A time‐marching collocation method based on quintic Hermite polynomials and adjustable acceleration and jerk constraints

Abstract

SUMMARYThis paper introduces a time‐marching algorithm for linear systems based on quintic Hermite polynomial displacement interpolation. This fully implicit, single‐step collocation method enforces acceleration and jerk constraints at a point in time that is selected by the user. The collocation point, then, is the parameter that determines accuracy, stability, and high‐frequency damping behavior. Numerical experiments indicate that the method converges at a rate of four, even for problems involving harmonic external forcing and viscous damping. Significantly, the convergence rate also does not deteriorate when algorithmic damping is employed. Period error is extremely small, on the order of a few percent at moderate frequencies even with algorithmic damping. The undamped algorithm is shown to have the highest accuracy and to be symplectic, but it is also conditionally stable and nondissipative—the exact stability limits are calculated. A wide range of parameter settings provide an unconditionally stable algorithm, which preserves low modes while rapidly dissipating high modes. Three finite element analysis numerical examples demonstrate the capabilities of the algorithm on one‐dimensional linear dynamic problems. This article is a US Government work and is in the public domain in the USA.