A model order reduction technique for speeding up computational homogenisation

A model order reduction technique for speeding up computational homogenisation Olivier Goury, Pierre Kerfriden, Wing Kam Liu, Stéphane Bordas Cardiff University

Outline Introduction Heterogeneous materials Computational Homogenisation Model order reduction in Computational Homogenisation Proper Orthogonal Decomposition (POD) Optimal snapshot selection System approximation Results Conclusion

Heterogeneous materials Many natural or engineered materials are heterogeneous Homogeneous at the macroscopic length scale 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: Direct numerical simulation: brute force! Multiscale methods: when modelling a non-linear materials = Computational Homogenisation

Semi-concurrent Computational Homogenisation (FE 2,...) Macrostructure Effective Stress Effective strain tensor Effective Stress Effective strain tensor

Problem 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! Need parallel programming

Strategy Use model order reduction to make the solving of the RVE boundary value problems computationally achievable Linear displacement: ( ) ɛ M ɛxx (t) ɛ (t) = xy (t) ɛ xy (t) ɛ yy (t) u(t) = ɛ M (t)(x x) + ũ with ũ Γ = 0 Fluctuation ũ approximated by: ũ i φ iα i

Projection-based model order reduction The RVE problem can be written: F int (ũ(ɛ M (t)), ɛ M (t)) + F }{{} ext (ɛ M (t)) = 0 (1) Non-linear We are interested in the solution ũ(ɛ M ) for many different values of ɛ M (t [0, T ]) ɛ xx, ɛ xy, ɛ yy. Projection-based model order reduction assumption: Solutions ũ(ɛ M ) for different parameters ɛ M are contained in a space of small dimension span((φ i ) i 1,n )

RVE boundary value problem Matrix Inclusions

Proper Orthogonal Decomposition (POD) How to choose the basis [φ 1, φ 2,...] = Φ?

Proper Orthogonal Decomposition (POD) How to choose the basis [φ 1, φ 2,...] = Φ? Offline Stage Learning stage : Solve the RVE problem for a certain number of chosen values of ɛ M

Proper Orthogonal Decomposition (POD) How to choose the basis [φ 1, φ 2,...] = Φ? Offline Stage Learning stage : Solve the RVE problem for a certain number of chosen values of ɛ M We obtain a base of solutions (the snapshot): (u 1, u 2,..., u ns ) = S

Find the basis [φ 1, φ 2,...] = Φ that minimises the cost function: J(Φ) = n POD u i φ k. φ k, u i 2 (2) µ P s with the constraint φ i, φ j = δij Use SVD (Singular Value Decomposition) k

Reduced equations Reduced system after linearisation: min α KΦ α + F ext

Reduced equations Reduced system after linearisation: min α KΦ α + F ext In the Galerkin framework: Φ T KΦ α + Φ T F ext = 0

Reduced equations Reduced system after linearisation: min α KΦ α + F ext In the Galerkin framework: Φ T KΦ α + Φ T F ext = 0 That s it! In the online stage, this much smaller system will be solved.

Arbitrary sampling unsatisfactory Problem: parameter space is HUGE! No guarantee that the arbitrary sampling explores the parameter space well enough

Load of worst prediction Rather than an arbitrary sampling, iteratively add the path of worst prediction:

Load of worst prediction find the step increment ɛ i that maximises: u exact (t i, ɛ(t i ) + ɛ i ) u approx (t i, ɛ(t i ) + ɛ i ) ɛ(t i+1 ) = ɛ(t i ) + ɛ i max

First paths generated 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.6 0 0.4 0.2 0 0.2 0.1 0 0.1 Initial loading 0.4 0.3 0.2

Example Snapshot selection using the load of worst prediction algorithm (36 load paths generated) First 3 modes:

Coarse contribution Fluctuation.. + + =. + +. +..

Is that good enough? Speed-up actually poor Equation Φ T KΦ α + Φ T F ext = 0 quicker to solve but Φ T KΦ still expensive to evaluate Need to do something more = system approximation

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

Idea Define a surrogate structure that retains only very few elements of the original one Reconstruct the operators using a second POD basis representing the internal forces

Gappy technique Originally used to reconstruct altered signals F int (Φ α) approximated by F int (Φ α) Ψ β

Gappy technique Originally used to reconstruct altered signals F int (Φ α) approximated by F int (Φ α) Ψ β F int (Φ α) is evaluated exactly only on a few selected nodes: Fint (Φ α)

Gappy technique Originally used to reconstruct altered signals F int (Φ α) approximated by F int (Φ α) Ψ β F int (Φ α) is evaluated exactly only on a few selected nodes: Fint (Φ α) β found through: min Ψ β Fint (Φ α) β 2

Controlled elements locations Mode 3 Mode 2 Mode 1

Error 120 1 Number of static basis vectors 100 80 60 40 20 0.3 0.1 0.03 0.01 0.003 0.001 Relative error in energy norm 0 0 2 4 6 8 10 12 14 16 18 20 Number of displacement basis vectors

Speedup 120 12 110 11 Number of static basis vectors 100 90 80 70 60 50 40 30 10 9 8 7 6 5 4 3 Speedup 20 2 4 6 8 10 12 14 16 18 20 Number of displacement basis vectors 1

0.035 Relative error in energy norm 0.03 0.025 0.02 0.015 0.01 0.005 Number of basis vectors increases 0 8 10 12 14 16 18 20 Speedup

Conclusion Model order reduction can be used to solved the RVE problem faster and with a reasonable accuracy An efficient snapshot selection algorithm can be developed when dealing with time-dependent parameters The controlled elements generated by the DEIM algorithm lie where damage is high 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!