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Radial basis reproducing kernel particle method for piezoelectric materials. (English) Zbl 1403.74331
Summary: To reduce the negative effect of different kernel functions on calculating accuracy, the radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is proposed, the corresponding governing equations are derived. The RRKPM is more efficient to solve the local problem domain, and can improve the accuracy and stability of the RKPM. Then the RRKPM is applied to the numerical simulation of piezoelectric materials, the corresponding formulae for piezoelectric materials are derived. The numerical results illustrate the proposed method is more stable and accurate than the RKPM.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
74F15 Electromagnetic effects in solid mechanics
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[1] Sun, XD; Yuan, WZ; Qiao, DY; Ren, S, The analysis of the self-oscillation system for resonant pressure sensor, Microsyst Technol, 23, 945-951, (2017)
[2] Chang, KT; Lee, CW, Fabrication and characteristics of thin disc piezoelectric transformers based on piezoelectric buzzers with gap circles, Ultrasonics, 48, 91-97, (2008)
[3] Bolborici, V; Dawson, FP; Pugh, MC, A finite volume method and experimental study of a stator of a piezoelectric traveling wave rotary ultrasonic motor, Ultrasonics, 54, 809-820, (2014)
[4] Li, C; Man, H; Song, CM; Gao, W, Fracture analysis of piezoelectric materials using the scaled boundary finite element method, Eng Fract Mech, 97, 52-71, (2013)
[5] Yang, CT, 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) · Zbl 06149123
[6] Peng, MJ; Cheng, YM, A boundary element-free Galerkin (IEFG) for two-dimensional potential problems, Eng Anal Boundary Elem, 33, 77-82, (2009)
[7] Li, PW; Fan, CM, Generalized finite difference method for two-dimensional shallow water equations, Eng Anal Boundary Elem, 80, 58-71, (2017) · Zbl 1403.76133
[8] Reutskiy, SY; Lin, J, A meshless radial basis function method for steady-state advection-diffusion-reaction equation in arbitrary 2D domains, Eng Anal Boundary Elem, 79, 49-61, (2017) · Zbl 1403.65172
[9] Ohs, RR; Aluru, NR, Meshless analysis of piezoelectric devices, Comput Mech, 27, 23-36, (2001) · Zbl 1005.74078
[10] Yang, XJ; Zheng, J; Long, SY, Topology optimization of continuum structures with displacement constraints based on meshless method, Int J Mech Mater Des, 13, 311-320, (2017)
[11] Chen, L; Ma, HP; Cheng, YM, Combining the complex variable reproducing kernel particle method and the finite element method for solving transient heat conduction problems, Chin Phys, 22, (2013)
[12] Wang, DD; Chen, PJ, Quasi-convex reproducing kernel meshfree method, Comput Mech, 54, 689-709, (2014) · Zbl 1311.65152
[13] Chen, L; Cheng, YM; Ma, HP, The complex variable reproducing kernel particle method for the analysis of Kirchhoff plates, Comput Mech, 55, 591-602, (2015) · Zbl 1311.74154
[14] Ĺ˝ilinskas, A., On similarities between two models of global optimization: statistical models and radial basis functions, J Global Optim, 48, 173-182, (2010) · Zbl 1202.90210
[15] Lin, SB; Liu, X; Rong, YH; Xu, ZB, Almost optimal estimates for approximation and learning by radial basis function networks, Mach Learn, 95, 147-164, (2014)
[16] Deng, YJ; Liu, C; Peng, MJ; Cheng, YM, The interpolating complex variable element-free Galerkin method for temperature field problems, Int J Appl Mech, 7, (2015)
[17] Cheng, YM; Bai, FN; Peng, MJ, A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity, Appl Math Model, 38, 5187-5197, (2014) · Zbl 1449.74196
[18] Cheng, YM; Bai, FN; Liu, C; Peng, MJ, 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, (2016)
[19] Liu, YH; Chen, J; Yu, S; Li, CX, Numerical simulation of three-dimensional bulk forming processes by the element-free Galerkin method, Int J Adv Manuf Technol, 36, 442-450, (2008)
