Algorithms by Design: Part II—A Novel Normalized Time Weighted Residual Methodology and Design of a Family of Symplectic-Momentum Conserving Algorithms for Nonlinear Structural Dynamics

REFERENCES

• Hoitink , A. , Masuri , S. , Zhou , X. and Tamma , K. K. 2008 . Algorithms by Design: Part I—On the Hidden Point Collocation Within LMS Methods and Implications for Nonlinear Dynamics Applications . Int. J. Comp. Meth. Eng. Sci. Mech. , vol. 8 : 383 – 407 .
   Google Scholar
• Masuri , S. , Hoitink , A. , Zhou , X. and Tamma , K. K. 2009 . Algorithms by Design: Part III - A Novel Normalized Time Weighted Residual Methodology and Design of Optimal Symplectic-Momentum Based Controllable Numerical Dissipative Algorithms for Nonlinear Structural Dynamics . Int. J. Comp. Meth. Eng. Sci. Mech. , vol. 10 : 57 – 90 .
   Google Scholar
• Zhou , X. and Tamma , K. K. 2004 . A New Unified Theory Underlying Time Dependent Linear First-Order Systems: A Prelude to Algorithms by Design . Int. J. Num. Meth. Eng. , vol. 60 : 1699 – 1740 .
   Web of Science ®Google Scholar
• Ge , Z. and Marsden , J. E. 1988 . Lie-Poisson Integrators and Lie-Poisson Hamilton-Jacobi Theory . Phys. Lett., A , 133 : 134 – 139 .
   Web of Science ®Google Scholar
• Simo , J. C. and Tarnow , N. 1992 . The Discrete Energy-Momentum Method. Conserving Algorithms for Nonlinear Elastodynamics . J. Appl. Math. and Phys. , vol. 43 : 757 – 792 .
   Web of Science ®Google Scholar
• Simo , J. C. and Tarnow , N. 1992 . Exact Energy-Momentum Conserving Algorithms and Symplectic Schemes for Nonlinear Dynamics . Comp. Meth. Appl. Mech. Eng. , vol. 100 : 63 – 116 .
   Web of Science ®Google Scholar
• Kuhl , D. and Ramm , E. 1996 . Constraint Energy Momentum Algorithm and Its Application to Nonlinear Dynamics of Shell . Comp. Meth. Appl. Mech. Engi. , vol. 136 : 293 – 315 .
   Web of Science ®Google Scholar
• Kuhl , D. and Crisfield , M. A. 1999 . Energy-Conserving and Decaying Algorithms in Non-Linear Structural Dynamics . Int. J. Num. Meth. Eng. , vol. 45 : 569 – 599 .
   Web of Science ®Google Scholar
• Masuri , S. , Hoitink , A. , Zhou , X. and Tamma , K. K. Algorithms By Design: A New Normalized Time Weighted Residual Methodology and Design of a Family of Energy-Momentum Conserving Algorithms for Nonlinear Structural Dynamics . Int. J. Num. Meth. Eng. , (In press)
   Google Scholar
• Masuri , S. , Hoitink , A. , Zhou , X. and Tamma , K. K. Algorithms by Design: Optimal Energy-Momentum Based Controllable Numerical Dissipative Schemes in the Sense of Linear Multi-Step (LMS) Methods for Nonlinear Structural Dynamics . Comp. Meth. Appl. Mech. Eng. , (In Review)
   Google Scholar
• Kane , C. , Marsden , J. E. and Ortiz , M. 1999 . Symplectic-Energy-Momentum Preserving Variational Integrators . J. Math. Phys. , vol. 40 : 3353 – 3371 .
   Web of Science ®Google Scholar
• Veselov , A. P. 1988 . Integrable Discrete-Time Systems and Difference Operators, Func.l Anal. and its Appli., vol. 22, pp. 83–93 . 1988. Translated from Funktsional'nyi Analiz i Ego Prilozheniya , vol. 22 : 1 – 13 .
