Time-multi-scale parameter identification of models describing - PowerPoint PPT Presentation

SMIP workshop Time-multi-scale 
 parameter identification 
 of models describing 
 material fatigue Guillaume PUEL 
 Denis AUBRY 
 Laboratoire MSSMat Ecole Centrale Paris / CNRS UMR 8579

Context • Rotor blade «Combined Cycle Fatigue» (CCF) Loading on the blades: � • aerodynamic forces 
 ↔ high frequency � • centrifugal force 
 ↔ low frequency Ratio LF/HF: ξ ∼ 10 − 4 EU Project 
 (2006-2011) PREdictive MEthods for Combined CYcle fatigue in gas turbines 
 G. Puel - SMIP workshop - 2 oct. 2014 - 2

Context • Necessity of a specific method � • complex interaction between low- and high- frequency loads, and dynamic effects 
 ➡ classical cumulative laws can be inadequate � • time-dependent simulations required 
 ➡ need to efficiently describe the `slow’ evolution of a structure withstanding `fast’ loading cycles � • use of a specific method to reduce the associated huge computation cost G. Puel - SMIP workshop - 2 oct. 2014 - 3

Context • Process: periodic time homogenization method 
 [Guennouni & Aubry 1986][Guennouni 1988] � • separation of two time scales � • asymptotic expansion 
 ➡ time-homogenized problem 
 solved on slow time steps only � • Similarities 
 with periodic space homogenization techniques 
 [Bensoussan et al. 1978, Sanchez-Palencia 1980, ...] G. Puel - SMIP workshop - 2 oct. 2014 - 4

Outline • Periodic time homogenization � • basic ingredients � • simple case of study � • towards industrial problems � • Time-multi-scale parameter identification � • Prospects G. Puel - SMIP workshop - 2 oct. 2014 - 5

Basic ingredients • Two independent time scales � τ = t • fast time scale � ξ = t ξ τ � 1 • slow time scale � t • Total differentiation rule: for � α ( t, τ ) d t α = ∂ t α + 1 ξ ∂ τ α � • time derivative w.r.t. slow time: � ∂ t α = ˙ α • time derivative w.r.t. fast time: ∂ τ α = α � G. Puel - SMIP workshop - 2 oct. 2014 - 6

Basic ingredients • Quasi-periodicity assumption � � ⇥ t, τ + 1 α ( t, τ ) = α ∀ t � F • periodicity w.r.t. the fast period � ξ /F • Fast-time average � 1 � F < α > ( t ) = F α ( t, τ ) d τ � 0 α ( t, τ ) • quasi-periodicity: < α > ( t ) < α � > = 0 τ + 1 F τ G. Puel - SMIP workshop - 2 oct. 2014 - 7

Basic ingredients • Time homogenization � • directly compute time-homogenized quantities � α ( t, τ ) � < α > ( t ) � � t t ξ = t • using asymptotic expansions w.r.t. τ � 1 ε p ( x , t, τ ) = ε p 0 ( x , t, τ ) + ξε p 1 ( x , t, τ ) + O ( ξ 2 ) ... G. Puel - SMIP workshop - 2 oct. 2014 - 8

Outline • Periodic time homogenization � • basic ingredients � • simple case of study � • towards industrial problems � • Time-multi-scale parameter identification � • Prospects G. Puel - SMIP workshop - 2 oct. 2014 - 9

Simple case: tensile test on a bar 1D • Description � x = 0 x = L fast • cylindrical bar � slow f s ( t, τ ) • two-frequency tensile load 
 0.129Hz / 1290Hz = 1 st mode 
 amplitudes ratio = 1/4 � • Material: titanium alloy � • viscoplastic flow rule with two hardenings � • Rayleigh damping (prop. to stiffness) G. Puel - SMIP workshop - 2 oct. 2014 - 10

