Analysis of hourglass instabilities and control in underintegrated finite element methods (Q796361)

The authors have divided the paper into two parts. In the first part a mathematical treatment of certain stabilization methods, which involve computing an underintegrated stiffness matrix (which is rank deficient) and then adding a stabilization matrix which effectively eliminates the spurious modes by \textit{T. Belytschko}, \textit{J. Sh.-J. Ong} and \textit{W. K. Liu} [ibid. 44, 269-295 (1984; Zbl 0525.73086)], is given. In the second part an a-posteriori stabilization method for hourglass control is investigated. In this method an approximate solution of the underintegrated approximation converges to the exact solution of a model problem, as the mesh is refined, at almost the same rate as the fully integrated solutions. All of these particular results are established for a simple model problem \((-\Delta u\neq f)\) defined on a square domain with approximations defined by bilinear shape functions on a uniform mesh.