- Critical parts are tested (PS) - The remaining parts are simulated (NS).

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2 Hybrid simulatio of a structural system Loads (Disp.) simulatig deck motio due to earthquake K V T B CBdV Deck modeled aalytically by F.E.M. Piers tested physically - Critical parts are tested (PS) - The remaiig parts are simulated (NS). 1) Physical Substrucutre should move cotiuously, with a costat timescale betwee the seismic evet ad the testig duratio; 2) I the curret applicatio, a refied FE model is itroduced for the Numerical Substructure.

3 Servo-Hydraulic Actuators The implemetatio scheme developed at JRC, Ispra Force trasducers Displacemet Trasducers Referece frame A digital implemetatio of PsD cotrol scheme: Dt s = 1 or 2 msec Substructurig capacity Measuremet of reactio force R... dt Imposed displacemet d.. M d Cd. R d MIa g t Accelerogram x g Numerical itegratio Numerical model

4 The classical moolithic time itegratio:.. M d Cd. Whe the NS is refied, a moolithic itegratio step caot be complete withi the cotrol samplig time. R d MIa g è 1) Use of Partitioed itegratio ad differet time steps betwee the Ns ad the PS. è 2) Explicit scheme is preferable for PS (without iteratio), whereas implicit scheme is suitable to the NS (stability cosideratio). t K. Park, C. Felippa, 1983; T. Hughes, W. Liu, 1978; W. Liu, T. Belytschko, 1982

5 A Multi-time-step Explicit-implicit Method (Gravouil ad Combescure 2001) Steps are sequetial. Steps A 1 ad B 1 are cocurret.

6 Time steppig schemes applied to liear problems ca be recast i a recursive form as where X is the state vector, A is the amplificatio matrix ad L is the load vector. For PM method, X reads with X AX L 1 X ( X, X, X ) A A B T 1 X ( u, u, u, u ), X ( u, u, u ) A A A A A T B B B B T, free X has 11 variables for a partitioed SDoF system. Boelli et.al, 2011

7 Photograph courtesy of JRC Isolator: the physical substructure. The emulated structure. Bridge deck: the umerical substructure.

8 Problems related to umerical dissipatio I FE aalysis, the higher modes are ofte spurious due to the artifacts of the umerical discretizatio process. These spurious modes could be excited due to experimetal errors durig hybrid simulatio. T. Hughes, 1987; Bursi, Shig, 1997; Hulbert, Chag, 1993/1996; C. Farhat, F.-X.Roux,

9 A ovel oe-step four-stage form of the G-alpha method Predictor: Equilibrium equatio (Im.): Corrector: u u tu t u t a 2 f 2 1 D D D ( ) 1 m 2 1 m f u u Dt u Dt(1 ) a 1 1 m m (Ex.): Mu Cu Ku F ( t ) ext Mu Cu Ku F ( t ) ext u u t u u u t u 2 f f 1 D 1; 1 D 1 1 m 1 m The equilibrium equatio is idetical to those of Newmark schemes. The itegratio parameters are chose as those of the origial implicit/explicit G-alpha method, that is, r as the spectral radius at the high frequecy limit for the implicit method ad r b as the spectral radius at the bifurcatio limit for the explicit method. He, 2008; Bursi, He 2012

10 Represetative Numerical Simulatio (I) 0.04 PM m2 u2 (rad) 0.00 m1 la ma u1 Aalytical Part mb time (s) lb Experimetal Part 0.04 PM-alpha (rad) 0.00 Partitioed three-dof system f 1 =0.24 Hz, f 2 =1.6 Hz, f 3 =35 Hz time (s) Rotatioal respose i free vibratio for the Three-DoF system (Dt A =5ms, ss = 100): (a) the PM method; (b) the PM-alpha method, r = r b = 0.5

11 A three-actuator testig setup of the previous 3DOF model Simulatio cosiderig the cotrol errors of the actuators i udershootig type (Mosqueda et.al 2007) ad measured errors of load cells as white oise (0.06%) Respose of partitioed three-dof system (Texp./Ts.e.=50; Dt A =2ms ; Dt B =0.02ms ;ss=100; )

12 This work presets partitioed time itegratio methods for solvig dyamic problems of hybrid structural systems, typical of pseudo-dyamic testig with substructurig or hybrid simulatio, as decomposed by computatioal/physical cosideratios. The methods implemet the iterfield parallel aalysis procedure suitable to maitai a costat time scale betwee the umerical ad physical substructures, by usig the implicit ad explicit schemes of differet time steps. The characteristics of the proposed time schemes are aalyzed with respect to their covergece (stability ad accuracy). High-frequecy umerical dissipatio based o a ew form of the G-alpha methods is itroduced, which is show to be helpful to hybrid simulatio with the presece of both the cotrol errors ad the measuremet errors.

13 Thak you for your attetio! Ackowledgemets: Prof. O.S. UNITN Dr. P. JRC, Ispra