Simple case: tensile test on a bar • BCs: � • Reference model � σ | x = L = f s u | x =0 = 0 ∂ x σ + c K d t ∂ x σ = ρ d 2 • PDEs: � � t u ∀ t, τ σ = E ( ∂ x u − ε p ) • zero initial values � ⇥ n � | σ − X | − R − k d t p = � K ∀ x ∈ (0 , L ) + d t ε p = d t p sign( σ − X ) � ∀ t, τ d t ε p = a ( σ ) ⇔ d t X = 2 � 3 C d t ε p − γ 0 d t p X [Lemaître & 
 � d t R = b ( Q − R )d t p Chaboche 1990] G. Puel - SMIP workshop - 2 oct. 2014 - 11

zeroth-order 
 time-homogenized pb. d t ε p = a ( σ ) • Evolution equation: � • asymptotic expansion: � 1 � + ( ˙ ξ ε p ε p 0 + ε p ε p 1 + ε p � ) + O ( ξ 2 ) � ) + ξ ( ˙ � 0 1 2 = a ( σ 0 ) + ξσ 1 D σ a ( σ 0 ) + O ( ξ 2 ) � 1 � = 0 ε p < ε p 0 > = ε p ξ ε p • order -1: 
 only 0 ( x, t ) ⇒ ⇒ 0 0 ➡ viscoplasticity is a slow-evolving phenomenon � = a ( σ 0 ) ε p 0 + ε p < ε p � > = 0 • order 0: ˙ avrg. ⇒ 1 1 ε p � ⇥ ⇒ ˙ 0 = a ( σ 0 ) G. Puel - SMIP workshop - 2 oct. 2014 - 12

zeroth-order 
 time-homogenized pb. • Equilibrium equation: � ∂ x σ + c K d t ∂ x σ = ρ d 2 t u • asymptotic expansion: � t u = 1 0 + 1 d 2 ξ 2 u 00 u 0 0 + u 00 u 0 1 + u 00 ξ (2 ˙ 1 ) + (¨ u 0 + 2 ˙ 2 ) + O ( ξ ) � u 00 u 0 ( x, t ) only • order -2: � 0 = 0 ⇒ u 1 ( x, t ) only • order -1: u 00 ⇒ 1 = 0 ... ➡ the two time scales are not separable! G. Puel - SMIP workshop - 2 oct. 2014 - 13


 
 zeroth-order 
 time-homogenized pb. • Equilibrium equation: � ∂ x σ + c K d t ∂ x σ = ρ d 2 t u ρ L 2 F 2 = βξ 2 • assumption: with � β ≤ O (1) E • equivalent physical criterion: 
 L L p p β or βξ = = λ F/ ξ λ F with , wavelengths of propagating waves 
 λ F λ F/ ξ F with frequency and respectively F ξ β E 0 + ξ β E ➡ ρ d 2 1 ) + O ( ξ 2 ) L 2 F 2 u �� u � 0 + u �� t u = L 2 F 2 (2 ˙ G. Puel - SMIP workshop - 2 oct. 2014 - 14

zeroth-order 
 time-homogenized pb. α ( t, τ ) ε p � ⇥ < α > ( t ) ˙ 0 = a ( σ 0 ) τ + 1 F α = < α > + α ∗ τ σ 0 ( x, t, τ ) = < σ 0 > ( x, t ) + σ ∗ 0 ( x, t, τ ) Slow elasto-viscoplastic q.s. pb. Fast elastic damped dyn. pb. < α > α ∗ 0 + γ β E t 0 = ∂ x σ ⇤ F ∂ x σ ⇤ L 2 F 2 u ⇤ 00 ∂ x < σ 0 > = 0 0 0 < σ 0 > = E ( ∂ x < u 0 > − ε p σ ∗ 0 = E ∂ x u ∗ 0 ) 0 u ∗ 0 | x =0 = 0 < u 0 > | x =0 = 0 σ ∗ 0 | x = L = f ∗ < σ 0 > | x = L = < f s > s G. Puel - SMIP workshop - 2 oct. 2014 - 15

