Research on Optimization of Insert Spatial Mounting Posture for Improved Tool Life and Surface Quality of an Indexable Shallow-Hole Drill 

Highlights

What are the main findings?

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A parametric model of insert posture was developed for quality and life improvement.

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Radial force and temperature were selected to assess surface integrity and wear.

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Optimal posture was determined using improved LO-NSGA-II and entropy-TOPSIS.

What are the implications of the main findings?

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Surface roughness reduced by 66%–74% after optimization.

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Tool life increased by 40%–60% with reduced hole enlargement.

Abstract

To address rapid tool wear and unstable hole surface quality during roughing and semi-finishing operations using indexable shallow-hole drills, an optimization study on the spatial mounting posture of the insert is conducted, aiming to improve tool life and machined surface quality. Considering that tool life and surface quality are significantly influenced by cutting force and cutting temperature, radial cutting force and cutting temperature are selected as the multi-objective optimization criteria. A mapping model between the insert mounting posture parameters and cutting performance metrics is established. An improved LO-NSGA-II algorithm is employed to perform multi-objective optimization, yielding a Pareto-optimal solution set, and the entropy weighted-TOPSIS method is subsequently applied to determine the optimal insert mounting posture. Experimental results demonstrate that the optimized spatial mounting posture significantly enhances the overall cutting performance of the tool. Compared with the non-optimized tool, the optimized configuration exhibits a significant extension in tool life and a notable improvement in machined hole surface quality. This study provides an effective methodology for the structural optimization design of indexable shallow-hole drills.

1. Introduction

As modern manufacturing continues to demand higher efficiency and quality in hole-making, drilling, as one of the core processes in mechanical manufacturing, plays a role in the machining efficiency and quality of critical components across sectors such as aerospace, energy equipment, and high-strength steel frames for new energy vehicles [1]. Since the introduction of indexable shallow hole drills in the 1970s, these instruments have become integral to industrial drilling applications. The advent of replaceable inserts, marked by exceptional wear resistance and adaptable cutting performance, has solidified the significance of these tools in metal material processing [2]. In the broader context of mechanical part manufacturing, drilling operations account for approximately 25% of the total process time. Statistical evidence demonstrates that within the automotive manufacturing industry, drilling operations represent approximately 50% of the mechanical processing workflow. Within the aerospace manufacturing sector, the proportion of drilling operations is even more significant. Drilling is one of the most complex machining processes; the indexable shallow hole drill and its drilling process are shown in Figure 1. Particularly in the machining of lightweight, high-strength materials such as aerospace aluminum alloys, drilling quality directly affects component assembly precision and service reliability. In automated drilling systems, fluctuating cutting forces, localized heat accumulation, and tool wear are critical factors influencing machining stability and production efficiency [3]. Meanwhile, drilling-induced mechanical loads and thermal effects may lead to microstructural evolution and damage progression within the material, thereby deteriorating hole quality and compromising component service performance [4]. It is precisely due to these factors that research into the improvement of drills and the drilling process has been a continuous area of investigation [5]. Indexable shallow hole drills, also referred to as shallow hole drills, U-drills, or heavy-duty drills, have been shown to achieve substantial improvements in machining efficiency, reduction in tooling costs, and augmentation in productivity through their implementation.

In recent years, research on indexable shallow-hole drills has mainly focused on numerical simulation and parameter optimization. Liu et al. [7,8] investigated the effects of insert geometric angles, such as the inclination angle and principal cutting edge angle, on cutting-force variation through geometric modeling and simulation. He et al. [9] conducted drilling experiments on 45 carbon steel to compare straight-edge and wavy-edge indexable shallow-hole drills. They found that the inner chips were predominantly transversely curled, whereas the outer chips mainly exhibited upward curling. Compared with the straight-edge drill, the wavy-edge drill produced chips with denser free-surface microtopography, greater equivalent thickness, and smaller curl diameters, resulting in better chip-breaking and chip evacuation performance. Lian et al. [10] investigated the effects of spiral flute cross-sectional geometry and helix angle on the stiffness and chip evacuation performance of indexable shallow-hole drills through theoretical analysis, numerical simulation, and experimental validation. A variable-helix-angle structure was optimized, resulting in significant improvements in drilling stability, chip evacuation efficiency, and hole quality. Ma et al. [11] replaced the conventional chip flute of a 5D indexable shallow-hole drill with a trapezoidal chip flute to enhance both stiffness and chip space. Simulation and drilling experiments on 45 steel showed that the modified drill improved hole accuracy, drilling stability, and chip evacuation performance, while producing short chips that were easier to evacuate. Xiang et al. [12] investigated hole diameter errors caused by tool deflection during drilling using radial force analysis. Employing geometric models, finite element simulations, and linear regression methods, they developed a hole diameter error prediction model, thereby enabling precise control of machined hole diameter through cutting parameter adjustments. In their study, Li et al. [13] investigated synchronized wear design for inner and outer inserts of indexable drills. They utilized finite element simulation and material constitutive models to achieve this objective. By optimizing cutting-edge parameters and initial radial force, synchronous wear of the inner and outer inserts was achieved during the drilling of Q345 steel. Ji [14] established a geometric model and a radial-force calculation model for a double-W indexable shallow-hole drill. The validity of the proposed models was verified through drilling experiments on aluminum alloy. In addition, optimization of the insert installation parameters further confirmed the resulting improvement in drilling performance. Okada et al. [15] discovered that when machining titanium alloys and nickel-based alloys, indexable shallow-hole drills exhibited lower cutting forces and cutting temperatures than conventional twist drills while achieving comparable surface roughness. However, severe coating delamination and insert wear were observed, indicating the need for lubrication optimization to extend tool life. Liu et al. [16] examined the influence of chip breaker groove parameters on cutting forces during the drilling of H13 steel. Optimal groove parameter combinations were identified through combined experimental and numerical optimization, significantly reducing axial cutting force and enhancing chip-breaking performance. Zheng [17] employed a systematic investigative approach to examine the drilling mechanism and cutting performance of indexable insert drill bits. This investigation encompassed three distinct approaches, namely cutting force modeling, chip deformation analysis, and tool wear experiments. Lu [18] derived the formula for the offset angle of the inner insert in an indexable shallow-hole drill and determined its theoretical value. Comparative experiments confirmed that increasing the offset angle led to higher cutting force, torque, and machined hole diameter. A proper selection of this angle can reduce cutting load and improve hole accuracy. Parsian et al. [19] advanced a method for the prediction of cutting forces, which was mechanically modeled, for indexable shallow hole drills. Cutting force coefficients were obtained through experimental means, followed by the modeling of both inner and outer inserts. This model enables the prediction of cutting forces and torque, providing a theoretical basis for the optimization of tool design and machining processes. Jiang et al. [20], based on chip curl radius analysis, optimized the groove parameters of indexable shallow-hole drill inserts through orthogonal experiments and finite element simulations. The optimal parameter combination was experimentally verified to significantly reduce chip curl radius and improve drilling efficiency and chip evacuation performance.

In summary, existing research has primarily focused on optimizing single parameters or localized structures, with limited systematic studies on the spatial mounting posture of inserts. However, for indexable shallow hole drills employing asymmetric installation configurations of inner and outer inserts, the spatial mounting posture parameters exhibit multidimensional and strongly coupled characteristics. Their rationality directly impacts radial force balance and cutting temperature distribution during the machining process. When these parameters are improperly set, excessive radial resultant forces and elevated cutting temperatures may occur. Since tool life and hole surface quality are influenced by the coupled effects of cutting forces, friction behavior, and thermal effects, establishing models directly targeting either life or surface quality presents significant challenges. Previous research [21] has modeled and analyzed the cutting process by focusing on intermediate physical quantities such as cutting forces, energy consumption, and cutting heat. Therefore, this study takes the spatial mounting posture parameters of the inserts as the optimization target. From a structural perspective, a parametric model of insert spatial mounting posture is established to reveal the mechanistic relationship between posture parameters and force-thermal coupling behavior during machining. On this basis, surrogate models for radial resultant force and cutting temperature are developed. A modified LO-NSGA-II algorithm is then employed for multi-objective optimization, and high-quality compromise candidate solutions are selected using the entropy weight-TOPSIS method. Finally, drilling experiments are conducted to verify the effectiveness of the optimized schemes in improving tool life, hole surface quality, and hole diameter accuracy, thereby achieving systematic optimization of the indexable shallow-hole drill structure.

2. Parametric Description of Insert Spatial Mounting Posture

Due to the complex structure of the cutting section in indexable shallow hole drills, any change in tool parameters may significantly impact the overall structural model, potentially leading to model failure. Simultaneously, adjustments to tool design parameters often necessitate redesigning production drawings, imposing substantial time costs on designers. Therefore, this paper systematically analyzes the structural characteristics of indexable shallow hole drills, focusing on their key components. This includes examining the geometric features and functions of the cutting section and chip flutes, ultimately establishing a mathematical model for the critical structural elements of indexable shallow hole drills.

2.1. Structural Analysis of the Indexable Shallow-Hole Drill

The analysis of the overall structure and key components of indexable shallow hole drills aims to identify the primary geometric features and structural constraints affecting the spatial mounting posture of inserts. This provides the structural foundation for subsequent parametric description and optimization analysis of the inserts’ spatial mounting posture. As shown in Figure 2, the overall structure of indexable shallow hole drills comprises a cutting head section, a chip-evacuation middle section, a shank section at the rear, and an internal cooling channel running throughout. The structural features of indexable insert drill are shown in Table 1.

Table 1. Structure feature table of indexable insert drill.

