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- Cosserat rod (1)
- Multi-Level Monte-Carlo, Stochastic Partial Differential Equations, Stochastic Finite Element Methods, Multi--Level Methods, Variational Inequalities (1)
- adaptivity (1)
- contact--stabilized Newmark method (1)
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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.
We formulate the static mechanical coupling of a geometrically exact Cosserat rod
to a nonlinearly elastic continuum. In this setting, appropriate coupling conditions have
to connect a one-dimensional model with director variables to a three-dimensional
model without directors.
Two alternative coupling conditions are proposed,
which correspond to two different configuration trace spaces.
For both we show existence of solutions of the coupled problems, using the direct
method of the calculus of variations. From the first-order optimality conditions
we also derive the corresponding conditions for the dual variables. These are
then interpreted in mechanical terms.
We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic
problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver.
Multi-Level Monte-Carlo Finite Element Methods for stochastic elliptic variational inequalities
(2013)
Multi-Level Monte-Carlo Finite Element (MLMC--FE) methods
for the solution of stochastic elliptic variational inequalities
are introduced, analyzed, and numerically investigated.
Under suitable assumptions on the random diffusion coefficient,
the random forcing function, and the deterministic obstacle,
we prove existence and uniqueness of solutions of ``mean-square''
and ``pathwise'' formulations.
Suitable regularity results for deterministic,
elliptic obstacle problems lead
to uniform pathwise error bounds, providing
optimal-order error estimates of the statistical error
and upper bounds for the
corresponding computational cost for
classical Monte--Carlo and novel MLMC--FE methods.
Utilizing suitable multigrid solvers for the occurring sample problems,
in two space dimensions
MLMC--FE methods then provide numerical
approximations of the expectation of the random solution
with the same order of efficiency as for a corresponding
deterministic problem, up to logarithmic terms.
Our theoretical findings are illustrated by numerical experiments.