Directly Self-Starting Second-Order Explicit Integration Methods with Dissipation Control and Adjustable Bifurcation Points for Second-Order Initial Value Problems

Abstract

This paper proposes a directly self-starting second-order [Formula: see text]-sub-step explicit method, within which the three novel members are developed and analyzed for demonstration. The novel explicit schemes can output second-order accelerations for general structures, which cannot be achieved for the directly self-starting single- and two-sub-step algorithms. Each explicit algorithm achieves controllable algorithmic dissipation and adjustable bifurcation points. As a result, the novel explicit algorithms impose two algorithmic parameters, [Formula: see text] and [Formula: see text], to control numerical dissipation and bifurcation point, respectively. The parameter [Formula: see text], denoting the spectral radius at the bifurcation point, controls numerical dissipation, whereas the parameter [Formula: see text], denoting the bifurcation point, adjusts the amount of dissipation imposed in the low-frequency range. This paper provides users with two recommended selections of [Formula: see text]. Apart from these desirable features, each novel explicit algorithm attains the maximum stability bound, [Formula: see text] where [Formula: see text] denotes the number of sub-steps. The four- and five-sub-step algorithms perform optimization to reduce numerical low-frequency dissipation. As the number of sub-steps increases, the novel explicit algorithms can have better numerical characteristics without an increase in the computational effort. Numerical examples are simulated to validate the performance and superiority of the novel explicit algorithms.