The family of Newmark and generalized α-methods is extended to constrained mechanical systems by using simultaneous position and velocity stabilization as key ideas. In this way, the acceleration constraints need not be evaluated, and the overall algorithm is about as expensive as the application of a BDF method to the GGL-stabilized equations of motion. Moreover, the RATTLE method of molecular dynamics is included as special case. A convergence analysis of the presented α-RATTLE algorithm shows global second order in both position and velocity variables while the Lagrange multipliers are computed to first order accuracy. Additonally, the property of adjustable numerical dissipation carries over from the unconstrained case.
Structural dynamics applications feature a particular type of second order stiff equations, often in combination with low smoothness of the right hand side, large dimension and non-linear force terms. As alternative to implicit schemes, explicit Runge-Kutta-Nyström methods are analysed, with focus on low order and maximized stability domain since spurious high frequency oscillations need not be resolved. It turns out that it is possible to construct methods with a stability domain that stretches up to hω = 2s on the imaginary axis where h is the stepsize, ω the largest frequency in the system, and s the stage number. Some numerical examples show the competitiveness of the proposed methods.
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