Stéphane P.A. Bordas - Pierre Kerfriden

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Model order reduction

POD and Homogenisation

Stéphane P.A. Bordas - Pierre Kerfriden

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bordasS@cardiff.ac.uk, stephane.bordas@alum.northwestern.edu

RealTcut

Ways to reduce the models

• Homogenisa+on (FE^2, etc.) - Hierarchical

• Concurrent and hybrid (bridging domain, ARLEQUIN, etc.)

• Enrichment (PUFEM, XFEM, GFEM)

• Model reduc+on (algebraic)

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Reduction methods based on homogenisation

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RealTcut

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Coupling of macroscopic and microscopic levels The volume averaging theorem is postulated for:

1) Strain tensor:

2) Virtual work (Hill-Mandel condition):

3) Stress tensor:

Definition of an RVE

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RealTcut Hierarchical multi-scale approaches (FE^2)

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In softening regime:

•   Lack of scale separation

•   Macroscale mesh dependence The macroscopic constitutive law is not

required

Non-linear material behaviour can be simulated Microscale behaviour of material is monitored at each load step

Advantages and abilities: Drawbacks:

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RealTcut

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RealTcut

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RealTcut

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Details in Phil. Magazine, 2015, Akbari, Kerfriden, Bordas

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Reduction methods based on algebraic reduction

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RealTcut Illustration of the method of separated representation

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0 20

40 60 80

100

0 200 400 600 800 1000

−1

−0.5 0 0.5 1

0 20

40 60 80

100

0 200 400 600 800 1000

−1

−0.5 0 0.5 1

X X

C¹ = sin(0.01x)

C² = (x 500)³

↵¹ = e ^0.02^t

↵² = cos(p t)

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RealTcut Illustration of the method of separated representation

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C¹ = sin(0.01x)

C² = (x 500)³

↵¹ = e ^0.02^t

↵² = cos(p t)

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RealTcut Illustration of the method of separated representation

19 +

=

C¹ = sin(0.01x)

C² = (x 500)³

↵¹ = e ^0.02^t

↵² = cos(p t)

C¹↵¹ +C²↵²

Very rich approximations!

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(C_xⁱ, C_yⁱ)_i_2J_1,n_c_K = argminX

xi

X

yj

(u(x_i, y_j) u(x¯ _i, y_j))²

RealTcut Data compression: get the nose with the POD!

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50 100 150 200 250 300 350 400 450 500 50

100 150 200 250 300 350 400 450

50 100 150 200 250 300 350 400 450 500

50 100 150 200 250 300 350 400 450

nc = 1

¯

u(xi, yi) =

nc

X

i=1

Cⁱ_x(xi)Cⁱ_y(yi)

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xi

X

yj

(u(x_i, y_j) u(x¯ _i, y_j))²

RealTcut Data compression: get the nose with the POD!

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100 150 200 250 300 350 400 450

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100 150 200 250 300 350 400 450

50 100 150 200 250 300 350 400 450 500

50 100 150 200 250 300 350 400 450

nc = 1 nc = 2

¯

u(xi, yi) =

nc

X

i=1

Got the nose (rectangle, approximation of order 2 is enough)

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xi

X

yj

(u(x_i, y_j) u(x¯ _i, y_j))²

RealTcut Data compression: get the nose with the POD!

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100 150 200 250 300 350 400 450

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100 150 200 250 300 350 400 450

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50 100 150 200 250 300 350 400 450

nc = 1 nc = 2

nc = 5 nc = 10

nc = 20

¯

u(xi, yi) =

nc

X

i=1

Converges slowly locally (idem fracture) Got the nose (rectangle, approximation of order 2 is enough)

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RealTcut

a posteriori model order reduction. Idea: search for the solution as a linear combination of a set of pre-calculated representative solutions

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−5 0 5 10 15 20 25

−3

−2

−1 0 1 2 3 4 5 6

U_D

F_D1 F_D2

S¹

nS loads

−5 0 5 10 15 20 25

−2 0 2 4 6 8 10

−5 0 5 10 15 20 25

−2 0 2 4 6 8 10

−5 0 5 10 15 20 25

−2 0 2 4 6 8 10

Reduced basis: family of representative solutions

C¹

C²

C³

C = U¹ U² ... Uⁿ^C

(1) Solve FINE for n_S parameters (EXPENSIVE!)

