an application of model order reduction to computational
TRANSCRIPT
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
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Heterogeneous materialsComputational Homogenisation
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Heterogeneous materialsComputational Homogenisation
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Heterogeneous materialsComputational Homogenisation
Heterogeneous materials
Many natural or engineered materials are heterogeneous
Homogeneous at the macroscopic length scaleHeterogeneous at the microscopic length scale
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Heterogeneous materialsComputational Homogenisation
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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Heterogeneous materialsComputational Homogenisation
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Heterogeneous materialsComputational Homogenisation
Semi-concurrent Computational Homogenisation(FE2, ...)
Effective strain
tensor
Macrostructure
Effective strain
tensorEffective
Stress
Effective
Stress
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Heterogeneous materialsComputational Homogenisation
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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
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)
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
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
, , , ...
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
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
, , , ...
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
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
, , , ...
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
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.
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Find the basis C that minimises the cost function:
J =1
nS
nS∑i=1
(‖ui − CCT ui‖2) (2)
Found through an SVD decompositionReduced system: min
α‖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.
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Find the basis C that minimises the cost function:
J =1
nS
nS∑i=1
(‖ui − CCT ui‖2) (2)
Found through an SVD decompositionReduced system: min
α‖Fint (Cα) + Fext‖
In the Galerkin framework: CT Fint (Cα) + CT Fext = 0
That’s it! In the online stage this much smaller system willbe solved.
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Find the basis C that minimises the cost function:
J =1
nS
nS∑i=1
(‖ui − CCT ui‖2) (2)
Found through an SVD decompositionReduced system: min
α‖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.
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Example
Snapshot selection:simplify to monotonic loading in εxx , εxy , εyy . 100 snapshots .First 2 modes:
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
FluctuationCoarse contribution
+
+
+
+
=
.
.
.
.
.
.
+
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Idea
Integrate only over some nodes of the domain
Reconstruct the operators using a second POD basisrepresenting the internal forces
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Idea
Integrate only over some nodes of the domain
Reconstruct the operators using a second POD basisrepresenting the internal forces
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
“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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
“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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
“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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Proper Orthogonal Decomposition (POD)System approximationResults
Speedup
Rel
ativ
e er
ror
Number of basis functions increases
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Outline
1 IntroductionHeterogeneous materialsComputational Homogenisation
2 Model order reduction in Computational HomogenisationProper Orthogonal Decomposition (POD)System approximationResults
3 Conclusion
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
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
USNCCM Institute of Mechanics and Advanced Materials
IntroductionModel order reduction in Computational Homogenisation
Conclusion
Thank you for your attention!
USNCCM Institute of Mechanics and Advanced Materials