ESAFORM 2022 - 25th International Conference on Material Forming R. Lourenço, A. Andrade-Campos, P. Georgieva Dept. of Mechanical Engineering Centre for Mechanical Technology and Automation, University of Aveiro, Portugal Dept. of Electronics Telecommunications and Informatics Institute of Electronics and Informatics Engineering, University of Aveiro, Portugal 27-29 April, Braga, Portugal

1. 25th International Conference on Material Forming
The Virtual Fields Method to indirectly train ANNs for
implicit constitutive modelling
ESAFORM 2022
R. Lourenço1,a, A. Andrade-Campos1,b, P. Georgieva2,c
1Dept. of Mechanical Engineering
Centre for Mechanical Technology and Automation, University of Aveiro, Portugal
2Dept. of Electronics Telecommunications and Informatics
Institute of Electronics and Informatics Engineering, University of Aveiro, Portugal
arubenl@ua.pt, bgilac@ua.pt, cpetia@ua.pt
27-29 April, Braga, Portugal
EUROPEAN UNION
European Social Fund

2. 25th International Conference on
Material Forming
Session outline
ESAFORM 2022
• Introduction
• Implicit constitutive modelling (direct and indirect training)
• Coupling the Virtual Fields Method with an ANN
• Numerical example
• Conclusions
1
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

3. Introduction
• Constitutive models rely on empirical parameters that need to be
calibrated
• Complex models with a higher number of parameters require
expensive and time-consuming experimental campaigns
• Artificial Neural Networks (ANNs) have the potential to provide a
radically different approach to the field
• ANNs can implicitly learn patterns directly from data, without assuming
a mathematical formulation or identifying parameters
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

4. Implicit constitutive modelling
Direct approach
• Most common approach in the literature
• Data is numerically generated
• Labelled data pairs (stress-strain)
• Easier to train with a ground-truth value
• Variable to predict not always obtainable in a real experimental setting
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

5. Implicit constitutive modelling
Indirect approach
• Not widely used
• Data numerically generated or from full-field measurements
• Relies only on directly measurable data (e.g. displacements, global force)
• Variable to predict is indirectly obtained from measurable or intermediate
variables
• Harder to train
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

6. Coupled FEM-ANN model1
1Liu, X, et al., Learning nonlinear constitutive laws using neural network models based on indirectly measurable data. J. Appl. Mech.,
Transactions ASME 2020, vol. 87, pp 1–8.
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

7. Coupled VFM-ANN model
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

8. Numerical example - heterogeneous test1
1J.M.P. Martins, A. Andrade-Campos, S. Thuillier. Comparison of inverse identification strategies for constitutive mechanical models
using full-field measurements, Int. J. Mech. Sci., vol. 145, pp 330–345, 2018
Swift’s hardening law
Surface traction distribution
Global force
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

9. • Numerically generated on Abaqus
• Mesh: 9x9 CPS4R elements
• 1000 time points
• Elastic and full-elastoplastic responses
Numerical example - data generation
Training set Validation set
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

10. Numerical example - ANN model
PReLU1
1. K. He, et al., Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet
Classification, 2015 IEEE International Conference on Computer Vision (ICCV). IEEE, 2015
• Adam optimizer
• LR scheduler (ReduceLROnPlateau)
• Early-stopping
• Epochs
• 20 epochs (elastic model)
• 77 epochs (elastoplastic model)
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

11. 25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

12. Numerical example – learning curves
Elastic model Elastoplastic model
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

13. Numerical example – validation results
• Overall, good predictions for stress along 𝑥-direction
• Notably lower sensitivity to the stress responses along 𝑦 and 𝑥𝑦
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

14. Numerical example – virtual fields
* *
* *
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
* Virtual fields used to train the elastic model

15. Numerical example – validation results
• Improved shear stress prediction for the elastic response, slight
improvement in the elastoplastic response
• Low sensitivity to stress along 𝑦-direction
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

16. Conclusions
• A coupled VFM-ANN system was proposed for implicit constitutive modelling
• Only directly measurable data is used. FEM not required for further calculations
• Elastic and elastoplastic response models were trained for a given material using
manually-defined virtual fields
• In general, model predictions were good for stress along x-direction
• Poor stress predictions along 𝑦 and 𝑥𝑦 -directions possibly related to:
• manually-defined VFs – tied to the user experience
• unconstrained optimization – multiple possible solutions, some of them with
no physical meaning
• mechanical test - not rich enough
25th International Conference on
Material Forming
ESAFORM 2022
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R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

17. Rúben Lourenço acknowledges the Portuguese Foundation for Science and Technology (FCT) for the financial support provided through the grant
2020.05279.BD, co-financed by the European Social Fund, through the Regional Operational Programme CENTRO 2020.
The authors also gratefully acknowledge the financial support of the Portuguese Foundation for Science and Technology (FCT) under the projects
CENTRO-01-0145-FEDER-029713, POCI-01-0145-FEDER-031243 and POCI-01-0145-FEDER-030592 by UE/FEDER through the programs CENTRO
2020 and COMPETE 2020, and UIDB/00481/2020 and UIDP/00481/2020-FCT under CENTRO-01-0145-FEDER-022083.
This project also supported by the Research Fund for Coal and Steel under grant agreement No 888153.
25th International Conference on Material Forming
ESAFORM 2022
27-29 April, Braga, Portugal
Thank you for your attention!
EUROPEAN UNION
European Social Fund