On Tuesday, May 28, scheduled routine maintenance may cause intermittent connectivity issues which could impact e-commerce, registration, and single sign-on. Thank you for your patience.

Technical Papers
Feb 29, 2016

Characterizing Errors and Evaluating Performance of Transient Simulations Using Multi-Time-Step Integration

Publication: Journal of Computing in Civil Engineering
Volume 30, Issue 5

Abstract

The multi-time-step method of time integration for problems in structural dynamics allows one to decompose the problem domain into small subdomains and use different time steps within each subdomain to reduce the computational cost of solving such problems. However, the number of possible decompositions and their associated time steps for a given model is huge and grows exponentially with the number of elements. To find an optimal decomposition that minimizes error in the solution while maintaining a bound on the computational cost is challenging. In this work, existing multi-time-step methods are used and, for the first time, a systematic approach for traversing the space of possible decompositions to characterize the nature of how solution errors and computational costs vary for different decompositions is devised. Through numerical examples for three different types of structures, trusses, frames, and continuum solid bodies, it is shown that the characteristics of these error and cost functions are similar across problem types. Based on these functions, optimal decompositions that maximize the benefits of multi-time-step methods are identified.

Get full access to this article

View all available purchase options and get full access to this article.

Acknowledgments

This material is based in part on work supported by the National Science Foundation under Grant CNS-1136075. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

