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The numerical analysis of piezoelectric ceramics based on the Hermite-type RPIM. (English) Zbl 1411.74062
Summary: In this paper, the Hermite-type radial point interpolation method (RPIM) is applied to analyze the property of piezoelectric ceramics in order to overcome the defects of finite element method. In this method, the inside and boundary of the problem domain are discreted by a distribution of nodes, and then the interpolation function of nodes are constructed to solve the displacement of the evaluation nodes. Compared with the finite element method, it is easier and faster for the Hermite-type RPIM to accurately achieve solution of the local regions. In contrast with the existing meshless methods, this method would not cause singularity in the process of evaluating the shape function. Furthermore, the shape function of the Hermite-type RPIM has a better stability and it can adapt to any distribution of nodes. In addition, the accuracy and stability of the method are proved by the numerical simulation.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74F15 Electromagnetic effects in solid mechanics
78A48 Composite media; random media in optics and electromagnetic theory
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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[1] He, Y. T.; Liu, J. H.; Li, L.; He, J. H., A novel capacitive pressure sensor and interface circuitry, Microsyst. Technol., 19, 25-30, (2013)
[2] Kawada, T.; Suzuki, H.; Shimizu, T.; Katsumata, M., Agreement in regard to total sleep time during a nap obtained via a sleep polygraph and accelerometer: a comparison of different sensitivity thresholds of the accelerometer, Int. J. Behav. Med., 19, 398-401, (2012)
[3] Liu, Y. X.; Liu, J. K.; Chen, W. S.; Feng, P. L., A square-type rotary ultrasonic motor using longitudinal modes, J. Electroceramics, 33, 69-74, (2014)
[4] Lee, H. Y.; Ohm, M. R.; Shin, J. Y., Fully discrete mixed finite element method for a quasilinear Stefan problem with a forcing term in non-divergence form, J. Appl. Math. Comput., 24, 191-207, (2007) · Zbl 1131.65087
[5] Guo, H., A splitting positive definite mixed finite element method for two classes of integro-differential equations, J. Appl. Math. Comput., 39, 271-301, (2012) · Zbl 1295.65126
[6] Monaghan, J. J., Particle method for hydrodynamics, Comput. Phys. Rep., 3, 71-124, (1985)
[7] Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics: theory and applications to non-spherical stars, Mon. Not. R. Astron. Soc., 18, 375-389, (1977) · Zbl 0421.76032
[8] Chen, L.; Ma, H. P.; Cheng, Y. M., Combining the complex variable reproducing kernel particle method and the finite element method for solving transient heat conduction problems, Chin. Phys. B, 22, 5, (2013)
[9] Zhang, X.; Lu, M. W.; Wegner, J. L., A 2-D meshless model for jointed rock structures, Int. J. Num. Meth. Eng., 47, 10, 1649-1661, (2000) · Zbl 0986.74080
[10] Yang, C. T., Application of reproducing kernel particle method and element-free Galerkin method on the simulation of the membrane of capacitive micromachined microphone in viscothermal air, Comput. Mech., 51, 295-308, (2013)
[11] Wang, D. D.; Chen, P. J., Quasi-convex reproducing kernel meshfree method, Comput. Mech., 54, 689-709, (2014) · Zbl 1311.65152
[12] Chen, L.; Cheng, Y. M.; Ma, H. P., The complex variable reproducing kernel particle method for the analysis of Kirchhoff plates, Comput. Mech., 55, 3, 591-602, (2015) · Zbl 1311.74154
[13] Lin, S. B.; Liu, X.; Rong, Y. H.; Xu, Z. B., Almost optimal estimates for approximation and learning by radial basis function networks, Mach. Learn., 95, 147-164, (2014) · Zbl 1320.68141
[14] Žilinskas, A., On similarities between two models of global optimization: statistical models and radial basis functions, J. Glob. Optim., 48, 173-182, (2010) · Zbl 1202.90210
[15] 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, (2015)
[16] Yin, Y.; Yao, L. Q.; Cao, Y., A 3D shell-like approach using element-free Galerkin method for analysis of thin and thick plate structures, Acta Mech. Sin., 29, 1, 85-98, (2013)
[17] Cheng, Y. M.; Bai, F. N.; Liu, C.; Peng, M. J., Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method, Int. J. Comput. Mater. Sci. Eng., 5, 4, (2016)
