Abstract
The dynamic condensation method has been recognized as an effective alternative for structural damage identification using spatially-incomplete modal measurements. However, comparative studies of different dynamic condensation techniques applied to the subject of structural damage identification have been scarcely found, especially for composite structures. In this regard, we conduct a comparative study of six typical dynamic condensation techniques utilized for addressing damage identification problems of composite plates made of functionally graded materials (FGM) and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) materials. Firstly, the six techniques consisting of Guyan’s method, Kidder’s method, Neumann series expansion-based second-order model reduction (NSEMR-II) method, improved reduced system (IRS) method, iterated IRS (IIRS) method, and iterative order reduction (IOR) method are reviewed. Then, their performance for reduced Eigen and optimization-damage identification problems are evaluated by studying two numerical examples of FGM plate and FG-CNTRC plate. For solving the optimization-damage identification problem of plate structures, the article proposes to use a hybrid global–local algorithm, Manta Ray Foraging Optimization—Sequential Quadratic Programming (MRFO-SQP), where the MRFO algorithm is utilized for global exploration and the SQP algorithm is used for the local searching process. The comparative study indicates that the IOR technique is the best dynamic condensation technique and is effective for addressing the structural damage identification problems when comparing with the other five techniques. It is also found that the damage identification approach based on the hybrid MRFO–SQP algorithm combined with the IOR technique can archive the high accuracy and low computational cost for damage localization and quantification.
Similar content being viewed by others
References
Noor AK (1994) Recent advances and applications of reduction methods. Appl Mech Rev 47:125–146. https://doi.org/10.1115/1.3111075
Koutsovasilis P, Beitelschmidt M (2008) Comparison of model reduction techniques for large mechanical systems. Multibody Syst Dyn 20:111–128. https://doi.org/10.1007/s11044-008-9116-4
Wagner MB, Younan A, Allaire P, Cogill R (2010) Model reduction methods for rotor dynamic analysis: a survey and review. Int J Rotating Mach. https://doi.org/10.1155/2010/273716
Ghannadi P, Kourehli SS (2018) Investigation of the accuracy of different finite element model reduction techniques. Struct Monit Maint 5:417–428. https://doi.org/10.12989/smm.2018.5.3.417
Thomas PV, Elsayed MSA, Walch D (2019) Review of model order reduction methods and their applications in aeroelasticity loads analysis for design optimization of complex airframes. J Aerosp Eng 32:1–17. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000972
Sehgal S, Kumar H (2016) Structural dynamic model updating techniques: a state of the art review. Arch Comput Methods Eng 23:515–533. https://doi.org/10.1007/s11831-015-9150-3
Sarmadi H, Karamodin A, Entezami A (2016) A new iterative model updating technique based on least squares minimal residual method using measured modal data. Appl Math Model. https://doi.org/10.1016/j.apm.2016.07.015
Mercer JF, Aglietti GS, Kiley AM (2016) Model reduction and sensor placement methods for spacecraft finite element model validation. AIAA J 1–15. doi: https://doi.org/10.2514/1.J054976
Dinh-Cong D, Dang-Trung H, Nguyen-Thoi T (2018) An efficient approach for optimal sensor placement and damage identification in laminated composite structures. Adv Eng Softw 119:48–59. https://doi.org/10.1016/j.advengsoft.2018.02.005
