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The complex variable meshless local Petrov-Galerkin method for elasticity problems of functionally graded materials. (English) Zbl 1410.74076
Summary: This paper proposed the complex variable meshless local Petrov-Galerkin (CVMLPG) method for the static analysis of functionally graded materials (FGMs). In the presented method, the complex variable moving least-square (CVMLS) approximation, which is established based on the moving least-square (MLS) approximation by introducing the complex variable theory, is adopted for construction of the field approximation function. Compared with the conventional MLS method, the number of the unknown coefficients in the trial function of the CVMLS method is less than that of the MLS approximation, thus higher efficiency and accuracy can be achieved under the same node distributions. One advantage of the CVMLPG method for the FGMs is that the variations of the functionally graded material properties are simulated by using material parameters at Gauss points, so it totally avoids the issue of the assumption of homogeneous in each element in the finite element method (FEM) for the FGMs. Some of selected benchmark examples are considered to confirm the validity and accuracy of the proposed method.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74E30 Composite and mixture properties
Software:
Mfree2D
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[1] Koizumi, M., The concept of FGM, Ceram. Trans. Func. Grad. Mater., 1-20, (1993)
[2] Liu, G. R., Mesh free methods: moving beyond the finite element method, (2009), CRC Press Inc.
[3] Belytschko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin methods, Int. J. Numer. Methods Eng., 37, 229-256, (1994) · Zbl 0796.73077
[4] Atluri, S. N.; Zhu, T. L., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22, 117-127, (1998) · Zbl 0932.76067
[5] Liu, W. K.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[6] Atluri, S. N.; Kim, H. G.; Cho, J. Y., A critical assessment of the truly meshless local Petrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods, Comput. Mech., 24, 348-372, (1999) · Zbl 0977.74593
[7] Atluri, S. N.; Shen, S. P., The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods, Comput. Model. Eng. Sci., 3, 11-51, (2002) · Zbl 0996.65116
[8] Liu, G. R.; Gu, Y. T., A local point interpolation method for stress analysis of two-dimensional solids, Struct. Eng. Mech., 11, 221-236, (2001)
[9] Liew, K. M.; Cheng, Y. M.; Kitipornchai, S., Boundary element-free method and its application to two-dimensional elasticity problems, Int. J. Numer. Methods Eng., 65, 1310-1332, (2006) · Zbl 1147.74047
[10] Gu, Y. T.; Liu, G. R., A meshless local Petrov-Galerkin method for free and forced vibration analyses for solids, Comput. Mech., 27, 188-198, (2001) · Zbl 1162.74498
[11] B.D. Dai, B.J. Zheng, Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method, Appl. Math. Comput. 219(2013) 10044-10052. · Zbl 1307.80008
[12] Cheng, Y. M.; Chen, M. J., A boundary element-free method for linear elasticity, Acta Mech. Sin., 35, 181-186, (2003)
[13] Cheng, Y. M.; Peng, M. J., Boundary element-free method for elastodynamics, Sci. China Ser. G Phys. Mech. Astron., 48, 641-657, (2005)
[14] Peng, M. J.; Cheng, Y. M., A boundary element-free method (BEFM) for two-dimensional potential problems, Eng. Anal. Bound. Elem., 33, 77-82, (2009) · Zbl 1160.65348
[15] Cheng, R. J.; Liew, K. M., Analyzing two-dimensional sine-Gordon equation with the mesh-free reproducing kernel particle Ritz method, Comput. Method Appl. Mech. Eng., 245-246, 132-143, (2012) · Zbl 1354.65202
[16] Cheng, R. J.; Cheng, Y. M., Error estimates for the finite point method, Appl. Numer. Math., 58, 884-898, (2008) · Zbl 1145.65086
[17] Cheng, R. J.; Liew, K. M., Analyzing modified equal width wave (MEW) equation using the improved element-free Galerkin method, Eng. Anal. Bound. Elem., 36, 1322-1330, (2012) · Zbl 1352.65482
[18] Cheng, R. J.; Liew, K. M., Modeling of biological population problems using the element-free KP-Ritz method, Appl. Math. Comput., 227, 274-290, (2014) · Zbl 1364.65193
[19] Liew, K. M.; Feng, C.; Cheng, Y. M.; Kitipornchai, S., Complex variable moving least-squares method: a meshless approximation technique, Int. J. Numer. Methods Eng., 70, 46-70, (2007) · Zbl 1194.74554
