Model order reduction for speeding up computational homogenisation methods of type FE2

Model order reduction for speeding up computational homogenisation methods of

type FE

Olivier Goury, Pierre Kerfriden, Stéphane Bordas Cardiff University

Outline

Introduction

Heterogeneous materials Computational Homogenisation

Model order reduction in Computational Homogenisation Proper Orthogonal Decomposition (POD)

System approximation Results

Conclusion

Heterogeneous materials

Many natural or engineered materials areheterogeneous

I Homogeneous at the macroscopic length scale

I Heterogeneous at the microscopic length scale

Heterogeneous materials

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

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

Two choices:

I Direct numerical simulation: brute force!

I Multiscale methods: when modelling a non-linear materials

=⇒Computational Homogenisation

Semi-concurrent Computational Homogenisation (FE

, ...)

Effective strain tensor

Macrostructure

Effective strain tensor Effective

Stress

Effective Stress

Problem

I For non-linear materials: Have to solve a RVE boundary value problem at each point of the macro-mesh where it is needed. Still expensive!

I Need parallel programming

Strategy

I Use model order reduction to make the solving of the RVE boundary value problems computationally achievable

I Linear displacement:

^M(t) =

xx(t) xy(t) _xy(t) _yy(t)

u(t) =^M(t)(x−¯x) + ˜u with u˜_|Γ=0 Fluctuationu˜approximated by: u˜≈P

iφ_iα_i

Projection-based model order reduction

The RVE problem can be written:

F_int(˜u(^M(t)), ^M(t))

| {z }

Non-linear

+F_ext(^M(t)) =0 (1)

We are interested in the solutionu(˜ ^M)for many different values of^M(t∈[0,T])≡xx, xy, yy.

Projection-based model order reduction assumption:

Solutionsu(˜ ^M)for different parameters^Mare contained in a space of small dimensionspan((φ_i)_i∈

J1,nK)

RVE boundary value problem

Matrix Inclusions

Proper Orthogonal Decomposition (POD)

How to choose the basis[φ₁,φ₂, . . .] =Φ?

I “Offline“ Stage≡Learning stage : Solve the RVE problem for a certain number of chosen values of^M

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

, , , ...

I That snapshot may be large and have linearly dependent components⇒Need to extract the core information from it

Proper Orthogonal Decomposition (POD)

How to choose the basis[φ₁,φ₂, . . .] =Φ?

I “Offline“ Stage≡Learning stage : Solve the RVE problem for a certain number of chosen values of^M

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

, , , ...

I That snapshot may be large and have linearly dependent components⇒Need to extract the core information from it

Proper Orthogonal Decomposition (POD)

How to choose the basis[φ₁,φ₂, . . .] =Φ?

I “Offline“ Stage≡Learning stage : Solve the RVE problem for a certain number of chosen values of^M

I We obtain a base of solutions (the snapshot):

(u₁,u₂, ...,un_S) =S

, , , ...

I That snapshot may be large and have linearly dependent components⇒Need to extract the core information from it

Proper Orthogonal Decomposition (POD)

How to choose the basis[φ₁,φ₂, . . .] =Φ?

I “Offline“ Stage≡Learning stage : Solve the RVE problem for a certain number of chosen values of^M

I We obtain a base of solutions (the snapshot):

(u₁,u₂, ...,un_S) =S

, , , ...

I That snapshot may be large and have linearly dependent components⇒Need to extract the core information from it

I Find the basis[φ₁,φ₂, . . .] =Φthat minimises the cost function:

J_h.i^s (Φ) = X

µ∈P^s

ku_i−

nPOD

X

k

φ_k.hφ_k,u_ii k² (2)

with the constraint φ_i,φ_j

=δ_ij

I What is the quantity of interest here? Which scalar product and norm should we choose?

I The canonic scalar product?⇒ hx,yi=x^Ty

I Some kind of energy scalar product makes more sense: hx,yi=x^TK{τ|τ <t}y

I Simplify tohx,yi=x^TK₀ysince we want a fixed basis with time

I One can prove analytically that the solution is given by the eigenvectors ofK₀S S^TK₀

I Find the basis[φ₁,φ₂, . . .] =Φthat minimises the cost function:

J_h.i^s (Φ) = X

µ∈P^s

ku_i−

nPOD

X

k

φ_k.hφ_k,u_ii k² (2)

with the constraint φ_i,φ_j

=δ_ij

I What is the quantity of interest here? Which scalar product and norm should we choose?

I The canonic scalar product?⇒ hx,yi=x^Ty

I Some kind of energy scalar product makes more sense: hx,yi=x^TK{τ|τ <t}y

I Simplify tohx,yi=x^TK₀ysince we want a fixed basis with time

I One can prove analytically that the solution is given by the eigenvectors ofK₀S S^TK₀

I Find the basis[φ₁,φ₂, . . .] =Φthat minimises the cost function:

J_h.i^s (Φ) = X

µ∈P^s

ku_i−

nPOD

X

k

φ_k.hφ_k,u_ii k² (2)

with the constraint φ_i,φ_j

=δ_ij

I What is the quantity of interest here? Which scalar product and norm should we choose?

