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©2012 Civil-Comp Ltd |
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J. Gwinner
Department of Aerospace Engineering, Universität der Bundeswehr München, Germany
Keywords: nonsmooth mechanics, contact, friction, hp-FEM, nonconforming approximation, Gauss-Lobatto quadrature.
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reference
This paper is concerned with the
hp-version of the finite element method (hp-FEM)
to treat a variational inequality in a vectorial Sobolev space
that models bilateral frictional contact in linear elastostatics.
Thus recent work is extended
[1] for the boundary element method to a larger class of nonlinear
variational problems that are treatable by the finite element method.
By the pioneering work of Babuška and co-workers, the
exponentially fast convergence of the hp-FEM
for linear elliptic problems is well-known.
Recently Maischak and Stephan [2,3], respectively
Dörsek and Melenk [4,5]
showed the superior convergence properties of adaptive hp boundary element methods (hp-BEM), respectively adaptive hp finite element methods in numerical experiments also for unilateral, nonsmooth problems compared to the standard h-version.
Such an approximation of higher order leads to a nonconforming
discretization scheme. Without any regularity assumptions,
the convergence of the
hp-FEM Galerkin solution in the energy norm is proved.
To this end the Glowinski convergence
for the friction-type functional is investigated.
The key to the norm convergence result for the hp-FEM
is the used Gauss-Lobatto integration rule with its high exactness order and
its positive weights together with a duality argument
in the sense of convex analysis. Thus the convergence analysis
complements prior work of Maischak and Stephan [3]
on the hp-BEM for frictionless unilateral contact and more recent work of
Dörsek and Melenk [4]
on a mixed hp-FEM for frictional bilateral contact.
Finally the convergence analysis can be further extended to
other nonlinear variational problems from nonsmooth mechanics.
In particular the Bingham fluid problem is considered and
a mixed hp-FEM discretization scheme is proposed with
analogous convergence properties.
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- 1
- J. Gwinner, "On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction", Appl. Numer. Math., 59, 2774-2784, 2009.
- 2
- M. Maischak, E.P. Stephan, "Adaptive hp-versions of BEM for Signorini problems", Appl. Numer. Math., 54, 425-449, 2005.
- 3
- M. Maischak, E.P. Stephan, "Adaptive hp-versions of boundary element methods for elastic contact problems", Comput. Mech., 39, 597-607, 2007.
- 4
- P. Dörsek, J.M. Melenk, "Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: The primal - dual formulation and a posteriori error estimation", Appl. Numer. Math., 60, 689-704, 2010.
- 5
- P. Dörsek, J.M. Melenk, "Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: The primal formulation", in "Proceedings of ICOSAHOM 2009", Lecture Notes in Computational Science and Engineering, 76, Springer, Berlin, 1-17, 2011.
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