1 Introduction

Forgings are used in highly stressed areas such as engines or in vehicle construction due to their excellent mechanical properties (optimal fibre direction) [1]. The forging is produced in several forming steps (so-called forging sequence). Due to the advancing trend of e-mobility, new types of forgings are emerging for which many companies still have no or few reference processes. The effort for the design of an efficient forging sequence is very high. FE simulations are often used for the design of forging sequences, as they are a very good tool for determining and analysing quality criteria such as form filling, folds, forming forces and temperature curves. A disadvantage of FE simulations is the opposite direction of the simulation to the design direction of forging sequences [2]. The design direction of a forging sequence is backwards from the forging. This makes the design of efficient forging sequences a complex and iterative development process that can take several days or weeks, depending on the complexity of the forging. The quality of a forging sequence and the corresponding intermediate forms is characterised by a good mass distribution in relation to the finished forging. Therefore, the following requirements arise for an automated and backward design of forging sequences for closed die forging based on the forging geometry:

  • The forging sequence must be designed in a short time in a process-safe manner (< 60 min)

  • 3D multi-stage forging sequence for different forging geometries

  • No need for existing reference processes and empirical knowledge

2 State of the art

In addition to FE simulations, the mass distribution diagram according to Mathieu [3] is a well-known tool for the design of forging sequences [4], which is applied to long forms according to the shape order of Spies [5]. Most forged parts belong to this shape class (see Fig. 1) [5] and have a high potential for flash reduction according to Herbertz [6]. By means of the mass distribution diagram, the individual preforms can be generated by the global material impact along the longitudinal axis and optimized for different components [7]. To determine the forging sequences for forgings backwards from the forging, inverse material flow simulations were used by Wienströr et al. However, the approach was only developed for rotationally symmetric parts in 2D space without an extension of the approach to non-rotationally symmetric parts in 3D space [8].

Fig. 1
figure 1

Shape order according to SPIES [5]

Various stochastic and analytical methods are used for preform optimization and process design in die forging. The development of approaches to automated design of forging sequences has been the subject of research for a long time. Many of the approaches are limited to explicit components and geometries such as the computer-aided method for intermediate form design for axisymmetric forgings by Biswas et al. [9] or the approach by Subramanian et al. which is limited to specific cross-sectional shapes of forgings [10]. Furthermore, Biswas et al. have developed an approach for the design of preforms for long forms, but no multi-stage forging sequences are generated and the approach is valid for open die forging and not for closed die forging [11]. Mirsadeidi et al. [12] and Alimirzaloo et al. use Lagrange interpolations for the design of preforms [13]. Knust et al. have developed a method for preform optimization for rotationally symmetric cross-wedge rolled preforms based on evolutionary algorithms [14, 15]. Numthong and Butdee developed an experience-based forging process design system for which they combined FE simulations with a case-based reasoning algorithm [16]. There are several Artificial Neural Network (ANN) based approaches to reduce the effort of FE simulations in the context of process design of forming processes. ANN based predictions of forming forces and die filling have been performed by Chan [17] and Tang [18]. Ciancio has investigated combinations of ANN and optimization methods to improve the prediction and optimization of manufacturing processes [19, 20]. Shao presents an approach to predict the influence of preform geometry on the forging process for a turbine blade using genetic algorithms and particle swarm optimization [21, 22]. Naeimi et al. optimized the 3D preform for rotationally symmetric components by using neural networks to predict the material flow for different preforms and compare them to determine the most suitable variant [23]. Guan et al. determined a single suitable 3D preform for multi-stage sequences using response surface methods [24]. Torabi et al. used an ANN of the type "NSGA II" to improve the preform geometry of a turbine blade [25]. Seu et al. used an ANN to optimize the geometry of an engine piston [26]. Caspari et al. describe in [27] an approach to optimize a preform by an inverse shape finding approach, based on the work of Chenot [28]. Liu et al. use generic algorithms to generate an optimized preform from a forging in combination with a Fourier analysis and FE methods [29]. The procedure follows a conformal mapping approach to generate a 3D preform [30]. Biba et al. investigated methods for the preform of forgings by a Laplace equation based on a comparison of the isothermal surfaces between the geometries blank and finished part [31, 32]. The presented investigations have in common that they are limited to simple, mostly rotationally symmetric, components or explicitly apply to only one component and only one preform is designed. In many of the presented works, FE results are used. An application for the design of a multi-stage forging sequence for different forging part geometries is not known.

