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

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Time-multi-scale  parameter identification 

f models describing

material fatigue

Guillaume PUEL  Denis AUBRY  Laboratoire MSSMat Ecole Centrale Paris / CNRS UMR 8579 SMIP workshop

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G. Puel - SMIP workshop - 2 oct. 2014 -

Context

Rotor blade «Combined Cycle Fatigue» (CCF)

PREdictive MEthods for Combined CYcle fatigue in gas turbines 

Loading on the blades:

aerodynamic forces

↔ high frequency

centrifugal force

↔ low frequency Ratio LF/HF: ξ ∼ 10−4

2

EU Project  (2006-2011)

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G. Puel - SMIP workshop - 2 oct. 2014 -

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

f a structure withstanding `fast’ loading cycles
use of a specific method to reduce the associated

huge computation cost

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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, ...]

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Outline

Periodic time homogenization
basic ingredients
simple case of study
towards industrial problems
Time-multi-scale parameter identification
Prospects

5

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G. Puel - SMIP workshop - 2 oct. 2014 -

Basic ingredients

Two independent time scales
fast time scale
slow time scale
Total differentiation rule: for
time derivative w.r.t. slow time:
time derivative w.r.t. fast time:

ξ = t τ 1 α(t, τ)

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τ = t ξ t dtα = ∂tα + 1 ξ ∂τα ∂tα = ˙ α ∂τα = α

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G. Puel - SMIP workshop - 2 oct. 2014 -

Basic ingredients

Quasi-periodicity assumption
periodicity w.r.t. the fast period
Fast-time average
quasi-periodicity:

ξ/F

7

< α > (t) = F

1

F

α(t, τ) dτ α(t, τ) = α

t, τ + 1

F ⇥ ∀t

τ α(t, τ) < α > (t) τ + 1 F

< α > = 0

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G. Puel - SMIP workshop - 2 oct. 2014 -

Basic ingredients

Time homogenization
directly compute time-homogenized quantities
using asymptotic expansions w.r.t.

t t

α(t, τ) < α > (t)

ξ = t τ 1 εp(x, t, τ) = εp

0(x, t, τ) + ξεp 1(x, t, τ) + O(ξ2)

...

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G. Puel - SMIP workshop - 2 oct. 2014 -

Outline

Periodic time homogenization
basic ingredients
simple case of study
towards industrial problems
Time-multi-scale parameter identification
Prospects

9

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Simple case: tensile test on a bar

Description
cylindrical bar
two-frequency tensile load

0.129Hz / 1290Hz = 1st mode  amplitudes ratio = 1/4

Material: titanium alloy
viscoplastic flow rule with two hardenings
Rayleigh damping (prop. to stiffness)

slow fast 10

x = L x = 0 fs(t, τ) 1D

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G. Puel - SMIP workshop - 2 oct. 2014 -

Simple case: tensile test on a bar

Reference model
PDEs:
BCs:
zero initial values

11

∀x ∈ (0, L) ∀t, τ σ|x=L = fs u|x=0 = 0 ∀t, τ σ = E (∂xu − εp) dtp = |σ − X| − R − k K ⇥n

+

dtεp = dtp sign(σ − X) dtX = 2 3Cdtεp − γ0dtp X dtR = b(Q − R)dtp dtεp = a(σ)

⇔

∂xσ + cKdt∂xσ = ρd2

tu

[Lemaître &  Chaboche 1990]

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G. Puel - SMIP workshop - 2 oct. 2014 -
Evolution equation:
asymptotic expansion:
order -1: 
order 0:

zeroth-order  time-homogenized pb.

12

nly

εp

0(x, t)

⇒

< εp

0 > = εp

⇒

dtεp = a(σ) 1 ξ εp

= 0

˙ εp

0 + εp 1 = a(σ0)

1 ξ εp

+ ( ˙

εp

0 + εp 1 ) + ξ ( ˙

εp

1 + εp 2 ) + O(ξ2)

= a(σ0) + ξσ1Dσa(σ0) + O(ξ2) ➡ viscoplasticity is a slow-evolving phenomenon < εp

1 >= 0

avrg. ⇒

⇒ ˙

εp

0 =

a(σ0)

⇥

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G. Puel - SMIP workshop - 2 oct. 2014 -

zeroth-order  time-homogenized pb.

