The conversion of the surface loads was simulated by Nilsson and Birath [38] during the stamping process by means of time integration, where the process of lifting and stamping are taken into account. Structural optimization was then ap- plied to reduce the weight by keeping the structure strength and rigidity. In a similar manner, Xu and Tang [39] developed the inner structure of the stamping die with the structural op- timization method based on the LS-Dyna platform and Hyperworks software. Using the finite element method, Sheu and Yang [1] tried to predict the pressure on the die face of stamping dies. Then, with the size and shape optimization methods, the inner structure was designed. The optimal results are completely different from a uniform distribution of the ribs which can be seen in usual designs. Zhu et al. [2] used Abaqus software to simulate the process of skin stretch forming nu- merically. Then, structural optimization was performed to maximize the structural stiffness with the boundary conditions and the material properties properly defined. Finally, the com- parison of usual design and the numerical results shows that the structural design can meaningfully improve the strength and stiffness of the stretch forming die. A SIMP-based topol- ogy optimization methodology for stamping die was proposed by Xu and Chen [40]. Topology optimization results in this study showed that 28.1 % mass reduction was achieved with a slight difference of the die structure performance and blank forming quality. Stamping tool design was conducted using FE simulation and topology optimization techniques to in- crease its rigidity by Hamasaki [41]. In the first step of the procedure, stamping simulation was carried out with rigid tools, and contact pressure (nodal forces) was extracted. Topology optimization with the obtained nodal force boundary condition successfully determined the stiffest structure under the given  volume  fraction  constraint  in the next step. Based on thus  optimized  die  structure, new CAD model was redesigned. Azamirad and Arezoo developed a software package which can design an appropriate topology of  body  structure  of  stamping die components with a reduced weight. This is done by implementing the ESO algorithm, and the  results  show that the optimal die structure is completely  different from a uniform distribution of  the  ribs  which  can  be seen  in  standard  die  design [42].Despite these researches in literature, there is still not an effective and efficient method which can automatically opti- mize the structure of stamping dies according to boundary conditions and prescribed loads. In this paper, a software package based on structural optimization is presented. This software implements BESO algorithm to reduce the volume of the main components of the stamping dies, including die, punch, and blank holder, while maintaining the forces applied in sheet metal forming operations. So the main contribution of the present work to this field is the automation of structural design of stamping die components where the most pop- ular topology optimization (BESO) method  is  used  for the first time. This can be a novelty in theory, and the reconstruction  of  die  components  with  respect  to manufacturing constraints and accommodating for a    sim-BESO is an improvement of the ESO method which allows for efficient material to be added to the structure at the same time as the inefficient one is removed [12–14].

2.1 Bi-directional evolutionary structural optimization method

The BESO method allows the material of part to be removed and added simultaneously. The initial research on BESO for stiffness optimization was conducted by Yang et al. [12]. In their study, the sensitivity numbers of the void elements are estimated after the finite element analysis by a linear extrapo- lation of the displacement field. Afterward, the solid elements with the lowest sensitivity numbers are removed from the structure, and simultaneously, the void elements with the highest sensitivity numbers are changed into solid elements. The numbers of added and removed elements in each iteration are determined by two independent parameters, namely: the inclusion ratio (IR) and the rejection ratio (RR) respectively.