Gupta P, Giridhara G, Gopalakrishnan S (2008) Damage detection based on damage force indicator using reduced-order FE models. Int J Comput Methods Eng Sci Mech 9:154–170. https://doi.org/10.1080/15502280801909127
Dinh-Cong D, Pham-Toan T, Nguyen-Thai D, Nguyen-Thoi T (2019) Structural damage assessment with incomplete and noisy modal data using model reduction technique and LAPO algorithm. Struct Infrastruct Eng 15:1436–1449. https://doi.org/10.1080/15732479.2019.1624785
Guyan RJ (1965) Reduction of stiffness and mass matrices. AIAA J 3:380–380. https://doi.org/10.2514/3.2874
Kidder RL (1973) Reduction of structural frequency equations. AIAA J 11:892–892. https://doi.org/10.2514/3.6852
Miller CA (1980) Dynamic reduction of structural models. J Struct Div 106:2097–2108
Suarez LE, Singh MP (1992) Dynamic condensation method for structural eigenvalue analysis. AIAA J 30:1046–1054. https://doi.org/10.2514/3.11026
O’Callahan JC (1989) A procedure for an improved reduced system (IRS) model. In: Proc. 7th Int. modal Anal. Conf. pp 17–21
Friswell MI, Garvey SD, Penny JET (1995) Model reduction using dynamic and iterated IRS techniques. J Sound Vib 186:311–323. https://doi.org/10.1006/jsvi.1995.0451
Friswell MI, Garvey SD, Penny JET (1998) The convergence of the iterated IRS method. J Sound Vib 211:123–132. https://doi.org/10.1006/jsvi.1997.1368
Xia Y, Lin R-M (2004) Improvement on the iterated IRS method for structural eigensolutions. J Sound Vib 270:713–727. https://doi.org/10.1016/S0022-460X(03)00188-3
Xia Y, Lin R (2004) A new iterative order reduction (IOR) method for eigensolutions of large structures. Int J Numer Methods Eng 59:153–172. https://doi.org/10.1002/nme.876
Yang QW (2009) Model reduction by Neumann series expansion. Appl Math Model 33:4431–4434. https://doi.org/10.1016/j.apm.2009.02.012
Dinh-Cong D, Vo-Duy T, Nguyen-Thoi T (2018) Damage assessment in truss structures with limited sensors using a two-stage method and model reduction. Appl Soft Comput 66:264–277. https://doi.org/10.1016/j.asoc.2018.02.028
Yin T, Zhu HP, Fu SJ (2019) Model selection for dynamic reduction-based structural health monitoring following the Bayesian evidence approach. Mech Syst Signal Process 127:306–327. https://doi.org/10.1016/j.ymssp.2019.03.009
Yin T, Jiang Q-H, Yuen K-V (2017) Vibration-based damage detection for structural connections using incomplete modal data by Bayesian approach and model reduction technique. Eng Struct 132:260–277. https://doi.org/10.1016/j.engstruct.2016.11.035
Mousavi M, Gandomi AH (2016) A hybrid damage detection method using dynamic-reduction transformation matrix and modal force error. Eng Struct 111:425–434. https://doi.org/10.1016/j.engstruct.2015.12.033
Kao C-Y, Chen X-Z, Jan JC, Hung S-L (2016) Locating damage to structures using incomplete measurements. J Civ Struct Heal Monit 6:817–838. https://doi.org/10.1007/s13349-016-0202-7
Zare Hosseinzadeh A, Ghodrati Amiri G, Seyed Razzaghi SA et al (2016) Structural damage detection using sparse sensors installation by optimization procedure based on the modal flexibility matrix. J Sound Vib 381:65–82. https://doi.org/10.1016/j.jsv.2016.06.037
Kourehli SS (2015) LS-SVM Regression for structural damage diagnosis using the iterated improved reduction system. Int J Struct Stab Dyn 16:1550018. https://doi.org/10.1142/S0219455415500182
Zare Hosseinzadeh A, Bagheri A, Ghodrati Amiri G, Koo K-Y (2014) A flexibility-based method via the iterated improved reduction system and the cuckoo optimization algorithm for damage quantification with limited sensors. Smart Mater Struct 23:045019. https://doi.org/10.1088/0964-1726/23/4/045019
