×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Belytschko, T.; Ong, J.S.-J.; Liu, W.K., A consistent control of spurious singular modes in the 9-node Lagrange element for the Laplace and Mindlin plate equations, Comput. meths. appl. mech. engrg., 44, 269-295, (1984) · Zbl 0525.73086
[2] Belytschko, T.; Tsay, C.S.; Liu, W.K., A stabilization matrix for the bilinear Mindlin plate element, Comput. meths. appl. mech. engrg., 29, 313-327, (1981) · Zbl 0474.73091
[3] Flanagan, D.; Belytschko, T., A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Internat. J. numer. meths. engrg., 17, 679-706, (1981) · Zbl 0478.73049
[4] Kasloff, D.; Frazier, G.A., Treatment of hourglass patterns in low order finite element codes, Internat. J. numer. anal. meths. geom., 2, 57-72, (1978)
[5] W.K. Liu and T. Belytschko, Efficient linear and non-linear heat conduction with a quadrilateral element, Internat. J. Numer. Meths. Engrg., to appear. · Zbl 0542.65067
[6] Oden, J.T.; Jacquotte, O.-P., Stability of some mixed finite element methods for Stokesian flows, Comput. meths. appl. mech. engrg., 43, 231-247, (1984) · Zbl 0598.76033
[7] Oden, J.T.; Jacquotte, O.-P., Stable and unstable RIP/perturbed Lagrangian method for the two dimensional viscous flow problem, () · Zbl 0571.73094
[8] Oden, J.T.; Kikuchi, N., Penalty methods for constrained problems in elasticity, Internal. J. numer. meths. engrg., 18, 701-725, (1982) · Zbl 0486.73068
[9] Oden, J.T.; Kikuchi, N.; Song, Y.J., Penalty-finite elements methods for the analysis of Stokesian flows, Comput. meths. appl. mech. engrg., 31, 297-329, (1982) · Zbl 0478.76041
[10] Scott, R., Optimal L estimates for the finite element method on irregular meshes, Math. comput., 30, 136, 681-697, (1976) · Zbl 0349.65060
[11] Temam, R., Navier Stokes equations, (1979), North-Holland Amsterdam · Zbl 0454.35073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.