Finite Element Modeling-Assisted Deep Subdomain Adaptation Method for Tool Condition Monitoring

Abstract

To reduce the experimental costs associated with tool condition monitoring (TCM) under new cutting conditions, a finite element modeling (FEM)-assisted deep subdomain adaptive network (DSAN) approach is proposed. Initially, an FEM technique is employed to construct a cutting tool model for the new cutting condition (target domain), and the similarity between simulated and experimental data is assessed to obtain valid simulated samples for the target domain. Subsequently, the time–frequency Markov representation method is utilized to extract imaging features from the simulated samples, which serve as input features for the monitoring model. Then, a DSAN model is established to facilitate the transfer from simulation to reality, with the source domain comprising a simulated sample set under new cutting conditions that includes various types of tool conditions obtained through FEM, and the target domain containing only a limited number of normal tool condition samples under new cutting conditions. The application analysis has demonstrated the effectiveness of the proposed method, achieving a classification accuracy of 99%. The proposed approach can significantly reduce experimental costs and obtain high-precision diagnostics of tool conditions with a small sample size.

1. Introduction

Tool condition monitoring (TCM) plays an important role in machining by monitoring the conditions of tool wear, breakage, and failure and taking timely action to ensure machining quality and productivity [1]. With the increase in computer technology and the use of artificial intelligence, it has been found that the generalization and classification of information can be achieved by machine learning methods, such as artificial neural networks [2], support vector machines [3,4], and long short-term memory networks [5,6]. The objective function of the classification model is optimized using a large amount of labeled data, and the optimized model is used for test data [7,8]. Such a method should satisfy two basic assumptions: there is a large amount of high-quality labeled data, and the data to be tested have the same distribution as the training data, that is, they are independent and identically distributed. However, as the speed of product iteration and updating accelerates, the difference between new and historical data gradually increases, and a model constructed using historical data will less well fit these two basic assumptions, making it difficult to apply to new tasks [9,10,11]. As a new learning paradigm, transfer learning applies knowledge learned from historical source domain data to the target domain to solve practical problems. At the Neural Information Processing Systems Summit in 2016 (NIPS 2016), Enda Wu said, “after supervised learning, transfer learning will lead the next wave of machine learning technology.” Transfer learning has become widely used in natural language processing, machine vision, healthcare, and machinery fault diagnosis due to its unique advantages [12,13,14].

It is time-consuming and laborious to obtain a large amount of data from the life test of key mechanical components; fault data in the actual operation process are very scarce, and available labeled samples are even more scarce, which makes it difficult to train a fault diagnosis model with high accuracy [15,16,17,18]. To solve the problem of easy overfitting of small samples in fault diagnosis methods, as a subset of machine learning, transfer learning (TL) can be applied to downstream tasks by utilizing a network model with strong generalization ability obtained from pretraining on large datasets using the model parameter transfer-based approach, which achieves transfer by learning parameter information from the source domain, which is shared with the target domain. TL complements traditional machine learning by facilitating improved learning efficiency and performance on new tasks with limited data. For example, Tajbakhsh et al. constructed a convolutional neural network (CNN) by pretraining fine-tuning to analyze the results of four different medical images after modeling; the results showed that pretrained CNNs using fine-tuned pretuned CNNs performed more robustly [19]. Bai et al. combined a transfer learning pretrained model with a new representation of a time-series to achieve the fault diagnosis of wheel axle assemblies in railroad vehicles [20].

When a target domain with labeled data is not available, a method based on model parameter transfer will fail. Because of the inconsistency in data distribution between the training and testing phases, a well-trained model based on source domain data will be difficult to apply to the target domain task. As a popular method of transfer learning, domain adaptive transfer can use labeled data under similar domains when labeled data are not available in target scenes, improving the predictive effect of diagnostic models in such cases. In the deep domain adaptation-based fault diagnosis method, Lu et al. proposed a multi-core maximum mean difference domain adaptive module to further optimize the distributional differences between feature layers in neural networks [21]. Long et al. used the maximum mean difference to spatially align the distributions of features in two domains to achieve knowledge transfer between them [22].

Traditional experimental methods are often time-consuming and costly under new working conditions. A numerical simulation-based strategy has been attempted as a solution [23,24,25]. Metal cutting is affected by machining accuracy, surface roughness, high temperature and pressure, and cutting parameters and is a very complex process. With significant increases in computing speed, the finite element method (FEM) can be used to simulate the machining process and better analyze the tool wear process.

