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Multi-Objective Design Optimisation of Stamping Process for Advanced High Strength Steels

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The paper investigates the multi-objective design optimisation of stamping process to control both the shape and quality of final Advanced High Strength Steels (AHSSs) in terms of springback and safety using Distributed Multi-Objective Evolutionary Algorithm (DMOGA) coupled with Finite Element Analysis (FEA) based stamping analyser. The design problem of stamping process is formulated to minimise the difference between the desired shape and the final geometry obtained by a numerical simulation accounting elastic springback. In addition, the final product quality is maximised by improving safety factor without winkling, thinning, or failure. Numerical results show that a proposed methodology improves the final product quality while reducing its springback.
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Multi-Objective Design Optimisation of Stamping Process for Advanced High Strength Steels
DongSeop Lee1,4, Jordi Pons-Prats1, Hector Espinoza2, Oscar Fruitos2, Eugenio ate1,3
1 Centre Internacional de Metodes Numerics en Enginyeria (CIMNE), Barcelona 08034, Spain
2CIMNE - MetalForm: Technology Transfer Department, Castelldefels 08860, Spain
3Universitat Politecnica de Catalunya, Barcelona 08034, Spain
4Deloitte Analytics Deloitte Consulting LLC, Seoul, Korea
Abstract
The paper investigates the multi-objective design optimisation of stamping process to control both the
shape and quality of final Advanced High Strength Steels (AHSSs) in terms of springback and safety
using Distributed Multi-Objective Evolutionary Algorithm (DMOGA) coupled with Finite Element
Analysis (FEA) based stamping analyser. The design problem of stamping process is formulated to
minimise the difference between the desired shape and the final geometry obtained by a numerical
simulation accounting elastic springback. In addition, the final product quality is maximised by
improving safety factor without winkling, thinning, or failure. Numerical results show that a proposed
methodology improves the final product quality while reducing its springback.
1. Introduction
One of important challenges in mechanical and manufacturing fields is to guarantee the final product
quality while controlling the final shape. In particular, for stamping processes numerical simulations
are commonly used to design a part to ensure that it is possible to arrive to the final desired shape
without too much stress (failure zone), without too small material thickness (thinning zone) or without
strong wrinkles (wrinkling zone) [1, 2]. The springback behaviour depends on the geometric stiffness
of the part as well as on the inner residual stresses. Those are a complex function of the stretching and
bending history and of the material properties [3 -6]. In this paper, the springback behaviour of
Advanced High Strength Steels (AHSS) shown in NUMISHEET11 [7] is simulated and analysed by
using Finite Element Analysis (FEA) based stamping analyser; named Stampack [8].
Advanced High Strength Steels have been focused recently in the automotive industry. AHSS has
superiority of the strength to weight ratio that improves fuel efficiency and crashworthiness assessment
of vehicles. However, the major drawback of the automotive structural member with AHSS is the
tendency of the large amount of springback due to the high yield strength and the tensile strength. In
this paper, one AHSS - DP780 with two hardening models: Nadai Ludwik and Voce, with constant
thickness is considered [9, 10]. Stamping process of AHSS is optimised using Multi-Objective
Evolutionary Algorithms. The proposed method couples computational intelligence system; RMOP
developed at CIMNE [11 -13] and Stampack. The paper shows how stamping process design
optimization can reduce the global stringback and also how to improve the global quality of final ducts.
The rest of paper is organized as follows; Section 2 describes a methodology. Section 3 presents FEA
based stamping analyser; Stampack. Two AHSS stamping process design optimizations are conducted
in Section 4. Section 5 concludes overall numerical simulation and presents future research avenues.
2. Methodology
2.1 Multi-Objective Design Optimisation
Often, engineering design problems require a simultaneous optimisation of conflicting objectives
and an associated number of constraints. Unlike single objective optimisation problems, the solution is
a set of points known as Pareto optimal set. Solutions are compared to other solutions using the concept
of Pareto dominance. A multi-criteria optimisation problem can be formulated as:
Maximise/Minimise
fi(x)i i = 1,…,n
Subject to constraints:
gj(x) = 0 and hk(x) 0 j = 1,…,m k = 1,…,l
where fi, gj, hk are, respectively, the objective functions, the equality and the inequality constraints. n is
the number of objective functions and x is an n dimensional vector where its arguments are the
decision variables. For a minimisation problem, a vector x1 is said partially less than vector x2 if:
() ()  () ()
In this case the solution x1 dominates the solution x2.
As Genetic Algorithms (GAs) evaluate multiple populations of points, they are capable of finding a
number of solutions in a Pareto set. Pareto selection ranks the population and selects the non-
dominated individuals for the Pareto fronts. A Genetic Algorithm that has capabilities for multi-
objective optimisation is termed Multi-Objective Genetic Algorithms (MOGAs). Theory and
applications of MOGAs can be found in References [14, 15].
2.2 Robust Multi-objective Optimisation Platform (RMOP)
For the optimization, the Multi-Objective Genetic Algorithm (MOGA) module in Robust Multi-
objective Optimisation Platform (RMOP) developed in CIMNE is utilized to minimize global
springback and to maximize global safety zone under distributed/parallel computing environment.
Details of RMOP can be found in references [11 -13]. The design variable taken into account is the
holding force during holding and forming stages.
3. Stamping Analysis Tool: Stampack
The methodology used for the implementation of Basic Shell Triangle (BST) finite elements considers
an actualized Lagrangian movement description and a hipoelastic constitutive model [16]. The main
computational implementation aspects are considered as follows:
The acceleration is evaluated by:
= 
  