3 Development of the method for automated forging sequence design

To fulfil the requirements of an automated forging sequence design, a method was developed that can generate multi-stage forging sequences based on the 3D geometry of the component. In order to be applicable to the widest possible range of forgings, the method has been developed to take into account all geometric features of forgings as they exist in the Spies shape order (see Fig. 1 main axis curved, bifurcations, long forms, secondary shape elements, compact shape, disc shape). All forgings can be composed of these characteristics, some may occur more frequently, e.g., several secondary shape elements, several curved axes. The aim of the method is to provide an approximate solution for the first draft of a forging sequence. Explicit geometric characteristics such as draft angles and fillet edges are not considered, as these are only necessary in the detailed design of the forging sequence. The structure of the developed forging sequence design method (FSD method) is shown in Fig. 2. The input of the method is the STL file of a forging for a die forging process. Depending on the forging geometry, certain specifics for the design of the forging sequence are to be derived. For this purpose, the shape order according to Spies is used, which divides the forgings into different shape classes (see Fig. 1 SC3 = long forms; SC2 = disc shapes, SC1 = compact shape) [5]. In the first step of the method, the module for classifying the forging is used. A classification tool was developed that uses ANNs to classify the forgings into the shape classes according to Spies and identifies information relevant for the design, such as the cutting axis, the geometric dimensions, and characteristics such as bifurcations and bending [33]. The main functionality of the classification tool is presented in detail and discussed in [33]. After a pre-recognition of the main shape class (SC1, SC2, SC3) by analysing the spatial dimensions in the three spatial directions, the forging is passed on to ANNs specialised in the main shape classes and the classification into the specific sub-shape class according to SPIES takes place. As a result, the shape class with the highest probability after classification is output, as well as the automatically determined cutting axis.

The output values are then subsequently passed to further modules of the FSD method that require this information (see Fig. 2). In the next step, a module is used to determine a complexity score for the forging. The complexity of a forging influences both the flash ratio [15] and the number of forming steps in the design of forging sequences. Therefore, a complexity score model was developed that determines a complexity score \(K_{Ges}\) of the forging based on the STL file of the forging [34]. The main functions of this module for determining the complexity score are presented and discussed in detail in [34]. The complexity of the geometry is evaluated based on the course of the centre of gravity line, the changes in the cross-section and the calculation of an envelope volume. The calculated complexity score is passed on to further modules to calculate the number of stages with fuzzy logic and to the point-shift-algorithm for generating the intermediate forms.

Fig. 2
figure 2

Flowchart FSD method (grey = modules of the method, blue = output and input parameters)

The next step of the FSD method is to run a module to calculate the number of stages in the forging sequence. The number of forming stages depends on various design parameters, such as the batch size, the available installation space for the tools or the available forming machines and the necessary process forces. A direct calculation of the number of stages is not possible, as the design process for forging sequences is experience-based and company-specific. However, correlations and dependencies of relevant design parameters can be established, which enables the determination of the number of stages by means of fuzzy logic. Fuzzy logic allows fuzzy correlations to be mapped via a set of rules and a discrete solution value to be output through defuzzification.

The fuzzy system is composed of the input parameters “forming force”, “complexity”, “max. forming per stage”, and “batch size” and provides a suggestion for the number of stages as an output value (Fig. 3). In the set of rules of the fuzzy system, rules such as “Small batch size & low complexity & low forming force = low number of stages” are mapped out.

