Robust optimization of consistency in filling of rib-grooves for titanium alloy multi-rib eigenstructure

Isothermal forging is an effective method for forming and manufacturing large-scale titanium alloy components with multi-rib. However, successive filling of the rib-grooves and reverse flow of the material are prone to occur during the forming process, which makes those rib-grooves difficult to be filled then resulting in disturbed material flow and excessive die loading. The variability of billet sizes and fluctuation of uncertain parameters during the forging process have the great impacts on the forming results and stability. To this end, the eigenstructure with multi-rib from large titanium alloy rib-web components was extracted, and the combined method of finite element simulation and physical simulation experiment was used based on isothermal forging technique. Firstly, the finite element model for the eigenstructure under isothermal forging is established, and then the behavior of the material flow and rib-groove filling in the die cavity are analyzed. Secondly, the variation pattern and fluctuation range of rib-grooves filling are investigated by considering the deterministic factors of billet sizes, as well as the uncertainties of die draft angle, forming temperature, forming speed, billet manufacturing deviation, die manufacturing deviation, and friction factor. Subsequently, the significant deterministic factors and uncertainty factors are screened out, and the correlation between the mean value as well as the variance of the filling consistency of the rib grooves and the deterministic factors, i.e., billet sizes, are established by the dual response surface method. Then, a robust optimization model is constructed and solved. Finally, the reliability of the robust optimization solution is compared and verified to obtain the ideal and stable fully filling of the rib grooves by adjusting and regulating the deterministic factors to weaken the interference of the uncertainties and achieve the simultaneous filling of the rib grooves.


Introduction
Large-scale titanium alloy rib-web components have a series of advantages such as small occupation space, light weight, and high performance characteristics [1,2], being widely used in aerospace and other fields as key load-bearing components, such as titanium alloy aircraft bulkhead (as shown in Fig. 1), impeller blade, and actuator cylinder [3,4]. The rib width, rib height, and rib spacing are different, and the shape is exceptionally complicated. Controlling the material flow during the deformation process can reduce the material forming load and improve the material forming accuracy, which is an effective way to improve the forming quality of large-scale titanium alloy rib-web components. Nowadays, isothermal forging technology is capable to significantly reduce material deformation resistance, increase material plasticity, improve and enhance material microstructure and properties on the basis of precisely forming complex components [5,6]. It has become the crucial approach for producing large-scale titanium alloy rib-web components.
The isothermal forming process of large-scale titanium alloy rib-web components is carried out by coupling of multiple constraints and parameters. The intricate material flow and intense inhomogeneous deformation of the components in die cavity may lead to macrostructure defects such as underfilling, flow disturbance, and folding [7], which affect the service life of the components and confine the high-performance accurate plastic forming to be achieved.
In order to avoid abovementioned macrostructure defects during forging of such components, a large number of scholars have conducted research on improving material flow and adjusting forming load during isothermal forging of complex components. Li [8] proposed a volume pre-distribution and equidistant flow quantitative compensation method for the precision extrusion-forming process of complex magnesium alloy rib-web components, which controls the axial material flow, improves the radial material flow, and reduces the die cavity resistance. Thiyagarajan [9] used the response surface method to optimize the billets for components such as three-dimensional spring seats and steering linkages, respectively. A response surface model between basic shape combinations into billet shape coefficients was developed for optimal design. Torabi [10] combined the response surface method to optimize billets for blade forging by developing a response surface model and combining it with the NSGA-II multi-objective optimization algorithm to obtain satisfactory results. However, there is a certain error between the fitted value and the real value in each sample point. If more design variables and the influence of uncertainty factors are also taken into account, the computation and storage capacity will be significantly increased and the accuracy will be reduced, which is difficult to be applied in the design of billets with high complexity and many geometric parameters.
However, the robust optimization can effectively improve production efficiency and is gradually being applied to various production design areas [11]. It is basically defined as a product or process that is reduced or insensitive to the effects of a source of uncertainty, although the variability itself is not eliminated [12]. Therefore, the purpose of robust optimization is to make the system response close to the target and reduce the sensitivity of the target response to the uncertainty factor without eliminating the noise factor of the system.
There are also more studies reported on the application of robust design in metal forming. Hou et al. [13] constructed a dual response surface model of the mean and variance of fracture and wrinkling with the resistance of the drawing rib and the crimping force, respectively, using the materialhardening index, each anisotropy index, and the friction coefficient as uncertainty factors to achieve robust optimization of the forming process for the inner lid of luggage compartment. Li et al. [14,15] proposed a robust design method for computer numerical control bending and forming parameters of large-diameter thin-walled aluminum tubes, considering the effects of noise factors such as material parameters, friction parameters, and tube geometry parameter fluctuations. Repalle et al. [16] analyzed the stochastic fluctuations of factors such as material and process parameters in the forging process and the die life variation law and combined the simulation and response surface method for a robust and optimal design. Comparing with this work for the consistency of material filling in rib grooves, the above studies are differences in the process parameters and forming results. To improve the robustness of material filling in rib grooves simultaneously, the influences of uncertain factors are considered by this work.
Accordingly, this work incorporates the robust optimization theory into the design process of forging billet for titanium alloy component with multi-rib. A robust optimization design method is adopted by combining dual response surface method and genetic algorithm. Meanwhile, the material filling law of rib grooves can be revealed during isothermal forging. In addition, the deterministic factors of billet sizes, the material flow law, and the filling order of the rib grooves are taken into an overall consideration under the influence of uncertainties. Robust optimization solution is carried out to weaken the interference for the fluctuations of uncertainties and improve the consistency and stability of the rib-groove filling effectively so as to avoid the reverse flow of the material and reduce the forming load. This work can provide the basis for the high performance and accurate manufacturing of complex multi-rib components of titanium alloys.

