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Smoothing technique based beta FEM \((\beta\)FEM) for static and free vibration analyses of Reissner-Mindlin plates. (English) Zbl 07124746

Summary: Recently, the edge-based and node-based smoothed finite element method (ES-FEM and NS-FEM) has been proposed for Reissner-Mindlin plate problems. In this work, in order to utilize the numerical advantages of both ES-FEM and NS-FEM for static and vibration analysis, a hybrid smoothing technique based beta FEM (\(\beta\)FEM) is presented for Reissner-Mindlin plate problems. A tunable parameter \(\beta\) is introduced to tune the proportion of smoothing domains calculated by ES-FEM or NS-FEM, which controlled the accuracy of the results. Numerical illustrations in both static and free vibration analysis are conducted. The shear locking free property, converge property and dynamic stability are carefully examined via several well-known benchmark examples. Moreover, an experimental test is carefully designed and conducted for validations, in which the mode values and shape of a rectangular steel plate is tested. Numerical examples demonstrate the advantages of \(\beta\)FEM, in comparison with the standard FEM, ES-FEM and NS-FEM using the same meshes. The numerical and experimental results are in good agreement with each other and the \(\beta\)FEM achieves the best accuracy among all the methods for the static or free vibration analysis of plates.

MSC:

74-XX Mechanics of deformable solids
65-XX Numerical analysis
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[1] Bletzinger, K. U., Bischoff, M. and Ramm, E. [2000] “ A unified approach for shear-locking-free triangular and rectangular shell finite elements,” Comput. Struct.75(3), 321-334.
[2] Cardoso, R. P., Yoon, J. W., Mahardika, M., Choudhry, S., Alves de Sousa, R. J. and Fontes Valente, R. A. [2008] “ Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one-point quadrature solid-shell elements,” Int. J. Numer. Methods Eng.75(2), 156-187. · Zbl 1195.74165
[3] de Sá, C., Jorge, R. N., Valente, R. F. and Areias, P. A. [2002] “ Development of shear locking-free shell elements using an enhanced assumed strain formulation,” Int. J. Numer. Methods Eng.53(7), 1721-1750. · Zbl 1114.74484
[4] Everstine, G. C. [1997] “ Finite element formulations of structural acoustics problems,” Comput. Struct.65(3), 307-321. · Zbl 0918.73240
[5] He, Z. C., Li, E., Li, G. Y., Wu, F., Liu, G. R. and Nie, X. [2015] “ Acoustic simulation using \(<mml:math display=''inline`` overflow=''scroll``>\)-fem with a general approach for reducing dispersion error,” Eng. Anal. Boundary Elem.61, 241-253. · Zbl 1403.76044
[6] He, Z. C., Li, E., Liu, G. R., Li, G. Y. and Cheng, A. G. [2016] “ A mass-redistributed finite element method (MR-FEM) for acoustic problems using triangular mesh,” J. Comput. Phys.323, 149-170. · Zbl 1415.65257
[7] Li, E. and He, Z. C. [2017] “ Development of a perfect match system in the improvement of eigen frequencies of free vibration,” Appl. Math. Model.44, 614-639. · Zbl 1443.74077
[8] Li, E., He, Z. C., Jiang, Y. and Li, B. [2016] “ 3D mass-redistributed finite element method in structural-acoustic interaction problems,” Acta Mech.227, 857-879. · Zbl 1334.74082
[9] Liu, G. R. and Nguyen, T. T. [2010] Smoothed Finite Element Methods (CRC Press, Boca Raton).
[10] Liu, G. R., Nguyen-Thoi, T. and Lam, K. Y. [2009] “ An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids,” J. Sound Vib.320(4), 1100-1130.
[11] Liu, G. R., Nguyen-Thoi, T., Nguyen-Xuan, H. and Lam, K. Y. [2009] “ A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems,” Comput. Struct.87(1), 14-26.