Araújo dos Santos JV, Mota Soares CM, Mota Soares CA, Maia NMM (2003) Structural damage identification: Influence of model incompleteness and errors. Compos Struct 62:303–313. https://doi.org/10.1016/j.compstruct.2003.09.029
Dinh-Cong D, Pham-Duy S, Nguyen-Thoi T (2018) Damage detection of 2D frame structures using incomplete measurements by optimization procedure and model reduction. J Adv Eng Comput 2:164–173. https://doi.org/10.25073/jaec.201823.203
Saint Martin LB, Mendes RU, Cavalca KL (2020) Model reduction and dynamic matrices extraction from state-space representation applied to rotating machines. Mech Mach Theory 149:103804. https://doi.org/10.1016/j.mechmachtheory.2020.103804
Dinh-cong D, Nguyen-thoi T, Nguyen DT (2021) A two-stage multi-damage detection approach for composite structures using MKECR-Tikhonov regularization iterative method and model updating procedure. Appl Math Model 90:114–130. https://doi.org/10.1016/j.apm.2020.09.002
Kourehli S, Amiri GG, Ghafory-Ashtiany M, Bagheri A (2013) Structural damage detection based on incomplete modal data using pattern search algorithm. J Vib Control 19:821–833. https://doi.org/10.1177/1077546312438428
Zare Hosseinzadeh A, Ghodrati Amiri G, Seyed Razzaghi SA (2016) Model-based identification of damage from sparse sensor measurements using Neumann series expansion. Inverse Probl Sci Eng 5977:1–21. https://doi.org/10.1080/17415977.2016.1160393
Tani J, Liu G-R (1993) SH surface waves in functionally gradient piezoelectric plates. JSME Int journal Ser A Mech Mater Eng 36:152–155
Liu GR, Han X, Lam KY (1999) Stress waves in functionally gradient materials and its use for material characterization. Compos Part B Eng 30:383–394. https://doi.org/10.1016/S1359-8368(99)00010-4
Vinyas M, Harursampath D, Nguyen-Thoi T (2020) Influence of active constrained layer damping on the coupled vibration response of functionally graded magneto-electro-elastic plates with skewed edges. Def Technol 16:1019–1038. https://doi.org/10.1016/j.dt.2019.11.016
Vinyas M, Harursampath D, Kattimani SC (2020) On vibration analysis of functionally graded carbon nanotube reinforced magneto-electro-elastic plates with different electro-magnetic conditions using higher order finite element methods. Def Technol. https://doi.org/10.1016/j.dt.2020.03.012
Mahesh V, Harursampath D (2020) Nonlinear deflection analysis of CNT/magneto-electro-elastic smart shells under multi-physics loading. Mech Adv Mater Struct 0:1–25. doi: https://doi.org/10.1080/15376494.2020.1805059
Mahesh V, Harursampath D (2020) Nonlinear vibration of functionally graded magneto-electro-elastic higher order plates reinforced by CNTs using FEM. Eng Comput. https://doi.org/10.1007/s00366-020-01098-5
Vinyas M (2020) On frequency response of porous functionally graded magneto-electro- elastic circular and annular plates with different electro-magnetic conditions using HSDT. Compos Struct 240:112044. https://doi.org/10.1016/j.compstruct.2020.112044
Mahesh V (2020) Nonlinear deflection of carbon nanotube reinforced multiphase magneto-electro-elastic plates in thermal environment considering pyrocoupling effects. Math Methods Appl Sci 15–19. doi: https://doi.org/10.1002/mma.6858
Han X, Liu GR, Lam KY, Ohyoshi T (2000) Quadratic layer element for analyzing stress waves in FGMS and its application in material characterization. J Sound Vib 236:307–321. https://doi.org/10.1006/jsvi.2000.2966
Liu GR, Han X, Xu YG, Lam KY (2001) Material characterization of functionally graded material by means of elastic waves and a progressive-learning neural network. Compos Sci Technol 61:1401–1411. https://doi.org/10.1016/S0266-3538(01)00033-1
Han X, Liu GR, Xi ZC, Lam KY (2002) Characteristics of waves in a functionally graded cylinder. Int J Numer Methods Eng 53:653–676. https://doi.org/10.1002/nme.305
