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ORIGINAL ARTICLE

Preform optimal design of H-shaped forging based on bi-directional evolutionary structural optimization

Hang Yang1 & Xinwu Ma1 & Feng Jiao1 & Zheng Fang1

Received: 31 March 2018 /Accepted: 22 October 2018 /Published online: 29 October 2018 # Springer-Verlag London Ltd., part of Springer Nature 2018

Abstract Preform design is one of the most important aspects in forging process design. An automatic method of preform optimal design based on the bi-directional evolutionary structural optimization is presented in this paper. The complete filling of die cavity with less flash and uniform deformation of material are the goals of the preform optimal design. A new criterion of element addition/ deletion is proposed. The new criterion is associated with objective functions. The technique of element information tracking and transferring is introduced and the variables can be transmitted from the final mesh of the forging to the background grid that represents the preform. The method of boundary smooth fitting of the preform using the B-Spline curve is given. The preform optimal design for an H-shaped forging is carried out to demonstrate the effectiveness of the proposed method. A near net-shape forming is achieved meanwhile the uniform deformation is obtained.

Keywords Preformoptimaldesign .H-shapedforging .Elementaddition/deletioncriterion .Bi-directionalevolutionarystructural optimization

1 Introduction

Forgings with complex shapes are difficult to be forged in one single stage [1]. It generally needs multi-stage deformation to guarantee that the shape of billet gradually approaches the ideal shape of the forging. Therefore, the preform design is one of the most important aspects in forming process design. With the development of modern technologies of a variety of materials’ forming, the requirements for the forming efficien- cy and qualities of forgings are significantly improved [2]. The methods of preform design that have long been relying on experience to repair and test dies can no longer meet the needs of production now. Achieving automatization and intel- lectualization is a main trend of material forming. With the development of computer-aided technologies and the maturity of material forming theories, preform optimal design based on the finite element analysis (FEA) is widely studied in recent years [3]. Badrinarayanan and An [4, 5] proposed a method for the preform optimal design based on the sensitivity

analysis. Chung [6] utilized coordinate variable techniques to improve the sensitivity optimization–based preform design. However, the sensitivity equations are difficult to be established, and the algorithm is quite complicated. Roy and Knust [7, 8] proposed a method of the preform optimal design based on the micro genetic algorithm to avoid cumbersome derivation of sensitivity information, which could simplify the problems of optimization. Guan [9] further improved the the- ories, but still could not improve optimization efficiency. Ozcelik, Lu and Guan et al. [10–12] proposed a method of the preform optimal design based on the response surface method with high optimization efficiency, but the limitations for the range of the application and the large amount of tests were still its shortcomings.

In order to improve the efficiency of the preform optimal design, Naceur [13] applied the evolutionary structural opti- mization (ESO) to the preform optimal design of sheet metal forming. In his research, the region with smaller plastic defor- mation of sheet metal is gradually removed. The utilization ratio of the material can be improved by the preform optimal design. Shao [14–17] proposed a method of the preform opti- mal design based on bi-directional evolutionary structural op- timization (BESO). He did a series of studies on aero engine blades. However, there is no direct correlationship between the objective functions and element addition/deletion

* Xinwu Ma [email protected]

1 School of materials science and engineering, Shandong University, Jinan 250061, China

The International Journal of Advanced Manufacturing Technology (2019) 101:1–8 https://doi.org/10.1007/s00170-018-2906-9

criterion. It might be inefficient in the continuous optimization process.

In this paper, a new element addition/deletion criterion as- sociated with the objective functions is proposed in order to improve the rationality and efficiency of the preform optimal design based on the BESO.

2 Preform optimal design based on BESO

The BESO was firstly proposed by Xie [18], and it is mainly used in the field of engineering structural optimal design [19, 20]. The basic idea of the BESO is that materials are gradually added to or deleted from the structures based on some criteri- ons to make the structures conform to certain engineering requirements. After years of development, a series of mature algorithms and software have been developed for the BESO [21]. The BESO has been successfully applied in the actual engineering structure design in recent years [22, 23].

The algorithms for the BESO are simple and efficient, so the BESO has a good application prospect in the field of the preform optimal design of metal forming. A background grid is defined based on the idea of the BESO, which is used for adding or deleting elements. The elements in the background grid have two types of states: active state and inactive state. All elements in active state constitute the shape of the cross section of the part, as shown in Fig. 1.

The process of the preform optimal design based on the BESO is shown in Fig. 2. Firstly, the FEA for the initial billet is carried out. If the analysis result meets the goals of the optimization, then the optimization process is finished; and if not, the tracking and transferring of element information are carried out by using the interpolation method. Then back- ground elements are added to or deleted from the background grid according to the element addition/deletion criterion. After the addition/deletion of elements, a preform with a new shape is obtained and the FEA is performed again for the new

preform. The above steps are repeated until the goals of the optimization are met.

3 Key technologies and steps of BESO-based preform optimal design

3.1 Objective functions

The preform optimal design for complex forgings usually has two primary goals: (1) complete filling of die cavity with less flash and (2) uniform deformation of the material. The goal (1) means that the shape of forgings must be ensured; meanwhile, the forgings have less flash. Therefore, the utilization ratio of the material can be improved. The goal (2) means that the forgings can have high quality if the material has uniform deformation because the uniform deformation usually results in uniform microstructure. Therefore, the objective functions for the preform optimal design can be defined as follows:

f 1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ n

i¼1 εe;i−ε

a� �2 n−1

vuuut ð1Þ

f 2 ¼ 1− Ncontact Nsurface

ð2Þ

f ¼ ρ1⋅f 1 þ ρ2⋅f 2 ð3Þ where εe;i is the equivalent strain of element i, ε

a is the arith- metic average of equivalent strain for all elements, n is the number of elements, Ncontact is the number of elements that contact the die, Nsurface is the total number of surface elements of the forging, f1 and f2 are two sub-objective functions, ρ1 and ρ2 are weight factors, and f is the total objective function. The values of ρ1 and ρ2 are given by the significance of the sub- objective functions. In this paper, the value 0.5 is set for two weights.

Fig. 1 Cross section of the part represented by the elements of background grid

Fig. 2 The process of the preform optimal design

2 Int J Adv Manuf Technol (2019) 101:1–8