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Finite element stabilization matrices - A unification approach. (English) Zbl 0553.73065
In this paper, we show that the stabilization vectory \(\gamma\) can be obtained naturally by taking the partial derivatives with respect to the natural coordinates. Hence, the components of the strains and stresses can be expressed in terms of a set of orthogonal coordinates. With these definitions of stresses and strains, a general form of the finite element stabilization matrices is then constructed for under integrated, irregular shape and anisotropic elements. The explicit expressions of the stabilization matrices for the 4-, 9- and 8-node Laplace and continuum elements and the 4-node plate element are then derived from the general form. These are sufficiently general for the developments herein. The generalization of the stabilization-matrices concept to nonlinear problems which usually require the numerical integration of a fairly general class of nonlinear, finite-deformation constitutive equations is also presented.
The computer implementation aspects and numerical evaluation of these stabilized elements are also considered. Numerical tests confirm the stability and accuracy characteristics of the resulting elements. In particular, the numerical experiments reported here also show that the rates of convergence in the \(L_ 2\)-norm and in the energy norm agree well with the expected convergence rates.

74S05 Finite element methods applied to problems in solid mechanics
74S99 Numerical and other methods in solid mechanics
74-04 Software, source code, etc. for problems pertaining to mechanics of deformable solids
Full Text: DOI
[1] Goudreau, G.L.; Hallquist, J.O., Recent developments in large scale finite elements Lagragian hydrocode technology, Rept. UCRL-86460, Lawrence livermore laboratory, 1981, nos. 1-3, Vol. 33, (1982) · Zbl 0493.73072
[2] Kulak, R.F., A finite element formulation for fluid-structure interaction in three-dimensional space, J. pressure vessel technology, 103, 183-190, (1981)
[3] Irons, B.; Ahmad, S., Techniques of finite elements, (1980), Ellis Horwood Chichester
[4] Belytschko, T.; Kennedy, J.M., Computer models for subassembly simulation, Nucl. engrg. design, 49, 17-38, (1978)
[5] Key, S.W., A finite elements procedure for the large deformation dynamics response of axisymmetric solids, Comput. meths. appl. mech. engrg., 4, 195-218, (1974) · Zbl 0284.73047
[6] Kosloff, D.; Frazier, G.A., Treatment of hourglass patterns in low order finite element codes, Internat. J. numer. anal. meths. geomech., 2, 57-72, (1978)
[7] Flanagan, D.P.; Belytschko, T., A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Internat. J. numer. meths. engrg., 17, 679-706, (1981) · Zbl 0478.73049
[8] Belytschko, T., Correction of article by D.P. flanagan and T. belytschko, Internat. J. numer. meths. engrg., 19, 467-468, (1983)
[9] Strang, G., Variational crimes in the finite element methods, Internat. J. numer. meths. engrg., 19, 689-710, (1983)
[10] Liu, W.K.; Belytschko, T., Efficient linear and nonlinear heat condition with a quadrilateral element, Internat. J. numer. meths. engrg., 20, 931-948, (1984) · Zbl 0542.65067
[11] 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
[12] Belytschko, T.; Tsay, C.S., A stabilization procedure for the quadrilateral plate element with one-point quadrature, Internat. J. numer. meths. engrg., 19, 405-419, (1983) · Zbl 0502.73058
[13] Hughes, T.J.R.; Liu, W.K., Nonlinear finite element analysis of shells: part I, Comput. meths. appl. mech. engrg., 26, 331-362, (1981) · Zbl 0461.73061
[14] Liu, W.K., FLUSTR — a nonlinear fluid-structure interaction program — user manual, (1982), Northwestern University, Department of Mechanical Engineering Evanston, IL
[15] Belytschko, T.; Ong, J.S.-J.; Liu, W.K.; Kennedy, J.M., Hourglass control in linear and nonlinear problems, Comput. meths. appl. mech. engrg., 43, 251-276, (1984) · Zbl 0522.73063
[16] Belytschko, T.; Liu, W.K.; Ong, J.S.-J.; Lam, D., Implementation and application of a 9-node Lagrange shell element with spurious mode control, J. comput. & structures, (1985), to appear · Zbl 0581.73089
[17] 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
[18] Liu, W.K., Development of finite lement procedures fo fluid-structure interaction, ()
[19] Malkus, D.S.; Hughes, T.J.R., Mixed finite element methods — reduced and selective integration techniques: A unification of concepts, Comput. meths. appl. mech. engrg., 15, 63-82, (1978) · Zbl 0381.73075
[20] Jacquotte, O.P.; Oden, J.T., Analysis of hourglass instabilities and control in underintegrated finite element methods, Comput. meths. appl. mech. engrg., 44, 339-363, (1984) · Zbl 0543.73104
[21] Liu, W.K.; Zhang, Y.F., Improvement of mixed time implicit-explicit algorithms for thermal analysis of structures, Comput. meths. appl. mech. engrg., 37, 207-223, (1983)
[22] Liu, W.K.; Belytschko, T.; Zhang, Y.F., Implementation and accuracy of mixed-time implicit-explicit methods for structural dynamics, Comput. & structures, 19, 530-541, (1984)
[23] Liu, W.K.; Zhang, Y.F.; Belytschko, T., Implementation of mixed-time partition algorithms for nonlinear thermal analysis of structures, Comput. meths. appl. mech. engrg., 48, 245-263, (1985) · Zbl 0552.73070
[24] J.Y.T. Leung, Northwestern University, Department of Electrical Engineering and Computer Science, Private Communication, 1984.
[25] Gallagher, R.H., Finte element analysis fundamentals, (1975), Prentice-Hall Englewood Cliffs, NJ
[26] Hughes, T.J.R., Generalization of selective integration procedures to anisotropic and nonlinear media, internat, J. numer. meths. engrg., 15, 1413-1418, (1980) · Zbl 0437.73053
[27] Liu, W.K.; Belytschko, T.; Ong, J.S.-J.; Law, S.E., The use of stabilization matrices in nonlinear analysis, (), J. engrg. comput., 233-258, (1985), also:
[28] Liu, W.K.; Belytschko, T.; Law, S.E.; Lam, D.; Lam, D., Resultant stress degenerated shell element, progress report to NSF on dynamic and buckling analyses of liquid-filled tanks, CEE82-13739, Comput. meths. appl. mech. engrg., (1986), also:
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