Finite element modeling
As a common load-bearing component in the aviation field, the large-scale titanium alloy components with multi-rib possess the special structural characteristics that integrate the large, integral, thin-web and multi-rib. Gao [17] extracted the characteristic structure of large-scale rib-web component and controlled the forming defects in the isothermal local loading transition zone of titanium alloy rib-web component by studying the eigenstructure, which is a typical representation of a complex multi-rib component and can fully reflect the forming process and deformation characteristics of a large-scale multi-rib component in transition zone. The specific geometric parameters in this work are given in Table 1. Accordingly, the die geometry shown in Fig. 2(b) was designed to avoid streamlined defects during forming and to reduce the material loss by choosing a flashless die-forging process. A simple equal-thickness billet (ETB) was chosen to observe the flow of material in the die rib grooves based on the principle of volume constancy. Rigid-plastic finite element method is used in the finite element model. Since the plastic deformation of die is much smaller than the workpiece in the die forging, the billet is set as deformed body and the die is set as rigid body. Due to the low strain rate of isothermal forging, the temperature of the billet and the die are set to be uniform, and thermal effects are ignored, such as heat loss and heat conduction. The billet is made of tetrahedral discrete elements, which are accurately discrete and approximate complex geometries [18] so that the material can be better and flexibly filled into the deep and narrow rib grooves during the simulation. The specific parameters of the finite element simulation process are shown in Table 2.
The material selected for this study is TA15 near-α titanium alloy, which is widely used in aerospace load-bearing components because it can work stably for more than 3000 h at 500 °C. When working at 450 °C, its lifetime can reach 6000 h and it has good comprehensive mechanical properties [19]. Figure 3 shows the stress-strain relationship of TA15 at different temperatures [20]; the influence of fluctuating uncertainty factors of robust optimization is considered, so the near-β forging temperature range of 950 to 980 °C is chosen in this work, and the data are put into the simulation software as the form of scatter points. Then, the flow stress of the TA15 under the different strains, strain rates, and deformation temperatures can be achieved by interpolation function in the simulation software.

