Automatic preform design and optimization for aeroengine disk forgings

To ensure a more uniform forging distribution microstructure and improve the service performance of aeroengine disk parts, an automated preform design method is proposed for integrated preform shape design and optimization based on the non-uniform rational B-spline (NURBS) curve, finite element method, and genetic algorithm (GA). First, the random preform shape graph is automatically constructed by the NURBS curve design criterion. The volume and shape complexity are used as the constraints of the preform. Second, the ratio of the mesh area within the set strain range to the total mesh area is used as the fitness function for the uniformity of deformation, and the GA is used for optimization. Finally, a large disk forging is an example of its optimal design. Results show that the deformation uniformity of the forgings is excellent, its fitness value is as high as 99.59%, and problems, such as folding, underfilling, and limited distribution of flash, do not exist, thereby verifying the effectiveness of the method. In addition, the method has the advantage of strong universality, that is, it can find the preform shape with good deformation uniformity for any shape forgings.


Introduction
In aeroengines, many disk forgings are formed by forging [1,2]. The performance and life of disk forgings determine the performance and life of the engine and indirectly affect the performance of the whole engine [3]. Given that disk forgings need to work under severe service conditions, higher requirements are placed on the organizational and mechanical properties of their forgings [1,4]. A uniform deformation distribution is an essential requirement for aerospace forgings [3]. Deformation uniformity should also be guaranteed while reducing the forming load [5] and eliminating the forming defects, such as underfilling [6][7][8], folds [6,8,9], and cracks [10,11]. The deformation uniformity inside the forging directly determines the uniformity of forging organization [12]. Moreover, forging internal organization is uniformly distributed by the deformation uniformity decision, and non-uniform organization reduces the fatigue life of forgings or lead to premature failure while in use. Mixed crystal and coarse grains are the primary organizational defects [13][14][15]. To ensure high-quality forging, the design and optimization of the preform for disk forgings are critical.
Some methods are used to design preform shapes. The Upper Bound Elemental Technique (UBET) is commonly used in backward simulation. It is widely used for designing preforms for axisymmetric parts because of its simple principle and fast solution speed [16][17][18]. Another method used in the backward simulation is based on the boundary node release criterion and backtracking loading path to determine the appropriate precast design (i.e., backward tracing scheme (BTS)) [19,20], and this method has been successfully applied to design the preform die-shapes of a generic turbine-disk forging process [20]. The equipotential field method is also a preform design technique used in material forming [21]. Its design is based on the similarity principle, which determines the preform shape of the forging based on the equipotential line between the initial and final shapes [22]. In addition, the use of other physical fields is possible to describe the material flow and thus obtain the shape of the preform, for example, by integrating QForm metal forming simulation software with special CAD software that enables isothermal surface extraction to preform modeling [21]. With the development of computer technology, a preform shape design method based on optimal design is widely used to optimize the design variables related to the preform shape size to obtain the optimal solution with a particular index or a specific relative relationship as the objective function, such as response surface method (RSM) [23][24][25], sensitivity analysis [26,27], genetic algorithms (GA) [28][29][30], topological optimization [31,32], and reduced basis technique [33].
The preform shape, regardless of the design method used to obtain it, needs to be verified by the forward forming process [21,25,30,32]. The difficulty of forwarding process analysis is parameter setting and constraint control, which reduce the number of trials and errors and improves optimization efficiency. However, UBET, BTS, sensitivity analysis, and topological optimization require user technicians to consider the structural features and metal flow characteristics of forgings, which are tedious processes that consumes considerable time and effort. Moreover, the aforementioned methods require preliminary guesses on the preform design during the designing of variables. Inevitably, the optimization results have certain user biases and cannot search for potential preform shapes [5]. Based on the stated problems, the need for an engineering method that can be used by simply telling the system basic information about the forgings is growing. The system will automatically complete the design and optimization of the preforms.
After considering the limitations of current preform design methods, an integrated system is developed to obtain satisfactory preforms in this paper. First, the appropriate strain range is selected according to the forging requirements. A suitable fitness function is established. The corresponding solution algorithm is built on the basis of the direct communication between finite element analysis and the GA. Second, the non-uniform rational B-spline (NURBS) curves are used to describe the shape of the preform. The positions of the control points are defined as design variables, and the GA is used as a controller to control these design variables to obtain a satisfactory preform shape by crossover, variation, and competition among all individuals. Finally, the feasibility of the optimization algorithm is verified using two disk forges. In addition, the optimal preform shapes obtained by the algorithm are redesigned to derive shapes that can be used in actual production.

