This document presents a methodology for quantifying uncertainty in the inverse identification of plastic material behavior parameters using a heterogeneous mechanical test and digital image correlation. A finite element model updating approach is used to calibrate the Swift power law and Yld2000-2d yield function for DP600 steel. Sources of uncertainty considered include DIC parameters like subset size, step size, and strain window size. Derivatives of the cost function are estimated to analyze how uncertainties propagate and influence the identified material parameters. Preliminary results show smaller errors for Swift law parameters and that step size is the most influential DIC parameter.

1. Uncertainty quantification of inversely
identified plastic material behavior using an
innovative heterogeneous mechanical test
for sheet metal
M Conde1,*, S Coppieters2, A Andrade-Campos1
* marianaconde@ua.pt
1 TEMA, Department of Mechanical Engineering, University of
Aveiro, Portugal
2 Department of Materials Engineering, Ghent Technology
Campus, KU Leuven, Belgium

2. Introduction, framework and literature review
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Inherent calibration
uncertainties
•Numerical simulations
inaccuracy
•Impact the engineering
decision making
Full-field inverse
analysis
•Model calibration
•Material behaviour
reproduction
Heterogeneous
mechanical test
•Several strain and stress
states simultaneously
•DIC techniques
Images sources: https://www.veryst.com/services/testing/specialized-tools-and-techniques/digital-image-correlation ; https://unsplash.com/photos/t9DooibgMEk ; https://unsplash.com/photos/40rACsmtNp8
• Usually, FEMU and
VFM analysis from
the literature do
not account for
uncertainty
quantification

3. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Identification problem under analysis
FEMU approach driven by the minimization of:
𝜑(𝝌) =
1
𝑛t
𝑖=1
𝑛t 1
3𝑛p 𝑗=1
𝑛p ߝ𝑥𝑥
num(𝝌)−ߝ𝑥𝑥
exp
ߝmax
exp
2
+
ߝ𝑦𝑦
num(𝝌)−ߝ𝑦𝑦
exp
ߝmax
exp
2
+
ߝ𝑥𝑦
num(𝝌)−ߝ𝑥𝑦
exp
ߝmax
exp
2
𝑗
+
Fnum
(𝝌)−Fexp
Fmax
exp
2
𝑖
𝝌 =[𝐾, 𝜀0, ν, 𝛼1, 𝛼2, 𝛼3, 𝛼4 , 𝛼5 , 𝛼6, 𝛼7, 𝛼8, 𝑛]
For the calibration of the Swift law and Yld2000-2d function

4. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Identification problem under analysis
Identification of 11 material parameters
More than 1000 evaluations
Large computational time
Uncertainty quantification method
Monte Carlo simulations
Importance sampling
Polynomial chaos expansion (PCE)
Fuzzy theory
Problem simplification

5. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
KKT conditions for problem simplification and error estimation
𝑓(𝑥)
𝑓1
𝑥∗
𝜕𝑓1(𝑥)
𝜕𝑥 𝑥=𝑥∗
= 0
𝜕𝑓2(𝑥)
𝜕𝑥 𝑥=𝑥∗
≠ 0
𝑓2
𝑥
Errors
[1] Kuhn H and Tucker A 1956 Princeton University Press, Princeton, NJ
First-order necessary conditions for a solution to be optimal: ∇𝑓 𝑥∗
= 𝟎
Non-linear unconstrained optimization problem
[1]

6. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
KKT conditions for problem simplification and error estimation
For a simple optimization problem with cost function 𝑓(𝑥) and optimum solution 𝑥∗
and source of
uncertainty 𝜃:
𝜕𝑓(𝑥)
𝜕𝑥 𝑥=𝑥∗
= 0
𝜕
𝜕𝜃
𝜕𝑓(𝑥)
𝜕𝑥 𝑥=𝑥∗
≠ 0
Known optimum solution 𝑥∗
Error estimation due to the variation 𝜕𝜃
For the FEMU optimization problem with cost function 𝜑(𝝌) and reference (known) material
parameters 𝝌∗
and several sources of uncertainty 𝜽:
𝜕𝜑(𝝌)
𝜕𝝌 𝝌=𝝌∗
≈ 0
Known material parameters 𝝌∗
𝜕
𝜕𝜽
𝜕𝜑(𝝌)
𝜕𝝌 𝝌=𝝌∗
≠ 0
Error estimation due to the different variations 𝜕𝜽

7. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
FEMU cost function derivative estimation
Finite difference method of approximation with 10%, 5%, 1% and 0.1% of
perturbation for the material parameter
𝜕ߝ𝑥𝑥
num
(𝝌)
𝜕𝝌
𝜕ߝ𝑦𝑦
num
(𝝌)
𝜕𝝌
𝜕ߝ𝑥𝑦
num
(𝝌)
𝜕𝝌
𝜕Fnum
(𝝌)
𝜕𝝌
𝜕𝜑(𝝌)
𝜕𝝌 𝝌=𝝌∗
= ?

8. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Heterogeneous mechanical test under consideration
P1 P2 P3 P4 P5 P6
xx
[mm]
0.000 3.708 23.092 23.505 15.889 3.086
yy
[mm]
10.000 11.421 31.784 17.078 5.163 0.000
[2] M. Conde, A. Andrade-Campos, M. G. Oliveira, and J. M. P. Martins, “Design of heterogeneous interior notched specimens for material mechanical characterization,”
in Esaform 2021 - 24th International Conference on Material Forming, 2021.
[2]

9. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Material constitutive model under consideration
[2]
[3] F. Ozturk, S. Toros, and S. Kilic, “Effects of anisotropic yield functions on prediction of forming limit diagrams of DP600 advanced high strength steel,” Procedia Eng.,
vol. 81, no. October, pp. 760–765, 2014 .
Hooke’s E [GPa] ν
Law 210 0.3
Swift K [MP] ߝ0 n
Law 979.460 0.00535 0.194
Yld α1 α2 α3 α4
2000- 1.011 0.964 1.191 0.995
2d α5 α6 α7 α8 n
1.010 1.018 0.977 0.935 6
[3]
DP600 dual-phase steel
[2] M. Conde, A. Andrade-Campos, M. G. Oliveira, and J. M. P. Martins, “Design of heterogeneous interior notched specimens for material mechanical characterization,”
in Esaform 2021 - 24th International Conference on Material Forming, 2021.

10. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
FEA and DIC test simulation
Abaqus software
• Plane stress state conditions
• 2975 CPS4R elements
• UMMDp JANCAE
• Imposed vertical displacement up to
rupture
• Used 19 time intervals

11. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
FEA and DIC test simulation
MatchID 2D software
• Synthetically deformed speckle
pattern
• Pattern numerically generated with
3px dot size
• Performance analysis to access the
adequate DIC parameters

12. Methodology and implementation
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Sources of uncertainty
Subset
Virtual strain gauge
• Subset size = 21, 25 px
• Step size = 5, 6 px
• Strain window size = 11, 13 data points
Approximately 20% of perturbation

13. 41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Step
• Subset size = 21, 25 px
• Step size = 5, 6 px
• Strain window size = 11, 13 data points
Approximately 20% of perturbation
Methodology and implementation
Sources of uncertainty
Subset
Virtual strain gauge

14. 41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Step
Strain
window
• Subset size = 21, 25 px
• Step size = 5, 6 px
• Strain window size = 11, 13 data points
Approximately 20% of perturbation
Methodology and implementation
Sources of uncertainty
Subset
Virtual strain gauge

15. 41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Step
Strain
window
• Subset size = 21, 25 px
• Step size = 5, 6 px
• Strain window size = 11, 13 data points
Approximately 20% of perturbation
Methodology and implementation
Sources of uncertainty
Subset
Virtual strain gauge