S = S¹ S² ... Sⁿ^S

(3) Trunca+on

Solution

Coefficients Family of

representative solutions

Approximation of the solution in a space of small dimension (nc)

F

_Int

(U) + F

_Ext

= 0

Initial set of equations

(4) Galerkin orthogonality

S = U⌃V^T =

nS

X

k=1

⌃^k U^kV^kT

(⌃^k)_k_2J₁ _n_S_K

where in decreasing order

n_S solutions, sorted by relevance

(2) Singular value decomposi+on

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•

P. Kerfriden, P. Gosselet, S. Adhikari, and S. Bordas.

Bridging proper orthogonal decomposi2on methods and augmented Newton-Krylov

algorithms: an adap2ve model order reduc2on for highly nonlinear mechanical problems. Computer Methods in Applied Mechanics and Engineering, 200(5-8):850-866, 2011.

RealTcut Limitations: case of highly non-linear fracture mechanics pbs

−5 0 5 10 15 20 25

−2 0 2 4 6 8 10

−5 0 5 10 15 20 25

−2 0 2 4 6 8 10

−5 0 5 10 15 20 25

−2 0 2 4 6 8 10

Reduced Ritz basis

C¹

C²

C³

0 5 10 15 20 25 30 35 40 45 50

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Error

Time step

Maximum damage /20

−5 0 5 10 15 20 25

−3

−2

−1 0 1 2 3 4 5 6

F_D

U_D

24 This solution is not in the

snapshot !

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RealTcut Application to a parametric fracture problem

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RealTcut Application to a parametric fracture problem

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‣ The POD solu+on is not able to reproduce the

solu+on in the cracked area

‣ Due to lack of correla+on introduced by crack growth

‣ Leads to a local projec+on

error

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RealTcut Parametric / stochastic multiscale fracture mechanics

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➡

Reduced order modelling?

➡

Direct numerical simulation: efficient preconditioner?

➡

Adaptive coupling?

First realisation Second realisation

Highly correlated solution fields

Localisation of fracture, uncorrelated

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THE RETURN OF THE MONKEY!

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50 100 150 200 250 300 350 400 450 500

50 100 150 200 250 300 350 400 450

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What can we do to address this lack of separation of scales/reducibility?

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RealTcut How we got to this point...

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http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3672853/

http://orbilu.uni.lu/bitstream/10993/12454/2/presentationUSNCCM.pdf

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RealTcut Data compression: fracture

31 POD order 1

POD order 3

“Exact” solution

Snapshot POD (snapshot space is spanned by the ensemble of solutions at all time steps)

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RealTcut Partitioned POD/DDM

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RealTcut Reduced DDM-POD

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‣ Decompose the structure into subdomains

‣ Perform a reduc+on in the highly correlated region

‣ Couple the reduced to the non-

reduced region by a primal Schur

complement

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Order of the POD transforms

Cross-validation error estimate

RealTcut Choice of the reduced subdomains: local error estimation by “leave

one out cross-validation” (LOOCV)

•

Reduced subspaces are independent and we assume a snapshot is a priori available

‣

(1) Dimension of the local space for each subdomain?

‣

(2) Is a given subdomain is reducible?

•

(1) and (2) will be treated by cross-valida+on (e.g. W. J. Krzanowski. Cross- valida+on in principal component analysis. Biometrics, 43(3):575-584, 1987.)

‣

Training set: snapshot

‣

Valida2on set: set of addi+onal ﬁnescale solu+ons

‣

Independent training/valida+on avoids overﬁjng

‣

Cross valida+on emulates independence. Error calculated using the local reduced basis obtained by a snapshot POD transform of all the available snapshot solu+ons except the one corresponding to the value of the summa+on variable.

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NOTE: If the snapshot is not assumed a priori then

‣ Assess whether the snapshot contains suﬃcient informa+on, and generate addi+onal, suitable, data if required

‣ Most analysis (mostly by sta+s+cians) assume the snapshot is known a priori.

Recent review: Hervé Abdi and Lynne J. Williams. Principal component analysis.

Wiley Interdisciplinary Reviews: Computa+onal Sta+s+cs, 2(4):433{459, 2010.

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RealTcut

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Order of the POD transforms

Cross-validation error estimate

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RealTcut Performance: load angle 40 | 27 - 121 nodes

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Rela+ve error

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40 ^o 27 ^o

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RealTcut Performance: load angle 40 | 27 - 256 nodes

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40 ^o 27 ^o

•

Rela+ve error

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RealTcut Performance: load angle 40 | 27 - 441 nodes

•

Rela+ve error

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40 ^o 27 ^o

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RealTcut Performance: load angle 40 | 27 - 961 nodes

•

Rela+ve error

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40 ^o 27 ^o

•

Rela+ve error

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M A M

Institute of Mechanics

& Advanced Materials

I

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Applica+ons to surgical simula+on

with INRIA, France; Karol Miller, UWA.