The design process of robust optimization
In the actual production process, certain deviations between the actual and ideal values of the results are brought about by the fluctuation of uncertainty factors such as billet manufacturing tolerances, environmental changes, equipment wear, and human   operations. In the optimization of billets for filling consistency of titanium alloy eigenstructure with multi-rib, it is necessary to consider the influence of fluctuations in uncertainty factors on forming results [21]. In this work, the robust control of the rib-groove filling consistency for billet optimization is proposed based on robust optimization design, combining Box-Behnken design (BBD), finite element simulation method, and dual response surface method. The implementation of the robust optimization of the billet sizes for multi-rib components is given in Fig. 4. The detailed procedure can be executed as follows: 1. For large-scale components, the position of each rib is analyzed, and the billet is divided into various regions to set key parameters for billet size design. Then, the simulation works are carried out through numerical simulation software. 2. Considering the uncertainty factors affecting the forming quality of large-scale components, orthogonal experiments are designed to screen out the factors with great influence on the forming quality.
3. Based on the cross array design, the certain factors and uncertain factors are combined to conduct experiments and extract data for analysis. 4. The double response surface model of the mean and variance for the underfilling rate of multi-rib component is constructed, which is used as the objective function to establish the genetic algorithm optimization model. The billet with the best filling consistency effect is obtained through iterative optimization process. 5. The optimized billet is verified by simulation and experiment results. By this way, the robust optimization billet is obtained.

Physical simulation experimental validation
The deformation of titanium alloy at high temperatures was predicted by physical simulation using materials such as plastic clay and lead, which are made of softer materials, instead of titanium alloys [19], in order to analyze the material flow and forming defects, etc. It is an effective method to verify the accuracy of the finite element simulation results, as it can increase the experimental efficiency, reduce the experimental cost, and save human and financial resources while providing guidance to the analysis of the forming process. According to Sofuoglu [22], plasticine has a low flow stress (0.08-0.175 MPa) and can be used with simple and inexpensive tolling to be used for 2D and 3D physical modeling analysis. Kim et al. [23] used pure plasticine to optimize the design of extrusion dies for "L" and "U" profiles, and the results were verified to be more satisfactory. Sun [24] has physically verified the non-uniform deformation law of multi-cavity forming under complex loading paths by using plasticine. Then, he proposed a multi-cavity forming path to avoid defects such as folding. From two aspects of geometrical parameters and process parameters, Cao [25] analyzed the feasibility of forming complex multi-cavity parts of 316LN steel by multi-direction loading by physical simulation experiments of plasticine. Therefore, in this work, pure plasticine is also used to predict the deformation of titanium alloy at high temperatures, analyze the material flow, and observe the rib-groove filling in order to improve the experiment efficiency and reduce the cost, etc. The combined die used for the experiment is shown in Fig. 5(a) with the same parameters as the finite element model of Fig. 2, and the die material is 45 steel. The specific implementation was carried out on the 20-ton press. The grease is applied for lubrication during the experimental process. From the physical simulation experimental results, it is observed that the rib grooves on two sides are filled before the middle one, resulting the middle rib groove being underfilled, as the result shown in Fig. 5(b)(c).

Selection of deterministic factors
In order to better analyze the effect of different volume distributions on the filling consistency of rib grooves, the billet H equal is the thickness of the billet with equal thickness under the principle of volume consistency, which is the ratio of the desired workpiece volume to the upper die contact surface workpiece. Ensuring that the volume remains unchanged is realized by adjusting the values of H middle when the billet size changes. It can be observed that two parts of variable thickness are emerged and resulting in four types on unequal-thickness billet (UTB), as shown in Fig. 6(b)-(e). The folding may produce in the variable thickness region of the UTB [26]. Figure 7 illustrates that an inclined surface is set at the variable thickness region of the UTB. The results from finite element simulation and physical simulation experiment both show that folding is not produced in the work. To this end, the variable thickness region is designed as the form of beveling. Therefore, the folding was not considered in the optimization in this study. Under this circumstance, the key dimensional parameters (l left and l right ) of each part of the UTB are changed to adjust the length of each part. By this means, the volume distribution in different areas of the UTB is equivalently adjusted by the coupling effect of H left , H equal , H right , l left , and l right .
To variable the quantitative parameters in geometric modeling and then obtain geometric models with similar shapes and different dimensions of the UTB, the parametric design is adopted to establish the association between geometric parameters and dimensional data [27]. Besides, if experimental optimization design is used for optimal design of billet sizes, a large number of geometric models need to be constructed, which is time consuming in 3D modeling, so that the design efficiency and optimization results are difficult to guarantee [28]. In order to solve this out, the geometric parameters of the billets are adjusted based on UG software expression command to achieve the quick adjustment of volume distribution of the UTB.

