Int J Adv Manuf Technol DOI 10.1007/s00170-016-9209-9

ORIGINAL ARTICLE

Preform optimization for hot forging processes using genetic algorithms Johannes Knust 1 & Florian Podszus 1 & Malte Stonis 1 & Bernd-Arno Behrens 1 & Ludger Overmeyer 1 & Georg Ullmann 1

Received: 10 May 2016 / Accepted: 20 July 2016 # Springer-Verlag London 2016

Abstract In multi-stage hot forging processes, the preform shape is the parameter mainly influencing the final forging result. Nevertheless, the design of multi-stage hot forging processes is still a trial and error process and therefore time-consuming. The quality of developed forging sequences strongly depends on the engineer’s experience. To overcome these obstacles, this paper presents an algorithm for solving the multiobjective optimization problem when designing preforms. Cross-wedge-rolled (CWR) preforms were chosen as subject of investigation. An evolutionary algorithm is introduced to optimize the preform shape taking into account the mass distribution of the final part, the preform volume, and the shape complexity. The developed algorithm is tested using a connecting rod as a demonstration part. Based on finite element analysis, the implemented fitness function is evaluated, and thus the progressive optimization can be traced. Keywords Preforming optimization . Hot forging . Evolutionary algorithms . Cross-wedge rolling

1 Introduction Hot die forging is a manufacturing process enabling the production of high quantities with a favorable strength to mass ratio in comparison to other manufacturing processes. Complex shapes are usually produced in a multi-stage process chain including a preforming operation. Possible preforming * Johannes Knust [email protected]

1

Institut für Integrierte Produktion Hannover, Hollerithallee 6, 30419 Hannover, Germany

processes are open and closed die forging and rolling operations. The design of the preform shapes is of great importance for the forging sequences’ quality parameters like flash ratio, forming load, and energy consumption and hence for the economic efficiency of the forging sequence [1]. Currently, engineers create the mass distribution diagram of the forging part by virtually cutting it into a specific number of N planes (Fig. 1). Afterwards, different preforms are derived and evaluated via finite element analysis (FEA). However, the evaluation of the developed preforms is a forward-oriented process using FEA. The reverse flow of preform development and evaluation using FEA leads to high costs due to time and computing effort. In addition, process and result quality are strongly dependent on the engineer’s experience. An algorithm for the automated generation of preforms will likely increase the economic efficiency of multistage forging design due to the reduced development time [2, 3]. Various stochastic and analytical methods have been investigated for the preform optimization in hot forging processes. HATZENBICHLER and BUCHMAYER describe a preform optimization method using the backtracking tool of FEA software [4]. MISRAEDI presents an approach using linear Lagrange interpolations for describing the preform shape. The aim of the research is to improve form-filling and to avoid folds during forging [5]. STONIS investigated the formation of internal defects in preforming processes by use of an analytical algorithm [6]. SHAO introduces a new element removal and addition to obtain a preform shape with an improved material distribution [7]. The presented research works reveal the significance of the preform optimization problem. One possible way to meet the challenges in the design of preforms is the use of genetic algorithms (GAs). GAs represent a subgroup of evolutionary algorithms. The main idea of these algorithms is to imitate the process of

Int J Adv Manuf Technol Fig. 1 Mass distribution diagram with possible preforms

biological evolution with the main mechanisms selection, crossover, and mutation. GAs are mainly used for multicriteria optimization problems with a high number of possible solutions. Using conventional optimization techniques, this would lead to high computation times [8, 9]. Therefore, genetic algorithms are suitable for solving the multi-criteria optimization problem in preform design. SEDIGHI investigated the optimization of preforms in closed die forging by use of neural networks and genetic algorithms [10]. The results show the feasibility of genetic algorithms for solving the preform optimization problem. In contrast to the work presented in this paper, an initial preform shape has to be derived by the user. Furthermore, the flash has not been considered in this research work. The method was further developed by TORABI. With these modifications, even the consideration of flash is possible [11]. Nevertheless, the presented approach still requires profound modeling knowledge, as an initial preform has to be derived. CIANCIO introduces a genetic algorithm to map the input-output relationship for various forming processes [12]. This paper aims to describe the development of a genetic algorithm using cross-wedge-rolled (CWR) preforms. To assess the suitability of the derived preforms, analytical equations are derived. The improvement of the derived preforms is achieved by a genetic algorithm.

