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Optimization method for stamping tools under reliability constraints using genetic algorithms and

finite element simulations

Y. Ledouxa, P. Sébastiana and S. Samperb aUniversité de Bordeaux, TREFLE – UMR 8508

Esplanade des Arts et Métiers, 33405, Talence cedex, FRANCE bUniversité de Savoie, SYMME

B.P. 80439, 74944, Annecy le Vieux Cedex, FRANCE

e-mail: yann.ledoux@u-bordeaux1.fr

Abstract 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 don’t 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 parts. Keywords : Variability, Optimization, Robustness,Genetic algorithm, Design of experiments, Sheet metal forming.

1. Introduction In manufacturing process design, the objective is to find a production process which leads to produce parts as close as possible to the nominal values. This approach is commonly called the optimization process. Moreover, during production, various sources of variability may arise like temperature or material variations. These variations often lead to very significant changes in production and non-conform part. The major challenge is therefore to design a manufacturing process robust to these changes. In this paper, it is proposed a general approach applied to a deep drawing operation to find different optimal configuration and then to quantify their robustness. A selection of the best robust configuration is then possible. The deep drawing process consists in transforming flat sheet blanks into cups, boxes or particular profiles corresponding to non-developable shapes. These stamped parts are typically employed in automotive or aeronautic industries.

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Stamping process is mainly used for the production of large series of parts because the design and set up of the tool is difficult and time consuming. In press shop, one of the most important problems is to obtain formed parts with precise geometric characteristics corresponding to the specifications of the customers. During the past 20 years, many works have been focused on the subject, and lead to investigate different methodologies, which have improved stamping tool technologies according to a better satisfaction of geometric requirements. These methods are essentially based on numerical simulations (i.e. finite element) of the stress inside of sheet blanks coupled to the development of optimization strategies. Optimization strategies mainly aim at avoiding and limiting part defects through the stamping process. Many different approaches have been developed. Direct optimization is used to find effective configurations of the tools. It requires a great number of time consuming simulations and doesn’t converge on the global optimum of the tool configurations. Other approaches are based on the computation of local response surfaces and on the global optimization of the tool set up. Response surfaces are regarded as fast and accurate simulation tools involved in the numerical process of a global optimization method; they are often computed using design of experiment techniques. In the industry, dominant defects are springback and excessive localized strains that lead to wrinkling or tearing the parts. These main defects are directly influenced by the parameters of the stamping process and are usually corrected by improving the tool setups. During the production process, even if the stamping process has been optimally designed, parts may have significant dimensional scatter. This variability in the characteristics of the product is mainly due to some scattered input process parameters (force, blank dimension, friction conditions, material scattering, etc). Figure 1 shows a schematic representation of a stamping process. It displays the main parameters which are classically taken into account for the set up of the operation and the different scattering sources. Most of works concerned with variability in stamping don’t take into consideration these sources of scattering.

Parameters considering for set up a stamping operation added to the main sources of scattering

Scattering sources Blank holder force

Tool geometry

Friction

Thickness

Material properties

Figure 1 : stamping process, input, output and main scattering sources.

Some authors have recently investigated the effects of the coupling between the sources of process variability and the requirement quality of parts. De Souza et al. (2008) has developed an analytical model of bending part and quantified its variability in function of process scatter. Col (Col, 2003) has studied few forming process scatters and attended to identify their origins. Gantar and Kuzman (2002) have evaluated the process stability in spite of different sources of scatter. It is proposed an evolution of its work in Gantar and Kuzman (2005) by coupling a procedure of optimization to the previous stability analysis.

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These authors highlight the main influent input parameters of the stamping process: • Materials properties scatter (hardening coefficient, yield strength, friction coefficient,

etc.), • Variation of the blank thickness.

Investigations are proposed by Sigvant and Carleer (2006) to quantify the range of variations of the material characteristics.

All of these recent studies have underlined the significant influence of the process variability on the geometry of the parts, but few papers propose a global approach starting from modelling the behaviour of the parts and the variability sources and leading to process optimization. The objective of our work is to propose such an approach and validate it. Validation is achieved by performing the optimization of a stamping process, which leads to obtain parts conform to the customer specifications and robust to the sources of scattering. This approach is commonly regarded as a reliability method.

Reliability of processes is defined as the capacity of a product to maintain the value of its performances, in spite of the variations in the functioning conditions and the uncertainties relating to the parameters of the process. G. Taguchi is considered as the pioneer in robust design; more particularly, in the definition of loss functions and signal-to-noise ratios (Taguchi, 1993). According to Taguchi, there are two types of parameters: the parameters controlled by the process and the parameters generating noise (variations). Generally, for complex processes, interactions between these parameters and the controlled parameters are related to uncertainties affecting their target values.

1.1. Modelling reliability Approaches of reliability modelling aims at finding relations linking the geometric variables of parts to some process characteristics (values and their related variability); see figure 2. These relations are usually based on physical analysis leading to complex functions or implicit functions such as finite element simulations. The product variability is computed using integration techniques or stochastic approaches such as Monte-Carlo or Quasi-Monte-Carlo Simulations (respectively MCS or QMCS), (Niederreiter, 1992). The computation of the output distributions (i.e. geometrical parameters of the part resulting from the stamping process) depends on the estimated variability of every input parameter (i.e. process parameters and material scattering).