Fig. 3
figure 3

Fuzzy-Logic for calculating the number of stages of the forging sequence

The batch size and maximum forming per stage are entered manually by the user. The input parameter for the complexity of the forging is transferred from the complexity score model (\(K_{Ges}\) cf. [34]). To be able to conduct the approximate determination of the forming force F based on the forging geometry; the projection surface of the forging is determined from the STL file. Using the projection surface Ad and the resistance to deformation at the end of forming \(k_{we} ,\) the required forming force F can be calculated [4, 35]:

$$F = A_{d} *k_{we} .$$

The projection surface corresponds to the mean component cross-section perpendicular to the forming direction. The projection surface can be determined using the slicer-algorithm (Sect. 3.1). The resistance to deformation at the end of forming \(k_{we}\) is calculated via the yield stress kf and a shape factor that takes the workpiece shape into account [35]. The yield stress depends on the material and the corresponding material data are stored in the method and the data sets can be extended accordingly. The number of steps calculated by the fuzzy logic system is passed on to the point-shift-algorithm to generate the intermediate forms. With the determined parameters (cutting axis, shape class, complexity score, and number of stages), a developed algorithm first calculates the necessary flash ratio and the mass distribution around the center of gravity line of the forging [36]. Step by step, the mass distribution of the forging and the semi-finished product is then adjusted, and the corresponding intermediate forms are exported as an STL file. In the following, the division of the forging into cutting planes as well as the functioning of the algorithm for the generation of the intermediate forms through the cross-section adjustments and the volume compensation through a length change via the forging sequence are presented in detail.

3.1 Slicing the forging geometry

The basis for the method is the division of the forging (in STL format) into cross-sections (slices) by a slicer-algorithm. The number of cutting planes \(n\) is specified by the user of the method. The forging is divided in \(n\) cross-sections at a constant distance \(d\) along the cutting axis (can be any of the three spatial directions) of the forging. Figure 4 shows the slicing of a 2-cylinder crankshaft into cross-sections. Geometric information such as surface area, center of gravity and polygon chain of the cross-sectional area can be determined from the sectional planes.

Fig. 4
figure 4

Slicing a 2-cylinder crankshaft to obtain cross-sectional information

Every point of the polygon chain is described through its position vector \(\vec{x}\) and satisfies the following equation in which \(\vec{p}\) is a support vector and \(\vec{n}\) a normal vector to the plane:

$$\left( {\vec{x} - \vec{p}} \right)* \vec{n} = 0.$$

After slicing the component, a graph model of the component is created to save the neighborhood relations of the points. This is necessary for the later creation of the intermediate forms and checking the intermediate forms for forming rules. In the first step, all faces of the STL file are read in and the neighborhood relationships of the faces are determined. Then the cross-section points \(P_{i,j}\) for \(n\) cuts are read in. For each cross-section point \(P_{i,j}\) a face is searched in which the point is located. A tracing map is created for this purpose. For performance reasons, the search is aborted after too many levels. The search is conducted using the breadth-first search (BFS). In contrast to the depth search, all nodes that can be reached directly from the start node are searched first. A data structure in the form of a graph object is available, which makes it possible to export the intermediate forms as STL objects later in the method. Slicing provides relevant geometric information such as the area of the cross-sections and centres of gravity.

