Control of hourglass modes in prisms with pentagons or hexagons as bases

A Flanagan–Belytschko-type method [D.P. Flanagan, T. Belytschko, A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Int. J. Numer. Methods Eng. 17 (1981) 679–706] of hourglass control in prisms with pentagons or hexagons as bases is addressed. This hourglass control is based on the description of strain–displacement modes through an orthogonal basis set as it is described in the paper of Margolin and Pyun for 2D quadrilateral cells [L.G. Margolin, J.J. Pyun, A method for treating hourglass patterns, in: C.T. Taylor, N.M. Hafey (Eds.), Proceedings of the Fifth International Conference on Numerical Methods in Laminar and Turbulent Flow, Montreal, Canada, 1987]. In this generalized approach, using shape functions is not necessary anymore. It follows that elements for which regular isoparametric shape functions do not exist can be treated. The method may also be used to handle usual element classes such as pyramids, prisms with triangles as bases and hexahedra. Results for the spherical problem are given.