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Analysis of hourglass instabilities and control in underintegrated finite element methods. (English) Zbl 0543.73104
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 T. Belytschko, J. Sh.-J. Ong and 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.
Reviewer: P.Narain

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74S99 Numerical and other methods in solid mechanics 93B40 Computational methods in systems theory (MSC2010)
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##### References:
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