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Figure 2: Reliability parameters and variables of stamping operations.

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The statistical characterization of the input parameters is difficult to quantify and requires major investigations and feedback on experimental data. The identification of the statistical properties of the input process parameters is based on particular statistical modes (i.e. average, standard deviation, skewness, kurtosis …). Typically, Gaussian distributions are the statistical distribution being used for modelling variability. The average and standard deviations are required for the definition of every input parameter.

1.2. General approach Regarding the uncertainties in the stamping process, the complexity of the strain path during the stamping operations and the number of process parameters to optimize, the setup of stamping tool is difficult and time-consuming task. The engineers are using their expertise and experience feedback to design stamping tools. Different empirical rules and digital tools (simulations) are supporting designers in their technological choices. Generally, several tool configurations are leading to a geometry respecting the geometrical and structural specifications of the design requirements. However, these configurations are not equivalent since they may respond differently to the variability of the stamping process. Our approach is based on the selection of the best configurations, and then, on testing their reliability regarding the dispersion of the process. The objective of this method is to select the most reliable tool.

FE Modeling Optimization Robustness Analysis

Step 1 (see paragraph 2)

Abaqus soft. : Forming simulation

Matlab soft. : Genetic Algorithm

Matlab soft. : Design Of Experiments

Sensitivity analysis

Experimental Tests Comparison PRSA vs GA*

Deviation Assessment FEM vs DOE**

Methodology

Tools

Validation

*: Pure Random Search Algorithm versus Genetic Algorithm **: Finite Element Method versus Design Of Experiments

Step 2 (see paragraph 3) Step 3 (see paragraph 4)

Figure 3 : general approach.

The approach is based on three different stages illustrated in figure 3. The first stage consists in setting up a numerical simulation of the operation. The part is a classic U shaped part which has been obtained through a unique forming operation. The validation of the simulation is performed by comparing geometries simulated using a Finite Element model to experimental results. The second stage consists in searching different optimal tool configurations. For that, a genetic algorithm is used. This optimisation method is well-known to be efficient and to converge to a global optimum. At the end of this stage, the three best tool configurations are selected. Through the third stage of the method, an evaluation of their robustness is proposed based on the evaluation of the scattering of the part geometry compared to the variability of the material scattering and friction conditions (called Noise Parameters, NP). The reliability of the optimal solutions is evaluated through the number of rejected non-conform parts. Added to this computation, a sensibility analysis of the NP is proposed and allows the designer to identify the parameters which are the most influent on the process variability according to the optimal configuration selected.

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2. Part and stamping process

2.1. Part and associated stamping tool Figure 4 presents the U shape part investigated in this paper. This elementary part is regarded as a benchmark problem for the optimization method. The geometry of the part is, for instance, representative of an automotive panel or reinforcement, commonly used in the automotive industry. This part is prone to develop 2D springbacks due to bending in the punch radius and bending/unbending in the die radius. More to the point, this part is accurately simulated by using a Finite Element code. The process variability resulting from the variability of the process parameters is exclusively based on numerical simulations and requires accurate simulations; this accuracy has to be assessed using the experimental process. The dimensions of the part, the geometry of the stamping tool and the process conditions are stated respectively in figure 4 and 5.

Figure 4 : component dimensions

Die

Blank HolderPunch

Blank Holder Force = 90 kN

Figure 5 : initial tool configuration

The blanks are obtained from rolled sheets of 0.8mm thick, 300mm long and 300mm width. The accuracy of the length and width dimensions of the blanks is 0.5mm. The material used has an anisotropic mechanical behaviour. Figure 6 shows the experimental tool used for this stamping operation. Testings were performed in a 1600 kN triple action hydraulic press. The maximal dimensions of the tool are 450 x 450mm. The maximal height of the stamped part is 150mm. This experimental tool is used to validate the numerical simulation.

Figure 6 : Stamping tool.

Cut plane

Figure 7 : stamped part and profile

measured on the stamped part.

Several stamping parts have been fabricated with the tool presented in figure 6. Figure 8 compares three different part profiles measured with a coordinate measuring machine. According to the experimental conditions, it could be observed a good reproducibility of the process and the maximum deviation of the part profile between experiments has been assessed to 0.3mm. This value has to be compared to the springback deviation, which is about 14mm. Moreover, experimental measurements prove that part profiles remain symmetric through the fabrication process. The plane of symmetry is symbolized on figure 8 by the line A.

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This geometrical property is highly significant and justifies the study of only half of the profile to characterize the defect of the part.

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Figure 8 : illustration of the process reproducibility.

2.2. Part defects and associated parameters From observations of the shape of the part, it can be assumed that the initial straight line remains rectilinear at the end of the stamping. Therefore, the part springback, with these initial process conditions, is mainly located in the round shape (in the punch radius and in the die radius). Hence, the part defects are quantified by associating three lines according to a least square criterion. The global form defect of the part is supposed to be null. Three parameters are required to characterize the part position defects as shown in figure 9.

• Two angular parameters between lines, 1 and 2, and, 2 and 3, corresponding to A1 and A2 angles.

• One height parameter called H, determining the distance between line 1 and the intersection of lines 2 and 3.