Uncertainty factors
In the actual production process, there exist a number of uncertainties that cannot be wiped out. In order to recognize the uncertainties and their levels of fluctuations that have a great impact on the forming quality, the following six uncertainties and their fluctuation levels are considered in this work, as shown in Table 3.
(1) Uncertainty factor A As shown in Fig. 8(a), shear friction is generated on the surface of the die and the billet. It (2) Uncertainty factor B As there exists manufacturing tolerance on the surface of billet, allowing a certain range of variation to occur for the geometric dimensions of billets. Then, the actual billet dimensions may deviate from the ideal values.
(3) Uncertainty factor C In the actual process, it is undeniable that a certain range of fluctuation is generated in the stroke speed of upper die under the comprehensive influence of factors such as friction fluctuation, equipment wear, and human operation.
(4) Uncertainty factor D As shown in Fig. 8(b), a certain assembly clearance as well as manufacturing tolerances may be considered after the die is manufactured, because the die manufacturing deviation will also affect the forming accuracy.
(5) Uncertainty factor E The heat transfer can be produced among the dies, the billet, the environment as well as the shear friction between the billet and the die, which are generated during the forming process, allowing certain fluctuations of the forming temperature to be developed under the combined effect of frictional heat generation and heat transfer.
(6) Uncertainty factor F According to the reference [29,30], the friction factor can affect the material flow in the die forging; therefore, the fluctuation of the friction factor is considered in this work. It is notable that rib-groove draft angle is an uncertain factor because there is a range of errors in the manufacturing, resulting in a certain amount of fluctuation in draft angle of rib grooves. However, die manufacturing deviation can result in size change of the integral die cavity. Therefore, these two uncertainties are not in conflict, as the fluctuation of rib-groove draft angle and die manufacturing deviation shown in Fig. R7(a)

Description and quantitative statistics of the rib-groove filling consistency
This study discusses the consistency of material filling in the rib grooves, i.e., each rib groove is filled simultaneously during the forging to avoid defects such as backflow of material and excessive load caused by the non-simultaneous filling of the rib grooves. Yang et al. [31] quantitatively described the result of filling in the die forging during the forming process by using the ratio of the surface area of the die cavity to the uncontacted area of forming forging as an objective function. In order to analyze the filling characteristics, the quantitative statistics of filling ability for the rib grooves in the die cavity is described as: where V underfill is the volume of the remaining unfilled rib groove when the first rib groove is filled, as shown in the blue area of Fig. 9(a). V eigen is the volume of the die cavity that is completely filled, as shown in the blue area of Fig. 9(a).

Experimental design of uncertainty factors
Orthogonal experiment is an efficient, rapid, and economical method for analyzing multifactorial and multilevel experimental designs, which are commonly used to analyze the significance of influencing factors [32]. In order to reduce the number of experiments and calculation for robust optimization, the uncertainty factors that have a large influence on the consistency of rib-groove filling are screened in this work. Five levels of uncertainty factors are selected for the experimental arrangement, which uses the L25(5 6 ) orthogonal table. Subsequently, 25 sets of schemes as well as results shown in Table 4 are given in accordance with the finite element simulation and the orthogonal table.

