Model Updating of Nonproportionally Damped Structural Systems Using an Adapted Complex Sum-of-Squares Optimization Algorithm
Publication: Journal of Engineering Mechanics
Volume 149, Issue 8
Abstract
Various model updating approaches relying on solving nonconvex optimization problems have been developed; however, few of them can reach the global optimum. To obtain the global optimum, the real sum-of-squares optimization algorithm (SOSOA) was proposed to minimize the modal dynamic residuals of proportionally damped structural systems by recasting the nonconvex optimization into a convex optimization. However, this algorithm becomes inefficient for model updating of nonproportionally damped systems with complex-valued eigenvalues and eigvenvectors. This study proposes a model updating approach that exploits and adapts a complex SOSOA to minimize the modal dynamic residuals of nonproportionally damped systems. Numerical considerations unique to using the adapted complex SOSOA to update medium- or large-scale structural models are discussed. Finite-element models of three nonproportionally damped structures—including a base-isolated shear building, a frame building implemented with a tuned mass damper, and a frame building implemented with buckling-restrained braces—are updated to demonstrate that the adapted complex SOSOA finds the global optimum while being orders of magnitude more computationally efficient and far less demanding of computer memory than the real SOSOA.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request, including the codes for updating the finite-element model in the first and third numerical examples.
Acknowledgments
The authors gratefully acknowledge Dr. Erik A. Johnson and Dr. Mahmoud Kamalzare for kindly providing the finite-element model of the 99-DOF superstructure in the second numerical example.
References
Alkayem, N. F., M. Cao, Y. Zhang, M. Bayat, and Z. Su. 2018. “Structural damage detection using finite element model updating with evolutionary algorithms: A survey.” Neural Comput. Appl. 30 (2): 389–411. https://doi.org/10.1007/s00521-017-3284-1.
Bakir, P. G., E. Reynders, and G. De Roeck. 2008. “An improved finite element model updating method by the global optimization technique ‘Coupled Local Minimizers’.” Comput. Struct. 86 (11–12): 1339–1352. https://doi.org/10.1016/j.compstruc.2007.08.009.
Casciati, S. 2008. “Stiffness identification and damage localization via differential evolution algorithms.” Struct. Control Health Monit. 15 (3): 436–449. https://doi.org/10.1002/stc.236.
De, S., E. A. Johnson, S. F. Wojtkiewicz, and P. T. Brewick. 2018. “Computationally efficient Bayesian model selection for locally nonlinear structural dynamic systems.” J. Eng. Mech. 144 (5): 04018022. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001397.
Dourado, M., J. Meireles, and A. M. A. C. Rocha. 2014. “A global optimization approach applied to structural dynamic updating.” In Proc., Computational Science and Its Applications—ICCSA 2014, edited by B. Murgante, S. Misra, A. M. A. C. Rocha, C. Torre, J. G. Rocha, M. I. Falcão, D. Taniar, B. O. Apduhan, and O. Gervasi, 195–210. Cham, Switzerland: Springer.
Gilbert, J. C., and C. Josz. 2017. “Plea for a semidefinite optimization solver in complex numbers.” Accessed May 4, 2023. https://hal.inria.fr/hal-01422932v1.
Grant, M., and S. Boyd. 2008. “Graph implementations for nonsmooth convex programs.” In Recent advances in learning and control, lecture notes in control and information sciences, edited by V. Blondel, S. Boyd, and H. Kimura, 95–110. London: Springer-Verlag Limited.
Grant, M., and S. Boyd. 2014. “CVX: MATLAB software for disciplined convex programming, version 2.1 (March).” Accessed April 14, 2023. http://cvxr.com/cvx/.
Hao, H., and Y. Xia. 2002. “Vibration-based damage detection of structures by genetic algorithm.” J. Comput. Civ. Eng. 16 (3): 222–229. https://doi.org/10.1061/(ASCE)0887-3801(2002)16:3(222).
Jiang, B., Z. Li, and S. Zhang. 2014. “Approximation methods for complex polynomial optimization.” Comput. Optim. Appl. 59 (1): 219–248. https://doi.org/10.1007/s10589-014-9640-5.
Jiang, B., Z. Li, and S. Zhang. 2016. “Characterizing real-valued multivariate complex polynomials and their symmetric tensor representations.” SIAM J. Matrix Anal. Appl. 37 (1): 381–408. https://doi.org/10.1137/141002256.
Josz, C., and D. K. Molzahn. 2015. “Moment/sum-of-squares hierarchy for complex polynomial optimization.” Preprint, submitted September 9, 2015. https://arxiv.org/abs/1508.02068.
Josz, C., and D. K. Molzahn. 2018. “Lasserre hierarchy for large scale polynomial optimization in real and complex variables.” SIAM J. Optim. 28 (2): 1017–1048. https://doi.org/10.1137/15M1034386.
Kamalzare, M., E. A. Johnson, and S. F. Wojtkiewicz. 2015. “Efficient optimal design of passive structural control applied to isolator design.” Smart Strut. Syst. 15 (3): 847–862. https://doi.org/10.12989/sss.2015.15.3.847.
