Theoretische Elektrotechnik
0
Virtual Process Design for Coupled Quasi-‐
Static and Electromagnetic Forming
Marco Rozgic̀ and Marcus Stiemer
Theoretische Elektrotechnik
0 Quasi-‐Static Forming
• Quasi-‐static forming is restricted by the forming limit
• Forming beyond limit is possible by high speed forming
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Minor strain ϵ
2Ma jo r st ra in ϵ
1F~ F~
F~ F~
Theoretische Elektrotechnik
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Electromagnetic Impulse Forming
• Electromagnetic impulse forming with pulsed currents (e.g. 30kA within 10µs)
→ magnetic flux between tool coil and workpiece: 1-‐10 Tesla
• Induced current results in Lorentz forces
→ forming
Theoretische Elektrotechnik
0 Combined Forming
0.4 0.2 0 0.2 0.4
0 0.2 0.4 0.6
Minor Strain [-]
M a jor S tr ai n [- ]
Theoretische Elektrotechnik
0 Combined Forming
0.4 0.2 0 0.2 0.4
0 0.2 0.4 0.6
Minor Strain [-]
M a jor S tr ai n [- ]
• Combination of both technologies yields forming beyond quasi static forming limits
• Reduction of wear by tool integration
• Forming of high-‐strength materials
Theoretische Elektrotechnik
0 Combined Forming
0.4 0.2 0 0.2 0.4
0 0.2 0.4 0.6
Minor Strain [-]
M a jor S tr ai n [- ]
• Combination of both technologies yields forming beyond quasi static forming limits
• Reduction of wear by tool integration
• Forming of high-‐strength materials
• Process is subject to many parameters
• Only careful adjustments of involved parameters yield good results
• Economic process design necessary
Theoretische Elektrotechnik
0 Combined Forming
0.4 0.2 0 0.2 0.4
0 0.2 0.4 0.6
Minor Strain [-]
M a jor S tr ai n [- ]
• Combination of both technologies yields forming beyond quasi static forming limits
• Reduction of wear by tool integration
• Forming of high-‐strength materials
• Process is subject to many parameters
• Only careful adjustments of involved parameters yield good results
• Economic process design necessary
➡ Virtual process design to overcome drawbacks!
Theoretische Elektrotechnik
0 Virtual Process Design
Theoretische Elektrotechnik
0 Virtual Process Design
Enhancement by virtual process
design
Theoretische Elektrotechnik
0 Virtual Process Design
Enhancement by virtual process
design
Adjustment of
parameters defining the
process
Theoretische Elektrotechnik
0 Virtual Process Design
Enhancement by virtual process
design
Adjustment of
parameters defining the process
Tune material model for
good simulaXon
Theoretische Elektrotechnik
0 Virtual Process Design
Enhancement by virtual process
design
Adjustment of
parameters defining the process
MathemaXcal opXmizaXon Tune material model for
good simulaXon
Theoretische Elektrotechnik
0 Virtual Process Design
Enhancement by virtual process
design
Adjustment of
parameters defining the process
MathemaXcal opXmizaXon Tune material model for
good simulaXon
Theoretische Elektrotechnik
0 Constitutive Material Model
Theoretische Elektrotechnik
0 Constitutive Material Model
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S,
backstress tensor X and stress-‐like tensors Y , Y kin
Theoretische Elektrotechnik
0 Constitutive Material Model
Plastic flow rule
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S, backstress tensor X and stress-‐like tensors Y , Y kin
• Plastic flow rule, for the Cauchy-‐Green Tensor C
Theoretische Elektrotechnik
0 Constitutive Material Model
Plastic flow rule
Kinematic hardening
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S, backstress tensor X and stress-‐like tensors Y , Y kin
• Plastic flow rule, for the Cauchy-‐Green Tensor C