 (4.103)
The acceleration is integrated in order to obtain the velocity in t


=

+
t() (4.104)
The velocity is integrated in order to find the global displacements in the deformed
configuration t

 =
+

t
With the movement
 the actualization of geometry is possible. The displacement
increasing
and the new position of the finite elements mesh

= 

 = 
In each element, and considering the discretization in the thickness, the constitutive equation
integration is done:
o The increase of the global displacements in each element refers to the local
coordinate system in t + t configuration:
()
=()
 ()
The thickness is actualized using the incompressibility condition:
h()
A()
= h()
 A()

h()
 =A()
h()
A()

Where:
h()
Element thickness
A()
Element area
o The strain increase in each layer at z
respect medium surface is calculated:
(z
)= 

+ z

+

 
Where:

Membranal strain matrix

Flexural strain matrix
o The rate strain tensor

and Jaumann rate stresses are evaluated for each layer.
An elastic stress state is supposed:

=

(4.113)
Where: is the planar lineal elastic constitutive tensor.
The flow criteria is verified in [17], the plastic corrector is calculated and a return
algorithm is applied to obtain the stresses in intermediated configuration

The corrected or elastic stresses are transported to final configuration in t + t and
the local internal forces vector in final configuration in each layer is calculated.
The resultant internal forces vector is transformed to the global coordinated system and
assembled:

 = ()