Design of objective function
According to the general forming process, the forgings are formed by preform, and then the forgings are machined to obtain the finished part, as shown in the white box in Fig. 1. In this study, the equivalent strain indicates deformation uniformity. In the actual production, the equivalent strain variation value of the finished parts should be 0.43-1.02 to ensure that the mechanical properties and microstructure of the aeroengine rotary forgings meet the requirements. The value of this variable usually comes from production experience and can be changed according to requirements. To accurately judge the forming effect of forgings using a computer, the equivalent strain field of the finished parts must be quantified. The essence of an equivalent strain field is the combination of the equivalent strain values for each grid element [25]. Through extraction and calculation, the ratio of the grid area within the set strain range to the total grid area can accurately judge whether the forming effect is good or not. The larger the ratio, the better the forming product, and vice versa, as shown in Eq. (1): where F s is the objective function to achieve the ratio of the grid area to the total grid area in the set strain range, n is the number of grids that reach the set strain range, m is the total grid number, and S i is the corresponding grid area under a certain grid number. The range of fitness value F s is from zero to one.
Although die filling must be quantified to apply in the data analysis process, it is considered an optimization objective [29]. A filling ratio parameter was introduced to quantify die filling, as follows: where V forge and V actual are the volume of the designed forging and the actual forging after trimming the flash, respectively. There is no underfilling situation, and the filling rate can be idealized to 1. Here, V actual = V forge .
Flash volume is obtained by Eq. (3): where V pre is the preform volume. Here, the flash volume is only used as a constraint on the fitness function F s .

Design principle
After obtaining the evaluation index, a series of preform shapes must be constructed to find the optimal individual by using the optimistic algorithm. NURBS is a handy tool for geometric modeling, and it has gained increased attention from the engineering community because of its good properties [34]. Given that NURBS has the characteristics of manipulating control points and weighting factors, it can provide sufficient flexibility for the shape design of preforms with various complex shapes. NURBS modeling is always defined by curves and surfaces so that sharp corners will not be generated, which the preform expects. On the one hand, sharp corners will cause the folding of forgings. On the other hand, it will increase the number of remeshing in finite element simulation and reduce computational efficiency. The NURBS curve describes the outline of the preform (Fig. 2). The NURBS curve equation is expressed as where N i,k (u) mainly refers to the basic function of k times B-sample, p i denotes the control point, and i denotes the weight factor of the control vertex p i . The size of i value determines the degree of curve deviation from the control point, see Fig. 2a, where 1 is smaller than 2 and 2 is smaller than 3 . A larger i represents a more significant weight factor, and the closer the curve is to the control point p i means that the shape of the NURBS can be changed to approximate any shapes by adjusting its control points, which are extremely important for various aspects of subsequent graph redesign, analysis, and processing. Figure 2a shows that the x-coordinate of the data point is relatively fixed, and its length is limited by the maximum radius length of the forging. Here, its y-coordinate is defined as a design variable so that the shape of the preform can be changed in a wide range, thereby screening out some potential preform shapes (Fig. 2b). Therefore, the shape optimization problem can be transformed into a parameter optimization problem.