Equilibrium equation:
asymptotic expansion:
order -2:
order -1:

∂xσ + cKdt∂xσ = ρd2

tu

d2

tu = 1

ξ2 u00

0 + 1

ξ (2 ˙ u0

0 + u00 1) + (¨

u0 + 2 ˙ u0

1 + u00 2) + O(ξ)

u00

0 = 0

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u00

1 = 0

u0(x, t) only

⇒

u1(x, t) only

⇒

➡ the two time scales are not separable! ...

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zeroth-order  time-homogenized pb.

Equilibrium equation:
assumption: with
equivalent physical criterion:

    with , wavelengths of propagating waves  with frequency and respectively ∂xσ + cKdt∂xσ = ρd2

tu

ρL2F 2 E = βξ2 β ≤ O(1)

➡ ρd2

tu =

βE L2F 2 u

0 + ξ βE

L2F 2 (2 ˙ u

0 + u 1) + O(ξ2)

L λF = p βξ

F

14

λF/ξ λF L λF/ξ = p β F ξ

r

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G. Puel - SMIP workshop - 2 oct. 2014 -

zeroth-order  time-homogenized pb.

15

t

< α >

Slow elasto-viscoplastic q.s. pb. ∂x < σ0 > = 0 < σ0 > = E (∂x < u0 > − εp

0)

< u0 >|x=0 = 0 < σ0 >|x=L = < fs >

τ α(t, τ) < α > (t) τ + 1 F

˙ εp

0 =

a(σ0)

⇥ σ0(x, t, τ) = < σ0 > (x, t) + σ∗

0(x, t, τ)

α = < α > +α∗

α∗ Fast elastic damped dyn. pb. σ∗

0 = E∂xu∗

u∗

0 |x=0 = 0

σ∗

0 |x=L = f ∗ s

∂xσ⇤

0 + γ

F ∂xσ⇤

0 =

βE L2F 2 u⇤

00

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Validation of the method

Plastic strain:
comparison for the first slow cycle of εp

impact of   inertial terms

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x=0 reference homogenized x=L

[Puel & Aubry EJCM 2012]

slow fast

x=0 x=L

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Validation of the method

Simulation for a 1-hour time interval
180 slow cycles / 1 800 000 fast cycles

time-homogenized: 90 000 time steps 

nly

x400 classical condition: 36 000 000 time steps x=L x=0

17

0,05Hz / 500Hz

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Outline

Periodic time homogenization
basic ingredients
simple case of study
towards industrial problems
Time-multi-scale parameter identification
Prospects

18

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Other material laws

Material fatigue simulation: CCF
viscoplasticity + damage / dynamic case

x=L x=0 x=L x=0

Lemaître-Chaboche viscoplasticity model Lemaître isotropic damage model slow fast 19

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Back to context

PREMECCY tests

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G. Puel - SMIP workshop - 2 oct. 2014 -

Back to context

Blade-shaped specimen:
CCF testing: 0.14Hz / 1400Hz

slow fast 21

100 slow cycles 1 000 000 fast cycles longitudinal plastic strain mode @ 1400Hz  (bending II)

[Puel & Aubry IJMCE 2014]

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Outline

Periodic time homogenization
Time-multi-scale parameter identification
generic framework
application to a simple case
Prospects

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Generic nonlinear model

Time-dependent forward state equation:

 

= an ODE with initial conditions
Model with scalar parameters :
of size N = number of DOFs (FE discretization)

➡ forward state

p

F (u(t), v(t), a(t), p, t) = 0 u(0) = U0 u(t; p) v(t) = du dt (t) u

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v(0) = V0 a(t) = d2u dt2 (t)