 ()
Nodal accelerations in t + t configuration, returning to step one, are calculated and by
integration the deformed configuration in t + 2t is obtained.
The stamping process is simulated using Stampack V.6.2.4 developed in CIMNE and commercialized
by QUANTECH ATZ Technology [17]. Stampack is explicit advanced, multipurpose and multistage
simulation software based on FEA. It’s oriented to automotive, aeronautics/aerospace, transport, and
metal packaging.
Stampack offers a library of solid, beam and shell finite elements based on the latest and best
formulations developed at CIMNE as well as those adapted from scientific literature. Different types of
elements can be mixed in a model and special constraints are available to connect solid elements to
beam and shell elements and to link beam with shell elements. Initial conditions for displacements and
velocities can be specified and structural damping can be included in the analysis. The time integration
is performed by the central difference explicit method using automatic time stepping if required. The
program has interfaces to a geometric modeler and pre-processor enabling the easy creation of the
geometry and applying the conditions, constraints and properties, as well as to a graphical post-
processor with which the results can be quickly seen. The program has been validated on a large
number of test and industrial examples in which very good performance and efficiency have been
shown [16].
In this paper, the forming process is started from a flat blank, without considering drawbeads, through
reproducing the forming stage and then finally simulating springback effect. The blank is discretized
with Basic Shell Triangle (BST) consisting three-node triangular shell elements [9] and a penalty
contact algorithm is implemented. An elastoplastic material model is used with anisotropic plane stress
hypothesis based on Hill’s theory in plastic part. The viability and quality of the process determination
is based on metal failure, wrinkling and springback analysis.
Hill’s Theory
Rodney Hill proposed his first yield condition theory in 1948 to take into account anisotropy [8]. It is
basically a generalization of the Von Mises yield condition which is isotropic. The 90 Hill’s yield
criterion [17] is used to model the plastic zone of the steels considered in this work.
Hardening
Hardening is the strengthening of a metal by plastic deformation. Some materials like Carbon steel
generally get stronger when subjected to plastic deformation but with specific behaviors in function of
alloy and thermo mechanic treatment of fabrication or preprocessing press. Some materials loose
strength when deformed plastically (this is known as softening).
Several mathematical models are used to model these behaviors. In the present work, we have used two
hardening models: Ludwik-Nadai model and the Voce model.
Ludwik-Nadai Hardening Model
It is a non-linear hardening model which approximates the plastic stress as: costant times the total
strain with an exponent.
= (+)
Where C is the Ludwik constant, n is the hardening exponent, is the plastic strain, is a strain
needed to reproduce the yield stress when there is not plastic strain.
Voce Hardening Model
It is a hardening model that considers saturation of the hardening. That is the strength does not increase
indefinitely, but reaches a maximum value in an asymptotic regime.
=
Where A is the saturation stress, A-B is the yield stress and c is an exponential constant.
4. Multi-Objective Design Optimisation of Stamping Process
4.1 Formulation of Problem
All tools for die and punch are made of the hardened steel, S45C and the holding force proposed is 784
N (0.08 Tons). The 3D U-Shape model [7] is based on the reference 3. Fixed shape of the rigid die is
applied and one Advanced High Strength Steel (AHSS - DP780) with two hardening models: Nadai-
Ludwik and Voce. Two hardening models were chosen to test the optimization with two different
models. These models might not be the most suitable model for the given material, but as we are
concerned in testing the effects of the holding force on the process, this choice is irrelevant. 1.4 mm
thickness is considered.
This stamping process design problem is based on the benchmark 4 (BM4) considered in Numisheet
2011 conference. This problem is a non-linear and discontinuous therefore, the implementation of
optimisation technique is essential for effective manufacturing (stamping) system.
All tools are made of the hardened steel, S45C. The holding force proposed is around 784 N (0.08
Tons). Figure 1 and Table 1 shows the schematic view of tools and their dimentions in 2D. The
material properties of the Nadai-Ludwik and Voce are shown in Table 2.
Figure 1. Schematic view of tools and parameters for bending test in 2D [7].
Table 1. Dimensions for the 2D parameters bending test [7].
Parameters
W1
W2
W3
W4
R1
R2
Stroke
Dimensions
50.0
54.0
89.0
89.0
5.0
7.0
71.8
Table 2. Material (Steel) properties; HSS and DP780 (Note: Materials have been denoted as Material 1
for the Nadai-Ludwik model and Material 2 for the Voce model.).
Properties
Young’s Modulus (GPa)
Poisson ratio
Density (Kg/m3)
Yield Strength (MPa)
Material 1
198
0.3
7800
795
Material 2
199
0.3
7800
572
For the physical model, a 3D model with shell elements is be used without considering drawbeads. The
half of the geometry due to symmetric model is considered during the simulation.
For the optimization, the Genetic Algorithm (GA) module in Robust Multi-objective Optimisation
Platform (RMOP) developed in CIMNE is utilized to minimize global springback (stage displacement)
and to maximize global safety zone (shown in Table 3 and shown in Figure 2). The design variable
taken into account is the holding force during holding and forming process.
Table 3. Safety range.
Range
0.5 -1.5
1.5 -2.5
2.5 -3.5
3.5 -4.5
4.5 -5.5
5.5 6.5
6.5 -7.5
Safety
Factor
Strong
Wrkl
Wrinkling
Low
Strain
Safe Marginal Thinning Fail
Figure 2. Safety range.
4.2 Multi-Objective Design Optimisation of Springback and Safety Zone