[12] Liu, G. R., Zeng, W. and Nguyen-Xuan, H. [2013] “ Generalized stochastic cell-based smoothed finite element method (GS_CS-FEM) for solid mechanics,” Finite Elements Anal. Des.63, 51-61. · Zbl 1282.74090
[13] Liu, G. R. [2009] Meshfree Methods: Moving Beyond the Finite Element Method, 2nd edn. (CRC Press).
[14] Liu, G. R. [2009] Meshfree Methods: Moving Beyond the Finite Element Method, 2nd edn. (CRC Press, Boca Raton, USA).
[15] Nguyen-Thoi, T., Liu, G. R., Vu-Do, H. C. and Nguyen-Xuan, H. [2009] “ A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh,” Comput. Methods Appl. Mech. Eng.198(41), 3479-3498. · Zbl 1230.74193
[16] Phung-Van, P., Nguyen-Thoi, T., Le-Dinh, T. and Nguyen-Xuan, H. [2013] “ Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3),” Smart Mater. Struct.22(9), 095026.
[17] Taylor, R. L. and Auricchio, F. [1993] “ Linked interpolation for Reissner-Mindlin plate elements: Part ii — A simple triangle,” Int. J. Numer. Methods Eng.36(18), 3057-3066. · Zbl 0781.73071
[18] Tessler, A. [1985] “ A priori identification of shear locking and stiffening in triangular Mindlin elements,” Comput. Methods Appl. Mech. Eng.53(2), 183-200. · Zbl 0553.73064
[19] Wanji, C. and Cheung, Y. K. [2010] “ Refined 9-dof triangular Mindlin plate elements,” Int. J. Numer. Methods Eng.51(11), 1259-1281. · Zbl 1065.74606
[20] Wu, F., Liu, G. R., Li, G. Y., Cheng, A. G. and He, Z. C. [2014] “ A new hybrid smoothed FEM for static and free vibration analyses of Reissner-Mindlin plates,” Computat. Mech.54(3), 865-890. · Zbl 1311.74131
[21] Wu, F., Liu, G. R., Li, G. Y., Cheng, A. G., He, Z. C. and Hu, Z. H. [2015] “ A novel hybrid fs-fem/sea for the analysis of vibro-acoustic problems,” Int. J. Numer. Methods Eng.102(12), 1815-1829. · Zbl 1352.74453
[22] Wu, F., Yao, L. Y., Hu, M. and He, Z. C. [2017] “ A stochastic perturbation edge-based smoothed finite element method for the analysis of uncertain structural-acoustics problems with random variables,” Eng. Anal. Boundary Elem.80, 116-126. · Zbl 1403.76052
[23] Zeng, W., Liu, G. R., Kitamura, K. and Nguyen-Xuan, H. [2013] “ A three-dimensional ES-FEM for fracture mechanics problems in elastic solids,” Eng. Fract. Mech.114, 127-150.
[24] Zeng, W., Larsen, J. M. and Liu, G. R. [2015] “ Smoothing technique based crystal plasticity finite element modeling of crystalline materials,” Int. J. Plasticity65, 250-268.
[25] Zeng, W., Liu, G. R., Jiang, C., Dong, X. W., Chen, H. D., Bao, Y. and Jiang, Y. [2016] “ An effective fracture analysis method based on the virtual crack closure-integral technique implemented in CS-FEM,” Appl. Math. Model.40(5-6), 3783-3800. · Zbl 1459.74168
[26] Zeng, W., Liu, G. R., Jiang, C., Nguyen-Thoi, T. and Jiang, Y. [2016] “ A generalized beta finite element method with coupled smoothing techniques for solid mechanics,” Eng. Anal. Bound. Elem.73, 103-119. · Zbl 1403.74133
[27] Zeng, W., Liu, G. R., Li, D. and Dong, X. W. [2016] “ A smoothing technique based beta finite element method \((<mml:math display=''inline`` overflow=''scroll``>\) FEM) for crystal plasticity modeling,” Comput. Struct.162, 48-67.
[28] Zhang, Z. Q. and Liu, G. R. [2011] “ An edge-based smoothed finite element method (es-fem) using 3-node triangular elements for 3d non-linear analysis of spatial membrane structures,” Int. J. Numer. Methods Eng.86(2), 135-154. · Zbl 1235.74328
[29] Zienkiewicz, O. C., Taylor, R. L. and Too, J. M. [1971] “ Reduced integration technique in general analysis of plates and shells,” Int. J. Numer. Methods Eng.3(2), 275-290. · Zbl 0253.73048
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