2 Design of genetic algorithms for preform optimization of CWR parts Cross-wedge rolling is a preforming process for reshaping circular cylindrical billets into rotationally symmetrical workpieces with variable diameter in axial direction using two oppositely moving wedge-shaped tools. During the cross rolling process, the rolled material is passed through two or more rolls to reduce the overall thickness of the parts, during crosswedge rolling, the wedges of the tools lead to an unequal mass distribution along the main axis [3]. Cross-wedge rolling is a favorable preforming operation, especially due to the high material utilization [3].

The main parameter describing a cross-wedge-rolled part is the cross-section area reduction ΔA (Fig. 2, Eq. 1).


d △A ¼ 1− d0

2 ð1Þ

Although cross-wedge rolling offers lots of advantages, it has not been widely accepted throughout the forging community for a long time. One of the main reasons was the complexity of cross-wedge-rolling tools and process design. However, latest research in the field of cross-wedge rolling leads to a simplified development of cross-wedge-rolling processes, as suitable simulation parameters have been identified. Furthermore, the effort for experimental tests could be reduced by developing a method for downscaling experimental trials [13]. An automated approach for designing a CWR preform is not known. The design of CWR preform geometry can be described as a multi-criteria optimization problem: formfilling and avoidance of folds must be ensured after final forging, whereas the preform volume and the complexity of the part have to be as small as possible. In addition, a virtually indefinite number of solutions are possible, as the geometry of the preform is almost arbitrary. Thus, GAs are possibly suitable to solve this optimization problem. The main process flow for solving the described multi-criteria optimization problem by the use of GAs is shown in Fig. 3. The algorithm is initiated by creating

Fig. 2 Parameter of cross-wedge-rolled part

Int J Adv Manuf Technol Fig. 3 Process flow of evolutionary algorithm for optimizing cross-wedge-rolled preforms

an initial population with random properties. As explained later, each individual is described by a set of genes that represent the preform radiuses along the longitudinal axis. Afterwards, a fitness function is calculated that uses the parameters form-filling, preform volume, and complexity. During the optimization, the genes of the selected parents are changed by the use of crossover and mutation methods to create a new population. Afterwards, the new population is evaluated again until a termination criterion (fitness value) is met. In order to implement this GA, a cross-wedge-rolled part can be described by using N boundary points and interpolation techniques between these points. This simple modeling is possible, because cross-wedge-rolled parts are rotationally symmetric. Each boundary point of the preform contour thus represents a gene of an individual. A gene Gn contains the information radius r and the position n along the longitudinal axis of the part (Fig. 4). In Eqs. 2 and 3, the modeling of the cross-wedge-rolled parts is summarized mathematically. I ¼ ðG1 ; :::Gn Þ

ð2Þ

G n ¼ ðr n Þ

ð3Þ

The described algorithm is written in MATLAB code. Necessary input data are the number of boundary points N for the final part and the corresponding cross-section areas An along the longitudinal axis. Furthermore, the following process parameters for the GA need to be defined:

I s p m c rmin and rmax

population size survival rate selective pressure mutation rate termination criterion x, y in order to define radius.

The population size I describes how many individuals are part of each population. The survival rate s controls how many individuals are taken from the parent population to the child population without changes. The selective pressure p influences the choice of parents. In the present model, the selection probability of each individual is proportionally linked to the previously calculated fitness value and can be adjusted by the selection pressure parameter p. Mutation rate m is the factor describing the probability of mutating a gene. The mutation method prevents the algorithm from remaining in local optima. In the present model, the Gaussian mutation method is implemented, as it allows a defined deviation of the gene. Selected genes are altered by adding a random value to or subtracting it from the current value rn. For the recombination, the one-point crossover method has been used. A complementary solution is not possible, so the new individual is built up out of the parent genes. The position of the sections is randomly chosen (Fig. 5). Possible termination criteria c are for example a maximum number of generations, a defined fitness value or a change of fitness value that is below a limit value. In order to limit gene characteristics to reasonable values, x and y need to be defined to

