Finite element Model Updating by using Maximum Likelihood Estimation

Yıl 2021, Cilt: 32 Sayı: 5, 11175 - 11196, 01.09.2021

Çağlayan Hızal

https://doi.org/10.18400/tekderg.739610

Öz

In recent years, the problem of vibration-based model updating of structures has been increasingly attracting the attention of researchers. In general sense, the corresponding studies available in the literature can be classified as deterministic and probabilistic methods. In this context, various implementations are available for the deterministic and probabilistic approaches. This study, however, presents an alternative approach based on the maximum likelihood estimation. In the proposed methodology, the modelling errors are considered by using a non-dimensional Rayleigh ratio, in addition to the measurement errors. System model parameters are updated via a probability density function obtained by the assumption that the measurement and modelling errors follow normal distribution. The reliability of the proposed method has been verified by one numerical and one experimental application. According to the results, it is observed that the proposed method gives rather reasonable solution.

Anahtar Kelimeler

Model updating, maximum likelihood estimation, modal identification, multiple measurement sets

Sonlu Eleman Modellerinin Maksimum Olasılık Tahmini ile Güncellenmesi

Öz

Matematiksel yapı modellerinin titreşim verileri kullanılarak güncellenmesi konusu, son yıllarda giderek artan bir şekilde araştırmacıların ilgisini çekmektedir. Bu hususta literatüre sunulan yöntemler genel olarak deterministik ve olasılıksal olarak sınıflandırılmaktadır. Bu bağlamda hem deterministik hem de olasılıksal model güncelleme yöntemlerinin birçok varyasyonu yer almaktadır. Bu çalışmada ise maksimum olasılık tahminine dayalı alternatif bir yaklaşım sunulmaktadır. Önerilen yöntemde, modal tanılama sırasında öngörülen ölçüm hatalarının yanı sıra model hatası da boyutsuz bir Rayleigh oranı üzerinden dikkate alınmaktadır. Sisteme ait model parametreler, ölçüm ve modelleme hatalarının normal dağılım göstereceği kabulüyle oluşturulan bir olasılık yoğunluk fonksiyonu üzerinden hesaplanmaktadır. Sunulan yöntemin güvenirliği bir sayısal ve bir deneysel uygulama üzerinden değerlendirilmiştir. Elde edilen verilere göre önerilen yöntemin oldukça makul sonuçlar verdiği gözlemlenmektedir.

Anahtar Kelimeler

Model güncelleme, Maksimum olasılık kestirimi, modal tanılama, çoklu ölçüm setleri

