and unconditionally stable implicit algorithm with the same numerical properties as the Bathe algorithm. The detailed analysis of the stability and numerical accuracy is presented for the new algorithm, which shows that its numerical characteristics are identical to those of the Bathe algorithm. Hence, the new sub-steps scheme could be considered as an alternative to the Bathe algorithm. Meanwhile, the new algorithm possesses the following properties: (a) it produces the same accurate solutions as the Bathe algorithm for solving linear and nonlinear problems; (b) it does not involve any artificial parameters and additional variables, such as the Lagrange multipliers; (c) The identical effective stiffness matrices can be obtained inside two sub-steps; (d) it is a self-starting algorithm. Some numerical experiments are given to show the superiority of the new algorithm and the Bathe algorithm over the dissipative CH- α algorithm and the non-dissipative trapezoidal rule. [ABSTRACT FROM AUTHOR]

No literature has reported an explicit integration algorithm to achieve controllable numerical dissipation and identical third-order accuracy simultaneously. This paper develops two novel explicit integration algorithms to achieve this goal well without increasing computational complexity. In detail, two novel methods are identical third-order accuracy, so avoiding the order reduction for solving damped and forced vibrations, and both of them provide a full range of dissipation control via adjusting their unique algorithmic parameter. Apart from these two highlights, two novel methods embed another two enjoyably advantages. One is to present a significantly larger stability limit than the published third-order explicit schemes. Two novel methods provide the maximum stability limit, reaching 2 3 in the non-dissipative case, getting close to four. The other is to maintain a relatively little computational cost. Two novel methods require explicit solutions only twice within each time step. Therefore, two novel methods are significantly superior to other composite sub-step explicit schemes with respect to the accuracy, stability, dissipation control, and computational cost. Numerical examples are also performed to confirm the numerical performance and superiority of two novel explicit methods. • Two novel fully explicit methods are proposed based on the composite two-sub-step technique. • Two novel methods are identically third-order accurate for solving general structures. • Two novel methods can control numerical dissipation imposed at the bifurcation point. • Two novel methods provide a larger stability limit and require a relatively little computational cost. [ABSTRACT FROM AUTHOR]

An implicit family of composite s -sub-step integration algorithms is developed in this paper. The proposed composite s -sub-step scheme is firstly designed to satisfy two requirements. One is the directly self-starting property, eliminating any starting procedures and avoiding computing the initial acceleration vector. The other is identical effective stiffness matrices within each sub-step, embedding optimal spectral properties. The analysis reveals that the composite s -sub-step implicit schemes with s ≤ 6 can achieve s th-order of accuracy when embedding the dissipation control and unconditional stability simultaneously, and that in case of s ≥ 7 , the increase of accuracy requires more sub-steps. Then, only the first six economical composite multi-sub-step schemes are developed. Remarkably, two approaches are also constructed to output accurate accelerations, which is also regarded as another minor superiority. Unlike some published higher-order algorithms, the proposed methods do not suffer from the order reduction and they provide the designed order of accuracy for solving general structures. Linear and nonlinear examples are finally solved to confirm the numerical performance and superiority of the proposed methods. • Four high-order algorithms are proposed and do not suffer from the order reduction. • Each algorithm achieves identical effective stiffness matrices. • Each algorithm can control numerical high-frequency dissipation. • Each algorithm is directly self-starting and unconditionally stable. [ABSTRACT FROM AUTHOR]