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Experimental data

Measurements:
associated with some given points only
assumption:

with linear operator

Matching DOFs:
Misfit function:
discrepancy between model and experiments

A Auexp(t) Au(t; p)

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G. Puel - SMIP workshop - 2 oct. 2014 -

Inverse problem

Misfit function:
Constrained minimization:
equivalent to the stationarity of the Lagrangian:

➡ Tikhonov 

regularization

J (p) = 1 2 T |A(u(t; p) − uexp(t))|2 dt + α 2 |p − p0|2

➡independent 

variables

L(u, p, z) = 1 2 T |A(u(t) − uexp(t))|2 dt + α 2 |p − p0|2 − T F (u(t), v(t), a(t), p, t)T z(t) dt −(u(0) − U0)Tz(0) − (v(0) − V0)T dz dt (0)

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Adjoint state

Stationarity of with respect to :
= adjoint state
, and directional derivatives

(↔differentiated forward equation)

L u

⇤ T δu(t)T ATA(u(t) uexp(t)) ⇥ dt

⇤ T
⇥uFδu(t) + ⇥vFδv(t) + ⇥aFδa(t)

⇥T z(t) dt δu(0)Tz(0) δv(0)T dz dt (0) = 0 uF vF aF z

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Adjoint state

Adjoint state problem:
time-backward ODE

with two final conditions:

solved with a change in variables

⇥uFTz d dt

⇥vFTz

⇥ + d2 dt2

⇥aFTz

⇥ = ATA(u uexp) aFTz|t=T = 0 ⇥vFTz|t=T d dt

⇥aFTz

⇥

|t=T = 0

τ = T − t

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Optimality conditions

Misfit function’s gradient:
Solving the problem:
gradient-based

minimization  algorithm

optimal computational cost and accuracy

⇥pJ (p) = ⇥pL(u(t; p), p, z(t)) = α(p p0) T ⇥pFTz(t) dt

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u(t; pk) z(t) pJ (pk) ∆pk

1 ODE 1 ODE integrations

➡ ➡

P

➡

τ = T − t

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Outline

Periodic time homogenization
Time-multi-scale parameter identification
generic framework
application to a simple case
Prospects

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Synthetic data

Description:
cylindrical bar
two-frequency tensile load (0.05Hz / 500Hz)

amplitudes ratio = 1/4

quasistatic calculation
Material: steel wih Norton’s law
parameters:
‘Experimental’ data:

slow fast 30

(E, K, n) uexp(t) = usynth(L, t; E, K, n)

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misfit function: 1st choice

Homogenized misfit function:
Experimental data:
for the homogenized model:  
nly known on ‘macro’ time steps
definition of a homogenized experimental

quantity: for each

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tk tk J 0(E, K, n) = 1 2 T | < u0 > (L, t; E, K, n) − < uexp > (t)|2 dt < u0 > < u0 > < uexp > (tk) = F ξ tk+ ξ

F

tk

uexp(t) dt

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misfit function: 1st choice

Associated adjoint state:
solved on ‘macro’ time steps only
Misfit function gradient:
integrals computed on ‘macro’ time steps

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z0(T) = 0 ⇥pJ (p) =

T

⇥pFTz(t) dt dz0 dt (t) = (< u0 > (L, t; E, K, n) − < uexp > (t)) L

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misfit function: 2nd choice

‘Instantaneous’ misfit function:
using with
Evaluation:
should be at the fast time scale
really necessary?

(quasi-periodicity)

33

J 0∗(E, K, n) = 1 2 T | < u0 > (L, t; E, K, n) + f ∗

s (t/ξ)L

E − uexp(t)|2 dt

u∗

0(L, t, τ; E)

τ = t/ξ

τ

τ + ξ F α(t, τ) < α > (t)

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misfit function: 2nd choice

Associated adjoint state:
should be solved at the fast time scale
Time-homogenized version:
in order to use ‘macro’ time steps only

   

identical to the first choice!