Key Feature	Symbol	Key Feature	Symbol
Drill body diameter	Db	Chip flute length	L8
Cutting diameter	Dc	Helix angle of chip flute	γ
Flange diameter	Df	Principal cutting edge angle of inner insert	α1
Shank diameter	Dcon	Principal cutting edge angle of outer insert	α0
Overall length	L4	Height difference between inner and outer insert cutting edges	H3
Overhang length	L5	Center offset of inner insert cutting edge	E
Shank length	L6	Radial runout of outer insert	H2
Flange length	Lf	Radial runout of inner insert	H1
Functional length	L7	Included angle between inner and outer inserts	β

Table 1. Structure feature table of indexable insert drill.

Key Feature	Symbol	Key Feature	Symbol
Drill body diameter	Db	Chip flute length	L8
Cutting diameter	Dc	Helix angle of chip flute	γ
Flange diameter	Df	Principal cutting edge angle of inner insert	α1
Shank diameter	Dcon	Principal cutting edge angle of outer insert	α0
Overall length	L4	Height difference between inner and outer insert cutting edges	H3
Overhang length	L5	Center offset of inner insert cutting edge	E
Shank length	L6	Radial runout of outer insert	H2
Flange length	Lf	Radial runout of inner insert	H1
Functional length	L7	Included angle between inner and outer inserts	β

(1)

The cutting section is the most critical component of an indexable shallow-hole drill, and its cutting performance directly determines the overall performance of the tool. As shown in Figure 3, This section typically consists of two cemented carbide inserts, which are fixed onto the drill body by clamping screws according to a specific mounting posture. In the radial direction of the drill body, the inner insert is positioned below the rotational center. The parallel distance between the inner insert and the plane passing through the tool axis is defined as the radial offset of the inner insert, denoted as H₁. The presence of H₁ prevents the cutting edge from reaching zero cutting speed during feeding and avoids the formation of a central boss at the hole bottom, thereby providing a certain self-centering effect. The outer insert is located above the rotational center, and the corresponding parallel distance is defined as the radial offset of the outer insert, denoted as H₂. The introduction of H₂ mainly aims to enhance the structural strength of the drill head. To balance the cutting forces acting on the inner and outer inserts during the drilling process, an included angle β between the inner and outer inserts is designed to reduce friction between the drill head and the hole wall caused by radial force imbalance. In the axial direction of the drill body, the cutting edge of the inner insert is positioned higher than that of the outer insert by a certain distance, denoted as H₃. During drilling, the inner insert comes into contact with the workpiece first and initiates the cutting action. The offset of the inner insert cutting edge relative to the rotational axis is denoted as E. The selection of E is mainly intended to prevent the mechanically weaker insert nose from participating in cutting; therefore, E is typically chosen to be greater than three-quarters of the insert nose radius. In addition, the principal cutting edge angles of the inner and outer inserts are denoted as α₁ and α₀, respectively.

(2)

The chip flute is the chip-evacuation structure of indexable shallow hole drills, consisting of both inner and outer insert chip flutes. Owing to the asymmetric mounting posture of the cutting inserts, the corresponding chip flutes also exhibit an inherently asymmetric spatial configuration. The chip flute geometry consists of straight segments smoothly connected to curved segments. From a manufacturing perspective, the chip flutes of indexable shallow hole drills are machined on CNC machining centers using ball-nose end mills according to pre-programmed sequences. This machining strategy yields a geometrically continuous flute profile with a simple contour and allows the flute to be formed in one or two milling passes. As a result, machining steps and tool changes can be effectively reduced, thereby lowering tooling costs while maintaining tool performance. As shown in Figure 4, the cross-sectional profile of the chip flute is governed by the shape and cutting position of the milling cutter, it directly serves as the reference for insert support and positioning. Consequently, variations in the flute cross-sectional geometry influence the mounting posture of the inserts within the drill body, which in turn modify the effective rake angle, clearance angle, and spatial orientation of the cutting edges. These changes ultimately affect the distribution of cutting forces and the stability of the drilling process.

(3)

In the design of drilling tools, common internal coolant channel configurations include straight and S-shaped channels. Straight internal coolant channels feature a simple structural design, typically consisting of linear passages that facilitate stable coolant flow and dimensional control during machining. They are also associated with lower manufacturing costs and higher production efficiency. In contrast, S-shaped internal coolant channels are generally formed through a thermal torsion process, providing more uniform cooling performance; however, their manufacturing complexity is higher, leading to increased production costs.

Considering cost-effectiveness and the requirements of experimental validation, and given that this study primarily focuses on optimizing the spatial installation orientation of the inserts—aimed at improving cutting stability and machining quality by modifying force distribution and thermal behavior—the influence of coolant channel configuration on the optimization results is considered limited. Therefore, a straight internal coolant channel configuration was selected. The coolant passages were fabricated using electrical discharge machining (EDM) to ensure dimensional accuracy and to meet the basic cooling performance requirements of the experiments.

2.2. Mathematical Modeling of the Indexable Shallow-Hole Drill

To establish a unified coordinate system framework for the indexable shallow-hole drill, as illustrated in Figure 5, a right-handed Cartesian coordinate system denoted as PCS-XYZ is defined as the global coordinate system of the tool. The origin of this coordinate system is located at the center of the bottom surface of the drill flange, and the rotational axis of the drill is defined as the Z-axis. A coordinate system denoted as CIP-XYZ is further established for the cutting section of the tool. The origin of this coordinate system is positioned at the center of the radial plane where the shallow-hole drill first comes into contact with the workpiece along the Z-axis direction. In addition, the cutting insert coordinate system is defined as mcs-uvw. Based on this system, CSW1-uvw and CSW0-uvw are established as the mounting coordinate systems for the inner and outer inserts, respectively, which are used to define their installation positions.

One of the key mathematical models used to describe the chip flute and internal cooling channel of an indexable shallow-hole drill is the helical curve equation, as illustrated in Figure 6. A helical curve is generated by a point P on the surface of a cylinder undergoing helical motion around the central axis. The helix angle is defined as the angle between the tangential velocity vector of the cutting edge during its helical motion and the generatrix of the cylindrical surface.

When constructing the parametric model, the parametric equations of the helical curve with respect to the parameter t are defined in the modeling software as follows:

$$r=\left\{\begin{array}{l}\cos \phi D_{\mathrm{b}}/2\hfill \\ \sin \phi D_{\mathrm{b}}/2\hfill \\ (L_5-L_3)(t-1)\hfill \end{array}\right.,\phi \in [0,2\pi ],t\in [0,1]$$

When the spiral rotates about the Z-axis, the tangent vector d_r at point P and the direction vector z_r of the generatrix can be expressed as:

$$\left\{\begin{array}{l}{d}_{\mathrm{r}}=r_{\mathrm{t}}\mathrm{d}t+r_{\phi }\mathrm{d}\phi \hfill \\ z_{\mathrm{r}}=r_{\mathrm{t}}zt\hfill \\ {\mathrm{r}}_{\mathrm{t}}=\{0,0,L_5-L_3\}\hfill \\ r_{\phi }=\{-\sin \phi D_{\mathrm{b}}/2,\cos \phi D_{\mathrm{b}}/2,0\}\hfill \end{array}\right.$$

The helix angle γ can be determined using the first fundamental form of the surface:

$${\cos }^2\gamma ={\displaystyle \frac{{(L_5-L_3)}^{2}d{t}^2}{{(L_5-L_3)}^{2}d{t}^2+D_{\mathrm{b}}^2/4d{\phi }^2}}$$

$$d\phi ={\displaystyle \frac{2(L_5-L_3)tan\gamma }{D_b}}dt$$

Integrating the above equation and substituting $t = 0, \phi = 0$ into the equation yields the expression:

$$\phi ={\displaystyle \frac{2(L_5-L_3)\tan \gamma }{D_{\mathrm{b}}}}t$$

Accordingly, the parametric equations of the helical curve of the indexable shallow-hole drill are obtained as follows:

$$\left\{\begin{array}{c}{x}_t=\cos \left[360t\left(L_5-L_3\right)\tan \gamma /\left(\pi D_{\mathrm{b}}\right)\right]D_{\mathrm{b}}/2\\ y_t=\sin \left[360t\left(L_5-L_3\right)\tan \gamma /\left(\pi D_{\mathrm{b}}\right)\right]D_{\mathrm{b}}/2\\ z_t=\left(L_5-L_3\right)\left(t-1\right)\end{array}, t\in (0,1)\right.$$

During machining, the chip flute of an indexable shallow-hole drill is generated by a ball-end milling cutter following a helical tool path. To facilitate the parametric modeling of the indexable shallow-hole drill, a mathematical model of the key points on the chip flute cross section is established in this study based on existing research. As shown in Figure 7, the chip flute cross section perpendicular to the drill body, with its center located at (0, 0, L3–L5), corresponds to the initial cross section of the chip flute.

