ESAFORM 2023 P.A. Prates, J. Henriques, J. Pinto, N. Bastos and A. Andrade-Campos

1. Coupling machine learning and synthetic
image DIC-based techniques for the
calibration of elastoplastic constitutive
models
P.A. Prates, J. Henriques, J. Pinto, N. Bastos
and A. Andrade-Campos
This project has received funding from the Research Fund for Coal and Steel under grant agreement No 888153

2. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
3
Outline
Summary
• Introduction - Parameter identification as an inverse problem
• State-of-the-art: Inverse methodologies for full-field
measurements
 External methods and the Finite Element Model Updating (FEMU)
 Equilibrium methods and the Virtual Fields Method (VFM)
 Data-driven and Machine Learning (ML) methods
• Methodology and implementation: DIC-levelling approach
• Case study: the biaxial tensile test on a cruciform sample
• Results and Digital Virtual Testing (DVT) database
• Concluding remarks

3. Introduction
Coupling machine learning and synthetic image DIC-based
techniques for the calibration of elastoplastic constitutive models

4. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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Decades ago…

5. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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Today…

6. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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Introduction
Increase information &
reduce time in material testing
Realistic simulations and material
behavior reproduction
Faster and accurate
development & production

7. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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Introduction
Parameter identification as an inverse problem
The problem that every user of numerical simulation tools faces:
How to obtain the material parameters for the
selected constitutive model?

8. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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Introduction
Parameter identification as an inverse problem
Direct
Problem
Inverse
Problem
Known Unknown Solution
• Geometry
• Boundary conditions
• Material properties/
parameters
• Displacements
• Strains
• Stresses
• Geometry
• Boundary conditions
• Displacements
• Strains
• Material properties/
parameters
• Stresses
FEA
simulations
?
Inverse Problem: Search for the r parameters A that

9. State-of-the-art: inverse
methodologies
Coupling machine learning and synthetic image DIC-based
techniques for the calibration of elastoplastic constitutive models

10. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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The objective function can be written as
Inverse methodologies for full-field measurements
External methods and the Finite Element Model Updating (FEMU)
• comparison between finite
element (FE) simulations
and the measured
experimental results
• the parameters are found
iteratively by A = min L(A)
• Very flexible (e.g. it does not
require full-field
measurements)
• Easy to implement
• The experimental test must
be accurately modelled
• High computational cost
• Simulation software is
required

11. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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The objective function can be written as
Inverse methodologies for full-field measurements
Equilibrium methods and the virtual field method (VFM)
• Balance between the
external and internal works,
however…
• the parameters are found
iteratively by A = min L(A)
• Low computational cost
• A simulation software is not
required
• The force distribution is not
required
• The choice of the virtual
fields is not straightforward,
however…

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The objective function can be written as
• Builds the inverse problem model
as non-linear regressor (using
data)
• Build the inverse model LLM and
finds directly the parameters
solving A = LLM(εexp,Fexp).
• The inverse model LLM is
boundary conditions’ dependent
• High computational cost for the
creation of a suitable samples’
database (direct problem
database)
• A simulation software is required
The objective function can be written as
Inverse methodologies for full-field measurements
Data-driven and Machine Learning (ML) methods

13. Methodology and
implementation
Coupling machine learning and synthetic image DIC-based
techniques for the calibration of elastoplastic constitutive models

14. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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Synthetic image DIC-based techniques
DIC levelling: the power of digital virtual testing (DVT)

15. Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
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DIC-levelling in ML approach
New stage
New stage

16. Case-study: the biaxial
(cruciform) test
Coupling machine learning and synthetic image DIC-based
techniques for the calibration of elastoplastic constitutive models

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
ux = 2 mm
Symmetry boundary conditons
(uy = 0 mm)
Symmetry
boundary
conditons
(u
x
=
0
mm)
uy = 2 mm
The case of the biaxial cruciform test
ML approach for the Cruciform test (Swift’s hardening law + Hill’48 yield criterion)
Constitutive parameters Input Space
σ0 [MPa] 120-300
n 0.1-0.3
K [MPa] 280-700
R0 0.6-2.5
R45 0.6-2.5
R90 0.6-2.5
Cruciform tensile test
Total displacement ux=uy = 2 mm
405 CPS4R elements (reduced integration)
Anisotropic plastic behaviour (Hill 1948)
Latin Hypercube Sampling – 2000 simul.
R = 7 mm
30
mm
0x RD
0y TD
R = 2.5 mm
15 mm