   Google Scholar
• Veselov , A. P. 1991 . Integrable Lagrangian Correspondences and Factorization of Matrix Polynomials, Func. Anal. and Its Appl., vol. 25, pp. 112–122 . 1991. Translated from Funktsional'nyi Analiz i Ego Prilozheniya , vol. 25 : 38 – 49 .
   Google Scholar
• Moser , J. and Veselov , A. P. 1991 . Discrete Versions of Some Classical Integrable Systems and Factorization of Matrix Polynomials . Comm. Math. Phys. , vol. 139 : 217 – 243 .
   Web of Science ®Google Scholar
• Simo , J. C. and Hughes , T. J. R. 1998 . Computational Inelasticity , New York : Springer .
   Google Scholar
• Kane , C. , Marsden , J. E. , Ortiz , M. and West , M. 2000 . Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems . Int. J. Num. Meth. Eng. , vol. 49 : 1295 – 1325 .
   Web of Science ®Google Scholar
• Feng , K. 1986 . Difference Schemes for Hamiltonian Formalism and Symplectic Geometry . J. of Comp. Math. , vol. 4 : 279 – 289 .
   Web of Science ®Google Scholar
• Noels , L. , Stainier , L. and Ponthot , J. P. 2004 . An Energy-Momentum Conserving Algorithm for Non-Linear Hypoelastic Constitutive Models . Int. J. Num. Meth. Eng. , vol. 59 : 83 – 114 .
   Web of Science ®Google Scholar
• MacKay , R. S. 1992 . “ Some Aspects of the Dynamics and Numerics of Hamiltonian System ” . In The Dynamics of Numerics and the Numerics of Dynamics , Edited by: Broomhead , D. S. and Iserles , A. 137 – 193 . Oxford : Oxford University Press .
   Google Scholar
• Marsden , J. E. and West , M. 2001 . Discrete Mechanics and Variational Integrators . Acta. Numerica. , vol. 10 : 357 – 514 .
   Google Scholar
• Finlayson , B. A. 1971 . The Method of Weighted Residuals and Variational Principles , New York : Academic Press .
   Google Scholar
• Tamma , K. K. and Namburu , R. R. 1990 . Applicability and Evaluation of an Implicit Self-Starting Unconditionally Stable Methodology for the Dynamics of Structures . Comp. and Struct. , vol. 34 : 835 – 845 .
   Web of Science ®Google Scholar
• Tamma , K. K. , Zhou , X. and Sha , D. 2000 . The Time Dimension: A Theory of Development/Evolution, Classification, Characterization and Design of Computational Algorithms for Transient/Dynamic Applications . Archives in Computational Mechanics , vol. 7 : 67 – 290 .
   Web of Science ®Google Scholar
• Tamma , K. K. , Zhou , X. and Sha , D. 2000 . A Theory of Development and Design of Generalized Integration Operators for Computational Structural Dynamics . Int. J. Num. Meth. Eng. , vol. 50 : 1619 – 1664 .
   Web of Science ®Google Scholar
• Zhou , X. and Tamma , K. K. 2004 . Design, Analysis, and Synthesis of Generalized Single Step Single Solve and Optimal Algorithms for Structural Dynamics . Int. J. Num. Meth. Eng. , vol. 59 : 597 – 668 .
   Web of Science ®Google Scholar
• Zhou , X. and Tamma , K. K. 2006 . Algorithms by Design with Illustrations to Solid and Structural Mechanics/Dynamics . Int. J. Num. Meth. Eng. , vol. 66 : 1738 – 1790 .
   Web of Science ®Google Scholar
• Newmark , N. M. 1959 . A Method of Computation for Structural Dynamics . J. Amer. Soc. Civil Eng. , vol. 1 : 67 – 94 .
   Google Scholar
• Wood , W. L. , Bossak , M. and Zienkiewicz , O. C. 1980 . An Alpha Modification of Newmark's Method . Int. J. Num. Meth. Eng. , vol. 15 : 1562
   Web of Science ®Google Scholar
• Hilber , H. M. , Hughes , T. J. R. and Taylor , R. L. 1977 . Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics . Earthq. Eng. and Struc. Dyn. , vol. 5 : 283
   Web of Science ®Google Scholar