The coordinates of the key points A, B, C, D and F on the chip flute cross section can be expressed as follows:

$$\left\{\begin{array}{l}({\mathrm{X}}_{\mathrm{A}},{\mathrm{Y}}_{\mathrm{A}},{\mathrm{Z}}_{\mathrm{A}})=(\begin{array}{c}(L_1+d_1)\sin \lambda -d_1\times \sin \theta +L_2\cos \delta ,\\ -(L_1+d_1)\cos \lambda +d_1\times \cos \theta +L_2\sin \delta ,L_3-L_5\end{array})\\ ({\mathrm{X}}_{\mathrm{B}},{\mathrm{Y}}_{\mathrm{B}},{\mathrm{Z}}_{\mathrm{B}})=((\begin{array}{c}(L_1+d_1)\sin \lambda -d_1\times \sin \theta ,\\ -(L_1+d_1)\cos \lambda +d_1\times \cos \theta ,L_3-L_5\end{array})\\ ({\mathrm{X}}_{\mathrm{C}},{\mathrm{Y}}_{\mathrm{C}},{\mathrm{Z}}_{\mathrm{C}})=(\begin{array}{c}(L_1+d_1)\sin \lambda +d_1\times \sin \theta ,\\ -(L_1+d_1)\cos \lambda -d_1\times \cos \theta ,L_3-L_5\end{array})\\ ({\mathrm{X}}_{\mathrm{D}},{\mathrm{Y}}_{\mathrm{D}},{\mathrm{Z}}_{\mathrm{D}})=(\begin{array}{c}(L_1+d_1)\sin \lambda +d_1\times \sin \theta +L_2\cos \delta ,\\ -(L_1+d_1)\cos \lambda -d_1\times \cos \theta +L_2\sin \delta ,L_3-L_5\end{array})\\ ({\mathrm{X}}_{\mathrm{F}},{\mathrm{Y}}_{\mathrm{F}},{\mathrm{Z}}_{\mathrm{F}})=\left(\begin{array}{c}(L_1+d_1)\sin \lambda ,-(L_1+d_1)\cos \lambda ,L_3-L_5\end{array}\right)\end{array}\right.$$

After the spatial mounting posture parameters of the cutting insert are determined, coordinate transformations are required to mount the insert onto the drill body. During this process, the coordinate system of the cutting insert, denoted as mcs–uvw, is first aligned with the global tool coordinate system PCS–XYZ. The insert is then rotated about the X-, Y-, and Z-axes by the corresponding angles, with clockwise rotation defined as the positive direction. The rotation matrices about the X-, Y-, and Z-axes are expressed as R_X(θ₁), R_Y(θ₂), R_Z(θ₃), respectively [22]:

$${\mathrm{R}}_{\mathrm{X}}\left({\theta }_1\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\theta }_1& \sin {\theta }_1\\ 0& -\sin {\theta }_1& \cos {\theta }_1\end{array}\right]$$

$${\mathrm{R}}_{\mathrm{Y}}\left({\theta }_2\right)=\left[\begin{array}{ccc}\cos {\theta }_2& 0& -\sin {\theta }_2\\ 0& 1& 0\\ \sin {\theta }_2& 0& \cos {\theta }_2\end{array}\right]$$

$${\mathrm{R}}_{\mathrm{Z}}\left({\theta }_3\right)=\left[\begin{array}{ccc}\cos {\theta }_3& \sin {\theta }_3& 0\\ -\sin {\theta }_3& \cos {\theta }_3& 0\\ 0& 0& 1\end{array}\right]$$

First, the coordinate transformation for the installation of the inner insert is performed. The inner insert is initially rotated about the Z-axis by an angle of 3π/2 − 1, followed by a rotation about the X-axis by π/2. Finally, the inner insert is rotated about the Z-axis by the specified angle. The corresponding rotation matrices for the inner insert are given as follows:

$$\begin{array}{c}{\mathrm{R}}_1={\mathrm{R}}_Z\left(-\beta \right){\mathrm{R}}_{\mathrm{X}}\left(\pi /2\right){\mathrm{R}}_Z\left(3\pi /2-{\alpha }_1\right)=\\ \left[\begin{array}{ccc}-\cos \beta \sin {\alpha }_1& -\cos \beta \cos {\alpha }_1& -\sin \beta \\ -\sin \beta \sin {\alpha }_1& -\sin \beta \cos {\alpha }_1& \cos \beta \\ -\cos {\alpha }_1& \sin {\alpha }_1& 0\end{array}\right]=\\ \left[\begin{array}{ccc}{m}_{11}& m_{12}& m_{13}\\ m_{21}& m_{22}& m_{23}\\ m_{31}& m_{32}& m_{33}\end{array}\right]\end{array}$$

The coordinate transformation for the installation of the outer insert is performed as follows. The outer insert is first rotated about the Z-axis by an angle of α₀ − π/2, followed by a rotation about the X-axis by an angle of −π/2. The corresponding rotation matrices for the outer insert are given as follows:

$$\begin{array}{c}{\mathrm{R}}_0={\mathrm{R}}_{\mathrm{X}}\left(-\pi /2\right){\mathrm{R}}_{\mathrm{Z}}\left({\alpha }_0-\pi /2\right)=\\ \left[\begin{array}{ccc}\sin {\alpha }_0& -\cos {\alpha }_0& 0\\ 0& 0& -1\\ \cos {\alpha }_0& \sin {\alpha }_0& 0\end{array}\right]=\left[\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right]\end{array}$$

To determine the precise mounting position of the insert, a mathematical model describing the installation position of the insert relative to the drill body is established based on existing studies using a vector-based approach. As illustrated in Figure 5, a vector transformation is applied from the reference point of the insert before installation to the corresponding reference point after installation on the drill body. Taking the center point C₁ of the inner insert as the terminal point, the position vectors of points A₁, B₁ can be expressed as follows:

$$\overrightarrow{{\mathrm{A}}_1{\mathrm{C}}_1}=\left[\begin{array}{c}-\left(d/2-r+r\cos {\alpha }_1\right)\\ d/2-r+r\sin {\alpha }_1\\ 0\end{array}\right]=\left[\begin{array}{c}-a_1\\ b_1\\ 0\end{array}\right]$$

$$\overrightarrow{{\mathrm{B}}_1{\mathrm{C}}_1}={\left[\begin{array}{ccc}{b}_1& a_1& 0\end{array}\right]}^{\mathrm{T}}$$

Similarly, the position vectors of points A₀, B₀, taking the center point C₀ of the outer insert as the terminal point, can be obtained as follows:

$$\overrightarrow{{\mathrm{A}}_0{\mathrm{C}}_0}=\left[\begin{array}{c}d/2-r+r\cos {\alpha }_0\\ d/2-r+r\sin {\alpha }_0\\ 0\end{array}\right]=\left[\begin{array}{c}{a}_0\\ b_0\\ 0\end{array}\right]$$

$$\overrightarrow{{\mathrm{B}}_0{\mathrm{C}}_0}={\left[\begin{array}{ccc}-b_0& a_0& 0\end{array}\right]}^{\mathrm{T}}$$

To determine the coordinates of the insert mounting points C₁′ and C₀′ in the tool coordinate system, the inner and outer inserts are respectively transformed using the insert coordinate transformation matrices R₁ and R₀. Accordingly, the position vectors of points A₁′, B₁′, as well as A₀′, B₀′, with C₁′ and C₀′ taken as the terminal points, are obtained as follows:

$$\overrightarrow{{\mathrm{A}}_1^{\prime }{\mathrm{C}}_1^{\prime }}={\mathrm{R}}_1\overrightarrow{{\mathrm{A}}_1{\mathrm{C}}_1}=\left[\begin{array}{c}-m_{11}{a}_1+m_{12}{b}_1\\ -m_{21}{a}_1+m_{22}{b}_1\\ -m_{31}{a}_1+m_{32}{b}_1\end{array}\right]$$

$$\overrightarrow{{\mathrm{B}}_1^{\prime }{\mathrm{C}}_1^{\prime }}={\mathrm{R}}_1\overrightarrow{{\mathrm{B}}_1{\mathrm{C}}_1}=\left[\begin{array}{c}{m}_{11}{b}_1+m_{12}{a}_1\\ m_{21}{b}_1+m_{22}{a}_1\\ m_{31}{b}_1+m_{32}{a}_1\end{array}\right]$$

$$\overrightarrow{{\mathrm{A}}_0^{\prime }{\mathrm{C}}_0^{\prime }}={\mathrm{R}}_0\overrightarrow{{\mathrm{A}}_0{\mathrm{C}}_0}=\left[\begin{array}{c}{n}_{11}{a}_0+n_{12}{b}_0\\ n_{21}{a}_0+n_{22}{b}_0\\ n_{31}{a}_0+n_{32}{b}_0\end{array}\right]$$

$$\overrightarrow{{\mathrm{B}}_0^{\prime }{\mathrm{C}}_0^{\prime }}={\mathrm{R}}_0\overrightarrow{{\mathrm{B}}_0{\mathrm{C}}_0}=\left[\begin{array}{c}-n_{11}{b}_0+n_{12}{a}_0\\ -n_{21}{b}_0+n_{22}{a}_0\\ -n_{31}{b}_0+n_{32}{a}_0\end{array}\right]$$

Based on the structural analysis of the indexable shallow-hole drill shown in Figure 5, the Z-coordinate of point A₁′ in the global tool coordinate system can be expressed as -L₅, while the X-coordinate of point B₁′ is given by E. Accordingly, the coordinates of the inner insert mounting point C₁′ on the indexable shallow-hole drill can be obtained as follows:

$$\begin{array}{c}\left({\mathrm{X}}_{{\mathrm{C}\mathrm{l}}^{\prime }},{\mathrm{Y}}_{{\mathrm{C}\mathrm{l}}^{\prime }},{\mathrm{Z}}_{{\mathrm{C}\mathrm{l}}^{\prime }}\right)=(E+m_{11}{b}_1+m_{12}{a}_1,\\ -H_{\mathrm{l}}/\cos \beta +{\mathrm{X}}_{{\mathrm{C}\mathrm{l}}^{\prime }}\tan \beta ,-L_5-m_{31}{a}_1+m_{32}{b}_1)\end{array}$$