Target values for those three parameters and their corresponding tolerances are listed in table 1. These values are determined according to the functional requirement of the part: to reinforce the panel of automotive. As this component is welded to a panel, a more important geometrical deviation leads to impose a more important deviation of the final part and then to manufacture non-conform parts.

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1 2

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A2

H

Figure 9: Parameterization of the part defects

Name Target Tolerance A1 90° ± 5° A2 90° ± 5° H 60mm ± 0.2mm

Table 1 : Geometrical parameter: target and tolerance.

2.3. Numerical simulation of the operation Numerical simulations of the stamping operation are calculated by using the Abaqus® software. Stamping simulations are performed through two different steps. The first step aims at simulating the forming operation through an explicit calculation. The second step, corresponding to the simulating of springbacks is realised through an implicit formulation.

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Such a simulation process applied to forming simulation was proposed by Hibbit et al. (1994). Moreover, in order to limit the calculation duration, the punch speed was increased numerically. In this case, the kinetic energy remains under 5% of the strain energy required for the quasi-static problem. This speed increase has no influence on the final part geometry (after the springback phenomenon). Modelling the blank has been performed by using one layer of four node shell elements (reduced integration) with an initial thickness of 0.8mm (S4R from Abaqus library, see Hibbit et al., 2003). Analytical rigid surfaces have been considered for the model of the tool. The key characteristics of the numerical simulations are listed in the left side of the figure 10. The right part of figure 10 shows the boundary conditions applied on the blank which correspond to a planar strain (translation along axis 3 and rotations 1 and 2 are impossible). Moreover, symmetrical conditions are forced on the symmetrical boundary of the half part of the profile.

Blank discretization Element type Shell (4 nodes)

Element size 1.68mm × 5mm × 0.8mm

Integration points 7

Number of elements 89

Tool discretization Tool type Analytical rigid surface

Process parameters Punch travel 60mm Punch speed 1.2m/s Friction coefficient 0.15

Symmetry conditions

Planar strain conditions

Planar strain conditions

Figure 10 : Process parameters, numerical variables and definition of the limit condition

Concerning the material of the blank, the steel sheet material is considered as an elastoplastic material with normal anisotropy governed by Hill’s 48 yield criterion (see Hill, 1948). Standard tensile tests were carried out for raw material to determine its mechanical properties and anisotropy coefficients (see Simões et al., 2003). The relation between stress and strain has been modelled by the Swift model with isotropic hardening described by equation 1.

σ = K . ε n (1)

The fundamental parameters of the characterization of the material are listed in table 2. USB (DC04) material

Young’s modulus 206.62GPa Yield strength 175MPa Poisson’s ratio 0.298 Lankford’s coefficients r0° = 2.09 r45° = 1.56 r90° = 2.72 Density 7200kg/m3 Strain hardening’s coefficients K = 466MPa n = 0.2056

Table 2 : main USB steel material properties.

2.4. Experimental validation The validation process of the numerical simulations consists in a comparison between experimental parts stamped by the tool and observations of the profile obtained through numerical simulations; in both cases, equivalent numerical and experimental process conditions have been used. The tool setup is stated in table 3.

Experimental parameters Blank holder force (BHF) 90kN Die radius (Dr) 10mm Punch radius (Pr) 5mm

Table 3 : process parameters for the validation of finite element simulations.

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A comparison between numerical and experimental results is presented in figure 11. The complete numerical profile is rebuilt according to the symmetrical condition (symmetry through the plane 2, 3 is illustrated in figure 11, by line A).

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Figure 11 : experimental and numerical comparison.

As it can be seen in figure 11, numerical results appear to be close to experimental results. The right hand side of the figure presents the deviation between these experimental and simulated profiles as a function of the curvilinear abscissa from the line A to the positive x value. The deviations between experimental and numerical results are less than 1mm, which proves the correctness of the numerical model and material characterization. The good prediction of the springback of the final part geometry has to be highlighted. The part springback corresponds to the maximal deflection of the profile (about 14mm). The geometrical parameters corresponding to this tool configuration are listed in table 4. With this initial tool, neither the angular values nor the height of the part respects the customer specifications.

Name Computed on FEM Target (tolerance) Deviation to target A1 96.34° 90° (± 5°) +6.3° A2 86.53° 90° (± 5°) - 3.5° H 59.2mm 60mm (± 0.2mm) -0.8mm

Table 4 : initial part geometry

2.5. Process parameters to optimize The choice of the value and admissible variations of the process parameters are key elements of the optimization process. These values have a major influence on the variations of the shapes of the stamped part. According to different studies realized by Davies et al. (1985), it is possible to assume for U shaped parts, that:

• The die radius value has a significant effect on the vertical wall curvature, • The increase of the blank holder force leads to the homogenization of the residual

stress along the thickness, and, by this way, reduces the springback magnitude.

According to these previous works, it is particularly important to optimize the different tool radii (punch and die) and the blank holder force. Moreover, the punch travel is taken into consideration to adjust the final height of the part. The parameters selected are listed in the table 5 with their respective variation ranges.

