Blank Shape Optimization Considering 3D Space Targets for External and Internal Boundaries in Sheet Metal Forming Processes

Optimum design of the shape of the initial blank is an important task before sheet metal forming processes. It reduces production costs, material waste and improves the quality of the product significantly. In the present work, the blank shape optimization problem is considered based on the 3D space target contours and moreover the sheet metal can have an internal boundary. In other words, the shape optimization process can be applied on the internal boundary, external boundary or both. To the best knowledge of the authors, there are no published works in this subject. Using the iterative simulation-based optimization process, a special shape error index and also updating algorithm were proposed to modify the blank geometry in each iteration until capturing the optimum shape. The sheet forming process is highly nonlinear in nature due to plastic behaviors, large deformations and frictional contact surfaces. Therefore, the updating formula should be robust enough to have less sensitivity with respect to the initial guess. To evaluate the proposed updating formula and its robustness, some numerical examples were considered and the effects of different tools geometries, 2D and 3D target contours, internal and external boundaries and different initial guesses were examined.


Introduction
Deep drawing is a useful sheet metal forming process for shaping flat blanks into cup-like forms. The process begins by cutting the blank with a specific geometry and placing it on the die. The plunging of the punch into the die drives the blank in, forming it into the shape of the die. Often, the part of the blank that remains outside does not develop a favorable shape and should be recut. The second cutting process creates two main issues of increasing wastes and involving complexities that can raise the production cost especially in cases with curved and 3D space cutting paths. Therefore, production costs remain lower if additional cutting can be avoided or at least minimized. An optimum blank geometry can realize this objective. Blank optimization refers to selecting an initial blank geometry that develops the desired shape after deep drawing without or with minimal need for additional cutting. Regarding the time-consuming and costly nature of the trial and errorbased blank optimization methods, researchers have attempted to use numerical approaches to designing an optimum blank. The use of blanks with optimum geometries offers many advantages in deep drawings, such as reducing the production cost, improving the process quality, thickness distribution and formability of the part, minimizing forming defects such as wrinkling and rupture and decreasing the number of trial and error steps in the product development process. The numerical simulation of the sheet metal forming process is integral to the study of the feasibility of production by deep drawing and the initial design of a new part with complex three-dimensional geometry.
So far, various optimization algorithms have been proposed to design blanks, the most notable algorithms of which can be classified into four groups, namely the slip line field [1][2][3][4][5][6][7][8][9], geometric mapping [10][11][12][13], inverse approach [14][15][16][17][18][19][20][21][22][23][24] and numerical simulation-based iterative methods [25][26][27][28][29][30][31][32][33][34][35]. Meanwhile, methods of the fourth category have been widely used by researchers, due to their higher simulation accuracy and more capacity to solve complex problems. The iterative methods are capable of simulating complex geometries, large deformations, complex material behavior models, complex friction models, complex contact models, dynamic behaviors and other factors with high accuracy. However, demanding a large volume of calculations is the main drawback of numerical simulation-based iterative methods. Developing efficient optimization algorithms can reduce the number of iterations and computational complexity. The present study aims to propose a new, efficient, iteration-based algorithm and implement it by a finite element numerical method to resolve some of the issues related to such methods. Given that the proposed algorithm is classified as an iterative method and is based on numerical simulation, studies on similar algorithms are reviewed in the following paragraph.
