Multi-step time integration methods with desirable stability for structural dynamics with nonlinear stiffness

For structural dynamics systems with nonlinear stiffness, a two-step method with all parameters controlled by ρ_∞ which is the spectral radius for infinity frequency, called the ρ_∞-TSM, was constructed based on the parameter spectral analysis theory (Eur. J. Mech. A-Solid. 94: 104582 (2022)). The ρ_∞-TSM has unconditional stability, but it is second-order accurate only when ρ_∞ is equal to 1. To address this issue, a three-step method and a four-step method are designed in this work, and their numerical properties including stability, accuracy order, and calculation accuracy are investigated. Unlike the ρ_∞-TSM, when ρ_∞ is within a certain range less than 1, the three-step and four-step methods have second-order accuracy. Besides, they possess desirable stability and controllable high-frequency dissipation for both linear and nonlinear dynamic systems. Numerical experiments show that for nonlinear structural dynamics problems, the three-step and four-step methods have advantage in stability over the ρ_∞-Bathe method and the OALTS method, and which method has higher calculation accuracy depends on the problem to be solved.