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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]