Parameter to be optimized Min value Max value Punch radius (Pr) 2mm 10mm Die radius (Dr) 2mm 10mm Punch travel (Pt) 56mm 64mm Blank Holder Force (BHF) 90kN 300kN

Table 5 : list of optimized parameters

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3. Optimization

3.1. Desirability function Geometrical parameters (angles and dimensions) of U shaped parts have to be related to an objective function since the parts formed from the stamping process are related to a nominal geometry. The definition of the nominal geometry is part of the design requirements of the stamping process. This nominal definition is completed by the definition of allowable deviations between the actual and nominal geometries. We use desirability functions to qualify such deviations and a global desirability function as a global objective function. Through desirability functions, it is proposed to transform every geometric response relating to different physical units and variation range into a satisfaction index. This kind of desirability value was first introduced by Harrington, 1965.

The desirability function requires the transformation of every geometrical parameter PGPi into a desirability value di. In the presented case, di may vary from 0.1 (undesirable responses), to 1 (fully desired response). Between these two limits, a linear variation has been assumed (see figure 12). The introduction of 0.1 as a threshold value aims at penalizing non satisfactory solutions regarding several criteria (which is not possible with the 0 value).

In the literature, different variation types are proposed a linear one by Kim and Lin (2000) or an exponential one by Rethy et al. (2004). These choices allow to graduate the mathematical translation of the satisfaction level.

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UDL i

Figure 12 : desirability function used in this study

The desirability value is defined as a function of the geometrical tolerances. It follows:

di =

if PGPi < LDLi di = 0.1

if LDL i ≤ PGPi ≤ µi di = 0,9

µi − LDLi × PGPi +

0.1µi − LDLi

µi − LDLi

if µi ≤ PGPi ≤ UDL i di = − 0,9

UDL i − µi × PGPi +

UDL i − 0.1µi

UDL i − µi

if PGPi > UDLi di = 0.1

(2)

With PGPi: the ith part geometrical parameter LDL i: Lower Desirable Limit (85° for angles and 55mm for H) UDLi: Upper Desirable Limit (95° for angles and 65mm for H) µi: target value (90° for angles and 60mm for H).

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An overall desirability D is calculated by aggregating the individual desirability values (equation 3). The formulation of the aggregation was proposed by Derringer and allows keeping the requirement of every individual goal without any compensation (see Derringer and Suich, 1980).

D =

3

i =1Πdi

1/3

(3)

The goal of the optimization phase is to maximize the overall desirability D, which is the objective function of the optimisation process. The maximal value for D is equal to 1. In this optimal case, every geometrical goal is satisfied (ie. A1=A2=90° and H=60mm). Moreover, the introduction of the threshold of 0.1 for every desirability di allows keeping the ranking of the different solution even if one of the goals is not respected.

3.2. Genetic algorithm The search for optimal configurations of stamping tools has been based on the development of a Genetic Algorithm (GA). Genetic Algorithms are evolutionary algorithms (Davis, 1991) inspired by the mechanisms of evolutionary biology proposed by Darwin in 19st century (Holland, 1975). This approach of optimization has been recently developed in many design, fabrication or industrialization applications as global optimization tools, which avoid trapping the search for optimal configurations inside of local optima. These tools also appear to be robust since they remain effective for many types of applications; GA effectiveness is little problem dependent. GA based optimization of a design problem is based on the optimization of sets of tenths or hundreds of candidate solutions to the design problem. This set of candidate solutions is regarded as a group of individuals, which constitutes a population. Every candidate solution is defined through genes corresponding to the design variables of the problem. The major idea of Genetic Algorithms is to perform optimization through a simulated competition between these individuals (candidate solutions). The selection process of individuals is related to the objective function of the optimization process. The selection process of stamping tool configurations is equivalent to a design problem. Selecting tool configurations consists in defining four different values which point to the levels of four process variables, namely the punch radius (Pr), die radius (Dr), punch travel (Pt) and blank holder force (BHF). The genes of individuals correspond to the design variables of the problem being investigated. In this case, genes characterizing the selection of stamping tools are values assigned to these four process variables. The four variables are said to constitute the genotype of the individuals. The initial population (see figure 13) is defined randomly through random values allotted to every gene of every individual, wherever these values are consistent with the value domains of the variables (uniform probabilistic distribution of value domains). Next, every individual (candidate solution for the stamping tool configuration) is evaluated according to the objective function (overall desirability D) and ranked from its overall desirability among the population. From this, a new population is obtained by transforming the evaluated population by means of three operators; namely operators of selection/reproduction, crossing and mutation of individuals. From a probabilistic point of view, these three operators tend to improve the population and increase the performances of the individuals in the next iterations of the optimization process.

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Evaluation of the population

Pri Dr i Pti BHFi Individual i

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Figure 13 : main stages of the optimization method.

