Generalized finite difference method for electroelastic analysis of three-dimensional piezoelectric structures.(English)Zbl 1462.74165
Summary: This short communication makes the first attempt to apply the generalized finite difference method (GFDM), a newly-developed meshless collocation method, for the numerical solutions of three-dimensional (3D) piezoelectric problems. In the present method, the entire computational domain is divided into a set of overlapping subdomains in which the local Taylor series expansion and moving-least square approximation are applied to construct the local systems of linear equations. By satisfying the coupled mechanical and electrical governing equations, a sparse and banded stiffness matrix can be established which makes the method very attractive for large-scale engineering simulations. Preliminary numerical experiments are presented to demonstrate the applicability and accuracy of the present method, where the results obtained are compared with the analytical solutions with very good agreement.
MSC:
| 74S20 | Finite difference methods applied to problems in solid mechanics |
| 74F15 | Electromagnetic effects in solid mechanics |
Keywords:
meshless collocation method;Taylor series expansion;moving-least square approximation;sparse stiffness matrixCite
Full Text:DOI
References:
| [1] | Chen, S. S.; Li, Q. H.; Liu, Y. H.; Xue, Z. Q., A meshless local natural neighbour interpolation method for analysis of two-dimensional piezoelectric structures, Eng. Anal. Bound. Elem., 37, 2, 273-279 (2013) ·Zbl 1351.74079 |
| [2] | Liu, Y.; Fan, H., Analysis of thin piezoelectric solids by the boundary element method, Comput. Methods Appl. Mech. Eng., 191, 21-22, 2297-2315 (2002) ·Zbl 1131.74342 |
| [3] | Cao, C.; Qin, Q.-H.; Yu, A., Hybrid fundamental-solution-based FEM for piezoelectric materials, Comput. Mech., 50, 4, 397-412 (2012) ·Zbl 1300.74050 |
| [4] | Fu, Z.; Xi, Q.; Li, Y.; Huang, H.; Rabczuk, T., Hybrid FEM-SBM solver for structural vibration induced underwater acoustic radiation in shallow marine environment, Comput. Methods Appl. Mech. Eng., 369, Article 113236 pp. (2020) ·Zbl 1506.74137 |
| [5] | Benito, J. J.; Urena, F.; Gavete, L., Influence of several factors in the generalized finite difference method, Appl. Math. Model., 25, 12, 1039-1053 (2001) ·Zbl 0994.65111 |
| [6] | Gu, Y.; Qu, W.; Chen, W.; Song, L.; Zhang, C., The generalized finite difference method for long-time dynamic modeling of three-dimensional coupled thermoelasticity problems, J. Comput. Phys., 384, 42-59 (2019) ·Zbl 1451.74215 |
| [7] | Li, P.-W.; Fan, C.-M., Generalized finite difference method for two-dimensional shallow water equations, Eng. Anal. Bound. Elem., 80, 58-71 (2017) ·Zbl 1403.76133 |
| [8] | Qu, W.; He, H., A spatial-temporal GFDM with an additional condition for transient heat conduction analysis of FGMs, Appl. Math. Lett., 110, Article 106579 pp. (2020) ·Zbl 1452.80005 |
| [9] | Benito, J. J.; Ureña, F.; Gavete, L.; Salete, E.; Ureña, M., Implementations with generalized finite differences of the displacements and velocity-stress formulations of seismic wave propagation problem, Appl. Math. Model., 52, 1-14 (2017) ·Zbl 1480.65203 |
| [10] | Wang, Y.; Gu, Y.; Fan, C.-M.; Chen, W.; Zhang, C., Domain-decomposition generalized finite difference method for stress analysis in multi-layered elastic materials, Eng. Anal. Bound. Elem., 94, 94-102 (2018) ·Zbl 1403.74282 |
| [11] | Wang, F.; Wang, C.; Chen, Z., Local knot method for 2D and 3D convection-diffusion-reaction equations in arbitrary domains, Appl. Math. Lett., 105, Article 106308 pp. (2020) ·Zbl 1524.65915 |
| [12] | Fu, Z.-J.; Xie, Z.-Y.; Ji, S.-Y.; Tsai, C.-C.; Li, A.-L., Meshless generalized finite difference method for water wave interactions with multiple-bottom-seated-cylinder-array structures, Ocean Eng., 195, Article 106736 pp. (2020) |
| [13] | Xia, H.; Gu, Y., Short communication: The generalized finite difference method for electroelastic analysis of 2D piezoelectric structures, Eng. Anal. Bound. Elem., 124, 82-86 (2021) ·Zbl 1464.74365 |
| [14] | Li, P.-W., Space-time generalized finite difference nonlinear model for solving unsteady Burgers’ equations, Appl. Math. Lett., 114, Article 106896 pp. (2021) ·Zbl 1458.65110 |
| [15] | Benito, J. J.; Urena, F.; Gavete, L.; Alvarez, R., An h-adaptive method in the generalized finite differences, Comput. Methods Appl. Mech. Engrg., 192, 5-6, 735-759 (2003) ·Zbl 1024.65099 |
| [16] | Fu, Z.-J.; Tang, Z.-C.; Zhao, H.-T.; Li, P.-W.; Rabczuk, T., Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method, Eur. Phys. J. Plus, 134, 6, 272 (2019) |
| [17] | Ding, H. J.; Chen, W. Q.; Jiang, A. M., Green’s functions and boundary element method for transversely isotropic piezoelectric materials, Eng. Anal. Bound. Elem., 28, 8, 975-987 (2004) ·Zbl 1112.74550 |
| [18] | Wang, Y. Y.; Gu, Y.; Liu, J. L., A domain-decomposition generalized finite difference method for stress analysis in three-dimensional composite materials, Appl. Math. Lett., 104, 6 (2020) ·Zbl 1444.74053 |
| [19] | Ding, H.; Liang, J., The fundamental solutions for transversely isotropic piezoelectricity and boundary element method, Comput. Struct., 71, 4, 447-455 (1999) |
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