A self-adaptive firefly algorithm for structural damage detection

Transcription

1 A self-adaptve frefly algorthm for structural damage detecton Chu-Dong Pan 1) and *Lng Yu ) 1), ) Department of Mechancs and Cvl Engneerng, Jnan Unversty, Guangzhou 5163, Chna ) MOE Key Lab of Dsaster Forecast and Control n Engneerng, Jnan Unversty, Guangzhou 5163, Chna ) lyu1997@163.com ABSTRACT Structural damage detecton (SDD) s a challengng task n the fled of structural health montorng (SHM). As an explorng attempt to the SDD problem, a self-adaptve frefly algorthm (SAFA) based method s proposed for SDD n ths study. Frst of all, the basc prncple of FA s ntroduced. A self-adaptve control strategy on lght absorpton coeffcent s proposed. After that, the SAFA s ntroduced nto the SDD feld. Combned wth the character of SDD problem, the mproved behavor of the best frefly and the mult-step method are used to mprove the dentfed results. In order to assess the accuracy and the feasblty of the proposed method, -storey rgd frame structure s taken as an example for numercal smulaton on SDD. The llustrated results show that the proposed method can accurately dentfy the structural damage. Some valuable conclusons are made and related ssues dscussed as well. 1. ITRODUCTIO Structural damage detecton (SDD) based on vbraton s one of the core technques n the feld of structural health montorng (SHM) and has been wdely concerned by the researchers all over the world (Yan, et al., 7). Lots of the vbraton-based structural damage detecton methods have been proposed (Alvand and Cremona, 6). Generally, the methods can be dvded nto two groups, statstcs-based SDD methods (Hos and Fassos, 14) and models-based SDD methods (Frtzen, et al., 1998). The frst group methods are not based on structural models. The statstcs-based methods always dentfy the structural damage only based on the statstcal characters of the dynamc response sgnals. Another group methods are usually mplemented by fnte element analyss, therefore, the dentfed results are based on the accuracy of the structural model (Perera, et al., 1). The nature frequency and mode shapes are usually used to detect the damage n model-based method (Zhu and Xu, 5). The model-based methods are generally recognzed and wdely used n cvl engneerng. These are often transformed nto mathematcal problem solvng constraned ) Professor

2 optmzaton. However, the mathematcal model of the tradtonal constraned optmzaton methods s very complex and cannot be used to solve the hgh dmensonal and complex optmzaton problems. Fortunately, some swarm ntellgence (SI) optmzaton algorthms are adopted to solve large-scale cvl engneerng structural optmzaton problems (Yu, et al., 1), such as the PSO algorthm (Shraz, et al., 14), the ACO algorthm (Yu and Xu, 11), the GAFSA algorthm (Yu and L, 14), etc. The performances of SI algorthms are manly depended on the key parameters (Yang, 14). However, for the structural optmzaton problems, t s very hard to select the effectve key parameters for the SI algorthms. Inspred by the flashng patterns and behavor of the frefles, frefly algorthm (FA) s frst developed by Yang and wdely used (Fster, et al., 13). In all SI algorthms, there are two mportant components: explotaton and exploraton. Exploraton means that the search space s suffcently nvestgated on a rough level, whle explotaton means that nterestng areas are searched more ntensvely n order to allow for a good approxmaton to an optmum (Yang, et al,. 15). In classcal FA, all frefles have two key behavors, attracton and random walk. Therefore, the exploraton components s ensured by the random behavor, whle the attracton behavor enhance the explotaton component. However, the balance of explotaton and exploraton s manly depended by the key parameters of FA. Therefore, the performance of FA s manly depended by the key parameters tunng and control. At the moment, there s no effcent method n general for parameter tunng. The methods for parameter control can be dvded nto three groups, fxed control method, random control method and self-adaptve control. The fxed control method means that the parameter values often fxed durng teratons and the random control method means that the parameter values often vary by a random process, such as chaotc stochastc process (Gandom, et al., 13). Self-adaptve control method means that the parameter values wll vary accordng to the teratons and the characters of the swarm, therefore, t has been wdespread concerned by the researchers. In ths study, a novel self-adaptve frefly algorthm (SAFA) s proposed and used n the SDD problem.. SELF-ADAPTIVE FIREFLY ALGORITHM.1 Basc Frefly Algorthm Frefly algorthm s a new SI optmzaton algorthm nspred by nature frefles flashng behavor. In FA, the frefles abde by the followng three rules: 1) all frefles are unsex so that one frefly wll be attracted to other frefles regardless of ther sex; ) Attractveness s proportonal to ther brghtness, thus for any two flashng frefles, the less brghter one wll move towards the brghter one. The attractveness s proportonal to the brghtness and they both decrease as ther dstance ncreases; 3) For a specfc problem, the brghtness of frefly s assocated wth the obectve functon. The lght ntensty of a frefly wll decrease wth the ncreasng dstance of vewer. In addton, lght s also absorpton by the meda. Therefore, t can be defned as: (1) I r I e γr Where I s the orgnal lght ntensty, r s the dstance between any two frefles and