Similarly, in the global tool coordinate system, the Z-coordinate of point A₀′ can be expressed as H₃–L₅, the X-coordinate of point B₀′ as D_c/2, and the Y-coordinate of point C₀′ as -H₂. Based on these geometric relationships, the coordinates of the outer insert mounting point C₀′ can be obtained as follows:

$$\begin{array}{c}\left({\mathrm{X}}_{\mathrm{C}{0}^{\prime }},{\mathrm{Y}}_{\mathrm{C}{0}^{\prime }},{\mathrm{Z}}_{\mathrm{C}{0}^{\prime }}\right)=(D_{\mathrm{c}}/2-n_{11}{b}_0+n_{12}{a}_0,\\ -H_2,H_3-L_5+n_{31}{a}_0+n_{32}{b}_0)\end{array}$$

Accordingly, the translation matrices of the inner and outer inserts can be obtained as follows:

$$T_i={\left[\begin{array}{ccc}{\mathrm{X}}_{{Ci}^{\prime }}& {\mathrm{Y}}_{{Ci}^{\prime }}& {\mathrm{Z}}_{{Ci}^{\prime }}\end{array}\right]}^{\mathrm{T}}\left(\begin{array}{c}i=0,1\end{array}\right)$$

Therefore, the coordinate transformation matrices describing the installation process of the inner and outer inserts can be obtained as follows:

$$M_i=\left[\begin{array}{cc}{R}_i& T_i\\ 0& 1\end{array}\right]\left(i=0,1\right)$$

The aforementioned coordinate transformations and vector expressions enable a parametric description of the spatial installation orientation of the cutting insert. These geometric relationships are not only used to analyze variations in the insert’s spatial positioning but also provide the foundation for selecting decision variables in subsequent multi-objective optimization. During the optimization process, parameters such as the principal cutting edge angle, included angle, and eccentricity relative to the drill center are defined based on this mathematical model and introduced into the optimization algorithm as design variables. The optimized orientation parameters are subsequently implemented in the actual tool structure and validated through drilling experiments.

2.3. Parametric Modeling of the Indexable Shallow-Hole Drill

In the field of cutting tool design and manufacturing, parametric modeling provides an efficient means for rapidly constructing tool models through the adjustment of key geometric parameters. By adopting a parametric modeling approach, the geometry and dimensions of a tool are defined using a set of predefined parameters and design rules. Once established, these parameters and rules can be readily reused across different design cases, thereby avoiding redundant modeling work and significantly improving design efficiency. This parameter-driven approach not only accelerates the tool design process but also ensures consistency and accuracy in the resulting models.

The parametric design methodology for indexable shallow-hole drills is illustrated in Figure 8. To enable parametric modeling of the indexable shallow-hole drill, a systematic analysis of its structural features is first conducted to identify the geometric characteristics and functional attributes of each constituent component. Based on ISO 13399 and the practical constraints encountered during the design process, the key geometric parameters of the tool are identified and extracted. Subsequently, the coordinate system information is integrated with the geometric feature parameters to establish the mathematical model of the tool. Finally, in accordance with the relevant requirements of the ISO 13399 sub-standards, a three-dimensional parametric model and the corresponding two-dimensional engineering drawings are generated based on the developed mathematical model of the indexable shallow-hole drill.

Top-down assembly modeling is an approach that takes the overall assembly structure as its core. Beginning with the top-level assembly framework, the geometric features and spatial relationships of individual components and subassemblies are systematically defined within a global assembly environment. This method is typically based on a pre-established assembly layout model, in which assembly constraints are used to control the geometric characteristics and spatial positioning of all components.

ISO 13399 categorizes tool models into four categories: cutting, tool, adaptive, and assembly. ISO 13399-80:2017 specifies the color conventions and data exchange formats for three-dimensional tool models. As illustrated in Figure 9, during the design process of the indexable shallow-hole drill, the model assembly file (asm.prt) is first established. Using the ‘Create Part’ command, the tool item, cutting item, and assembly item are then sequentially generated. This process ultimately results in the creation of a three-dimensional parametric model for the entire tool assembly that complies with ISO 13399.

(1)

Tool item: As illustrated in Figure 10, parametric modeling of the tool item of the indexable shallow-hole drill starts with the construction of a cross-sectional sketch of the drill body, in which the key dimensions are constrained using parametric expressions. In this study, an indexable shallow-hole drill with an effective cutting diameter of 21 mm is selected as an example to develop the parametric model. The basic drill body geometry is first generated by applying a rotational feature to the cross-sectional sketch. Subsequently, fully constrained sketches of the insert pockets are created according to the geometric parameters of the insert slots, and material is removed from the drill body to form the mounting grooves for the inner and outer inserts, respectively. Based on the mathematical model of the key points on the chip flute cross section, the initial flute cross section and the corresponding helical curve are defined using parametric feature curves. The chip flute structure of the drill body is then generated by sweeping the flute cross section along the helical curve. Finally, the helical feature curve of the internal coolant channel is defined in a similar manner, and the spiral internal cooling channel of the indexable shallow-hole drill is established.

(2)

Cutting item: As shown in Figure 11, the cutting section of the indexable shallow-hole drill investigated in this study employs square cemented carbide inserts. After completing the modeling of the tool item, a new cutting item file is created in the assembly environment, inheriting the insert pocket features from the tool item. Based on these features, sketches are constructed in the cutting item to define the initial geometry of the insert. The basic geometric model of the insert is then generated by extrusion, and the detailed insert geometry is progressively refined by incorporating key parameters such as the rake angle, inscribed circle diameter, and chamfer radius.

(3)

Assembly item: During the modeling of the assembly item, the clamping screw is first constructed by inheriting the insert clamping hole features and establishing a cross-sectional sketch of the screw. This cross-sectional sketch serves as the foundation of the screw design and contains the primary dimensional information, including the head geometry, shank diameter, and thread profile. After the sketch is completed, the basic geometric model of the screw is generated using a rotational feature to define its overall form. Once the base model is established, key structural details are further refined through additional extrusion and rotational features.

3. Multi-Objective Optimization of the Spatial Mounting Posture of Inserts

Experiments on coating design optimization demonstrate that variations in interfacial friction conditions and cutting heat levels during machining are closely correlated with tool life and machined surface quality [23,24]. During drilling, the radial resultant force reflects the tool loading condition and cutting stability, whereas the cutting temperature represents the thermal level in the cutting zone. Both are closely related to tool wear evolution and hole machining quality. Since direct evaluation of tool life and surface roughness is costly and difficult to incorporate into iterative optimization, radial resultant force and cutting temperature were adopted in this study as surrogate indicators for multi-objective optimization, while tool life and hole surface quality were used as the final experimental validation criteria. In addition, excessively low cutting temperature or unreasonable loading conditions may also induce built-up edge formation or affect chip evacuation stability. Therefore, rather than simply pursuing a monotonic reduction in force and temperature, this study seeks a reasonable balance among drilling stability, thermal load control, and chip evacuation conditions.

3.1. Multi-Objective Optimization of Insert Spatial Mounting Posture Based on an Improved NSGA-II Algorithm

3.1.1. Constraints on the Spatial Mounting Posture of Inserts

In the design of the spatial mounting posture of inserts for indexable shallow-hole drills, the configuration of geometric parameters associated with the insert mounting posture has a significant influence on the dynamic characteristics and thermodynamic behavior of the cutting system. Appropriate optimization of these geometric parameters can effectively regulate the dynamic fluctuations of cutting forces and the distribution of the temperature field during the cutting process, thereby affecting tool life and machining accuracy [25]. Accordingly, the optimization of the insert spatial mounting posture in this study focuses on four parameters with sustained influence on cutting performance: the included angle between the inner and outer inserts β, the principal cutting edge angle of the inner insert α₁, the principal cutting edge angle of the outer insert α₀, and the over-center offset of the inner insert tip E.

To achieve systematic optimization of the spatial mounting posture of the inserts, geometric parameters that can comprehensively characterize the insert installation state are selected as the optimization design variables. The parameter vector describing the spatial mounting posture of the inserts is defined as follows:

$$X={\left[{\alpha }_1,{\alpha }_0,E,\beta \right]}^T$$

The optimization objective is to achieve optimal performance for indexable shallow hole drills under specified rotational speed, feed rate conditions, and given workpiece material, specifically manifested as minimizing the radial resultant force F and minimizing the drilling temperature T, as shown in the following equations:

$$F=\sqrt{{(F}_x^2{+F}_y^2)/2}$$

$$\left\{\begin{array}{l}\min \mathrm{T}\left(x_1,x_2,x_3,x_4\right)\hfill \\ \min \mathrm{F}\left(x_1,x_2,x_3,x_4\right)\hfill \end{array}\right.$$

where T denotes the maximum cutting temperature, F represents the radial resultant force during drilling, and x₁, x₂, x₃, x₄, correspond to the four spatial mounting posture parameters of the indexable shallow-hole drill inserts, namely the included angle between the inner and outer inserts β, the principal cutting edge angle of the inner insert α₁, the principal cutting edge angle of the outer insert α₀, and the over-center offset of the inner insert tip E.

During the optimization process, each design variable must satisfy engineering constraints such as tool structure design, assembly feasibility, and actual machining conditions. Their value ranges can be expressed as:

$$\left\{\begin{array}{c}{\alpha }_{1min}\le {\alpha }_1\le {\alpha }_{1max}\\ {\alpha }_{0min}\le {\alpha }_0\le {\alpha }_{0max}\\ E_{min}\le E\le E_{max}\\ {\beta }_{min}\le \beta \le {\beta }_{max}\end{array}\right.$$

According to recommendations from industrial design practice, the constraint ranges of the four spatial mounting posture parameters for a 21 mm diameter indexable shallow-hole drill are summarized in

Table 2. Optimization variables and their value ranges.