3.2 Point-shift-algorithm for generating the intermediate forms

To generate the intermediate forms of the forging sequence backwards from the forging to the semi-finished product, an algorithm was developed that shifts the mass distribution and center of gravity line of the forging step by step in the direction of the semi-finished product geometry [36]. For this purpose, forming factors fi are defined, which indicate what percentage of the total forming takes place within a stage of the forging sequence. The following must apply to the entire forging sequence: \(\sum\nolimits_{i = 1}^{n} {f_{i} } = 1\). The cross-section adjustment is done by adjusting the polygon courses of the cross-section contours. The cross-sectional contours of the forging are determined in all cutting planes \(A_{j}\) using the slicer algorithm presented in Sect. 3.1. The contour is described by a polygon consisting of the corner points \(P_{1}\) to \(P_{m - 1} .\) It is \(\left[ {P_{1} ,P_{2} } \right] \cup \left[ {P_{2} ,P_{3} } \right] \cup \ldots \cup \left[ {P_{m - 1} ,P_{m} } \right]\) and thus there is a closed polygon (see Fig. 4). For each point \(P_{i}\) the target coordinates on the target contour (semi-finished cross-section) are determined for displacement in the direction of the semi-finished product. The semi-finished product can be round or rectangular. Starting from the center of gravity of the forging and the semi-finished product, beams are generated through the polygon points \(P_{i}\) in the cross-section. The geometry of the semi-finished product is described by means of individual lines. For this purpose, a defined number of points is evenly distributed on the semi-finished product geometry and a line segment is defined between two points. To calculate the target coordinates for the point \(P_{i}\), the intersection point between the beam and line segment on the target contour is calculated by solving a linear system of equations. The system of equations has a solution on the straight line between two points on the target geometry (semi-finished product) and on the beam starting from the center of gravity to the point \(P_{i}\) (see Fig. 5):

$$x\left( 1 \right) \le 1 \cup x \left( 1 \right) \ge 0 \cup x\left( 2 \right) \ge 0$$
Fig. 5
figure 5

Target coordinates for determining the shift vector

A shift vector \(S_{v}\) is defined between the target coordinates determined in this way and the polygon point \(P_{i } .\)

Along this shift vector \(\left( {S_{v} } \right)\) the point \(P_{i }\) is shifted stepwise according to the number of forming steps and the forming factors \(f_{i}\)(see Fig. 6). If the centres of gravity of the semi-finished product and the forging are not identical in the cross-section, first all points of the polygon chain of the cross-section of the forging are virtually shifted by the offset of the centres of gravity of the semi-finished product and the forging (see Fig. 6 Case 2). Then the cross-section contour is adjusted according to the displacement vectors. Finally, the points corresponding to the virtual displacement at the beginning are displaced back.

Fig. 6
figure 6

Point-shift-algorithm for the cases centre of gravity for forged part = centre of gravity semi-finished product (Case 1) and forged part ≠ centre of gravity semi-finished product (Case 2)

The forming factors \(f_{i}\) can be entered individually for cross-section classes depending on the cross-section size (e.g., cross-sections larger than semi-finished product diameter or smaller than semi-finished product diameter). The number of classes \(x\) can be entered manually and the cross-sections are evenly sorted into the selected classes. This allows the mass distribution in the intermediate forms to be set more finely and individually. The point clouds of the intermediate forms \(\left( {IMF} \right)\) over all iterations (s) of the forging sequence, result from the sum of the multiplication of all shift vectors \(S_{v}\) for all points \(P_{i}\) in all cross-sections \(A_{j}\) with the forming factor matrix:

$$IMF,\,\, iter\left( {1, \ldots ,s} \right) = \mathop \sum \limits_{j = 1}^{n} S_{v} \left( {A_{j} } \right)*\left( {\begin{array}{*{20}c} {f_{1,1} } \\ {f_{2,1} } \\ {f_{x,1} } \\ \end{array} \begin{array}{*{20}c} {f_{ \ldots } } \\ {f_{ \ldots } } \\ {f_{ \ldots } } \\ \end{array} \begin{array}{*{20}c} {f_{1,s} } \\ {f_{2,s} } \\ {f_{x,s} } \\ \end{array} } \right).$$

The number of shift vectors in a cross-section \(A_{j}\) results from the amount of points Pi in the cross-section.

