An application of model order reduction to computational

IntroductionModel order reduction in Computational Homogenisation

Conclusion

An application of model order reduction tocomputational homogenisation

O. Goury1, P. Kerfriden1, S. P. A. Bordas1

1Institute of Mechanics and Advance Materials, CardiffUniversity, Wales, United Kingdom

USNCCM Institute of Mechanics and Advanced Materials


Conclusion

Outline

1 IntroductionHeterogeneous materialsComputational Homogenisation

2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults

3 Conclusion



Conclusion

Heterogeneous materialsComputational Homogenisation

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3 Conclusion



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3 Conclusion



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Heterogeneous materials

Many natural or engineered materials are heterogeneous

Homogeneous at the macroscopic length scaleHeterogeneous at the microscopic length scale



Conclusion


Heterogeneous materials

Need to model the macro-structure while taking themicro-structures into account

=⇒ better understanding of material behaviour, design, etc..

Two choices:Direct numerical simulation: brute force!Multiscale methods: when modelling a non-linear materials=⇒ Computational Homogenisation



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3 Conclusion



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Semi-concurrent Computational Homogenisation(FE2, ...)

Effective strain

tensor

Macrostructure

Effective strain

tensorEffective

Stress

Effective

Stress



Conclusion


Problem

For non-linear materials: Have to solve a RVE boundaryvalue problem at each point of the macro-mesh where it isneeded. Still expensive!Need parallel programming



Conclusion

Proper Orthogonal Decomposition (POD)System approximationResults

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3 Conclusion



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3 Conclusion



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Strategy

Use model order reduction to make the solving of the RVEboundary value problems computationally achievableLinear displacement:

εM(t) =

(εxx (t) εxy (t)εxy (t) εyy (t)

)u(t) = εM(t)(x− x) + u with u|Γ = 0

Fluctuation u approximated by: u ≈∑

i ciαi



Conclusion


Projection-based model order reduction

The RVE problem can be written:

Fint(u(εM(t)), εM(t))︸︷︷︸Non-linear

+ Fext(εM(t)) = 0 (1)

We are interested in the solution u(εM) for many differentvalues of εM(t ∈ [0,T ]) ≡ εxx , εxy , εyy .

Projection-based model order reduction assumption:

Solutions u(εM) for different parameters εM are contained in aspace of small dimension span((ci)i∈J1,ncK)



Conclusion


Proper Orthogonal Decomposition (POD)

How to choose the basis [c1,c2,c3, ...] = C ?

“Offline“ Stage: Solve the RVE problem for a certainnumber of chosen values of εM

We obtain a base of solutions (the snapshot):(u1,u2, ...,unS ) = S

, , , ...



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, , , ...



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, , , ...



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Find the basis C that minimises the cost function:

J =1

nS

nS∑i=1

(‖ui − CCT ui‖2) (2)

Found through an SVD decomposition

Reduced system: minα

‖Fint (Cα) + Fext‖

In the Galerkin framework: CT Fint (Cα) + CT Fext = 0That’s it! In the online stage this much smaller system willbe solved.



Conclusion



J =1

nS

nS∑i=1

(‖ui − CCT ui‖2) (2)

Found through an SVD decompositionReduced system: min

α‖Fint (Cα) + Fext‖




Conclusion



J =1

nS

nS∑i=1

(‖ui − CCT ui‖2) (2)



In the Galerkin framework: CT Fint (Cα) + CT Fext = 0

That’s it! In the online stage this much smaller system willbe solved.



Conclusion



J =1

nS

nS∑i=1

(‖ui − CCT ui‖2) (2)






Conclusion


Example

Snapshot selection:simplify to monotonic loading in εxx , εxy , εyy . 100 snapshots .First 2 modes:



Conclusion


FluctuationCoarse contribution

+

+

+

+

=

.

.

.

.

.

.

+



Conclusion


Is that good enough?

Speed-up actually poorEquation “CT Fint (Cα) + CT Fext = 0“ quicker to solve butCT Fint (Cα) still expensive to evaluateNeed to do something more =⇒ system approximation



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3 Conclusion



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Idea

Integrate only over some nodes of the domain

Reconstruct the operators using a second POD basisrepresenting the internal forces



Conclusion


Idea

Integrate only over some nodes of the domain

Reconstruct the operators using a second POD basisrepresenting the internal forces



Conclusion


“Gappy“ technique

Originally used to reconstruct altered signals

Fint (Cα) approximated by Fint (Cα) ≈ Dβ

Fint (Cα) is evaluated exactly only on a few selectednodes: Fint (Cα)

β found through: minβ

∥∥∥Dβ − Fint (Cα)∥∥∥

2



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2



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2



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3 Conclusion



Conclusion


0 5 10 150.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Number of displacement basis vectors

Rel

ative

erro

r

At time step 5At time step 10At time step 15



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Speedup

Rel

ativ

e er

ror

Number of basis functions increases



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3 Conclusion



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Conclusion

Model order reduction can be used to solved the RVEproblem faster and with a reasonable accuracyCan be thought of as a bridge between analytical andcomputational homogenisation:the reduced bases are pseudo-analytical solutions of theRVE problem that is still computationally solved at veryreduced cost



Conclusion

Thank you for your attention!


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An application of model order reduction to computational