Significance analysis of uncertainty factors
In the process of analyzing the test results, extreme difference analysis was used to obtain the influence and significance of uncertainty factors on the target values [33]. Uncertainty factor A is taken as an example.
where K Ai is the ith level of factor A (I = 1, 2, 3); Y(P) is the Pth test in Table 3 (P = 1, 2, ……, 9). It can be seen that K ji is the sum of each factor j (j = A, B, C, D) and K ji is the average of each factor at the same level where a higher R j indicates a more significant factor.
In order to better compare the effects of uncertainties as well as the contribution rate, the underfilling rate is normalized as in Eq. (4).
where Φ(p) is the underfilling rate of the No.P test in Table 4, max Φ(p) and min Φ(p) are the maximum and minimum values of underfilling rate, respectively. The normalized results are shown in the normalized value in the rightmost column of Table 4.
The range analysis result of the uncertainty factors on the underfilling rate given in Fig. 10 shows that the billet manufacturing deviation, the die manufacturing deviation, and friction factor have some influence on the rib-groove underfilling rate of eigenstructure. The die angle, forming speed, and forming temperature have little effect on the underfilling rate. Due to the billet manufacturing deviation, the die manufacturing deviation and friction factor all affect the material flow in rib grooves, which affects the underfilling rate of rib grooves. The specific reasons are as follows: Fig. 9 Eigenstructure of multi-rib and thin-web material filling: a a rib groove is filled; b all ribs are fully filled (1) Friction is related to the normal stress, but the normal stress change caused by the deviation of rib-groove die draft angle is negligible, so the effect on the rib-groove underfilling rate is small.
(2) Billet manufacturing deviations can lead to changes in the volume distribution of the billet, which, in turn, affects the underfilling rate of the rib grooves.
(3) The forming speed basically does not affect the material flow in the rib grooves, so it has less effect on the underfilling rate of rib grooves.
(4) The dimensional tolerance of the die manufacturing makes its geometry fluctuate in a certain range, which leads to changes in the volume distribution of the billet, and the rib-groove underfilling rate is also affected. (5) The material flow in the rib grooves is basically unaffected by the temperature change. Gao et al. [17] compared the material flow in the rib grooves at different temperatures and showed that the relationship between temperature and material flow in the rib grooves is not significant. (6) The increase in friction factor will increase the resistance to material flow and prevent material from flowing into the rib grooves. Under the effect of volume distribution, the material of the larger volume part flows into the rib grooves first due to the material always flows to the region with less resistance, resulting in cross-flow of material in the web region, which accelerates the other rib-groove filling with low efficiency. Eventually, it contributes to reducing the rib-groove underfilling rate.
This study also used the contribution margin to quantify the importance of individual uncertainty factors on each shaping indicator, which is given by (5) = SS I ∕SS T Fig. 11 Contribution rate of uncertainty factors on underfilling rate Ф u  Table 6 The cross array with geometric parameters of UTB and uncertainty factors  where SS I is the sum of squared deviations of the Ith uncertainty factor (I = 1, 2, 3, 4); SS T is the total sum of squared deviations.
The contribution rate of each uncertainty to the filling consistency is given in Fig. 11. It can be seen from the figure that the influence of forming temperature, forming speed, and die draft angle on the underfilling rate is very small, and the contribution rate is less than 4%, which can be ignored in the robust optimization of UTB. While billet manufacturing deviation, die manufacturing deviation, and friction factor have great significant effects on the underfilling rate. Accordingly, the billet manufacturing deviation, die manufacturing deviation, and friction factor are considered the key parameters of the uncertain factors for robust optimization of the UTB.

