An element-free IMLS-Ritz method for numerical solution of three-dimensional wave equations. (English) Zbl 1425.65122
Summary: This paper presents an element-free based numerical approach for solving three-dimensional wave equations based on the Ritz minimization procedure. The method involves the use of a set of orthogonal shape functions to approximate its field variables. In this study, an improved moving least-squares (IMLS) is used to generate sets of orthogonal shape functions that reduced the number of unknown coefficients in the trial functions. The entire approximation procedure can be easily implemented numerically. The accuracy of the approximation can be enhanced by increasing the number of nodes used in the computation. As the result of the above procedures, a final algebraic equation system is derived through discretizing the constructed functional. The functional is established by enforcing the Dirichlet boundary conditions via the penalty approach. The simplicity and applicability of the element-free method are demonstrated by solving several selected linear and nonlinear three-dimensional wave equations. Besides the convergence study, the present results are compared with the available published solutions from the literature, where possible, verifying the accuracy, efficiency and reliability of the IMLS-Ritz method.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
Keywords:
element-free method; improved moving least-squares; Ritz method; three-dimensional wave equationReferences:
| [1] | Hirota, R., Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14, 805-809, (1973) · Zbl 0257.35052 |
| [2] | Velasco, M. P.; Vázquez, L., On the fractional Newton and wave equation in one space dimension, Appl. Math. Model., 38, 3314-3324, (2014) · Zbl 1449.35451 |
| [3] | Liu, S.; Fu, Z.; Liu, S.; Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289, 69-74, (2001) · Zbl 0972.35062 |
| [4] | Whitham, G. B., Linear and nonlinear waves, (2011), John Wiley & Sons · Zbl 0373.76001 |
| [5] | Vitanov, N. K.; Dimitrova, Z. I., Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics, Commun. Nonlinear Sci. Numer. Simul., 15, 2836-2845, (2010) · Zbl 1222.35201 |
| [6] | Dehghan, M.; Salehi, R., A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation, Appl. Math. Model., 36, 1939-1956, (2012) · Zbl 1243.76069 |
| [7] | Liew, K. M.; Cheng, R. J., Numerical study of the three-dimensional wave equation using the mesh-free KP-Ritz method, Eng. Anal. Bound. Elem., 37, 977-989, (2012) · Zbl 1287.65080 |
| [8] | Zhang, Z.; Li, D. M.; Cheng, Y. M.; Liew, K. M., The improved element-free Galerkin method for three-dimensional wave equation, Acta Mech. Sin., 28, 808-818, (2012) · Zbl 1345.65059 |
| [9] | Zhang, L. W.; Lei, Z. X.; Liew, K. M.; Yu, J. L., Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels, Comput. Methods Appl. Mech. Engrg., 273, 1-18, (2014) · Zbl 1296.76116 |
| [10] | Zhang, L. W.; Zhu, P.; Liew, K. M., Thermal buckling of functionally graded plates using a local Kriging meshless method, Compos. Struct., 108, 472-492, (2014) |
| [11] | Zhang, L. W.; Deng, Y. J.; Liew, K. M., An improved element-free Galerkin method for numerical modeling of the biological population problems, Eng. Anal. Bound. Elem., 40, 181-188, (2014) · Zbl 1297.65123 |
| [12] | Zakeri, A.; Shayegan, A. H.S., Gradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problems, Appl. Math. Model., 38, 775-783, (2014) · Zbl 1427.65385 |
| [13] | Keramidas, G. A.; Ramberg, S. E.; Griffin, O. M., Finite element modelling of surface waves in a channel, Appl. Math. Model., 8, 298-304, (1984) · Zbl 0555.76020 |
| [14] | Jin, J., The finite element method in electromagnetics, (2014), John Wiley & Sons · Zbl 1419.78001 |
| [15] | Johnson, C., Numerical solution of partial differential equations by the finite element method, (2012), Courier Dover Publications |
| [16] | Schanz, M., Wave propagation in viscoelastic and poroelastic continua: A boundary element approach, (2012), Springer Science & Business |
| [17] | Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Internat. J. Numer. Methods Engrg., 37, 229-256, (1994) · Zbl 0796.73077 |
| [18] | Li, D. M.; Zhang, Z.; Liew, K. M., A numerical framework for two-dimensional large deformation of inhomogeneous swelling of gels using the improved complex variable element-free Galerkin method, Comput. Methods Appl. Mech. Engrg., 274, 84-102, (2014) · Zbl 1296.74120 |