Machining simulations have been a significant area of research in the field of manufacturing and materials science [26,27], as they allow for the understanding of the complex physical phenomena that occur during the machining process [28], the prediction of machining performance [29], and the optimization of machining parameters [30]. Wu et al. employed a Taguchi-FEM approach to evaluate the significant contributing boundary conditions of Turning and milling machines to tool center point deformation [31]. Zhu et al. constructed a 3D model through SolidWorks and imported it into Deform for cutting force simulation, verified the validity of the simulated data by calculating the cosine similarity and KL dispersion between the simulated and experimental data, and combined the data obtained from simulation with experimental data to form a complete dataset. The features in the time and frequency domains were extracted from the simulation data, and experimental data were input into the machine learning model to achieve a better tool condition monitoring effect [32].

The finite element method can analyze the cutting force, cutting temperature, cutting speed, and other aspects of the metal cutting process by numerical simulation and has received increasing attention due to its availability and low construction cost. However, there are inevitable errors between the simulation results and the real situation, and how to ensure the validity of the simulation data is a problem that we need to consider. The transfer learning method described in the previous section could not be conducted due to the lack of labeled source domain data.

Therefore, to reduce the requirement for training sample size in monitoring models and lower the cost of TCM experiments, a new TCM method is proposed, combining FEM and domain adaptive methods. The FEM is employed to obtain simulated samples under target cutting conditions, and a deep subdomain adaptation method is established to achieve the transfer from simulation to reality. The proposed method can realize high precision under target domain cutting conditions with a small number of unlabeled samples, significantly reducing the cost of TCM experiments in the target domain (new cutting conditions). The main innovations and contributions of this article are as follows:

(1)

FEM is adopted and verified to obtain simulated samples under target cutting conditions.

(2)

Deep subdomain adaptation is established to transfer simulated samples that are very similar to measured samples into available samples.

(3)

The proposed method can reduce experimental costs significantly for TCM under the condition of small samples.

2. Related Studies

2.1. Time–Frequency Markov Transition Field

The Markov transition field (MTF) was proposed based on a Markov chain to encode one-dimensional time-series data into two-dimensional images [33,34]. Given a time-series $X=\{x_1,x_2,\cdots ,x_n\}$ and its corresponding Markov chain with q states $S=\{s_1,s_2,\cdots ,s_q\}$, the MTF quantifies the multi-step transition probabilities between pairs of data points within the dataset X during the creation of the two-dimensional data representation. The MTF matrix of X with S is as follows:

$$M_{n\times n}=\left[\begin{array}{cccc}\left.p_{ij}\right|x_1\in s_i,x_1\in s_j& \left.p_{ij}\right|x_1\in s_i,x_2\in s_j& \cdots & \left.p_{ij}\right|x_1\in s_i,x_n\in s_j\\ \left.p_{ij}\right|x_2\in s_i,x_1\in s_j& \left.p_{ij}\right|x_2\in s_i,x_2\in s_j& \cdots & \left.p_{ij}\right|x_2\in s_i,x_n\in s_j\\ ⋮& ⋮& \ddots & ⋮\\ \left.p_{ij}\right|x_n\in s_i,x_1\in s_j& \left.p_{ij}\right|x_n\in s_i,x_2\in s_j& \cdots & \left.p_{ij}\right|x_n\in s_i,x_n\in s_j\end{array}\right]$$

(1)

where p_ij is the transition probability from state s_i of x_k to state s_j of x_l.

Given the escalating complexity of mechanical systems, the generation of non-stationary signals is a common occurrence. Traditional signal analysis techniques typically perform statistical average analyses in either the time or frequency domain when dealing with intricate non-stationary sensor data. Such methods often struggle to elucidate the integrated characteristics of these two domains [35], rendering them inadequate for engineering applications involving non-stationary signals. Time–frequency analysis can enable one to discern the frequency components of a signal and unveil its time-varying attributes and proves to be an effective approach for extracting mechanical health information embedded within non-stationary signals. Zhou et al. introduced a time–frequency Markov transform field (TFMTF) algorithm and contrasted it with conventional methods such as the CWT, short-time Fourier transform, and MTF [36]. TFMTF constructed using cutting force signals was found to obtain better classification results. The flowchart of TFMTF is shown in Figure 1.