Theoretische Elektrotechnik
0 Constitutive Material Model
Plastic flow rule
Kinematic hardening
Isotropic hardening
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S, backstress tensor X and stress-‐like tensors Y , Y kin
• Plastic flow rule, for the Cauchy-‐Green Tensor C
• Evolution equations for kinematic and isotropic hardening
Theoretische Elektrotechnik
0 Constitutive Material Model
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S, backstress tensor X and stress-‐like tensors Y , Y kin
• Plastic flow rule, for the Cauchy-‐Green Tensor C
• Evolution equations for kinematic and isotropic hardening
• Yield function of Hill-‐type
Theoretische Elektrotechnik
0 Constitutive Material Model
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S, backstress tensor X and stress-‐like tensors Y , Y kin
• Plastic flow rule, for the Cauchy-‐Green Tensor C
• Evolution equations for kinematic and isotropic hardening
• Yield function of Hill-‐type
• Rate dependent Perzyna formulation (high-‐speed part)
High-‐speed part
Theoretische Elektrotechnik
0 Constitutive Material Model
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S, backstress tensor X and stress-‐like tensors Y , Y kin
• Plastic flow rule, for the Cauchy-‐Green Tensor C
• Evolution equations for kinematic and isotropic hardening
• Yield function of Hill-‐type
• Rate dependent Perzyna formulation (high-‐speed part)
• Kuhn-‐Tucker conditions for the plastic multipliers (quasi-‐static part)
Quasi-‐static part
High-‐speed part
Theoretische Elektrotechnik
0 Constitutive Material Model
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S, backstress tensor X and stress-‐like tensors Y , Y kin
• Plastic flow rule, for the Cauchy-‐Green Tensor C
• Evolution equations for kinematic and isotropic hardening
• Yield function of Hill-‐type
• Rate dependent Perzyna formulation (high-‐speed part)
• Kuhn-‐Tucker conditions for the plastic multipliers (quasi-‐static part)
• Scalar damage variable (Lamaitre type)
Theoretische Elektrotechnik
0 Constitutive Material Model
Ingredients:
• Equations for second order Piola-‐Kirchhoff stress tensor S, backstress tensor X and stress-‐like tensors Y , Y kin
• Plastic flow rule, for the Cauchy-‐Green Tensor C
• Evolution equations for kinematic and isotropic hardening
• Yield function of Hill-‐type
• Rate dependent Perzyna formulation (high-‐speed part)
• Kuhn-‐Tucker conditions for the plastic multipliers (quasi-‐static part)
• Scalar damage variable (Lamaitre type)
• Effective stress contributions
Theoretische Elektrotechnik
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Parameter of the Constitutive Material Model
• Isotropic hardening parameters in the yield function
Theoretische Elektrotechnik
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Parameter of the Constitutive Material Model
• Isotropic hardening parameters in the yield function
y
Q
true strain [-]
true stress [MP a]
Theoretische Elektrotechnik
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Parameter of the Constitutive Material Model
• Isotropic hardening parameters in the yield function
• Kinematic hardening parameters
Theoretische Elektrotechnik
0
Parameter of the Constitutive Material Model
• Isotropic hardening parameters in the yield function
• Kinematic hardening parameters
Theoretische Elektrotechnik
0
Parameter of the Constitutive Material Model
• Isotropic hardening parameters in the yield function
• Kinematic hardening parameters
• Damage rate and threshold parameters
Theoretische Elektrotechnik
0
Parameter of the Constitutive Material Model
