## Abstract

This paper focuses on the difficulties in high-quality and high-efficiency micro-milling nickel-based superalloy micro thin-walled parts. The second-generation Non-dominated Sorting Genetic Algorithm (NSGA-II) is improved. A central composite experiment is designed, and a surface roughness prediction model is developed for micro-milling thin-walled parts. A prediction model for surface residual stress on thin-walled parts is developed using an L9(34) orthogonal simulation experiment. Using the NSGA-II algorithm, the four cutting parameters (spindle speed, feed per tooth, axial cutting depth, and radial cutting depth) are optimized to achieve low surface roughness and high material removal rate, while stable cutting and surface compressive residual stress are considered constraints. Finally, the high-quality and high-efficiency micro-milling of the Inconel 718 cross-shaped thin-walled parts is realized.

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## 1 Introduction

Aerospace, energy and power, and biomedicine have seen an increase in demand for micro thin-walled parts. These small parts exhibit mesoscale [1] thin-walled features typically require high machining accuracy and surface quality. Microfluidic chip hot-pressing molds, for example, not only have requirements for the size and surface roughness but also have requirements for residual stress [2].

At high temperature, some materials with high strength and good thermal stability can be used to produce the above parts. Nickel-based superalloy Inconel 718 is a typical material of this kind. Due to its wide variety of materials, high machining accuracy, and ability to create three-dimensional surfaces, micro-milling is to machine Inconel 718 micro parts.

However, the mesoscale thin-walled components generated by micro-milling processing encounter difficulties in undergoing subsequent processes due to their scale, and the surface integrity is difficult to predict and control. Their performance depends on surface integrity indicators such as residual stress and surface roughness. Residual stress is an important cause of microcracks. Uneven stress affects the fatigue resistance characteristics of parts. Micro thin-walled parts usually require specific surface roughness on the sidewall, but there is a contradiction between low surface roughness and high material removal rate (MRR), making it difficult to select micro-milling parameters. At present, the systematic and in-depth research on residual stress and surface roughness in thin-walled micro-milling has not yet generated. It is urgent to explore the technical difficulties of high-quality and high-efficiency micro-milling processing for small thin-walled nickel-based superalloy.

Many experts and scholars have conducted extensive exploration and research on high-quality and high-efficiency micro-milling. Zhou et al. [3] designed a micro-milling groove experiment for Al7075-T6 material. A regression coefficient estimation model was established using the least squares method for the prediction of the relationship between spindle speed, feed per tooth, axial cutting depth, tool overhang, micro-milling force, and top burr width. The optimal cutting parameter combination for micro-milling was obtained to minimize the width of the top burrs. Kumar et al. [4] established the relationship model between surface roughness and spindle speed and feed per tooth through micro-milling experiments. To minimize processing time and surface roughness, the spindle speed and feed per tooth were optimized by a genetic algorithm. Saeed et al. [5] carried out micro-milling experiments with three factors (spindle speed, feed per tooth, and axial cutting depth). Using principal component analysis and the simulated annealing algorithm, the optimal cutting parameters were determined to minimize burr size. Through the literature review, we can see that experts and scholars selected spindle speed, feed per tooth, and axial cutting depth for process parameters optimization. However, thin-walled machining mostly uses the side milling method, and the thin-walled features itself has poor rigidity. The improper radial cutting depth will cause serious machining deformation. Therefore, in this paper, four process parameters (spindle speed *n*, feed per tooth *f*_{z}, axial cutting depth *a*_{p}, and radial cutting depth *a*_{e}) are optimized.