Orthogonal design by inside and outside array
Orthogonal design by inside and outside arrays is an experimental design, in which two experimental design tables are arranged vertically to examine the influence of one factor on another, taking into account both deterministic and uncertain factors [19,33], as shown in Table 5. Based on the inner array and the outer array of robust design, the geometric parameters of UTB are arranged in the outer array with Box-Behnken design (BBD), as shown in Table 6. Control factors in are taken as examples to combine the key dimension parameters of the cross array and adjust the dependent variable to achieve a constant volume of the billet. The uncertainty factors, such as billet manufacturing deviation, die manufacturing error, and friction factor are arranged in the inner array with uniform design (UD); when the variation range of factors is large and more levels need to be taken, the number of works can be greatly reduced by combining BBD with UD, which is conducive to the statistical mean and variance to evaluate the nonlinear relationships between indicators and factors. In this study, a quadratic polynomial is considered in the process of response surface establishment to ensure the accuracy of the model, as in Eq. (6).
where y is the response (a composite indicator of the rib underfilling rate), x i and Xj are input variables (a = H left /H equal , b = H right /H equal , c = l left /L left , d = l right /L right ), and k are these variables. The number of 0 , jj , and ij is regression coefficients.
A dual response surface model of the mean and variance of the underfilling rate Ф is constructed from the experimental sample data in Table 5.
Whether the fitted model can reflect the relationship between the targets and variables needs to be tested by the deterministic coefficient R 2 a , and the criteria for evaluating the goodness of the model are expressed in Eq. (9).
is the residual sum of squares, Y i is the response value at point i, Ŷ i is the predicted value of point i, Y is the mean value of the experiment at point I, n is the number of experiments, k is the number of regression coefficients in the model.
This study evaluates the accuracy of the dual response surface by means of the experiment method as described above. If the value of the correlation coefficient adjust R 2 is higher than 0.9, then the constructed response surface model is close enough to the actual model. The closer its value is to 1, the better the response surface model reflects the actual problem. The obtained results are shown in Table 7. It can be seen that the adjust R 2 of the mean and variance are all greater than 0.9. These results indicate that the dual response surface model has high accuracy, which (7)  can be used for the replacement of finite element simulation to obtain the response targets for the rib-groove filling consistency. Figure 12 gives the dual response surface graphs of the mean and variance of underfilling rate and the geometric parameters of UTB. Under the influence of fluctuated uncertainties, the mean and variance of underfilling rate show a trend that decreases and then increases with H left and H right increasing. The best point of filling consistency is the point where the mean and variance of the underfilling rate are minimal, as the result of P 0 shown in Fig. 12(d). The first fully filled rib groove is gradually converted into rib 3 with H right increasing, as the result of P 1 shown in Fig. 12(c). Alternatively, the filling sequence of rib grooves is switched to rib 1 fully filled firstly when H left is increased to a certain value, as the result of P 2 shown in Fig. 12(e).
To sum up, in order to attain the desirable and steady filling consistency, the minimum value under the combined effect of mean and variance of the underfilling rate should be considered in the optimization of UTB. On the basis of establishing the correlation between the underfilling rate of rib grooves and the deterministic factors under the fluctuation of the uncertainty factors, further robust optimization solution for UTB is required to weaken the effect of fluctuation of uncertainties.

Optimization objectives
The goal of filling consistency is to adjust the material flow in different rib grooves of complex multi-rib components, which reduces forming load, and to obtain obvious neutral lay and fully filled forging without folding defect. Specifically, the following objectives need to be achieved for the rib-groove filling consistency. The geometric shape conforms as much as possible to the requirements of the ribgroove filling sequence, i.e., to ensure that the material contacts with end surface of the rib grooves simultaneously and minimize the underfilling rate Ф during the filling process of the rib grooves. Through robust optimization of UTB, the influence of uncertain factor fluctuations can be weakened to reduce the forming load.

Building a robust optimization model
To establish the robust optimization model for the rib-groove filling consistency, the meaning of the objective function is that the smaller the sum of the mean and variance of the rib-groove underfilling rate, the better the optimization result and the less interference with uncertain factors during the filling process of titanium alloy multi-rib eigenstructure. The forming temperature is defined as 970 °C, considering the effect of uncertainty factors. The robust optimization model for UTB is shown in the following.
where f (a, b, c, d) is the objective function to be optimized, V unequal (a, b, c, d) represents the equality constraints, a, b, c and d are four-variable parameters, and V eigen is ideal volume of eigenstructure. (10)