Validation of the method • Plastic strain: � [Puel & Aubry EJCM 2012] • comparison for the first slow cycle of ε p 0 homogenized x=0 reference x=L fast slow x=0 x=L impact of 
 inertial terms G. Puel - SMIP workshop - 2 oct. 2014 - 16

Validation of the method • Simulation for a 1-hour time interval � • 180 slow cycles / 1 800 000 fast cycles x=0 classical condition: � 36 000 000 time steps x=L x400 time-homogenized: � 0,05Hz / 500Hz 90 000 time steps 
 only G. Puel - SMIP workshop - 2 oct. 2014 - 17

Outline • Periodic time homogenization � • basic ingredients � • simple case of study � • towards industrial problems � • Time-multi-scale parameter identification � • Prospects G. Puel - SMIP workshop - 2 oct. 2014 - 18

Other material laws • Material fatigue simulation: CCF � • viscoplasticity + damage / dynamic case Lemaître-Chaboche Lemaître isotropic viscoplasticity model damage model x=0 x=0 x=L x=L fast slow G. Puel - SMIP workshop - 2 oct. 2014 - 19

Back to context • PREMECCY tests G. Puel - SMIP workshop - 2 oct. 2014 - 20

Back to context • Blade-shaped specimen: � [Puel & Aubry IJMCE 2014] • CCF testing: 0.14Hz / 1400Hz fast mode @ 1400Hz 
 slow (bending II) longitudinal plastic strain 100 slow cycles � 1 000 000 fast cycles G. Puel - SMIP workshop - 2 oct. 2014 - 21

Outline • Periodic time homogenization � • Time-multi-scale parameter identification � • generic framework � • application to a simple case � • Prospects G. Puel - SMIP workshop - 2 oct. 2014 - 22


 Generic nonlinear model • Time-dependent forward state equation: 
 F ( u ( t ) , v ( t ) , a ( t ) , p , t ) = 0 a ( t ) = d 2 u v ( t ) = d u d t ( t ) d t 2 ( t ) • = an ODE with initial conditions � � u (0) = U 0 v (0) = V 0 • Model with scalar parameters : � p ➡ forward state � u ( t ; p ) • of size N = number of DOFs (FE discretization) u G. Puel - SMIP workshop - 2 oct. 2014 - 23

Experimental data • Measurements: � • associated with some given points only � • assumption: 
 Au exp ( t ) with linear operator � A • Matching DOFs: � Au ( t ; p ) • Misfit function: � • discrepancy between model and experiments G. Puel - SMIP workshop - 2 oct. 2014 - 24

Inverse problem • Misfit function: � � T � J ( p ) = 1 | A ( u ( t ; p ) − u exp ( t )) | 2 d t + α 2 | p − p 0 | 2 � 2 0 ➡ Tikhonov 
 • Constrained minimization: � regularization • equivalent to the stationarity of the Lagrangian: � T L ( u , p , z ) = 1 | A ( u ( t ) − u exp ( t )) | 2 d t + α 2 | p − p 0 | 2 2 0 ➡ independent 
 � T F ( u ( t ) , v ( t ) , a ( t ) , p , t ) T z ( t ) d t variables − 0 − ( u (0) − U 0 ) T z (0) − ( v (0) − V 0 ) T d z d t (0) G. Puel - SMIP workshop - 2 oct. 2014 - 25

Adjoint state • Stationarity of with respect to : � L u ⇤ T δ u ( t ) T � ⇥ A T A ( u ( t ) � u exp ( t )) d t � 0 ⇤ T ⇥ T � � ⇥ u F δ u ( t ) + ⇥ v F δ v ( t ) + ⇥ a F δ a ( t ) z ( t ) d t � 0 � � δ u (0) T z (0) � δ v (0) T d z � d t (0) = 0 • = adjoint state � z • , and directional derivatives 
 � u F � v F � a F ( ↔ differentiated forward equation) G. Puel - SMIP workshop - 2 oct. 2014 - 26