Han X, Liu GR (2003) Computational inverse technique for material characterization of functionally graded materials. AIAA J 41:288–295. https://doi.org/10.2514/2.1942
Han X, Liu GR, Ohyoshi T (2004) Dispersion and characteristic surfaces of waves in hybrid multilayered piezoelectric circular cylinders. Comput Mech 33:334–344. https://doi.org/10.1007/s00466-003-0536-y
Dai KY, Liu GR, Lim KM et al (2004) A meshfree radial point interpolation method for analysis of functionally graded material (FGM) plates. Comput Mech 34:213–223. https://doi.org/10.1007/s00466-004-0566-0
Dai KY, Liu GR, Han X, Lim KM (2005) Thermomechanical analysis of functionally graded material (FGM) plates using element-free Galerkin method. Comput Struct 83:1487–1502. https://doi.org/10.1016/j.compstruc.2004.09.020
Vinyas M (2019) A higher-order free vibration analysis of carbon nanotube-reinforced magneto-electro-elastic plates using finite element methods. Compos Part B Eng 158:286–301. https://doi.org/10.1016/j.compositesb.2018.09.086
Zhao W, Zhang Z, Wang L (2020) Manta ray foraging optimization: An effective bio-inspired optimizer for engineering applications. Eng Appl Artif Intell 87:103300. https://doi.org/10.1016/j.engappai.2019.103300
Boggs PT, Tolle JW (1995) Sequential quadratic programming. Acta Numer 4:1–51. https://doi.org/10.1017/S0962492900002518
Dinh-Cong D, Nguyen-Thoi T, Vinyas M, Nguyen DT (2019) Two-stage structural damage assessment by combining modal kinetic energy change with symbiotic organisms search. Int J Struct Stab Dyn 19:1950120. https://doi.org/10.1142/S0219455419501207
Dinh-Cong D, Nguyen-Thoi T, Nguyen DT (2020) A FE model updating technique based on SAP2000-OAPI and enhanced SOS algorithm for damage assessment of full-scale structures. Appl Soft Comput 89:106100. https://doi.org/10.1016/j.asoc.2020.106100
Selem SI, Hasanien HM, El‐Fergany AA (2020) Parameters extraction of PEMFC’s model using manta rays foraging optimizer. Int J Energy Res 1–12. doi: https://doi.org/10.1002/er.5244
Zhao X, Lee YY, Liew KM (2009) Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. J Sound Vib 319:918–939. https://doi.org/10.1016/j.jsv.2008.06.025
Allemang RJ, Brown DL (1982) A correlation coefficient for modal vector analysis. In: Proc. 1st Int. modal Anal. Conf. SEM, Orlando, pp 110–116
Yang X-S, Suash Deb (2009) Cuckoo Search via Lévy flights,. In: 2009 World Congr. Nat. Biol. Inspired Comput. IEEE, pp 210–214
Zhao W, Wang L, Zhang Z (2019) Artificial ecosystem-based optimization: a novel nature-inspired meta-heuristic algorithm. Neural Comput Appl. https://doi.org/10.1007/s00521-019-04452-x
Zhu P, Lei ZX, Liew KM (2012) Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory. Compos Struct 94:1450–1460. https://doi.org/10.1016/j.compstruct.2011.11.010
Han Y, Elliott J (2007) Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput Mater Sci 39:315–323. https://doi.org/10.1016/j.commatsci.2006.06.011
Zhang CL, Shen HS (2006) Temperature-dependent elastic properties of single-walled carbon nanotubes: Prediction from molecular dynamics simulation. Appl Phys Lett 89:2004–2007. https://doi.org/10.1063/1.2336622
Acknowledgements
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.02-2019.330.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dinh-Cong, D., Truong, T.T. & Nguyen-Thoi, T. A comparative study of different dynamic condensation techniques applied to multi-damage identification of FGM and FG-CNTRC plates. Engineering with Computers 38 (Suppl 5), 3951–3975 (2022). https://doi.org/10.1007/s00366-021-01312-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-021-01312-y