Multi-Objective Optimization Algorithms for Finite Element Model Updating

Transcription

1 Multi-Objective Optimization Algoithms fo Finite Element Model Updating E. Ntotsios, C. Papadimitiou Univesity of Thessaly Geece

2 Outline STRUCTURAL IDENTIFICATION USING MEASURED MODAL DATA Weighted Modal Residuals Famewok Multi-Objective Famewok Optimally Weighted Modal Residuals Method COMPUTATIONAL ISSUES Single-Objective Optimization Multi-Objective Optimization Gadient and Hessian of Objectives ILUSTATIVE EXAMPLE Stuctual Identification of a Full Scale Bidge Using Ambient Vibation Measuements CONCLUSIONS

3 Model Updating Issues MODELLING ERROR Assumptions used to descibe a physical system by a model Numeical eos (e.g. discetization of patial diffeential equations of motion) MEASUREMENT AND PROCESSING ERROR Measuement of esponse time histoies Modal estimation fom esponse time histoies

4 Stuctual Identification - Fomulation D= { ˆ ω, fˆ, =, L, m; k =, L, N } D = Available Measued Modal Data Μ = Class of Linea Models, q = stuctual paamete set to be identified ω ( θ), f ( θ), =, L, m = Modal data pedicted by Model, solving the Eigenvalue Poblem K( θ ) ωm( θ ) f = Poblem: Find q values so that model pedicted modal data ae close to the measued modal data Measue of fit (Modal Residuals): Modal Fequencies J ω ( θ ) ND [ ω ( ) ˆ ω ] = θ N [ ˆ ω ] D k= ( k ) ( k ) =, K, m Modeshapes J φ ( θ ) ND = N D k = β φ ( θ) ˆ φ ( k) ( k) ˆ φ ( k ) N D = Numbe of available Data sets Fo m modesà maximum m objectives

5 Gouping of Modal Popeties Geneal Case Modal Popeties ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 L ˆm f f f 3 L f m ω ω ω ω Modal Goups g g K gn Modal Residuals ( θ) ( θ) K ( θ) J J J n

6 Gouping of Modal Popeties Special Case Modal Popeties ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 L ˆm f f f 3 L f m ω ω ω ω Modal Goups g g Modal Residuals J ( θ) J ( θ) J m ( θ) = J ω ( θ) = J m ( θ) = Jφ ( θ) = Modal Fequencies Modeshapes

7 Weighted Modal Residuals Famewok Find θ that minimizes the weighted modal esiduals: n J( θ; w) = wj ( θ) n = numbe of modal goups n i= w i i= = i i Optimal Solution θ ˆ( w) depends on the value of the weights w θ ˆ( w) Values of weight factos affect optimal which in tun affects model pedictions wi Poblem: Find the most pobable (optimal) weight values, based on the measued data and the noms used to measue the fit between measued and model pedicted modal popeties ŵ

8 Multi-Objective Famewok Find θ that simultaneously minimizes the objectives ( J ( ) J ( ) K J n( )) J ( θ) = θ, θ,, θ Paeto optimal solutions (Set of altenative solutions). Paeto solutions Objective Space J Paeto font x Paamete Space J All Paeto solutions ae acceptable: The chaacteistics of the Paeto solutions ae that the modal esiduals cannot be impoved in any modal popety without deteioating the modal esiduals in at least one othe modal popety. x Relation to Weight Modal Residuals Famewok : Vaying the values of the weights fom to, Paeto optimal solutions ae altenatively obtained. w i Equivalent Poblem: Find the most pobable Paeto point and optimal solution to be used fo model pedictions, based on the measued data

9 Optimally Weighted Modal Residuals Method Most pefeed model Find ˆ θ p = ˆ θ( wˆ ) that minimizes the weighted modal esiduals: J( θ; wˆ ) = wj ˆ ( θ) n i= i i selecting the most pefeed values of weights to be invesely popotional to the optimal values of the modal esiduals αi wˆ i =, i=, K, n J ( ˆ θ ( wˆ )) i The most pefeed model fo the most pefeed weights ae obtained by simultaneously solving the above set of equality equations and the optimization poblem Efficient Solution Stategy: Most pefeed model minimizes the sum of the logaithms of the esiduals (Chistodoulou and Papadimitiou 7) ˆ θ = ag min I( θ) p θ n I( θ) = α ln J ( θ) i= i i

