Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

Abstract

SummaryThis paper reviews the published composite three‐sub‐step implicit algorithms all of which adopt the trapezoidal rule in the first sub‐step. Three optimal families of three‐sub‐step implicit algorithms are developed to achieve second, third, and fourth‐order accuracy. The present second‐ and third‐order methods achieve identical effective stiffness matrices, thus embedding optimal spectral characteristics. Besides, both of them impose two dissipative parameters ( and ) to control numerical dissipation imposed in the second and third sub‐steps. The parameter adjusts overall numerical dissipation in the whole integration schemes, while the firstly used parameter can change numerical dissipation in the second sub‐step. The numerical example has shown the superiority of controlling middle sub‐step dissipation via . The present fourth‐order method is moderately dissipative due to achieving higher‐order accuracy, but it presents a more reasonable sub‐step division than the published fourth‐order three‐sub‐step trapezoidal rule. Linear and nonlinear examples are simulated to show the numerical performance and superiority of the three novel methods. This paper recommends using the proposed second‐ and third‐order three‐sub‐step methods to solve various dynamic problems.