[18] Cheng, Y. M.; Liu, C.; Bai, F. N.; Peng, M. J., Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method, Chin. Phys. B, 24, 10, (2015)
[19] Tatari, M.; Kamranian, M.; Dehghan, M., The finite point method for the p-Laplace equation, Comput. Mech., 48, 689-697, (2011) · Zbl 1239.65072
[20] Huang, Z. Y.; Yang, X., Tailored finite point method for first order wave equation, J. Sci. Comput., 49, 351-366, (2011) · Zbl 1368.65193
[21] Shi, J. P.; Ma, W. T.; Li, N., Extended meshless method based on partition of unity for solving multiple crack problems, Meccanica, 48, 2263-2270, (2013) · Zbl 1293.74427
[22] Christensen, O.; Massopust, P., Exponential B-splines and the partition of unity property, Adv. Comput. Math., 37, 301-318, (2012) · Zbl 1260.42021
[23] Varsamis, D. N.; Karampetakis, N. P., On the Newton bivariate polynomial interpolation with applications, Multidimens. Syst. Sign. Process., 25, 179-209, (2014) · Zbl 1292.65008
[24] Strozecki, Y., On enumerating monomials and other combinatorial structures by polynomial interpolation, Theory Comput. Syst., 53, 532-568, (2013) · Zbl 1298.68096
[25] Svalina, I.; Sabo, K.; Šimunović, G., Machined surface quality prediction models based on moving least squares and moving least absolute deviations methods, Int. J. Adv. Manuf. Technol., 57, 1099-1106, (2011)
[26] Song, C. Y.; Choi, H. Y.; Lee, J. S., Approximate multi-objective optimization using conservative and feasible moving least squares method: application to automotive knuckle design, Struct. Multidisc. Optim., 49, 851-861, (2014)
[27] Sun, F. X.; Wang, J. F.; Cheng, Y. M.; Huang, A. X., Error estimates for the interpolating moving least-squares method in n-dimensional space, Appl. Numer. Math., 98, 79-105, (2015) · Zbl 1329.65280
[28] Mirzaei, D.; Schaback, R., Solving heat conduction problems by the direct meshless local petrov−galerkin (DMLPG) method, Numer. Algor., 65, 275-291, (2014) · Zbl 1292.65107
[29] Sheu, G. Y., Prediction of probabilistic settlements by the perturbation based spectral stochastic meshless local petrov−galerkin method, Geotech. Geol. Eng., 31, 1453-1464, (2013)
[30] Dai, B. D.; Zheng, B. J., Numerical solution of transient heat conduction problems using improved meshless local petrov−galerkin method, Appl. Math. Comput., 219, 10044-10052, (2013) · Zbl 1307.80008
[31] Dai, B. D.; Cheng, J.; Zheng, B. J., A moving Kriging interpolation-based meshless local petrov−galerkin method for elastodynamic analysis, Int. J. Appl. Mech., 5, 1, (2013)
[32] Liu, M. B.; Liu, G. R., Smoothed particle hydrodynamics (SPH): an overview and recent developments arch, Comput. Meth. Eng., 17, 25-76, (2010) · Zbl 1348.76117
[33] Han, Y. W.; Qiang, H. F.; Liu, H.; Gao, W. R., An enhanced treatment of boundary conditions in implicit smoothed particle hydrodynamics, Acta Mech. Sin., 30, 1, 37-49, (2014) · Zbl 1346.76005
[34] Mantegh, I.; Jenkin, M. R.M.; Goldenberg, A. A., Path planning for autonomous mobile robots using the boundary integral equation method, J. Intell. Robot. Syst., 59, 191-220, (2010) · Zbl 1203.68254
[35] Xie, G. Z.; Zhang, J. M.; Huang, C.; Lu, C. J.; Li, G. Y., A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains, Comput. Mech., 53, 575-586, (2014) · Zbl 1398.74445
[36] Cui, X. Y.; Liu, G. R.; Li, G. Y., A smoothed Hermite radial point interpolation method for thin plate analysis, Arch. Appl. Mech., 81, 1-18, (2011) · Zbl 1271.74425
[37] Liu, Y.; Hon, Y. C.; Liew, K. M., A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems, Int. J. Numer. Methods Eng., 66, 1153-1178, (2006) · Zbl 1110.74871
[38] Rocca, A. L.; Power, H., A Hermite radial basis function collocation approach for the numerical simulation of crystallization processes in a channel, Commun. Numer. Meth. Eng., 22, 119-135, (2006) · Zbl 1229.82151
[39] Gao, H. F.; Wei, G. F., Stress intensity factor for interface cracks in bimaterials using complex variable meshless manifold method, Math. Probl. Eng., (2014)
[40] Gao, H. F.; Wei, G. F., Complex variable meshless manifold method for elastic dynamic problems, Math. Probl. Eng., (2016) · Zbl 1400.74118
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