• Isotropic hardening parameters in the yield function
• Kinematic hardening parameters
• Damage rate and threshold parameters
0 100 200 300 400 500 600 700 800 900
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Cauchy stress σ11 [MPa]
Logarithmic strain lnV11 [-]
Flow curves - dependence on s (k=1, pd=0)
s=0.01 s=0.1 s=1 s=10 s=100 undamaged s
0 100 200 300 400 500 600 700 800 900
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Cauchy stress σ11 [MPa]
Logarithmic strain lnV11 [-]
Flow curves - dependence on k (s=1, pd=0)
k=0.01 k=0.1 k=1 k=10 k=100 undamaged
k
Theoretische Elektrotechnik
0
Parameter of the Constitutive Material Model
• Isotropic hardening parameters in the yield function
• Kinematic hardening parameters
• Damage rate and threshold parameters
• Challenge: Also identify the elastic modulus E of the material under consideration
Theoretische Elektrotechnik
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Parameter of the Constitutive Material Model
• Isotropic hardening parameters in the yield function
• Kinematic hardening parameters
• Damage rate and threshold parameters
• Challenge: Also identify the elastic modulus E of the material under consideration
→End up with a total of 9 parameters to be identified
Theoretische Elektrotechnik
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Identification by Non-‐Linear Optimization
• Parameters are identified by fitting the model to experimental force-‐displacement curves
Theoretische Elektrotechnik
0
Identification by Non-‐Linear Optimization
• Parameters are identified by fitting the model to experimental force-‐displacement curves
Theoretische Elektrotechnik
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Identification by Non-‐Linear Optimization
• Parameters are identified by fitting the model to experimental force-‐displacement curves
0.5 0 0.5 1 1.5 2 2.5
0 500 1,000
u x [mm]
f tot [N ]
0.5 0 0.5 1 1.5 2 2.5
0 500 1,000
u x [mm]
f tot [N ]
Theoretische Elektrotechnik
0
Identification by Non-‐Linear Optimization
• Parameters are identified by fitting the model to experimental force-‐displacement curves
Theoretische Elektrotechnik
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Identification by Non-‐Linear Optimization
• Parameters are identified by fitting the model to experimental force-‐displacement curves
• Non-‐linear objective function to identify optimal parameter vector p
F (p) = 1
2 (u N u 1 )
N X 1
i =1
(u i +1 u i ) h
f i sim +1 (p) f i exp +1 2 + f i sim (p) f i exp 2 i
Theoretische Elektrotechnik
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Identification by Non-‐Linear Optimization
• Parameters are identified by fitting the model to experimental force-‐displacement curves
• Non-‐linear objective function to identify optimal parameter vector p
F (p) = 1
2 (u N u 1 )
N X 1
i =1
(u i +1 u i ) h
f i sim +1 (p) f i exp +1 2 + f i sim (p) f i exp 2 i Initial parameter set
Simulation of tensile test with LS-DYNA
Evaluate F (p) with ⇣
f sim, u ⌘ from simulation
Scripts that process the LS-
DYNA output
Optimal solution?
EXIT Choose new parameter set
Numerical derivatives LBFGS update
for Hessian approximation
IPOPT
n
y
Theoretische Elektrotechnik
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Identification by Non-‐Linear Optimization
0.5 0 0.5 1 1.5 2 2.5
0 500 1,000
u
x[mm]
f
tot[N ]
Experimental Data Optimized Simulation
0.5 0 0.5 1 1.5 2 2.5
0 500 1,000
u
x[mm]
f
tot[N ]
Experimental Data Optimized Simulation After 3 iteration steps After 12 iteration steps
0 10 20 30 40
45 50 55
Iteration No.
