By Bernadette Miara, Georgios E. Stavroulakis, Vanda Valente
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Extra resources for Topics on Mathematics for Smart Systems: Proceedings of the European Conference Rome, Italy, 26-28 October,2006
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5. The limit problem with Lagmnge multipliers Here we formulate the Kirchhoff problem as the limit of the ReissnerMindlin problem by taking into account the essential free boundary condition with Lagrange multipliers: Find (w, f3,, q, A) E W x V x S x A 51 for all (v, q, r, p ) E W x form V x S x A, with the additional symmetric bilinear corresponding to the free boundary FF. 10)1. 10)3. For the free boundary, we combine the boundary terms in M above with the ones in C and obtain This implies the following connections: First, by testing with qes, and then with v, we obtain, respectively, mns(P)-X=O and q n - - ax = O .
T. Ideka, Fundamentals of piezoelectricity, Oxford University Press, Oxford, 1990. 3. D. Mindlin, Polarisation gradient in elastic dielectrics, Internat. J. Solids Structures 4 (1968), 637-663. 4. A. Toupin, The elastic dielectrics, J. Rational Mech. Anal. 5 (1956), 84% 915. 29 5. M. R. FernBndez and Y. Ouafik, Numerical analysis of two frictionless elastic-piezoelectric contact problems, Preprint (2006). 6. S. Hiieber, A. I. Wohlmuth, A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity, Bull.
Therefore, as already mentioned, our stabilization is actually a potential hindrance for the method in bending dominated problems. As we can appreciate in Fig. 11, the MITCGa gives fairly satisfactory results. Regarding the stabilized MITCGa, the convergence curves of Figs. 1213 clearly show that the element behaves almost as well as the original MITCGa element, for both choices of the stabilization parameter C. 42 Fig. 11. Convergence curves associated with the s-norm and Ab + A m norm for the free hyperboloid shell problem and MITCGa shell finite element.