Int J Adv Manuf Technol Fig. 4 Phenotypic representation of a cross-wedge-rolled preform

calculate minimum and maximum values for the radii by equations (Eqs. 4 and 5). The evaluation of the individuals is described in the following chapter. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi minðAn Þ ð4Þ rmin ¼ x* π

rmax

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi maxðAn Þ ¼ y* π

ð5Þ

3 Design of fitness function To enable the optimization, each individual i has to be evaluated using a fitness value. The fitness value depends on the suitability of the rated individual to be used as a preform for the chosen final part. The suitability of a preform depends on the three parameters form-filling, preform-volume, and complexity (see Fig. 3). Fig. 5 Selection, crossover, and mutation techniques

3.1 Form-filling Form-filling is the main quality parameter for forging parts. Depending on the complexity of the part, a certain amount of flash is necessary for allowing form-filling. In the present model, the form-filling fitness Ffillingi, n for each cutting plane n of an individual i is calculated by using the following equation: Ai;n Ai;n for a ≤ ≤1 þ a AFp;n AFp;n Ai;n 1þa F filling;n ¼ 1for AFp;n

F fillingi;n ¼ −a þ

ð6Þ

Ai, n and AFp, n represent the cross-section area of the individual and the final part at cutting plane n. The factor a is describing the material surplus necessary for form-filling and is defined in advance for the whole part. The form-filling fitness Ffillingi, n is limited between 0 and 1 as shown in Fig. 6 because neither a negative form-filling is possible nor a form-filling >1.

Int J Adv Manuf Technol

Fvolumei. The minimum Fvolumei and therefore the best preform volume are reached if Vi is equal VFp. F volume;i ¼

V i −V Fp Vi for ≤1 V min −V Fp V Fp

ð11Þ

F volume;i ¼

V Fp −V i Vi for ≥1 V Fp −V max V Fp

ð12Þ

Figure 7 shows the course of the volume fitness value Fvolumei in dependency of the calculated individual preform volume Vi.

Fig. 6 Normalization of form-filling value Ffillingi,n

To describe the form-filling fitness of the whole individual Ffillingi, Eq. (7) can be used. F fillingi

N 1X ¼ F fillingi;n N n¼1

3.3 Complexity ð7Þ

3.2 Preform volume The overall preform volume is a parameter mainly influencing the economy of a forging process. Increasing preform volume leads to higher material costs and also to increasing processing costs as the total forming force increases due to a higher flash amount. Thus, the consideration of the preform volume is necessary. The total volume of each individual Vi is being calculated by summing up the volume between the boundary points which can be described as a truncated cone. Therefore, the total volume is calculated by the following equation: Vi ¼

X N −1 n−1

 L*π  2 rn þ rnþ1 þ rnþ1 2 ðN −1Þ*3

ð8Þ

In order to calculate the volume fitness of an individual Fvolumei, the individual volume Vi is set in relation to the volume of the final part VFp, the minimum possible preform volume Vmin, and the maximum possible preform volume Vmax. V min ¼ π*rmin 2 *L

ð9Þ

V max ¼ π*rmax *L

ð10Þ

2

The volume fitness value Fvolumei is a discontinuous function. In preforming design, the aim is to reduce the preform volume as much as possible. Therefore, the following relationships between Vi and VFp are derived. In case Vi is smaller VFp, this would lead to scrap parts as form-filling cannot be achieved. Therefore, if Vi is smaller VFp, a decreasing Vi leads to a higher Fvolumei. In case Vi is bigger VFp, this leads to higher manufacturing costs due to increasing material and processing costs. Therefore, in case Vi is bigger VFp, an increasing Vi leads to a higher

The parameter complexity for each individual Ci is used for describing the suitability of a preform regarding the manufacturing costs and process constraints. In cross-wedge rolling, the manufacturing costs increase for every wedge as more material needs to be processed. Furthermore, sharp edges need to be avoided as this may lead to folds in the final part. Both aspects can be considered by calculating the length of the spline connecting all boundary points. Thus, the complexity Ci of an individual is described by using the Euclidean distance of the boundary points (Eq. 13). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 N −1 X  2 L 2 Ci ¼ ri;n −ri;nþ1 þ ð13Þ N −1 n¼1 In order to calculate the complexity fitness value of an individual Fcomplexityi, the individual complexity value Ci is normalized between 0 and 1. The minimum possible complexity Cmin has a complexity fitness value Fcomplexityi of 0 whereas the maximum possible complexity value Cmax has a relative complexity fitness value F complexityi of 1 because the complexity should be as small as possible. C min ¼ L sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi L C max ¼ ðrmax −rmin Þ2 þ N −1

