In this paper, the modified BESO algorithm by Huang and Xie [43] is used for the structural optimization of stamping dies. In this method, many problems related to structural op- timization of continuum structures such as a proper statement of the optimization problem, mesh-dependency, checkerboard pattern, and convergence of solution are resolved.

2.2 Sensitivity number

The purpose of structural optimization is searching for the stiffest structure with a given volume of material. In the BESO method, a structure is optimized by removing and adding elements. The optimization problem with the con- straint of volume is stated as Eq. 1 [43]: pler casting method could also be classed as novelty in manufacturing  techniques.

2 Method

Structural optimization has attracted notable attention in the last three decades, and several methods have been developed based on the finite element analysis [3–18, 28]. The ESO and the SIMP are two commonly used methods [28]. In the SIMP method, a density of material is defined for each element which varies between 0 and 1. The elastic property for each element is stated in terms of its density [18]. The ESO method is based on the simple idea that by progressively removing inefficient material from a part, the topology of the remaining design will evolve towards an optimum structure [10,   11].

Where f and u are the applied load and displacement vec- tors and C is known as the mean compliance. Vi is the element volume, and V* is the prescribed total structural volume. N is the total number of elements in the model of part. The binary design variable (x) expresses the absence (0) or presence (1) of an element.

When a solid element is removed from a structure, the change of total strain energy or the mean compliance is equal to the elemental strain energy [44]. This change is defined as the elemental sensitivity number. When the mesh of part is not homogeneous, the sensitivity number should consider the ef- fect of element volume. In such a case, the sensitivity number can be replaced with the strain energy density of element as Eq. 2 [43]: calculation of the improved sensitivity number of the ith ele- ment as Eqs. 5 and 6 [43].

Where Ki is the elemental stiffness matrix, and ui is the nodal displacement vector of the ith element.

2.3 Filter scheme and improved sensitivity number

A filter scheme is used to obtain the sensitivity number for the void elements to add material into the design space and to smooth the sensitivity number in the whole design space. More importantly, by using the filter scheme, the problems of mesh dependency and checkerboard pattern will be re- solved at once. Nodal sensitivity numbers are defined before applying the filter scheme by averaging the elemental sensi- tivity numbers as Eqs. 3 and 4 [43]:

where K is the total number of nodes in the sub-domain Ωi. The sensitivity numbers of void elements are automatically obtained. They may have high values due to high-sensitivity numbers of solid elements within the sub-domain Ωi. Conse- quently, in the next iteration, some of the void elements may be changed to solid ones.

摘要：目前，冲压模具的铸造结构是根据模具设计标准设计的。这些标准通常不基于结构优化算法，并且经常依赖于导致模具部件的重量超过所需的高安全系数。这又要求更高的模具价格和每个零件所需的生产能量。因此，需要替代方法来减少这些部件的重量。在本文中，提出了一种软件包，其可以设计冲压模具的改进的结构，具有显着减小的重量。这个软件包实现了Abaqus软件，并使用双向演化结构优化（BESO）方法来创建一个新的更轻的结构，类似于钣金零件的形状和操作中施加的力。它通过从模具部件结构去除和添加材料获得期望的最佳设计。这种方法包括将材料添加到部件的结构过应力的部分，同时移除材料，其中结构被挤压。该过程一次又一次地执行，直到目标函数最小化。最后，所提出的结构也可以由设计者重建以适应更简单的铸造方法。通过研究用于钣金零件的模具的示例来说明软件的操作。模具部件最初被设计，分析并与标准模具（当前通常使用的模具）进行比较。最终结果表明，当模具的最大位移和应力不减小时，体积减少了31％。该软件包是在Microsoft Visual C＃编程环境中开发的，具有指向Abaqus软件的链接，用于分析有限元模拟过程。