Consistency results on Newmark methods for dynamical contact problems

Abstract

The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the classical Newmark method, the improved energy dissipative version due to Kane et al., and the contact-stabilized Newmark method recently suggested by Deuflhard et al. In the absence of contact, any such variant is equivalent to the Störmer-Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper. Mathematics Subject Classification (2000)35L85 · 74M15 · 74H15 · 65J15 Supported by the DFG Research Center Matheon, "Mathematics for key technologies: modelling, simulation, and optimization of real-world processes", Berlin.

Adaptive timestep control for the contact-stabilized Newmark method

“…3, we will work out an asymptotic error expansion of the contact-stabilized Newmark method. For the convenience of the reader, we recall known results concerning the sensitivity and consistency of the scheme from [21] and [29].…”

“…Remark 1 In [21], the authors have proven that the contact-stabilized Newmark method coincides with the modified Newmark scheme by Kane et al [15] in function space. Since we use the method of time layers, the numerical analysis in this paper will cover both Newmark methods simultaneously.…”

“…In view of existing perturbation and consistency results, see [20] and [21], we consider linear viscoelastic bodies fulfilling the Kelvin-Voigt constitutive law. For the convenience of the reader, here we merely collect the notation used therein.…”

“…The latter feature is achieved by performing a discrete L 2 -projection at contact interfaces at each timestep. In [21], the authors have given the generalization of the contact-stabilized Newmark method to the viscoelastic case. In order to fix notation, let the continuous time interval [t 0 , T ] be subdivided by N τ + 1 discrete time points t 0 < t 1 < · · · < t N τ = T which are forming an equidistant mesh τ = {t 0 , t 1 , .…”

“…In the unconstrained situation, the symmetric Newmark scheme is equivalent to the Störmer-Verlet scheme which is well-known to be second order consistent (see, e.g., [14]). In the constrained situation, we have proven an estimate for the consistency error of the classical Newmark method, the modified Newmark method by Kane et al, and the contact-stabilized Newmark method under the assumption of bounded total variation of the solution [21].…”