As shown in Fig. 2, the structure of the workpiece before the finishing stage is defined as the transition structure, and it is critical for the overall machining distortion process and final-part distortion control. The newly increased distortion is defined as the difference in distortion before and after the finishing stage (Wang et al., 2021; Fan et al., 2023). Based on the aforementioned analysis, this paper proposes a new idea to control the machining distortion by optimizing the design of the transition structure: the transition structure is designed so that its distortion is equal to the expected final-part distortion. At this point in the machining process, loosening the fixture, cutting off the existing distortion, and obtaining a new datum can make the newly increased distortion at the finishing stage almost zero.

Controlling machining distortion through optimal transition structure design includes creating an appropriate transition structure that matches the load and stiffness distribution, forming the expected machining distortion and removing it before the finishing stage. Establish the optimization model with the expected distortion as the target, give the unit material residual stress and other properties, minimize the difference between the transition structure's distortion and the final part's distortion as the objective function, and constrain the material amount to achieve the optimal design of the transition structure.

The research object of this paper is the long beam part of identical section and its left–right symmetric structure; the optimization design of the transition structure can be optimized only for half of the cross-section, which will significantly improve the computational efficiency. As shown in Fig. 3, the cross-section of the blank structure is discretized into several units, and the length and width of each unit are set as a and b, respectively. The blue area in the figure shows the area where the final part is located. The total number of units contained in the transition structure needs to be nominated before optimization, and its ratio to the total number of units in the section is the volume retention ratio. The cross-section along the thickness of the blank is divided into n layers. Since the initial residual stress of the blank is distributed along the thickness direction, the initial residual stress contained in each cell in the same layer contributes equally to the machining distortion. In the transition structure, the number of units finally retained in layer i is defined as xi.

For the final part, its expected distortion wfinal can be calculated according to Eq. (1). For the distortion of the transition structure in the optimization model, the detailed derivation process is as follows.

The distance between the centre of the transition structure and the bottom surface can be expressed as follows: 3y‾2=∑i=1nxi∫∫A(i)ydxdy/∑i=1nxiab, where Ai is the area where the ith cell is located.

The bending moment of the transition structure section can be expressed as the algebraic sum of the equivalent residual stresses in all units over the neutral axis moment. 4M=∑i=1nxi∫∫A(i)σeq(y-y‾2)dxdy, where σeq is the equivalent residual stress distribution along the thickness of the blank.

The moment of inertia of the transitional structure section can be expressed as the algebraic sum of the quadratic product of the area of each unit and the distance from each unit to the neutral axis. 5Itransition=∑i=1nxi∫∫A(i)(y-y‾2)2dxdy Combining Eqs. (2), (4) and (5), the distortion of the transition structure wtransitioncan be solved. Based on the above analysis, the optimization model can be described as 6min⁡f=|wfinal-wtransition|s.t.∑i=1nxi=N1xi≤xi+1,(1≤i≤s)xi≥xi+1,(t≤i≤n)xi=0orB1,(t≤i≤n)li≤xi≤ui,(i=1,2…,n), where 1≤i≤n and N1 is the total number of disposable units remaining, based on the volume removal ratio. Considering the actual machining situation, in the area below the web, the entire layer is set to be retained or removed during optimization design. Therefore, in the area below the web(t≤i≤n), xi=0 or B1, while in the area above the web(1≤i≤s), xi≤xi+1 is set to avoid units that are half overhanging and observe the units below. In addition, in order to prevent “empty” units in the optimized section profile, it is necessary to manually set all units in each layer to be contiguous, and the right side is adjacent to the final-part sidewall. Besides, ui is the upper bound of xi and li is the lower bound of xi.

Figure 4 depicts the algorithm flow proposed in this paper. The following are the steps in the entire method:

The blank section is created and divided into several equally sized units. The position of the part in the blank is entered, and the initial residual stress distribution curve inside the blank is introduced.

A simplified U-shaped beam part is built as illustrated in Fig. 5 to validate the proposed method in this study. The blank is a pre-stretched aluminium alloy plate made from AA6061-T651, and the part dimensions are shown in Fig. 5. The material properties are listed in Table 1 (Fan et al., 2021).

The crack compliance method (Yang et al., 2019) was used to determine the initial residual stress distribution. As is shown in Fig. 6, an electrode made of 0.18 mm molybdenum wire was used for crack making, and the specimen was cut 1 mm along the thickness direction each time. A KD7016 static strain gauge with a precision of 5±10-6ε was chosen for strain measurement, and the corresponding strain was recorded. Figure 7 depicts the calculated equivalent residual stress distribution along the thickness direction of the blank. The equivalent residual stress takes into account the effect of transverse stress on bending distortion (Ye et al., 2020). It is used in the topology optimization model and the finite-element simulation that follows.

As shown in Fig. 8, half of the blank cross-section was selected for optimization. The position of the bottom surface of the part with respect to the bottom surface of the blank was set to be 8 mm. At this point, the predicted distortion of the final part is -0.1782 mm. The entire blank cross-sectional area is discretely divided into n=110×30 rectangular units, each with a size of 1 mm × 1 mm. In traditional machining process design, the allowance after rough machining is generally selected as 2 mm in the thickness and height direction (Yang et al., 2019). Based on this, a volume retention ratio of 20 % was chosen for this study.