The convergence effectiveness of Genetic Algorithms is submitted to the pertinence of the selection of three parameters. These parameters are the selection pressure (φ), the crossover probability (pc) and the mutation probability (pm). The selection pressure parameter influences the effectiveness of the selection/reproduction operator since every individual “i” is duplicated in the new generation according to the following probability:

Where Nind corresponds to the number of individuals of the population, ri to the rank of the individual “i” and φ to the selection pressure. φ is ranging between 1 (no selective pressure) to 2 (extreme selective pressure). The crossover operation consists in selecting two individuals, selecting sequences of genes and performing a permutation of the gene sequences between the individuals. The crossover probability determines the percentage of the population affected by this operator. The higher the value of this parameter, the higher the amount of candidate configurations of the stamping process mixing their respective values of Pr, Dr, Pt and BHF inside of the new population. In the same way, the mutation probability determines the percentage of the population affected by the mutation of their individual genes. Mutation consists in selecting a gene inside of the gene collection of an individual and performing a reinitialization of the value of the gene (random selection inside the domain of feasible values of the corresponding variable). Genetic algorithms effectiveness emerges from the combination of the three operators. Selection/reproduction aims at guiding the evolution of the population towards more and more competitive individuals regarding the optimization objective function. Crossover steers the exploration process by combining genes of a priori competitive individuals, whereas mutation disturb the exploration process by generating original individuals. This perturbation is essential to the global optimization since it avoids trapping the search inside of local optima.

pi =

1Nind

φ −

(r i − 1)(2φ − 2)Nind − 1

(4)

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GA based optimization methods are iterative methods since the population has to be improved through the computation of several tenths or hundreds of generations before being ended. The ending condition cannot rely on the determination of the global optimum of the problem since the optimization process cannot prove that the best individual of a population is a global optimum. As being numerical stochastic methods, GA based optimization methods may improve a population of candidate solutions but cannot perform a complete exploration of search space. Therefore, in the following, the ending condition is achieved as soon as a target generation number has been reached (NGmax). This type of ending condition has been chosen because it is strongly correlated to the number of individuals evaluated through the optimization. Individual evaluations require much computation time since every individual evaluation is essentially performed through a Finite Element simulation code (Abaqus), which is very slow compared with other components of the optimization code. In other words, the ending condition is linked to the computation time of the global optimization code. The computation time required for the evaluation of every candidate solution is constant and has been assessed to 480 seconds (realized on a PC running a 2.26 GHz Pentium Centrino2 under Windows XP Pro). This computation time also bounds the validation process of GA based optimization. As a stochastic method, it requires a lot of computations to investigate the effectiveness and adjust the parameters of the algorithm. In the following, the influences of the crossover and mutation probabilities are investigated by comparing the performance of the algorithm with two different values of both the crossover and mutation probabilities. These values have been selected from the experience derived from previous applications as follows:

pc = {0.5, 0.8} pm = {0.1, 0.2} (5)

Likewise, the selection pressure as been fixed to: 1.5 =φ (6) More to the point, the number of individuals of the population has been limited to 40. 40 N ind = (7)

The maximum number of generations of the ending condition has been limited to 20. 20 NGmax = (8)

Five tests have been realized to set up the configuration of the GA algorithm with different probability of mutation and crossover (respectively 10 to 20% and 50 to 80%). The efficiency of the GA is compared to a pure random approach. The best configuration between the convergence and the number of tested individuals is obtained with a probability of mutation of 10% and a crossover of 80%. This configuration of GA has been used in this paper. Figure 14 illustrated the results.

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Figure 14 : comparison between GA configurations and pure random search.

3.3. Optimization process A GA based algorithm has been developed in the Matlab® environment to drive the optimization process. This optimization code generates candidate solutions corresponding to different stamping tool configurations. These configurations are simulated using the Abaqus® software and the optimization code derives the evaluation of every candidate solution from the simulations. From these evaluations, the optimization code generates new candidate solutions (new generation) and iterates the optimization process until the ending condition has been validated. Several translation routines have been developed in the Matlab environment to generate or interpret simulation files matching the language of the Abaqus solver. These routines are used to translate the definition of a stamping tool configuration into an inlet file for Abaqus (upstream translation) or an Abaqus output file into data (downstream translation) suitable for the computation of the overall desirability D of the tool. The upstream translation routine translates data relating to the part (geometry) and tool (geometry, force) into data suitable for Abaqus. The mesh of the part is generated by the Abaqus software before starting the simulation process. Simulations concern the forming phase of the part while the punch goes down into the part and the springback phenomenon as soon as the punch and the tool are removed. The boundary conditions of the simulations have been developed in paragraph 2.3. Geometrical computations are performed, through the Abaqus software, by associating 3 straight lines according to the least squares method (see figure 9). The overall desirability “D” is computed by the downstream translation routine from the simulation output file resulting from Abaqus. The output file contains the geometry of the part shaped by the stamping process. These steps are repeated until every tool configuration of the population has been evaluated and until the global iterative process of the genetic algorithm is stopped.

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Figure 15 presents the evolution of the overall desirability of the population evolving through 20 generations. The maximum overall desirability of the population starting from 0.75 increases and reaches a maximum desirability value of 0.95 at the eighth generation. This maximum value (denoted by “+”) corresponds to the best individual of the population. Other individuals correspond to a lower desirability (denoted by “o”) and emerge from the crossover or the mutation of the genes of the individuals.

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Number of generation Figure 15 : evolution of the overall desirability D of the population

Height hundred individuals have been generated or reproduced through the twenty generations. However, only 240 of them correspond to different individuals and have been simulated as being different tool configurations. Among these 240 simulated configurations, only 23 of them respect every geometrical constraint of the problem, namely A1 and A2 equal 90 ±5° and H equals 60 ±0.2mm. In the following, we consider the three best configurations (see table 6).

