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Locking-free stabilized conforming nodal integration for meshfree Mindlin–Reissner plate formulation. (English) Zbl 1060.74675

Summary: The cause of shear locking in Mindlin-Reissner plate formulation is due to the inability of the numerical formulation in representing pure bending mode without producing parasitic shear deformation (lack of Kirchhoff mode). To resolve shear locking in meshfree formulation of Mindlin-Reissner plates, the following two issues are addressed: (1) construction of approximation functions capable of reproducing Kirchhoff modes, and (2) formulation of domain integration of Galerkin weak form capable of producing exact solution under pure bending condition. In this study, we first identify the Kirchhoff mode reproducing conditions (KMRC), and show that the employment of a second order monomial basis in the reproducing kernel or moving least-square approximation of translational and rotational degrees of freedom is an effective means to meet KMRC. Next, the integration constraints that fulfill bending exactness (BE) in the Galerkin meshfree discretization of Mindlin-Reissner plate are derived. A nodal integration with curvature smoothing stabilization that fulfills BE is then formulated for Mindlin-Reissner plate. The curvature smoothing stabilization is introduced in the nodally integrated Galerkin weak form. The resulting meshfree formulation is stable and free of shear locking in the limit of thin plate. Both computational efficiency and accuracy are achieved in the proposed meshfree Mindlin-Reissner plate formulation.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74K20 Plates
Full Text: DOI

References:

[1] Beissel, S.; Belytschko, T., Nodal integration of the element-free Galerkin method, Comput. Methods Appl. Mech. Eng., 139, 49-74 (1996) · Zbl 0918.73329
[2] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Eng., 37, 229-256 (1994) · Zbl 0796.73077
[3] Belytschko, T.; Kronggauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Eng., 139, 3-47 (1996) · Zbl 0891.73075
[4] Chen, J. S.; Wang, H. P., Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact, Comput. Mech., 25, 137-156 (2000) · Zbl 0978.74085
[5] Chen, J. S.; Yoon, S.; Wang, H. P.; Liu, W. K., An improved reproducing kernel particle method for nearly incompressible hyperelastic solids, Comput. Methods Appl. Mech. Eng., 181, 117-145 (2000) · Zbl 0973.74088
[6] Chen, J. S.; Wu, C. T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin meshfree methods, Int. J. Numer. Methods Eng., 50, 435-466 (2001) · Zbl 1011.74081
[7] Chen, J. S.; Wu, C. T.; Yoon, S., Nonlinear version of stabilized conforming nodal integration for Galerkin meshfree methods, Int. J. Numer. Methods Eng., 53, 2587-2615 (2002) · Zbl 1098.74732
[8] Dolbow, J.; Belytschko, T., Volumetric locking in the element free Galerkin method, Int. J. Numer. Methods Eng., 46, 925-942 (1999) · Zbl 0967.74079
[9] Donning, B.; Liu, W. K., Meshless methods for shear-deformable beams and plates, Comput. Methods Appl. Mech. Eng., 152, 47-72 (1998) · Zbl 0959.74079
[10] Garcia, O.; Fancello, E. A.; Barcellos, C. S.; Duarte, C. A., Hp-clouds in Mindlin’s thick plate model, Int. J. Numer. Methods Eng., 47, 1381-1400 (2000) · Zbl 0987.74067
[11] Huerta, A.; Fernández-Méndez, S., Locking in the incompressible limit for the element free Galerkin method, Int. J. Numer. Methods Eng., 51, 1361-1383 (2001) · Zbl 1065.74635
[12] Hughes, T. J.R.; Cohen, M., The heterosis finite element for plate bending, Comput. & Structures, 9, 445-450 (1978) · Zbl 0394.73076
[13] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2000), Dover publications: Dover publications Mineola, NY · Zbl 1191.74002
[14] Kanok-Nukulchai, W.; Barry, W.; Saran-Yasoontorn, K.; Bouillard, P. H., On elimination of shear locking in the element-free Galerkin method, Int. J. Numer. Methods Eng., 52, 705-725 (2001) · Zbl 1128.74347
[15] Krysl, P.; Belytschko, T., Analysis of thin plates by the element-free Galerkin method, Comput. Mech., 16, 1-10 (1995) · Zbl 0841.73064
[16] Krysl, P.; Belytschko, T., Analysis of thin shells by the element-free Galerkin method, Int. J. Numer. Methods Eng., 33, 3057-3080 (1996) · Zbl 0929.74126
[17] Kim, S. H.; Choi, C. K., Improvement of quadratic finite element for Mindlin plate bending, Int. J. Numer. Methods Eng., 34, 197-208 (1992) · Zbl 0825.73834
[18] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. Comput., 37, 141-158 (1981) · Zbl 0469.41005
[19] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 20, 1081-1106 (1995) · Zbl 0881.76072
[20] Liu, W. K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. Numer. Methods Eng., 38, 1655-1679 (1995) · Zbl 0840.73078
[21] Liu, W. K.; Li, S.; Belytschko, T., Moving least square kernel Galerkin method, Part I: methodology and convergence, Comput. Methods Appl. Mech. Eng., 143, 422-433 (1997)
[22] Pugh, E. D.L.; Hinton, E.; Zienkiewicz, O. C., A study of quadrilateral plate bending elements with reduced integration, Int. J. Numer. Methods Eng., 12, 1059-1079 (1978) · Zbl 0377.73065
[23] Simo, J. C.; Hughes, T. J.R., On the variational foundation of assumed strain method, J. Appl. Mech., 53, 51-54 (1986) · Zbl 0592.73019
[24] Timoshenko, S. P.; Woinowsky-Krieger, S., Theory of Plates and Shells (1959), McGraw-Hill: McGraw-Hill New York · Zbl 0114.40801
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