| [19] | Duarte, C. A.; Oden, J. T., Hp clouds—a meshless method to solve boundary-value problems, technical report 95-05, (1995), Texas Institute for Computational and Applied Mathematics, University of Texas at Austin |
| [20] | Liu, W. K.; Chen, Y.; Uras, R. A.; Chang, C. T., Generalized multiple scale reproducing kernel particle, Comput. Methods Appl. Mech. Engrg., 139, 91-157, (1996) · Zbl 0896.76069 |
| [21] | Liu, W. K.; Chen, Y. J., Wavelet and multiple scale reproducing kernel methods, Internat. J. Numer. Methods Fluids, 21, 901-931, (1995) · Zbl 0885.76078 |
| [22] | Salehi, R.; Dehghan, M., A generalized moving least square reproducing kernel method, J. Comput. Appl. Math., 249, 120-132, (2013) · Zbl 1285.65080 |
| [23] | Salehi, R.; Dehghan, M., A moving least square reproducing polynomial meshless method, Appl. Numer. Math., 69, 34-58, (2013) · Zbl 1284.65137 |
| [24] | Mirzaei, D.; Schaback, R.; Dehghan, M., 3-on generalized moving least squares and diffuse derivatives, IMA J. Numer. Anal., 32, 983-1000, (2012) · Zbl 1252.65037 |
| [25] | Dehghan, M.; Ghesmati, A., 4-combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation, Eng. Anal. Bound. Elem., 34, 324-336, (2010) · Zbl 1244.65147 |
| [26] | Liew, K. M.; Chen, X. L., Buckling of rectangular Mindlin plates subjected to partial in-plane edge loads using the radial point interpolation method, Int. J. Solids Struct., 41, 1677-1695, (2004) · Zbl 1075.74533 |
| [27] | Ritz, W., Über eine neue methode zur Lösung gewisser variationsprobleme der mathematischen physik, J. Reine Angew. Math., 135, 1-61, (1909) · JFM 39.0449.01 |
| [28] | Cheung, Y. K.; Zhou, D., Free vibrations of rectangular unsymmetrically laminated composite plates with internal line supports, Comput. Struct., 79, 1923-1932, (2001) |
| [29] | Cheung, Y. K.; Zhou, D., Three-dimensional vibration analysis of cantilevered and completely free isosceles triangular plates, Int. J. Solids Struct., 39, 673-687, (2002) · Zbl 0991.74511 |
| [30] | Liew, K. M.; Kitipornchai, S.; Zhang, X. Z.; Lim, C. W., Analysis of the thermal stress behaviour of functionally graded hollow circular cylinders, Int. J. Solids Struct., 40, 2355-2380, (2003) · Zbl 1087.74529 |
| [31] | Liew, K. M.; Yang, J.; Kitipornchai, S., Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading, Internat. J. Solids Structures, 40, 3869-3892, (2003) · Zbl 1038.74546 |
| [32] | Liew, K. M.; Zhao, X.; Ferreira, A. J.M., A review of meshless methods for laminated and functionally graded plates and shells, Int. J. Solids Struct., 93, 2031-2041, (2011) |
| [33] | Liew, K. M.; Feng, Z. C., Three-dimensional free vibration analysis of perforated superelliptical plates via the p-Ritz method, Int. J. Mech. Sci., 43, 2613-2630, (2001) · Zbl 1045.74027 |
| [34] | Lim, C. W.; Liew, K. M., A pb-2 Ritz formulation for flexural vibration of shallow cylindrical shells of rectangular platform, J. Sound Vib., 173, 343-375, (1994) · Zbl 0925.73926 |
| [35] | Liew, K. M.; Lei, Z. X.; Yu, J. L.; Zhang, L. W., Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach, Comput. Methods Appl. Mech. Engrg., 268, 1-17, (2014) · Zbl 1295.74062 |
| [36] | Lei, Z. X.; Zhang, L. W.; Liew, K. M.; Yu, J. L., Dynamic stability analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the element-free KP-Ritz method, Compos. Struct., 113, 328-338, (2014) |
| [37] | Lei, Z. X.; Liew, K. M.; Yu, J. L., Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free KP-Ritz method in thermal environment, Compos. Struct., 106, 128-138, (2013) |
| [38] | Lei, Z. X.; Liew, K. M.; Yu, J. L., Large deflection analysis of functionally graded carbon nanotube-reinforced composite plates by the element-free KP-Ritz method, Comput. Methods Appl. Mech. Engrg., 256, 189-199, (2013) · Zbl 1352.74165 |
| [39] | Lei, Z. X.; Liew, K. M.; Yu, J. L., Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free KP-Ritz method, Compos. Struct., 98, 160-168, (2013) · Zbl 1352.74165 |
| [40] | Peng, M. J.; Li, R. X.; Cheng, Y. M., Analyzing three-dimensional viscoelasticity problems via the improved element-free Galerkin (IEFG) method, Eng. Anal. Bound. Elem., 40, 104-113, (2014) · Zbl 1297.74153 |
| [41] | Mohanty, R. K.; Gopal, V., A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations, Appl. Math. Model., 37, 2802-2815, (2013) · Zbl 1352.65254 |
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