Hybrid Reduced-Order Modeling and Particle-Kalman Filtering for the - PowerPoint PPT Presentation

Hybrid Reduced-Order Modeling and Particle-Kalman Filtering for the Health Monitoring of Flexible Structures Giovanni Capellari 1 , Saeed Eftekhar Azam 2 , Stefano Mariani 1 1 Politecnico di Milano, Dipartimento di Ingegneria Civile e Ambientale 2 University of Thessaly, Department of Mechanical Engineering

Str tructural Hea ealth Mon onitoring 2 Singapore: reduce risk related to damage Göta Bridge (Sweden): safely extend the assessment after natural events lifetime of the ageing bridge Halifax Metro Center (Canada): Making use of I35W Bridge (USA): reassure public on the safety existing structural reserves to allow increased of the new bridge, support the rapid construction snow and equipment loads on the roof schedule, provide data to local researchers

Dam amage Id Iden entification 3 1. Observation of the system through periodically spaced measurements 2. Selection of a certain number of features and indexes in order to identify the damage 3. Estimation of the aforementioned indexes using an inverse identification method based on the observations Balageas et al. 2006

Ob Objective an and moti otivations 4 Dam amage e identi tific icati tion an and locali lizatio ion Observations Damage indexes Req equir irements Model order reduction Reduced computational cost • Hybrid Extended Kalman Particle Filter • On-line tracking Dual estimation Coupling with FE commercial • Use of reference substructures code

Mod odel Or Order Red eduction 5 Linear dynamic equation: Full order n model Reduced order l<n model Proper orthogonal decomposition based methods (POD) • Optimization statement find the projection such that is minimized :P :Proper Ort rthogonal l Modes (POM)

Mod odel Or Order Red eduction 6 Calculation of POMs: Sing ngular Valu lue Decomposit itio ion Any given matrix U can be decomposed by: : Snapshot matrix : Singular values : Left singular vectors The i-th POM can be calculated through: Level of information:

Mod odel Or Order Red eduction 7 Galerkin in-based Pro roje jectio ion The vector can be expressed as a linear combination of : From the orthogonality condition , we get:

Dual Es Estimation Prob oblem 8 Discrete-time s sta tate s space eq equatio ions: State vector Process noise Process model Measurement model Observations Measurement noise Dua ual l es estim imati tion: Dynamic process Parameters

Kalm alman Fil Filter an and Ex Extended Kalm alman Fil Filter 9 Hypothesis: If

Ex Extended Kalm alman Fil Filter vs vs. . Particle Fil Filter 10 Dra rawbacks of f the e Exte tended Kalm alman Par arti ticle le Filt lter Filt lter no assumptions on the probability • linearization error distribution function are required • • computational cost of the Jacobian • generation of samples and relative matrix weights from non-holonomic systems • Dra rawbacks of f the e Par arti ticle e Filt lter Solu lutio ions number of samples sub-optimal importance function • • degeneracy of the weights • re-sampling •

Par article Fil Filter 11

Application to to str tructural dyn ynamics 12 State vector: Coordinates of the reduced system Stiffness reduction Damage indexes: Newmark explicit integration method Process model:

Application to to str tructural dyn ynamics 13 Stiffness matrix: coupling with any FE Abaqus: use of keywords ELEMENT • commercial code MATRIX OUTPUT applied to a fictiotious substructure the parametric formulation • of the stiffness matrix is not required Measurement model: POMs

Application to to str tructural dyn ynamics 14 Previous works: Bruggi, Mar ariani, Optimization of sensor placement to detect damage in flexible plates (Engineering Optimization, • 2012) Mar ariani, Bruggi, Cai Caimmi, Ben endiscioli li, Optimal placement of MEMS sensors for damage detection in flexible plates • (Structural Longevity, 2014)

Res esults 15 Elements S4R (Mindlin-Reissner)

Res esults: : Ben enchmark Anal alysis 16 The damage identification method is evaluated in function of the following features: order of the reduced system • initial conditions • • measurement noise process noise • number of observations • mesh refinement • POMs convergence •

Res esults 17 Mod odel Or Order Red eduction – Undamaged vs vs Dam amaged Cas ase Displacement Error - Node 2 Displacement - Node 2 Displacement - Node 2 Displacement Error - Node 2

Res esults 18 Dam amage par arameters es estimation Num umber of f POMs re reta tain ined

Res esults 19 Dam amage par arameters es estimation Ini Initi tial l condit itions

Res esults 20 Dam amage par arameters es estimation Mea easurement no nois ise Maximum amplitude

Res esults 21 Dam amage par arameters es estimation Pro roces ess no nois ise

Res esults 22 Dam amage par arameters es estimation Num umber of f obs bserved deg egree ees of f free freedom

Res esults 23 Dam amage par arameters es estimation Mes esh re refin finement 3x3 nodes 11x11 nodes: 2182 d.o.f. Spee eed-up up 2 POMs: 10 d.o.f. ≃ 400 3 POMs: 13 d.o.f. ≃ 250

Res esults 24 Dam amage Id Iden entification Non Non-sta tatio ionary cas ase variation of stiffness

Con onclusions 25 We have introduced several innovations with respect to previous works: Identification and estimation of damage indexes related to the reduction of • stiffness • Localization of damage Coupling with commercial FE code • We assessed the effects on the algorithmic performance of: Number of POMs retained  Initial conditions  Measurement noise  Process noise  Number of observations  Mesh refinement  On-line variation of the structural health 