This test case is a multi-objective design optimisaiton using RMOP coupled to Stampack. The
objectives are to minimise global springback and to maxmise global safety factor. The fitness functions
are shown in Equations (1) and (2);
(Global/average stage displacement) (1)
(Global/average safety factor) (2)
where Di and SFj represent the locale strage displacement and local safety factor. The local control
points over physical model; n and m are 92 and 90 respectively.
The reason why the global/average stage dispalcement and safety factor over 90 control points is
considered instead of only one control point (conventional) is to make sure that the optimal design has
lower springback and higher safety factor all over the phycial model.
Numerical Results with Nadai Ludwik Model
The optimization ran for 20 hours with 520 function evaluations in Intel 2 × 3.6GHz processors. Figure
3 shows Pareto optimal front obtained by RMOP. It can be seen that Pareto optimal front produce
lower springback and better safety factor. Pareto optimal members 1 (the best solution for springback),
5 (compromised solution) and 6 (the best solution for safety factor) are selected and compared to the
baseline design.
Figure 3. Pareto optimal front compared to the baseline design for Material 1 case.
Table 3 compares the fitness values obtained by the baseline design and Pareto optimal solutions and
also illustrates the optimal force factors during the holding and forming process. Even though the
optimal solutions have high force factor during the holding and forming, they reduce the springback by
more than 90% and more than 60% closer to the perfect safety factor 4 when compared to the baseline
design. The main reason why the optimal solutions have higher force factor is that the physical model
did not take into account of the drawbeads between holder and die.
Table 3. Comparison of springback and safety factor obtained by the baseline design and Pareto
optimal members 1, 5 and 6.
Models (mm)
Force Factor
(Holding)
Force Factor
(Forming)
Baseline
22.693
1.0
1.0
1.0-1.0
Pareto M1
0.294 (- 99%)
0.37 (- 63%)
1.0
37.11-27.50
Pareto M5
0.491 (- 98%)
0.33 (- 67%)
1.0
36.44-33.28
Pareto M6
2.365 (- 90%)
0.31 (- 69%)
1.0
36.88-31.12
(Note: Pareto Mi represents the ith Pareto optimal member. The value of represents how close
to the perfect safety factor 4.)
Figures 4 and 5 compares the stage displacement (springback) obtained by the baseline design and
Pareto optimal member 6 (the best solution for safety factor) in 2D and 3D isometric view. It can be
seen that Pareto member 6 keeps the shape even after stamping while the baseline design tries to
springback to the original shape as shown in Figure 5. As consequence, Pareto member 6 (max local
displacement of 0.87 mm) has 98% lower local max displacement when compared to the baseline
design (max local displacement of 57 mm).
Figure 4. 2D (x and y axis) view of the stage displacement obtained by the baseline design (top: max
local displacement of 57 mm) and Pareto member 6 (bottom: max local displacement of 0.87 mm) for
Material 1 test case.
Figure 5. 3D isometric view of stage displacement contour obtained by the baseline design (top: max
local displacement of 57 mm) and Pareto member 6 (bottom: max local displacement of 0.87 mm) for
Material 1 test case.
Figure 6 compares the safety factor contour obtained by the baseline design and Pareto member 6 (the
best solution for safety factor). It can be seen that more than 70% of the physical model of Pareto
member 6 has safety factor 4 while the baseline design has only 3 all over the physical model.
Figure 6. 3D isometric view of the safety factor contour obtained by the baseline design (top: max local
safety factor of 3) and Pareto member 6 (bottom: max local safety factor of 4) for Material 1 test case.
Numerical Results with Voce Model
The optimization ran for 20 hours with 3700 function evaluations in Intel 4 × 3.6GHz processors.
Figure 7 shows Pareto optimal front obtained by RMOP. It can be seen that Pareto optimal members 2
-5 produce lower springback and better safety factor. Pareto optimal members 1 (the best solution for
springback) and 5 (the best solution for safety factor and a compromised solution) are selected and
compared to the baseline design.
Figure 7. Pareto optimal front compared to the baseline design for Material 2 case.
Table 4 compares the fitness values obtained by the baseline design and Pareto optimal solutions and
also illustrates the optimal force factors during the holding and forming process. Even though Pareto
member 5 has high force factor during the holding and forming, it reduces the springback by more than
86% and more than 43% closer to the perfect safety factor 4 when compared to the baseline design.
The main reason why the optimal solutions have higher force factor is that the physical model did not
take into account of the drawbeads between holder and die.
Table 4. Comparison of springback and safety factor obtained by the baseline design and Pareto
optimal members 1, 5 and 6.
Models (mm)
Force Factor
(Holding)
Force Factor
(Forming)
Baseline
14.505
1.0
1.0
1.0-1.0
Pareto M1
1.465 (- 90%)
1.0
1.0
24.72-21.10
Pareto M5
2.088 (- 86%)
0.57 (- 43%)
1.0
18.97-24.19
(Note: Pareto Mi represents the ith Pareto optimal member. The value of represents how close
to the perfect safety factor 4.)
Figures 8 and 9 compares the stage displacement (springback) obtained by the baseline design and
Pareto optimal member 5 (the best solution for safety factor) in 2D and 3D view. It can be seen that
Pareto member 5 keeps the shape even after stamping while the baseline design tries to springback to
the original shape as shown in Figure 9. As a consequence, Pareto member 5 (max local displacement
of 3.8 mm) has 90% lower local max displacement when compared to the baseline design (max local
displacement of 37 mm).
Figure 8. 2D (x and y axis) view of the stage displacement obtained by the baseline design (top: max
