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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 59
Error Estimates for Finite Element Approximation of Hemivariational Inequalities M.A. Noor+ and M.H.B.M. Shariff+*
+Etisalat University, United Arab Emirates
Full Bibliographic Reference for this paper
M.A. Noor, M.H.B.M. Shariff, "Error Estimates for Finite Element Approximation of Hemivariational Inequalities", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 59, 2004. doi:10.4203/ccp.79.59
Keywords: hemivariational inequalities, nonconvex functions error estimates, finite element, material modelling.
Summary
In this paper we consider a class of variational inequalities known as hemivariational
inequalities involving nonconvex functions. Hemivariational inequalities have very important
and novel applications in structural analysis, material modelling, transportation and etc. In a
hemivariatonal inequality formulation the presence of "friction" leads to nonconservative
"forces" which gives rise to nondifferentiable and nonconvex forms. In most cases the issue
of the existence of solutions to such forms is an open problem. However, in recent years numerical
techniques have been applied to address this problem. Error estimates for various types of variational
inequalities
involving second order linear and nonlinear elliptic operators have been
derived by many authors under sufficient regular solutions. To the best of our knowledge,
finite element method has not been considered for the hemivariational
inequalities. Hemivariational inequalities are much more complicated due to
the presence of the nonlinear terms involving the nonlinear nonlinear terms.
This represents a major difficulty in obtaining the error estimates for the
finite element approximation of variational inequalities involving the
nonlinear terms. In this paper, we consider the finite element approximation
of the hemivariational inequalities and derive the error estimate which is
of order h in the energy norm. Our result represent a substantial
generalization and improvement of the error analysis of finite element
approximation of hemivariational inequalities.
To convey the main idea we consider the problem of finding where ![]() ![]() ![]() ![]() ![]() ![]() ![]()
We derive the errors estimate for the finite element
approximation of the hemivariational inequalities. For this purpose, we
first consider a finite dimensional subspace
The error estimate by the following theorem:
Theorem Let the operator
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