S P . Ladevze ,D. Nron , S.Rodriguez and R.Scanff LMT (ENS - - PowerPoint PPT Presentation

P . Ladevèze ,D. Néron , S.Rodriguez and R.Scanff LMT (ENS Paris-Saclay / CNRS / Université Paris-Saclay)

A TIME-MULTISCALE MODEL ORDER REDUCTION METHOD IN NONLINEAR SOLID MECHANICS

S

coll G. Nahas , P .E Charbonnel CEA , IRSN coll U.Nackenhorst, A. Fau, M. Bhattacharyya, S.Alameddin , Hanover University

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Motivation

fatigue loadings with large number of cycles

Motivation

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fatigue loadings with large number of cycles

FINAL ROM

To compute time-dependent nonlinear problems (viscoplasticity + fatigue damage) initiation of a macrocrack

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Motivation

seismic loadings problems

Motivation

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seismic loadings

FINAL ROM

To compute time-dependent nonlinear problems (viscoplasticity + damage)

Worst earthquake : Chili 1960 —Richter scale:9.5 —duration : 10mn

damage evaluation

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Motivation

Our answer : two levels of complexity reduction ♦ A signal theory with two time scales ♦ The time-multiscale PGD

MOR methods : intrusive ➡ engineering diffusion➘➘➘➘ LATIN-PGD : version non intrusive

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—Very general: + —Performance: — —Applications (in progress) in SAMCEF (coll SIEMENS) and in CASTEM (coll CEA)

Another motivation

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Outline

• 1. A signal theory :a ROM for the loading
• 2. A new PGD approach : multiscale in time and non

intrusive

• 3. First illustrations
• 4. Conclusion-Prospects

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Outline

• 1. A signal theory :a ROM for the loading
• 2. A new PGD approach : multiscale in time and non

intrusive

• 3. First illustrations
• 4. Conclusion-Prospects

Characteristics:

A signal theory: a ROM-loading

Two time-scale approximation ( micro : harmonic funct.)

t : « macro » time τ : « micro » time

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Specific aim: minimum of time shape functions (wavelet theory: too much terms) sd(t) :

m

X

i=0

µ ϕR

i (t)cos2π τ

τi +ϕI

i(t)sin2π τ

τi ∂

|τ=t

τi = ∞

(τi, ϕR

i , ϕI i

| {z }

macro functions

| i ∈ 0,1,...,m)

τi < 1 4macro-time scale

(τi -mode(H))

N

≤ ( N = 3)

A signal theory: a ROM-loading

Two time-scale approximation (micro : harmonic funct.)

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sd(t) :

m

X

i=0

n(τ,τi)T S S S(t)Ai

|τ=t

n(τ,τi) :      cos2π τ τi sin2π τ τi     

S S S(t) = [ψ1,...,ψm](t)

FE basis

A =    aR

1

aI

1

. . . . . . aR

m

aI

m

  

A signal theory: a ROM-loading

Best approximation (micro : harmonic funct.)

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sm(t) : (τi, Ai | i ∈ 0,1,...,m) S S S(0,T )

m

sm(t) = arg min

s0

m2S

S S(0,T )

m

1 T ZT (sd(t)° s0

m(t))2dt

Computational method: « greedy » (micro : harmonic funct.)

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(sm+1 − sm) : (τ, A) n(τ,τ)T S S SA

1 τ = argmax

τ0 0 RT M°1R

Mi j = 1 T ZT ψiψj dt

New correction:

A signal theory: a ROM-loading

R j = FT[(sd − sm)ψj ]× 1 T

A = M−1R|τ

≈

≈ ≈

Error : 1/ (6,6xN2) for P1 1/ (165xN3) for P2

( N = 3)

≈ + strict separation of computed periods

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A signal theory: a ROM-loading

Im

• Convergence proof : data = finite sum of

modes (H)

Computational method: « greedy » (micro : harmonic funct.)

Other writing Theory extension

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sm(t) =

n

X

j=1 FE shape function

z }| { ψj (t) × (

m

X

i=0

θi

j (τ,τi)

)

|τ=t

| {z }

j−nodal contribution

A signal theory: a ROM-loading

(harmonic functions) harmonic function sum of τk-modes(H) arbitrary periodic function sum of τk-modes(P ) q

I * *

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A signal theory: a ROM-loading

τk −mode(P) =

q

X

j=1

Ψj (t) h a j

Rh j R(τ,τk)+ a j J h j J (τ,τj )

i

h j

R

h j

J

• dd

even

τk/2

X

−τk/2

(h j

R)2dτ = τk/2

X

−τk/2

(h j

J )2dτ = τk/2

⎨

h j

R

: mother periodic function

— — — — — — —

I I I

Theory extension ( micro: arbitrary periodic funct.)

k

Theory extension ( micro: arbitrary periodic funct.)

