2S-ML: A simulation-based classification and regression approach for drawability assessment in deep drawing

The development of new deep-drawn sheet metal parts is a complex engineering task. Low costs, structural durability, improved crash properties, aesthetics, and sustainability are contrasting requirements. Another constraint in part development is ensuring manufacturability, which has to be guaranteed from the beginning of the early design phase until a part is produced in a press. It needs to be re-assessed for design changes, uncertainties in material and sheet metal properties, and changes in the drawing configuration. To reduce the complexity in development, engineers define the shape and drawing configuration for a part iteratively. This experience-driven process leads to a compromise in the aforementioned requirements and lacks reproducibility. On the other hand, existing methods like optimization are often impractical for manufacturability assessment. This is due to the limitations of the corresponding large-scale simulation models, which are computationally expensive [5].

As we focus on the deep drawing production step, we investigate drawability rather than full manufacturability. There have been ongoing scientific contributions to improve drawability assessment. Baseline formulas to estimate drawability are given in [12]. The method is based on a cylindrical cup drawing setup. It rather aims to give feedback on whether a part should be deep-drawn or not than to give reliable feedback in part development. A simplified two-dimensional approach of a stamping analysis is presented in [14]. We employ the finite element method (FEM). It is a widely used numerical technique for simulating the behavior of complex systems, such as the deep drawing of sheet metals. The method has become an essential tool in this field, as it allows engineers and scientists to predict the behavior of a system under different loading conditions, material properties, and addendum geometries. The main advantage of FEM is the flexibility and low cost in comparison to real-world experiments.

Machine learning (ML) has shown promising results for hybrid modeling in engineering environments. Ambrogio et al. [1] proposed a kriging meta-model on incremental sheet forming to predict sheet thicknesses. A surrogate model for textile forming applications was presented in [25]. Here, simulation results are evaluated with an image-based data representation to investigate the influence of different base geometries on the surrogate model extrapolation quality. Morand et al. [16] presented a surrogate model that predicts the final geometry and field variables of a deep-drawn cup, focusing on selected process and geometry parameters. Slimani et al. [20] developed an artificial neural network surrogate for the flat rolling process based on a combination of the slab method and FEM to predict the rolling force. A dimension-reduced neural network based on features is used to predict the springback behavior in [11].

In our hybrid approach, we utilize the concept of multi-fidelity (MF) modeling to create surrogate models. Illustrated by Kennedy et al. [13], the goal is to achieve high-fidelity (HF) model accuracy at the computational cost of a low-fidelity (LF) model. Accordingly, a combined MF surrogate with decreased evaluation time is set up. Song et al. [22] mixed an LF and an HF model using radial basis functions to decrease computational cost for surrogate setup. It was tested on numerical and engineering problems. For a comprehensive overview of MF modeling, readers are referred to [24].

To decrease evaluation times and save expensive simulation runs while preserving the prediction quality of sophisticated simulation models, we propose a novel approach named 2S-ML here, an ML-based MF-inspired architecture. It incorporates information from both, LF and HF simulation schemes as well as non-simulated information to train low-dimensional surrogate models based on a data set. Our approach differs from existing MF methods in two ways. First, our surrogate models do not depend solely on simulation parameters but also on additional ML features that contain complementary drawability information. These features improve applicability to a more diverse set of shapes. Second, unlike MF scaling functions, space mappings, and difference mappings, 2S-ML does not explicitly use mixing or tuning factors, as the mixing is performed implicitly during the training of the ML surrogate models. Third, as our method is intended to work on an established historic data set, we focus on surrogate performance, not computational efficiency in their setup.

Using the 2S-ML architecture, we build a regression surrogate that allows us to continuously assess drawability and rank drawable designs in a large, industry-relevant design space. Subsequently, the same architecture is used to train a classification model that can exclude non-drawable designs. In that regard, we introduce an extension to an existing drawability measure to be able to distinguish different drawable states of drawable designs, which enables continuous ranking of designs. Furthermore, this approach assesses drawability without specific knowledge of tooling. Therefore, it can map the drawability assessment of a later design stage to the early design stage.

The paper is structured in the following manner. In Sect. "2S-ML Machine learning architecture ", two FEM simulation schemes involved in the generation of the data set are presented. Their material model is introduced in Sect. 3. The measure used for drawability assessment is outlined in Sect. 4. Section 5 describes the Design of Experiments (DOE) used for data set generation. It serves as the basis for the development of the regression and classification model presented in Sect. 6. Afterward, the results of the ML models are presented and discussed in Sect. 7. Finally, conclusions are drawn in Sect. 8.

The cross-die geometry used in the simulation schemes is similar to the ones presented in [4, 10]. Its shape contains convex, concave, and planar faces that yield a variety of representative multiaxial stress and strain states.

