1. Introduction
Metal forming processes are widely used in automotive, aerospace, and metalworking industries due to their ability to produce complex shapes with high precision and efficiency [
1]. These processes cover a wide range of techniques, including stamping, forging and extrusion, each tailored to specific applications and material properties. With the increasing demand for lightweight components with improved mechanical properties, the optimization of metal forming processes has become critical to maintaining competitiveness in the global marketplace.
Finite Element Analysis (FEA) plays a key role in predicting deformation processes and identifying factors that limit component formability. It enables the simulation of complex forming processes and the performance evaluation of different tool configurations and process parameters. However, conventional FEA approaches often overlook the inevitable uncertainties present in real-world industrial environments, a concern highlighted by several researchers [
2,
3]. These uncertainties arise from a variety of sources, including material variability, geometric imperfections, and process variations, and can have a significant impact on the reliability and robustness of forming simulations [
4,
5,
6]. In addition, the authors in [
2] pointed out that these uncertainties can change over time (e.g., due to tool wear). Ignoring these uncertainties leads to sub-optimal or unreliable forming process designs [
7], which in turn has a significant impact on component quality, scrap rates and manufacturing costs, for example due to downtime for troubleshooting and maintenance [
8].
With the increasing availability of large amounts of data and improved computational capabilities, interest in stochastic modelling and quantification of uncertainties in sheet metal forming processes have been growing [
9,
10,
11,
12]. Several researchers have modelled and quantified the influence of different uncertainty sources on the final product, by resorting to distinct methods, such as Monte Carlo simulation [
13,
14], the design of experimental techniques [
15,
16] and metamodels [
17,
18]. In [
12], an uncertainty analysis combining FEA and Monte Carlo simulation was designed to evaluate the influence of uncertainty sources on the end product. Also, in [
16], the influence of the material and friction variability on the geometry of a U-channel was studied using an analysis of variance (ANOVA), with them reaching the conclusion that the parameters
and
of the Swift hardening law were the most influential factors in the springback and maximum thinning results. In another piece of research, [
18], a metamodel was used based on machine learning techniques to predict the occurrence of defects, considering the variability in the material properties and process parameters. In [
17], the authors compared the performance of various metamodel techniques by considering the uncertainties in the material behaviour and sheet thickness. The authors concluded that some metamodel techniques are able to accurately predict the forming process results. The application of any of the presented methods requires the execution of several numerical simulations of the forming process. To reduce the computational cost of these strategies, screening techniques can be used to select the main sources of uncertainty to be considered in the stochastic model [
16].
A sensitivity analysis is traditionally performed using analysis of variance (factorial ANOVA). However, this approach assumes a unimodal distribution of results, which is not necessarily true. Factorial ANOVA also requires a model to be predefined to establish a correlation between the factors and the results [
16]. Sobol indices (also referred to as functional ANOVA [
19]) can address this problem by not requiring a hypothesis in the form of a model, but have the limitation of only being reliable for unimodal data. The PAWN indices appear as a technique that can overcome both of these vulnerabilities [
20,
21], as they do not rely on a statistical second-order moment (i.e., variance) [
22]. Instead, PAWN indices use the Cumulative Distribution Functions to evaluate the influence of the different factors on the results.
Although PAWN sensitivity analysis can present a promising alternative to the abovementioned approaches, its application in the sheet metal forming process is still unexplored. The goal of this work is to explore the use of PAWN sensitivity analysis, by investigating the sensitivity of the square cup forming process to material properties, friction, and process conditions. The results of the PAWN indices will be compared to the Sobol indices, obtained in a previous sensitivity analysis [
23]. By employing sensitivity analysis techniques such as PAWN and Sobol indices, the aim is to identify key parameters influencing the forming process of a square cup and provide insights for optimizing process parameters to mitigate the effects of uncertainties.