Variable	Description	Value Range
x1	Included angle between inner and outer inserts β	−5°~10°
x2	Principal cutting edge angle of inner insert α1	80°~90°
x3	Principal cutting edge angle of outer insert α0	80°~90°
x4	Over-center offset of inner insert tip E	0.3~1 mm

. The allowable range of the over-center offset of the inner insert tip should not compromise the complete engagement between the inner and outer inserts and is therefore typically set between 0.3 and 1.0 mm. During the design of indexable shallow-hole drills, the principal cutting edge angles of both the inner and outer inserts are generally selected within the range of 80° to 90°, while the recommended range for the included angle between the inner and outer inserts is −5° to 10°.

The spatial mounting posture parameters of cutting inserts exert a comprehensive influence on the mechanical and thermal behavior of the drilling process by altering the spatial force state of the cutting edges, the distribution of instantaneous uncut chip thickness, and the generation and transfer characteristics of cutting heat. The included angle between the inner and outer inserts and their principal cutting edge angles primarily affect the distribution and resultant characteristics of cutting forces in the radial and tangential directions. In contrast, the over-center offset of the insert tip influences local load levels and friction-induced heat generation by regulating the participation of cutting edges in low cutting speed regions.

Therefore, by treating the spatial mounting posture parameters of the inserts as optimization design variables and adopting the radial resultant force and cutting temperature as multi-objective optimization criteria, the influence of posture configuration on cutting stability and tool wear behavior can be effectively characterized from both mechanical and thermal perspectives. This approach further provides a solid basis for subsequent analysis of how posture optimization extends tool life and improves hole surface quality.

3.1.2. Multi-Objective Optimization of Insert Spatial Mounting Posture

The NSGA-II algorithm is a classical multi-objective optimization method and has been widely applied due to its elite selection strategy and fast non-dominated sorting mechanism. However, the standard NSGA-II algorithm still exhibits several limitations. First, randomness in population initialization: NSGA-II relies on random generation to initialize the population, which often leads to insufficient uniformity and diversity in the solution distribution. Second, limited local search capability: the offspring evolution process lacks effective directional guidance, making the algorithm prone to premature convergence to suboptimal solutions due to inadequate local exploitation ability. To overcome these limitations, an improved algorithm, termed LO-NSGA-II, is proposed by integrating Latin Hypercube Sampling (LHS) and Opposition-Based Learning (OBL) strategies. The flowchart of the LO-NSGA-II multi-objective optimization algorithm is shown in Figure 12.

• Population initialization. Latin Hypercube Sampling (LHS) has been widely adopted across numerous domains due to its high sampling efficiency. By partitioning each dimension of the multidimensional space into equiprobable intervals, LHS ensures a uniform distribution of sampling points along each dimension, thereby avoiding the clustering effects commonly observed in conventional random sampling. As a result, LHS improves the coverage of samples in the parameter space and enhances the comprehensiveness, uniformity, and diversity of the initial population. Opposition-Based Learning (OBL) generates solutions that are opposite to those in the current population, further enhancing the exploration capability and coverage of the search space. In this study, the population X generated by LHS and the opposite population Y generated by OBL are merged to form an expanded population R. Subsequently, the expanded population is ranked according to fitness values, and the top N individuals with better fitness are selected as the final initial population.
• Population update. During the population update stage, Latin Hypercube Sampling (LHS) and Opposition-Based Learning (OBL) strategies are again incorporated to enhance the exploration capability of the algorithm, facilitate escape from local optima, and maintain global search performance. The specific procedure is described as follows:

  (1)

  Elite population $E$ generation. The current-generation population $P$ is produced using the basic operators of NSGA-II. An elite preservation strategy is then employed to select individuals with higher fitness, thereby forming the elite population $E$.

  (2)

  Latin Hypercube Sampling (LHS). A new population $C$ is generated by applying Latin Hypercube Sampling to individuals from the current generation $E$. Owing to its dynamically uniform distribution property, LHS effectively guides the algorithm to explore regions of the search space that have not yet been covered, thereby expanding the search scope and reducing the likelihood of the algorithm becoming trapped in local optima.

  (3)

  Reverse Learning Strategy Generates Adversarial Populations: For each individual in the elite population $E$, an opposition-based learning strategy is applied to generate its corresponding opposite individual, thereby forming an opposite population $Q$. The introduction of the opposite population further expands the search space and enhances population diversity.

  (4)

  Merging and selection. The elite population $E$, the Latin Hypercube Sampling (LHS) population $C$, and the opposite population $Q$ are merged to form an expanded population $R=E\cup C\cup Q$. Subsequently, the expanded population is ranked according to fitness values, and the top $N$ individuals with superior fitness are retained as the next- generation population.

  (5)

  As the population gradually approaches the Pareto front, the contribution of the population generated by the LO-based update strategy to further exploration of the search space progressively diminishes. To reduce computational complexity, the LO-based update strategy is applied only during the early n generations of the population evolution (typically n = 5–10).

Figure 12. Flowchart of the LO-NSGA-II multi-objective optimization algorithm.

Figure 12. Flowchart of the LO-NSGA-II multi-objective optimization algorithm.

To further validate the effectiveness of the improved LO-NSGA-II algorithm in optimizing the spatial installation orientation of the cutting insert, a comparative analysis was conducted between the conventional NSGA-II algorithm and the improved LO-NSGA-II algorithm in terms of convergence performance and solution distribution uniformity. The algorithm parameter settings are listed in Table 3. Based on commonly used settings reported in previous studies, the parameters were further adjusted according to preliminary optimization results to achieve a balance between solution accuracy and computational efficiency. The Generational Distance (GD) metric was employed to evaluate the convergence of the algorithms toward the Pareto front, while the Spread (SP) metric was used to assess the uniformity of the obtained solution distribution. The comparison of SP values and GD values between the LO-NSGA-II algorithm and the NSGA-II algorithm is shown in Table 4 and Table 5.

Table 3. Parameter settings of the LO-NSGA-II algorithm and the NSGA-II algorithm.

	LO-NSGA-II	NSGA-II
Population size	100	100
Number of generations	100	100
Crossover rate	0.8	0.8
Mutation rate	0.2	0.2
n	6	-

Table 3. Parameter settings of the LO-NSGA-II algorithm and the NSGA-II algorithm.

	LO-NSGA-II	NSGA-II
Population size	100	100
Number of generations	100	100
Crossover rate	0.8	0.8
Mutation rate	0.2	0.2
n	6	-

Table 4. Comparison of SP values between the LO-NSGA-II algorithm and the NSGA-II algorithm.

Test Function	LO-NSGA-II			NSGA-II
	SP Max	SP Min	SP Mean	SP Max	SP Min	SP Mean
ZDT1	0.215842	0.010096	0.054121	0.325010	0.011754	0.100689
ZDT2	0.354121	0.012920	0.031242	0.410984	0.002348	0.064926
ZDT3	0.472145	0.008325	0.063258	0.453302	0.019654	0.071245
ZDT6	0.631241	0.006551	0.102144	0.638906	0.010315	0.124582
Viennet2	0.210110	0.045214	0.105866	0.150912	0.056752	0.113258
Viennet3	0.203251	0.038845	0.088254	0.294523	0.042543	0.142351

Table 4. Comparison of SP values between the LO-NSGA-II algorithm and the NSGA-II algorithm.

Test Function	LO-NSGA-II			NSGA-II
	SP Max	SP Min	SP Mean	SP Max	SP Min	SP Mean
ZDT1	0.215842	0.010096	0.054121	0.325010	0.011754	0.100689
ZDT2	0.354121	0.012920	0.031242	0.410984	0.002348	0.064926
ZDT3	0.472145	0.008325	0.063258	0.453302	0.019654	0.071245
ZDT6	0.631241	0.006551	0.102144	0.638906	0.010315	0.124582
Viennet2	0.210110	0.045214	0.105866	0.150912	0.056752	0.113258
Viennet3	0.203251	0.038845	0.088254	0.294523	0.042543	0.142351

Table 5. Comparison of GD values between the LO-NSGA-II algorithm and the NSGA-II algorithm.

Test Function	LO-NSGA-II			NSGA-II
	GD Max	GD Min	GD Mean	GD Max	GD Min	GD Mean
ZDT1	0.028521	0.001476	0.011251	0.033521	0.001176	0.018436
ZDT2	0.019132	0.001408	0.001087	0.022132	0.001608	0.001217
ZDT3	0.019021	0.000951	0.000964	0.021021	0.001051	0.001156
ZDT6	0.056256	0.002362	0.002365	0.063256	0.003162	0.003479
Viennet2	0.027251	0.001062	0.013218	0.031251	0.001562	0.017188
Viennet3	0.024522	0.006311	0.011281	0.025522	0.007611	0.014031

Table 5. Comparison of GD values between the LO-NSGA-II algorithm and the NSGA-II algorithm.

Test Function	LO-NSGA-II			NSGA-II
	GD Max	GD Min	GD Mean	GD Max	GD Min	GD Mean
ZDT1	0.028521	0.001476	0.011251	0.033521	0.001176	0.018436
ZDT2	0.019132	0.001408	0.001087	0.022132	0.001608	0.001217
ZDT3	0.019021	0.000951	0.000964	0.021021	0.001051	0.001156
ZDT6	0.056256	0.002362	0.002365	0.063256	0.003162	0.003479
Viennet2	0.027251	0.001062	0.013218	0.031251	0.001562	0.017188
Viennet3	0.024522	0.006311	0.011281	0.025522	0.007611	0.014031

Multiple independent runs of both algorithms under identical population sizes and generation numbers revealed that the improved LO-NSGA-II algorithm achieved significantly lower GD values than the conventional NSGA-II at the same number of generations, demonstrating superior convergence capability. Meanwhile, the SP metric indicates that the improved algorithm produces a more uniformly distributed Pareto solution set, effectively avoiding the solution clustering phenomenon associated with conventional random initialization. These results demonstrate that the incorporation of Latin hypercube sampling and opposition-based learning strategies enhances the global search capability and population diversity of the algorithm, thereby providing a more stable and reliable solution foundation for optimizing the spatial installation orientation of cutting inserts.

