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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
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
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
2
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
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
3
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
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
4
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
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
5
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
Coupled VFM-ANN model
25th International Conference on
Material Forming
ESAFORM 2022
6
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
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
7
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
• 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
8
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
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
9
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
25th International Conference on
Material Forming
ESAFORM 2022
10
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
Numerical example – learning curves
Elastic model Elastoplastic model
25th International Conference on
Material Forming
ESAFORM 2022
11
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
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
12
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
Numerical example – virtual fields
* *
* *
25th International Conference on
Material Forming
ESAFORM 2022
13
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
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
14
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
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
15
R. Lourenço, A. Andrade-Campos, P. Georgieva
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling
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
The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

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The Virtual Fields Method to indirectly train ANNs for implicit constitutive modelling

  • 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 2 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 3 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 4 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 5 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 6 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 7 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 8 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 9 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 10 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 11 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 12 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 13 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 14 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 15 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