Enhancement of Precise Integration Method for Dynamic Structural Analysis using Inversion of State Matrix
Authors
Abstract
For solving the dynamic equilibrium equation of structures, several second-order numerical methods have so far
been proposed. In these algorithms, conditional stability, period elongation, amplitude error, appearance of spurious frequencies
and dependency of the algorithms to the time steps are the crucial problems. Among the numerical methods, Newmark average
acceleration algorithm, regardless of existence of spurious frequencies, is very popular in the structural dynamics due to its
unconditionally stability status of the method. Recently, several first-order methods have been introduced for resolving the
accuracy and stability issues. However, in these methods stability, accuracy and error in inversion of the state matrix are known
as major issues. When the state matrix became singular or ill conditioned, numerical errors will occure in the computational
process. Many of the available first-order methods were to improve the stability and accuracy and also to remove the error of
inversion. Even though the introduced methods are conditionally stable, no investigation on errors, occuring during dynamic
loading, has been reported for them. The main purpose of this paper is to utilize a specific decomposition method based on
Singular Value Decomposition (SVD) for modifying PIM algorithm. Using the SVD inversion technique, the singularity problem
of the state matrix has been resolved. In this paper, the modified method is called PIMS. As well, by applying the developed
method for dynamic loading, the error of responses has been investigated. The results show that PIMS algorithm is stable and,
comparing with secoend order Newmark and other available first order methods, has more accuracy.
been proposed. In these algorithms, conditional stability, period elongation, amplitude error, appearance of spurious frequencies
and dependency of the algorithms to the time steps are the crucial problems. Among the numerical methods, Newmark average
acceleration algorithm, regardless of existence of spurious frequencies, is very popular in the structural dynamics due to its
unconditionally stability status of the method. Recently, several first-order methods have been introduced for resolving the
accuracy and stability issues. However, in these methods stability, accuracy and error in inversion of the state matrix are known
as major issues. When the state matrix became singular or ill conditioned, numerical errors will occure in the computational
process. Many of the available first-order methods were to improve the stability and accuracy and also to remove the error of
inversion. Even though the introduced methods are conditionally stable, no investigation on errors, occuring during dynamic
loading, has been reported for them. The main purpose of this paper is to utilize a specific decomposition method based on
Singular Value Decomposition (SVD) for modifying PIM algorithm. Using the SVD inversion technique, the singularity problem
of the state matrix has been resolved. In this paper, the modified method is called PIMS. As well, by applying the developed
method for dynamic loading, the error of responses has been investigated. The results show that PIMS algorithm is stable and,
comparing with secoend order Newmark and other available first order methods, has more accuracy.
Keywords
1. Hart, G. C, and Wong, K., Structural Dynamic for
Structural Engineering, John Wily & Sons, New
York, 1975.
2. Bathe, K. J., and Wilson, E. L., Numerical Methods
in Finite Element Analaysis, Prentice Hall, New
York, 1976.
3. Wood, W. L., Practical Time-Stepping Schemes,
Clarendon Press, Oxford, 1990.
4. Hughes, T. J. R., The Finite Element Method: Linear
Static and Dynamic Finite Element Analysis,
Prentice-Hall, Englewood Cliffs, 1978.
5. Dokainish, M. A., and Subbaraj, K., “A Survey of
Direct Time Integration Methods in Computational
Structural Dynamics. II. Implicit methods”,
Computer & Structures, Vol. 32, No. 6, pp. 1387-
1401, 1989.
6. Zhong, W. X., and Williams, F. W., “A Precise Time
Step Integration Method”. Proceedings of the
Institution of Mechanical Engineers, Vol. 208, No. 6,
pp. 427-430, 1994.
7. Shen, W, Lin, J, and Williams, F. W., “Parallel
Computing for High Precision Direct Integration
Method”, Computer Methods in Applied Mechanics
and Engineering, Vol. 26, pp. 315-331, 1995.
8. Yuanxian, G., Biaosong, C., and Hongwu, Z.
“Precise Time-Integration with Dimension
Expanding Method”, Acta Mechanica Sinica, Vol.
32, No. 4, pp. 447, 2000.
9. Wang, M, and Zhou, X., “Modified Precise Time
Step Integration Method of Structural Dynamic
Analysis”, Journal of Earthquake Engineering and
Engineering Vibration, Vol. 4, No. 2, pp. 287-293,
2005.
10.Wang, M., and Au, F. T. K., “Assessment and
Improvement of Precise Time Step Integration
Method”, Computer & Structures, Vol. 84, No. 12,
pp. 779-786, 2006.
11. Wu, C. L., and Chuang, C. C., “Application of an
Innovative Precise Integration Method in Solving
Equilibrium Equation of Motion for Structural
Dynamic Problems”, 15th World Congress of
Earthquake Engineering (15WCEE), Lisbon,
Portugal, 2012.
12. Chen, C. T., Linear System Theory and Design,
Oxford University Press, New York, 1999.
13. Molar, C., and Loan, C. V., “Nineteen Dubious Ways
to Compute the Exponential of a Matrix, Twenty-five
Years Later”, SIAM Review, Vol. 45, No. 1, pp. 1-46,
2003.
14.Drof, C. T, and Bishop, R. H, Modern Control
System Solution Manual, Prentic Hall, New Jersey,
2008.
15. Franklin, G. F, Da Powell. J., and Emami-Naeini, A.,
Feedback Control of Dynamic Systems, Vol. 3.
