×

The static and free vibration analysis of a nonhomogeneous moderately thick plate using the meshless local radial point interpolation method. (English) Zbl 1244.74237

Summary: A meshless local radial point interpolation method (LRPIM) for the bending and free vibration analysis of a nonhomogeneous moderately thick plate is presented in this paper. It uses a radial basis function coupled with a quadratic polynomial basis function as a trail function and a quartic spline function as a test function of the weighted residual method. The shape functions obtained in the trail function have the Kronecker delta function property, and the essential boundary conditions can be easily imposed. The present method is a true meshless method as it does not need any grids and all integrals can be easily evaluated over regularly shaped domains and their boundaries. In computational procedures, variations of material properties in the considered domain are modelled by adopting proper material parameters at Gauss points in integrations. Examples show that results obtained by the presented method are found to agree well with the existing solutions in the literature and with the results obtained by the finite element method, and the presented method has a number of advantages, such as high efficiency, quite good accuracy and easy implementation.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74K20 Plates
74H45 Vibrations in dynamical problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Lu, Y.Y.; Belytschko, T.; Gu, L., A new implementation of the element-free Galerkin method, Computer methods in applied mechanics and engineering, 113, 397-414, (1994) · Zbl 0847.73064
[2] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach in computational mechanics, Computational mechanics, 22, 117-127, (1998) · Zbl 0932.76067
[3] Gu, Y.T.; Liu, G.R., A meshless local petrov – galerkin (MLPG) formulation for static and free vibration analyses of thin plates, Computer modeling in engineering & sciences, 2, 4, 463-476, (2001) · Zbl 1102.74310
[4] Long, S.Y., A local petrov – galerkin method for the elasticity problem, Acta mechanical sinica, 33, 4, 508-518, (2001)
[5] Liu, G.R.; Gu, Y.T., A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids, Journal of sound and vibration, 246, 1, 29-46, (2001)
[6] Liu, G.R.; Yan, L.; Wang, J.G.; Gu, Y.T., Point interpolation method based on local residual formulation using radial basis functions, Structural engineering and mechanics, 14, 6, 713-732, (2002)
[7] Long, S.Y.; Liu, K.Y.; Hu, D.A., A new meshless method based on MLPG for elastic dynamic problems, Engineering analysis with boundary elements, 30, 43-48, (2006) · Zbl 1195.74291
[8] Sun, J.D.; Zhang, W.X.; Tong, L.W., Bending problem of moderately thick plates solved by mesh-free method, Chinese quarterly of mechanics, 27, 2, 348-353, (2006)
[9] Sun, J.D.; Zhang, W.X.; Tong, L.W., Modal analysis of moderately thick plates by element-free method, China civil engineering journal, 39, 10, 29-33, (2006)
[10] Qian, L.F.; Batra, R.C.; Chen, L.M., Elastostatic deformations of a thick plate by using a higher-order shear and normal deformable plate theory and two meshless local petrov – galerkin (MLPG) methods, Computer modeling in engineering and sciences, 4, 161-176, (2003) · Zbl 1148.74344
[11] Qian, L.F.; Batra, R.C.; Chen, L.M., Free and forced vibrations of thick rectangular plates by using higher-order shear and normal deformable plate theory and meshless petrov – galerkin (MLPG) method, Computer modeling in engineering and sciences, 4, 519-534, (2003) · Zbl 1108.74388
[12] Xiao, J.R.; Batra, R.C.; Gilhooley, D.F., Analysis of thick plates by using a higher-order shear and normal deformable plate theory and MLPG method with radial basis functions, Computer methods in applied mechanics and engineering, 196, 979-987, (2007) · Zbl 1120.74865
[13] He, P.X.; Li, Z.R.; Feng, M.L.; Wu, C.C., Element-free Galerkin method for nonhomogeneous medium, Journal of mechanical strength, 24, 1, 70-72, (2002)
[14] Liu, G.R.; Gu, Y.T., An introduction to meshfree methods and their programming, (2005), Springer Dordrecht, The Netherlands
[15] Wang, J.G.; Liu, G.R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer methods in applied mechanics and engineering, 191, 2611-2630, (2002) · Zbl 1065.74074
[16] Cao, Z.Y.; Yang, S.T., Dynamic theory and application of thick plates, (1983), Science Press Beijing
[17] Cheng, X.S., Application of plates and shells, (1989), Shandong Science and Technology Press Jinan
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.