Umer et al. [6] optimized process parameters in the micro-milling alumina ceramics with Nd:YAG laser employing the Multi-Objective Genetic Algorithm II. Kuram et al. [7] used the Taguchi method to optimize the cutting force, surface roughness, and tool wear in micro-milling Inconel 71. Aslantas et al. [8] minimize burr width and surface roughness from studying the parameters optimization of micro-milling Ti-6Al-4 V alloy. Zhang et al. [9] combined cuckoo search and gray wolf algorithms to reduce energy consumption in micro-milling. Chen et al. [10] aimed to minimize the surface roughness of micro-milling brass mold inserts and determined the spindle speed, feed speed, and cutting depth through orthogonal experiments and factor analysis. La Fé-Perdomo et al. [11] designed a decision system for micro-milling based on the fuzzy inference system to select the optimal process parameters. For determining the optimal process parameters, the system used the cross-entropy method for multi-objective optimization. Bhavsar et al. [12] utilized a genetic algorithm toolbox to optimize micro-milling parameters with the goal of high MRR and low surface roughness. Lauro et al. [13] consolidated the least squares method and genetic optimization algorithm, to minimize micro-milling force, obtaining the optimal cutting parameters of micro-milling steel. Joshi et al. [14] employed a metaheuristic multi-objective optimization algorithm to optimize micro-milling cutting parameters for high material removal rate (MRR) and low surface roughness. Yi et al. [15] established a surface roughness prediction model. Then, contour maps of MRR and surface roughness were established, and cutting parameters were optimized to raise cutting efficiency and get lower surface roughness. Experts usually perform parameter optimization with low surface roughness as the optimization objective, which provides ideas for the research in this paper. Micro-milling cutter size and micro-milling cutting dimensions are significantly smaller than conventional milling, resulting in lower machining efficiency. Moreover, high MRR and low surface roughness are often contradictory goals, making the selection of micro-milling cutting parameters a challenge. Therefore, to achieve high-quality and high-efficiency micro-milling nickel-based superalloy micro thin-walled parts, process parameters are optimized with high MRR and low surface roughness as optimization objectives in this paper.

Owing to the weak stiffness, improper selection of cutting parameters can lead to chattering during micro-milling, increasing deformation and affecting machining accuracy. The processing requirements for thin-walled features are numerous and high, and the cutting parameters cannot be directly determined by experience. Genetic algorithm (GA) transforms the optimal solution search problem into processes such as selection, crossover, and mutation operators of chromosome gene. It is an evolutionary search algorithm formed by simulating biological genetic processes and is often used to achieve multi-objective optimization [16]. The Non-dominated Sorting Genetic Algorithm (NSGA) performs well in handling multi-objective problems [17]. The improved NSGA-II is to be a better solution to multi-objective and multi-constraint problems. Cutting parameters are optimized with the objectives of achieving low surface roughness and high MRR. Additionally, surface compressive residual stress and stable cutting are considered constraints. Then, the high-quality and high-efficiency micro-milling processing of nickel-based superalloy micro thin-walled structures is realized.

## 2 Improved second-generation non-dominated sorting genetic algorithm

NSGA-II has the following advantages: (1) Introducing fast non-dominated sorting criteria and making the algorithm simpler; (2) the introduction of elite strategy enables some individuals of the excellent population to be preserved during the evolutionary process, improving the superiority of the population; (3) the use of crowding distance calculation reduces the impact of human subjective factors on the distribution of the solution set, and crowding distance is regarded as a measure of distance, allowing individuals set to be evenly distributed, ensuring the diversity of individual populations [18].

The NSGA-II also has two shortcomings: (1) The NSGA-II initializes the population at the beginning, without considering the ergodicity of individual population in space; (2) the mutation operators used in the traditional NSGA-II are fixed values, making the algorithm’s global search ability relatively insufficient in the entire space, which makes the solution set easy to converge unevenly and fall into local optima. To address these two issues, this paper improves population initialization and mutation operators. Figure 1 shows the process of the improved NSGA-II.

### 2.1 Chaos initialization based on logistic mapping equation

To prevent it from premature convergence and falling into local optima, this paper introduces the logistic mapping method to perform chaotic optimization on the initial position [15], making it evenly distributed in the initial space, thereby improving the global search range. The logistic mapping equation is a typical chaotic system, as shown in Eq. (1):

where \(\mu \in [0,4],x_{k} \in [0,1]\).

This model possesses the ability to monitor in advance and also satisfies the structural transitivity of chaotic motion within the population. Minor perturbations to the initial value can result in significant alterations across the entirety of the sequence.

### 2.2 Hybrid mutation strategy based on composite mutation operator

Traditional NSGA-II has a fixed crossover probability and mutation probability. The general range of values for crossover probability is 0.4 ~ 0.8, while the range of values for mutation probability is generally 0.001 ~ 0.1, respectively, representing the number of individuals participating in crossover and mutation operations. This paper makes the population evolution divided into different stages based on evolutionary algebra, using different mutation operators to perform mutation operations at different stages, dynamically adjusting the mutation probability of each generation, thereby achieving dynamic changes of the mutation operator and improving the fitness of NSGA-II.