Reading, MA: Addison-Wesley, 1994.
16. Clough, R. W, and Penzien, J., Dynamic of Structure,
McGraw-Hill, New York, 1975.
17. Bathe, K. J., Finite Element Procedures, Prentice
Hall Inc., Englewood Cliffs, 2006.
18. Wood, W. L., “Numerical Integration of Structural
Dynamics Equations Including Natural Damping and
Periodic Forcing Terms”, International Journal for
Numerical Methods in Engineering, Vol. 17, pp. 281-
289, 1981.
19. Penry, S. N., and Wood, W. L., “Comparison of
Some Single-Step Methods for Numerical Solution
of the Structural Dynamics Equation”, International
Journal for Numerical Methods in Engineering, Vol.
21, No. 11, pp. 1941-1955, 1985.
20. Preumont, A, “Frequency Domain Analysis of Time
Integration Operator”, Earthquake Engineering &
Structural Dynamics, Vol. 10, No. 5, PP. 691-697,
1982.
21. Paultre, P., Dynamics of Structure, John Wiley &
Sons, New York, 2006.
22. Menun, C., “A Response-Spectrum-Based Envelop
for Mohr,s Circle”, Earthquake Engineering &
Structural Dynamics, Vol. 32, pp. 1917-1935, 2003.
Structural Engineering, John Wily & Sons, New
York, 1975.
2. Bathe, K. J., and Wilson, E. L., Numerical Methods
in Finite Element Analaysis, Prentice Hall, New
York, 1976.
3. Wood, W. L., Practical Time-Stepping Schemes,
Clarendon Press, Oxford, 1990.
4. Hughes, T. J. R., The Finite Element Method: Linear
Static and Dynamic Finite Element Analysis,
Prentice-Hall, Englewood Cliffs, 1978.
5. Dokainish, M. A., and Subbaraj, K., “A Survey of
Direct Time Integration Methods in Computational
Structural Dynamics. II. Implicit methods”,
Computer & Structures, Vol. 32, No. 6, pp. 1387-
1401, 1989.
6. Zhong, W. X., and Williams, F. W., “A Precise Time
Step Integration Method”. Proceedings of the
Institution of Mechanical Engineers, Vol. 208, No. 6,
pp. 427-430, 1994.
7. Shen, W, Lin, J, and Williams, F. W., “Parallel
Computing for High Precision Direct Integration
Method”, Computer Methods in Applied Mechanics
and Engineering, Vol. 26, pp. 315-331, 1995.
8. Yuanxian, G., Biaosong, C., and Hongwu, Z.
“Precise Time-Integration with Dimension
Expanding Method”, Acta Mechanica Sinica, Vol.
32, No. 4, pp. 447, 2000.
9. Wang, M, and Zhou, X., “Modified Precise Time
Step Integration Method of Structural Dynamic
Analysis”, Journal of Earthquake Engineering and
Engineering Vibration, Vol. 4, No. 2, pp. 287-293,
2005.
10.Wang, M., and Au, F. T. K., “Assessment and
Improvement of Precise Time Step Integration
Method”, Computer & Structures, Vol. 84, No. 12,
pp. 779-786, 2006.
11. Wu, C. L., and Chuang, C. C., “Application of an
Innovative Precise Integration Method in Solving
Equilibrium Equation of Motion for Structural
Dynamic Problems”, 15th World Congress of
Earthquake Engineering (15WCEE), Lisbon,
Portugal, 2012.
12. Chen, C. T., Linear System Theory and Design,
Oxford University Press, New York, 1999.
13. Molar, C., and Loan, C. V., “Nineteen Dubious Ways
to Compute the Exponential of a Matrix, Twenty-five
Years Later”, SIAM Review, Vol. 45, No. 1, pp. 1-46,
2003.
14.Drof, C. T, and Bishop, R. H, Modern Control
System Solution Manual, Prentic Hall, New Jersey,
2008.
15. Franklin, G. F, Da Powell. J., and Emami-Naeini, A.,
Feedback Control of Dynamic Systems, Vol. 3.
Reading, MA: Addison-Wesley, 1994.
16. Clough, R. W, and Penzien, J., Dynamic of Structure,
McGraw-Hill, New York, 1975.
17. Bathe, K. J., Finite Element Procedures, Prentice
Hall Inc., Englewood Cliffs, 2006.
18. Wood, W. L., “Numerical Integration of Structural
Dynamics Equations Including Natural Damping and
Periodic Forcing Terms”, International Journal for
Numerical Methods in Engineering, Vol. 17, pp. 281-
289, 1981.
19. Penry, S. N., and Wood, W. L., “Comparison of
Some Single-Step Methods for Numerical Solution
of the Structural Dynamics Equation”, International
Journal for Numerical Methods in Engineering, Vol.
21, No. 11, pp. 1941-1955, 1985.
20. Preumont, A, “Frequency Domain Analysis of Time
Integration Operator”, Earthquake Engineering &
Structural Dynamics, Vol. 10, No. 5, PP. 691-697,
1982.
21. Paultre, P., Dynamics of Structure, John Wiley &
Sons, New York, 2006.
22. Menun, C., “A Response-Spectrum-Based Envelop
for Mohr,s Circle”, Earthquake Engineering &
Structural Dynamics, Vol. 32, pp. 1917-1935, 2003.
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