In 1985, Toh and Kobayashi [25] were one of the earliest studies conducted on the application of numerical simulation-based iterative methods for blank optimization in deep drawing. The authors relied on a finite element method in the numerical simulation of the deep drawing process, using a geometric correction algorithm based on the material flow. The main shortcoming of this method is that, material flow patterns are not a reliable reference for developing the geometric correction algorithm, due to the nonlinear behavior prevalent in deep drawing. Further, this method fails to converge when the initial guess is far from the optimum blank. In 1997, Chung, Barlat, Brem and Lege [26] developed a sequential design method for drawing the ideal forming theory, by using finite element analysis and experimental investigation. They applied the method to design blanks from a severely anisotropic aluminum sheet, aiming to minimize corners. In 1999, Park, Yoon, Yang and Kim [28] proposed a blank design method by combining the ideal forming method with a deformation path iteration method based on the finite element analysis. First, a test blank was prepared based on the ideal forming theory. Then, the optimum blank was obtained by the iterative deformation path iteration method. Although this method is applicable in cases where the depth of drawing and sheet deformation are limited, it reduces the convergence rate in case of a high depth of drawing and severe deformation. In 2002, Pegada, Chun and Santhanam [29] used the iterative design procedure in an algorithm to detect optimum blank contours, considering anisotropy and friction in the deep drawing of an aluminum cup. In the same year, Shim and Son [30] applied the iterative sensitivity method for blank optimization, in which sensitivity was calculated by the finite element analysis of deformation using both main and offset blanks. In 2003, Son and Shim [31] proposed the initial velocity of the boundary nodes (INOV) method, in which the ratio of the initial velocity boundary nodes to the entire path was used in the optimized blank algorithm. In 2008, Vafaeesefat [32] utilized iteration and the finite element simulation for blank optimization. The blank correction algorithm was based on the projection of the target contour over the deformed blank (boundary projection method). In the same year, Azaouzi, Naceur, Delamézière, Batoz and Belouettar [33] introduced an initial blank estimation method based on the one-step inverse approach and optimized the blank geometry by drawing iterations and integrating heuristic optimization algorithms into the finite element analysis. In 2009, Hammami, Padmanabhan, Oliveira, BelHadjSalah, Alves and Menezes [34] developed an iteration-based method with an experimental initial blank using the Push-Pull design optimization technique. They also studied the effect of initial anisotropy in obtaining the initial blank shape [35]. In 2012, Fazli and Arezoo [27] proposed a novel iterationbased blank optimization technique and improved the method employed by Son and Shim [31], recommending to correct the initial blank based on the difference between the final shape and the target shape to adjust boundary nodes. Comparing the results with INOV and Push-Pull reveals that the final products of the algorithms are practically similar with no apparent distinction, and the proposed method is more efficient. In 2020, Zhanga, Gaob and Caob [36] designed a blank geometry for reinforced carbon fiber using finite element analysis and presented an innovative approach to automated network-tuning integration. They showed that this newly developed approach could accurately design blank geometry under different process conditions. In 2020, Yaghoubi and Fereshteh-Saniee [37] optimized the geometrical parameters for elevated temperature hydro-mechanical deep drawing process of 2024 aluminum alloy using a group data management method and bee algorithm to achieve optimal values for process variables.
It must be noted that all of the above-mentioned works considered 2D target contours and also just a single external boundary. On the best knowledge of the authors, no works are published considering 3D target contours or internal boundaries in the blank shape optimization problem in deep drawing. The main objective of the present study is to achieve a new algorithm that can solve such blank optimization problems. To do this, an iterative simulation-based optimization approach is followed using finite element code ABAQUS. To modify the initial geometry, a specific shape error formula and also updating formula are proposed to improve the blank geometry iteratively by starting from an initial guess. Due to highly nonlinear nature of the deep drawing process, a shape smoothing formula is also proposed to stabilize the method and prevent sharp zigzag patterns from the optimum contour. To evaluate the proposed algorithm, five different numerical examples were solved. In these examples, different tool geometries, different target contours including 2D and 3D ones with different cases with and without internal boundaries are considered. It is also worth to note that due to highly nonlinear nature of the deep drawing process, robustness of the method with respect to the different initial guesses is important. To evaluate this, different initial guesses are considered. It shows that the method has less sensitivity to the initial guess.