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A signal theory: a ROM-loading

τi < 1 4macro-time scale

≤ ( N = 3)

N

τ — mode (P) =

𝑜 :

—

hR

n(τ,τ)T S S SA

hI

A signal theory: a ROM-loading Illustration 1

Data

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A signal theory: a ROM-loading Illustration 1

ROM- loading with P2 macro-elements

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number of components : 8 modes(H) accuracy : 5 10—2

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Macro functions

Modes

A signal theory: a ROM-loading Illustration 1

. . . . .

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A signal theory: a ROM-loading Illustration 1

classical time discretisation :15 000 dofs for 8 modes : 690

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A signal theory: a ROM-loading Illustration 1

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A signal theory: a ROM-loading Illustration 1

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A signal theory: a ROM-loading Illustration 2

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A signal theory: a ROM-loading Illustration 2

classical time discretisation :15 000 dofs for 8 modes : 720

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A signal theory: a ROM-loading Illustration 2

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A signal theory: a ROM-loading Illustration 2

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Outline

• 1. A signal theory :a ROM for the loading
• 2. A new PGD approach : multiscale in time and non

intrusive

• 3. First illustrations
• 4. Conclusion-Prospects

Discretized time-space domain [0, T] × Ω + fields- approximation Fundamental quantities

29

Reference discretized problem to be solved over [0, T] × Ω

FE-framework

—nodal points :

xj j ∈ 1...n

—time discretization : or 𝑢i ∈ I

ti i ∈ 1...m

—nodal « displacement » vector : u(t) t ∈ I —nodal « force » vector : F(t) t ∈ I Time T

Ud (∂1Ω) Fd (∂2Ω) f d (Ω)

Work : F(t)T u(t) t ∈ I

Equilibrium equation Constitutive relation

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Reference discretized problem to be solved over [0, T] × Ω

Formulation FE-software

∀t ∈ I F(t) = Ĥ (t, u(τ) τ ≤ t ) ∈ I F(t) = Fd(t,u(t)) (given) Time T

Ud (∂1Ω) Fd (∂2Ω) f d (Ω)

∀ t

Find (u(t), F(t)) t ∈ I such that: with Ĥ (t, u(τ) τ ≤ t ) = ∑ Ĥe(t, ue(τ) τ ≤ t

e ∈ E

e : element

Ad Γ

quasi-static

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Reference discretized problem to be solved over [0, T] × Ω

Global re-formulation

Sexact

f

𝒇

= {F(t) ; t ∈ I} = {u(t) ; t ∈ I}

Hypothesis Ad :linear

• Ad

Ad Γ

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Reference problem to be solved over [0, T] × Ω

Data : complex loading history (fd,Fd,Ud)

• ver [0,T ]

Data-ROM:

Micro-harmonic functions

F d(t) =

r

X

`=1

∞`(t,ø)×F `

∞`(t,ø) = √

m

X

k=1

øk −mode(P) !

`

H

ROM in nonlinear Solid Mechanics

Idea: iterative process (time/space separation)

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Succession of linear global problems

• ver [0,T]xΩ (parameters)

SOLVER ROM for each linear problem

• ver [0,T]xΩ (parameters)

RB, POD, PGD FINAL ROM

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Classical step-by-step methods LATIN method LATIN method: designed for the PGD ( Ladeveze 1985) LATIN method: mature computational tool book [Springer NY 1999] + many works

The solver LATIN

• Ad

Search directions

The solver LATIN

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Principe :iterative and alternative scheme

Ad Γ

··· − → sn ∈ Ad − →

local stage

• ˆ

sn+1/2 ∈ Γ − →

linear stage

• sn+1 ∈ Ad
• iteration n +1

− → ˆ sn+3/2 − → ···

Linear stage Local stage

Iteration n+1

sn

sn+1

ˆ sn+1/2

Sexact

f

𝒇

= {F(t) ; t ∈ I} = {u(t) ; t ∈ I}

search directions search directions

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Linear stage at Iteration n+1: ➙ Sex

sn+1

ˆ sn+1/2 ˆ sn+1/2

sn+1

• admissibility

conditions

Ad

Search direction

Problem to be solved

Linear

The solver LATIN

Sexact

H H H−(un+1 − ˆ un+1/2) = F n+1 − ˆ F n+1/2

𝒇 = {u(t) ; t ∈ I}

= {F(t) ; t ∈ I}

f

• Ad

Ad Γ

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Local stage at Iteration n+1 : ➙ Sex

ˆ sn+1/2 ˆ sn+1/2

state evolution laws

Γ

Search direction

Problem to be solved

Local in space Nonlinear

sn sn

very suitable for parallel computing!