16. Obtained results and analysis
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
Different perturbation sizes

17. Obtained results and analysis
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
1% material parameter perturbation

18. Obtained results and analysis
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
1% material parameter perturbation

19. Main conclusions
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
• A methodology was developed to relatively quantify the uncertainties in material parameter
identification
• Smaller errors were observed with the Swift law parameters probably because the used mechanical
test was design for this material model
• The KKT conditions are not verified with all the material parameters due to errors coming from
measurements, interpolation, discretization of the problem, or others.
• The uncertainties estimated in the identification process are a cause of the propagation of errors
previously mentioned and of the influence of each source of uncertainty (subset, step and strain
window sizes) in the material parameters
• The step size is the analyzed DIC parameter that most influences the FEMU problem

20. Main conclusions
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
• A methodology was developed to relatively quantify the uncertainties in material parameter
identification
• Smaller errors were observed with the Swift law parameters probably because the used mechanical
test was design for this material model
• The KKT conditions are not verified with all the material parameters due to errors coming from
measurements, interpolation, discretization of the problem, or others.
• The uncertainties estimated in the identification process are a cause of the propagation of errors
previously mentioned and of the influence of each source of uncertainty (subset, step and strain
window sizes) in the material parameters
• The step size is the analyzed DIC parameter that most influences the FEMU problem

21. Main conclusions
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
• A methodology was developed to relatively quantify the uncertainties in material parameter
identification
• Smaller errors were observed with the Swift law parameters probably because the used mechanical
test was design for this material model
• The KKT conditions are not verified with all the material parameters due to errors coming from
measurements, interpolation, discretization of the problem, or others.
• The uncertainties estimated in the identification process are a cause of the propagation of errors
previously mentioned and of the influence of each source of uncertainty (subset, step and strain
window sizes) in the material parameters
• The step size is the analyzed DIC parameter that most influences the FEMU problem

22. Main conclusions
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
• A methodology was developed to relatively quantify the uncertainties in material parameter
identification
• Smaller errors were observed with the Swift law parameters probably because the used mechanical
test was design for this material model
• The KKT conditions are not verified with all the material parameters due to errors coming from
measurements, interpolation, discretization of the problem, or others.
• The uncertainties estimated in the identification process are a cause of the propagation of errors
previously mentioned and of the influence of each source of uncertainty (subset, step and strain
window sizes) in the material parameters
• The step size was the analyzed DIC parameter that most influences the FEMU problem

23. Main conclusions
41st International Deep Drawing Research Group Conference – IDDRG 2022
M Conde, S Coppieters and A Andrade- Campos, Uncertainty quantification of inversely identified plastic material behavior using an innovative heterogeneous mechanical test for sheet metal
• A methodology was developed to relatively quantify the uncertainties in material parameter
identification
• Smaller errors were observed with the Swift law parameters probably because the used mechanical
test was design for this material model
• The KKT conditions are not verified with all the material parameters due to errors coming from
measurements, interpolation, discretization of the problem, or others.
• The uncertainties estimated in the identification process are a cause of the propagation of errors
previously mentioned and of the influence of each source of uncertainty (subset, step and strain
window sizes) in the material parameters
• The step size was the analyzed DIC parameter that most influences the FEMU problem

24. Uncertainty quantification of inversely identified
plastic material behavior using an innovative
heterogeneous mechanical test for sheet metal
M Conde1,*, S Coppieters2, A Andrade-Campos1
* marianaconde@ua.pt
This project has received funding from the Research Fund for Coal and Steel under
grant agreement No 888153. The authors also acknowledge the financial support of
the Portuguese Foundation for Science and Technology (FCT) under the project
PTDC/EME-APL/29713/2017 by UE/FEDER through the programs CENTRO 2020 and
COMPETE 2020, and UID/EMS/00481/2013-FCT under CENTRO-01-0145-FEDER-
022083. Mariana Conde is grateful to the Portuguese Foundation for Science and
Technology (FCT) for the PhD grant 2021.06115.BD.
Thank you!