Kosmatka, J. B., and J. M. Ricles. 1999. “Damage detection in structures by modal vibration characterization.” J. Struct. Eng. 125 (12): 1384–1392. https://doi.org/10.1061/(ASCE)0733-9445(1999)125:12(1384).
Lasserre, J. B. 2001. “Global optimization with polynomials and the problem of moments.” SIAM J. Optim. 11 (3): 796–817. https://doi.org/10.1137/S1052623400366802.
Li, D., X. Dong, and Y. Wang. 2018. “Model updating using sum of squares (SOS) optimization to minimize modal dynamic residuals.” Struct. Control Health Monit. 25 (12): e2263. https://doi.org/10.1002/stc.2263.
Li, D., and Y. Wang. 2019. “Sparse sum-of-squares optimization for model updating through minimization of modal dynamic residuals.” J. Nondestr. Eval. Diagn. Progn. Eng. Syst. 2 (1): 011005. https://doi.org/10.1115/1.4042176.
MathWorks. 2018. MATLAB user’s guide (R2018b). Natick, MA: MathWorks.
Mellinger, P., M. Döhler, and L. Mevel. 2016. “Variance estimation of modal parameters from output-only and input/output subspace-based system identification.” J. Sound Vib. 379 (Sep): 1–27. https://doi.org/10.1016/j.jsv.2016.05.037.
MOSEK ApS (MOSEK Application Server). 2019. “The MOSEK optimization toolbox for MATLAB manual, version 9.0.” Accessed April 14, 2023. https://docs.mosek.com/9.0/toolbox/index.html.
Nie, J., and J. Demmel. 2009. “Sparse SOS relaxations for minimizing functions that are summations of small polynomials.” SIAM J. Optim. 19 (4): 1534–1558. https://doi.org/10.1137/060668791.
Otsuki, Y., P. Lander, X. Dong, and Y. Wang. 2022. “Formulation and application of SMU: An open-source MATLAB package for structural model updating.” Adv. Struct. Eng. 25 (4): 698–715. https://doi.org/10.1177/13694332211022066.
Parrilo, P. A. 2000. “Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization.” Ph.D. thesis, Division of Engineering and Applied Science, California Institute of Technology.
Parrilo, P. A. 2003. “Semidefinite programming relaxations for semialgebraic problems.” Math. Program. 96 (2): 293–320. https://doi.org/10.1007/s10107-003-0387-5.
Perera, R., and R. Torres. 2006. “Structural damage detection via modal data with genetic algorithms.” J. Struct. Eng. 132 (9): 1491–1501. https://doi.org/10.1061/(ASCE)0733-9445(2006)132:9(1491).
Saada, M. M., M. H. Arafa, and A. O. Nassef. 2013. “Finite element model updating approach to damage identification in beams using particle swarm optimization.” Eng. Optim. 45 (6): 677–696. https://doi.org/10.1080/0305215X.2012.704026.
Sorber, L., M. V. Barel, and L. D. Lathauwer. 2012. “Unconstrained optimization of real functions in complex variables.” SIAM J. Optim. 22 (3): 879–898. https://doi.org/10.1137/110832124.
Sturm, J. F. 1999. “Using SeDuMi 1.02: A MATLAB toolbox for optimization over symmetric cones.” Optim. Methods Software 11 (1–4): 625–653. https://doi.org/10.1080/10556789908805766.
Suita, K., Y. Suzuki, and M. Takahashi. 2015. “Collapse behavior of an 18-story steel moment frame during a shaking table test.” Int. J. High-Rise Build. 4 (3): 171–180. https://doi.org/10.21022/IJHRB.2015.4.3.171.
Teughels, A., G. De Roeck, and J. A. Suykens. 2003. “Global optimization by coupled local minimizers and its application to FE model updating.” Comput. Struct. 81 (24–25): 2337–2351. https://doi.org/10.1016/S0045-7949(03)00313-4.
Toh, K.-C., M. J. Todd, and R. H. Tütüncü. 1999. “SDPT3—A MATLAB software package for semidefinite programming, version 1.3.” Optim. Methods Software 11 (1–4): 545–581. https://doi.org/10.1080/10556789908805762.
Van Overschee, P., and B. De Moor. 1994. “N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems.” Automatica 30 (1): 75–93. https://doi.org/10.1016/0005-1098(94)90230-5.
Yu, T., and E. A. Johnson. 2023. “CTLS-based CMCM model updating method.” Eng. Struct.
Zhu, D., X. Dong, and Y. Wang. 2016. “Substructure stiffness and mass updating through minimization of modal dynamic residuals.” J. Eng. Mech. 142 (5): 04016013. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001063.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 149 • Issue 8 • August 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Sep 4, 2022
Accepted: Feb 18, 2023
Published online: May 17, 2023
Published in print: Aug 1, 2023
Discussion open until: Oct 17, 2023
ASCE Technical Topics:
- Algorithms
- Buildings
- Computer models
- Continuum mechanics
- Damping
- Dynamics (solid mechanics)
- Engineering fundamentals
- Engineering mechanics
- Finite element method
- Mathematics
- Methodology (by type)
- Models (by type)
- Numerical methods
- Optimization models
- Solid mechanics
- Structural dynamics
- Structural engineering
- Structural models
- Structural systems
- Structures (by type)
Authors
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.