10 Computational Issues Single Objective Optimization " Gadient Based Methods Local methods - Cannot guaantee the estimation of global optimum Requie use-defined initial estimates Fast convegence exploit gadient infomation " Evolution Stategies (ES) Global Methods Do not equie use-defined initial estimates Vey slow convegence in the neighbohood of the global optimum " Hybid Algoithms (Combine ES and Gadient methods) Exploit the advantages of ES and Gadient methods ES exploe the paamete space and detect the neighbohood of the global optimum Gadient methods stat fom the best estimate of ES and use gadient infomation to acceleate convegence to the global optimum

11 Computational Issues Multi Objective Optimization " Stength Paeto Evolutionay Algoithms (SPEA) [Zitzle and Thiele 999] Random initialized population of seach points in the paamete space which by means of selection, mutation and ecombination evolves towads bette and bette egions in the seach space Clusteing techniques ae used to unifomly distibute points along the Paeto font, povided that the values of objectives ae of the same ode of magnitude along the Paeto font Requie use-defined initial estimates Slow convegence in the neighbohood of the Paeto font " Nomal Bounday Intesection Method (NBI) [Das and Dennis 998] Deteministic algoithms based on gadient methods Poduces an evenly spead of points along the Paeto font Fast convegence Computationally expensive fo moe than 3 objectives

12 Computational Issues Gadients of Objectives In ode to guaantee the convegence of the gadient-based optimization methods, the gadients of the objective functions with espect to the paamete set θ has to be estimated accuately Nelson s Method Gadient of eigenvalue and eigenvecto of a mode is computed using infomation fom the eigenvalue and eigenvecto of the same mode Adjoint Fomulation The computational cost is independent of numbe of paametes. Fo each mode a solution of a linea system of algebaic equations is equied Hessian of objectives

13 Polymylos Bidge - Instumentation Instumented with an aay of 4 acceleometes optimally placed on the deck and the base of the columns and beaings 4 3 T3RT U3LL U3LV U3RT Aκρόβαθρο T ΠΟΛΥΜΥΛΟΣ 3m BRV 4 BRT 4 BLV 4 35m MRV M LL ULV 4 ULT 4 MLV MRT ULL 4 35m ARV4 Βάση Πυλώνα M ALV 4 ART 4 3m SRV TRT SRT ULV ULL SLV URT Aκρόβαθρο T z-axis y-axis x-axis

14 Polymylos Bidge Opeational Modal Analysis Modal Identification Softwae No Identified Modes Hz Damping Ratios (%) st longitudinal nd tansvese st bending (deck) nd bending (deck) th tansvese d bending (deck) Mode :.3 Hz, zeta=.6% Mode :.5 Hz, zeta=3.737% z-axis - -3 z-axis x-axis x-axis y-axis y-axis Mode 3: 3.74 Hz, zeta=.35% Mode 4: 4.95 Hz, zeta=.435% z-axis - -3 z-axis x-axis x-axis y-axis y-axis

15 Polymylos Bidge Finite Element Model Finite Element Model Updating using Multi-Objective Identification Finite element model of 8 beam elements (38 DOF) 3 paamete FE model z x y J θθ θ θ θ 3 θ Model Updating Softwae

16 Polymylos Bidge - Model Updating Results 3 paametes Multi-Objective model updating using 3 modes J Paeto Solutions w= Objective and paametes space θ Paeto Solutions w= J J [ ω ( ) ˆ ω ] = q ND ( k ) ( q ) ( k ) N k [ ˆ D = ω ] ( k) ˆ( k) ND β φ( q ) φ ( q ) = N k ˆ( k ) D = φ Equally weighted method paamete values θ (E beaings) θ (E deck).93 θ 3 (E pie).645 J (mode fequencies).347 J (mode shapes).635 θ J θ 9 Paeto Solutions w= θ θ θ 6 Paeto Solutions w= θ value # of solutions θ θ θ 3 θ w =

17 Conclusions Model updating algoithms wee poposed to compute all Paeto optimal models consistent with measued data and the noms used to measue the fit between the measued and model pedicted modal popeties. The equivalence between the multi-objective identification and the weighted modal esiduals method was established. Hybid algoithms based on evolution stategies and gadient methods ae well-suited optimization tools fo solving the esulting optimization poblem and identifying the global optimum fom multiple local ones. NBI algoithms ae well-suited multi-objective optimization tools fo solving the multiobjective identification poblem. NBI effectively computes the useful identifiable pat of the Paeto font. The computational cost fo estimating analytically the gadients of the objectives is shown to be independent of the numbe of stuctual model paametes.