Configuration 1 2 3 Pr (mm) 3.1 3 4.4 Dr (mm) 2.1 4.7 2.1 Pt (mm) 59.8 60.2 59.9 BHF (kN) 124.1 276.2 170.2 A1 (°) 89.6 89.3 89.6 A2 (°) 89.9 90.3 89.0 H (mm) 59.94 60.00 59.99

Tableau 6 : Geometries of the 3 best tool configurations.

The profiles of the parts are displayed in figure 16.

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Figure 16 : Profiles of the 3 best tool configurations.

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The die and punch radii are small (ranging from 3 to 4.4mm) and the punch travel is very close to the height of the part resulting from the simulation. However, the blank holder force is drastically different and can double (ranging from 1 kN to more than 2 kN). The die radius is about 2mm for the configuration indexes 1 and 3 and about 5mm for the configuration index 2.

4. Process variability

4.1. Variability analysis of the stamping operation Previous work realized by Janssona et al. (2008) has investigated several methods for assessing process variabilities and analyzing the influence of materials and friction phenomena in the neighbourhood of optimal solutions. Chiefly, two different approaches have been developed in the literature. The first method consists in directly estimating the variability of the process through numerous random configurations (materials and friction conditions) and then, in measuring the corresponding geometry of the part. This approach is commonly regarded as a Monte-Carlo simulation oriented method. This approach results in the computation of the variability of the output geometrical parameters assuming a Gaussian distribution of the input parameters; the distribution of the output parameters isn’t Gaussian. The second approach aims at limiting the number of computations necessary for investigating the process variability by using a response surface. The response surface is an approximation model of the process variability developed through design of experiments (DOE). Several authors have established that the best approximations are obtained from quadratic multivariate response surfaces (see Ledoux et al., 2007). The design of experiments (DOE) is performed in the domain surrounding the configuration being studied. Prior studies have proven that Monte Carlo techniques and DOE methods give solution in close agreement; typically, the relative deviations between the two models don’t exceed a few percent. In the following paragraphs, we use this type of approach.

4.2. Choice of the noise parameters The assessment of the noise characteristic parameter (NP) related to the stamping process is difficult since it derives from several variability sources. The main variability sources are:

• Material scattering and lack of knowledge relating to materials • Friction phenomena evolution • Process parameter dispersion (blank holder force, punch velocity, etc.).

Some authors propose several methods to identify and quantify the dispersion sources in laboratory or industrial environments. Blumel et al. (1988) has proposed to identify the effect of the process variations on the final stamped part. Majeske et al. (2003) has quantified the stamping process capability for automobile body side panel. In Auto/Steel report, (Auto/Steel report, 1999), it was analysed data from more than 1750 coils and 50,000 individual tests for a range of current steel grades. The more recent work of Karthik et al. (2006) has studied material properties scatter of tensile tests from three laboratories using more than 45 coils of material. These quantitative analyses highlight the effects of the intrinsic variability of stamped parts, at batch to batch and run to run levels during production. Furthermore, lubrication is a well known key parameter of stamping processes, which has a major influence on the achievement of forming operations.

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The estimation of the variability range of materials and friction parameters has been exposed in a technical report developed in the framework of the European project Inetformep (see Simões et al., 2003). In the following paragraphs, these data are related to other works (see Auto/Steel report, 1999 or Karthik et al. 2006), which confirms the order of magnitude of the scattering parameters. Material scattering is related to material yield stress (Re) and some parameters (K and n) of the behavior law of the material defined in equation 1. The friction coefficient is taken into account through a global coefficient F according to the Coulomb model: T = N . F (9) Table 7 lists the nominal values and standard deviations of the preceding parameters. We assume that the scattering of the parameters follow a Gaussian distribution.

Re K n F

Nominal value 175MPa 466MPa 0.2056 0.15 Standard deviation value 6MPa 15MPa 0.03 0.036

Table 7 : mean and standard deviation for each noise parameters

Here, only material and friction parameters variability have been considered for limiting the complexity of the analysis. For having a global reliable approach it could be necessary to complete these noise parameters with, at least, the process optimized parameters (Pr, Dr, Pt, BHF).

4.3. Reliability of the solutions In the following paragraphs, the influence of scattering parameters on stamping processes are modelled using design of experiments based response surfaces. Previous work realised by Ledoux et al. (2007) proves the effectiveness of response surface models based on the development of Box-Wilson central composite design (CCD); see reference Box and Draper, 1987. This kind of DOE uses a Full Factorial Design (FFD) with a central point. In table 8, experiment 17 corresponds to the central point and experiments 1 to 16 correspond to FFD points. The variation range of every noise parameter is a standard deviation. The FFD has been improved by means of a group of star points used for appraising the quadratic curvatures of the response surfaces. In table 8, star points correspond to experiments 18 to 25. The optimal point position of the star points is defined according to the literature (see Goupy, 2001). The response surface is a second degree polynomial corresponding to a quadratic model of the geometrical output variables (GO). This polynomial is a function of every noise process input (NP).

GO= a0+a1NP1+ a2NP2+…+ a12NP1NP2+ aijNPiNPj +…+ a11NP12 + annNPn

2 (10)

Each line of table 8 corresponds to a numerical simulation. The geometrical parameters have been derived from each optimal tool configuration listed in table 6.