ð14Þ ð15Þ

Between the minimum and maximum complexity value, a linear course has been used for calculating the complexity fitness value Fcomplexityi. F complexityi ¼

C i −C min C max −C min

ð16Þ

Figure 8 shows the course of the complexity fitness value F complexityi in dependency of the calculated individual complexity Ci.

Int J Adv Manuf Technol Fig. 7 Volume fitness value Fvolumei for the parameter preform volume

& &

3.4 Fitness evaluation The total fitness value for each individual Fi is being calculated by using the fitness values Ffillingi, Fvolumei, and Fcomplexityi. Within the optimization, the parameter form-filling needs to be maximized whereas the parameter preform volume and complexity will be minimized (Eq. 17). The genetic algorithm aims to maximize this function. F i ¼ wi *F fillingi −w2 * F volumei −w3 * F complexityi

ð17Þ

4 Results In the following investigations, a connecting rod with a total length of 217 mm is being used as demonstration part. The part has been divided into N = 23 cutting planes. Unless otherwise mentioned, the following parameters for the algorithm have been used during the investigation: & & & & & & & &

Population size I: 500 Mutation rate m: 1 % Survival rate s: 0 % Selective pressure p: 1.5 Termination criterion c: 250 generations w1 = 2 w2 = 1 w3 = 1

Fig. 8 Complexity fitness value Fcomplexityi for the parameter complexity

x = 0.5 y=2

FEA simulations were performed by use of the software Forge NxT. The temperature of the parts was set to 1200 °C, and the material data file of C45k (DIN 1.0503), provided in the database of Forge NxT, was used. The model of friction used in the FEA simulations for hot forging is a combination of the coulomb friction (τR = μ ⋅ σN) and the friction law (τR = m ⋅ k, where k is the shear yield strength). In the combined model, the shear stress first increases linearly with the coefficient of friction μ and is limited to the top by the maximum value of the shear yield strength from the friction law [14, 15]. The coefficient of friction μ and the factor of friction m are dimensionless. Friction coefficients were set to μ = 0.15 and m = 0.3, which represents a water graphite lubrication in Forge NxT. The mesh size of the parts was about 2.6 mm (mesh option “fine” was used). As forming speed, a velocity of 600 mm/s was assumed. Further, the data listed in Table 1 for the experimental trials were used. For all, other parameters not specifically indicated standard parameters (initial parameters which are set by Forge NxT) for a hot forging process were assumed. The aim of the investigations is to identify the influence of the parameters of the fitness function as defined in Chapter 5 on the quality of the derived preform. This enables the determination of the best combination of fitness function parameters. To quantify the quality of the derived preforms, the following indicators have been used because these are the parameters mainly influencing the rating of a forging process [16]: the form-filling, folds, preform volume, forming force, and number of wedges. The following hypothesis is proposed for the investigations: H0: Each of the fitness function parameters form-filling, preform volume, and complexity has at least one significant influence on a forging quality indicator. The significance factor α has been chosen as 0.05 to decide whether an effect is significant or not. In order to check the derived hypothesis, the experimental design shown in Table 1 has been used. The optimization has

Int J Adv Manuf Technol Experimental design for parameter investigation

Table 1

Consideration fitness function parameter

Measured variable

Run 1 2

Form-filling + −

Preform volume − +

complexity − −

Form-filling 1.00 0.00

Folds 1.00 1.00

Preform volume [mm3] 954,170.00 214,970.00

Forming force [to]