2. Numerical Model
A schematic representation of the square cup forming process is shown in
Figure 1, with tool geometries based on the benchmark proposed by the NUMISHEET’ 93 conference [
24]. Throughout the process, a constant blank holder force (
) is used to control the material flow. The punch is then advanced to a total displacement of 40 mm. At the end, the square cup is extracted from the tools, promoting the springback of the square cup. To optimize the computational efficiency, only a quarter of the model is simulated, taking advantage of material and geometric symmetries. The square blank, with an initial thickness of
and a side length of 75 mm, is discretized into 1800 elements using 8-node hexahedral solids, with 2 elements in thickness and 30 elements per side. This mesh guarantees an error inferior to 1% in the maximum yield stress and a simulation time inferior to 5 min (as can be concluded from
Figure 1c). The interaction between the blank and the tools is governed by Coulomb’s law, using a constant coefficient of friction
. The contact with friction problem is solved using the augmented Lagrangian method. The tools are rigid, represented using Nagata patches [
25], and are only allowed to displace on the vertical direction, being the displacement of the blank holder controlled by the imposed force (
). Computational simulations were performed using the in-house code DD3IMP (Deep Drawing 3D Implicit Code) [
26], which uses an updated Lagrangian scheme to integrate the constitutive law in an implicit way. All numerical simulations were performed on computers equipped with an Intel
® Core™ i7-8700K Hexa-Core processor 4.7 GHz. On average, each simulation took approximately 4 min and 34 s to complete.
A low-carbon steel (DC06) was considered as the material of the sheet. The plastic behaviour is described by the Hill’48 orthotropic yield criterion [
27] and the Swift work hardening law [
28]. The Hill’48 orthotropic yield criterion is defined as follows:
where
Y is the yield stress;
,
,
,
,
and
are the parameters that define the shape of the yield surface; and
,
,
,
,
and
are components of the Cauchy stress tensor, written in the orthotropic coordinate system 0
xyz. In this work, it is assumed that
(identical to von Mises) and the condition
, meaning that the yield stress,
Y, is comparable to the uniaxial tensile stress aligned with the rolling direction.
The anisotropy coefficients for 0°, 45° and 90° (relative to the rolling direction), respectively denoted by
,
and
, can be determined by following the equations:
The yield stress
) evolution with plastic deformation is described by the Swift hardening law:
where
,
and
n are material constants, and
represents the equivalent plastic strain derived from the Hill’48 orthotropic yield criterion by assuming an associated flow rule. The initial yield stress,
, is given by:
This study will focus on investigating the influence of the uncertainty in the material properties, friction and the forming process conditions on the stamping of the square cup. Therefore, the input parameters of the numerical model whose influence will be studied are Young’s modulus (
), Poisson’s ratio (
), anisotropy coefficients (
,
and
,), the initial yield strength (
), the hardening coefficient (
), the parameter
of the Swift law, the initial sheet thickness (
), the coefficient of friction (
) and the blank holder force (
). The uncertainty in the input parameters, associated with the material properties, friction, and process conditions, is assumed to follow a normal distribution with a given mean (
) and standard deviation (
), as shown in
Table 1. The mean values of the constitutive parameters, blank thickness and friction coefficient were obtained from [
16] based on the NUMISHEET’ 93 Benchmark. The mean value of the blank holder force was optimized in a previous work [
17]. The standard deviations were obtained from [
16] based on a literature review, except the Poisson ratio, friction coefficient and blank holder force that were based on empirical assumptions made in [
17].
The influence of the uncertainty in these input parameters was studied on the numerical results of the square cup, i.e., the output parameters of the model: punch force (PF), equivalent plastic strain (EPS), thickness reduction (TR), geometry changes (GC) and springback (SB). The PF and the EPS values are directly obtained from the numerical simulation, while the TR, the GC and SB are defined by:
where
and
are the initial and final sheet thickness, respectively, evaluated for each node of the square cup numerical model;
and
are, respectively, the final spatial position of a given node for the numerical simulation with and without uncertainty (i.e., using the mean values of
Table 1); and
is the spatial position of a given node before the springback, i.e., before removing the tools. The GC quantifies the positional difference of a given node between the deterministic and the stochastic simulation, i.e., it is a measure of the shape accuracy between the desired cup shape and the final shape that is affected by the uncertainty. The SB quantifies the positional difference of a given node before and after removing the tools, i.e., it is a measure of the springback.
The variability in the forming results was evaluated using a quasi-Monte Carlo method coupled with Sobol sequence to generate the sample of input parameters according to the normal distributions given in
Table 1. A sample size of 3000 simulations of the square cup was chosen to ensure the convergence of the statistical measures, mean and standard deviation.