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No of expts. Re

(MPa) K

(MPa) n F 1 169 451 0.176 0.11 2 169 451 0.18 0.19 3 169 451 0.24 0.11 4 169 451 0.24 0.19 5 169 481 0.18 0.11 6 169 481 0.18 0.19 7 169 481 0.24 0.11 8 169 481 0.24 0.19 9 181 451 0.18 0.11

10 181 451 0.18 0.19 11 181 451 0.24 0.11 12 181 451 0.24 0.19 13 181 481 0.18 0.11 14 181 481 0.18 0.19 15 181 481 0.24 0.11

FFD points

16 181 481 0.24 0.19 Central Point 17 175 466 0.21 0.15

18 165 466 0.21 0.15 19 185 466 0.21 0.15 20 175 440 0.21 0.15 21 175 492 0.21 0.15 22 175 466 0.15 0.15 23 175 466 0.26 0.15 24 175 466 0.21 0.09

Star Points

25 175 466 0.21 0.21

Table 8: Central composite experimental design.

4.4. Variability of every solution Here, once the three optima have been selected, 25 FEM simulations have been performed for each of them with noisy material parameters. Using eq.10, response surfaces have been constructed. Then, 100 000 individuals (Pr, Dr, Pt, BHF) have been calculated using GO (and not FEM) to build the probability distributions of table 9. For every geometrical parameter, the corresponding average and standard deviation are computed.

Figures in table 9 highlight the differences between the target values resulting from the search for optimal solutions (optimal according to non stochastic simulations) and the stochastic simulations in the neighbourhood of optimal solutions. Mean and target values appear to be different since the distributions of the output variables are non-Gaussian. More to the point, according to the preceding results, there are important differences between the output variable variabilities although the process inputs are alike. All the three optimal tool configurations seem to be very close to one another. The angular deviations of angles A1 and A2 are very sensitive since their relative variations are ranging between 1.3° (A1, configuration 3) to 3° (A2, configuration 2). The height parameter is relatively less sensitive to the variations of the input parameters; it is ranging between 0.07 to 0.25mm. These quantitative results illustrate the major influence of the noise parameters on the resulting part geometry and the importance of tool configurations in the geometrical dispersions. In order to define the most relevant configurations of stamping tools from the robustness point of view, the non-conformity rates of particular tool and setup configurations are estimated through Monte Carlo simulations. The maximal allowable deviation for angular and height values are defined in the design requirements of the U shaped parts. The maximum deviation for angular values is ±5° and the maximum deviation for the part height is ±0.2mm. Non-conformity rates are listed at the end of table 9. As it can be seen in the table, the best tool configuration is configuration 1 and corresponds to a rate of non-conform geometry of about 18%.

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No Opt. 1 2 3

Pr 3.1mm 3mm 4.4mm Dr 2.1mm 4.7mm 2.1mm Pt 59.8mm 60.2mm 59.9mm

BHF 124.1kN 276.2kN 170.2kN A1 Opt. value=89.6°;

Mean=87.7° ; Std=2.8°

75 80 85 87.68 900

1000

2000

3000

4000

5000

6000

7000

87.7

Opt. value=89.3°; Mean=87.8° ; Std=1.3°

75 80 85 87.8 900

500

1000

1500

2000

2500

3000

Opt. value=89.6°; Mean=87.7° ; Std=2.1°

75 80 85 87.66 900

500

1000

1500

2000

2500

3000

3500

87.7

A2 Opt. value=89.9°; Mean=91.8° ; Std=2.0°

85 90 91.79 95 100 1050

500

1000

1500

2000

2500

3000

3500

91.8

Opt. value=90.3°; Mean=92.9° ; Std=2.9°

85 90 92.86 95 100 1050

1000

2000

3000

4000

5000

92.9

Opt. value=89.0°; Mean=90.4° ; Std=2.2°

85 9090.45 95 100 1050

500

1000

1500

2000

2500

90.4

H Opt. value=59.94mm; Mean=59.92mm ; Std=0.07mm

59.5 59.7 59.959.92 60.1 60.3 60.5 60.7 60.90

1000

2000

3000

4000

5000

6000

Opt. value=60.00mm; Mean=60.14mm ; Std=0.25mm

59.5 59.7 59.9 60.160.14 60.3 60.5 60.7 60.90

1000

2000

3000

4000

5000

60.14

Opt. value=59.99mm; Mean=59.91mm ; Std=0.16mm

59.5 59.7 59.959.91 60.1 60.3 60.5 60.7 60.90

500

1000

1500

2000

2500

Rate of non-

conform geometry

17.72% 30% 28.04%

Table 9 : mean and standard deviation comparison for every optimal configuration

4.5. Sensitivity analysis From the polynomial models of geometrical output variables (GO polynomials are functions of noise parameters NP) characteristic of the most robust tool configuration (configuration 1), it is proposed to discuss the sensitivity of the part geometry to process variabilities. Sensitivity functions are calculated with equation 11:

Si =

∂PGPi

∂xi xo

(11)

Where xo corresponds to angular (A1 and A2) or dimensional (H) variables of the observed configuration. As we work using normalized variations: xo = {0,0,0,0}

Configuration A1 A2 H

F 1.72 -0.96 -0.02 K -0.24 0.28 -0.01 n 0.31 -0.4 0

Re 0.02 0.06 0 Max variation 2.29° 1.70° 0.03mm

Table 10 : Sensitivity of the noise parameters of the part geometry for configuration 1