To take different geometry-specific features into account when designing the forging sequence with the point-shift-algorithm, special cases of the algorithm were defined, which are selected at the beginning based on the ANN-based geometric analysis and classification (cf. [33]). Figure 7 shows the different special cases for the point-shift-algorithm. In a first query, it is checked whether there are gaps in the component. If there are gaps, the gaps or bifurcations are closed by a thin material layer between the two cross-sections in a cutting plane so that there is only one continuous cross-section per cutting plane (CloseGap- algorithm in Fig. 7). The second query checks whether the component is a rotationally symmetrical component (e.g., flanged shaft, upright forged). If this is the case, the generation of the intermediate forms is only conducted by observing the cross-section in the center of the component in 2D. Subsequently, the cross-section contour is rotated by 360 degrees to obtain a 3D contour of the intermediate forms. If the part is not rotationally symmetrical, it is queried whether it is a long form or a compact shape or a disc shape. In the case of a bended long form, a variant of the point-shift-algorithm is called up in which a bending operation is integrated. In the bending operation, only the centers of gravity of the cross-sections are shifted and there is no adjustment of the cross-section contour. This bending line is determined by a linear regression. The differences in calling the point-shift-algorithm for long forms and disc shapes lie in the orientation of the semi-finished product and the length changes over the forging sequence.

Fig. 7
figure 7

Queries for the appropriate point-shift-algorithm depending on geometry-specific features and the shape class

3.3 Change in length of the intermediate forms over the forging sequence

By means of the point-shift-algorithm and the resulting change in cross-section (change in surface area), the material flow is mapped transversely to the forming direction. In order to equalise a possible volume difference due to the cross-section changes, a length change of the intermediate forms must be carried out and thus material flow in the longitudinal direction to the forming direction must be considered. The boundary condition for a length change is the volume constancy valid over the entire forging sequence for the closed die forging process, so that the following applies for each volume \(V_{iter, i}\) of the intermediate forms:

$$V_{iter, i} = V_{semifinished} = V_{forging} + V_{flash}$$

Knust et al. have developed a method for the calculation of an adaptive flash ratio for the preform optimisation of rotationally symmetric and cross-wedge rolled preforms [15]. This approach was extended to calculate the flash ratio for an entire forging sequence for closed die forging. An additional factor \(m\) was integrated into the calculation and the complexity value calculated by the complexity score model [34] was used used to calculate the flash ratio \(V_{{flash, A_{j} }} :\)

$$V_{{flash, A_{j} }} = m + 0.0059*e^{{5.6249*K_{Ges} }} .$$

The factor \(m\) is calculated as a function of the shape class [5] as follows:

$$m = \left\{ \begin{aligned} 0.05*\left( {1 + 0.3*e^{{K_{Ges} }} } \right)\, \, & if\,\,SC = 1 \cap 2 \\ 0.1*\left( {1 + 0.3*e^{{K_{Ges} }} } \right)\, \, & if\,\,SC = 3 \\ \end{aligned} \right..$$

An exception to the volume constancy in the method is a calibration stroke before the final form, in which only very little forming takes place \(\left( {f_{1} \le 10 \% } \right)\). Here it is assumed that a large part of the material has already flowed into the flash in the previous stages. Therefore, the volume of the forging without flash is used as the reference volume for the volume adjustment in the calibration stroke. The reference volume over the iterations of the forging sequence is given by:

$$V_{Iter, i} = \left\{ \begin{aligned} V_{forging} \,\, & if\,\,f_{i} \le 0.1 \cup i = 1 \\ V_{semifinished} \,\, & if\,\,f_{i} > 0.1 \cup i > 1 \\ \end{aligned} \right.$$

In the presented method, only die forging processes are considered and only intermediate forms are generated for die forging. Therefore, the following conditions for long forms (SC 3 according to Spies) apply to the length changes over the forging sequence:

$$\begin{aligned} l_{forging} \ge l_{Iter, 1} \ge & \ldots \ge l_{i} \ge l_{semifinished} . \\ & \Delta l < f_{l} \\ \end{aligned}$$