local displacement of 37 mm) and Pareto member 5 (bottom: max local displacement of 3.8 mm) for
Material 2 test case.
Figure 9. 3D isometric view of stage displacement contour obtained by the baseline design (top: max
local displacement of 37 mm) and Pareto member 5 (bottom: max local displacement of 3.8 mm) for
Material 2 test case.
Figure 10 compares the safety factor contour obtained by the baseline design and Pareto member 5 (the
best solution for safety factor). It can be seen that more than 40% of the physical model of Pareto
member 5 has safety factor 4 while the baseline design has only 3 all over the physical model.
Figure 10. 3D isometric view of the safety factor contour obtained by the baseline design (top: max
local safety factor of 3) and Pareto member 5 (bottom: max local displacement of 4) for Material 2 test
case.
Conclusion
In this paper, a methodology for the stamping process design optimisation for advanced high
strength steels has been described and investigated. The methodology couples a robust multi-
objective evolutionary algorithm and a finite element analysis based stamping analysis tool
under a parallel computing system. It has been implemented to improve the quality of final
products in terms of both the global springback and the global safety factor. Analytical research
shows that Pareto optimal solutions obtained from the optimisation offers a set of selections to
design engineers so that they may proceed into more detail phases of the stamping design
process. Future work will focus on the detailed design optimisation of a stamping process
including drawbeads shape, initial cutting for automobile parts.
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A theory is suggested which describes, on a macroscopic scale, the yielding and plastic flow of an anisotropic metal. The type of anisotropy considered is that resulting from preferred orientation. A yield criterion is postulated on general grounds which is similar in form to the Huber-Mises criterion for isotropic metals, but which contains six parameters specifying the state of anisotropy. By using von Mises' concept (1928) of a plastic potential, associated relations are then found between the stress and strain-increment tensors. The theory is applied to experiments of Korber & Hoff (1928) on the necking under uniaxial tension of thin strips cut from rolled sheet. It is shown, in full agreement with experimental data, that there are generally two, equally possible, necking directions whose orientation depends on the angle between the strip axis and the rolling direction. As a second example, pure torsion of a thin-walled cylinder is analyzed. With increasing twist anisotropy is developed. In accordance with recent observations by Swift (1947), the theory predicts changes in length of the cylinder. The theory is also applied to determine the earing positions in cups deep-drawn from rolled sheet.
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Controlling variability and process optimization are major issues of manufacturing processes which should be tackled together since optimal processes must be robust. There is a lack of numerical tool combining optimization and robustness. In this paper, a complete approach starting from modelling and leading to the selection of robust optimal process parameters is proposed. A model of stamping part is developed through Finite Element simulation codes and validated by experimental methods. The search for optimal tool configurations is performed by optimizing a desirability function and by means of a genetic algorithm based optimization code. Several tool configurations are selected from the resulting solutions and are observed through robustness analysis. Noise parameters relating to friction and material mechanical properties are taken into consideration during this analysis. A quadratic response surface developed with design of experiments (DOE) links noise parameters to geometrical variations of parts. For every optimal configuration, the rate of non-conform parts which do not satisfy the design requirements is assessed and the more robust tool configuration is selected. Finally, a sensitivity analysis is performed on this ultimate configuration to observe the respective influence of noise parameters on the process scattering. The method has been applied on a U-shape part.
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Nowadays, sheet metal stamping processes design is not a trivial task due to the complex issues to be taken into account (complex shapes forming, conflicting design goals and so on). Therefore, proper design methodologies to reduce times and costs have to be developed mostly based on computer aided procedures. In this paper, a computer aided approach is proposed with the aim to offer a methodology able to solve very complex sheet metal stamping processes, in particular a progressive design approach based on the integration between numerical simulations and optimization methodologies is presented. In particular, Response Surface Method, Moving Least Squares approximation and Pareto optimal solutions search techniques were applied in order to design two different complex 3D stamping operations. The proposed design procedure is able to verify the necessity of a spatially differentiated restraining forces approach and to design the best policy for them. In particular, different part “quality” indicators were monitored such as springback occurrence and thinning. An explicit/forming-implicit/springback approach was utilized to develop the numerical simulations. To sum up, a new and flexible design methodology is proposed, able to: deal with complex sheet metal stamping processes; investigate many possible technological scenarios; carry out a set of reliable solutions able to satisfy different design requirements; offer different optimization possibilities in order to take in to account all the sheet metal stamping design issues. KeywordsSheet stamping-Response surface methodology-Moving least squares approximation-Restraining forces