2.2. Subdomain Adaptation Methods

In the absence of available labeled samples for the new classification task, domain adaptation can transfer the knowledge learned in the source domain to the target domain. Current deep domain adaptation methods mainly consider aligning the global distribution of the two domains using the idea of distance or are adversarial, while ignoring the relationship between different subdomains within the same category, which makes it difficult to capture more detailed and important information between two domains, thus causing the degradation of learning performance. In recent years, researchers have increasingly focused on subdomain adaptive approaches, that is, knowledge transfer by precisely aligning the distributions of relevant subdomains. Most such approaches are adversarial in nature, incorporating multiple loss functions and exhibiting slow convergence. Zhu et al. designed a nonparametric distribution distance estimation method with local maximal mean difference (Local MMD) and constructed a Deep Subdomain Adaptation Network (DSAN), with a simple and effective training process that does not require adversarial training and hence converges quickly [37]. Formally, Local MMD can be defined as follows:

$$d_{\mathcal{H}}(p,q)\triangleq E_c\parallel E_{p^{(c)}}[\varphi (x_s)]-E_{q^{(c)}}[\varphi (x_t)]{\parallel }_{\mathcal{H}}^2$$

(2)

where x_s and x_t are samples of instances from the source domain D_s and target domain D_t, respectively, and p and q are the respective sample distributions of D_s and D_t. By inputting the labeled source domain data and unlabeled target domain data into the neural network for training to obtain the minimum loss, the distribution of relevant subdomains within the same category is approximated using fine-grained information between domains for domain adaptation. Assuming that each sample is classified into the corresponding category according to the weight w, Equation (2) can be expressed as follows:

$${\widehat{d}}_{\mathcal{H}}(p,q)={\displaystyle \frac{1}{C}}{{\displaystyle \sum }}_{c=1}^C\parallel {{\displaystyle \sum }}_{x_i\in {\mathcal{D}}_i}{w}_i^{sc}\varphi (x_i^s)-{{\displaystyle \sum }}_{x_j^{\prime }\in {\mathcal{D}}_i}{w}_j^{tc}\varphi (x_j^t){\parallel }_{\mathcal{H}}^2$$

(3)

where $w_i^{sc}$ and $w_j^{tc}$ are the weights of respective samples $x_i^s$ and $x_j^t$ belonging to a category c, and

$$w_i^c={\displaystyle \frac{y_{ic}}{{{\displaystyle \sum }}_{(x_j,y_j)\in \mathcal{D}}{y}_{jc}}}$$

(4)

where $y_{ic}$ is an element of vector $y_i$ that is similar to the highest score in one-hot coding. Because there is no true label in the unsupervised case of the target domain, ${\widehat{y}}_i^t$ is used to assign a sample $x^t$ to a certain category c.

2.3. Numerical Simulation Based on Deform

(1)

Three-Dimensional Model Construction

Since Deform is not professional three-dimensional drawing software, in the face of the simulation and analysis of complex structures, they need to be premodeled with the help of professional three-dimensional modeling software and then imported into Deform for analysis. In this paper, SolidWorks is employed to minimize the difference between the 3D model and the actual cutting machining process. The dimensions of the workpiece used in the experimental process are 300, 100, and 80 mm, and the diameter and length of the tool are 10 and 75 mm, respectively. To save computation time and storage space, the dimensions of the workpiece in the simulation do not need to be the same as those in the experiments; the dimensions of the workpiece in the simulation are 40, 20, and 10 mm, and the milling cutter model has the same diameter and length as the actual tool. Figure 2 shows the SolidWorks 3D assembly model before and after importing Deform.

(2)

Creating material properties and meshing

Deform V 14.0 software comes with a material library for workpieces and tools, with about 300 materials with relevant parameter attribute data, which can satisfy the deformation or cutting simulation needs of almost every industry while providing convenience to the user. Users can manually add special materials and their corresponding properties according to their needs. The materials used for the tool and workpiece models during the experiments are carbide steel and AISI-1045, respectively. The material properties of the tool and workpiece are shown in

Table 1. Material properties of cutting tools and workpieces.