The commonly used mutation operators mainly include uniform mutation operators, Gaussian mutation operators, Cauchy mutation operators, etc. This paper combines the mutation operators of these three distributions and proposes a hybrid mutation operator that dynamically changes with evolutionary algebra. The population evolution in the algorithm is divided into three evenly divided stages, namely early, mid, and late stages of evolution. In the early stages, uniform distribution mutation operators are used to improve global search capabilities. In the mid-term of evolution, the use of Cauchy distribution mutation operators enables the algorithm to have relatively balanced search capabilities of local and global. The Gaussian distribution mutation operator is used in the later stage because the algorithm has already solved and obtained most of the excellent solutions.

## 3 Performance verification of improved NSGA-II algorithm

### 3.1 Performance verification based on standard test functions

The standard test function is suitable for verifying the performance of numerous algorithms. Taking the standard test function as the objective function, compare the Pareto frontier solution generated by the improved algorithm with the ideal Pareto frontier solution. The gap between the solution sets produced prior to and following the enhancement and the optimal solution set are accurately assessed. There are various types of standard test functions, and this paper selects test functions ZDT1 ~ ZDT4. By evaluating the Pareto frontier solutions of the improved NSGA-II across four different dual objective functions, the levels of superiority and inferiority are assessed. The ZDT1 ~ ZDT4 functions are all double objective functions, and the ZDT function problem is uniformly described in Eq. (2):

The ZDT1 test function expression is:

The ZDT2 test function expression is:

The ZDT3 test function expression is

The ZDT4 test function expression is:

Before the improvement, the mutation probability is 0.1. To ignore the impact of different parameters and maintain parameter consistency, the parameters before and after improvement are the same: population size 300, iteration number 300, and crossover probability 0.5.

The overall trend of Pareto frontier solutions solved by the traditional NSGA-II is relatively close to the ideal as shown in Fig. 2. However, in the early stages of evolution, some solution results are abnormal, and the distribution of Pareto solutions is uneven within some ranges. The improved NSGA-II has a good fit between the solution and the ideal solution set, with a more uniform distribution of points and no discontinuity, proving the improved performance of the NSGA-II and achieving good solution results.

### 3.2 Performance verification based on performance indicators

Section 3.1 visually and qualitatively verifies the performance improvement effect in the form of images. This section will analyze and calculate the performance indicators before and after the algorithm improvement. To estimate the performance of the algorithm, we show the convergence index and the distribution index.

To describe the convergence of the algorithm, this paper chooses to calculate the generation distance (GD). The smaller the GD, the better the convergence of the algorithm. When completely close, GD tends to be 0. The expression for GD is shown in Eq. (7):

where *GD* is the distance between generations,* D*_{i} is the shortest Euclidean distance between individuals and the ideal frontier solution, and *n* is the number of individuals.

This article selects spacing (SP) to describe and evaluate the distribution of the algorithm. The expression for the distance is shown in Eq. (8), where the distance deviation represents the average distance of individuals distributed on the leading edge. When the spacing is smaller, the deviation of the distribution of the algorithm in solving the solution set will be smaller. It will be closer to the uniform distribution of the ideal solution set. The expression for spacing is shown in Eq. (8)

where *SP* is the spacing and \(\overline{D}\)is the mean of individual values.

As Table 1 and Fig. 3, the mean generation distance and mean distance decrease with the improved NSGA-II. This improved algorithm is suitable for tackling the dual-objective optimization challenge in thin-walled micro-milling discussed in this study.

## 4 Multi-objective optimization of micro-milling thin-walled parts

### 4.1 Optimization objective

#### 4.1.1 The model of MRR

The cutting quantity in micro-milling is very small, and improving the processing efficiency is an important factor to reduce the time cost. The calculation of MRR is shown in Eq. (9) [19]:

where *n* is the spindle rotational speed (r/min), *f*_{z} is the feed per tooth (µm/z), *a*_{p} is the axial cutting depth (µm), *a*_{e} is the radial cutting depth (µm), and *Z* is the number of teeth of the milling cutter, and in this paper, *Z* = 2.