I The canonic scalar product?⇒ hx,yi=x^Ty

I Some kind of energy scalar product makes more sense: hx,yi=x^TK{τ|τ <t}y

I Simplify tohx,yi=x^TK₀ysince we want a fixed basis with time

I One can prove analytically that the solution is given by the eigenvectors ofK₀S S^TK₀

I Find the basis[φ₁,φ₂, . . .] =Φthat minimises the cost function:

J_h.i^s (Φ) = X

µ∈P^s

ku_i−

nPOD

X

k

φ_k.hφ_k,u_ii k² (2)

with the constraint φ_i,φ_j

=δ_ij

I What is the quantity of interest here? Which scalar product and norm should we choose?

I The canonic scalar product?⇒ hx,yi=x^Ty

I Some kind of energy scalar product makes more sense:

hx,yi=x^TK{τ|τ <t}y

I Simplify tohx,yi=x^TK₀ysince we want a fixed basis with time

I One can prove analytically that the solution is given by the eigenvectors ofK₀S S^TK₀

I Find the basis[φ₁,φ₂, . . .] =Φthat minimises the cost function:

J_h.i^s (Φ) = X

µ∈P^s

ku_i−

nPOD

X

k

φ_k.hφ_k,u_ii k² (2)

with the constraint φ_i,φ_j

=δ_ij

I What is the quantity of interest here? Which scalar product and norm should we choose?

I The canonic scalar product?⇒ hx,yi=x^Ty

I Some kind of energy scalar product makes more sense:

hx,yi=x^TK{τ|τ <t}y

I Simplify tohx,yi=x^TK₀ysince we want a fixed basis with time

I One can prove analytically that the solution is given by the eigenvectors ofK₀S S^TK₀

I Find the basis[φ₁,φ₂, . . .] =Φthat minimises the cost function:

J_h.i^s (Φ) = X

µ∈P^s

ku_i−

nPOD

X

k

φ_k.hφ_k,u_ii k² (2)

with the constraint φ_i,φ_j

=δ_ij

I What is the quantity of interest here? Which scalar product and norm should we choose?

I The canonic scalar product?⇒ hx,yi=x^Ty

I Some kind of energy scalar product makes more sense:

hx,yi=x^TK{τ|τ <t}y

I Simplify tohx,yi=x^TK₀ysince we want a fixed basis with time

I One can prove analytically that the solution is given by the eigenvectors ofK₀S S^TK₀

Next question: how many vectors should we pick?

0 50 100 150 200 250 300

10⁻¹⁴ 10⁻¹² 10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰ 10²

Eigenvalue Number

Eigenvalues

99% accuracy

99.99999%

accuracy

Reduced equations

I Reduced system after linearisation: min

α kKΦα+F_extk

I In the Galerkin framework: Φ^TKΦα+Φ^TFext =0

I That’s it! In the online stage, this much smaller system will be solved.

Reduced equations

I Reduced system after linearisation: min

α kKΦα+F_extk

I In the Galerkin framework: Φ^TKΦα+Φ^TFext =0

I That’s it! In the online stage, this much smaller system will be solved.

Reduced equations

I Reduced system after linearisation: min

α kKΦα+F_extk

I In the Galerkin framework: Φ^TKΦα+Φ^TFext =0

I That’s it! In the online stage, this much smaller system will be solved.

Example

Snapshot selection:

simplify to monotonic loading in_xx, _xy, _yy. 100 snapshots . First 2 modes:

Fluctuation Coarse contribution

+

+ +

= +

.

.

.

.

.

.

+

Is that good enough?

I Speed-up actually poor

I Equation “Φ^TKΦα+Φ^T F_ext=0“ quicker to solve but Φ^T KΦstill expensive to evaluate

I Need to do something more=⇒system approximation

Idea

I Define a surrogate structure that retains only very few elements of the original one

I Reconstruct the operators using a second POD basis representing the internal forces

Idea

I Define a surrogate structure that retains only very few elements of the original one

I Reconstruct the operators using a second POD basis representing the internal forces

“Gappy“ technique

Originally used to reconstruct altered signals

I F_int(Φα)approximated byF_int(Φα)≈Ψβ

I F_int(Φα)is evaluated exactly only on a few selected nodes: F_int\(Φα)

I βfound through: min

β

Ψbβ−F_int\(Φα) 2

“Gappy“ technique

Originally used to reconstruct altered signals

I F_int(Φα)approximated byF_int(Φα)≈Ψβ

I F_int(Φα)is evaluated exactly only on a few selected nodes: F_int\(Φα)

I βfound through: min

β

Ψbβ−F_int\(Φα) 2

“Gappy“ technique

Originally used to reconstruct altered signals

I F_int(Φα)approximated byF_int(Φα)≈Ψβ

I F_int(Φα)is evaluated exactly only on a few selected nodes: F_int\(Φα)

I βfound through: min

β

Ψbβ−F_int\(Φα) 2

4 2 8 6

10 0 20

40 60

80 100 10⁻⁵

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Relative error in energy norm

Number

of vectors in the stati

c basis Number of vectors in the displacement basis

5 6 7 8 9 10 11 12 13 14 4.5

5 5.5 6 6.5 7 7.5 8 8.5

9x 10⁻³

Speedup

Relative error

Number of basis functions increases

Conclusion

I Model order reduction can be used to solved the RVE problem faster and with a reasonable accuracy

I Can be thought of as a bridge between analytical and computational homogenisation:

the reduced bases are pseudo-analytical solutions of the RVE problem that is still computationally solved at very reduced cost

Thank you for your attention!