3 γ s the lght absorpton coeffcent. The attractveness s proportonal to the lght ntensty, whch s defned as: () β r β e γr where β s the attractveness at r. Then the -th frefly s attracted to the -th frefly, and the movement s formulated by: d d γr d d x t1 x t βe x (t) x (t) α Ld( rand.5) (3) where α s the randomzaton parameter, L d s the length of d-th dmenson, rand s a random number generator unformly dstrbuted n [, 1]. r s the dstance between -th frefly and -th frefly, whch s defned as the Cartesan dstance as: r x x (4). Self-adaptve Frefly Algorthm There are three key parameters to be tuned for better performance of FA. They are ntal attractveness β, lght absorpton coeffcent γ and randomzaton parameter α. The lght absorpton coeffcent controls the decrease of lght ntensty and s very mportant n balancng the exploraton and the explotaton (Chen and Dng, 15). Generally, the lght absorpton coeffcent can be selected between, and s commonly set to be 1 as done n (Yang, 1). However, t s dffcult to select the lght absorpton coeffcent for dfferent problems to enhance the performance of FA, therefore, a self-adaptve control of γ s necessary. A varyng lght absorpton coeffcent γ s desgned based on both the teratons process and the maxmum dstances among the frefles n ths paper and can be expressed as: γ t 1 averageχ1, χ, χ3,, χt 1, t γ Gen mn Gen γt, t γmax where Gen Gen s the max generatons, 45 ln 1 χ and 1 1 smax other parameters can be obtaned as follows: (5) ln.9,. The γ γ γ mn 1 max Gen s nt( ) max

4 t 1 ln 1 χ s χ t 1 max 45 average χ, χ, χ,, χ, χ.5χ χ, others 1 3 t 1 (6) t where s max s the maxmum dstance among all frefles at t-th teraton. average () s the functon for calculatng the mean value. The ntal postons of swarm populatons are randomly located n the searchng area whch s requred for the self-control strategy proposed above. In order to balance the exploraton and the explotaton durng the latter half teraton. All frefles has a probablty P to randomly rebuld except the best frefly. The basc steps of SAFA s shown n Fg. 1. Fg. 1 The basc steps of SAFA 3. STRUCTURAL DAMAGE DETECTIO SIMULATIO Generally, for a SDD problem, only the stffness s consdered to reduce whle gnorng the change of the mass, as: K K αk (7) where K, K s the global stffness matrx of the damaged and undamaged structures, respectvely. α s the coeffcent of -th element stffness damage, K s the expanded stffness matrx of the -th element n the global coordnate system, s the number of elements. The moton equaton of the damaged structure can be expressed as follows: Kφ λmφ (8) Where M s the mass matrx of the structure, φ s the -th mode shapes of the