Shape constraint
As shown in Fig. 2, the random preform graph generated by the NURBS curve consists of two curves on the top and bottom and a raised arc on the right side, where the two unexpected curves are divided by the x-axis and are generated by interpolating two or more reference points. Given that the values of the reference points are strongly random, they also need to be subjected to certain constraints.
In the actual forging process, the length of the preform shape is shorter than the forging length. The design length of the preform is highly related to the maximum forging length. This work sets the maximum length of the curve in the x-axis direction to "m times the maximum radius length of the forging (5)." The distance between each reference point is assigned according to the number of reference points (6). The distance between each reference point can be customized.
where x pre in Eq. (5) is equal is the length of forging, l max is the maximum radius length of the forging, and m is the coefficient between them. x distance in Eq. (6) is equal is the distance between each reference point, and n is the number of reference points.
The reference points are random only in the y-axis direction, and the ranges of the upper and lower parts are set to [LB, UB] T , where UB is the "maximum height of the forging (maxh)" and LB is a value greater than 0. The values of LB and UB can be changed according to the requirements, thereby narrowing and increasing the efficiency of the scope of the graphical search. The feasible region of the design variables of the two-stage curve shall be defined as follows: where y 11 , y 12 ⋅ ⋅y 1n are the shape parameters of the upper-part-preform and y 21 , y 22 ⋅ ⋅y 2n are the shape parameters of the lower-part-preform.
The preform volume is a parameter that is worthy of attention in the field of forging. The increment in volume causes production costs and more flash volume to increase in forming load. Thus, the preform volume must be considered. According to the volume invariance principle of plastic forming principle, the volume of the preform should be the same as the volume of the forging plus the flash. Therefore, volume constraints must also be imposed. Specifically, a penalty function is established for the random preform graph. When the accumulated differential volume ( V pre ) is smaller than the forging volume ( V forge ) or larger than the set threshold ( v * V forge ), the combination of such variables or the preform shape is eliminated, and fitness F s = 0.
As we learned from the previous section, the characteristics of NURBS determine whether the graphs constructed by NURBS will not produce a sharp angle structure, so when the upper and lower curves are closed to construct the graphs, a natural rounded section will be created between the two points, which corresponds to the "bulging" phenomenon caused by the pier diameter in the preform, as seen in the rightmost shaded part in Fig. 3, which can be used as a flash control area. Its size could be adjusted to achieve near-net forming in the subsequent design process.
Then, in the investigation process, the generation of preform shape has strong randomness. Very complex shapes easily appear, and such forgings are easy to fold [35], thereby resulting in the scrapping of forgings. As shown in Fig. 3a, b, the shape of the forging fluctuates violently, the height difference between the peak and the valley is large, and the peak and valley regions are prone to folding. The two simulations also show that processing or forging these preforms is impossible. The shape complexity factor is used to design preforms [36], especially for more complex mass distribution parts, which becomes more important. Hence, the shape complexity factor should be considered in the design of preforms. Euclidean distances have also been used to describe the complexity of preforms [35]. However, in the present study, the method is less applicable, based on which the thesis proposes a height difference shape control method. Through the analysis of three consecutive control points, a Δh is calculated, and Δh is used as a complexity factor with following Expression (9): where Δh is related to the values of LB, UB, and x distance , as shown in Fig. 3b, c, d, and the larger the value of |∆h|, the higher the probability of folding (here the value of ∆h is obtained from the calculation of y 2,1 , y 2,2 , y 2,3 ). Δh is calculated by the following empirical formula, which is obtained based on a previous survey.
In summary, the constraint model for preform shape design is established as follows:

FE simulation
The hot forging process was simulated by the Deform-3D commercial package. DEFORM, as a finite element simulation software, is generally operated by the user through the graphical user interface (GUI) of the software. The GUI mode has the advantage of being intuitive and concise.
However, the DEFORM software can also be run in another mode, "Text mode," in which the user can use their edited command stream files to operate DEFORM's secondary simulation software. The text mode allows the graphical data in the '.key' file to be updated and simulation tasks to be submitted, which provides automated simulation and optimization feasibility.