Iterative Simulation-based Shape Optimization
Shape optimization problems are classified as variabledomain problems because the geometry of the problem domain should be determined as a part of the solution process. In these problems, the unknown boundaries are often parameterized for searching in a space with a limited number of dimensions, which is performed by selecting some key points on the boundary and connecting them together. Founding the coordinates of the key points consolidates unknown boundaries. Figure 1 illustrates a schematic representation of a blank shape optimization problem in the sheet metal forming process.
As observed, the initial blank (undeformed part) and the final product (deformed part) are shown. The problem is to find that the geometry of the initial contour is such that the final contour matches the specified target after deformation.
Solving the shape optimization problems by simulationbased optimization methods consists of the following steps. First, an initial guess is considered for the blank shape (initial contour). Then, the sheet metal forming process is simulated based on the initial guess to obtain the final shape of the part (final contour). The final contour does not match the target contour at this stage. As a result, the mismatch between the final and target contours can be quantified by defining a shape error parameter. The geometry of the initial contour is then modified, and the forming process is simulated again. The algorithm continues and reduces the shape error iteratively until the error drops below a set level, and consequently, the process stops. Figure 2 provides the general flow chart of the simulation-based optimization steps.
The finite element method is used here for numerical simulation of sheet metal forming process. The four-node bilinear thin-thick shell elements with six degree of freedom at each node are considered here. This element has in plane and out of plane deformations considering large deformation and plastic behaviors. The surface-to-surface contact with penalty formulation is used to model tools to sheet plate contacts. The problem is solved using dynamic explicit solver with automatic time incrementation and

Shape Error Estimation
The objective of the optimum design problem is to minimize the difference between the final contour and the target. With this aim, the target shape is represented using a narrow ribbon in the 3D space. Figure 3a demonstrates a schematic representation of the initial contour, final contour and the target ribbon. It is worth noting that CAD software facilitates generating such a target ribbon. In the first iteration, an initial guess should be considered for the blank geometry and the final contour would obviously be far from the target ribbon in this stage. Figure 3b shows the flow path of the material points that start from the initial contour and end at the final contour.
Some material points pass the target, while some others fail to reach the target. As shown in Fig. 2, the optimization procedure should be followed to capture the optimum initial contour. The optimization procedure has two important steps of shape error estimation and modification formula. The two following sections explain these two steps.
For each point of the initial contour, the shape error is defined as the shortest distance between the corresponding point on the final contour and the target ribbon. In Fig. 4, two representative points A and B are shown on the initial contour, and the corresponding points on the final contour are shown by A' and B', respectively.
The shortest distance between final points A' and B' and the target ribbon are considered here as the absolute value of the shape error. If the flow path fails to pass the target ribbon, the shape error is considered as positive and if the flow path passes the target ribbon, the shape error is considered as negative. The shape error will be used in the modification formula to update the shape of the initial blank.

Shape Modification
In this section, a robust and simple shape modification formula is presented to update the location of the key points on the initial contour in each step. As illustrated in Fig. 5, two representative points A and B on the initial contour should be considered to explain the modification formula.
The shape error corresponding to each boundary point is obtained as explained in Sect. 3. For each point of the initial contour, unit vector n is considered normal and outward with respect to the initial contour. The unit vectors corresponding to two representative points A and B are shown in Fig. 5. The coordinate updating formula of the points of the initial contour is proposed as follow: where P and P new represent the current and updated coordinate vectors of the points on the initial contour, respectively, e indicates the shape error, as explained in the previous section (see Fig. 4), and a is the under-relaxation factor. This factor should be selected empirically as less than unity to increase the stability of the optimization iterative procedure, due to the nonlinear behavior of the sheet metal forming process. Figure 5 also displays a schematic representation of the updating formula. As shown, the new coordinates corresponding to two representative points A and B are shown with hollow circles. The point A has a positive shape error (see Fig. 4). In other words, the corresponding point A on the final contour fails to reach the target ribbon, meaning that there are some excessive materials near this point and these redundant materials should be removed from this region in the next iteration. Therefore, point A should move to the opposite direction of the normal vector n in this point.    Further, a similar procedure can be followed for point B. This point passes the target ribbon during the sheet forming process and has a negative shape error (see Fig. 4), implying that there is less material near this point and the updating procedure should add some materials in this region. Therefore, point B should move along the unit Fig. 8 Initial and final contours for the first four iterations starting from a hexagonal initial guess for example 1 Fig. 9 Initial and final contours for successive iteration of example 1; a starting from a squared initial guess; b starting from a circular initial guess vector n. The updating formula given in Eq. (1) can be applied for all key points on the initial contour to obtain the new initial contour, as shown schematically in Fig. 5.