When the LO-NSGA-II algorithm is used for multi-objective optimization, the outcome is not a single optimal solution but a set of Pareto-optimal solutions forming the Pareto front. These solutions reflect different trade-offs among the objectives and provide diverse candidate schemes for decision-making. The objective values of the Pareto front in Figure 13 were predicted by the BP neural network surrogate models. The distribution of the Pareto front indicates a certain conflict between radial resultant force and cutting temperature, suggesting that single-objective optimization cannot simultaneously satisfy the requirements of cutting stability and thermal control. Therefore, multi-objective optimization provides a more flexible basis for posture-parameter configuration.

To validate the reliability of the surrogate models, five-fold cross-validation was performed for the prediction of radial resultant force F and cutting temperature T. The results in

Table 6. Predictive performance of the surrogate models.

Response Variables	R2	RMSE
F	0.9659	26.36
T	0.8531	39.25

show that the surrogate models achieve satisfactory predictive accuracy for both response variables and are therefore adequate for subsequent multi-objective optimization analysis.

3.2. Parameter Selection Based on the Entropy Weight-TOPSIS Hybrid Method

The entropy weight-TOPSIS method is a multi-criteria decision-making approach that combines the entropy weight method with the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS). The entropy weight method is mainly used to determine the objective weight of each indicator according to its data distribution characteristics, thereby reducing the influence of subjective factors on weight assignment. The TOPSIS method evaluates candidate solutions by calculating their relative distances from the ideal solution and the negative ideal solution, and ranks them according to these distances to identify the optimal solution. By considering both indicator weighting and alternative ranking, this method ensures both objectivity and scientific rigor. Therefore, radial resultant force and cutting temperature were adopted as surrogate indicators of tool life and machining quality for multi-objective optimization using the entropy weight-TOPSIS method. Based on this method, the weights of radial resultant force and cutting temperature were determined to be 54.09% and 45.91%, respectively. This weighting scheme makes the allocation of weights for radial resultant force F and cutting temperature T more objective and reasonable, while enabling a better compromise between tool life and machining accuracy. The weight coefficients of radial resultant force and cutting temperature calculated by the entropy weight method are listed in

Table 7. Weight coefficients of radial force and cutting temperature.

Weight Value	F	T
ω	54.09%	45.91%

.

The entropy weight method objectively determines weights based on data dispersion. During the optimization process, it enables a reasonable allocation of weights to radial resultant force F and cutting temperature T, thereby achieving a balance between tool life and machining accuracy. As shown in the table, the weight of radial resultant force F is higher than that of cutting temperature T, indicating greater variation in F among different schemes and, consequently, a more significant effect on the comprehensive evaluation results. The obtained Pareto-optimal solutions were then combined with the corresponding weights and evaluated using the TOPSIS method, with the results shown in

Table 8. Results of the entropy weight TOPSIS method.

Rank	${\mathit{x}}_1$ (°)	${\mathit{x}}_2$ (°)	${\mathit{x}}_3$ (°)	${\mathit{x}}_4$ (mm)	Radial Resultant Force F (N)	Cutting Temperature T (°C)	Closeness
1	4.1	85.6	86.1	0.32	23.963	554.370	0.531
2	1.7	87.3	86.4	0.35	36.880	533.230	0.521

. The closeness coefficient of the entropy weight-TOPSIS method reflects the compromise level of each scheme with respect to the two surrogate indicators, namely radial resultant force and cutting temperature. Since the closeness coefficients of Solutions 1 and 2 differ only slightly, both can be regarded as representative Pareto-optimal solutions, with Solution 1 being relatively more balanced. Therefore, the top two ranked solutions, 1 and 2, were selected for drilling validation, and the final decision was made based on tool life and hole surface quality.

To assess the sensitivity of the TOPSIS decision-making results to weight assignment, a weight sensitivity analysis was performed for Solutions 1 and 2. As shown in

Table 9. Weight sensitivity analysis.

Weight (F/T)	1	2
60/40	0.556	0.468
50/50	0.513	0.558
40/60	0.471	0.650

, Solutions 1 and 2 are adjacent Pareto-optimal solutions with similar overall performance on the Pareto front, and their relative ranking is somewhat sensitive to the decision weights. Because the original weights were objectively determined by the entropy weight method according to the dispersion of the sample data, they can represent the discriminating power of each indicator in the current study. Accordingly, the ranking results under the original weights were taken as the main basis for decision-making.

The above analysis demonstrates that the spatial mounting posture parameters of indexable shallow-hole drill inserts exert a significant regulatory effect on the radial resultant force and cutting temperature during the drilling process. Different combinations of posture parameters exhibit a clear trade-off between mechanical stability and thermal behavior control, indicating that reliance on a single parameter or empirical configuration is insufficient to simultaneously optimize both mechanical and thermal performance. By adopting the radial resultant force and cutting temperature as multi-objective optimization criteria, the spatial mounting posture of the inserts was optimized. Among multiple feasible posture configurations, two parameter combinations with superior comprehensive performance were identified.

4. Experimental Validation and Results Analysis

4.1. Experimental Conditions and Protocol for Indexable Shallow Hole Drilling

4.1.1. Experimental Conditions

The experimental machine tool used is the VMC1600B vertical machining center (SMTCL, Shenyang, China), which features a maximum spindle speed of 8000 rpm, a spindle power of 15 kW, and feed axis travel of XYZ = 1600 mm × 800 mm × 800 mm. For this experiment, 42CrMo steel was selected as the material, and two distinct workpiece geometries were designed to facilitate cutting force measurement and subsequent testing of other parameters. Prior to machining, the material underwent preliminary turning and milling processes to ensure that the workpiece geometry met the required machining specifications. One workpiece was machined into a Φ170 mm × 150 mm shape with a clamping step, while the other was in the form of a Φ230 mm × 200 mm bar. The body is made of hot-work tool steel. The actual measured dimensions of each tool are provided in Table 8.

The indexable shallow-hole drill used in this experiment is shown in Figure 14. Based on the multi-objective optimization model for the spatial mounting posture of the inserts established in Section 3, the design variable space was explored and screened, yielding several representative insert mounting posture schemes. According to the optimization results, two optimized tools, denoted as U2 and U3, were selected for comparative experiments with the unoptimized tool U1. Since the TOPSIS closeness coefficients of U2 and U3 differed only slightly, both were taken as optimized tools for comparison. SPGT060204 inserts were employed, with YGB205 for the outer insert and YGB212 for the inner insert. The drill body was made of hot-work die steel, and the actual measured dimensions of each tool are listed in

Table 10. Measured parameter table of the indexable insert drill used in the experiment.

Tool Number	Β (°)	α1 (°)	α0 (°)	E (mm)	Dc (mm)	H1 = 0.15 mm, H2 = 0.65 mm, H3 = 0.17 mm
U1	5	84.2	85.1	0.4	21.06
U2	4.1	85.5	86	0.32	21.03
U3	1.8	87.2	86.3	0.35	21.08

.

This experiment measured the forces acting on the workpiece in the Fx and Fy directions using a Kistler 9139AA triaxial dynamic piezoelectric force sensor (Kistler Group, Winterthur, Switzerland)., a Kistler 5167A charge amplifier (fixed-mount DAQ), and Kistler DynoWare software (Version 2825A). The surface roughness of the bore wall was measured using a MarSurf VD 280 profilometer (Mahr, Göttingen, Germany). The deviation in the drilled bore diameter was measured with an internal micrometer. Tool insert wear was inspected using a Dino-Lite AM7915 industrial microscope (Dino-Lite, New Taipei City, Taiwan, China).. The experimental setup is shown in Figure 15.

4.1.2. Experimental Protocol

All drilling experiments were conducted on the same CNC machine tool under identical tool holder conditions, with a constant tool overhang maintained throughout the tests. To eliminate the influence of spindle system errors and clamping inaccuracies, the radial runout of the tool was measured using a dial indicator after installation. All tool groups were mounted using identical clamping methods, and no significant abnormal runout was observed. Therefore, the radial force variations discussed in this study are primarily attributed to differences in the spatial installation orientation of the cutting inserts rather than machine tool system errors.

The workpiece material used in this study was forged 42CrMo steel, a commonly used medium-carbon alloy structural steel in engineering applications, with a hardness of approximately 42 HRC. Based on the recommended tool parameters, the spindle speed and feed rate were set to n = 2500 r/min and f = 0.07 mm/r, respectively. Under identical cutting conditions, dry drilling experiments were conducted with a drilling depth of 105 mm. This was intended to minimize the influence of cutting-parameter variation on the experimental results, thereby highlighting the structural effect of insert spatial mounting posture on machining performance. The workpiece was pre-marked for hole positioning prior to machining. To ensure data reliability and repeatability, each indexable shallow-hole drill machined three holes under identical cutting parameters, and the averaged results were used as representative values. Surface roughness was measured at three uniformly distributed positions along the hole circumference, and the mean value was calculated to minimize single-point measurement error.

Tool life was evaluated according to internationally accepted criteria, with failure defined when the flank wear width VB reached 0.3 mm. Wear was inspected after every two drilled holes to document the wear evolution process. Insert wear was monitored using an industrial camera until the specified wear criterion was reached.