No. of wedges

n.a n.a

11.00 10.00

3 4

− +

− +

+ −

1.00 1.00

0.00 0.00

601,636.00 267,260.00

909.46 390.19

1.00 6.00

5

+

−

+

1.00

0.00

751,736.00

953.81

1.00

6 7

− +

+ +

+ +

0.00 1.00

0.00 0.00

212,466.00 281,206.00

n.a 389.71

3.00 4.00

8 9

+ −

− +

− −

1.00 0.00

1.00 1.00

797,624.00 214,960.00

n.a n.a

10.00 8.00

10

−

−

+

1.00

0.00

727,016.00

932.57

2.00

11 12

+ +

+ −

− +

1.00 1.00

0.00 0.00

266,330.00 803,192.00

379.02 961.38

5.00 1.00

13 14 15 16

− + + −

+ + − +

+ + − −

0.00 1.00 1.00 0.00

0.00 0.00 1.00 1.00

212,176.00 279,150.00 979,212.00 213,240.00

n.a 397.85 n.a. n.a

1.00 4.00 11.00 10.00

17 18

− +

− +

+ −

1.00 1.00

0.00 0.00

664,866.00 269,258.00

909.46 384.19

2.00 5.00

19 20 21

+ − +

− + +

+ + +

1.00 0.00 1.00

0.00 0.00 0.00

790,848.00 215,182.00 279,380.00

975.50 205.74 389.85

1.00 2.00 4.00

n.a not applicable

been performed for every possible parameter combination to determine the influence of each parameter. In Table 1, a “+” means that the parameter was considered in this run whereas a “−” means the parameter has not been taken into account (see Table 1, consideration of fitness function parameter). As a genetic algorithm is a stochastic optimization method, all parameter combinations have been repeated three times. The presented experimental design enables a quantification of the effect of each parameter on the optimization result. To check whether the parameter is significant, a t test has been performed with the α value 0.05. In case form-filling could not be achieved or folds occurred during forging, the forming force was not measured as these parts are scrap parts. The measured variable form-filling is mainly influenced by the fitness function parameter form-filling and preform volume. The effect of the fitness function parameter complexity is not significant for this measured variable. Table 2 shows the effects for the significant parameters form-filling and preform volume. As Table 2 shows, taking into account the fitness function parameter form-filling leads to a form-filling of 1. Therefore, it is not possible to neglect the fitness function parameter form-filling as this may lead to incomplete formfilling in final forging which can be observed in run 6 for example (Fig. 9). Taking into account, the fitness function

parameter volume may lead to incomplete form-filling, because the algorithm aims to reduce the preform volume as the effects for the measured variable preform volume reveal (Table 2 and run 2 in Fig. 9). Therefore, it is necessary to take into account both parameters during the optimization to reach form-filling with a reduced billet volume. The fitness function parameter preform volume has also a significant effect on the measured variable forming force. Due to the reduced billet volume, the forming force is decreasing as well by taking into account the parameter preform volume. The fitness function parameter complexity has a significant effect on two measured variables. Very important is the significant effect on the measured variable folds. As shown in Table 2, taking into account the fitness function parameter complexity leads to 0 fold in the final part. This means it is necessary to take this parameter into account as a part with folds would be scrap. This effect can also be observed in Fig. 9. Especially, run 1 and run 2 show the necessity of taking into account this parameter because neglecting this parameter leads to parts with many sharp edges which will cause folds in the final part. The parameter complexity has a significant influence on the measured variable number of wedges as well. The number of wedges determines the effort for producing the cross-

