Directly self-starting higher-order implicit integration algorithms with flexible dissipation control for structural dynamics

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 sth-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.