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z0∗

0 (T) = 0

z0∗(T) = 0 dz0∗ dt (t) =

< u0 > (L, t; E, K, n) + f ∗

s (t/ξ)L

E − uexp(t) ⇥ L ˙ z0∗

0 (t) = (< u0 > (L, t; E, K, n) − < uexp > (t)) L

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Time-homogenized inverse problem

Solving the id. problem: (for the two choices)
using the general strategy presented before
identical adjoint states eventually:

adjoint state of homogenized inverse problem  =  homogenized adjoint state of original inv. pb.

similar to optimal control problems

with space periodic homogenization 

[Kesavan & Saint Jean Paulin 1997], [Mahadevan & Muthukumar 2009] 35

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Identification results

Parameter identification: (for the two choices)
no regularization, ~20 iterations for each optim.
exp. init. id.

E (GPa) 200 220 200 K (MPa) 100 110 97.6 n 10 11 10.2

36

exp. init. id.

E (GPa) 200 220 200 K (MPa) 100 110 97.9 n 10 11 10.1

J 0 J 0∗ 8 000 time steps for:

each ODE solution
each integral evaluation

8 000 time steps for:

each ODE solution
each integral evaluation
8 000 time steps for each ODE
800 000 time steps for each integral

➡ use of the ‘macro’ time steps instead

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Identification results

Parameter identification: (for the two choices)
no regularization, ~20 iterations

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Less-separated scales

What if a bigger ?
ex. with : need of an additional order

with order 1 verifying:

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ξ ξ = 1/5 ∂τu1(L, t, τ) L = ✓|fs(t, τ)| K ◆n∗ ∀t ∈ [0; T] u1(L, 0, 0) = 0 < u0 > (L, t) + u∗

0(L, t, τ) + u1(L, t, τ)

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Less-separated scales

Associated parameter identification
misfit function

     

additional adjoint state to be used

along with for the gradient estimates

39

J1(E, K, n) = 1 2 Z T

< u0 > (L, t; E, K, n) + f ∗

s (t/ξ)L

E +ξu1(L, t, t/ξ; E, K, n) − uexp(t)

2

dt z1(t) z0 ∗

0 (t)

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Identification results

Multi-order parameter identification:

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Summary

Time homog. for material fatigue simulation
reduced cost for high numbers of cycles
various possible material laws
complex loading history taken into account
Parameter identification of homog. models
based on a robust strategy
first encouraging methods

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Prospects

Fatigue life estimation:
predictions using the identified model

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

D0°

slow stress fast stress contour plots of  limit damage

42

Goodman diagrams  for CCF tests  (N cycles)

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Prospects

Use of a 3rd time scale:
to further speed up the calculations
‘meso’ time scale θ
periodicity w.r.t. fast time τ and ‘meso’ time θ

(CCF loading)

starting point: initial time-homogenized model

43

ξ = θ τ 1 η = t θ 1

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Prospects

Validation: for 3 time scales
comparison for a 1-day sim.

εp

2 time scales  ~200 000 time steps 3 time scales  ~500 time steps reference  ~109 time steps ~4 000 slow cycles  ~40 000 000 fast cycles

0.05Hz / 500Hz

44 slow fast

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Prospects

Towards predictive maintenance
for structures in operation
no periodicity assumption
use alternate frameworks 
e. g. stochastic homogenization ➡ in time
associated identification process

45

[Sab 1992]

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References

A. Devulder, D. Aubry and G. Puel : Two-time scale fatigue

modelling: application to damage. Computational Mechanics, 45:6, pp. 637–646, 2010.

G. Puel and D. Aubry : Material fatigue simulation using a

periodic time homogenization method. European Journal of Computational Mechanics, 21:3-6, pp. 312–324, 2012.

G. Puel and D. Aubry : Efficient fatigue simulation using

periodic homogenization with multiple time scales. International Journal for Multiscale Computational Engineering, 12:4, pp. 291–318, 2014.

