Abstract
We present and analyze a method for thin plates based on cut Bogner-Fox-Schmit elements, which are C^1 elements obtained by taking tensor products of Hermite splines. The formulation is based on Nitsche’s method for weak enforcement of essential boundary conditions together with addition of certain stabilization terms that enable us to establish coercivity and stability of the resulting system of linear equations. We also take geometric approximation of the boundary into account and we focus our presentation on the simply supported boundary conditions which is the most sensitive case for geometric approximation of the boundary.
Highlights
The Bogner-Fox-Schmit (BFS) element [6] is a classical C1 thin plate element obtained by taking tensor products of cubic Hermite splines and removing the interior degrees of freedom that are zero on the boundary
We consider a variant where we retain these degrees of freedom to obtain a C1 version of the Q3 approximation [23]. This element is only C1 on tensor product elements, which is a serious drawback since it severely limits the applicability of the resulting finite element method
In this paper we present an alternative idea where C1 continuity is retained: we develop a cut finite element version, allowing for discretizing a smooth boundary which may cut through the tensor product mesh in an arbitrary manner
Summary
The Bogner-Fox-Schmit (BFS) element [6] is a classical C1 thin plate element obtained by taking tensor products of cubic Hermite splines and removing the interior degrees of freedom that are zero on the boundary. We consider a variant where we retain these degrees of freedom to obtain a C1 version of the Q3 approximation [23] This element is only C1 on tensor product (rectangular) elements, which is a serious drawback since it severely limits the applicability of the resulting finite element method. An alternative to C1 approximations for Kirchhoff plates is to either use discontinuous Galerkin methods [8,15,17,22], or to use mixed finite elements for the Reissner–Mindlin model with small plate thickness [2,4,12]. In this paper we present an alternative idea where C1 continuity is retained: we develop a cut finite element version, allowing for discretizing a smooth boundary which may cut through the tensor product mesh in an arbitrary manner. For smooth boundary and f ∈ L2 we have the elliptic regularity u H4( ) f (9)
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