Figure 2 shows the mean and standard deviation for the results of EPS, TR, GC and SB along the square cup [
23]. From this figure, it can be seen that the equivalent plastic strain reached its maximum at the wall and side edges of the flange, with the most pronounced dispersion of its values occurring in the latter region. In the case of thickness reduction, the highest mean values occurred at the cup wall, with significant standard deviation values only at the flange edge. The mean and standard deviation value for the geometry changes were higher at the flange edge and wall. The highest mean and standard deviation values for the springback occurred at the flange edge. The bottom of the cup is the only area where the uncertainty of the input parameters did not affect the cup results, making a sensitivity analysis in this area unnecessary.
Figure 3 shows the evolution of the mean and standard deviation of the force applied by the punch as a function of its displacement. The force reaches maximum values between 17 and 30 mm of displacement and the standard deviation was higher within this range of values, indicating that it is during this phase of displacement that the uncertainty of the input parameters has the greatest effect on the force variability of the punch.
Figure 4 shows the distributions of the maximum values of each output parameter (i.e., forming results). All outputs follow central tendency distributions, except for the springback (
Figure 4e). It is important to consider the distributions of each output variable because, as noted above, the distributions of the results limit the sensitivity analysis that can be applied. In particular, the distribution associated with springback (
Figure 4e) is an obstacle to the use of variance-based approaches (e.g., Sobol indices and factorial ANOVA), as these are not appropriate for data that do not follow a distribution with central tendency.
3. Sensitivity Analysis
The PAWN indices are a sensitivity analysis technique based on the cumulative distribution function of a given data sample, as opposed to other techniques (Sobol and factorial ANOVA) based on the variance of the probability density functions [
20]. This allows PAWN indices to be applied to samples that follow more complex distributions, i.e., multimodal distributions. The basic concept behind these indices is that the influence of an input is related to the change produced in the cumulative distribution function of an output.
The first approach proposed for deriving PAWN indices, for a given output
, is to compare cumulative distribution functions [
20]. Specifically, the cumulative distribution function obtained by assuming that all input parameters vary,
, is compared with other
functions,
, obtained by assuming a fixed value for a particular input parameter,
, while varying the remaining input parameters. The
functions
are derived for different fixed values of
. The PAWN index,
, for the parameter
was obtained through a statistical measure, such as the average of the Kolmogorov–Smirnov statistics between the function
and the
functions
with the input parameter
fixed. The PAWN index,
, is defined by the following equation:
where
is the Kolmogorov–Smirnov statistic that quantifies the difference between the cumulative distribution functions
and one of the
cumulative distribution functions
when the input parameter
is fixed at a given value
. The statistic
is obtained by the following expression:
Subsequently, a more efficient approach [
21] was proposed that allows the use of generic sampling (i.e., it is not mandatory to assume fixed values for the input parameter
). The new approach allows PAWN indices to be calculated, starting by dividing the input parameter domain
into
equal intervals and obtaining the cumulative distribution functions for each of the
subdomains,
. In addition, the cumulative distribution,
, was calculated for the entire data set, i.e., the entire domain of
. Similarly to the first approach, the maximum vertical difference was calculated (using the Kolmogorov–Smirnov statistic) between the cumulative distribution curve of each data interval,
, and the cumulative distribution curve of the entire data set,
. The value of the PAWN index,
, for the parameter
was calculated using Equation (9).
The Sobol indices are a sensitivity analysis technique based on the calculation of the variance (i.e., based on the probability distribution function), unlike the PAWN indices which are based on the calculation of the cumulative distribution function. The variance is a parameter of central tendency that represents the average of the squares of the difference between the value of the mean of a given distribution and the value of each of its constituent points. The Sobol Indices (total sensitivity indices) were calculated using the following equation [
29]:
where
is the unconditional variance of the output parameter
and
is the conditional variance of the expected value of the output parameter
when varying all the input parameters except
.
The fact that Sobol indices are based on variance means that they only have accurate results for unimodal distributions. Bearing this in mind, it is expected that the results obtained by the PAWN indices for the springback output parameter will be different from the results obtained by the Sobol indices for the same parameter. On the other hand, PAWN indices have the disadvantage of requiring the calculation of cumulative distribution functions, whereas Sobol indices only require the calculation of variance. Another disadvantage of PAWN indices is that it is necessary to define the number of intervals into which the data are divided, which is another factor that can influence the results. Both indices quantify the influence of the parameter uncertainty on the forming results in the same way; i.e., the higher their values are, the higher is the influence of a given input uncertainty on the result variability. For indices close to zero there is no influence on the result.