4.2. Comparison of Cutting Performance Before and After Spatial Mounting Posture Optimization of the Tool

4.2.1. Mechanism Analysis of Cutting Force and Temperature

In the multi-objective optimization model, the radial force and cutting temperature were selected as the optimization objectives. To clarify the physical basis for this objective selection, it is necessary to experimentally elucidate the mechanism of cutting force formation and temperature distribution characteristics during the drilling process of indexable shallow-hole drills prior to analyzing the optimization results.

Drilling experiments were conducted at a spindle speed of n = 2500 r/min and a feed rate of f = 0.07 mm/r. To minimize experimental uncertainty, three repeated tests were carried out under identical cutting parameters, and the average values were used for subsequent analysis. Among them, the force curves are shown in Figure 16.

Figure 16. Mechanical property curves of the Indexable insert drill in drilling experiments.

Figure 16. Mechanical property curves of the Indexable insert drill in drilling experiments.

Experimental results on cutting forces indicate that the drilling process generally progresses through three distinct stages: an initial stage dominated by the inner insert cutting alone, a transitional stage involving both inner and outer inserts, and a steady-state cutting stage. During the steady-state stage, the axial force and torque tend to stabilize, whereas the radial force component is primarily determined by the difference in force contributions generated by the inner and outer inserts at different cutting radii. Due to the differences in the principal cutting edge angle and included angle between the inner and outer inserts, their force decomposition patterns differ, resulting in a non-negligible resultant radial force and a tendency toward force imbalance.

Since the temperature at the tool-chip interface is difficult to measure directly, workpiece temperature was adopted in this study as an indirect indicator of cutting thermal level to reflect the relative thermal effects under different posture-parameter conditions. The arrangement of the four thermocouples is shown in Figure 17. The temperature curves are shown in Figure 18.

Figure 17. Thermocouple placement location.

Figure 17. Thermocouple placement location.

Figure 18. Cutting temperature experiment curve.

Figure 18. Cutting temperature experiment curve.

Experimental results indicate that as the drilling depth increases, the workpiece temperature exhibits a pronounced upward trend from point A1 to point B1, indicating progressive heat accumulation along the feed direction. In the bottom cutting region, the temperature at point C1 is significantly higher than that at point D1. This is because point C1 is located closer to the outer insert edge, corresponding to a larger cutting radius and thus a higher cutting speed, leading to more intense friction-induced heat generation. In contrast, point D1 is situated nearer to the drill center, where the cutting speed is lower, resulting in a smaller temperature rise.

These results demonstrate that the temperature distribution in the drilling zone exhibits a distinct radial gradient, with the outer insert edge region serving as the primary heat concentration area. This non-uniform temperature distribution is closely related to the spatial installation orientation of the inserts and directly affects the location and rate of tool wear.

From a machining mechanics perspective, the radial force can be expressed as:

$$F_r=F_{1}sin{\alpha }_1-F_{0}sin{\alpha }_0$$

where $F_1$ and $F_0$ denote the cutting forces of the inner and outer inserts, respectively, and ${\alpha }_1$ and ${\alpha }_0$ represent their corresponding principal cutting edge angles. When the matching relationship between the principal cutting edge angles is inappropriate, the radial force components generated by the inner and outer inserts cannot effectively offset each other, resulting in an increased resultant radial force and elevated peak contact stress at the tool–workpiece interface. According to the cutting heat generation mechanism, the heat generation rate is proportional to the cutting force and relative sliding velocity. Therefore, in regions with both high radial force and high cutting speed, localized heat concentration becomes more pronounced. The temperature rise accelerates thermal softening of the tool material as well as diffusion and oxidation wear, thereby promoting the expansion of flank wear.

Therefore, by adjusting the spatial installation orientation parameters of the inserts, radial force balancing and thermal distribution optimization can be achieved at the structural level, thereby improving drilling stability and retarding tool wear evolution. Based on this understanding, the following section systematically analyzes the changes in resultant radial force, surface quality, and tool life before and after optimization.

4.2.2. Elliptical Hole Analysis in the Drilling Process

Based on the aforementioned mechanistic understanding, a comparative analysis of the resultant radial force before and after optimization was conducted. During the drilling process with indexable shallow hole drills, the asymmetric installation of the inner and outer inserts inevitably generates radial forces and their fluctuations, which result in machining errors such as axial displacement and out-of-roundness. The ellipticity or diameter deviation of the hole not only reflects the stability of the drilling process but is also closely linked to the tool’s force balance, vibration characteristics, and the smoothness of the cutting process. To further assess the impact of the optimized spatial mounting posture of the inserts on drilling stability and machining accuracy, it is essential to analyze the diameter deviation characteristics of holes drilled by indexable shallow hole drills before and after optimization.

As shown in Figure 19, the radial component of the force on the indexable shallow hole drill U1 exhibits significant fluctuations throughout the drilling process before optimization. This suggests that U1 experiences considerable radial force disturbances during machining, potentially leading to unstable interactions between the drill and the workpiece. In contrast, the radial forces of the optimized indexable shallow hole drills U2 and U3 are significantly reduced. Particularly for U2, the amplitude of the radial force is notably smoother and lower, indicating that the optimization measures effectively minimized lateral interactions between the drill and the workpiece, thereby enhancing machining stability.

As shown in Figure 20, the radial resultant force of U1 is the highest, reaching 235 N. In contrast, the radial forces of the optimized indexable shallow hole drills, U2 and U3, are reduced to 25.3 N and 40.8 N, respectively, demonstrating a significant decrease. Through optimization of the spatial installation posture of the inserts, the radial force during drilling was effectively minimized. This indicates that optimizing the spatial installation posture allows the radial forces of the inner and outer inserts to counterbalance each other, significantly improving the machining performance of the indexable shallow hole drill. The optimized tool exhibits lower radial force and a more stable distribution of component forces.

Based on the characteristics of indexable shallow hole drilling, radial deviation values were used to analyze elliptical holes. To evaluate the drilling performance of indexable shallow hole drills under identical parameters, an internal micrometer was employed to measure and compare the hole characteristics produced by each tool. Measurements were taken for each hole within the same plane, with diameter deviation measured at each doubling of the bore diameter.

As shown in Figure 21, the diameter deviation of the indexable shallow hole drill U1 before optimization is generally large, especially at locations with three times the diameter or greater, where the deviation significantly increases, reaching a maximum of 0.256 mm. This indicates that, before optimization, tool U1 exhibited noticeable machining instability during hole drilling due to excessive radial force, with diameter deviation worsening as drilling depth increased. Diameter deviation reduces the assembly accuracy between the workpiece and its mating shaft, causing accelerated fatigue damage in moving parts due to stress concentration and increasing surface wear. In contrast, the optimized indexable shallow hole drills U2 and U3 show clear advantages in controlling diameter deviation. Experimental data show that their drilling diameter deviation is significantly reduced compared to U1. Furthermore, the diameter deviations at different drilling depths for U2 and U3 are more consistent and uniform. This suggests that the optimized tools effectively reduced the impact of radial force on the machining process, improving the force balance of the tools and significantly enhancing drilling accuracy and machining stability.

4.2.3. Hole Surface Quality Comparison

To assess the improvement in the surface quality of machined holes before and after tool optimization, a comparative analysis was conducted. Surface roughness measurements were taken using roughness measurement equipment to evaluate the machined holes, allowing for the determination of whether the optimized tools demonstrate superior machining performance. The experiment utilized the MarSurf VD 280 surface roughness measuring instrument (Mahr, Göttingen, Germany) as shown in Figure 22 along with the BFWA 4-90-27676 probe (Mahr, Göttingen, Germany) to measure the surface roughness of holes machined by the tools before and after optimization.

A comparative analysis was conducted between holes drilled with the unoptimized indexable shallow hole drill U1 and those drilled with the optimized indexable shallow hole drills U2 and U3. The measurement results show significant differences in surface quality between the holes machined with unoptimized and optimized tools, as shown in Figure 23. The measured parameters included roughness values Ra and Rz, with units in μm. Measurements were taken at three uniformly distributed points along the bore diameter, and the average value was calculated. A comparison of the surface quality of holes drilled by different tools before and after optimization is provided in

Table 11. Surface Roughness Ra of Indexable Shallow-Hole Drills Before and After Optimization.

Tool Type	Point 1 (µm)	Point 2 (µm)	Point 3 (µm)	Mean (µm)	Standard Deviation (µm)
U1	22.742	22.915	23.031	22.896	0.145
U2	6.320	6.395	6.485	6.400	0.083
U3	7.590	7.650	7.734	7.658	0.072

and

Table 12. Surface Roughness Rz of Indexable Shallow-Hole Drills Before and After Optimization.

Tool Type	Point 1 (µm)	Point 2 (µm)	Point 3 (µm)	Mean (µm)	Standard Deviation (µm)
U1	108.850	109.306	109.660	109.272	0.406
U2	28.215	28.360	28.472	28.349	0.129
U3	36.825	36.941	37.093	36.953	0.134

, showing the measured mean roughness parameters.

4.2.4. Analysis of Hole Diameter Accuracy

To further evaluate the effect of optimizing the spatial installation orientation of the inserts on hole machining accuracy, hole diameter measurements were carried out for holes machined by different tools. Considering potential variations in tool manufacturing dimensions, the actual measured tool diameter was used as the reference to calculate hole diameter enlargement. The measured diameters of the three comparison tools were φ21.06 mm (U1), φ21.03 mm (U2), and φ21.08 mm (U3), respectively.