   

RealTcut 

1

Interactive multiscale 

cutting simulations

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Surgical simulation (real time/interactivity)

Learning PlanningSim Assistance

Precision

RealTcut

iMAM

The ERC RealTcut project

‣ Reduce the problem size while controlling error in solving very large multiscale mechanics problems

complex microstructure

Courtecuisse et al. PBMB 2011 Discretise

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ERC Star+ng Grant - RealTcut - 2012-2016

‣ reduce the problem size but preserve relevant mechanical information, control the error

complex

microstructure

discretization

Real-2me/interac2vity for non-linear problems involving topological changes

appren+ssage planiﬁca+on

Sim

assistance

précision requise

complexité des calculs

RealT ^cut

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Interac2ve simula2on of cuEng in soF 2ssue

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Approach

Concrete objec2ve: compute the response of organs during surgical procedures (including cuts) in real +me (50-500 solu+ons per second)

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Two schools of thought

‣ constant +me

➡ accuracy oqen controlled visually only

‣ model reduc+on or “learning”

➡ scarce development for biomedical problems

➡ no results available for cujng

Proposed approach: maximize accuracy   for given computa+onal +me. Error control

A

4.30 A

10

⁷

2.6 ⇥

U Q

0 0.02 0.04 0.06 0.08 0.1

0 20 40 60 80 100 120 140

U Q

15%

3.4 U

Q

x

^c

W

^c

1%

25 U

[Courtecuisse et al., MICCAI, 2013]

Collabora+on INRIA

First implicit, interac2ve method   for cuEng with contact

Model

reduc2on

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Error control

• interac+vity

• space-+me discre+za+on?

• op+mize use of compute resources

Complex geometries from

medical images Topological changes & contact

Veriﬁca2on & Valida2on Four main diﬃcul+es

RoI MRI

RoI

segmenta2on

-

RdI Region of interest (RoI)

in té gr a+on e n t em ps, e rr eur

discre2za2on

adap

2vit é

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Model reduc2on

Advanced discre2za2on

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6 =

enriched zone

calculs oﬄine calculs online: interac+vité

ac+on de l’instrument généra2on solu+ons par+culières

tri

pré-opératoire

“mapping”

spéciﬁque pa+ent

~10^3 snapshots

POD

O(10) fonc2ons

espace réduit de pe+te

dimension

approxima+on POD globale

Local (FE) Local (FE)

~10^6 snapshots

enrichissement “pointe de coupe”

r u

calcul champs asympto+ques

représenta+on locale

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Results - Dr Hadrien Courtecuisse, PhD INRIA

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NUMERICAL SOLUTION GEOMETRICAL MODEL DISCRETISATION

MATERIAL MODELS Phenomenological Elasticity/Plasticity Crack growth law (Paris…)

Fracture energy

Maximum tensile strength Multi-scale

Debonding, Fibre pull-out Fibre breakage, interface fracture, grains, dislocations, MD, quantum…

…

A POSTERIORI ERROR CONTROL

EXPERIMENTS

Validation & parameter identification

Verification

CONVENTIONAL APPROACH

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NUMERICAL SOLUTION GEOMETRICAL

MODEL

DISCRETISATION

LEARN MATERIAL MODELS

which scales?

what models?

what parameters?

what scale transition?

what data is missing?

A POSTERIORI ERROR

REAL SYSTEM

DIGITAL TWIN OF THE SYSTEM

DATA INFORMATION

Strain

Structural Health

Cracks

Environment Conditions Scales of

interest

Crack growth rate

Worst load combination

Inspection

interval

Mission?

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a-priori knowledge

upscaling techniques

automatic model selection

data &

measurements

VISION

Scales++

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Machine Learning

data Models

General approach  No predefined model

fracture patterns in a composite panel

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Digital Twins…

⁵¹

Characterisation Monitoring

Multiscale models are unreliable

Quantitative predictions

?

Fracture/lack of scale separation

Learn better models

Measure Analyse & Learn

from data Improved model

Identify missing

data Validate

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10010110100011110100101010100010100100101 10101010101010101101111100011010100001110 1010000100000001110101111001010010111101 01110101011110100000011111000010000000111

10100001000000011101011110010100101111010101010101110100101010010010101010000001111010101010000000111010101010010100100101001001010110111111100011 1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100101011011111110001100010100

1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100101011011111110001100010100

10100001000000011101011110010100101111010101010101110100101010010010101010000001111010101010000000111010101010010100100101001001010110111111100011000101 1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100101011011111110001100 1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100101011011111110001100101000010000000111010111100101001011110101010101011101001010100100101010100000011110101010100000001110101010100101001001010010010101101111111

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10100001000000011101011110010100101111010101010101110100101010010010101010000001111010101010000000111010101010010100100101001001010110111111100011 1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100101011011111110001100010100

1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100101011011111110001100010100