Automatic optimization algorithms for preform
The automatic optimization algorithm takes the reference points [y 11 , y 12 ⋅ ⋅y 1n , y 21 , y 22 ⋅ ⋅y 2n ] T as the optimization variables and the deformation uniformity as the evaluation index. The number of reference points can be set according to the complexity of the forgings. The more complex the forgings are, the more reference points can be set. However, the more complex the random preform shape will be, the longer the computation time is. In addition, a virtually indefinite number of solutions are possible. Thus, selecting an appropriate optimization algorithm is necessary to solve this problem. However, finite element simulation is a typical black box, where each individual generated by the design variables has to be calculated by FEM to obtain the fitness value, which cannot be obtained by solving the mathematical model, and some optimization algorithms, such as gradient descent, Newton's method, and conjugate gradient method, are no longer applicable. The use of GAs is possibly suitable to solve this optimization problem because they are population-based heuristics [37]. They are particularly suitable to deal with black-box problems that do not require auxiliary information: differentiable, derivable, and continuous. GA is also a stochastic optimization algorithm that excels at searching for problems with large search spaces. It can effectively use existing information to search for individuals that show promise for improving the quality of the solution.
GA is similar to natural evolution. It simulates replication, crossover, and mutation in natural selection and inheritance. An automatic, collaborative, real-time dynamic optimization process is implemented by writing the corresponding interface program using the direct communication between the finite element method (FEM) and the GA. The GA module is used as a controller to send design variables to the FEM module and receive the individual fitness returned by the FEM module. As shown in Fig. 4, the core modules in the algorithm flow chart are the preparation, Auto Fitness, and GA module. The Auto Fitness module includes several sub-modules: the preform shape modeling sub-module, the  The specific process is presented as follows.

Module_1 Preparation
The forging forming is divided into three work steps, namely, the heat transfer step, forming step, and Boolean cutting step. The heat transfer step represents the calculation of the heat transfer between the forging and the air during the removal of the preform from the furnace and its transfer to the final forging die. The forming step is the forming process of the preform in the cavity of the final forging die. The Boolean cutting step is the cutting down of the finished part from the forming forging part, which is used to observe and calculate the strain. After setting the various parameters, the '.key' file in the first step is exported. Subsequently, the forging contour graph '.dxf' file is imported in MATLAB. Data are extracted to calculate the maximum height, maximum radius length, and volume of the forging to provide constraint data for the random preform shape. After selecting a suitable reference point variable matrix [LB, UB] T , the appropriate preform length coefficient m and height difference Δh are evaluated according to the variable matrix to further constrain the shape of the preform and reduce its complexity.

Module_2 Auto fitness
In this module, an automated integrated program was developed to complete the three steps of graphical modeling, numerical simulation, and feature extraction to obtain the fitness values of individuals. The operation of the whole program is based on the communication between MATLAB and Deform software. The MATLAB core commands for automated integration programs are as follows:

Module_3 GA
The GA is used to create preforms with random characteristics as an initial population, where each individual is described by a set of genes that represents the control variables for the preform shape. Then, the fitness function is calculated using the deformation uniformity parameter, and the termination condition is the preset genetic algebra. In the optimization process, the vectors of individuals with larger fitness values in the retained population are selected by changing the genes of the selected parents using crossover and variation methods, and their variation variable vectors are compared with the original variable vectors and passed to the next generation. Subsequently, the new population is evaluated again until the termination condition is satisfied. The specific algorithmic procedure can be displayed in the GA module in Fig. 4. Finally, all the optimizations have been completed, and the preforms with the above key dimensions will be determined.