f

The solver LATIN

Sexact

H H H+(ˆ un+1/2 − un)+(ˆ F n+1/2 −F n) = 0

= {F(t) ; t ∈ I} = {u(t) ; t ∈ I}

𝒇

• Ad

Ad Γ

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Initialisation

• Ad

Γ Ad

solution of the problem

The solver LATIN

elastic calculation (in dynamics) S0

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Error indicator

• Ad

Γ Ad

solution of the problem

sn

sn+1

ˆ sn+1/2

in+1 in+1 = sn+1 ⇥ ˆ sn+1/2 1

2(sn+1 + ˆ

sn+1/2)

The solver LATIN

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A word about convergence

• Ad

Γ Ad

solution of the problem

sn

sn+1

ˆ sn+1/2

The solver LATIN

[s, s]t = Z t Z

Ω

f · ˙ epdΩdt

Dissipation bilinear form: Theorem

( Ladeveze 1999)

Convergence of the LATIN method if

• material operator : monotone
• search directions : H+ = H- positive definite,

,symmetric

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Updating of the given force

The solver LATIN

Fd(t,un(t)) : given

Linear stage at Iteration n+1: ➙

ˆ sn+1/2

sn+1

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Reinforced concrete structure Seismic excita+on: Imposed displacement at only one side.

Illustration

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Illustration

Approximation with 7 modes (H)-(P1 macro element-error 8%)

Unilateral damage model (Vassaux et al EFM 2015)

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Illustration

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Comeback: Linear stage at Iteration n +1: find

sn+1

Problem defined

• ver [0,T]×Ω

Find

=

Time-multiscale PGD construction at iteration n +1

s(t) = (un+1,F n+1) ∈ S S S(0,T )

F n+1(t) = F d(t) t ∈ I I I

H H H−(un+1 − ˆ un+1/2) = F n+1 − ˆ F n+1/2

f

𝒇

• Ad

sn+1

ˆ sn+1/2

= {u(t) ; t ∈ I} = {F(t) ; t ∈ I}

• ver [0,T]×Ω

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Corrections Find

∆u ≡ un+1 − un

∆F = F n+1 −F n = 0

H H H−∆u = Rd = (F d − ˆ F n+1/2)−H H H−(un − ˆ un+1/2)

∆u(t) ∈ R R Rn t ∈ I I I

Time-multiscale PGD construction at iteration n+1

47

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Point1 : The PGD-global Residual

∆u(t) ∈ R R Rn t ∈ I I I

∆u = argmin

∆u0

ZT (H H H°∆u0 °Rd)T (H H H°)°1(H H H°∆u0 °Rd)dt

Find

Rd : known

Iteration n+1

Time-multiscale PGD construction at iteration n +1

classical PGD = λ(t) x G

∆uk

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Illustration

Vector norm

49

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Point 2 : Time-multiscale PGD

.

__

Step 1: Rd —decomposition

Rd :

r

X

k=1

Rk

d

Hypothesis : Period τk = given loading period τk

∆u :

r

X

k=1

∆uk

Iteration n+1

Time-multiscale PGD construction at iteration n +1

x Z k

—

—

Rdk : τk — mode (P)

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Extension of the signal theory

Rk

d = nk(τ,τk)SAk ×Z k

with

nk(τ,τk) = ∑hk

R(τ,τk) (even)

hk

I (τ,τk) (odd)

∏

Zτk/2

−τk/2

(hk

R)2dτ =

Zτk/2

−τk/2

(hk

I )2dτ = τk

Time-multiscale PGD construction at iteration n +1

— — — —

/2

Unknowns : Z k ,hR , hI ,A

k k k

ℎI (

) = 0

τk/2

k

Point 2 : Time-multiscale PGD

Step 1: Rd —decomposition

51

Rk

d = nk(τ,τk)SAk ×Z k

Time-multiscale PGD construction at iteration n +1

— — —

Rk

d = argmin Rk

d

1 T ZT (Rd °Rk

d 0)·(Rd °Rk d 0)dt

Unknowns : Z k ,hR , hI ,A

k k k

Point 2 : Time-multiscale PGD

Step 1: Rd —decomposition

Rk

d = nk(τ,τk)SAk ×Z k

Z k

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Time-multiscale PGD construction at iteration n +1

(Ak,hk

R,hk I ) = arg max A,hR,hI

AT 〈ST n RkT

d 〉〈Rk dnT S〉A

AT MA

〈•〉 = Zτk/2

−τk/2

• dτ

—

Rd / ⟨ ⟩

AkT𝕅 Ak

ƒ( ) = ƒ(t + )

Σ

τ τ

j j = 1

m

∼ ∼ ∼ ∼ ∼ ∼ ∼

Point 2 : Time-multiscale PGD

Step 1: Rd —decomposition Rk

d = nk(τ,τk)SAk ×Z k

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Illustration

Error : 4,8%

Approximated Residual function Vector Vector

increases a little bit the number of iterations

Illustration

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Macro-amplitudes Residual modes

55

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Point 2 : Time-multiscale PGD

.