\(f_{l}\) is defined as the elongation factor in a stage. Values between 0 and 10% can be specified here. The limit value of 10% results from expert interviews with project partners, since a length change of more than 10% in a forming step during die forging occurs only very rarely in practice. However, a change of the limit value is feasible with little effort. To perform the length change over the forging sequence, the distance between the cutting planes generated by the slicer-algorithm is changed. The initial distance \(d\) is calculated as follows:

$$d = \frac{{l_{forging} }}{n}.$$

By comparing the volume after the execution of the point-shift-algorithm with the volume of the previous iteration, the volume difference that has to be compensated by a length change can be determined. Using the cross-sectional areas \(A_{j,iter}\) and the distances \(d_{i,iter}\) between the cross-sectional areas, the volume segments and the volume difference can be calculated:

$$V = \left| {V_{i} - V_{i + 1} } \right| = \left| {\sum\limits_{j = 1}^{n - 1} {A_{j,iter,i} *d_{j,i} } - \sum\limits_{j = 1}^{n - 1} {A_{j,iter,i + 1} *d_{j,i + 1} } } \right|.$$

A variable length change of the distance \(d_{i,iter}\) in different areas of the component leads to an improved mass distribution and a better form filling behaviour than a uniform adjustment over all cross-sections [37]. The individual volume segments are divided into six (number can be varied) different classes (V+ +, V+, V+ = , V− = , V−, V−) according to their volume (Fig. 8). Segments of class V+ + have the largest volume and segments V− the smallest volume. A volume segment \(V_{i}\) is formed by a cut \(i\), the cut \(i + 1\) and the distance \(d_{i}\). For each volume segment, a volume change factor \(V_{L}\) is calculated:

$$V_{L,i} = \frac{{V_{i} }}{{d_{i} }}$$
Fig. 8
figure 8

Division of a connecting rod into volume segments and classification into the six volume classes

In order to carry out small length changes with a simultaneous significant change in volume, large cross-sections (V+ + and V+) in the forging are suitable. Large length changes with small volume adjustment are carried out with small cross-sections (V− and V−).

In addition to the \(\Delta V\) to be compensated, the maximum permissible length change \(f_{l}\) in the forming stage is another boundary condition for the length change. In order to bring \(\Delta V\) to zero and at the same time to maintain and achieve the maximum length change in a stage, first a change in the distances in V− (low \(V_{L,i}\)) is made to achieve the length change, then the volume is balanced via V+ + (high \(V_{L,i}\)). If the volume constancy is not achieved or the changed length of the intermediate form does not correspond to the boundary conditions (\(l_{iter} \le l_{iter - 1}\) and \(\Delta l \le f_{l}\)), the adjustment is repeated with the next volume segments (V+ and V−, then V+ = and V− =). For forgings from the disc shape class, the change in length is defined differently, since in this case the forging has the shortest length (or the lowest height) and the semi-finished product is significantly longer. Here, a maximum compression ratio is taken into account when balancing the volume to achieve volume constancy.

3.4 Rules library

In order to ensure the process reliability of the generated forging sequences, a rule library was developed and implemented in the FSD method. This allows design rules and forming boundary conditions to be implemented in formulas in the method. Three rule classes were defined: Point rules, area rules and volume rules (see Table 1). A grammar for the creation of rules was developed to enable the users to integrate their own rules into the FSD method.

Table 1 Rule Categories

An executable tree generator is used for parsing (checking whether the rules can be mapped with the grammar). After an intermediate form has been generated by the point-shift-algorithm, the intermediate form is checked for the rules activated in the rule library. It is possible to activate and deactivate the rules. If a rule is not respected, an adjustment is made, e.g., by shifting the points to a target point by which the rule is respected. For this purpose, tolerance ranges are defined for the point shift.

In this way, process-related boundary conditions can be checked and adhered to. This prevents, for example, undercuts from occurring in a cross-section contour. Figure 9 shows an example of how to check a cross-section for undercuts. Each cross section is divided into four quarters (Q1 and Q2 above the plane of division, Q3 and Q4 below the plane of division) and it is checked whether an undercut is present. The condition for point P2 in Q3 that there is no undercut is as follows:

$$x\left( {P_{1} } \right) \le x\left( {P_{2} } \right) \le x\left( {P_{3} } \right) \le x\left( {P_{4} } \right)$$
Fig. 9
figure 9

Checking a cross-section of an intermediate form for undercuts (areas marked in red show an undercut)

If this condition is violated, point P2 is moved to the position of point P2* in Fig. 9 and the undercut is resolved. This check after each iteration in each cross-section ensures that no faulty intermediate forms can be generated.

