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UNSUPERVISED LEARNING BASED MODEL ORDER REDUCTION FOR HYPERELASTOPLASTICITY

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UNSUPERVISED LEARNING BASED MODEL

ORDER REDUCTION FOR

HYPERELASTOPLASTICITY

Soumianarayanan Vijayaraghavan

University of Luxembourg University of Li`ege

Supervisors: Dr. Lars Beex Prof. St´ephane Bordas assoc-Prof. Ludovic Noels

November 29, 2020

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Why Model Order Reduction?

Figure:Metal lattice in the form of periodic mesostructural unit cells

• Real time surgical

simulations requires quick computational results. • Expensive simulations for

problems dealing with micro-scale phenomena

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How model order reduction helps?

• Parameterised non-linear mechanical problem: ˜fint(˜u(µ), µ)−˜fext =˜0 ˜

u is the unknown that has to be computed for any value of

parameter µ.

• Ansatz: Use precomputed solutions to speed up the online simulation.

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How model order reduction helps?

Projection based model order reduction

• Use Galerkin framework, solve for a reduced system: φT ˜fint(φ˜α)− φ T ˜fext =˜0 • Interpolation: ˜ u = φ ˜ α

• Reduced stiffness matrix: Kr = φTKφ

• Reduced force vector: ˜f

r

int = φT

˜fint • Solve using Newton Raphson: ∆

˜ α =−(φTKφ)−1φT ˜ R ˜ R = ˜fint(φα) +˜fext

Reduced number of unknowns: ˜ α <<

˜ u

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How model order reduction helps?

X Reduced number of unknowns × Number of Gauss points

Hyper-reduction strategy to reduce the number of Gauss points

• Utilize the modes to obtain an adaptive non confirming mesh with less number of elements

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Hyperreduction: Modes and number of elements

X Y Z X Y Z X Y Z X Y Z X Y Z X Y Z

1:530 Elements 2:668 Elements 3:728 Elements

4:737 Elements 5:821 Elements 6:872 Elements

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Hyperreduction: Modes and number of elements

• Hyper-reduction:Number of elements (Gauss points) increases with increase in number of modes

(8)

RVE model with multiple voids

Large deformation hyperelastoplastic material model. Enforced periodic boundary condition using Lagrange multipliers.

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Offline stage: Generation of snapshots

• Input parameters: The coefficients of stretch tensor (UM).

FM= RMUM

Monotonic loading training parameters for full training set.

Random loading training path for one simulation.

Red lines are training parameters and black lines are test case parameters

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Additive split of the snapshot solutions

˜u = φ˜α + ψ˜β

Additively split the snapshot solution into fluctuating and homogenised deformation

Homogeneous deformation Fluctuating deformation

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Drawback of POD

POD solution with additive split of snapshots will be used as a reference to compare the results obtained using novel approach.

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Why unsupervised learning method?

Different snapshot solutions

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The proposed clustering strategy

Clustering algorithm

RB: Centroid of each cluster as basis POD: Incorporate SVD to individual clusters

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Clustering: The methodology

INPUT CLUSTERINGALGORITHM OUTPUT

Snapshot solutions

• Fluctuating displacements • Cumulative plastic strains • Plastic deformation gradient

• Centroid based • Connectivity based Snapshots grouped based on deformation pattern Which feature?

Scaling the snapshots?

Which algorithm?

Number of clusters?

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Clustering: The methodology

Clustering approaches

Centroid based clustering

× Partition the snapshots based on similarity × Prespecify the number of

clusters

Connectivity based clustering

× Establish connectivity with nearby snapshots

× Specify the measure for search radius

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Clustering: The methodology

Evaluation metric and preprocessing

Similarity measure

× Euclidien distance × Absolute projection

Normalization

× Scale Features from 0 to 1 × Scale snapshots to unit norm

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Snapshot solutions

Three snapshot matrices

• Cumulative plastic strain values at multiple increments ( ep)

• Displacement values at multiple increments ( ˜u)