__

Iteration n+1

Step 2 : Two time-scales PGD computation of ∆uk

Time-multiscale PGD construction at iteration n +1 Rd

k

— — —

= τk — mode (P) x gk with

∆uk

Unknowns : g k , A

k k k

hR , hI

56

15

Point 2 : Time-multiscale PGD

.

__

Step 2 :Two time-scales PGD computation of

Iteration n+1

Find

∆uk

∆uk = argmin

∆u0k

ZT dt[H H H°∆u0k °Rd]T H H H°°1[H H H°∆u0k °Rd]

∆uk

Time-multiscale PGD construction at iteration n +1

k

57

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Principle : alternative minimization on time functions and space functions (greedy calculation) time-problem:(A ) space -problem: standard FE problem ( g k)

Iteration n+1

Point 2 : Time-multiscale PGD

∆uk

Step 2 :Two time-scales PGD computation of

Time-multiscale PGD construction at iteration n +1

k

—

Final step :

∆uk

=

∆uk

Σ

k=1

m

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Illustration

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Outline

• 1. A signal theory :a ROM for the loading
• 2. A new PGD approach : multiscale in time and non

intrusive

• 3. First illustrations
• 4. Conclusion-Prospects

60

Reinforced concrete structure Seismic excita+on: Imposed displacement at only one side.

Seismic illustration

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Approximation with 7 modes (H)

Seismic illustration

Unilateral damage model (Vassaux et al EFM 2015)

62

Illustration

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Seismic illustration

Space dofs : 3 087 Space integration points : 34 792 Classical time discretisation : 15 000 dofs with the signal theory : 885

Research FE software (Rodriguez-Neron 2018)

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Seismic illustration

65

Seismic illustration

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Seismic illustration

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Fatigue illustrations

level of loading

Low-cycle fatigue High-cycle fatigue Very-high-cycle fatigue ≈ 105 ≈108 yield stress

macro-plasticity micro-plasticity

Very-low-cycle fatigue ≈ 102

number of cycles

Fatigue computation (quasi-periodic loading) : vast literature

sd(t) =

m

X

i=0

n(τ,τi)S S S(t)Ai

|τ=t

Fd

1 period

68

Cycle jumping method —Cailletaud 1986, Chaboche 1986, Lesne-Savalle 1989, Hayburst 1994, Lemaitre-Doghri 1994,Van Paepegem et al 2001, Karlsson 2006 , Burlon et al 2014, Saanouni 2015,… Time- homogenization —Guennouni - Aubry 1986, Oskay-Fish 2004,Devulder et al 2010, Haoula-Doghri 2015,… Time -multiscale LATIN-PGD (periodic loadings) —Cognard-Ladeveze IJP 1993, Ladeveze1996, Cognard et al AIS1999, Maitournan et al CRAS 2002, Comte et al CRAS 2006 ,Ammar et al 2017 ,Bhattacharyya et al CM 2018 , CMAME 2018, Alameddin et al EJM 2019, …

Fatigue illustrations

69

15

Numerical test-Case1 : low cycle fatigue

level of loading

Low-cycle fatigue High-cycle fatigue Very-high-cycle fatigue ≈ 105 ≈108 yield stress

macro-plasticity micro-plasticity

Very-low-cycle fatigue ≈ 102

number of cycles

Fatigue illustrations

Academic example : 3D -plate with hole

70

Viscoplasticity(Chaboche law) + unilateral fatigue damage (Lemaitre)

Alameddin et al 2018

Transient zone

71

Time-multiscale PGD construction at iteration n +1

time T0 T

Standard PGD Time-multiscale PGD: semi-incremental Macro amplitudes: linear

• n each sub-interval

T2 T1 T3

Computation scenario

72

The loading (fatigue)

Training stage

Nodal cycle 1 Nodal cycle 0 Nodal cycle 2

Time-multiscale PGD construction at iteration n +1

Macrotime element 1 Macrotime element 2

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Low cycle fatigue

number of cycles :738 715 semi-incremental approach accuracy :10—5 (86 time elements) macrotime : piecewise linear

74

Low cycle fatigue

number of computed cycles :86 (738 715) number of PGD modes / per cycle : much less than10

Low cycle fatigue

75

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Outline

• 1. A signal theory :a ROM for the loading
• 2. A new PGD approach : multiscale in time and non

intrusive

• 3. First illustrations
• 4. Conclusion-Prospects

Conclusion-Prospects

77

Associated challenging questions family of loadings with parameters (random)

sd(t) =

m

X

i=0

n(τ,τi)S S S(t)Ai

|τ=t

Fd

x Z i

Complexity reduction ? Engineering virtual charts ?

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