5 α α and f s the -th frequency of the structure. Therefore, λ, φ can be obtaned by substtutng Eq. (7) nto Eq. (8) wth the coeffcent vector of element stffness damage α α1, α, α3,, α. Then the SDD problem can be transformed nto the followng constraned optmzaton problem: structure, λ πf s t a a t φ φ f f mn f α 1 MAC, ER,, subect to : α.9, 1,,3,, (9) Where s s the number of measured modes, f t and φ t are the -th test nature frequency and the correspondng mode shape of the structure. f α and φ α are the analytcal nature frequency and he correspondng mode shape, respectvely. The t a, a t ER f, f are defned as follows: functon of MAC φ φ and t MAC φ, φ ER f tt a φ φ a tt t at a φ φ φ φ a t f f, f 1% f a t t (1) (11) 4. SPECIAL ASPECTS I IMPLEMETATIO In FA, the best frefly wll try to mprove tself by flyng randomly. Ths behavor wll be mproved for the SDD problem. Actually, comparng to the number of undamaged elements, the number of damaged elements s much smaller. Therefore, for the global best locaton of the SDD problem, most of the elements n coeffcent vector αbest α1, α, α3,, α should be. And the best frefly wll try to mprove tself by settng coeffcent vector elements as. Ths can be expressed as: α new a, ε 1 max a a a, ε 1 max best abest 1 3 ε rand(), α α, α, α,, α, max a, 1,,3,, best best (1) where ε rand() s the rand number dstrbuted n [, 1], αbest α1, α, α3,, α s the locaton of the best frefly n the current teraton. If the orgnal lght ntensty at locaton new α s better than one at locaton new α best, then the best frefly wll move to locaton α. If max a, the best frefly wll try to mprove tself by flyng randomly. the best In order to dentfy the structural damage more exactly, the mult-step method wll be proposed n ths paper. Frst of all, the SAFA wll be used to solve the SDD problem. Then the undamaged elements wll be pcked out by the threshold value ξ. Ths

6 process can be expressed as: damage, α ξ element undamage, α ξ (13) where the threshold value ξ can be determned by the percent of the maxmum damage factor n the front step results.then the undamaged elements wll be settng as kn condtons for the next step. It means that the Eq. (9) can be rewrote as: s t a a t mn f α 1 MAC, ER, φ φ f f, α.9, damaged elements subect to : α, undamaged elements (14) The above problem can be solved by SAFA and ths process can be redone as a next step. For a specal case, f the results of the frst step satsfy α ξ, 1,,3,,, whch show that all the elements are undamaged, then the more exactly search wll be done by settng the stffness factors as α Up, 1,,3,,. Here the Up.9 and usually can be set to UMERICAL SIMULATIOS Fg. Fnte element model of -storey rgd frame The fnte element model of -storey rgd frame s shown n Fg.. Both the heght and wdth at each storey are 1.41m and the structure s dvded nto 18 elements. The structure s smulaton wth the followng parameters as shown n Table. 1. There cases are studed as: 1) 5% damage at element 17; ) 15% damage at element 8 and 15%

7 damage at element 17; 3) 1% damage at element 8, % damage at element 11 and 15% damage at element 17. Two nose levels % and 1% are added to the modes shapes, respectvely. Table. 1 Materal propertes of -storey rgd frame Type Elastc modulus Moment of nerta Cross-sectonal area Densty 11 Column 1 m m m 859 kg m 11 Beam 1 m m 3. 1 m 7593 kg m 3 3 The parameters of SAFA are set as follows: ntal attractveness β 1 ; randomzaton parameter α.1, max teraton Gen 1 and swarm populaton swarm 3, the threshold value ξ max( α ) 5% and α s the results of the front step. The ntal step dentfed results s taken as the mean value of 5-tmes runnng results. Fve mult-steps are used to mproved the dentfed results. The rebuld probablty P for the frst half teraton, whle P.5 for the latter half teraton. One frefly wll occuped the pont where all the damage factors are when ntalzng the swarm populaton poston for the ntal step. (a). 5%@8 (b). 15%@8 and 15%@17 (c). 1%@8, %@11 and 15@17 Fg. 3 SDD results only by SAFA The SDD results, whch are dentfed only by usng 5-tmes SAFA, are shown n Fg. 3. It clearly shows that: 1) There has a large devaton between the real structure damages and detecton damages for all the cases. ) The magntudes of the detecton results at the damaged locatons are always much bgger than others. Actually, the proposed SAFA has a ablty of parameter self-tunng and self-control. But for hgh dmensonal optmzaton problem, the SAFA s very hard to fnd the global best locaton