References

Belytschko, T., and Mullen, R. (1978). “Stability of explicit-implicit mesh partitions in time integration.” Int. J. Numer. Methods Eng., 12(10), 1575–1586.
Belytschko, T., Yen, H.-J., and Mullen, R. (1979). “Mixed methods for time integration.” Comput. Methods Appl. Mech. Eng., 17–18, 259–275.
Bergan, P. G., and Mollestad, E. (1985). “An automatic time-stepping algorithm for dynamic problems.” Comput. Methods Appl. Mech. Eng., 49(3), 299–318.
Choi, C.-K., and Chung, H.-J. (1996). “Error estimates and adaptive time stepping for various direct time integration methods.” Comput. Struct., 60(6), 923–944.
Chopra, A. K. (2012). Dynamics of structures: Theory and applications to earthquake engineering, 4th Ed., Prentice-Hall, Upper Saddle River, NJ.
Combescure, A., and Gravouil, A. (2002). “A numerical scheme to couple subdomains with different time-steps for predominantly linear transient analysis.” Comput. Methods Appl. Mech. Eng., 191(11–12), 1129–1157.
Combescure, A., Mahjoubi, N., Gravouil, A., and Greffet, N. (2010). “A time variational method to couple heterogeneous time integrators.” Eur. J. Comput. Mech., 19(1–3), 11–24.
Daniel, W. J. T. (1997). “The subcycled Newmark algorithm.” Comput. Mech., 20(3), 272–281.
Demkowicz, L. (2006). Computing with hp-adaptive finite elements. Vol. 1: One and two dimensional elliptic and Maxwell problems, CRC Press, Boca Raton, FL.
Farhat, C., and Roux, F. X. (1991). “A method for finite element tearing and interconnecting and its parallel solution algorithm.” Int. J. Numer. Methods Eng., 32(6), 1205–1227.
Felippa, C., Park, K., and Farhat, C. (2001). “Partitioned analysis of coupled mechanical systems.” Comput. Methods Appl. Mech. Eng., 190(24), 3247–3270.
Fragakis, Y., and Papadrakakis, M. (2003). “The mosaic of high performance domain decomposition methods for structural mechanics: Formulation, interrelation and numerical efficiency of primal and dual methods.” Comput. Methods Appl. Mech. Eng., 192(35–36), 3799–3830.
Fragakis, Y., and Papadrakakis, M. (2004). “The mosaic of high-performance domain decomposition methods for structural mechanics—Part II: Formulation enhancements, multiple right-hand sides and implicit dynamics.” Comput. Methods Appl. Mech. Eng., 193(42–44), 4611–4662.
Gravouil, A., and Combescure, A. (2001). “Multi-time-step explicit-implicit method for non-linear structural dynamics.” Int. J. Numer. Methods Eng., 50(1), 199–225.
Gravouil, A., Combescure, A., and Brun, M. (2015). “Heterogeneous asynchronous time integrators for computational structural dynamics.” Int. J. Numer. Methods Eng., 102(3-4), 202–232.
Hilber, H. M., Hughes, T. J. R., and Taylor, R. L. (1977). “Improved numerical dissipation for time integration algorithms in structural dynamics.” Earthquake Eng. Struct. Dyn., 5(3), 283–292.
Komatitsch, D., and Vilotte, J.-P. (1998). “The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures.” Bull. Seismol. Soc. Am., 88(2), 368–392.
Komatitsch, D., Vilotte, J.-P., Vai, R., Castillo-Covarrubias, J. M., and Sanchez-Sesma, F. J. (1999). “The spectral element method for elastic wave equations—Application to 2D and 3D seismic problems.” Int. J. Numer. Methods Eng., 45(9), 1139–1164.
Kuhn, M. (1985). “A numerical study of Lamb’s problem.” Geophys. Prospect., 33(8), 1103–1137.
Lamb, H. (1904). “On the propagation of tremors over the surface of an elastic solid.” Philos. Trans. R. Soc. London Ser. A, 203(359–371), 1–42.
Levy, D., and Tadmor, E. (1998). “From semidiscrete to fully discrete: Stability of Runge-Kutta schemes by the energy method.” SIAM Rev., 40(1), 40–73.
Liu, C., Jamal, M. H., Kulkarni, M., Prakash, A., and Pai, V. (2013). “Exploiting domain knowledge to optimize parallel computational mechanics codes.” Proc., 27th Int. ACM Conf. on Supercomputing, ACM, New York, 25–36.
Mahjoubi, N., Gravouil, A., Combescure, A., and Greffet, N. (2011). “A monolithic energy conserving method to couple heterogeneous time integrators with incompatible time steps in structural dynamics.” Comput. Methods Appl. Mech. Eng., 200(9), 1069–1086.
MATLAB [Computer software]. MathWorks, Natick, MA.
Neal, M. O., and Belytschko, T. (1989). “Explicit-explicit subcycling with non-integer time step ratios for structural dynamic systems.” Comput. Struct., 31(6), 871–880.
Newmark, N. M. (1959). “A method of computation for structural dynamics.” J. Eng. Mech., 85, 67–94.
Ohtori, Y., Christenson, R., Spencer, B., Jr., and Dyke, S. (2004). “Benchmark control problems for seismically excited nonlinear buildings.” J. Eng. Mech., 366–385.
Park, K., and Underwood, P. (1980). “A variable-step central difference method for structural dynamics analysis—Part 1. Theoretical aspects.” Comput. Methods Appl. Mech. Eng., 22(2), 241–258.
Prakash, A., and Hjelmstad, K. D. (2004). “A FETI based multi-time-step coupling method for Newmark schemes in structural dynamics.” Int. J. Numer. Methods Eng., 61(13), 2183–2204.
Prakash, A., Taciroglu, E., and Hjelmstad, K. D. (2014). “Computationally efficient multi-time-step method for partitioned time integration of highly nonlinear structural dynamics.” Comput. Struct., 133, 51–63.
Red Hat Enterprise Linux 6 [Computer software]. Red Hat, Raleigh, NC.
Romero, I., and Lacoma, L. M. (2006). “Analysis of error estimators for the semidiscrete equations of linear solid and structural dynamics.” Comput. Methods Appl. Mech. Eng., 195(19–22), 2674–2696.
Sanchez-Gasca, J., D’Aquila, R., Price, W., and Paserba, J. (1995). “Variable time step, implicit integration for extended-term power system dynamic simulation.” 1995 IEEE Power Industry Computer Application Conf., IEEE, Salt Lake City, 183–189.
Smolinski, P. (1996). “Subcycling integration with non-integer time steps for structural dynamics problems.” Comput. Struct., 59(2), 273–281.
Smolinski, P., Belytschko, T., and Liu, W. K. (1987). “Stability of multitime-step partitioned integrators for first-order systems of equations.” Comput. Methods Appl. Mech. Eng., 65(2), 115–125.
Smolinski, P., Sleith, S., and Belytschko, T. (1996). “Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations.” Comput. Mech., 18(3), 236–244.
Toselli, A., and Widlund, O. (2005). Domain decomposition methods—Algorithms and theory, Springer, New York.
Wiberg, N.-E., and Li, X. (1993). “A post-processing technique and an a posteriori error estimate for the Newmark method in dynamic analysis.” Earthquake Eng. Struct. Dyn., 22(6), 465–489.
Wilson, E., Farhoomand, I., and Bathe, K. (1972). “Nonlinear dynamic analysis of complex structures.” Earthquake Eng. Struct. Dyn., 1(3), 241–252.
Wood, W., Bossak, M., and Zienkiewicz, O. (1981). “An alpha modification of Newmark’s method.” Int. J. Numer. Methods Eng., 15(10), 1562–1566.
Zienkiewicz, O., and Xie, Y. (1991). “A simple error estimator and adaptive time stepping procedure for dynamic analysis.” Earthquake Eng. Struct. Dyn., 20(9), 871–887.

Information & Authors

Information

Published In

Go to Journal of Computing in Civil Engineering
Journal of Computing in Civil Engineering
Volume 30Issue 5September 2016

History

Received: Sep 7, 2015
Accepted: Dec 21, 2015
Published online: Feb 29, 2016
Discussion open until: Jul 29, 2016
Published in print: Sep 1, 2016

Permissions

Request permissions for this article.

Authors

Affiliations

Gregory Bunting [email protected]
Ph.D. Candidate, Lyles School of Civil Engineering, Purdue Univ., West Lafayette, IN 47907 (corresponding author). E-mail: [email protected]
Arun Prakash, A.M.ASCE [email protected]
Assistant Professor, Lyles School of Civil Engineering, Purdue Univ., West Lafayette, IN 47907. E-mail: [email protected]
Shirley Dyke [email protected]
Professor, School of Mechanical Engineering, Lyles School of Civil Engineering, Purdue Univ., West Lafayette, IN 47907. E-mail: [email protected]
Amin Maghareh, A.M.ASCE [email protected]
Ph.D. Candidate, Lyles School of Civil Engineering, Purdue Univ., West Lafayette, IN 47907. E-mail: [email protected]

Metrics & Citations

Metrics

Citations

Download citation

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited by

View Options

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share