Int J Adv Manuf Technol Table 2

Experimental design for parameter investigation

Measured variable

Form-filling

Folds

Preform volume

Forming force

No. of wedges

Fitness function parameter

Form-filling

Volume

Complexity

Volume

Volume

Complexity

Complexity

Mean +

1.00

0.50

0.00

243,798.17

284.20

617.40

2.17

Mean − Effect

0.33 0.67

1.00 −0.50

0.67 −0.67

785,588.89 −541,790.72

940.36 −656.16

273.96 343.44

8.44 −6.28

Significance

0.00

0.01

0.00

0.00

0.05

0.00

0.00

wedge-rolling tools. With an increasing number of wedges, the effort is increasing because the tool is getting longer. As Table 2 reveals, taking into account the fitness function factor complexity leads to a decreasing number of wedges for the cross-wedge-rolling tool. The investigations performed show that all of the fitness function parameters developed in Chapter 5 have a significant influence on at least one measured variable. The parameter form-filling is influencing the measured variable form-filling. The fitness function parameter preform volume is influencing the measured variable form-filling, preform volume, and forming force. The parameter complexity is influencing the measured variable folds, forming force, and number of wedges. Therefore, it is necessary to take all three parameter into account for a holistic optimization approach. Figure 9 shows the derived preform for the first seven runs as shown in Table 1. Especially, the first two runs show the necessity of taking into account the complexity parameter as with these preforms no forging would be possible. Run 3 and run 5 reveal the necessity of taking into account the preform volume parameter because otherwise the preform volume increases, and this leads to a high amount of flash. Run 6 is the preform derived by taking into account the variable preform volume and complexity which leads to incomplete filling due to neglecting the form-filling parameter. Run 4 is also a possible preform. Form-filling is achieved, and the flash amount is not too big. Nevertheless, the investigations show that it is necessary to take into account the complexity factor as this factor reduces the risk of fold and decreases the number of wedges. Especially for parts with a more complex mass distribution, this parameter becomes more important. In order to check the suitability of the developed algorithm, the calculation was interrupted periodically, and the corresponding current preform was checked with the help of FEA (Fig. 10) by simulating the forging process. Because of the previous investigations, all three fitness function parameters have been taken into account during this investigation. Thus, the development of the best preform can be traced. Furthermore, the development of the maximum, minimum, and average fitness value was analyzed. Figure 10 shows the development of the best rated individual within the population for 250 generations. The optimization has been performed on an Intel core I5 processor with

2.8 MHz and took 40 s. The subsequent evaluation of the FEA results of the respective forged connecting rod was done visually by experienced forging engineers. The best preform in the first generation has totally randomly distributed diameters along the longitudinal axis as the start population is initialized randomly. During the optimization, the randomly distributed diameters are substituted, and better-shaped cross-wedge-rolling preforms are being developed. Already after 50 generations, a general tendency can be anticipated with two mass allocations at the end and a narrow bridge in the middle of the cross-wedge-rolled part. Nevertheless, this part still has some sharp edges and would require many wedges to manufacture this part which results in high manufacturing costs. In the further course of the optimization, the sharp edges disappear. Already after 125 iterations, most of these edges have disappeared. But especially, at the narrow bridge in the middle of the part are still some irregularities left, which would make the cross-wedge-rolling process very difficult. These irregularities disappear after 250 iterations. The development of the best preform within each generation can be explained as follows. The irregularities of the part disappear during the optimization due to the complexity factor. Taking into account, this factor leads to a worse fitness evaluation for those parts and decreases their probability of inheriting their characteristics to the next population. Furthermore, it can be observed that the optimization is aiming to a flash amount of 30 %. This is because the formfilling factor has a higher weighting factor as the preform volume. The optimization algorithm aims to create preforms which ensure form-filling with the minimum possible preform volume. The development of the maximum fitness value within each population is shown in Fig. 11. For a selection pressure of 1.5 the maximum fitness value is reached after 245 iterations and does not increase anymore afterwards. Until generation 245, there is still a small increase in the fitness value. Furthermore, it can be seen that neglecting the selection pressure by setting this parameter to 1 does not lead to an optimization. This is because the search remains randomly for a selective pressure p = 1.0. Therefore, it can be stated that for the present algorithm, the selective pressure is

Int J Adv Manuf Technol Fig. 9 Preform design in dependency of consideration of fitness function parameters and forging result

run

preform design

forged conneectingrod

1

2

3

4

5

6

7

an important parameter as it influences the selection of suitable parts significantly. The higher the selective pressure p is chosen, the higher is the probability of choosing those parts with the best fitness value. A selective pressure of 1 is not suitable because all parts of the generation have the same selection probability regardless of their fitness value. Furthermore, it can be observed that for all investigated parameters, a

deterioration of the maximum fitness value remains possible. This is because the survival rate s was chosen 0. So in every generation, there are only new parts present. The crossover and mutation method aim to create better preforms, but nevertheless, it remains possible that a worse part is created, and the fitness value from the previous generation will not be reached. But, as Fig. 10 shows in general, the fitness value is increasing.