8 wth lmted teraton and lmted swarm populaton. That s because the searchng area s too large. However, the SAFA has good ablty for global searchng, therefore, the SAFA can always fnd a better locaton close to the best locaton. So the SDD results by SAFA can always locate the damage but hard to estmate extent of damage. The SDD results, mproved by usng the mult-step method, are shown n Fg. 4. It can be seen that the damage can be accurately detected for all the three cases, even though 1% nose level are consdered. Actually, as the elements are estmated to be an undamaged elements, the searchng area wll be smaller and smaller durng the mult-step processng, therefore, the SAFA can fnd the global best locaton wth more possbltes. (a). 5%@8 6. COCLUSIOS (b). 15%@8 and 15%@17 (c). 1%@8, %@11 and 15@17 Fg. 4 SDD results mproved by usng proposed mult-step method A novel self-adaptve frefly algorthm based structural damage detecton (SDD) method s proposed n ths study. The basc prncple of FA s ntroduced. A self-adaptve control strategy s used to mprove the FA. Combned wth the character of SDD problem, the new best frefly behavor s used to mprove the SAFA whle the mult-step method s proposed to mprove the SDD results. A -storey rgd frame structure s taken as an example for numercal smulatons on SDD. The followng conclusons can be made. 1) The performance of SAFA s well wth the self-adaptve control strategy based on both the dstance among the frefles and the teraton. ) For the SDD problem, the new behavor of best frefly and the mult-step method can clearly mprove the accuracy of SDD result. 3) The proposed method can accurately dentfy the structural damage and has a strong robustness to noses. ACKOWLEDGMETS The proect s ontly supported by the atonal atural Scence Foundaton of Chna

9 wth Grant umbers and REFERECES Alvand, A. and Cremona, C. (6), Assessment of vbraton-based damage dentfcaton technques, J. Sound. Vb., 9(1), Chen, G. and Dng, X. (15), Optmal economc dspatch wth valve loadng effect usng self-adaptve frefly algorthm, Appl. Intell., 4(), Fster, I., Yang, X. S. and Brest, J. (13), A comprehensve revew of frefly algorthms, Swarm Evol. Comput., 13, Frtzen, C. P., Jennewen, D. and Kefer, T. (1998), Damage detecton based on model updatng methods, Mech. Syst. Sgnal. Process., 1(1), Gandom, A. H., Yang, X. S., Talatahar, S. and Alav, A. H. (13), Frefly algorthm wth chaos, Comm. onlnear Sc. umer. Smul., 18(1), Hos, J. D. and Fassos, S. D. (14), A global statstcal model based approach for vbraton response-only damage detecton under varous temperatures: A proof-of-concept study, Mech. Syst. Sgnal. Process., 49(1), Perera, R., Fang, S. E. and Ruz, A. (1), Applcaton of partcle swarm optmzaton and genetc algorthms to multobectve damage dentfcaton nverse problems wth modellng errors, Meccanca, 45(5), Shraz, M.., Mollamahmoud, H. and Seyedpoor, S. M. (14). Structural damage dentfcaton usng an adaptve mult-stage optmzaton method based on a modfed partcle swarm algorthm. J. Optm. Theory. Appl., 16(3), Yan, Y. J., Cheng, L., Wu, Z. Y. and Yam, L. H. (7), Development n vbraton-based structural damage detecton technque, Mech. Syst. Sgnal. Process., 1(5), Yang, X. S. (1), Frefly algorthm, stochastc test functons and desgn optmzaton, Int. J. Bo-nspre. Com., (), Yang, X. S. (14), Swarm ntellgence based algorthms: a crtcal analyss, Evol. Intellgence, 7(1), Yang, X. S., Deb, S., Hanne, T. and He, X. (15), Attracton and dffuson n nature-nspred optmzaton algorthms. eural Comput. Appl., 1-8. Yu, L and Xu, P. (11), Structural health montorng based on contnuous ACO method, Mcroelectron. Relab., 51(), Yu, L. and L, C. (14), A Global Artfcal Fsh Swarm Algorthm for Structural Damage Detecton. Adv. Struct. Eng., 17(3), Yu, L., Xu, P. and Chen, X. (1). A SI-based algorthm for structural damage detecton. Proceedngs of Advances n Swarm Intellgence - Thrd Internatonal Conference, Shenzhen, Chna. Zhu, H. P. and Xu, Y. L. (5), Damage detecton of mono-coupled perodc structures based on senstvty analyss of modal parameters, J. Sound. Vb., 85(1),