Kaynakça

• Overschee, P.V., De Moor, B., Subspace algorithms for the stochastic identification problem, Automatica., 29, 649–660, 1993.
• Brincker, R., Zhang, L., Andersen, P., Modal identification of output-only systems using frequency domain decomposition, Smart Mater. Struct., 10, 441–445, 2001.
• Hermans, L., Van Der Auweraer, H., Guillaume, P., A frequency-domain maximum likelihood approach for the extraction of modal parameters from output-only data, Proc. 23rd Int. Conf. Noise Vib. Eng. ISMA. 963–972, 1998.
• Yuen, K.V., Katafygiotis, L.S., Bayesian Fast Fourier Transform Approach for Modal Updating Using Ambient Data, Adv. Struct. Eng., 6, 81–95, 2003.
• Katafygiotis, L.S., Yuen, K.V., Bayesian spectral density approach for modal updating using ambient data, Earthq. Eng. Struct. Dyn., 30, 1103–1123, 2001.
• Mevel, L., Basseville, M., Benveniste, A., Goursat, M., Merging sensor data from multiple measurement set-ups for non-stationary subspace-based modal analysis, J. Sound Vib., 249,719–741, 2002.
• Hızal, Ç., Turan, G., Aktaş, E., Ceylan, H., A mode shape assembly algorithm by using two stage Bayesian Fast Fourier Transform Approach, Mech. Syst. Signal Process., 134, 106328, 2019.
• Au, S.K., Fast Bayesian ambient modal identification in the frequency domain, Part I: Posterior most probable value, Mech. Syst. Signal Process., 26, 60–75, 2012.
• Yuen, K.V., Bayesian Methods for Structural Dynamics and Civil Engineering, 2010.
• Hızal, Ç., Modal identification of structures by using Bayesian statistics, Ph.D. Thesis, Izmir Institute of Technology, 2019.
• Beck, J.L., Au, S.K., Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation, J. Eng. Mech., 128, 380–391, 2002.
• Ching, J., Chen, Y., Transitional Markov Chain Monte Carlo Method for Bayesian Model Updating, Model Class Selection, and Model Averaging, J. Eng. Mech., 133, 816–832, 2007.
• Bansal, S., A New Gibbs Sampling Based Bayesian Model Updating Approach Using Modal Data From Multiple Setups, Int. J. Uncertain. Quantif., 5, 361–374, 2015.
• Cheung, S.H., Bansal, S., A new Gibbs sampling based algorithm for Bayesian model updating with incomplete complex modal data, Mech. Syst. Signal Process., 92, 156–172, 2017.
• Khodaparast, H.H., Mottershead, J.E., Friswell, M.I., Perturbation methods for the estimation of parameter variability in stochastic model updating, Mech. Syst. Signal Process., 22, 1751–1773, 2008.
• Abu Husain, N., Khodaparast H.H., Ouyang, H., Parameter selection and stochastic model updating using perturbation methods with parameter weighting matrix assignment, Mech. Syst. Signal Process., 32, 135–152,2012.
• Hua, X.G., Ni, Y.Q., Chen, Z.Q., Ko, J.M., An improved perturbation method for stochastic finite element model updating, Int. J. Numer. Methods Eng., 73, 1845–1864, 2008
• Behmanesh, I., Moaveni, B., Lombaert, G., Papadimitriou, C., Hierarchical Bayesian model updating for structural identification, Mech. Syst. Signal Process., 64–65, 360–376, 2015.
• Sedehi, O., Papadimitriou, C., Katafygiotis, L.S., Probabilistic hierarchical Bayesian framework for time-domain model updating and robust predictions, Mech. Syst. Signal Process., 123, 648–673, 2019.
• Yan, W.J., Katafygiotis, L.S., A novel Bayesian approach for structural model updating utilizing statistical modal information from multiple setups, Struct. Saf., 52, 260–271, 2015.
• Zhang, F.L., Ni, Y.C., Lam, H.F., Bayesian structural model updating using ambient vibration data collected by multiple setups, Struct. Control Heal. Monit., 24, 1–18, 2017.
• Hızal, Ç., Turan, G., A two-stage Bayesian algorithm for finite element model updating by using ambient response data from multiple measurement setups, J. Sound Vib., 469, 115139, 2020.
• Au, S.K., Zhang, F.L., Fundamental two-stage formulation for Bayesian system identification, Part I: General theory, Mech. Syst. Signal Process., 66–67, 31–42, 2016.
• Zhang, F.L., Au, S.K., Fundamental two-stage formulation for Bayesian system identification, Part II: Application to ambient vibration data, Mech. Syst. Signal Process., 66–67, 43–61, 2016.
• Shiradhonkar, S.R., Shrikhande, M., Seismic damage detection in a building frame via finite element model updating, Comput. Struct. 89, 2425–2438, 2011.
• MATLAB 2018a (Computer Software), MathWorks, Natick, MA, (2018).
• Zhang, L., Wang, T., Tamura, Y., A frequency-spatial domain decomposition (FSDD) method for operational modal analysis, Mech. Syst. Signal Process., 24, 1227–1239, 2010.
• Ceylan, H., Modal parameter identification of civil engineering structures by using an Output-Only System Identification Technique, M.Sc. Thesis, Izmir Institute of Technology, 2015.
• Ceylan, H., Turan, G., Hızal, Ç., Pre-Identification Data Merging for Multiple Setup Measurements with Roving References, Exp. Tech. 44, 435-456, 2020.
• Bansal, S., Bayesian model updating using modal data based on dynamic condensation, J. Eng. Mech. 146(2), 04019123, 2020.
• Papadimitriou, C., Lombaert, G., The effect of prediction error correlation on optimal sensor placement in structural dynamics, Mech. Syst. Signal Process. 28, 105–127, 2012.
• Papadimitriou, C., Optimal sensor placement methodology for parametric identification of structural systems, J. Sound Vib. 278, 923–947, 2004.