Materials	Young’s Modulus/Gpa	Density/kg/m3	$\mathbf{Poisson}’\mathbf{s} \mathbf{Ratio}/\mathsf{\mu }$	$\mathbf{Heat} \mathbf{Conductivity}/\mathbf{W}/\mathbf{m}·\mathbf{K}$	$\mathbf{Specific} \mathbf{Heat} \mathbf{Capacity}/\mathbf{J}/\mathbf{kg}·$10−6/K	Coefficient of Thermal Expansion/10−6/K
AISI-1045	200	7800	0.3	47.7	732.6	11
Carbide steel	600	5600	0.25	25	12	9.4

.

The meshing is critical to the results of FEA. A larger number of meshes in a model means a smaller mesh size, a longer computation time, and more accurate simulation results. To balance between calculation accuracy and time, we choose the number of meshes recommended by the software, that is, the tool is divided into 32,000 meshes, and the workpiece into 50,000 meshes. The tool cuts only a part of the surface of the workpiece, and the stress received by this part of the mesh should be increased. To enhance the precision of the contact area, local refinement of the machined surface is applied. The refinement ratio is set to 0.001 after the division of the simulation model shown in Figure 3.

(3)

Tool wear setting

Since tool wear is a process of gradual degradation, simulating it every moment through finite element simulation will greatly increase the time cost of the method. In this paper, the wear length of the lower surface of the tool wear is used as the index, and three different wear states of the tool are constructed for numerical simulation. Figure 4a shows the initial stage of tool wear, Figure 4b shows the stable wear stage of the tool, and Figure 4c shows the severe wear stage.

2.4. Proposed TCM Method

To obtain labeled samples in both the source and target domains during actual processing, this paper proposes an FEM-enhanced adaptive network method for TCM, as shown in Figure 5.

(1)

Deform-based finite element simulation is used to obtain simulation samples. This includes 3D modeling of milling tools with different tool wear levels, with the range of tool wear values set in SolidWorks and imported into Deform to obtain the data under different wear levels by simulation. This paper studies the analysis of tool condition under three wear states.

(2)

The time–frequency Markov representation method is used to characterize the simulation data. The simulation data here are essentially the same operating conditions as the real data, except with different data sources, so the method is equally adapted in this case.

(3)

Model training stage: The data are trained and tested based on the DSAN, which contains modules for feature extraction and subdomain adaptation, through which network learning can further distribute the labeled simulation data closer to the unlabeled real data on the subdomain and realize the effective transfer from virtual to experimental data.

3. TCM Experiment Investigation

3.1. Data Description

The milling experiment uses a CNC milling machine center as the experimental vehicle. During the cutting process, three-directional cutting force signals are collected simultaneously by a charge amplifier and a data acquisition instrument, and a force measuring instrument (Kistler 9139AA) is installed under the workpiece, as shown in Figure 6a. AISI 1045 steel was employed as the machining workpiece, with dimensions of 300 mm × 100 mm × 80 mm.

The cutting tool is a three-flute uncoated carbide steel milling cutter with a diameter of 10 mm. The cutting process is dry slot milling, and the cutting flow is shown in Figure 6b. During the experiment, the spindle speed was adjusted to 2400 rpm, the feed rate was set at 400 mm/min, and the axial depth of cut was maintained at 0.4 mm.

The data acquisition instrument used in this experiment is the MI-70xx series data collector and analyzer provided by Hangzhou Yiheng Technology Co., Ltd. (ECON, Hangzhou, China), as shown in Figure 6c. After the milling cutter completes one cutting task on the workpiece surface, pause the milling operation and position the end milling cutter vertically under the microscope to measure the wear length of each blade (shown in Figure 6d). Take the maximum tool wear value of three blades, VB = Max (VB1, VB2, and VB3), as the current tool wear value. Figure 7 depicts the progression of tool wear, categorized into three groups: slight wear (VB < 0.8 mm), stable wear (0.8 mm ≤ VB ≤ 1.6 mm), and sharp wear (VB > 1.6 mm) [38].

3.2. Results and Analysis

(1)

Verification of similarity between simulation data and experimental data

Deform is used to obtain transferable data under the new working conditions to further realize the virtual-to-real transfer. To assess the validity of the simulated data, the simulation data under normal operating conditions of the tool in the target domain was compared with experimental data. Figure 8a displays the time-series diagrams for the three directions (X, Y, and Z) of the sensor, while Figure 8b presents the spectrograms for these directions. It can be found that, in the frequency domain, the simulated data closely matches the experimental data in terms of frequency features, with only minor differences in amplitude.

