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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 99
PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping
Paper 149

On Higher Order Approximation for Nonlinear Variational Problems in Nonsmooth Mechanics

J. Gwinner

Department of Aerospace Engineering, Universität der Bundeswehr München, Germany

Full Bibliographic Reference for this paper
J. Gwinner, "On Higher Order Approximation for Nonlinear Variational Problems in Nonsmooth Mechanics", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 149, 2012. doi:10.4203/ccp.99.149
Keywords: nonsmooth mechanics, contact, friction, hp-FEM, nonconforming approximation, Gauss-Lobatto quadrature.

Summary
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.

References
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. doi:10.1016/j.apnum.2008.12.027
2
M. Maischak, E.P. Stephan, "Adaptive hp-versions of BEM for Signorini problems", Appl. Numer. Math., 54, 425-449, 2005. doi:10.1016/j.apnum.2004.09.012
3
M. Maischak, E.P. Stephan, "Adaptive hp-versions of boundary element methods for elastic contact problems", Comput. Mech., 39, 597-607, 2007. doi:10.1007/s00466-006-0109-y
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. doi:10.1016/j.apnum.2010.03.011
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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