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A locking-free finite element method for Naghdi shells. (English) Zbl 0913.73065

Summary: A locking-free method, using mixed finite elements, is introduced to approximate the solution of Naghdi shell problems with small parameter \(t\), the thickness of the shell. The approach of D. N. Arnold and F. Brezzi [Math. Comput. 66, No. 217, 1-14 (1997; Zbl 0854.65095)] is employed with some important changes. Instead of proving the discrete inf-sup condition for arbitrary geometric coefficients, which does not seem possible, we prove a weaker stability condition for smooth enough geometrically dependent coefficients, which is sufficient for deducing the optimal error estimate as long as \(h^2/t\) is uniformly bounded. For extremely small \(t\), we can relax this condition either using a larger bubble space or stabilizing the problem by replacing \(t^2\) by \(t^2+ h^4\). In either case an optimal error estimate still holds.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74K15 Membranes
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 0854.65095
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Full Text: DOI

References:

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