10100001000000011101011110010100101111010101010101110100101010010010101010000001111010101010000000111010101010010100100101001001010110111111100011000101 1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100101011011111110001100 1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100101011011111110001100101000010000000111010111100101001011110101010101011101001010100100101010100000011110101010100000001110101010100101001001010010010101101111111

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10010110100011110100101010 10101010101010101101111100 10100001000000011101011110 01110101011110100000011111

1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100 1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100

1010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100 10100001000000011101011110010100101111010101010101110100101010010010101010000001111010101010000000111010101010010100100101001001010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100100

10100001000000011101011110010100101111010101010101110100101010010010101010000001111010101010000000111010101010010100100101001001010000100000001110101111001010010111101010101010111010010101001001010101000000111101010101000000011101010101001010010010100

• Experience every event that its ﬂying twin experiences

• Will revolu+onise cer+ﬁca+on, ﬂeet management and support (mirrors life of the “as-built” state)

• Will decrease weight

• no reliance on sta+s+cal distribu+on of material proper+es

• no reliance on heuris+c design methods

• less reliance on physical tes+ng (environment?)

• no assumed similitude between tes+ng and opera+onal condi+ons

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NUMERICAL SOLUTION GEOMETRICAL MODEL DISCRETISATION

MATERIAL MODELS Phenomenological Elasticity/Plasticity Crack growth law (Paris…)

Fracture energy

Maximum tensile strength Multi-scale

Debonding,Fibre pull-out Fibre breakage, interface fracture, grains, dislocations, MD, quantum…

…

A POSTERIORI ERROR CONTROL

EXPERIMENTS

Verification

CONVENTIONAL APPROACH

(57)

57

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NUMERICAL SOLUTION

IMAGE/MODEL DISCRETISATION

MATERIAL MODELS Phenomenological Neo-Hookean, Ogden, …

Multi-scale cutting, fracture,

???

Patient specific ???

A POSTERIORI ERROR CONTROL

EXPERIMENTS ???

Verification

Alex Bilger

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REAL PATIENT

DIGITAL TWIN OF THE PATIENT

DATA INFORMATION

Organ state

Health

Disease

Environment Conditions Scales of

interest

Disease evolution

“Inspection”int

erval

Fitness

Treatment

simulation

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Model

Data

Global error

Local errors

Multi-physics Multi-scale Data-enriched

> E _g

< E _g

Global single scale   model selection

h, p refinement

analytical enrichment experimental enrichment atomistics

meso-scopic continuum

a priori

a posteriori  real-time

Data-aware

mechanics

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Papers on fracture

http://orbilu.uni.lu/handle/10993/26421 http://orbilu.uni.lu/handle/10993/22289 http://orbilu.uni.lu/handle/10993/20721 http://orbilu.uni.lu/handle/10993/24170 http://orbilu.uni.lu/handle/10993/21427 http://orbilu.uni.lu/handle/10993/21295 http://orbilu.uni.lu/handle/10993/16323

http://orbilu.uni.lu/handle/10993/22420 http://orbilu.uni.lu/handle/10993/19535 http://orbilu.uni.lu/handle/10993/21330 http://orbilu.uni.lu/handle/10993/18262

http://orbilu.uni.lu/handle/10993/19509 http://orbilu.uni.lu/handle/10993/19371 http://orbilu.uni.lu/handle/10993/17536 http://orbilu.uni.lu/handle/10993/17647 http://orbilu.uni.lu/handle/10993/14135 http://orbilu.uni.lu/handle/10993/16842

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Papers on fracture

https://orbilu.uni.lu/bitstream/10993/22331/2/paper.pdf http://orbilu.uni.lu/handle/10993/25048

http://orbilu.uni.lu/handle/10993/20721 http://orbilu.uni.lu/handle/10993/22420 http://orbilu.uni.lu/handle/10993/19960 http://orbilu.uni.lu/handle/10993/12316 http://orbilu.uni.lu/handle/10993/15109 http://orbilu.uni.lu/handle/10993/14067 http://orbilu.uni.lu/handle/10993/13879 http://orbilu.uni.lu/handle/10993/13876 http://orbilu.uni.lu/handle/10993/12523 http://orbilu.uni.lu/handle/10993/10965 http://orbilu.uni.lu/handle/10993/21442 http://orbilu.uni.lu/handle/10993/12107 http://orbilu.uni.lu/handle/10993/12026 http://orbilu.uni.lu/handle/10993/12026 http://orbilu.uni.lu/handle/10993/12089 http://orbilu.uni.lu/handle/10993/12113 http://orbilu.uni.lu/handle/10993/12116 http://orbilu.uni.lu/handle/10993/21337 http://orbilu.uni.lu/handle/10993/15234 http://orbilu.uni.lu/handle/10993/19960

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Stéphane P.A. Bordas - Pierre Kerfriden

Model order reduction

POD and Homogenisation