Exact Implementation of Multiple Initial Conditions in the DQ Solution of Higher-Order ODEs

Subject Areas : Engineering

S.A Eftekhari ¹ ( Young Researchers and Elite Club, Karaj Branch, Islamic Azad University )

Received: 2016-06-10 Accepted : 2016-08-10 Published : 2016-09-20

Keywords: Rectangular plates, Beams, New differential quadrature methodology, Imposing multiple initial conditions, Higher-order initial-value problems, CBCGE approach, MWCM approach,

Abstract :

The differential quadrature method (DQM) is one of the most elegant and useful approximate methods for solving initial and/or boundary value problems. It is easy to use and also straightforward to implement. However, the conventional DQM is well-known to have some difficulty in implementing multiple initial and/or boundary conditions at a given discrete point. To overcome this difficulty, this paper presents a simple and accurate differential quadrature methodology in which the higher-order initial conditions are exactly implemented. The proposed methodology is very elegant and uses a set of simple polynomials with a simple transformation to incorporate the higher-order initial conditions at the initial discrete time point. The order of accuracy of the proposed method for solving an rth order ordinary differential equation is “m + r – 1,” where m being the number of discrete time points. This is better than the accuracy of the CBCGE (direct Coupling the Boundary/initial Conditions with the discrete Governing Equations) and MWCM (Modifying Weighting Coefficient Matrices) approaches whose order is in general “m – 1.” Some test problems are also provided to highlight the superiority of the proposed method over the CBCGE and MWCM approaches.

References:

