A local radial basis functions – finite difference technique for the analysis of composite plates.

*(English)*Zbl 1259.74078Summary: Radial basis functions are a very accurate means of solving interpolation and partial differential equations problems. The global radial basis functions collocation technique produces ill-conditioning matrices when using multiquadrics, making the choice of the shape parameter a crucial issue. The use of local numerical schemes, such as finite differences produces much better conditioned matrices. However, finite difference schemes are limited to special grids. For scattered points, a combination of finite differences and radial basis functions would be a possible solution. In this paper, we use a higher-order shear deformation plate theory and a radial basis function – finite difference technique for predicting the static behavior of thin and thick composite plates. Through numerical experiments on square and L-shaped plates, the accuracy and efficiency of this collocation technique is demonstrated, and the numerical accuracy and convergence are thoughtfully examined. This technique shows great potential to solve large engineering problems without the issue of ill-conditioning.

##### MSC:

74S20 | Finite difference methods applied to problems in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74K20 | Plates |

74E30 | Composite and mixture properties |

##### Software:

Matlab
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\textit{C. M. C. Roque} et al., Eng. Anal. Bound. Elem. 35, No. 3, 363--374 (2011; Zbl 1259.74078)

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##### References:

[1] | Carrera, E., Historical review of zig-zag theories for multilayered plates and shells, Applied mechanics reviews, 56, 287-308, (2003) |

[2] | Carrera, E., Developments, ideas, and evaluations based upon Reissner’s mixed variational theorem in the modelling of multilayered plates and shells, Applied mechanics reviews, 54, 301-329, (2001) |

[3] | Carrera, E., C^{0}z requirements—models for the two dimensional analysis of multilayered structures, Composite structures, 37, 373-384, (1997) |

[4] | Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Computer methods in applied mechanics and engineering, 139, 3-47, (1996) · Zbl 0891.73075 |

[5] | Belytschko, T.; Lu, Y.Y.; Gu, L., Element free Galerkin methods, International journal for numerical methods in engineering, 37, 229-256, (1994) · Zbl 0796.73077 |

[6] | Atluri, S.N.; Shen, S.P., The meshless local petrov – galerkin (MLPG) method, (2002), Tech Science Press · Zbl 1012.65116 |

[7] | 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 |

[8] | Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, Journal of geophysical research, 176, 1905-1915, (1971) |

[9] | Hardy, R.L., Research results in the application of multiquadric equations to surveying and mapping problems, Surveying and mapping, 35, 4, 321-332, (1975) |

[10] | Kansa, E.J., Multiquadrics. A scattered data approximation scheme with applications to computational fluid-dynamics. I. surface approximations and partial derivative estimates, Computers & mathematics with applications, 19, 8-9, 127-145, (1990) · Zbl 0692.76003 |

[11] | Kansa, E.J., Multiquadrics. A scattered data approximation scheme with applications to computational fluid-dynamics. II. solutions to parabolic hyperbolic and elliptic partial differential equations, Computers & mathematics with applications, 19, 8-9, 147-161, (1990) · Zbl 0850.76048 |

[12] | Roque, C.M.C.; Ferreira, A.J.M.; Jorge, R.M.N., A radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory, Journal of sound and vibration, 300, 3-5, 1048-1070, (2007) |

[13] | Roque, C.M.C.; Ferreira, A.J.M.; Jorge, R.M.N., Free vibration analysis of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions, Journal of sandwich structures and materials, 8, 6, 497-515, (2006) |

[14] | Roque, C.M.C.; Ferreira, A.J.M.; Jorge, R.M.N., Modelling of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions, Composites part B: engineering, 36, 8, 559-572, (2005) |

[15] | Ferreira, A.J.M.; Roque, C.M.C.; Martins, P.A.L.S., Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method, Composites part B: engineering, 34, 7, 627-636, (2003) |

[16] | Ferreira, A.J.M.; Roque, C.M.C.; Jorge, R.M.N., Natural frequencies of fsdt cross-ply composite shells by multiquadrics, Composite structures, 77, 3, 296-305, (2007) |

[17] | Ferreira, A.J.M.; Roque, C.M.C.; Jorge, R.M.N., Modelling cross-ply laminated elastic shells by a higher-order theory and multiquadrics, Computers and structures, 84, 19-20, 1288-1299, (2006) |

[18] | Ferreira, A.J.M.; Roque, C.M.C.; Jorge, R.M.N., Static and free vibration analysis of composite shells by radial basis functions, Engineering analysis with boundary elements, 30, 9, 719-733, (2006) · Zbl 1195.74280 |

[19] | Ferreira, A.J.M.; Roque, C.M.C.; Jorge, R.M.N., Analysis of composite plates by trigonometric shear deformation theory and multiquadrics, Computers and structures, 83, 27, 2225-2237, (2005) |

[20] | Ferreira, A.J.M.; Roque, C.M.C.; Jorge, R.M.N., Free vibration analysis of symmetric laminated composite plates by fsdt and radial basis functions, Computer methods in applied mechanics and engineering, 194, 39-41, 4265-4278, (2005) · Zbl 1151.74431 |

[21] | Ferreira, A.J.M.; Roque, C.M.C.; Fasshauer, G.E.; Jorge, R.M.N.; Batra, R.C., Analysis of functionally graded plates by a robust meshless method, Journal of mechanics of advanced materials and structures, 14, 8, 577-587, (2007) |