Int J Adv Manuf Technol Fig. 10 Development of preform design and forging result

No

preform design

forgedconneectingrod

1

50

125

250

The selective pressure determines how much the best individuals are preferred against the worse ones. The higher the selective pressure, the more the best individuals are preferred. A selective pressure of 1.5 is suitable as better individuals are preferred. However, it is still possible that parts with a worse rating are being selected for recombination. In this way, the probability of reaching the best preform increases. However, if the

Fig. 11 Development of maximum fitness value using two different selection pressures

selective pressure becomes too high, the risk of remaining in local optima increases especially for smaller populations. The development of the minimum fitness value within each population is analogous to the development of the maximum fitness value (Fig. 12). However, the fluctuation of the minimum fitness value within population is lower. The course of the minimum fitness value can be explained using the survival rate and the selective pressure. The survival rate was chosen 0 which means in every new generation the worse parts have been substituted. The new parts have a higher probability of being derived from good parts of the previous generation due to the selective pressure of 1.5 which leads to an increasing minimum fitness value. The development of the average fitness value is also analogous to the maximum and minimum fitness value (Fig. 13). Within this graph, no deterioration can be noticed. Thus the overall, the fitness value is increasing for each population which is because all parts in every generation are being optimized.

Int J Adv Manuf Technol

Fig. 12 Development of minimum fitness value using two different selection pressures

5 Summary and outlook 5.1 Summary The aim of the presented research was to develop a genetic algorithm for CWR preforms meeting the requirements for preforming optimization algorithms. A modular approach was used for the problem definition. The approach for solving the 3D multi-criteria optimization problem was to divide the preforming optimization problem in a certain number N of 2D problems with equidistant spacing l between the cutting planes. The results of the research show the feasibility of using l = 10 mm as a suitable equidistant spacing between the cutting planes, as this spacing enables sufficient results accuracy (see Fig. 10). However, for the optimization of the preform volume, the N 2D problems could easily transform into a 3D description by using the equation for truncated cons. The results enable the conclusion that dividing the preform optimization problem in N 2D problems is a suitable approach for long flat pieces. Regarding the design of the fitness function, three parameters have been derived to rate the suitability of the preforms. It has been shown that these parameters are suitable for enabling an optimization of CWR preforms for long flat pieces. The form-filling criterion is necessary because this parameter is mainly influencing the measured variable form-filling. The preform volume criterion is necessary because it is the main economic parameter as it influences the measured variable forming force and preform volume significantly. For CWR

preforms, the complexity factor is very important as this parameter enables the generation of preforms which can be produced by CWR. This parameter has been greatly improved during the optimization. Therefore, regarding the fitness function, the results enable the conclusion that the parameters chosen are suitable for a CWR preform optimization. The results show the suitability of the GA for optimizing CWR preforms as less preforms have to be investigated compared to the established, manual preform design process. The GA is able to create suitable preforms in about 40 s. 5.2 Outlook Overall, it can be stated that the chosen approach is suitable for solving the multi-criteria optimization problem of designing CWR preforms. However, further research is necessary for improving the computation results and the computation efficiency. Therefore, possible investigations are for example the influence of the algorithm parameters on the results quality and results efficiency. Furthermore, investigations are possible on improving the fitness function. Although the research shows the parameters taken into account for the fitness function are suitable for the present model, further improvement is possible. Regarding the form-filling criterion, Fig. 10 shows that form-filling is reached after the optimization. Nevertheless, for improving the users acceptance of the results, a flash reduction is necessary. This also requires further research, as the flash must be reduced but form-filling has to be ensured. A possible approach could be used to calculate the parameter a (Eq. 6) in dependency of the characteristics of the corresponding cutting planes.

Acknowledgments The authors thank the German Research Foundation (Deutsche Forschungsgemeinschaft) for the funding of the research project “Entwurf optimaler Vorformstufen zum Herstellen von Schmiedebauteilen unter Anwendung von stochastischen Optimierungsverfahren” (DFG Be 1691/177-1 and DFG OV 36/22-1). The authors declare that they have no conflict of interest.

References 1.

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3. 4. Fig. 13 Development of average fitness value using two different selection pressures

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