[1] Bellman R.E., Casti J., 1971, Differential quadrature and long term integrations, Journal of Mathematical Analysis and Applications 34: 235-238.
[2] Bellman R.E., Kashef B.G., Casti J., 1972, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics 10: 40-52.
[3] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics: A review, ASME Applied Mechanics Reviews 49: 1-28.
[4] Shu C., 2000, Differential Quadrature and Its Application in Engineering, Springer, NY, USA.
[5] Bert C.W., Jang S.K., Striz A.G., 1988, Two new approximate methods for analysing free vibration of structural components, AIAA Journal 26: 612-618.
[6] Jang S.K., Bert C.W., 1989, Application of differential quadrature to static analysis of structural components, International Journal for Numerical Methods in Engineering 28: 561-577.
[7] Wang X., Bert C.W., 1993, A new approach in applying differential quadrature to static and free vibration of beams and plates, Journal of Sound and Vibration 162: 566-572.
[8] Wang X., Bert C.W., Striz A.G., 1993, Differential quadrature analysis of deflection, buckling, and free vibration of beams and rectangular plates, Computers & Structures 48(3): 473-479.
[9] Malik M., Bert C.W., 1996, Implementing multiple boundary conditions in the DQ solution of higher-order PDE’s: application to free vibration of plates, International Journal for Numerical Methods in Engineering 39: 1237-1258.
[10] Shu C., Du H., 1997, Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates, International Journal of Solids and Structures 34(7): 819-835.
[11] Shu C., Du H., 1997, A generalized approach for implementing general boundary conditions in the GDQ free vibration analyses of plates, International Journal of Solids and Structures 34(7): 837-846.
[12] Eftekhari S.A., 2015, A simple and systematic approach for implementing boundary conditions in the differential quadrature free and forced vibration analysis of beams and rectangular plates, Journal of Solid Mechanics 7(4): 374-399.
[13] Tanaka M., Chen W., 2001, Dual reciprocity BEM applied to transient elastodynamic problems with differential quadrature method in time, Computer Methods in Applied Mechanics and Engineering 190: 2331-2347.
[14] Shu C., Yao K.S., 2002, Block-marching in time with DQ discretization: an efficient method for time-dependent problems, Computer Methods in Applied Mechanics and Engineering 191: 4587-4597.
[15] Hashemi M.R., Abedini M.J., Malekzadeh P., 2006, Numerical modeling of long waves in shallow water using Incremental Differential Quadrature Method, Ocean Engineering 33: 1749-1764.
[16] Malekzadeh P., Rahideh H., 2007, IDQ two-dimensional nonlinear transient heat transfer analysis of variable section annular fines, Energy conversion & Management 48: 269-276.
[17] Civalek Ö., 2007, Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods, Applied Mathematical Modelling 31: 606-624.
[18] Civalek Ö., Oztürk B., 2009, Discrete singular convolution algorithm for non-linear transient response of circular plates resting on Winkler-Pasternak elastic foundations with different types of dynamic loading, Indian Journal of Engineering and Material Sciences 16: 259-268.
[19] Golfam B., Rezaie F., 2013, A new generalized approach for implementing any homogeneous and non-homogeneous boundary conditions in the generalized differential quadrature analysis of beams, Scientia Iranica: Transaction A, Civil Engineering 20(4): 1114-1123.
[20] Wu T.Y., Liu G.R., 1999, The differential quadrature as a numerical method to solve the differential equation, Computational Mechanics 24: 197-205.
[21] Wu T.Y., Liu G.R., 2000, The generalized differential quadrature rule for initial-value differential equations, Journal of Sound and Vibration 233: 195-213.
[22] Wu T.Y., Liu G.R., 2001, Application of the generalized differential quadrature rule to eighth-order differential equations, Communications in Numerical Methods in Engineering 17: 355-364.
[23] Wu T.Y., Liu G.R., Wang Y.Y., 2003, Application of the generalized differential quadrature rule to initial-boundary-value problems, Journal of Sound and Vibration 264: 883-891.
[24] Fung T.C., 2001, Solving initial value problems by differential quadrature method-Part 2: second- and higher-order equations, International Journal for Numerical Methods in Engineering 50: 1429-1454.
[25] Fung T.C., 2002, Stability and accuracy of differential quadrature method in solving dynamic problems, Computer Methods in Applied Mechanics and Engineering 191: 1311-1331.
[26] Zong Z., Zhang Y., 2009, Advanced Differential Quadrature Methods, Chapman & Hall, London, UK.
[27] Reddy J.N., 1993, An Introduction to the Finite Element Method, McGraw-Hill, NY, USA.
[28] Quan J.R., Chang C.T., 1989, New insights in solving distributed system equations by the quadrature method, Part I: Analysis, Computers & Chemical Engineering 13: 779-788.
[29] Quan J.R., Chang C.T., 1989, New insights in solving distributed system equations by the quadrature methods, Part II: Numerical experiments, Computers & Chemical Engineering 13: 1017-1024.
[30] Eftekhari S.A., Farid M., Khani M., 2009, Dynamic analysis of laminated composite coated beams carrying multiple accelerating oscillators using a coupled finite element-differential quadrature method, ASME Journal of Applied Mechanics 76(6): 061001.
[31] Eftekhari S.A., Khani M., 2010, A coupled finite element-differential quadrature element method and its accuracy for moving load problem, Applied Mathematical Modelling 34: 228-237.
[32] Khalili S.M.R., Jafari A.A., Eftekhari S.A., 2010, A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads, Composite Structures 92(10): 2497-2511.
[33] Jafari A.A., Eftekhari S.A., 2011, A new mixed finite element-differential quadrature formulation for forced vibration of beams carrying moving loads, ASME Journal of Applied Mechanics 78(1): 011020.
[34] Eftekhari S.A., Jafari A.A., 2012, Numerical simulation of chaotic dynamical systems by the method of differential quadrature, Scientia Iranica: Transaction B, Mechanical Engineering 19(5): 1299-1315.
[35] Jordan D.W., Smith P., 1999, Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, Oxford University Press, NY, USA.
[36] Bathe K. J., Wilson E. L., 1976, Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA.
[37] Eftekhari S.A., Jafari A.A., 2013, Numerical solution of general boundary layer problems by the method of differential quadrature, Scientia Iranica: Transaction B, Mechanical Engineering 20(4): 1278-1301.
[38] Hughes T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentic-Hall, Englewood Cliffs, NJ, USA.
[39] Zienkiewicz O.C., Taylor R.L., 2000, The Finite Element Method, McGraw-Hill, NY, USA.
[40] Fryba L., 1972, Vibration of Solids and Structures Under Moving Loads, Noordhoff International, Groningen, the Netherlands.
[41] Meirovitch L., 1967, Analytical Methods in Vibrations, Macmillan, NY, USA.
[42] Rao S.S., 2000, Vibration of Continuous Systems, John Wiley & Sons, Inc. NJ, USA.
[43] Eftekhari S.A., Jafari A.A., 2012, A novel and accurate Ritz formulation for free vibration of rectangular and skew plates, ASME Journal of Applied Mechanics 79(6): 064504.
[44] Eftekhari S.A., Jafari A.A., 2014, Accurate variational approach for free vibration of simply supported anisotropic rectangular plates, Archive of Applied Mechanics 84: 607-614.