[20] Cheng, Y. M.; Peng, M. J.; Li, J. H., The complex variable moving least-square approximation and its application, Chin. J. Theor. Appl. Mech., 37, 719-723, (2005), (in Chinese)
[21] Cheng, Y. M.; Li, J. H., A meshless method with complex variables for elasticity, Acta Phys. Sin., 54, 10, 4463-4471, (2005) · Zbl 1202.74163
[22] Peng, M. J.; Li, D. M.; Liew, K. M.; Cheng, Y. M., The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems, Eng. Struct., 33, 127-135, (2011)
[23] Peng, M. J.; Liu, P.; Cheng, Y. M., The complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems, Int. J. Appl. Mech., 1, 2, 367-385, (2009)
[24] Cheng, Y. M.; Wang, J. F.; Li, R. X., The complex variable element-free Galerkin (CVEFG) method for two-dimensional elastodynamics problems, Int. J. Appl. Mech., 4, 1250042, (2012)
[25] Cheng, Y. M.; Wang, J. F.; Bai, F. N., A new complex variable element-free Galerkin method for two-dimensional potential problems, Chin. Phys. B, 21, 090203, (2012)
[26] Cheng, Y. M.; Li, R. X.; Peng, M. J., Complex variable element-free Galerkin method for (CVEFG) for viscoelasticity problems, Chin. Phys. B, 21, 090205, (2012)
[27] Wang, J. F.; Cheng, Y. M., A new complex variable meshless method for transient heat conduction problems, Chin. Phys. B, 21, 120206, (2012)
[28] Deng, Y. J.; Liu, C.; Peng, M. J.; Cheng, Y. M., The interpolating complex variable element-free Galerkin method for temperature field problems, Int. J. Appl. Mech., 7, 2, 1550017, (2015)
[29] Bai, F. N.; Li, D. M.; Wang, J. F.; Cheng, Y. M., An improved complex variable element-free Galerkin (ICVEFG) method for two-dimensional elasticity problems, Chin. Phys. B, 21, 020204, (2012)
[30] Li, D. M.; Bai, F. N.; Cheng, Y. M.; Liew, K. M., A novel complex variable element-free Galerkin method for two-dimensional large deformation problems, Comput. Methods Appl. Mech. Eng., 233-236, 1-10, (2012) · Zbl 1253.74106
[31] Liew, K. M.; Cheng, Y. M., Complex variable boundary element-free method for two-dimensional elastodynamic problems, Comput. Methods Appl. Mech. Eng., 198, 3925-3933, (2009) · Zbl 1231.74502
[32] Li, D. M.; Liew, K. M.; Cheng, Y. M., Analyzing elastoplastic large deformation problems with the complex variable element-free Galerkin method, Comput. Mech., 53, 1149-1162, (2014)
[33] Li, D. M.; Liew, K. M.; Cheng, Y. M., An improved complex variable element-free Galerkin method for two-dimensional large deformation elasto-plasticity problems, Comput. Methods Appl. Mech. Eng., 269, 72-86, (2014) · Zbl 1296.74015
[34] Chen, L.; Cheng, Y. M., Reproducing kernel particle method with complex variables for elasticity, Acta Phys. Sin., 57, 1-10, (2008), (in Chinese)
[35] Chen, L.; Cheng, Y. M., The complex variable reproducing kernel particle method for transient heat conduction, Acta Phys. Sin., 57, 1-9, (2008), (in Chinese)
[36] Chen, L.; Zhu, Y. J.; Cheng, Y. M., The complex variable reproducing kernel particle method for potential problems, Chin. J. Appl. Mech., 26, 1-6, (2009), (in Chinese)
[37] Chen, L.; Cheng, Y. M., The complex variable reproducing kernel particle method for elastoplasticity problems, Sci. China Ser. G-Phys. Mech. Astron., 53, 954-965, (2010)
[38] Yang, X. L.; Dai, B. D.; Li, Z. F., Meshless local Petrov-Galerkin method with complex variable for elasticity, Acta. Phys. Sin., 61, 050204, (2012), (in Chinese)
[39] Yang, X. L.; Dai, B. D.; Zhang, W. W., The complex variable meshless local Petrov-Galerkin method of solving two-dimensional potential problems, Chin. Phys. B, 21, 100208, (2012)
[40] Wang, Q. F.; Dai, B. D.; Li, Z. F., The complex variable meshless local Petrov-Galerkin method for transient heat conduction problems, Chin. Phys. B, 21, 080203, (2013)
[41] Dai, B. D.; Wang, Q. F.; Zhang, W. W.; Wang, L. H., The complex variable meshless local Petrov-Galerkin method for elastodynamic problems, Appl. Math. Comput., 243, 311-321, (2014) · Zbl 1335.74054
[42] Chen, L. M.; Yang, B. N., Mechanical analysis for composite materials, Chin. Sci. and Tech. Press, 78-98, (2006), (in Chinese)
[43] Huang, D. J.; Ding, H. J.; Chen, W. Q., Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load, Appl. Math. Mech., 28, 855-860, (2007) · Zbl 1231.74028
[44] Huang, D. J.; Ding, H. J.; Chen, W. Q., Analytical solution and semi-analytical solution for anisotropic functionally graded beam subjected to arbitrary loading, Sci. China Ser. G-Phys. Mech. Astron., 52, 1244-1256, (2009)
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