In addition, we quantitatively evaluate the similarity between simulation data and experimental data. Due to the limited sample size of the simulated data, only 800 data points are selected as a sample, and the data after the tool cuts into the workpiece by one cutter diameter during the simulation are taken as valid data.

The similarity between the simulated data X and experimental data Y in the three directions is calculated using cosine similarity, as follows:

$$\cos (\theta )={\displaystyle \frac{X·Y}{‖X‖·‖Y‖}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{X}_i\times Y_i}}{\sqrt{{\displaystyle \sum _{i=1}^n{\left(X_i\right)}^2}}\times \sqrt{{\displaystyle \sum _{i=1}^n{\left(Y_i\right)}^2}}}}$$

(5)

The closer the value of cos($\theta$) is to 1, the higher the similarity between the two sets of signals. In practical engineering, the cosine value is generally considered greater than 0.6 to meet requirements [32,39].

The simulated data of 800 points are compared with the experimental data by means of a data sliding window, whose step size is set to 800, and the values of cosine similarity are recorded once for each sliding window. Figure 9 shows the cosine similarity of simulated data and experimental data in the Z direction, which can be seen to reflect high similarity, thus reflecting the validity of the simulated data.

(2)

Analysis and discussion of results

In the experiment, the sample sizes of both the source domain dataset and the target domain dataset were 300. Each sample contains 800 sampling data points, and the original cutting force signal is converted into a two-dimensional image with a size of 224 × 224 through TFMTF and input into the DSAN network for training and testing.

To test the effectiveness of the proposed method, three other methods are introduced for comparison. AlexNet and ResNet50 are both effective methods in the field of CNN, which train simulations and test them directly on experimental data.

Deep domain adaptive network (DDAN) jointly learns simulated data (source domain) and unlabeled experimental data (target domain) using the multicore maximum mean difference as a loss function between the two domains and continuously optimizing the distance between them through network updates so that they are distributed as consistently as possible in the high-dimensional feature space [39].

Deep Subdomain Adaptation Network (DSAN) is an improved DDAN method. Most traditional domain adaptive networks are globally adaptive, thus ignoring the relationship between the subdomains within the domain, leading to confusion when classifying data from each subdomain. DSAN takes into account the relational subdomains between subdomains and introduces a Local MMD domain loss function [37]. The comparison of the classification outcomes for the six methods using single-channel data and the comparison of results in the scenario of three-channel fusion are presented in

Table 2. Classification accuracy of four methods.

Dataset	AlexNet	ResNet50	DDAN	DSAN
Single channel	67.54%	66.77%	94.63%	97.57%
Three channel	74.33%	73.61%	95.67%	99%

.

From Table 2, it can be seen that when using only single-channel data, AlexNet and ResNet50 are trained directly on simulated data, and the experimental data are used for testing. It can be seen that the classification results differ greatly from those of the methods using domain adaptation, that is, DDAN and DSAN. DSAN, as a subdomain adaptive method, can use the simulated data as the source domain data and, through transfer learning, can realize tool status monitoring without available labeled data in both domains. The classification accuracy can reach 97.57% in the case of a single channel and 99% in the case of three-channel fusion.

4. Conclusions

To reduce TCM experimental costs under new cutting conditions, an FEM-enhanced DSAN method is proposed. This method generates simulated signals of different tool conditions via FEM modeling, eliminating the need for many experimental samples. It employs time–frequency domain feature representation to characterize the original signals, thereby forming training samples without the laborious process of feature extraction. The simulated samples are then transferred to unlabeled experimental samples using DSAN, thereby reducing the costs associated with machining experiments. With a limited number of unlabeled experimental samples under new cutting conditions, the proposed method achieved a classification accuracy of 97.57% utilizing a single-channel sensing signal and an impressive 99% accuracy with a three-channel sensing signal.

The contribution of the proposed method is that, in new cutting conditions (target domain), only a small amount of sample data for the normal condition of the tool needs to be collected to verify the effectiveness of simulated sample data through FEM, without the need to collect sensing data for different wear conditions of the tool (these sample data for different wear conditions can be obtained through simulation through an FEM model that has passed effectiveness testing). In this way, the number of experiments and data collection time have been reduced. Thus, the proposed method provides a feasible way to realize high-precision tool condition monitoring with low experimental cost.