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
ML approach for the Cruciform test (Swift’s hardening law + Hill’48 yield criterion)

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
ML approach for the Cruciform test (Swift’s hardening law + Hill’48 yield criterion)
In/out features definition for the ML inverse model
XGBoost
Model
σ0
n
K
Strain xy
(20 time steps *
405 features per time step)
Strain yy
(20 time steps *
405 features per time step)
Strain xx
(20 time steps *
405 features per time step)
Total force Ox
(20 time steps *
1 feature per time step)
Total force Oy
(20 time steps *
1 feature per time step)
R0
R45
R90
24340 features 6 Outputs

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
Testing result for original ML approach
σ0 n K
R0 R45 R90
Testing – 900 samples
5900 samples were generated, 5000 for training and 900 for testing

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
Features importance analysis: SHAP
σ0
n K

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
R0 R45 R90

23. Results with the DIC-
levelling methodology
Coupling machine learning and synthetic image DIC-based
techniques for the calibration of elastoplastic constitutive models

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
Digital virtual tests (DVT) database
ML architecture
Synthetic
image
generator
?
?
?
?
?
Each sample was divided into 6455 subsets. In each subset, the
location in pixels and the strains εxx, εyy, and εxy were recorded.
2000 samples were generated, 1800 for training and 200 for
testing.

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
Digital virtual tests (DVT) database
Synthetic
image
generator
?
?
?
?
?

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
Digital virtual tests (DVT) database
… resulting in more than
400k features for each
sample.
To validate the models,
2000 images were
generated (1800 for training
and 200 for test).
only the regions of interest found with the SHAP values
were used to train the models (just 10% of the data)
number of
features was too
large
impossible to train
the model with
every feature

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
Performance evaluation
σ0 n K
R0 R45 R90
Predictive
Simulated
R2=0.9980 R2=0.9610
R2=0.9779
R2=0.9779
R2=0.9923 R2=0.9925
Predicted (vertical axis) vs. real (horizontal axis) values of the constitutive parameters
2000 samples were generated, 1800 for training and 200 for testing

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
DVT database results
R2 and MAE for Abaqus vs MatchID database
The results are
still robust
even with DIC
noise/errors!

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
The case of the biaxial cruciform test
Robustness analysis: material rotation
0
2
4
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45
Relative
error
(%)
Rotation ( )
Y0
K
n
r0
r45
r90
Relative error in the
prediction of the material
parameters as a function of
the rotation of the material
axes w.r.t. the global axes
system.
In this context, one
of the samples from
the testing set was
chosen (Y0=172 MPa,
n=0.16; K=486 MPa;
r0=2.38; r45=1.8;
r90=1.06) and the
orientation of the
material axes was
varied w.r.t. the
global axes system
0xy

30. Conclusions
Coupling machine learning and synthetic image DIC-based
techniques for the calibration of elastoplastic constitutive models

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
Closing remarks
The work presented …
• Feature importance enabled the identification of the most
important features of the database for identifying constitutive
parameters. ML training was greatly accelerated, without
compromising predictive performance;
• Virtual experiments allow for accurate model predictions of
the constitutive parameters, bridging the gap between
simulation results and experimental measurements;
• Sample misalignments do not significantly influence the
performance of the ML model.

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Coupling machine learning and synthetic image DIC-based techniques for the calibration of elastoplastic constitutive models, Prates et al. 2023
Acknowledgments
• This research has received funding from the Research Fund for
Coal and Steel under grant agreement No 888153.
• This research was also sponsored by FEDER funds through the
program COMPETE (Programa Operacional Factores de
Competitividade), by national funds through FCT (Fundação
para a Ciência e a Tecnologia) under the projects
UIDB/00481/2020, UIDP/00481/2020, CENTRO-01-0145-FEDER-
022083 and LA/P/0104/2020.
• J. Henriques was supported by a grant for scientific research
from the Portuguese Foundation for Science and Technology
(ref. 2021.05692.BD).

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