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Consistency Results for the Contact-Stabilized Newmark Method

Please always quote using this URN: urn:nbn:de:0297-zib-11164
  • 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 contact-stabilized Newmark method recently suggested by Deuflhard et al., which is compared to the classical Newmark method and an improved energy dissipative version due to Kane 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.

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Metadaten
Author:Corinna Klapproth, Anton Schiela, Peter Deuflhard
Document Type:ZIB-Report
Tag:Dynamical contact problems; Newmark method; Signorini condition; consistency; viscoelasticity
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Lxx Hyperbolic equations and systems [See also 58J45] / 35L85 Linear hyperbolic unilateral problems and linear hyperbolic variational inequalities [See also 35R35, 49J40]
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Hxx Dynamical problems / 74H15 Numerical approximation of solutions
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Mxx Special kinds of problems / 74M15 Contact
Date of first Publication:2009/02/23
Series (Serial Number):ZIB-Report (09-06)
ISSN:1438-0064
ZIB-Reportnumber:09-06
Published in:Appeared under the title "Consistency Results on Newmark Methods for Dynamical Contact Problems" in: Numer. Math., 116/1:65-94 (2010)
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