In addition to undercuts, potential forming faults such as folds are also checked. For this purpose, the criterion developed by Behrens et al. for checking folds in mesh structures [38] is adapted to the developed FSD method and transferred to the corresponding grammar. Within the control loop for checking the rules, rule category 1 is checked first. Here, for example, undercuts and potential folds [34] are checked and corrected. In the second step, the surfaces are checked (e.g., compliance with a maximum cross-section change) and the surface is scaled without changing the cross-section contour. In the third step, the component volume is checked with a change in length, if necessary, to adjust the volume. Formula 13 shows an example of the structure of a rule, as entered in the method:

$$iter = 1 | A\left( {iter - 1, cut} \right) < A\left( {iter, cut} \right)*3$$

The example rule describes the cross-section change over the iterations (intermediate forms). Here it is checked that a cross-sectional change may not be greater than a factor of three.

4 Results of the FSD method

In order to validate the method for the automated design of multi-stage forging sequences, the developed method is applied to various forgings. Following the shape order according to Spies, forgings with different geometric characteristics are selected for validation. The selected forgings and the corresponding geometric characteristics are listed in Table 2. The selected geometries contain all geometric characteristics of the Spies shape order from which the individual shapes are composed. This ensures that they can be applied to different forgings.

Table 2 Selected forgings for the validation of the FSD method

The FSD method is implemented in Matlab. In this study, the method is applied to a Windows computer with 16 GB RAM, an SSD and an I7 processor. The selected forgings were loaded and processed in all presented modules of the method (cf. chapter 3). The design parameters determined by the FSD method are listed in Tables 3 and 4. The used material in this study is 1.7225. The detection of the shape class for forging A gave an incorrect result and was corrected manually.

Table 3 Result parameters from automated stage planning (complexity score, forming force and shape class)
Table 4 Result parameters from automated stage planning (Number of stages, flash ratio and forming factors)

The detection of the shape class for forging A gave an incorrect result and was corrected manually.

All forging sequences contain a calibration stroke, which means \(f_{1} \le 10 \% .\) The semi-finished product diameters are selected manually. Boundary conditions are the maximum permissible change in length that can be achieved over the entire forging sequence, as well as the proposed flash ratio. In the selection, a diameter is aimed for that is as close as possible to the largest cross-section of the forging. This corresponds to the procedure for design in practice. Table 5 shows the generated forging sequences and the information on the selected semi-finished product.

Table 5 Forging sequences generated by the method for the validation forgings A–F

The runtime of the method is dependent on both the number of facets of the STL file and the number of cuts, as these two parameters significantly influence the number of points to be shifted by the point-shift-algorithm. The runtimes for the forging sequences (Table 5) is visualized in Fig. 10.

Fig. 10
figure 10

Runtime and number of facets of the forgings A–F

For all forgings \(\left( {n = 60} \right)\), the forging sequence was generated in less than one hour. The process is particularly fast for part D, as the points for the rotationally symmetrical forging only need to be moved in one cross-section. For a sufficient quality of the result export, however, points are added during the generation of forging sequences if the initial geometry was not sufficiently finely meshed.