Preform optimal process and results
In the following investigation, a disk forging is being used as demonstration parts, as shown in Fig. 5. Then the preform shape of this forging will be designed and optimized automatically.   [20,20,20,20,20

Parameter definition
Finite element simulation was performed using Deform-2D. The part temperature was set to 1010 °C, and the material data file IN718 (1650-2200 F (900-1200 ℃)) provided in the Deform database was used. The friction coefficient was set to 0.3, and the heat transfer coefficient is 5 N/sec/mum/C. For all parameters, other parameters not explicitly stated as standard (initial parameters set by Deform) were assumed for the hot forging process.
In this research, an individual consists of 10 genes that represent the design variables. The population size should contain 10 times the number of individual genes [38].
Therefore, the population size is assigned as 100, and the number of evolutionary generations is 20, as a stopping condition to reduce the total simulation time. The crossover rate is set to 0.8, and the mutation rate is set to 0.2.
The forging drawing shown in Fig. 5 is imported, the calculated maximum radius length l is 323 mm, and the maximum height maxh is 173 mm, and the forging volume V forge is 3.5225e + 07 mm 3 . A total of 10 reference points is taken in the example, and the range of the values of the reference point of the NURBS curve [LB, UB] T can be obtained according to maxh. Through preliminary exploration, constraints are defined to improve the convergence rate and avoid folding during the deformation. Considering that   : a 1st generation, b 3rd generation, c 5th generation, d 9th generation, e 13th geneation, f 15th generation, g 20th generation, h forming load and flash rate the irregularity of the parts will disappear during the optimization process, the selection range should be appropriately relaxed when setting the constraints to avoid the optimization algorithm from falling into a local optimum and losing many potential combinations of variables. The specific data are shown in Table 1. Figure 6 shows the development of the best-rated and average individuals within the population for 20 generations. A total of 2100 simulations were run for this preform optimization algorithm. Considering the time cost, the optimization was executed on an Intel Core i7 processor with four processors. Parallel operation was implemented by MATLAB software to improve the computing efficiency. The total time consumed from the shape design of the preform to the FEM simulation and then to the completion of the optimization was approximately 55 h. Figure 6 shows that the optimal fitness value converges quickly and surges to a higher fitness value in the 5th generation. In contrast, the average fitness is not equal to the former until after the 15th generation, which indicates that the design parameters of the preforms have a large convergence rate before the 15th generation. Then, after the 15th generation, the variation of the best fitness value and the average fitness tends to a constant value (0.9959), as shown in Table 2. This finding indicates that the convergence rate gradually decreases, and the search range approaches the optimal individual as the number of evolutionary generations increases. Among them, 0.9959 indicates that the area of the grid element satisfying deformation 0.43-1.02 in the finished part accounts for 99.59% of their total area, which shows that the algorithm searches for the preform shape with very good results.

Results and discussions
The best preform shape at each generation is illustrated in Fig. 7a-g to easily understand the evolutionary process. With the increase in the evolutionary generation, the shape of the preform becomes increasingly similar, and all of them show the shape of "bulging in the middle, flat on both sides." The data in Table 2 are not difficult to observe, and the numerical changes in the reference points are also becoming smaller and smaller. From the equivalent strain evolution diagram in Fig. 7a-g, gray, light blue, and green gradually dominate from the 1st generation to the 15th generations, thereby exhibiting a more uniform strain distribution in forging. Moreover, Fig. 7h Fig. 9 Redesign of optimal preform shape: a original optimal preform shape, b data point modification, c control point modification, and d locating structure shows that the flash rate and forming load can be reduced in the continuous evolution process, which again proves that the shape of the preform has been optimized significantly.
The histogram of the distribution of the equivalent strain (Fig. 8a) and the forming load-stroke diagram (Fig. 8b) of the finished part at the 20th generation are presented. The distribution of the equivalent strain of the forging is nearly normal. The strain is mainly concentrated in the range of 0.65-0.85 (Fig. 8a), where the average value of the strain distribution (Avg) is 0.8, and the standard deviation (Stdev) is 0.1. The minimum and maximum values are 0.4 and 1.0, respectively. The maximum load during the forging process is approximately 30,227 tons from Fig. 8b, which meets the production requirements. The optimization results show that the GA can automatically converge to the optimal design parameters and obtain the optimal combination of the design parameters.