[20] Huang, ZY; Yang, X, Tailored finite point method for first order wave equation, J Sci Comput, 49, 351-366, (2011) · Zbl 1368.65193
[21] Tatari, M; Kamranian, M; Dehghan, M, The finite point method for the p-Laplace equation, Comput Mech, 48, 689-697, (2011) · Zbl 1239.65072
[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] Shi, JP; Ma, WT; Li, N, Extended meshless method based on partition of unity for solving multiple crack problems, Meccanica, 48, 2263-2270, (2013) · Zbl 1293.74427
[24] Ren, HP; Cheng, YM, The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems, Eng Anal Boundary Elem, 36, 873-880, (2012) · Zbl 1352.65539
[25] Strozecki, Y., On enumerating monomials and other combinatorial structures by polynomial interpolation, Theory Comput Syst, 53, 532-568, (2013) · Zbl 1298.68096
[26] Li, XL; Zhang, SG, Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving least-square approximation, Appl Math Model, 40, 2875-2896, (2016)
[27] Song, CY; Choi, HY; Lee, JS, Approximate multi-objective optimization using conservative and feasible moving least squares method: application to automotive knuckle design, Struct Multidiscip Optim, 49, 851-861, (2014)
[28] Sun, FX; Wang, JF; Cheng, YM; Huang, AX, Error estimates for the interpolating moving least-squares method in n-dimensional space, Appl Numer Math, 98, 79-105, (2015)
[29] Chen, L; Liu, C; Ma, HP; Cheng, YM, An interpolating local Petrov-Galerkin method for potential problems, Int J Appl Mech, 6, (2014)
[30] Sheu, GY, Prediction of probabilistic settlements by the perturbation based spectral stochastic meshless local Petrov-Galerkin method, Geotech Geol Eng, 31, 1453-1464, (2013)
[31] Dai, BD; Zheng, BJ; Liang, QX; Wang, LH, Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method, Appl Math Comput, 219, 10044-10052, (2013) · Zbl 1307.80008
[32] Darani, MA, Direct meshless local Petrov-Galerkin method for the two-dimensional Klein-Gordon equation, Eng Anal Boundary Elem, 74, 1-13, (2017) · Zbl 1403.65066
[33] Liu, MB; Liu, GR, Smoothed particle hydrodynamics (SPH): an overview and recent developments arch, Arch Comput Methods Eng, 17, 25-76, (2010) · Zbl 1348.76117
[34] Han, YW; Qiang, HF; Liu, H; Gao, WR, An enhanced treatment of boundary conditions in implicit smoothed particle hydrodynamics, Acta Mech Sin, 30, 37-49, (2014)
[35] Mantegh, I; Jenkin, MRM; Goldenberg, AA, Path planning for autonomous mobile robots using the boundary integral equation method, J Intell Rob Syst, 59, 191-220, (2010) · Zbl 1203.68254
[36] Xie, GZ; Zhang, JM; Huang, C; Lu, CJ; Li, GY, 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
[37] Cui, XY; Liu, GR; Li, GY, A smoothed Hermite radial point interpolation method for thin plate analysis, Arch Appl Mech, 81, 1-18, (2011) · Zbl 1271.74425
[38] Liu, Y; Hon, YC; Liew, KM, A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems, Int J Numer Methods Eng, 66, 1153-1178, (2006) · Zbl 1110.74871
[39] Rocca, AL; Power, H, A Hermite radial basis function collocation approach for the numerical simulation of crystallization processes in a channel, Commun Numer Methods Eng, 22, 119-135, (2006) · Zbl 1229.82151
[40] Ma, JC; Wei, GF; Liu, DD; Liu, GT, The numerical analysis of piezoelectric ceramics based on the Hermite-type RPIM, Appl Math Comput, 309, 170-182, (2017) · Zbl 1411.74062
[41] Gao, HF; Wei, GF, Complex variable meshless manifold method for transient heat conduction problems, Int J Appl Mech, 9, (2017)
[42] Gao, HF; Wei, GF, Complex variable meshless manifold method for elastic dynamic problems, Math Prob Eng, 2016, (2016)
[43] Liew, KM; Lim, HK; Tan, MJ; He, XQ, Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method, Comput Mech, 29, 486-497, (2002) · Zbl 1146.74370
[44] Gaudnzi, P; Bathe, KJ, An iterative finite element procedure for the analysis of piezoelectric continua, J Intell Mater Syst Struct, 6, 266-273, (1995)
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