Shape Smoothing
The sheet metal forming is a highly nonlinear process, due to complex material and geometric behaviors. Thus, the shape modification algorithm may show some instabilities. The under-relaxation factor, \alpha, which was introduced in the previous section, is a controlling parameter that tries to increase the stability of the optimization process. The under-relaxation factor also seeks to limit large movements of the initial contours in successive iterations. The other source of instability is zigzag patterns that may appear in the shape modification step. To remove the zigzag patterns, a smoothing step is considered in the optimization procedure (see Fig. 2).
To explain the smoothing step, Fig. 6 is considered. As shown, a part of the modified initial blank is shown schematically with a thick solid line. By considering two adjacent points P i and P iþ1 , the directions of the angles h i and h iþ1 are addressed as an indicator to show whether a zigzag pattern appears at this part of the contour. If both angles h i and h iþ1 are in the same direction (both CW and both CCW), no zigzag pattern appears in this part of the contour. Otherwise, this part of the contour is found to show zigzag pattern. In this case, the following smoothing formula is used to update the location of boundary points P i and P iþ1 .
where b represents a smoothing parameter. It should be noted that value b = 0 means no smoothing, while b = 1 leads to the movement of each point to the midpoint of left and right adjacent points. Figure 6 depicts the smoothed initial contour with a thick dashed line. This smoothing procedure should be applied in all parts of the modified initial contours showing a zigzag pattern. However, selecting a relatively small value for the smoothing parameter requires implementing the smoothing operator several times to whole points of the initial contour.

Numerical Examples
In the following, some numerical examples have been solved, and the results have been analyzed to evaluate the proposed algorithm. The sheet metal forming process was numerically simulated by using the finite element method and the commercial software, ABAQUS. Additionally, deformations were assumed large, and nonlinear geometric effects were taken into account. The frictional contact Fig. 10 The optimum blanks obtained using three different initial guesses and compared with the shape given in Reference [27] Fig . 11 Dimensions of the tools for example 2 (mm) constraint was considered between all connected surfaces, and the friction Coulomb model was applied by the penalty method. The material was assumed to have an elasticplastic behavior, regardless of strain rate and temperature effects. A Python program was developed in ABAQUS software to implement the whole optimization process.
In all examples, the sheet metal was assumed to have a thickness of 0.85 mm. The stress-strain behavior in the plastic region was characterized by using a power-law relation as follows.
where p and r p represent the plastic strain and stress, respectively. Table 1 presents the elastic properties and the coefficients of plastic behavior [27]. The die, punch and plate holder were modeled as rigid parts. The only deformable part was the plate, and the four-node shell elements were used to simulate the sheet metal forming process.

Example 1: L-Shaped Cup
For the first example, the deep drawing of an L-shaped cup has been considered [27]. Figure 7 displays all geometric details and dimensions (in millimeters) of the tools. The depth of the drawing was considered as 20 mm, and a 9800 N blank holder force was applied, which remained constant throughout the process. Table 2 reports the coefficients of friction between contacting surfaces [27]. The target contour was considered such that a 2 mm uniform flange was formed all around the part. A regular hexagonal shape was considered as the initial guess for the initial contour. Figure 8 depicts the first four steps of the optimization process. The selected initial guess is far from the optimum shape, and therefore, the final contour is also far from the target. The initial contour was modified iteratively in subsequent steps, and the final contour converged to the target shape. The proposed optimization procedure could capture the optimum shape only in four modification steps. In other words, only five numerical simulations were required to capture the optimum shape, regardless of the initial guess. The small number of iterations is an advantage of the proposed method as few iterations lead to lower computational costs and high efficiency. After four iterations, the maximum shape error has been found to be at 0.45 mm, and the average shape error across all key points is at 0.11 mm. Fig. 12 Results of example 2; a three different initial guesses for the initial blank; b the optimum initial blank obtained with three different initial guesses; c the final contours obtained from three different initial guesses Fig. 13 The geometric details of the tools for example 3 (mm) To investigate the effect of initial guess, two different initial guesses were selected, the results of which are given in Fig. 9. Figure 9a and b shows the initial and final contours starting from the square and circular initial guesses, respectively. In both cases, the optimum shape is captured in four iterations. Figure 10 compares the optimum initial blanks, which are obtained from different initial guesses, and the results given in Reference [27]. As observed, the selected initial guesses fail to affect the proposed algorithm, indicating the high accuracy and consistency of the results.

