Accelerating multi-scale microstructure simulations by exploiting …congress.cimne.com/icme2016/admin/files/filepaper/p29.pdf · 2016-05-02 · rm 1 Accelerating multi-scale microstructure

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Accelerating multi-scale microstructure simulationsby exploiting local macroscopic quasi-homogeneities

J. Gawad1 Md. Khairullah1 D. Roose1 A. Van Bael2

1Dept. of Computer Science, KU Leuven, [email protected]

2Dept. of Metallurgy and Materials Engineering, KU Leuven, Belgium

ICME workshop13 April 2016, Barcelona

J. Gawad et al. Accelerating multi-scale microstructure simulations . . . ICME 2016, Barcelona

[email protected]

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Outline of the HMS modelling framework

Macroscopic FE

Mesoscopic Crystal Plasticity

Microscopic substructural hardeningm

μm

Shape of the yield locus

Size of the yield locus

SLOW

FAST

nm

e.g. state variables: 2x24 CRSS

Discrete ODFe.g. 5,000 orientations24 slip systems (BCC)

e.g. 100,000 time increments x 20,000 elements


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Outline of the HMS modelling framework

Macroscopic FE

Mesoscopic Crystal Plasticity

Microscopic substructural hardeningm

μm

nm

e.g. state variables: 2x24 CRSS

Discrete ODFe.g. 5,000 orientations24 slip systems (BCC)

e.g. 100,000 time increments x 20,000 elements

Plastic potential orYield locusFacet, BBC2008, YLD200x, …

Adaptive hardeningAnalytical formula


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Computational advantage of the HMS

How much can we save in terms of the simulation time?

For example:Crystal Plasticity fully embedded in an explicit time integration FE:

I 1 call/update to the fine-scale CP model per increment of thecoarse-scale FE model

tCPFEM ≈ 100.000 updates× 20.000 integration points× tCP + tFE

HMS: Crystal Plasticity hierarchically coupled with FEM:tHMS ≈ 30 updates× 20.000 integration points× tCP + tFE


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Contributions to the computational cost of multi-scaleCP-FEM

evolutionof material

state & properties

eachintegration

point

largesimulation

timeX =X

every time increment

The HMS addresses the issue:

I in the temporal dimension: reconstruction of the plastic potentialfunction is adaptively conducted only at the moments when agiven deformation-based criterion is satisfied.

I in the spatial dimension: another acceleration is possible!


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Acceleration of the multi-scale CP models by exploitingspatial quasi-homogeneities

Fundamental assumptions:I Similar microstructural state variables subjected to a similar

deformation history would evolve along nearly identicaltrajectories.

I Derived macroscopic plastic properties would be similar as well.

Additional assumption:I The accumulated plastic strain determines the evolution of the

material properties.

Example:

field of total

plastic slip0.000.060.130.190.260.320.380.450.510.580.640.700.770.830.890.961.021.091.151.211.281.341.411.471.53


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Exploiting spatial quasi-homogeneities

control field variables: total plastic slip, accumulated plastic strain,equivalent plastic strain . . .

Integration points grouped together (clustered):

material state: microstructure (texture, substructure, . . . ), plasticpotential function, . . .

material properties: plastic potential function, hardening, . . .J. Gawad et al. Accelerating multi-scale microstructure simulations . . . ICME 2016, Barcelona

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Overview of the spatial clustering method in HMS

A B C D E

F G

But it has to be decided when and how the clusters should be formed.


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A B C D E

F G



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A B C D E

F G



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x √ x √ √

x √√ updates (representative point)

x does not update (propagated by the representative point)



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Clustering schemes and clustering criteria

Clustering scheme decides when the clusters are formed.Several schemes can be considered, including:

1. static: the clusters are constructed once and later usedthroughout the simulation.

2. dynamic: the cluster are adaptively rebuilt as the HMSsimulation advances.

Clustering criterion decides how the integration points are groupedinto the clusters.

I primary criterion: spatial proximity of the integration pointsI secondary criteria:

1. magnitude of total plastic slip, or2. plastic strain tensor,3. sum of the absolute values of plastic strain increment tensors,4. . . .


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Static spatial clustering

The algorithm in a nutshell:

I Parameter: number of clusters k .

I Initialization: all integration points belong to a single cluster

I Given the control field variable and the Euclidean distance matrix,construct k clusters and use these clusters throughout thesimulation.

