Abstract
In this paper, the localized method of approximate particular solutions using polynomial basis functions is proposed to solve plate bending problems with complex domains. The closed-form particular solutions of fourth-order differential equations can be reduced to the linear combination of the particular solutions of Helmholtz and modified Helmholtz equations. To alleviate the difficulty of solving overdetermined fourth-order plate bending problems using the localized collocation method, additional ghost points outside the computational domain are introduced to improve the stability and accuracy. Three examples are illustrated to validate the feasibility of the proposed localized MAPS.
Similar content being viewed by others
References
Bai, Y.H., Wu, Y.K., Xie, X.P.: Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method. Sci. China Math. 59, 1835–1850 (2016)
Bengzon, F.: The Finite Element Method: Theory, Implementation, and Applications (2013)
Chang, W., Chen, C.S., Li, W.: Solving fourth order differential equations using particular solutions of Helmholtz-type equations. Appl. Math. Lett. 86, 179–185 (2018)
Chen, W., Ye, L., Sun, H.: Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59(5), 1614–1620 (2010)
Chen, C.S., Fan, C.M., Wen, P.H.: The method of approximate particular solutions for solving certain partial differential equations. Numer. Methods Partial Differ. Equ. 28(2), 506–522 (2012)
Dangal, T., Chen, C.S., Ji, Lin: Polynomial particular solution for solving elliptic partial differential equations. Comput. Math. Appl. 73, 60–70 (2017)
Fu, Z.J., Chen, W., Yang, W.: Winkler plate bending problems by a truly boundary-only boundary particle method. Comput. Mech. 44(6), 757–763 (2009)
Fu, Z.J., Xi, Q., Ling, L., Cao, C.Y.: Numerical investigation on the effect of tumor on the thermal behavior inside the skin tissue. Int. J. Heat Mass Transf. 108, 1154–1163 (2017)
Fu, Z.J., Xi, Q., Chen, W., Cheng, A.H.-D.: A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations. Comput. Math. Appl. 76(4), 760–773 (2018)
Golberg, M.A., Muleshkov, A.S., Chen, C.S., Cheng, A.H.-D.: Polynomial particular solutions for certain kind of partial differential operators. Numer. Methods Partial Differ. Equ. 19, 112–133 (2003)
Jiang, Y., Xu, X.: A monotone finite volume method for time fractional Fokker–Planck equations. Sci. China-Math. 62(4), 783–794 (2019)
Le, J., Jin, H., Lv, X.G., Cheng, Q.S.: A preconditioned method for the solution of the Robbins problem for the Helmholtz equation. Anziam J. 52, 87–100 (2010)
Li, H.B., Huang, T.Z., Zhang, Y., Liu, X.P., Gu, T.X.: Chebyshev-type methods and preconditioning techniques. Appl. Math. Comput. 218, 260–270 (2011)
Li, M., Amazzarb, G., Naji, A., Chen, C.S.: Solving biharmonic equation using the localized method of approximate particular solutions. Int. J. Comput. Math. 91, 1790–1801 (2014)
Li, J., Liu, F., Feng, L., Turner, I.: A novel finite volume method for the Riesz space distributed-order diffusion equation. Comput. Math. Appl. 74(4), 772–783 (2017)
Liang, J.W., Liu, Z.X., Huang, L., Yang, G.G.: The indirect boundary integral equation method for the broadband scattering of plane P, SV and Rayleigh waves by a hill topography. Eng. Anal. Bound. Elem. 98, 184–202 (2019)
Lin, J., Chen, C.S., Wang, F., Dangal, T.: Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems. Appl. Math. Model. 49, 452–469 (2017)
Liu, C.: A multiple-scale Trefftz method for an incomplete Cauchy problem of biharmonic equation. Eng. Anal. Bound. Elem. 37, 1445–1456 (2013)
Liu, X., Wu, X.: Differential quadrature Trefftz method for irregular plate problems. Eng. Anal. Bound. Elem. 33(3), 363–367 (2009)
Liu, F., Feng, L., Vo, A., Li, J.: Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch–Torrey equations on irregular convex domains. Comput. Math. Appl. 78(5), 1637–1650 (2019)
Qu, W.Z., Fan, C.M., Li, X.L.: Analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions. Comput. Math. Appl. 80(1), 13–30 (2020)
Tang, Z.C., Fu, Z.J., Zheng, D.J., Huang, J.D.: Singular boundary method to simulate scattering of SH wave by the canyon topography. Adv. Appl. Math. Mech. 10(4), 912–924 (2018)
Tian, Z., Li, X., Fan, C.M., Chen, C.S.: The method of particular solutions using trigonometric basis functions. J. Comput. Appl. Math. 335, 20–32 (2018)
Ventsel, E., Krauthammer, T., Carrera, E.: Thin plates and shells: theory, analysis, and applications. Appl. Mech. Rev. 55(4), 1813–1831 (2001)
Wang, C., Huang, T.Z., Wen, C.: A new preconditioner for indefinite and asymmetric matrices. Appl. Math. Comput. 219(23), 11036–11043 (2013)
Wang, F.J., Fan, C.M., Hua, Q.S., Gu, Y.: Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations. Appl. Math. Comput. 364, 124658 (2020)
Wei, X., Sun, L., Yin, S., Chen, B.: A boundary-only treatment by singular boundary method for two-dimensional inhomogeneous problems. Appl. Math. Model. 62, 338–351 (2018)
Wu, H.Y., Duan, Y.: Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis–Procesi equation. Appl. Math. Comput. 274, 83–92 (2016)
Xi, Q., Fu, Z.J., Rabczuk, T.: An efficient boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials under heat source load. Comput. Mech. 1–15 (2019)
Yan, L., Yang, F.: The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition. Comput. Math. Appl. 70(3), 254–264 (2015)
Yao, G., Chen, C.S., Tsai, C.C.: A revisit on the derivation of the particular solution for the differential operator \(\Delta ^{2} \pm \lambda ^{2}\). Adv. Appl. Math. Mech. 1, 750–768 (2009)
Yao, G., Chen, C.S., Kolibal, J.: A localized approach for the method of approximate particular solutions. Comput. Math. Appl. 61, 2376–2387 (2011)
Zheng, H., Yang, Z., Zhang, C., Tyrer, M.: A local radial basis function collocation method for band structure computation of phononic crystals with scatterers of arbitrary geometry. Appl. Math. Model. 60, 447–459 (2018)
Zhou, F.L., You, Y.L., Li, G., Xie, G.Z.: The precise integration method for semi-discretized equation in the dual reciprocity method to solve three-dimensional transient heat conduction problems. Eng. Anal. Bound. Elem. 95, 160–166 (2018)
Acknowledgements
The work described in this paper was supported by the National Science Funds of China (Grant Nos. 11772119), the Fundamental Research Funds for the Central Universities (Grant No. B200202124), Alexander von Humboldt Research Fellowship (ID: 1195938), the Six Talent Peaks Project in Jiangsu Province of China (Grant No. 2019-KTHY-009) and the Programme B18019 of Discipline Expertise to Universities MOE & MST. The third author acknowledges HPC at the University of Southern Mississippi supported by the National Science Foundation under the Major Research Instrumentation (MRI) program via Grant # ACI 1626217.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tang, ZC., Fu, ZJ. & Chen, C.S. A localized MAPS using polynomial basis functions for the fourth-order complex-shape plate bending problems. Arch Appl Mech 90, 2241–2253 (2020). https://doi.org/10.1007/s00419-020-01718-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-020-01718-y