We use an elasto-plastic material model with Hill48 [8] as yield criterion, assuming transverse anisotropy and plane stress condition in the thin-walled sheet. To embed a variety of different hardening properties in our upcoming surrogates, we need a tunable and representative hardening law for deep drawing steels. The isotropic hardening is calculated based on the Hockett-Sherby saturation law [9]with \({\sigma }_{s}\) as saturation and \({\sigma }_{y}\) as yield stress, \(N\) and \(p\) as material constants, and \(\varepsilon\) as the plastic strain. Strain rate-dependent effects are considered negligible. To generate only monotonically increasing hardening curves and to represent different hardening trends, a stress ratiobetween the true stress equivalent of the tensile strength \({R}_{m}\), converted with the elongation without necking \({A}_{g}\) and yield stress \({\sigma }_{y}\) is introduced. By setting \({\sigma }_{y}\) and \({i}_{{\sigma }_{c}}\), the computation of the hardening curve can be conducted.

By adjusting the parameter ranges (see Table 4), we generate representative material properties. Figure 3 shows a comparison of our sampling range to standardized cold rolled (CR) forming steels. All flow curves used afterward are drawn from the sampling range.

The HF simulation model itself calculates nodal deformations, but no direct measure for drawability. Therefore, there is a need to further evaluate the FEM model to describe and calculate drawability. A commonly used method for this is using the forming limit diagram (FLD). It allows for element-wise subdivision of the minor and major true principal strains of a formed sheet metal into regions of cracks, wrinkles, and good (drawable) points. Sun et al. [23] calculate a weighted sum of element-wise vertical distances from crack points to the forming limit curve (FLC) and wrinkle points to the wrinkling limit curve (WLC). In [15], a strain path-dependent distance between the FLC and each point in the crack region is presented.

We select the function proposed by Sun et al. [23] as an outset because of its sophisticated drawability representation. It is originally defined in a positive semidefinite manner, where non-drawable configurations can be distinguished from one another. Drawable setups cannot, because their distance inherently is 0. Therefore, we extend the drawability measurewith \({f}_{\mathrm{n}-\mathrm{d}}\) for non-drawable configurations and \({f}_{\mathrm{d}}\) for drawable ones. This allows ranking of different levels of drawable designs. As most drawing setups tend to have some minor wrinkling, especially near the blankholder region where the mesh is relatively coarse, we introduce a setup-specific, user-adjustable drawability threshold of \({f}_{\mathrm{dt}}=1.395\cdot 1{0}^{-6}\). Setups with values below the threshold are considered drawable. The calculation of the two domainsis conducted with \({A}^{\mathrm{e}}\) as the surface area of an element normalized with the total surface area of the formed blank, \({d}^{\mathrm{e}}\) as an element’s minimum Euclidean distance to the FLC or WLC, respectively, \(n\) as the number of elements, and \(w\) as weights. The indices \(c\), \(w\), and \(g\) denote the affiliation with the crack, wrinkle, or good (drawable) regions in the FLD. We set the weights for wrinkling at \({w}_{\mathrm{w}}=0.1\) and for good points at \({w}_{\mathrm{g}}=0.2\).

We make four modifications to the original function to improve robustness and simulation-specific mechanics. First, we use the minimum Euclidean distance because it more accurately represents the deformation process, especially for wrinkles. Second, by element-wise area weighting, we mitigate the effect of different element sizes in the mesh to minimize overall mesh dependence. Third, as we do not delete cracked elements during simulation, each cracked element’s surface area is set to the surface area at the time step at which the crack occurred. Otherwise, the weighted distances calculation would be affected by non-realistically deformed crack elements with a high surface area. Fourth, a maximum element-wise distance for crack elements \({d}_{c-\mathrm{flc},\mathrm{max}}^{\mathrm{e}}=2.35\) is introduced. This limits the effects of heavily distorted or failed elements on distance calculation. The cumulative distance approach adopted here is preferred over a perhaps more intuitive maximum distance approach for two reasons. First, it mitigates the impact of single heavily distorted elements on the drawability measure. Second, cumulative distances provide more consistent feedback, which benefits surrogate training and optimization tasks conducted with them.

The resulting weighted distribution in the drawable domain is depicted in Fig. 4.

To embed a learnable drawability threshold, a data set containing drawable and non-drawable parts is needed. To avoid imbalanced learning, the ratio between drawable and non-drawable setups should not exceed about 85% instances of one label. We use a Sobol sequence, originally published in [21], to on the one hand sample material parameters and on the other hand sample the remaining parameters separately. Parameters are then combined randomly. The parameters and their ranges are shown in Table 4. The resulting parameter combinations are depicted in Fig. 5.