The inputs may be taken from:

I a plain FE simulation (non-HMS) of the process (off-line), or

I the HMS simulation itself, for instance when the first update ofproperties is calculated (on-line)

I a reference HMS simulation (off-line, very impractical)


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Static spatial clustering in HMS: performance gain vsaccuracyExample: tensile test on a complex geometry specimen

1 10 100 1000 100000

5

10

15

20

25

30

0

1

2

3

4

5

6

7

8

9

error (clustering at the first update) error (clustering at the final state)

speedup

# clusters

spe

ed

up

% e

rro

r in

pla

stic

slip

speedup(#clusters) =tHMS

tHMS,#clusters

Note: the deformation is nearlymonotonic in this example.


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Static spatial clustering in HMSExample: torque bar, 9 clusters

X

Y

Z X

Y

Z

I Example: a bar subjected to torsion followed by tension.

I The two deformation steps introduce a non-monotonic strainpath.


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X

Y

Z X

Y

Z




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X

Y

Z X

Y

Z




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I Static clustering does not follow the evolution of the control fieldvariable.

I Not much can be gained by simply increasing the number ofclusters – an adaptive approach is needed.


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Dynamic adaptive clustering

I Parameter: threshold value tsplit .

I Initialization: all points belong to a single clusterI At each updating event, consider splitting each existing cluster C :

1. ∆maxv = maxi∈C

(vi )−mini∈C

(vi ), where v is the control field variable.

2. If ∆maxv ≥ tsplit , re-cluster C into new clusters C1, . . . ,Cn thatinherit the extended material states from the representative of Cto the new representatives of C1, . . . ,Cn.

I A high threshold value permits the clusters to persist longer;fewer clusters are generated overall, resulting in a high speedupbut lower accuracy.


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Dynamic clustering: the “split” operation

I Local operation on a limited number of integration pointsI think of divisive hierarchical clustering: O(N2), or DBSCAN or

OPTICS: O(n log n))

I Direct back-traceability of the fine-scale model state variables


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Dynamic adaptive clustering: resultsExample: torque bar, high threshold for splitting


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Dynamic adaptive clustering: resultsExample: torque bar, low threshold for splitting


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Approximation error: material propertiesReference HMS vs. clustering-enabled HMS

In terms of plastic potential function ψ, therelative error at integration point i :

eψi =1

m

m∑j=1

|ψ(Dj)i − ψ′(Dj)cr(i)

ψ(Dj)i| × 100%

Error for the whole model:

eψ =1

n

n∑i=1

eψi ,

A B C D E

F G

A B C D E

F G

Reference HMS

Improved HMS


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Approximation error: field variablesReference HMS vs. clustering-enabled HMS

With respect to the field variable γ, therelative error at integration point i :

eγi =|γi − γ′i |

γi× 100%

Error for the whole model:

eγ =1

n

n∑i=1

eγi

A B C D E

F G

A B C D E

F G

Reference HMS

Improved HMS

Note: γ is the control variable used in clustering.


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Accuracy of static and dynamic clustering schemesMaterial property: plastic potential

static clustering scheme dynamic clustering scheme


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Accuracy of static and dynamic clustering schemesPlastic strain

static clustering scheme dynamic clustering scheme


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Performance gain: static vs. dynamic clustering scheme

Overall gain in total computation time(speedup):

gt =trtc

Reduction in the number of calls to thefine-scale model (“speedup in updates”):

gu =nrnc


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Microstructural similarity within clustersExample: dynamic clustering scheme, torque bar

Recall the assumption we made:similar deformation in adjacent points ⇒ similar microstructures ⇒ similarproperties

cluster representative point (RP) corresponding point in the reference HMS

Do actually similar microstructures develop if neighboring points are subjected tosimilar deformation?→ Yes, see deformation textures at several points of the reference HMS that wouldbelong to the same cluster:


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Summary and take-aways

1. The HMS adaptively approximates mechanical responses of theCP model by much simpler analytical models.

2. Concurrent evolution of texture and plastic anisotropy can bethen handled in component-scale FE models.

3. Further acceleration of the HMS is attained if spatialquasi-homogeneities are exploited.

4. Static and dynamic clustering schemes restrict tracking theevolution of the fine-scale variables to a number of representativematerial points.

5. Large performance gains (e.g. a speedup of 25) are attained atthe expense of minor modelling error.

6. Dynamic clustering scheme in the HMS allows following the fieldof interest.


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Advanced solutions for component-scale crystal plasticitysimulations by KU Leuven Knowledge Platform M2Form:

HMS : Hierarchical Multi-Scale framework

VEF : Virtual Experimentation Framework

Web: https://set.kuleuven.be/m2form/projects

e-mail: [email protected]


https://set.kuleuven.be/m2form/projects

[email protected]

Documents

Accelerating multi-scale microstructure simulations by exploiting …congress.cimne.com/icme2016/admin/files/filepaper/p29.pdf · 2016-05-02 · rm 1 Accelerating multi-scale microstructure