p F [N ]
Theoretische Elektrotechnik
0 Validation and Verification
Theoretische Elektrotechnik
0 Validation and Verification
• Comparison to stress-‐strain curves
(with evolution model for damage
threshold)
Theoretische Elektrotechnik
0 Validation and Verification
• Comparison to stress-‐strain curves (with evolution model for damage threshold)
• Application to complex situation
(cup drawing)
Theoretische Elektrotechnik
0
Combined Deep Drawing of a Cup
Theoretische Elektrotechnik
0
Combined Deep Drawing of a Cup
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0
0.1 0.2 0.3 0.4 0.5
Minor Strain [-]
Ma jo r St ra in [- ]
Theoretische Elektrotechnik
0 Virtual Process Design
Enhancement by virtual process
design
Adjustment of
parameters defining the process
MathemaXcal opXmizaXon Tune material model for
good simulaXon
Theoretische Elektrotechnik
0
Process Optimization in Cup Forming
Theoretische Elektrotechnik
0
Process Optimization in Cup Forming
• Only the first half wave is relevant for forming
Theoretische Elektrotechnik
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Process Optimization in Cup Forming
• Only the first half wave is relevant for forming
→ Remaining energy absorbed by coils
• Try novel approach to reduce wear and energy consumption
Theoretische Elektrotechnik
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Process Optimization in Cup Forming
• Only the first half wave is relevant for forming
→ Remaining energy absorbed by coils
• Try novel approach to reduce wear and energy consumption
→ Double exponential pulse
0 0.4 0.8 1.2
· 10
20
0.5 1
· 10
4Time t in [s]
I ( t )i n[ A ]
Theoretische Elektrotechnik
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Process Optimization in Cup Forming
Theoretische Elektrotechnik
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Process Optimization in Cup Forming
• Maximize the radius at bottom edge
Theoretische Elektrotechnik
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Process Optimization in Cup Forming
• Maximize the radius at bottom edge
→ Maximize the first principle strain
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0
0.1 0.2 0.3 0.4 0.5
Minor Strain [-]
Ma jo r St ra in [- ]
Theoretische Elektrotechnik
0
Process Optimization in Cup Forming
• Maximize the radius at bottom edge
→ Maximize the first principle strain
• No damage occurs
→ Constrain the damage variable in all elements
Theoretische Elektrotechnik
0
Process Optimization in Cup Forming
• Maximize the radius at bottom edge
→ Maximize the first principle strain
• No damage occurs
→ Constrain the damage variable in all elements
• Current must be technically reasonable
→ Constrain the current at each time step (i.e. 125 000 A)
Theoretische Elektrotechnik
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Process Optimization in Cup Forming
• Maximize the radius at bottom edge
→ Maximize the first principle strain
• No damage occurs
→ Constrain the damage variable in all elements
• Current must be technically reasonable
→ Constrain the current at each time step (i.e. 125 000 A)
Theoretische Elektrotechnik
0 Results
1 1.01 1.02
· 10
20
1 2 3 4
· 10
4Time t in [s]
Cu rr en t I ( t )i n[ A ]
Optimized Pulse
Starting Pulse
Theoretische Elektrotechnik
0 Conclusion & Outlook
Theoretische Elektrotechnik
0 Conclusion & Outlook
• Introduction of a framework for virtual process design in the context of combined quasi-‐static
and electromagnetic impulse forming, featuring:
Theoretische Elektrotechnik
0 Conclusion & Outlook
• Introduction of a framework for virtual process design in the context of combined quasi-‐static and electromagnetic impulse forming, featuring:
✓ Automatic scheme for parameter identification in material models based on experimental data
Theoretische Elektrotechnik
0 Conclusion & Outlook
• Introduction of a framework for virtual process design in the context of combined quasi-‐static and electromagnetic impulse forming, featuring:
✓ Automatic scheme for parameter identification in material models based on experimental data
✓ Linearization based scheme for process parameter identification
Theoretische Elektrotechnik
0 Conclusion & Outlook
• Introduction of a framework for virtual process design in the context of combined quasi-‐static and electromagnetic impulse forming, featuring:
✓ Automatic scheme for parameter identification in material models based on experimental data
✓ Linearization based scheme for process parameter identification
• First steps have been taken, but:
Theoretische Elektrotechnik
0 Conclusion & Outlook
• Introduction of a framework for virtual process design in the context of combined quasi-‐static and electromagnetic impulse forming, featuring:
✓ Automatic scheme for parameter identification in material models based on experimental data
✓ Linearization based scheme for process parameter identification
• First steps have been taken, but:
➡ Verification of computed process parameters by experiments
Theoretische Elektrotechnik
0 Conclusion & Outlook
• Introduction of a framework for virtual process design in the context of combined quasi-‐static and electromagnetic impulse forming, featuring:
✓ Automatic scheme for parameter identification in material models based on experimental data
✓ Linearization based scheme for process parameter identification
• First steps have been taken, but:
➡ Verification of computed process parameters by experiments
➡ Taking into account more process parameters at the same time, control of quasi-‐static part and
electromagnetic part simultaneously
Theoretische Elektrotechnik
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Special thanks to:
Theoretische Elektrotechnik
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