Robust optimization solution based on genetic algorithm
Genetic algorithm has excellent robustness, randomness, and global optimization ability [34,35]. In this work, a genetic algorithm is used to solve the nonlinear and multiparametric optimization problem of rib groove filling consistency for titanium alloy multi-rib eigenstructure without any requirement for the continuity and directability of the design space. The MATLAB genetic algorithm optimization toolbox function is adopted to punish the individuals who do not meet the constraints of the robust optimization model for the rib-groove filling consistency. Combined with MATLAB and simulation results, the population evolution of the genetic algorithm is developed, as shown in Fig. 13.  Fig. 13. According to the optimization results, the middle part of the billet is the thinnest. The robust optimal UTB geometry is illustrated in Fig. 13 and belongs to type III in Fig. 6(c), which has a better filling consistency. Before optimization, the filling order of the rib grooves of the billet is filled from the left to the right. After the iterative optimization of the genetic algorithm, the filling order of the rib grooves is changed to fill all rib-groves at the same time, which meets the robust optimization requirements of the rib-groove consistent filling. Figure 14 compares the forming loads of the three billets before and after optimization, with different underfilling rates during the forming process. From the diagram in Fig. 14(a), it can be seen that, in the first stage, i.e., the basic forming stage, the billet is deformed from the initial shape until it basically fills the rib grooves, and the forming load increases relatively slowly. When the underfilling rate reaches about 25%, the billet deformation enters the second stage; meanwhile, the forming load increases sharply, but its deformation is much smaller compared to that of the first stage. When the underfilling rate reaches about 5% or below, the billet has basically filled the die groove and the forming load reaches the maximum. The maximum forming load is 147.9 tons for the ETB, 106.4 tons for the UTB, and 80.5 tons for the UTB after robust optimization. The maximum reduction of forming load after robust optimization is 45.57%.
It is well known that the material flow is always in the direction of least resistance. The unfavorable phenomenon of material cross-flow exists in the die cavity, as shown in Fig. 14(b), where the material cross-flow at the web exists in the ETB and the neutral layer disappears. The ETB suffers from the material cross-flow at the web, and the neutral layer I is disappeared. After the robust optimization, the volume distribution of the billet is more reasonable. The phenomenon of material cross-flow at the web is less, the neutral layer I is always existing, and the forming load of the optimized billet is higher. When the underfilling rate reaches 5% or less, the die cavity is filled, and the material cross-flow at the web is eliminated. In this optimization process, the volume distribution of the UTB is reasonable; the ribs grooves are filled nearly simultaneously. In addition, it is not observed material backflow in the billet-filling process after optimization, as shown in Fig. 14(c). Therefore, the maximum forming load of the robust optimized billet is smaller than that of ETB and UTB, and the forming load shows a decrease of 5%.