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SLIDE 47

Time-multi-scale  parameter identification 

f models describing

Time-multi-scale parameter identification

material fatigue

Guillaume PUEL Denis AUBRY Laboratoire MSSMat Ecole Centrale Paris / CNRS UMR 8579 SMIP workshop

Context

Loading on the blades:

↔ high frequency

↔ low frequency Ratio LF/HF: ξ ∼ 10−4

Context

frequency loads, and dynamic effects

➡ classical cumulative laws can be inadequate

➡ need to efficiently describe the `slow’ evolution

huge computation cost

Context

➡ time-homogenized problem solved on slow time steps only

with periodic space homogenization techniques

Outline

Basic ingredients

ξ = t τ 1 α(t, τ)

τ = t ξ t dtα = ∂tα + 1 ξ ∂τα ∂tα = ˙ α ∂τα = α

Basic ingredients

ξ/F

< α > (t) = F

α(t, τ) dτ α(t, τ) = α

F ⇥ ∀t

< α > = 0

Basic ingredients

ξ = t τ 1 εp(x, t, τ) = εp

...

Outline

Simple case: tensile test on a bar

0.129Hz / 1290Hz = 1st mode amplitudes ratio = 1/4

x = L x = 0 fs(t, τ) 1D

Simple case: tensile test on a bar

∀x ∈ (0, L) ∀t, τ σ|x=L = fs u|x=0 = 0 ∀t, τ σ = E (∂xu − εp) dtp = |σ − X| − R − k K ⇥n

dtεp = dtp sign(σ − X) dtX = 2 3Cdtεp − γ0dtp X dtR = b(Q − R)dtp dtεp = a(σ)

⇔

∂xσ + cKdt∂xσ = ρd2

zeroth-order time-homogenized pb.

εp

⇒

< εp

⇒

dtεp = a(σ) 1 ξ εp

˙ εp

1 ξ εp

εp

εp

= a(σ0) + ξσ1Dσa(σ0) + O(ξ2) ➡ viscoplasticity is a slow-evolving phenomenon < εp

⇒ ˙

εp

⇥

zeroth-order time-homogenized pb.

∂xσ + cKdt∂xσ = ρd2

d2

ξ2 u00

ξ (2 ˙ u0

u0 + 2 ˙ u0

u00

u00

u0(x, t) only

⇒

u1(x, t) only

⇒

➡ the two time scales are not separable! ...

zeroth-order time-homogenized pb.

with , wavelengths of propagating waves with frequency and respectively ∂xσ + cKdt∂xσ = ρd2

ρL2F 2 E = βξ2 β ≤ O(1)

➡ ρd2

βE L2F 2 u

L2F 2 (2 ˙ u

L λF = p βξ

F

λF/ξ λF L λF/ξ = p β F ξ

zeroth-order time-homogenized pb.

< α >

Slow elasto-viscoplastic q.s. pb. ∂x < σ0 > = 0 < σ0 > = E (∂x < u0 > − εp

< u0 >|x=0 = 0 < σ0 >|x=L = < fs >

˙ εp

⇥ σ0(x, t, τ) = < σ0 > (x, t) + σ∗

α∗ Fast elastic damped dyn. pb. σ∗

Time-multi-scale  parameter identification 

Guillaume PUEL  Denis AUBRY  Laboratoire MSSMat Ecole Centrale Paris / CNRS UMR 8579 SMIP workshop

frequency loads, and dynamic effects 

➡ time-homogenized problem  solved on slow time steps only

with periodic space homogenization techniques 

0.129Hz / 1290Hz = 1st mode  amplitudes ratio = 1/4

zeroth-order  time-homogenized pb.

zeroth-order  time-homogenized pb.

zeroth-order  time-homogenized pb.

    with , wavelengths of propagating waves  with frequency and respectively ∂xσ + cKdt∂xσ = ρd2

zeroth-order  time-homogenized pb.

impact of   inertial terms

time-homogenized: 90 000 time steps 

100 slow cycles 1 000 000 fast cycles longitudinal plastic strain mode @ 1400Hz  (bending II)

➡ Tikhonov 

➡independent 

minimization  algorithm