Redesign of optimizing the shape of preform
The optimal preform shape is obtained by using the above method, and the numerical simulations are good and satisfy the optimization objectives. However, some problems in practical applications, such as sharp corners caused by control    [20,20,20,20,20 points, difficult processing area, and billet positioning structures, must still be considered. Therefore, this study improves the original optimization results based on the concept of the redesign proposed in the literature [39], to find a more feasible preform shape to meet the practical application requirements while ensuring that the optimization objectives will be met.
To eliminate the complexity of the original optimal preform shape, the preform shape is simplified using the approximation method while maintaining the original shape characteristics (Fig. 9). First, the shape of the preform was fine-tuned by changing the NURBS data points to make the preform more machinable (Fig. 9b). Subsequently, the number of control points was increased, and the weight values of the control points were adjusted (Fig. 9c) to avoid severe sharp corners, which can cause mold penetration during simulation and stress concentration during plastic deformation. Finally, the preform shape's locating structure was designed using CAD software at the blue-shaded position in Fig. 9d.
The positioning of the workpiece in the die and the results are shown in Fig. 10a-c; it can be concluded that the deformation uniformity of the redesigned preform inherits the characteristics of the original optimal one, and the equivalent strain in 99% of the area of the finished part reaches 0.43-1.02. More importantly, the complexity of the preform shape is sharply lowered, thereby making it easier to be manufactured.
In our previous work [40], the preform shape of the forging was designed by the sectional design method. A total of 96.55% of the finished product was in the equivalent strain range of 0.43-1.02 after forging. The actual forging production was carried out for the forgings, and the forgings met the quality requirements in terms of physicochemical detection and property analysis. The grain grade in each deformation zone was above grade 10, as shown in Fig. 10d. This observation shows that the preform design method and the integrated system constructed in the paper are accurate and reliable.

Case study
To better reflect the generality of the automation optimization algorithm, an aeroengine disk forging is identified as a case study in this research. The diagram of the forging is shown in Fig. 11.
The operation process is the same as that in the previous section and will not be repeated here. The maximum radius length l of the forging is 297 mm, the maximum height maxh is 94 mm, and the forging volume V forge is 1.8511e + 07mm 3 . Other detailed data are shown in Table 3.
The forging shape contour is simple, so it takes 42 h to complete the design and optimization of the preform. The number of finite element simulations is 2100, and the optimization results are shown as follows.
The preform shape of the aeroengine disk was obtained by the automatic optimization algorithm in the case study. After 20 generations of evolution, the fitness value of the optimal individual reached 98.30% (Table 4). From the evolution curve in Fig. 12, the fitness value changes very little after the 10th generation. The optimization process can be ended at this point to improve efficiency. The shape and equivalent strain of the best individual at each generation is illustrated in Fig. 13a-d.
After obtaining the optimal shape of the preform, the redesign method is used to simplify the shape of the performed part (Fig. 14a). The results show that the uniform deformation of the redesigned preform part is still realized well. The equivalent strain in 96% of the area of the finished part reaches 0.43-1.02 (Fig. 14a,b). Moreover, the  12 Evolution curve for optimization design of preform shape using GA maximum forming load is only 22,082t (Fig. 14d), which is suitable for production.

Conclusion
To improve the quality of forgings, this paper proposes an automated preform design method based on NURBS curve modeling, FEM analysis, and GA optimization.
(1) The proposed preform design method can automatically complete the design and optimization of preform shapes for various types of disk forgings without human intervention during the design and optimization processes. The optimization process and results can be observed in real time.
(2) The design method can optimize the preform according to specific target values. The optimization procedure can be stopped in practical use to reduce the computational cost while meeting the deformation requirements. (3) To verify the effectiveness of the method, two different engine disks are investigated. The results show that the forgings have good deformation uniformity. The equivalent strain in 96.55% and above areas of the finished parts can meet the requirements, thereby proving the practicability and effectiveness of this method in engineering applications. Data availability Not applicable.

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Competing interests
The authors declare no competing interests.