Comparison of robust optimization and deterministic optimization for rib-groove filling consistency results
The optimized UTB is obtained by iterative optimization according to the genetic algorithm. In order to better verify the effect of robust optimization of UTB, those obtained by the traditional optimization method without considering the influence of uncertainty factors are adopted in this work so that the two optimization solutions can be compared. The billet geometry parameters are included in Table 8.
The mean and variance of the underfilling rate under the robust optimization solution and the deterministic optimization solution are calculated using Eqs. (6) and (7). From the calculation results in Table 8, it can be concluded that the calculated mean underfilling rate of the robust optimization solution is slightly larger than that of the deterministic optimization solution. Nevertheless, the variance of the robust optimization solution is much smaller than that of the deterministic optimization solution. This indicates that the filling consistency results are better for both objectives, and the stability of the robust optimization solution is preferred.
Under the fluctuation of uncertainty factors, five schemes with UD method of the outer array in Table 6 were used for finite element simulation. The underfilling rate is normalized with reference to Eq. (5), and the result obtained is shown in Fig. 15. It can be seen that the mean value and fluctuation range obtained from the robust optimization are smaller than the results of the deterministic optimization. Accordingly, it is shown that the rib-groove filling consistency robust optimization of titanium alloy multi-rib eigenstructure is better than deterministic optimization.
Combined with the finite element simulation results of the ETB, a P-diagram is given robust optimization of the ribgroove filling consistency of titanium alloy multi-rib eigenstructure, as shown in Fig. 16. From Fig. 16(a), it is observed that the probability density function of the ETB fluctuates in a wide range, and the distribution of the results is far from the optimization target, and its robustness is poor. The analysis of the minimum distance for the filling of the rib grooves is shown in Fig. 16(d). The probability density function of the rib-groove underfilling rate of the deterministic optimized billet fluctuates widely, but the results are distributed around the optimization target. Under this circumstance, the rib groove filling consistency effect is good but not enough to meet the requirements of robust optimization, as shown in Fig. 16(b). It can be seen from Fig. 16(c) that the probability density function of the rib-groove underfilling rate of the robustly optimized billet fluctuates in a small range, and the results are distributed in the optimization target, so the rib-groove filling consistency effect and robustness are preferred. The minimum distance analysis of material filling is shown in Fig. 16(e).
The die stress analysis of ETB shows that, when all three ribs are filled, the rib 3 groove and the region from rib 3 to rib 2 have a strong stress concentration, and the maximum stress value is close to 700 MPa, which has exceeded the strength limit of 565 MPa of K403 alloy at 970 ℃, as shown in Fig. 17(a). It is highly likely to produce crack. From the stress analysis of the robust optimized UTB, it can be seen that the stress in each area of the die cavity is relatively uniform, and the maximum value does not exceed 500 MPa, so the risk of cracking is low. It further confirms that the billet Fig. 19 Physical experimental verifications of the deterministic optimization solution and robust optimization solution robust optimization is effective and feasible in this work. Figure 18 is given to analyze the forming load of Fig. 16 (d) and 16 (e), clarifying the variation trend of the forming load. The forming load of ETB reaches 134.2 tons, while the optimized UTB forming load is only 90.3 tons, which decreases by 32.7% compared with the ETB outcomes.
The optimization results are experimentally verified by physical simulation and finite element simulation, which are based on deterministic, robust optimization solutions. From the experimental results, for the deterministic optimization solution, as the deformation increases, the rib 2 is filled first and the underfilling rate is in a small range. As for the optimal billet obtained by robust optimization, the three rib grooves are filled simultaneously under increasing deformation, as shown in Fig. 19. The robust optimized billet obtained in this work greatly improves the forming quality of titanium alloy multi-rib eigenstructure with consistent filling, weakening the influence of uncertainty factors on the forming quality and making the billet fill the rib grooves almost simultaneously.

Conclusion
In this paper, the effect of billet sizes and uncertainty factors on the consistent filling of rib grooves on titanium alloy complex multi-rib eigenstructure are investigated, and the billet sizes are optimized by robust optimization. The following conclusions are then drawn: (1) By using numerical simulation of isothermal forging, the material flow characteristics and the filling sequence in the rib grooves of titanium alloy complex multi-rib eigenstructure have been investigated. It is observed that cross-flow at the web. And longitudinal flow of material at the rib-grooves can also be observed. When the first rib groove is filled and the upper die continues to stroke down, the cross-flow behavior of the material across the rib at the web is developed, resulting in a longitudinal backflow of the material in that first rib groove to be produced.
(2) The L25(5 6 ) orthogonal design was used for significance analysis of the uncertainty factors such as die manufacturing deviation, billet manufacturing deviation, and friction factor were screened out. By arranging the inside and outside arrays of experimental scheme with the Box-Behnken design combined with Uniform design, the robust optimization model based on the dual response surface method was established for the correlation between the mean and variance of the filling consist-ency of multi-rib eigenstructure and the billet sizes under the fluctuation of uncertainty factors.
(3) The robust optimal solution for billet sizes is solved by genetic algorithm. It is found that the maximum forming load of UTB is reduced by 45.57% after robust optimization. In this forming stage, the phenomenon of material cross-flow at the web is less; the neutral layer is always existing. (4) By comparing with the deterministic optimization, the robust optimized billet obtained in this work greatly improves the forming quality of titanium alloy multi-rib eigenstructure with consistent filling of rib grooves, weakening the influence of uncertainty factors on the forming quality and making the billet fill the rib grooves almost simultaneously.
Author contribution Tong Ding: investigation, software, data curation, experimental operation, writing of the original draft preparation, reviewing, editing, and partial fund acquisition. Ke Wei: funding acquisition and visualization. Yang Chao: investigation and validation. Haibing Tang: experimental support.

Declarations
Consent for publication All the authors agreed to publish the manuscript as a journal article.

Conflict of interest
The authors declare no competing interests.