Dual-Rate Integration Using Partitioned Runge-Kutta Methods for Mechanical Systems with Interacting Subsystems

Abstract

A framework is presented allowing dual-rate numerical integration of the equations of mechanical system dynamics to be considered as a form of Partitioned Runge-Kutta (PRK) integration. Certain coefficients of a PRK integrator are set to zero, so that Runge-Kutta integrators that constitute the PRK integrator can be made to have different numbers of stages. As a result, one Runge-Kutta integrator requires fewer function evaluations than the other does, which is a form of dual-rate integration. Well-established order of accuracy theory for PRK integrators is used to develop a rigorous methodology for designing explicit PRK dual-rate integrators. Stabilized Runge-Kutta theory developed for single-rate RungeKutta integrators is combined with PRK integrator theory to design PRK dual-rate integrators with the largest possible stability regions. Dual-rate PRK integrators created using these approaches are used to simulate the dynamics of vehicle systems that contain subsystems with higher frequency response characteristics than do the basic vehicle subsystems.

On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems

“…It has been also considered for HBVMs [14] and is implemented in the Matlab code HBVM [12]. The novelty, in the present case, is due to the approximation (45), which makes it extremely efficient. As a result, the iteration (46) modifies as follows: initialize ψ 0 for ℓ = 0, 1, .…”

“…Multi-frequency highly-oscillatory Hamiltonian problems appear often in mathematical models of real life applications such as molecular dynamics [48] or multibody mechanical systems [45,46]. They also occur when solving Hamiltonian PDEs by means of a proper space-semidiscretization.…”

Rosenbrock‐based algorithms and subcycling strategies for real‐time nonlinear substructure testing

“…In numerical analysis, it is custom to deal with these requirements by means of subcycling strategies. For instance, see Weiner et al [35] and Shome et al [36], among others, in the context of partitioned Runge-Kutta methods. Similarly, interfield parallel algorithms without and with numerical dissipation were developed for linear multistep methods [18,37].…”

Efficient coupling of multibody software with numerical computing environments and block diagram simulators

“…A closely related open field of research in the simulation of multidisciplinary systems is the use of multirate integration schemes, which improves the numerical efficiency during the simulation of interacting subsystems with very different time scales. Multirate algorithms have been developed ( [10,11]), while, however, the implementation of these techniques in the communication between software packages, specially when block-diagram software is involved, is still in progress. It is noteworthy that the numerical performance of multirate algorithms dependents greatly of the co-simulation strategy selected for solving the problem.…”