On the Obstacle Problem for a Naghdi Shell

Belgacem, Faker Ben; Bernardi, Christine; Blouza, Adel; Frekh, Taallah

doi:10.1007/s10659-010-9269-2

Cited by 5 publications

(2 citation statements)

References 13 publications

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“…Theorem 4.9. For any f ∈ L 2 (ω) 3 , the discrete problem (4.5) has a unique solution in V h × M h . Moreover, this solution satisfies (4.31)…”

Section: Furthermore the Bilinear Form B( ) Is Uniformly Continuous Onmentioning

confidence: 99%

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Untitled

2017

Math. Comp.

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In this work a mixed method and its DG formulation are proposed to solve Naghdi's equations for a thin linearly elastic shell. The unknowns of the problem are the displacement of the points of the middle surface, the rotation field of the normal vector to the middle surface of the shell and a Lagrange multiplier which is introduced in order to enforce the tangency requirement on the rotation. In addition to existence and uniqueness results of solutions of the continuous and the discrete problems we derive an a priori error estimate. We further propose an a posteriori estimator that yields an upper bound and a lower bound of the error. Numerical tests that validate and illustrate our approach are given.

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“…Theorem 4.9. For any f ∈ L 2 (ω) 3 , the discrete problem (4.5) has a unique solution in V h × M h . Moreover, this solution satisfies (4.31)…”

Section: Furthermore the Bilinear Form B( ) Is Uniformly Continuous Onmentioning

confidence: 99%

“…The flow of a viscous incompressible fluid in a deformable shell of Naghdi type was considered in [7]. The obstacle problem for a Naghdi's shell was considered in [3]. Numerical stability for a dynamic Naghdi shell with little regularity was studied in [19].…”

Section: Introductionmentioning

confidence: 99%

Untitled

2017

Math. Comp.

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Models of Elastic Shells in Contact with a Rigid Foundation: An Asymptotic Approach

Rodríguez-Arós

2017

J Elast

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We consider a family of linearly elastic shells with thickness 2ε (where ε is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface S, and may enter in contact with a rigid foundation along the bottom face.We are interested in studying the limit behavior of both the threedimensional problems, given in curvilinear coordinates, and their solutions (displacements u ε of covariant components u ε i ) when ε tends to zero. To do that, we use asymptotic analysis methods. On one hand, we find that if the applied body force density is O(1) with respect to ε and surface tractions density is O(ε), a suitable approximation of the variational formulation of the contact problem is a two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem for an elastic membrane. On the other hand, if the applied body force density is O(ε 2 ) and surface tractions density is O(ε 3 ), the corresponding approximation is a different two-dimensional inequality which can be identified as the variational formulation of the obstacle problem for an elastic flexural shell. We finally discuss the existence and uniqueness of solution for the limit two-dimensional variational problems found.

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