Based on actual machining conditions, at least one layer of units must be retained around the final part after rough machining. This part of the units, like the final part, belongs to the frozen area during optimization. After removing the units in the frozen area, a total of 164 units were used as optimization objects. Based on the actual machining, the diagonal area in the figure is selected as the design area. Based on the limit of the total number of optimization units and combined with the part structure, the optimization model of this part can be expressed as follows: 7min⁡f=|-0.1782-wtransition|s.t.∑i=110xi=164xi≤xi+1(i=1,2…,9)x1=0or5≤x1≤180≤x2≤200≤x3≤230≤x4≤270≤x5≤320≤x6≤410≤x7≤4211≤x8≤7111≤x9≤71x10=0or76. Here, the initial value of optimization is taken as x1=0, x2=11, x3=11, x4=11, x5=11, x6=11, x7=11, x8=11, x9=11 and x10=76. The optimization calculation process was performed using MATLAB 2022b software.

The optimization iteration process is shown in Fig. 9, and the objective function f decreases as the number of iterations increases. After 48 iterations, the algorithm converges. The minimum value of the objective function f is 0.0046 mm, and the corresponding distortion of the transition structure wtransition is -0.1737 mm. The optimal solution is x1=0, x2=6, x3=6, x4=6, x5=6, x6=16, x7=16, x8=16, x9=16 and x10=1. The shape of the transition structure corresponding to the optimal solution is shown in Fig. 10.

The ABAQUS software was used to validate the proposed methodology through finite-element method (FEM). The simulation model's material characteristics were defined in accordance with Table 1. In the simulation, the material's properties were set to be homogeneous and isotropic. The “birth and death” method was used to remove material in FEM (Su et al., 2013; Arrazola et al., 2013). C3D8R is used to mesh the blank, which has 19 358 cells and 386 626 nodes. The boundary condition, which constrained the workpiece's rigid motion, was adopted as the 3–2–1 constraint condition. The corresponding initial residual stresses were loaded into the finite-element model. The transition structure's simulation results are displayed in Fig. 11b, while the final part's simulation results are displayed in Fig. 11c. It can be seen from Fig. 10a that the initial residual stress added to the blank is uniformly distributed along the thickness direction, and the distortion trend of the transition structure and the final part is completely consistent. According to the simulation results, the transition structure's distortion value is -0.17 mm. Likewise, the final-part distortion value is -0.167 mm. For the final distortion of the part and the distortion value of the transition structure, the simulation results are highly consistent with the theoretical calculation values, with relative errors of 4.6 % and 3.8 %, respectively.

For further verification, machining experiments were carried out. The experiments were carried out on a VC-3016G high-speed vertical milling machine. A three-tooth carbide end mill with a diameter of 12  mm was used for the machining experiment. The machining was carried out at high speed with a spindle speed of 18 000 rpm, a feed of 7000 mm min-1 and a depth of cut of 2 mm. The machining process was cooled by cutting fluid to minimize the effect of cutting heat on the machining distortion. The clamping method is vise clamping. The machining process is shown in Fig. 12a. The corresponding transition structure was machined in the experiment, and the maximum distortion of the transition structure of the part was measured using a micrometer, as shown in Fig. 12b.

The part was re-clamped after the distortion of the transition structure was measured, and it was then machined to the final-part shape. To verify the validity of the experimental results, the machining tests were conducted three times with identical clamping settings and cutting parameters for each cut. The final distortion is determined as the average of the measured values after three machining operations.

A MISTRAL coordinate measuring machine (CMM) with 0.005 mm precision was used to measure the final part's contour. As indicated in Fig. 12c, distortion is measured every 5 mm along the midline of the part's bottom.

The experimental distortion is compared to the finite-element simulation and theoretical calculations in Fig. 13. The distortion curves of the parts from theoretical analysis and finite-element simulation are approximately smooth, as shown in the figure. The largest distortion is found at 175 mm in the midpoint of the theoretical analysis and finite-element simulation results. It can be seen that the finite-element simulation value is highly consistent with the theoretical calculation value. The distortion value and distribution of the actual experimental measurement value are in good agreement when compared to the first two; however, there is a little deviation at each measurement point. The comparison of maximum distortion values of the transition structure and final part is shown in Table 2. The maximum experimental distortion values of the transition structure and final part are -0.20 and -0.21 mm, respectively. The maximum distortion value of the experiment is slightly larger than the theoretical calculation value and simulation value. The transition structure and final part have acceptable maximum relative errors of 24.6 % and 23.1 %, respectively, between theoretical calculation results and experimental results. Error generation is primarily caused by three factors (Ye et al., 2020; Weber et al., 2022; Huang et al., 2022): (1) the residual stress brought on by machining is not considered; (2) the effects of clamping and cutting load on machining distortion are not considered; (3) a measurement error. The results of residual stress measurements and CMM measurements are influenced by the instrument's accuracy.