Hole diameter measurements were performed using a dial indicator, as shown in Figure 24. During measurement, the workpiece was fixed on the machine table. After calibration with standard gauge blocks, the dial indicator probe was positioned against the inner surface of the hole wall. Measurements were taken at three axial locations, defined as upper, middle, and lower, in order to analyze the axial variation in hole diameter. The hole-diameter measurement data are shown in

Table 13. Comparison of Hole Diameter for Different Tool Configurations.

Tool Type	Upper (mm)		Middle (mm)		Lower (mm)		Actual Tool Diameter (mm)	Mean Hole Diameter (mm)	Hole Enlargement (mm)
U1	21.10	21.17	21.16	21.16	21.16	21.15	21.06	21.15	0.09
U2	21.05	21.04	21.06	21.05	21.06	21.06	21.03	21.053	0.023
U3	21.10	21.09	21.11	21.11	21.11	21.11	21.08	21.105	0.025

. The comparison of hole diameter enlargement is shown in Figure 25.

Figure 25. Comparison of Hole Diameter Enlargement.

Figure 25. Comparison of Hole Diameter Enlargement.

The measurement results indicate that the mean hole diameter of the holes machined by U1 was approximately 21.15 mm, with a hole enlargement of about 0.09 mm. In contrast, the optimized tools U2 and U3 produced mean hole diameters of 21.053 mm and 21.105 mm, corresponding to hole enlargements of approximately 0.023 mm and 0.025 mm, respectively. Compared with U1, the optimized configurations significantly reduced hole enlargement, resulting in hole diameters closer to the actual tool diameters. To compare the effects of different tool schemes on hole diameter, one-way analysis of variance (ANOVA) was performed on the hole-diameter measurements obtained at different positions along the hole for the three tool schemes (U1, U2, and U3). The results indicated significant differences in hole-diameter measurements among the different tool schemes (p < 0.001). These results suggest that optimizing the spatial installation orientation of the inserts effectively reduces the resultant radial force and lateral vibration amplitude, thereby improving hole diameter stability. The observed reduction in diameter deviation is consistent with the decrease in resultant radial force, further confirming the positive impact of structural optimization on drilling accuracy.

4.2.5. Tool Life Comparison

By recording the tool life of indexable shallow hole drills before and after optimization during machining, and comparing the wear patterns of three drills under identical operating conditions, this study evaluates the impact of the optimized spatial installation posture design on tool life.

The three indexable shallow hole drills, U1, U2, and U3, were each mounted on the machining center, with a new insert of the same type installed on each tool. After each tool drilled two holes, the inserts were photographed and the wear value was measured using the Dino-Lite AM7915 industrial camera, as shown in Figure 26. Each set of tools continued to drill until the insert wear reached the failure threshold (wear amount VB = 0.3 mm), and the number of holes drilled by each tool set was recorded. The wear evaluation indicators were the wear amount VB (width of wear on the main cutting face, in mm) and tool life (total number of holes drilled).

The measurement results are shown in Figure 27. The comparison of tool life is shown in

Table 14. Comparison of Tool Life.

Tool Number	Inner Insert Wear Width (mm)	Tool Life (Number of Holes)	Life Improvement Percentage (%)
U1	0.117	20	-
U2	0.094	28	40%
U3	0.114	32	60%

. The outer insert primarily handles the cutting of the outer diameter of the hole, and its wear is significantly influenced by cutting forces and cutting temperature.

A comparison of the wear width and tool life data for the indexable shallow hole drills U1, U2, and U3 reveals that the pre-optimized indexable shallow hole drill U1 reached the wear failure standard after drilling 20 holes. In contrast, the tool life of the optimized indexable shallow hole drills U2 and U3 was significantly extended, with U2 and U3 drilling 28 and 32 holes, respectively, before reaching the wear failure threshold. The tool life of U2 and U3 was improved by 40% and 60%, respectively, compared to U1. The wear rate of the outer insert was significantly reduced after optimization. The inserts of U2 and U3 exhibited improved force uniformity and heat distribution during cutting, resulting in more stable wear patterns. Optimizing the spatial installation posture of the insert alleviated the problem of heat concentration, and the reduction in cutting temperature effectively slowed down the thermal softening and wear rate of the outer insert. The wear rate of the outer insert was higher than that of the inner insert, primarily due to the higher cutting speed in the outer insert region, which subjected it to greater cutting forces and friction-induced heat. Therefore, when the tool reached the wear failure threshold, the inner insert generally showed minimal wear.

Based on surface observations and measurements, the non-optimized indexable shallow-hole drill U1 produced holes with relatively high surface roughness, with mean Ra and Rz values of 22.896 µm and 109.272 µm, respectively. Minor tool marks and waviness were observed on the machined surfaces, indicating limited cutting performance and difficulty in meeting high surface quality requirements. In contrast, the optimized drills U2 and U3 exhibited significantly improved surface integrity, with substantially reduced roughness values. The mean Ra values decreased to 6.400 µm and 7.658 µm, corresponding to reductions of 72.05% and 66.56%, respectively. Similarly, the mean Rz values were reduced to 28.349 µm and 36.953 µm, representing decreases of 74.05% and 66.18%. To further compare the effects of different tool schemes on hole surface quality, one-way analysis of variance (ANOVA) was conducted based on the Ra and Rz values measured at three uniformly distributed locations on the machined hole surface for each of the three tool schemes (U1, U2, and U3). The results indicated significant differences in both Ra and Rz among the different tool schemes (p < 0.001). These improvements suggest enhanced cutting stability, characterized by reduced friction and vibration at the tool–workpiece interface. Consequently, surface defects were markedly diminished, and the hole surfaces became smoother and more uniform, with fewer tool marks and no evident waviness or scratches, leading to a significant improvement in machining quality. The improved performance can be attributed to the optimized spatial installation orientation of the inserts, which promotes radial force balancing and mitigates unstable tool–workpiece interactions during cutting.

Chip evacuation conditions also play an important role in drilling stability. Continuous chips combined with poor evacuation may accumulate within the flutes, increasing force fluctuations and compromising process stability. Meanwhile, insufficient heat dissipation from the cutting zone can accelerate insert wear. These observations indicate an intrinsic relationship among the mechanical state, thermal distribution, and chip evacuation behavior during drilling. Under the selected cutting parameters and tool geometry, no significant burr formation was observed at the hole exit, further confirming that the drilling process maintained a stable machining condition.

5. Conclusions

(1)

To address the complex influence mechanism of the spatial installation orientation of inserts in indexable shallow-hole drills on cutting force distribution and thermal behavior—an issue that remains largely dependent on empirical design—a parametric mathematical model of the insert spatial installation orientation was established. By constructing a unified coordinate system and corresponding spatial transformation relationships, key parameters—including the principal cutting edge angles of the inner and outer inserts, the included angle, and the insert eccentricity—were modeled in a unified framework, thereby providing a geometric and mechanical foundation for subsequent structural optimization and performance analysis.

(2)

Using radial resultant force and cutting temperature as key indicators of drilling stability and tool life, this study established a multi-objective optimization model for insert spatial mounting posture. An improved LO-NSGA-II algorithm was employed to obtain a Pareto-optimal solution set representing the trade-off between radial resultant force and cutting temperature under different posture-parameter combinations. The Pareto-optimal solutions were further evaluated using the entropy weight-TOPSIS method, through which two representative high-quality compromise schemes for insert spatial mounting posture parameters were selected. The results show that insert spatial mounting posture plays a significant role in regulating radial force balance and cutting thermal behavior. The proposed optimization framework is capable of effectively identifying high-quality compromise candidate solutions and provides a systematic decision-making basis for tool posture design under different performance requirements.

(3)

Comparative drilling experiments were conducted to evaluate the cutting performance of indexable shallow-hole drills before and after posture optimization. Based on multi-objective optimization and multi-criteria decision-making methods, representative high-quality compromise candidate solutions were selected from the Pareto solution set, and their actual machining performance was further compared through drilling experiments. The results showed that the optimized tools exhibited significantly reduced radial resultant force and markedly improved drilling stability. Compared with the unoptimized tool, the optimized tools produced substantially lower Ra and Rz values, indicating a significant improvement in hole surface quality. In addition, the tool lives of the two candidate solutions, U2 and U3, increased by approximately 40% and 60%, respectively. These findings indicate that posture optimization can effectively improve the load uniformity of the inner and outer inserts and the distribution of cutting heat during drilling, thereby slowing insert wear. The results are generally consistent with those reported in previous studies, and further demonstrate that, in addition to cutting-parameter optimization and local structural improvement, optimization of insert spatial mounting posture is also an important approach for enhancing the machining performance of indexable shallow-hole drills.

(4)

Experimental validation confirmed the effectiveness of the proposed optimization framework in identifying preferred candidate solutions, extending tool life, and improving hole machining quality. Furthermore, the cross-validation results, weight sensitivity analysis, and drilling experiments consistently showed that the relative ranking of the optimized candidate solutions agreed with the overall performance trade-offs, indicating that the established surrogate models can effectively support the multi-objective optimization analysis in this study. However, the present work was conducted for 42CrMo workpiece material under fixed cutting conditions. Extending the proposed method to other workpiece materials or machining conditions would require the collection of new sample data and retraining of the surrogate models. In addition, the evaluation of hole machining quality in this study mainly focused on surface roughness and dimensional accuracy, without further considering surface integrity characteristics such as residual stress and subsurface microstructural evolution. Moreover, the sensitivity of the LO-NSGA-II algorithm parameters and the statistical significance of the experimental results still require further investigation. Future work will therefore focus on extending the method to multiple materials and machining conditions, conducting deeper characterization of surface integrity, analyzing the sensitivity of algorithm parameters, and optimizing the synchronized wear behavior of the inner and outer inserts.