Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

DOI: 10.4236/ajcm.2011.12009   PDF   HTML     3,370 Downloads   7,847 Views   Citations

Abstract

The present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method (GM ), Galerkin method with weak form (GM / WF ), Petrov-Galerkin method ( PGM), weighted residual method (WRY ), and least squares method or process ( LSM or LSP ) to construct finite element approximations in time. A correspondence is established between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) computational processes for which types of operators and, 2) to establish which integral forms do not yield unconditionally stable computations (variationally inconsistent integral forms, VIC ). It is shown that variationally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approximations as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson’s θ method as well as Newmark method to demonstrate highly meritorious features of the proposed methodology.

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K. Surana, L. Euler, J. Reddy and A. Romkes, "Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs," American Journal of Computational Mathematics, Vol. 1 No. 2, 2011, pp. 83-103. doi: 10.4236/ajcm.2011.12009.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. Gelfand and S. V. Foming, “Calculus of Variations,” Dover, New York, 2000.
[2] S. G. Mikhlin. “Variational Methods in Mathematical Physics,” Pergamon Press, New York, 1964.
[3] J. N. Reddy, “Functional Analysis and Variational Methods in Engineering,” McGraw-Hill, New York, 1986.
[4] J. N. Reddy, “Energy Principles and Variational Methods in Applied Mechanics,” 2nd Edition, John Wiley, New York, 2002.
[5] C. Johnson, “Numerical Solution of Partial Differential Equations by Finite Element Method,” Cambridge University Press, Cambridge, 1987.
[6] T. Be-lytschko and T. J. R. Hughes, “Computational Methods in Transient Analysis,” North-Holland, 1983.
[7] T. J. R. Hughes and T. Belytschko. “A Precis of Developments in Computational Methods for Transient Analysis,” Journal of Applied Mechanics, Vol. 50, No. 4a, 1983, 1033-1041. doi:10.1115/1.3167186
[8] Nathan M. Newmark. “A Method of Computation for Structural Dynamics,” Journal of the En-gineering Mechanics Division, Vol. 85, No. EM3, 1959, pp. 67-94.
[9] K. J. Bathe and E. L. Wilson, “Numerical Methods in Finite Element Analysis,” Prentice Hall, New Jersey, 1976.
[10] W. L. Wood, M. Bossak, and O. C. Zienkiewicz, “An Alpha Modification of Newmark’s Method,” International Journal for Numerical Methods in Engineering, Vol. 15, No. 10, 1980, pp. 1562-1566. doi:10.1002/nme.1620151011
[11] M. Hans, J. Hilber, Tho-mas, R. Hughes and L. Robert Taylor, “Improved Numerical Dissipation for Time Integration Algorithms in Structural Dy-namics,” Earthquake Engineering and Structural Dynamics, Vol. 5, No. 3, 1977, 283-292. doi:10.1002/eqe.4290050306
[12] J. Chung and G. M. Hulbert, “A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized- Method,” Journal of Applied Mechanics, Vol. 60, No. 2, 1993, pp. 371-375. doi:10.1115/1.2900803
[13] V. A. Leontiev, “Extension of LMS Formulations for L- Stable Optimal Integration Methods with U0-V0 Overshoot Properties in Structural Dynamics: The Level-Symmetric (LS) Integration Methods,” International Journal for Numerical Methods in Engineering, Vol. 71, No. 13, 2007,pp. 1598-1632. doi:10.1002/nme.2008
[14] S. Erlicher, L. Bonaventura and O. S. Bursi, “The Analysis of the Generalized- Method for Non-Linear Dynamic Problems,” Computational Mechanics, Vol. 28, No. 2, 2002, pp. 83-104. doi:10.1007/s00466-001-0273-z
[15] D. Kuhl and M. A. Cris-field, “Energy-Conserving and Decaying Algorithms in Non-Linear Structural Dynamics,” International Journal for Numerical Methods in Engineering, Vol. 45, No. 5, 1999, pp. 569-599. doi:10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.CO;2-A
[16] K. K. Tamma, X. Zhou and D. Sha. “A Theory of Development and Design of Generalized Integration Operators for Computational Structural Dynamics,” Interna-tional Journal for Numerical Methods in Engineering, Vol. 50, No. 7, 2001, pp. 1619-1664. doi:10.1002/nme.89
[17] K. K. Tamma and R. R. Namburu. “Applicability and Evaluation of an Implicit Self-Starting Unconditionally Stable Methodology for the Dynamics of Structures,” Computers and Structures, Vol. 34, No. 6, 1990, pp. 835-842. doi:10.1016/0045-7949(90)90354-5