The mixing of lines to the left of the \(r\) axis shows the random combinations of material parameters with respect to the remaining parameters. Otherwise, the uniformity of the Sobol sequence can be seen from evenly intersecting parallel lines. Regions of thicker blue lines indicate that there are more feasible combinations of geometric parameters (e.g. without undercuts). Regions with fewer lines are prone to less accurate predictions. As a result of feasible parameter combinations and error-free simulation runs, we end up generating a data set of 2541 samples. This relatively large design space represents a variety of setups, making it harder to obtain high-quality results with a limited number of samples.

Noteworthy industrial appearance of complex deep drawn parts with multiple development iterations is mostly found in car structures. A chassis contains approximately 200—500 sheet metal parts, of which only a varying fraction is deep drawn due to cost implications [3]. Same-part strategies, symmetric structures, inconsistent documentation, and changes in the state-of-the-art of the deep drawing process further narrow the number of usable sheet metals for ML. In this limited data environment, deep learning is unlikely to provide the best model performance. To promote applicability, we use domain knowledge to set up the 2S-ML architecture with application-dependent feature and label engineering.

We interpret drawability assessment in two ways:

We use supervised learning to ensure the best possible model performance. An overview of the proposed 2S-ML architecture is given in Fig. 6.

In the training phase, the first step involves setting parameter ranges. Next, the DOE is employed using the Sobol sequence to sample parameter combinations. For each parameter set representing a drawing configuration, all cross-die-related geometry is created using the geometric parameters.² The punch geometry here is derived by offsetting the die geometry. Analytical drawbead sets are used directly in the incremental simulation to apply restraining forces to nearby blank nodes. The material parameters are used to specify the Hockett-Sherby hardening and the anisotropic elasto-plastic material model. An HF simulation is then set up using the die, drawbead, and blank geometry, along with the material model and process parameters. The drawability measure evaluates the incremental simulation to generate a label for classification or a target value for regression. Additionally, an LF simulation is set up and run using the cross, the material model, and process parameters. Here, restraining forces are mapped from the drawbeads to the boundary nodes of the cross. Features are generated based on the material model, the one-step simulation, and the parameter combinations. These features are then used as input for a classification or regression ML model, while the label or target value is used for the supervision of the training.

In the inference phase, the first step involves defining the parameter set of interest. Next, a representative feature set is generated as described above. Finally, the label or target value is predicted using the feature set on the trained ML model. It should be noted that no incremental forming simulation needs to be run during the inference phase.

We establish a set of 40 initial features that contain baseline simulation parameters and supplementary features. An in-depth overview of their calculation is given in the Appendix 2. The goal of the features is to capture all necessary information for drawability assessment. Features associated with the shape of the part are inspired by differential geometry. Most material features are intended to embed the hardening behavior. The simulation-based features are application-dependent and should contain relevant drawability information. We do not propose features to characterize tooling geometry, as it is generated parametrically here.

Using this combination of information, a surrogate model captures the representative underlying drawability problem, which helps to generalize to new configurations. Additionally, the supplementary features are independent of the cross-die geometry. They can therefore be applied to a variety of shapes. The assignment of all computed features to their source domain is shown in Fig. 7.

Computing the feature set for every configuration in the whole data set includes five aberrations. These are further excluded from the upcoming training and testing process. The features of the resulting data set are shown in Fig. 17.

Further investigation of the data set is done by calculating the Pearson correlation coefficientbetween two features \(X\) and \(Y\), \(cov(X,Y)\) as the covariance between the features, and the standard deviation of the features \({\sigma }_{X}\) and \({\sigma }_{Y}\). A high correlation of features can lead to worse prediction quality. The features \({h}_{min}\), \({h}_{\sigma }\), \({h}_{\mu }\), \({h}_{max}\), \({K}_{max}\), and \(p{s}_{m1}\) are neglected due to one or more Pearson correlation coefficients greater than 0.9.

Despite feature deletion, there remains redundant information (colinearities) in the feature set, which can limit surrogate quality. Still, further deletion would reduce relevant information and likewise decrease surrogate quality. To address this issue, principal component analysis (PCA) is utilized. While losing the interpretability of a feature’s influence, PCA enhances the model’s generalization ability and here guarantees the applicability of the proposed approach to non-parametric data sets. To find the most optimal compromise between small dimensionality and variance loss, we make the number of principal components available for hyperparameter optimization.

For regression, the drawability domains are min–max-normalized each. The resulting distribution is shown in Fig. 8.

Due to the piecewise definition of the drawability measure, there exist two separable histograms. The drawable setups are quite evenly distributed, while the non-drawable setups show an agglomeration near the drawability threshold. The data set is sparsely filled for values greater than 0.75. The resulting label ratio for classification is within the limits of imbalanced learning.