5 Validation of the FSD method with FE simulations

The generated forging sequences, with the respective intermediate forms and semi-finished product, are checked by means of FE simulations in order to verify quality criteria such as form filling and freedom from folds. The material modelling in the simulations is done with the Hensel-Spittel equation implemented in Forge® NxT 3.2 [39]. With this equation, the forming behavior is determined by means of a function dependent on the temperature T, the effective strain ε, the flow stress \(\sigma_{f}\), the strain rate \(\dot{\varepsilon }\), and the coefficients A and m1–4. The Hensel-Spittel equation used by Forge® NxT 3.2 is represented in the following equation:

$$\sigma_{f} = A e^{{m_{1} T}} \varepsilon^{{m_{2} }} e^{{m_{4} /\varepsilon }} \dot{\varepsilon }^{{m_{3} }} .$$

The flow curve used for hot bulk metal forming processes is sufficiently precise by means of the coefficients \(A,\,\,m_{1} ,\,\,m_{2} ,\,\,m_{3} ,\,\,m_{4}\). The individual regression coefficients of the Hensel-Spittel equation of 1.7225 are shown in Table 6.

Table 6 Regression coefficients of Hensel-Spittel equation of 1.7225

The parameters from Table 7 were used for the simulations for validation.

Table 7 Simulation parameter in Forge Nxt 3.2

The evaluation of the FE simulations is summarized in Fig. 11. The quality criteria form filling and freedom from folds are checked. The form filling was analyzed via the contact between the workpiece and the form. The blue areas represent areas with form filling. For the forgings A, B, D, E and F, form filling can be achieved in the final form, but often complete form filling is not achieved in the first stage. The forging F was simulated with a flash barrier in stage 1. This was done to adapt the forging sequence closer to reality, since an initial mass distribution is usually achieved by specific preforming processes such as stretch rolling or cross wedge rolling, which are significantly more material-efficient than a closed-die forging process. Here, other length changes can also be mapped with process reliability. For the forging C, no form filling was achieved in the first two intermediate forms. Folds also occurred in the process, which suggests that the algorithm, which places a thin layer of material in the bifurcation, still needs to be adapted and further rules are necessary to ensure freedom from folds for this class of forgings. In the other forging sequences, no folds occurred.

Fig. 11
figure 11

Simulation results for the validation forgings A–F to check the quality criterium form filling by the contact in the FE simulations

The results for forging E show that the FSD method can also process bended forgings and realistically represents bending operations in the forging sequence. Besides long forms, other shape classes (stocky part D) can also be processed and formable approximate solutions for forging sequences are generated. Overall, the simulation results show that the design parameters determined automatically by the FSD method (number of stages, flash ratio, etc.) lead to process-reliable forging sequences for a wide range of forgings. The results are generated in a short time and represent a very good approximation solution for an initial design of the forging sequences.

6 Summary and outlook

A method was presented that enables the automated generation of multi-stage forging sequences for die forging based on the forging geometry (FSD method). The FSD method is composed of different modules (ANN classification tool, complexity score model, fuzzy design, intermediate form generation algorithm), which were developed to be able to fulfil the requirements of a method for automated forging sequence design. It was shown that the FSD method generates multi-stage forging sequences for different forgings based on the 3D model (STL file) of the geometry in a short time (less than 60 min). The individual intermediate forms of the forging sequence are exported as STL files. Based on the STL files, initial FE simulations can be set up to check quality criteria of the forging sequences (freedom from folds, form filling). The presented simulation results for the validation of the developed method show that the quality criteria are very well met and the suggestions for the forging sequence provide a very good approximate solution for design support. With the current method, there are still limitations in the applicability of the method for strongly bended geometries (SC 331 and higher). Such parts can currently only be processed segment by segment. This means that the forging would have to be divided into segments with a curved main axis for the method and the partial results would then have to be joined together. In future development work, this limitation is to be solved through adjustments, e.g., by calculating the neutral fiber of the forging and slicing along the neutral fiber to determine the cross sections. For complex forgings with bifurcations, adjustments are still necessary to obtain process-reliable forging sequences. In a next development step, preforming processes could be implemented in the FSD method (e.g., stretch rolling and cross wedge rolling) to further increase the realism of the results. This would allow new boundary conditions for the geometry generation, such as significantly greater length changes or process-specific cross-sectional shapes. In addition, the forging sequences could be designed to be even more material-efficient if preforming processes are considered.