Example 2: Two-Level Pan
In this example, a deep drawing problem with two-level die and punch was considered, and two different drawing depths were obtained [27]. The proposed algorithm was used to solve this problem with three different initial guesses. Figure 11 shows the geometrical dimensions of the die, blank holder and the punch. All dimensions shown in Fig. 11 are in millimeters.
The punch stroke was 30 mm, and the target contour was a uniform edge of 1.75 mm width across the workpiece. A 17,800 N blank holder force was also considered, which was assumed to remain constant during the process. Table 2 reports the coefficients of friction between all surfaces. Further, the under-relaxation factor was assumed at 0.6 mm. Figure 12a displays the three different initial guesses: circular, rectangular and zigzag shape. As mentioned earlier, the number of iterations to reach the required accuracy increases to five iterations in this problem, due to the more complex behavior resulting from the two-level punch. The optimization problem was solved using different initial guesses, and the optimum initial blank was obtained in each case after five iterations (see Fig. 12b). The final contours are also plotted in Fig. 12c and compared with the target contour. The final contours are aligned with the target contour for all initial guesses. In other words, using different and far-initial guesses have little impact on the final optimum blank configuration.
Although the initial guesses are considerably different from the optimum in this example, the proposed algorithm manages to achieve the optimum in less than five iterations.

Example 3: Symmetric Flange
It should be noted that the target contours are flat and are completely located in a plane in the previous two examples. However, the targets are 3D space contours in the present and the next examples. In the current example, the punch and the die are circular, and the edges are rounded. The geometric details of the tools are given in Fig. 13.
The material, coefficients of friction and blank holder force are similar to example 1. The geometric details of the target shape are also provided in Fig. 14. Figure 14a shows the front and top view of the target final product (mm),  a the initial guess for the blank; b the material flow paths of the boundary points; c the modified initial blank and the deformed plate; d material flow path of the boundary points; e to h the second and third iterations Fig. 16 The optimum initial blank for example 3 Fig. 17 The maximum shape error for different iterations for example 3 123 while Fig. 14b demonstrates the 3D view of the target product. As explained before, the target shape is defined using the target ribbon, which can be generated simply in CAD software. The target ribbon is then used in the optimization procedure to compute the shape error and update the initial contour. Figure 14c provides the target contour for the present example.
The problem is solved using the proposed optimization procedure, and the first three steps are given in Fig. 15. In Fig. 15a, the initial guess for the blank is shown, which is selected as a rectangular shape with round corners. The deformed plate is also shown in this figure. It is visible that the final product is far from the target shape for the initial guess. The material flow paths of the boundary points are shown in Fig. 15b. As observed, the material flow paths cross the target ribbon in some points and there are other points where the flow path fails to reach the target. The shape errors are calculated for all boundary points, and the initial blank is modified. Figure 15c shows the modified initial blank, in which the first iteration starts. The deformed plate and the material flow path of the boundary points are illustrated in Fig. 15c and d, respectively. As shown, the deformed plate is now closer to the target ribbon, although it fails to match the target completely. The second and third iterations are demonstrated in Fig. 15e-h. The final contours converge to the target shape.
Finally, the optimum initial blank is obtained after 20 iterations, as shown in Fig. 16. Figure 17 displays the maximum shape error for different iterations. The maximum shape error for the initial guess is more than 13 mm, which decreases rapidly as the optimization steps proceeds.
The maximum shape errors reach less than 1 mm at the 12th iteration and are about 0.4 mm for 20 iterations.

