74M15 Contact
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- Dynamical contact problems (2)
- Newmark method (2)
- Signorini condition (2)
- (visco-)elasticity (1)
- Signorini contact (1)
- Tresca friction (1)
- a priori error analysis (1)
- adaptivity (1)
- consistency (1)
- contact--stabilized Newmark method (1)
We present a globally convergent method for the solution of frictionless large deformation contact problems involving hyperelastic materials.
For the discretisation we apply the dual mortar method which is known to be more stable than node-to-segment approaches. The resulting non-convex constrained minimisation problems are solved using a filter–trust-region scheme.
This method combines several techniques from non-linear optimisation to achieve global convergence towards first-order optimal points. To speed up the method inexact linearisations of the non-penetration constraint are used
whenever the current iterate is far away from a critical point. A monotone multigrid method is applied for the fast solution of the constrained Newton
problems.
Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints
(2012)
A global higher differentiability result in Besov
spaces is proved for the displacement fields of linear elastic models
with self contact.
Domains with cracks are studied, where nonpenetration
conditions/Signorini conditions are imposed on the crack faces.
It is shown that
in a neighborhood of crack tips (in 2D) or
crack fronts (3D) the displacement fields are
$B^{3/2}_{2,\infty}$ regular.
The proof relies on a difference
quotient argument for the directions tangential to the crack. In order
to obtain the regularity estimates also in the normal direction, an
argument due to
Ebmeyer/Frehse/Kassmann is modified.
The methods are then applied to further examples like
contact problems with nonsmooth rigid foundations, to a model with
Tresca friction and
to minimization problems with
nonsmooth energies and constraints as they occur for instance in the modeling of
shape memory alloys.
Based on Falk's approximation Theorem for variational
inequalities, convergence rates for FE-discretizations of contact
problems are derived relying on the proven regularity properties.
Several numerical examples illustrate the theoretical results.
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.
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.
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.