High-order accurate multi-sub-step implicit integration algorithms with dissipation control for second-order hyperbolic problems

Abstract

This paper develops an implicit family of sub-step integration algorithms, which firstly requires identical effective stiffness matrices and third-order consistency within each sub-step. Consequently, the trapezoidal rule has to be employed in the first sub-step and optimal spectral properties are naturally embedded into the proposed algorithms. The analysis reveals at the first time that the constructed s-sub-step implicit schemes with s ≤ 6 can reach sth-order accuracy when achieving dissipation control and unconditional stability simultaneously. Hence, only four cost-optimal high-order implicit algorithms corresponding to three, four, five, and six sub-steps are developed. Unlike some published high-order algorithms, four novel methods do not suffer from order reduction for solving forced vibrations. Moreover, the novel methods overcome the defect that the authors' previous high-order algorithms require an additional solution to obtain accurate accelerations. Linear and nonlinear examples are solved to confirm the numerical performance and superiority of four novel high-order algorithms.Keywords implicit time integration • composite multi-sub-step • dissipation control • high-order accuracy • optimal spectral features