• Components of plastic deformation gradient tensor (FP)

F Pincr ii1 F P incr ii1 F P incr ii1 F P ... incr ii2 ... F Pincr iin

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Clustering: Steps involved at the offline stage

Snapshots: Solutions of multiple load increments from DNS

Choose the appropriate modes from each cluster and orthonormalize Additively split snapshots

into homogeneous and fluctuating part

SVD to fluctuating snapshot solutions in individual clusters

Group snapshots into clusters based on uf luc e ✏p e ✏pand FP e ✏p, FPand uf luc ~ ~

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Selecting the optimal modes

1c1

2c1

3c1

4c1

4c2

3c2

2c2

1c2

1c3

2c3

3c3

4c3 ⌃ = Singular values c = Cluster

• Top singular vectors of each cluster are the first 3 modes • For 4th mode, Normalize

singular values for all clusters and compute:

Σ1ci − Σ2

ci i = 1, 2, 3

• Select the one with minimum difference.

• Continue until decided number of modes are reached.

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Result of POD based on centroid clustering with Euclidean

distance measure by scaling feature between 0 and 1

Sum of error in stress values for all test cases with 10 modes

Clusters1: Clustering based on ˜

ufluc; Clusters2: Clustering based onep; Clusters3: Clustering based on

e

pand FP; Clusters4: Clustering based onep, FPand ˜ufluc

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Result of POD based on centroid clustering with Euclidean

distance measure by scaling snapshots to unit norm

Sum of error in stress values for all test cases with 10 modes

Clusters1: Clustering based on ˜

ufluc; Clusters2: Clustering based onep; Clusters3: Clustering based onepand FP; Clusters

4: Clustering based on

e p, FPand

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Result of POD based on centroid clustering with projection

measure by scaling snapshots to unit norm

Sum of error in stress values for all test cases with 10 modes

Clusters1: Clustering based on

˜

ufluc; Clusters2: Clustering based onep; Clusters3: Clustering based on

e

pand FP; Clusters4: Clustering based onep, FPand

(23)

Result of RB based on centroid clustering with Euclidean

distance measure by scaling feature between 0 and 1

Sum of error in stress values for all test cases with 10 modes

Clusters1: Clustering based on ˜

ufluc; Clusters2: Clustering based onep; Clusters3: Clustering based onepand FP; Clusters4: Clustering based onep, FPand

(24)

Result of RB based on centroid clustering with Euclidean

distance measure by scaling snapshots to unit norm

Sum of error in stress values for all test cases with 10 modes

Clusters1: Clustering based on

˜

ufluc; Clusters2: Clustering based on

e p;

Clusters3: Clustering based onepand FP; Clusters4: Clustering based onep, FPand

(25)

Result of RB based on centroid clustering with projection

measure by scaling snapshots to unit norm

Sum of error in stress values for all test cases with 10 modes

Clusters1: Clustering based on

˜

ufluc; Clusters2: Clustering based on

e p;

Clusters3: Clustering based on

e

pand FP; Clusters4: Clustering based onep, FPand

(26)

Result of POD based on connectivity clustering with

distance measure by scaling feature between 0 and 1

Sum of error in stress values for all test cases with 10 modes

Clusters1: Clustering based on ˜

ufluc; Clusters2: Clustering based onep; Clusters3: Clustering based onepand FP; Clusters

4: Clustering based on

e p, FPand

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Inferences from test cases

• Centroid clustering approach with distance measure improves online prediction with two clusters of snapshots grouped based onep.

• Connectivity clustering approach with distance measure improves online prediction with three clusters of snapshots grouped based on ep.

• Clustering based on FP and

˜ufluc does not facilitate to improve online prediction for monotonic loading.

• Measure of similarity based on Euclidean distances works better than the measure using absolute projection.

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Future work

• Investigate more on clustering approaches to have a significant improvement in online prediction with less number of modes. • Incorporate clustering approach for random load path.

• Utilize the results of clustering approach to obtain an adaptive mesh for hyper-reduction strategy.

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