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Locking in the incompressible limit for the element-free Galerkin method. (English) Zbl 1065.74635
Summary: Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element-free Galerkin method. The modal analysis developed here shows that the number of non-physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non-physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element-free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
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[1] The Finite Element Method. Prentice-Hall: Englewood Cliffs, NJ, 1987.
[2] On the locking of standard finite elements. Classroom Notes. University of California, Berkeley, 1999.
[3] Suri, Computer Methods in Applied Mechanics and Engineering 133 pp 347– (1996) · Zbl 0893.73070 · doi:10.1016/0045-7825(95)00947-7
[4] Belytschko, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077 · doi:10.1002/nme.1620370205
[5] Zhu, Computational Mechanics 21 pp 211– (1998) · Zbl 0947.74080 · doi:10.1007/s004660050296
[6] Dolbow, International Journal for Numerical Methods in Engineering 46 pp 925– (1999) · Zbl 0967.74079 · doi:10.1002/(SICI)1097-0207(19991030)46:6<925::AID-NME729>3.0.CO;2-Y
[7] Askes, Computer Methods in Applied Mechanics and Engineering 173 pp 99– (1999) · Zbl 0962.74076 · doi:10.1016/S0045-7825(98)00259-X
[8] Chen, Computer Methods in Applied Mechanics and Engineering 181 pp 117– (2000) · Zbl 0973.74088 · doi:10.1016/S0045-7825(99)00067-5
[9] Belytschko, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996) · Zbl 0891.73075 · doi:10.1016/S0045-7825(96)01078-X
[10] Liu, Computer Methods in Applied Mechanics and Engineering 139 pp 1– (1996) · doi:10.1016/S0045-7825(96)90021-3
[11] Liu, Archives of Computational Methods in Engineering, State of the Art Reviews 3 pp 3– (1996) · doi:10.1007/BF02736130
[12] Liu, Computer Methods in Applied Mechanics and Engineering 143 pp 113– (1997) · Zbl 0883.65088 · doi:10.1016/S0045-7825(96)01132-2
[13] Nayroles, Computational Mechanics 10 pp 307– (1992) · Zbl 0764.65068 · doi:10.1007/BF00364252
[14] Liu, International Journal for Numerical Methods in Fluids 20 pp 1081– (1995) · Zbl 0881.76072 · doi:10.1002/fld.1650200824
[15] Liu, International Journal for Numerical Methods in Fluids 21 pp 901– (1995) · Zbl 0885.76078 · doi:10.1002/fld.1650211010
[16] Belytschko, Computational Mechanics 17 pp 186– (1995) · Zbl 0840.73058 · doi:10.1007/BF00364080
[17] Element-free Galerkin methods for dynamic fracture in concrete. In Comp. Plasticity. Fundamentals and Applications, (eds). CIMNE, Barcelona, 1997; 304-321.
[18] Belytschko, International Journal for Numerical Methods in Engineering 39 pp 923– (1996) · Zbl 0953.74077 · doi:10.1002/(SICI)1097-0207(19960330)39:6<923::AID-NME887>3.0.CO;2-W
[19] Lu, Computer Methods in Applied Mechanics and Engineering 113 pp 397– (1994) · Zbl 0847.73064 · doi:10.1016/0045-7825(94)90056-6
[20] Organ, Computational Mechanics 18 pp 225– (1996) · Zbl 0864.73076 · doi:10.1007/BF00369940
[21] Huerta, International Journal for Numerical Methods in Engineering 48 pp 1615– (2000) · Zbl 0976.74067 · doi:10.1002/1097-0207(20000820)48:11<1615::AID-NME883>3.0.CO;2-S
[22] Theory of Elasticity (3rd edn). McGraw-Hill: New York, 1987.
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