Example 4: Unsymmetrical Flange
This example is another problem with a 3D target contour. The shape of the target product in the present example is different relative to the previous example, and the flanges are not symmetric. The geometry of the punch and die, as well as other parameters, is similar to example 3. The geometric details of the target shape are illustrated in Fig. 18. Specifically, Fig. 18a shows the front and top view of the target final product, while Fig. 18b presents the 3D view of the target product. The target ribbon for the present example is also shown in Fig. 18c.
The problem is solved using the proposed optimization procedure, the first three steps of which are given in Fig. 19. Figure 19a depicts the initial guess for the blank. The initial blank is selected as a rectangular shape with round corners. The deformed plate is also shown in this figure. It is observed that the final product is far from the target shape for the initial guess. The material flow paths of the boundary points are plotted in Fig. 0.19b. The material flow paths cross the target ribbon in some points, while it fails to reach the target in some other points. The shape errors are calculated for all boundary points, and the initial blank is modified, which is shown in Fig. 19c and the first iteration is started. The deformed plate and the material flow path of the boundary points are given in Fig. 19c and d, respectively. As observed, the deformed plate is now closer to the target ribbon, although it fails to match the target completely. The second and third iterations are given Fig. 18 The geometric details of the target shape for example 4; a the front and top view of the target final product (mm); b the 3D view of the target product; c the target ribbon in Fig. 19e-h, in which the final contours converge to the target shape. Finally, the optimum initial blank is obtained after 20 iterations, as shown in Fig. 20. Figure 21 provides the maximum shape error for different iterations. As concluded, the maximum shape error for the initial guess is more than 26 mm, but it decreases rapidly as the optimization steps proceeds. The maximum shape errors reach less than 1 mm at the 12th iteration and are about 0.4 mm for 20 iterations.

Example 5: Blank with Internal Boundary
The last example addresses the sheet metal forming of a perforated plate. The blank has an internal boundary and the optimization problem should obtain both internal and external boundaries in such a way that the final product matches the internal and external targets. The geometry of the punch and the die and other parameters is similar to example 3. The geometric details of the target shape and internal and external targets are given in Fig. 22. Specifically, Fig. 22a shows the front and top view of the target final product, while Fig. 22b illustrates the 3D view of the target product. Ultimately, Fig. 22c depicts the target ribbons for the internal and external targets.
The problem is solved using the proposed optimization procedure. Figure 23 provides the three iterations 2, 4 and 6. Figure 23a shows the initial guess for the blank, which is selected as an annulus with circular internal and external It is observed that the final product is far from the target shape for the initial guess. The material flow paths of the boundary points are illustrated in Fig. 23b. The shape errors are calculated for all boundary points on both internal and external boundaries and the initial blank is modified, which is shown in Fig. 23c and is started in the first iteration. The deformed plate and the material flow path of the boundary points are depicted in Fig. 23c and d, respectively. Accordingly, the deformed plate is now closer to the target ribbon although it fails to match the target completely. The fourth and sixth iterations are provided in Fig. 23e-h, in which the final contours converge to the target shape.
Finally, the optimum initial blank is obtained after 20 iterations, as shown in Fig. 24. The maximum shape error for different iterations is given in Fig. 25. This figure shows that the maximum shape error for the initial guess is 13 mm for internal boundary and more than 7 mm for external boundary although it decreases rapidly as the optimization steps proceeds. Based on the results, the maximum shape errors reach less than 1 mm at the 12th iteration and are about 0.4 mm for 20 iterations.

Conclusion
The present study proposes a new blank optimization algorithm for deep drawing process based on 3D space target contours and also domains with internal and external boundaries. The efficiency of the proposed method has been evaluated by solving five examples. Briefly, the results show that: • The optimal blank design algorithm based on 3D space target contours in the deep drawing process, which has been first introduced in this paper, can appropriately predict the optimal blank shape.
• It is shown that the proposed algorithm can well be used to design the internal boundaries of parts in addition to the external boundaries. • The results show that the proposed algorithm is sufficiently robust against the initial guesses for the blank, which is an advantage of the present algorithm compared to other algorithms. • Finally, the present study reveals that the proposed algorithm can be effectively used to solve blank external and internal boundaries optimization for the deep drawing process.