#### 4.1.2 Prediction model of surface roughness

The response surface method for designing experimental methods can not only avoid an excessive number of experiments but also consider the interaction between cutting parameters. The CCD experimental factors-levels table which includes four factors and five levels was designed with *n*, *f*_{z}, *a*_{p}, and *a*_{e} as influencing factors. Five levels were selected for each factor. *n* is 40,000–80,000 r/min, *f*_{z} is 0.6–1.2 µm/z, *a*_{p} is 20–100 µm, and *a*_{e} is 10–30 µm.

Micro-milling experiments on thin-walled parts were conducted using a CNC micro-milling machine, as illustrated in Fig. 4. Before the thin wall is processed, the length is 5 mm, the height is 1 mm, and the thickness is 0.16 mm. The experiments use the micro-milling tool MSE230. Their simplified structure diagram is shown in Fig. 5. The thin wall is machined by cyclic layer-by-layer climb milling on both sides. The photo after processing is shown in Fig. 6.

We use the Zygo NewView™ 9000 3D surface optical profilometer to measure the surface roughness *S*_{a}. Five different locations are selected for each thin wall. The average value of the five measurements is taken in Table 2.

A quadratic regression prediction model for *S*_{a} is established:

To verify the accuracy of the model of Eq. (10), three additional thin-walled surfaces are machined using cutting parameters different from those in the response surface experimental design mentioned above. The predicted and experimental values are compared in Table 3.

Table 4 illustrates that the average relative error is 5.90%. The maximum relative error is 6.67%. The accuracy of roughness prediction is verified.

### 4.2 Constraint condition

#### 4.2.1 Surface residual stress

The cutting parameters are controlled by the constraint condition of avoiding residual tensile stress on the surface of Inconel 718. Before processing, the thin-walled parts are 1 mm high, 5 mm long, and 0.16 mm thick, as shown in Fig. 7.

The experiment is designed as shown in Table 4. As shown in Fig. 8, we choose the X-ray diffraction method to measure the surface residual stress. The experimental measurement equipment is an iXRD X-ray stress meter from Proto, Canada. Mn is selected as the target material according to the nickel-based superalloy material tested. The voltage is 20 kV, the current is 4 mA, and the diameter of the collimator is \(\phi\) 0.5 mm. The final measurement results of surface residual stress value in X-direction \(\sigma_{{x{\text{ - s}}}}\) and Z-direction \(\sigma_{{z{\text{ - s}}}}\) are as Table 4.

Using the first-order model fitting, the residual stress prediction models in the X- and Z-direction \(\sigma_{{\text{x}}}\) and \(\sigma_{{\text{z}}}\) are shown in Eqs. (11) and (12) based on the experimental data in Table 4:

The constraint conditions are as Eqs. (13) and (14).

The variance analysis results of Eqs. (11) and (12) are as Tables 5 and 6.

Tables 5 and 6 illustrate that the *R*^{2} value of Eq. (11) is 97.13% and the *P* value is 0.002. The *R*^{2} value of Eq. (12) is 96.77%, and the *P* value is 0.003. The *R*^{2} values of the prediction model are all above 95%, and the *P* values are all below 0.01. The significance and reliability of the model are illustrated.

Selecting three sets of different parameter combinations and measuring the surface residual stress after processing. The measurements are compared with the predictive values of Eqs. (11) and (12), as Tables 7 and 8.

The maximum and average relative error of the prediction are 14.18% and 12.20% in the X-direction and 19.32% and 17.38% in the Z-direction. Reliable surface residual stress models are obtained.

#### 4.2.2 Stable cutting

Due to the thin wall’s low stiffness, it is susceptible to flutter and deformation. To maintain machining accuracy, it is essential to control the cutting parameters within a stable range. Based on the stability lobe diagram developed by our group in the previous, the *a*_{p} and *n* within the stable cutting area are chosen, as Fig. 9. As shown in Eq. (15), the stable cutting constraints are established segmentally.

#### 4.2.3 The range of cutting parameters

Micro-milling cutting parameters must also align with the processing capabilities of the micro-milling machine: *n* is 40,000–80,000 r/min,* f*_{z} is 0.6–1.2 µm/z, *a*_{p} is 20–100 µm, and *a*_{e} is 10–30 µm.

