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• 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]

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]