@misc{KlapprothSchielaDeuflhard, author = {Klapproth, Corinna and Schiela, Anton and Deuflhard, Peter}, title = {Consistency Results for the Contact-Stabilized Newmark Method}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11164}, number = {09-06}, 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 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 St{\"o}rmer-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.}, language = {en} } @misc{KlapprothDeuflhardSchiela, author = {Klapproth, Corinna and Deuflhard, Peter and Schiela, Anton}, title = {A Perturbation Result for Dynamical Contact Problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10793}, number = {08-27}, abstract = {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.}, language = {en} } @misc{Klapproth, author = {Klapproth, Corinna}, title = {The Contact-Stabilized Newmark Method - Consistency Error of a Spatiotemporal Discretization}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-15198}, abstract = {The paper considers an improved variant of the contact-stabilized Newmark method by Deuflhard et al., which provides a spatiotemporal numerical integration of dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Up no now, the question of consistency in the case of contact constraints has been discussed for time integrators in function space under the assumption of bounded total variation of the solution. Here, interest focusses on the consistency error of the Newmark scheme in physical energy norm after discretization both in time and in space. The resulting estimate for the local discretization error allows to prove global convergence of the Newmark scheme under an additional assumption on the active contact boundaries.}, language = {en} } @misc{SanderKlapprothYouettetal., author = {Sander, Oliver and Klapproth, Corinna and Youett, Jonathan and Kornhuber, Ralf and Deuflhard, Peter}, title = {Towards an Efficient Numerical Simulation of Complex 3D Knee Joint Motion}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14512}, number = {12-06}, abstract = {We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity. Its efficient numerical simulation requires advanced numerical algorithms that have been developed just recently. Up to now, the model comprises bones, cartilage, and the major ligaments (patella and menisci are still missing). Bones (femur, tibia, and fibula) are modelled by linear elastic materials, cartilage by viscoelastic materials, ligaments by one-dimensional so-called Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity in the presence of strict contact inequalities. For the total spatio-temporal discretization we apply a method of layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts, we apply a new contact-stabilized Newmark method, while for the Cosserat rods we choose an energy-momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet-Neumann method, and the arising large algebraic systems are solved by a recent fast multigrid solver, the truncated non-smooth Newton multigrid method.}, language = {en} } @misc{KlapprothSchielaDeuflhard, author = {Klapproth, Corinna and Schiela, Anton and Deuflhard, Peter}, title = {Adaptive Timestep Control for the Contact-Stabilized Newmark Method}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11714}, number = {10-09}, abstract = {The aim of this paper is to devise an adaptive timestep control in the contact--stabilized Newmark method (CONTACX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini's condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact--stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example.}, language = {en} } @phdthesis{Klapproth2012, author = {Klapproth, Corinna}, title = {Adaptive numerical integration for dynamical contact problems}, year = {2012}, language = {en} } @article{KlapprothDeuflhardSchiela2009, author = {Klapproth, Corinna and Deuflhard, Peter and Schiela, Anton}, title = {A Perturbation Result for Dynamical Contact Problems}, series = {Numer. Math}, volume = {2}, journal = {Numer. Math}, pages = {237 -- 257}, year = {2009}, language = {en} }