[22] | Ferreira, A.J.M.; Batra, R.C.; Roque, C.M.C.; Qian, L.F.; Martins, P.A.L.S., Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method, Composite structures, 69, 4, 449-457, (2005) |

[23] | Ferreira, A.J.M.; Batra, R.C.; Roque, C.M.C.; Qian, L.F.; Jorge, R.M.N., Natural frequencies of functionally graded plates by a meshless method, Composite structures, 75, 1-4, 593-600, (2006) |

[24] | Tolstykh, A.I.; Lipavskii, M.V.; Shirobokov, D.A., High-accuracy discretization methods for solid mechanics, Archives of mechanics, 55, 5-6, 531-553, (2003) · Zbl 1064.74173 |

[25] | Shu, C.; Ding, H.; Yeo, K.S., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible navier – stokes equations, Computer methods in applied mechanics and engineering, 192, 7-8, 941-954, (2003) · Zbl 1025.76036 |

[26] | Cecil, T.; Qian, J.; Osher, S., Numerical methods for high dimensional hamilton – jacobi equations using radial basis functions, Journal of computational physics, 196, 1, 327-347, (2004) · Zbl 1053.65086 |

[27] | Wright, G.B.; Fornberg, B., Scattered node compact finite difference-type formulas generated from radial basis functions, Journal of computational physics, 212, 1, 99-123, (2006) · Zbl 1089.65020 |

[28] | Shan, Y.Y.; Shu, C.; Qin, N., Multiquadric finite difference (mq-fd) methods and its application, Advances in applied mathematics and mechanics, 1, 5, 615-638, (2009) |

[29] | Reddy, J.N., Mechanics of laminated composite plates: theory and analysis, (1997), CRC Press Boca Raton · Zbl 0899.73002 |

[30] | Reddy, J.N., Simple higher-order theory for laminated composite plates, Journal of applied mechanics, transactions ASME, 51, 4, 745-752, (1984) · Zbl 0549.73062 |

[31] | Micchelli, C.A., Interpolation of scattered data distance matrices and conditionally positive definite functions, Constructive approximation, 2, 1, 11-22, (1986) · Zbl 0625.41005 |

[32] | Fornberg, B.; Wright, G.; Larsson, E., Some observations regarding interpolants in the limit of flat radial basis functions, Computers & mathematics with applications, 47, 37-55, (2004) · Zbl 1048.41017 |

[33] | Reddy, J.N., An introduction to the finite element method, (1993), McGraw-Hill International Editions New York |

[34] | Reddy, J.N., Energy and variational methods in applied mechanics, (1984), John Wiley New York · Zbl 0635.73017 |

[35] | Ferreira, A.J.M., MATLAB codes for finite element analysis: solids and structures, (2008), Springer |

[36] | Akhras, G.; Cheung, M.S.; Li, W., Finite strip analysis for anisotropic laminated composite plates using higher-order deformation theory, Computers & structures, 52, 3, 471-477, (1994) · Zbl 0872.73074 |

[37] | Akhras, G.; Cheung, M.S.; Li, W., Static and vibrations analysis of anisotropic laminated plates by finite strip method, International journal of solids and structures, 30, 22, 3129-3137, (1993) · Zbl 0790.73078 |

[38] | Reddy, J.N., A simple higher-order theory for laminated composite plates, Journal of applied mechanics, 51, 745-752, (1984) · Zbl 0549.73062 |

[39] | Pagano, N.J., Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of composite materials, 4, 20-34, (1970) |

[40] | Pandya, B.N.; Kant, T., Higher-order shear deformable theories for flexure of sandwich plates-finite element evaluations, International journal of solids and structures, 24, 419-451, (1988) · Zbl 0676.73044 |

[41] | Ferreira, A.J.M., A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates, Composite structures, 59, 3, 385-392, (2003) |

[42] | Srinivas, S., A refined analysis of composite laminates, Journal of sound and vibration, 30, 495-507, (1973) · Zbl 0267.73050 |

[43] | Reddy, J.N.; Phan, N.D., Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation-theory, Journal of sound and vibration, 98, 2, 157-170, (1985) · Zbl 0558.73031 |

[44] | Liew, K.M.; Huang, Y.Q.; Reddy, J.N., Vibration analysis of symmetrically laminated plates based on fsdt using the moving least squares differential quadrature method, Computer methods in applied mechanics and engineering, 192, 19, 2203-2222, (2003) · Zbl 1119.74628 |

[45] | Reddy, J.N., Mechanics of laminated composite plates, (1997), CRC Press New York · Zbl 0899.73002 |

[46] | Khdeir, A.A.; Librescu, L., Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory. 2. buckling and free-vibration, Composite structures, 9, 4, 259-277, (1988) |

[47] | Nayak, A.K.; Moy, S.S.J.; Shenoi, R.A., Free vibration analysis of composite sandwich plates based on Reddy’s higher-order theory, Composites part B—engineering, 33, 7, 505-519, (2002) |

[48] | Srinivas, S.; Rao, C.V.J.; Rao, A.K., An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates, Journal of sound and vibration, 12, 2, 187, (1970) · Zbl 0212.57801 |

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.