### 4.3 Results of multi-objective optimization

Figure 10 shows the Pareto frontier solution obtained with this algorithm. The improved NSGA-II algorithm uses 200 iterations, a population size of 100, and a crossover probability of 0.6. The optimal parameters are as follows: *n* = 80,000 rpm, *f*_{z} = 1.2 µm/z, *a*_{p} = 50 µm, and *a*_{e} = 20 µm. The cutting parameters are within the stable cutting domain. The \(\sigma_{{\text{x}}}\) and \(\sigma_{{\text{z}}}\) are − 112.16 MPa and − 10.74 MPa. After optimization, the MRR is 0.2 mm^{3}/min, and the *S*_{a} is 0.24 µm, fulfilling the requirements for high-quality and efficient machining of thin-walled parts for microfluidic chip molds.

### 4.4 Experiment of thin-wall micro-milling of nickel-based superalloy

The Inconel 718 hot-pressing mold for microfluidic chips is processed using the optimal cutting parameters identified in Section 4.3. A hot-pressing mold for a microfluidic chip with a cross-shaped thin-walled structure was designed. Its length is 5 mm, thickness is 0.1 mm, and height is 1 mm. The thin wall is initially rough-milled to 0.16 mm thickness, leaving a machining allowance of 0.06 mm for micro-milling. Figure 11 shows the micro-milling process.

Figure 12 shows the microfluidic chip hot-pressing mold produced by micro-milling, which took 35 min to complete. The dimensions of the processed cross-shaped thin-walled parts are measured using an ultra-depth of field microscope. Widths are measured at three different positions in four directions on the thin wall, and an average value is calculated. Compared to the theoretical thickness of 0.1 mm, the maximum relative error in the micro-milled thin-walled thickness is 1.94%, with an average relative error of 1.36%. The final optimization target results are as follows: the *S*_{a} is 0.24 µm, the \(\sigma_{{\text{x}}}\) and \(\sigma_{{\text{z}}}\) are − 112.16 MPa and − 10.74 MPa.

## 5 Conclusions

The prediction model of surface roughness with respect to *n*, *f*_{z}, *a*_{p}, and *a*_{e} is established. The maximum relative error is 6.67%, with an average relative error of 5.90%, confirming the model’s effectiveness and accuracy.

The prediction model of surface residual stress of micro-milling thin-walled parts is established. The maximum and average relative error of the prediction are 14.18% and 12.20% in the X-direction and 19.32% and 17.38% in the Z-direction. The validity and accuracy are verified.

The NSGA-II algorithm is improved with chaotic initialization using a logistic map equation and a hybrid mutation strategy employing composite operators. This improvement significantly increases the diversity and uniformity of the Pareto frontier solutions. Through the improved NSGA-II algorithm, the surface residual compressive stress and stable cutting are used as constraints to minimize the surface roughness and maximize the MRR. The optimal parameter combination is as follows: *n* = 80,000 rpm, *f*_{z} = 1.2 µm/z, *a*_{p} = 50 µm, and *a*_{e} = 20 µm. The optimized cutting parameters are applied to subsequent processing. The maximum relative error is 1.94%, with an average error of 1.36%. \(\sigma_{{\text{x}}}\) and \(\sigma_{{\text{z}}}\) are − 112.16 MPa and − 10.74 MPa. The *S*_{a} is 0.24 µm, and the MRR is 0.2 mm^{3}/min after optimizing. High-quality and efficient machining of thin-walled parts for nickel-based superalloy microfluidic chip molds is reached.

## Data availability

All data generated or analyzed during this study are included in this manuscript.

## Code availability

The code is available on request.

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## Funding

The research was supported by the National Natural Science Foundation of China (Grant No. 52275410).

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Xiaohong Lu: Conceptualization, supervision, methodology, funding acquisition, project administration.

Yu Zhang: Conceptualization, methodology, software, validation, writing—original draft, investigation, writing—review and editing.

Zhuo Sun: Writing—review and editing, formal analysis, investigation.

Han Gu: Methodology, writing—original draft.

Chao Jiang: Writing—review and editing.

Steven Y. Liang: Resources.

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Lu, X., Zhang, Y., Sun, Z. *et al.* Multi-objective optimization of cutting parameters for micro-milling nickel-based superalloy thin-walled parts based on improved NSGA-II algorithm.
*Int J Adv Manuf Technol* (2024). https://doi.org/10.1007/s00170-024-14478-8

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DOI: https://doi.org/10.1007/s00170-024-14478-8