Because there is no prior knowledge of which ML algorithm will build the most suitable surrogate for drawability assessment, we heuristically select a subset of training algorithms (compare Figs. 9 and 12) published in [19]. For an in-depth overview of low-dimensional ML algorithms, interested readers are referred to [17]. As ensemble methods usually provide gains in prediction quality, we train bagging meta-models that use the selected algorithms as base estimators. Also, a stacking and a voting ensemble model are built based on the aforementioned meta-models. We tune the hyperparameters of the PCA, the base estimators, the bagging meta-model, and of the voting and stacking models via cross-validated grid search optimization. The same ensemble approach is applied to both, regressors and classifiers.

To measure surrogate performance, appropriate metrics must be defined. For classification, the most costly misjudgments are false positive predictions, as this means continuing to develop a part that is not drawable. Therefore, we usewith the weight factor \(\beta =0.5\), \(TP\) for true positive, \(TN\) for true negative, \(FP\) for false positive, and \(FN\) for false negative predictions as our binary classification metric for training.is used subsidiarily to measure overall performance. For regression, we utilize the coefficient of determinationwhere \(n\) is the number of data points, \({y}_{i}\) is the true value, \({\widehat{y}}_{i}\) is the predicted value, and \({\overline{y} }_{i}\) is the observed mean value to investigate the overall regression quality. The relative maximum absolute erroris used to indicate local accuracy.

For regression and classification, the data is randomly divided into 80% training data and 20% test data. Cross-validation with five iterations eliminates the need for validation data.

In this paper, an ML approach to developing a regression and a classification model for drawability assessment in the early design stage in deep drawing is presented. To generate the data sets for their training, a DOE with a Sobol sequence is followed by a one-step and incremental forming simulation. Feature computation is performed to obtain a drawability representation. An evaluation using an enhanced area-weighted minimum distance drawability measure is applied to the HF model to obtain supervision for the training.

The purpose of the classifier is to distinguish between drawable and non-drawable configurations to detect scrap configurations. It is shown that high prediction scores of \({F}_{0.5}\)-score of 0.943 and an \(Accuracy\) of 94.5% can be achieved using LSVC. A major finding is that even for a much smaller data set, almost identical prediction quality can be achieved for the classifier. Small potential lies in adding one-step-based features at the cost of running the simulation. Very accurate, late design phase FEM models will remain irreplaceable in a limited data environment. However, the calculation of many small to medium-sized simulations during the early design process can be significantly reduced. Another benefit of our method is that it can be used by component developers who are not simulation engineers. Reproducible evaluations by the classifier allow decision pathways to be traced for process improvements. Saving calculation time and licensing costs is another advantage.

The regressor can be used to rank the drawing setups at hand. The goal to accelerate assessment while keeping its prediction quality is achieved by the regressor, which is split into a regression model for the drawable and one for the non-drawable domain for reasons of higher prediction quality. The stacking regressors have \({R}^{2}\)-scores of 0.916 for drawable and 0.843 for non-drawable configurations for a data set size of 2536 samples. If the data set size is limited, \({R}^{2}\)-scores of 0.835 and 0.744 can be achieved for about 240 samples (\(20\cdot d\)) each. The classification model can be used to decide which model to choose for prediction.

The use of classification and regression for drawability assessment allows saving and speeding up design iteration loops, decoupling part design and simulation to some extent. We can conclude that our cross-die showcase illustrates the applicability of the method to sheet metal deep drawing. Moreover, the 2S-ML architecture is not limited to deep drawing. The application-dependent features and labels can be adapted to assess other manufacturing processes if a hierarchy between different sources (e.g. simulation, measurement) of information can be defined. Other manufacturing process surrogates trained with this architecture, especially forming applications, can benefit from the presented features.

To apply the proposed method, a data set of previously investigated drawing configurations has to exist. Given these setups have diverse shapes, cross-die-specific features \({z}_{\mathrm{slant}}\), \({\Psi }_{\mathrm{slant}}\), \({r}_{\mathrm{die}}\), \({x}_{\mathrm{die}}\), and \({y}_{\mathrm{die}}\) (12.5% of proposed features) cannot be utilized. Also, there typically is more than one feasible tooling configuration per part. To account for this influence, we note that further features to describe the tooling geometry have to be complemented to the feature set to ensure the applicability of our proposed method. As most documented configurations are likely to be drawable, there is a risk of imbalanced learning. Another aspect of our method is the loss of local space-resolved feedback. There is no way to determine where, and to what extent drawability is given on a part. The surrogate models only allow for a global assessment. As no measured data from a press is included in the architecture, the prediction can only be as accurate as the incremental deep drawing simulation. Further influences like tailor-rolled or welded blanks, simultaneous multiple-part drawing, edge crack sensitivity, and springback are not considered in the presented workflow. Furthermore, additional criteria for drawability such as dimensional accuracy and surface irregularities have to be taken into account once automotive body parts are assessed. The 2S-ML architecture has the flexibility in its feature set and labeling process to consider these influences.

Future investigations could focus on overcoming the current limitations. Furthermore, we identified the following promising aspects: