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  <doc>
    <id>1116</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2009-02-23</completedDate>
    <publishedDate>2009-02-23</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Consistency Results for the Contact-Stabilized Newmark Method</title>
    <abstract language="eng">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.</abstract>
    <identifier type="serial">09-06</identifier>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="opus3-id">1165</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-11164</identifier>
    <enrichment key="SourceTitle">Appeared under the title "Consistency Results on Newmark Methods for Dynamical Contact Problems" in: Numer. Math., 116/1:65-94 (2010)</enrichment>
    <author>Corinna Klapproth</author>
    <submitter>unknown unknown</submitter>
    <author>Anton Schiela</author>
    <author>Peter Deuflhard</author>
    <series>
      <title>ZIB-Report</title>
      <number>09-06</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Dynamical contact problems</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>viscoelasticity</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Signorini condition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>consistency</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Newmark method</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="35L85">Linear hyperbolic unilateral problems and linear hyperbolic variational inequalities [See also 35R35, 49J40]</collection>
    <collection role="msc" number="74H15">Numerical approximation of solutions</collection>
    <collection role="msc" number="74M15">Contact</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="deuflhard">Deuflhard, Peter</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/1116/consistency.pdf</file>
    <file>https://opus4.kobv.de/opus4-zib/files/1116/consistency.ps</file>
  </doc>
  <doc>
    <id>1079</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
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    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2008-07-10</completedDate>
    <publishedDate>2008-07-10</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A Perturbation Result for Dynamical Contact Problems</title>
    <abstract language="eng">This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.</abstract>
    <identifier type="serial">08-27</identifier>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="opus3-id">1113</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-10793</identifier>
    <enrichment key="SourceTitle">Appeared in: Numer. Math. Theor. Meth. Appl. 2 (2009)</enrichment>
    <author>Corinna Klapproth</author>
    <submitter>unknown unknown</submitter>
    <author>Peter Deuflhard</author>
    <author>Anton Schiela</author>
    <series>
      <title>ZIB-Report</title>
      <number>08-27</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Dynamical contact problems</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>stability</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>(visco-)elasticity</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Signorini condition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Newmark method</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="35L85">Linear hyperbolic unilateral problems and linear hyperbolic variational inequalities [See also 35R35, 49J40]</collection>
    <collection role="msc" number="74H55">Stability</collection>
    <collection role="msc" number="74M15">Contact</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compmed">Computational Medicine</collection>
    <collection role="persons" number="deuflhard">Deuflhard, Peter</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/1079/ZR_08_27.pdf</file>
    <file>https://opus4.kobv.de/opus4-zib/files/1079/ZR_08_27.ps</file>
  </doc>
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