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Variation rates of sensitivities must be compared to the corresponding allowable variation ranges (∆a). Therefore, we compute relative sensitivities by dividing sensitivities with intervals of allowable variations (see figure 17). ∆a are derived from the maximal variations of noise parameters within a standard centred range of [-1;+1]. Numerical results displayed by figure 17 attest that allowable variation ranges of angles are about ±2°, which is high regarding the geometry characteristics of the parts. On the contrary, the sensitivity of the part height is about ±0.03mm, which is low. The main influent noise parameter in the variability of the part geometry corresponds to the friction coefficient F. This coefficient represents more than 60% of the overall variation. The material characteristics, the behaviour law coefficients (K and n) and the yield strength (Re) have a lower influence (respectively between 10 to 20% and some percents for Re). These relative influences are illustrated in figure 17 by circle charts.

A1

∆a = ±2.29°

F :75%

K :10%

n :14%Re :< 1%

A2

∆a = ±1.70°

F :56%

K :16%

n :24%

Re :4%

H

∆a = ±0.03mm

F :67%

K :33%

Figure 17 : sensitivity results of the NP to the part geometry.

The preceding results highlight that the key parameter for controlling the production process should be the friction coefficient. This phenomenon is familiar to production engineers taking an interest in stamping processes and is proved to be difficult to manage throughout industrial processes. Friction mainly is sensitive to the temperature, lubrication, material roughness and tool wear. Material properties are also influent parameters (K and n) and their variabilities provoke significant variabilities of the geometry of the part.

5. Discussion The sensibility analysis of the previous paragraph corroborates experimental knowledge on stamping processes. The key phenomenon influencing stamping processes is friction and, consequently, lubrication since this phenomenon mainly is controlled through lubrication. The influence of materials properties scatter is also significant and may be controlled through quality processes in material industrialisation. Solely 4 noise parameters have been taken into account in the framework of this study. Real processes are subjected to other sources of scattering. These sources originate in:

• tooling variability due to wear like radii variation, • clearance variation between die and punch, • random variability arises from incorrect tool or sheet positioning or malfunctions of

mechanical devices. Tooling wear may be taken into account through probabilistic models. Authors propose a particular model for the wear of the die like Wang et al. (2007) or by Hambli et al. (2003) in the case of punch wear.

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The proposed approach mainly is based on numerical simulations. This approach presumes the suitability of stamping and variability models. A comparison between geometries resulting from numerical simulation and actual stamping parts have been performed through experiments to validate the numerical model. The reliability of numerical simulations has been observed by testing numerical parameters of simulations. The size or type of meshing elements and simulation time steps have been considered and adjusted to match simulation requirements. The influence of these numerical treatment parameters has been proven to be insignificant compared with the variability of the process.

Numerical simulations are predictive and result in accurate estimations of the actual processes. More to the point, every parameter may be modified separately. Experimental approaches are much more restrictive since noise parameters generally are difficult to measure. However, the present probabilistic model assumes no correlation between the input scatterings. Our approach attempts to simplify the complexity of real processes by classifying the known sources of variations and disregarding the effect of unknown variables.

6. Conclusion Due to the increasing requirements for quality on stamped parts, it becomes imperative to take into account the process variability during the design phase of a new stamping tool. Usual optimization strategies are not sufficient to guarantee the quality of production series and must be completed by new optimization methods taking into account robustness; otherwise, the number of parts rejected from the process may be significant. The scattering sources in stamping operations mainly are material properties, friction phenomena and tooling wear. In this paper, we propose a quantitative approach to classify and rank these phenomena through robustness. More to the point, a method is proposed to design stamping tools which are less sensitive to material and friction variations. The method is illustrated on a U shape stamped part formed in one single stage. This kind of part is representative of parts used in automotive reinforcements prone to high springback deviations. The first step of the method begins by defining the allowable variation ranges of the stamped parts from geometrical specifications. These values are derived from customer specifications. Next, a stamping model has to be developed through Finite Element simulation codes and validated by experimental methods. The second step is concerned with the search for optimal tool configurations matching the specified geometry of the part. This optimization phase is performed by defining a desirability function relating to the geometrical specifications and variabilities of the part. This desirability is used as an objective function by a genetic algorithm based optimization code. Several optimal tool configurations are selected from the solution set resulting from the optimization code. The second step consists in developing a robustness analysis of every optimal configuration. Four different noise parameters have been considered; namely the friction coefficient, two parameters characterizing the material mechanical behaviour and yield stress. Dispersions have been assessed from measurements of material scattering resulting from scientific literature. Next, a quadratic response surface is used to relate noise parameters to geometrical part variations, which allows quantifying noise influence on the part geometry. Finally, the most relevant optimal configurations is selected through the assessment of the rate of rejected parts; these parts don’t respect the customer requirements. The more robust tool configuration generates the lowest amount of non-conform parts.

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A sensitivity analysis is performed on this best optimal configuration and leads to classification of the noise parameters on the process scattering. The friction coefficient is the most significant parameter responsible for more than 60% of the process scattering, whereas the material variability is responsible for more than 30% of this scattering. The variability relating to yield strength seems to have a quasi null influence on this kind of stamping operation.

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