[18] X. Zhou and K. K. Tamma, “Design, Analysis, and Synthesis of Generalized Sin-gle Step Single Solve and Optimal Algorithms for Structural Dynamics,” International Journal for Numerical Methods in Engineering, Vol. 59: No. 5, 2004, pp. 597-668. doi:10.1002/nme.873
[19] X. Zhou and K. K. Tamma. “Algo-rithms by Design With Illustrations to Solid and Structural Mechanics/Dyna- mics,” International Journal for Numerical Methods in Engineering, Vol. 66, No. 11, 2006, pp. 1738-1790. doi:10.1002/nme.1559
[20] X. Zhou and K. K. Tamma. “A New Unified Theory Underlying Time Dependent Linear First-Order Systems: A Prelude to Algorithms by Design,” International Journal for Numerical Methods in Engineering, Vol. 60, No. 10, 2004, pp. 1699-1740. doi:10.1002/nme.1019
[21] A. Hoitink, S. Masuri, X. Zhou and K. K. Tamma, “Algorithms by Design: Part I – on the Hidden Point Collocation within LMS Methods and Implica-tions for Nonlinear Dynamics Applications,” International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 9, No. 6, 2008, pp. 383-407. doi:10.1080/15502280802365873
[22] K. K. Tamma, X. Zhou and D. Sha, “The Time Dimension: A Theory Towards the Evolution, Classification, Characterization and Design of Computational Algorithms for Transient/Dynamic Applica-tions,” Archives of Computational Methods in Engineering, Vol. 7, No. 2, 2000, pp. 67-290. doi:10.1007/BF02736209
[23] K. S. Surana and J. N. Reddy, “Mathematics of Computations and Finite Element Method for Boundary Value Problems,” ME 861 class notes. University of Kansas, Department of Me-chanical Engineering, Manuscript of textbook in preparation, 2010.
[24] K. S. Surana, “Mathematics of Computations and Finite Element Method for Initial Value Problems,” ME 862 class notes, University of Kansas, Department of Mechanical Engineering, Manuscript of textbook in preparation, 2010.
[25] K. S. Surana, A. Ahmadi and J. N. Reddy, “k-Version of Finite Element Method for Non-Linear Differen-tial Operators in BVP,” International Journal of Computational Engineering Science, Vol. 5, No. 1, 2004, pp. 133-207. doi:10.1142/S1465876304002307
[26] K.S. Surana, J. N. Reddy and S. Allu, “The k-Version of Finite Element Method for Initial Value Problems: Mathematical and Computational Framework,” International Journal of Computational Methods in Engineering Science and Mechanics, Vol. 8, No. 3, 2007, pp. 123-136. doi:10.1080/15502280701252321
[27] K. S. Surana, S. Allu, J. N. Reddy and P. W. Tenpas, “Least Squares Finite Element Processes In Hpk Mathematical Framework for Non-Linear Conservation Law,” International Journal of Numerical Meth-ods in Fluids, Vol. 57, No. 10, 2008, pp. 1545-1568. doi:10.1002/fld.1695
[28] B. C. Bell and K. S. Surana, “p-Version Least Squares Finite Element Formulation of Two Dimensional Incompressible Non-Newtonian Isothermal and Non-Isothermal Fluid Flow,” International Journal of Numeri-cal Methods in Fluids, Vol. 18, No. 2, 1994, pp. 127-167. doi:10.1002/fld.1650180202
[29] B. C. Bell and K. S. Surana. A Space-Time Coupled P-Version LSFEF for Unsteady Fluid Dynamics, International Journal for Numerical Methods in Engineering, Vol. 37, No. 20, 1994, pp. 3545-3569. doi:10.1002/nme.1620372008
[30] D. L. Winterscheidt and K. S. Surana, “p-Version Least Squares Finite Element Formula-tion for Two Dimensional Incompressible Fluid Flow,” Inter-national Journal of Numerical Methods in Fluids, Vol. 18, No. 1, 1994, pp. 43-69. doi:10.1002/fld.1650180104
[31] D. L. Winterscheidt and K. S. Surana, “p-Version Least Squares Finite Element Formulation for Burgers Equation,” Interna-tional Journal for Numerical Methods in Engineering, Vol. 36, No. 21, 1993, pp. 3629-3646. doi:10.1002/nme.1620362105
[32] K. S. Surana, A. Ahmadi and J. N. Reddy, “k-Version of Finite Element Method for Self-Adjoint Operators in BVP,” International Journal of Computational Engineering Science, Vol. 3, No. 2, 2002, pp. 155-218. doi:10.1142/S1465876302000605
[33] K. S. Surana, A. Ahmadi, and J. N. Reddy, “k-Version Of Finite Element Method for Non-self-adjoint Operators in BVP,” International Journal of Computational Engineering Science, Vol. 4, No. 4, 2003, pp. 737-812. doi:10.1142/S1465876303002179

  
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