A mixed finite element for shell model with free edge boundary conditions. I: The mixed variational formulation. II: The numerical scheme
After a very brief recall on general shell theory, we construct a mixed variational formulation based on the introduction of a new unknown: the rotation of the normal to the medium surface. In Koiter shell theory (for instance), this rotation can be expressed with respect to the three components of the displacement field of the medium surface and their derivatives. The Lagrange multiplier corresponding to this relation (known as the Kirchhoff-Love kinematical assumption), is also introduced as an independent unknown. There are two main difficulties: one is due to the differential geometry of surfaces and is rather technical; the other is to define correctly the dual space for the Kirchhoff-Love relation. From the mixed variational formulation suggested in Part I, we construct a finite element scheme. Then, a mathematical justification concerning the error estimate is developed. The most important point is to justify two compatibility conditions between the approximation of the kinematics and the stresses. Because of the structure of the shell operator they are not obvious at all, and several restrictions on the mesh are necessary. Therefore, the classical result known in fluid mechanics for Stokes equations cannot be applied directly.
scientific article; zbMATH DE number 881052
scientific article