1. Introduction
The virtual simulation of the manufacturing process is an interesting topic for many researchers. The conventional machining process is necessary in the manufacturing industry to produce high-precision parts. The wide application of machined parts and the high complexity involved in the machining of materials to achieve the desired shapes and properties has motivated researchers for decades [
1]. The analytical and experimental investigations provide certain information to develop the process, but still, this information are inadequate. Experimental analysis is highly expensive and time-consuming, and it is difficult to measure variables such as stresses, strain, and temperature distribution [
2]. The significant advancement in computation technologies enables us to develop a numerical model of the orthogonal cutting process [
1].
Finite Element (FE) modeling is the most prominent numerical modeling technique for the simulation of the orthogonal metal cutting process [
3] and the high-speed grinding of titanium alloy [
4]. The FE simulation of the chip-formation process replaces the expensive experimental test and predicts the difficult-to-measure variables and results with higher accuracy than an analytical model [
5]. The modeling of the complex machining process by the FE model is quite challenging as it involves various inputs. The efficiency of the cutting model is dependent on the numerical parameters such as the formulation type (Lagrangian, Eulerian, Arbitrary Lagrangian-Eulerian, or Coupled Eulerian–Lagrangian), the quality of the mesh [
6], boundary conditions, constitutive models [
7], and contact conditions [
8]. Accurate material models and friction conditions between the tool-chip are essential to obtain accurate and reliable results from the simulation [
9]. A reliable flow of stress data which relates the large plastic strains (1–6) at the very high strain rates (
s
) and very high temperatures (800 K to 1400 K) observed during the machining process is necessary to frame the model. In this work, the Ti6Al4V alloy, an expensive alloy commonly used alloy for its excellent properties in the aerospace, biomedical, and marine fields, is considered for cutting process simulation.
In numerical modeling of the machining process, many different material models are employed, and they are classified as empirical/phenomenological, physical-based, and hybrid models [
8]. The empirical models are highly recommended for their robustness, lower number of parameters, and the large availability of data when compared with physical-based and hybrid models [
8]. Likewise, many friction models are available which are directly associated with the behavior of the material [
10]. However, the credibility of the material model and the friction model depends on the pertinent parameters involved in defining the behavior of the material during machining process. These material model parameters are determined using a direct method or an inverse method [
1]. The direct method is the dedicated experimental tests to obtain information. The experimental methods use curve fitting techniques to describe the experimental data from quasi-static and dynamic material tests such as the Split Hopkinson Pressure Bar (SHPB) test [
11].
Nevertheless, these experiments can reach a maximum strain of 0.5 and a strain rate near
s
, which is well below the strain of 3 and the strain rate above
s
that are encountered during the cutting process, which makes the extrapolation of data necessary [
12]. Although the pin-on ring, open and closed tribometers [
13] friction test is available to determine the friction characteristics during the cutting process, the information is uncertain due to the phenomena taking place at the tool–chip contact area [
14]. In [
15], Sahoo et al. worked on tool coating and its relation with friction coefficient. They stated that, due to the lower friction coefficient of the coating material, the stress, strain, and temperature generation and tool wear rate in the case of TiAlN-coated tools are comparatively lower than uncoated WC tools. This adds further arguments for considering optimizing the friction coefficient with the constitutive model parameters in the identification framework.
The inverse methods are mainly used to overcome the drawback of extrapolation. In this context, Ozel and Altan developed the earliest approach to inversely identify the parameters from the cutting process [
16]. In their study, the authors claim that the method can achieve less than 10% deviation from the measured and simulated cutting forces by using the flow stress data from the low-strain and strain rate tests as an initial starting point. In [
17], Shrot and Baeker employed the Levenberg–Marquardt algorithm to re-identify some of the parameters of the Johnson–Cook model.
In [
18], Klocke et al. proposed an inverse approach to determine the JC material and damage parameters for AISI 316L stainless steel. To determine the model parameters, the lower and upper values that underestimated and overestimated the experimental results were guessed, and the material model parameters were interpolated to find the best fit with the experimental data. Later, they adopted the same approach to determine the material model parameters of AISI 1045 and Inconel 718.
In [
19], Bosetti et al. compared Pure NMM (Nelder–Mead method), r-NMM, and Hybrid (Genetic Algorithm and NMM) approaches to determine the five JC parameters and Tresca friction parameter of AISI 304 stainless steel. Deviations up to 113% have been observed between the simulated and experimental observables. The authors state that the range of validity is limited, and the number of simulations and computation time depend on the initial guess for the parameters. Denkena et al. utilized Particle Swarm Optimization (PSO) algorithm in conjunction with Oxley’s machining theory [
20]. This approach suffers from the drawback of using the assumption of Oxley’s machining theory.
In recent times, Bergs et al. [
21] adapted a gradient-free approach called the Downhill Simplex Algorithm (DSA) for the inverse identification of the material model parameters. In this approach, they investigated the method with the inverse reidentification of a set of initial material model parameters taken from the literature. He observed a close match between the target and the simulated process observables. The authors applied this approach to experimental data from AISI 1045 and claim that the results are in agreement with the experimental results [
22]. Hardt et al. [
23] investigated this approach to evaluate the robustness of the algorithm to determine the parameter set and revealed the drawbacks associated with the algorithm, such as the algorithm being stuck in local minima.
In [
24], the authors also used the PSO meta-heuristic to determine the parameters of the JC model for AISI 1045 steel. In this approach, the authors bound the upper limit and lower limit with an empirical value and investigated different swarm sizes of the PSO algorithm. The authors claim the approach can inversely re-identify the parameters of the constitutive model in a few iterations when compared with his earlier work [
23]. In [
25], Hardt et al. extended his work to include the automation of the post-processing of the results and increased the number of observables in the objective function. The authors concluded that different parameter sets identified by the algorithm result in the prediction of identical temperature, stress, and strain profiles, highlighting the non-uniqueness.
In the literature, mostly the parameters of the constitutive model are considered for the inverse identification process. The optimization algorithms implemented in the literature for the inverse identification procedure is computationally expensive (9 to 30 days minimum in parallel computation domain [
24]). In addition, automatized optimization procedures need to be improved to transfer and interpret the data more efficiently.
In this present work, an identification procedure is proposed to inversely identify the value of the Johnson–Cook (JC) parameters and the Coulomb’s friction coefficient correlatively, with the objective function being to minimize the error difference between experimental and numerical results. The inverse identification problem is tackled with the Finite Element simulation-based optimization concept. This work is a novel approach to overcome the drawbacks stated in the literature for the inverse identification of parameters. The first novelty of the work is the implementation of a complete automatized routine to perform the surrogate-guided optimization procedure to identify the model parameters values in the context of machining process simulation. The optimization process is based on the Bayesian Optimization (BO) algorithm named Efficient Global Optimization (EGO).
The EGO algorithm, first introduced by Jones et al. [
26], uses a surrogate model to approximate the objective function (i.e., the difference between simulation and experiment) and determine the value of a candidate point to be exactly evaluated with the simulator (i.e., the FE model). It consequently alleviates the time cost associated with optimization since only valuable candidate points are simulated. EGO rests on Gaussian Process (GP) surrogate models for their ability to provide both a prediction and a measure of uncertainty around it. This characteristic allows one to define an Acquisition Function (AF) that assesses the value of a candidate point before evaluation (with the time-consuming simulator). Intuitively, the AF can be seen as an agent deciding if a region of the search space is worth sampling or not. It is then responsible for the Exploration/Exploitation trade-off in the optimization process. In the context of time-consuming objective function with low budget and no place for parallel evaluations, the sequential EGO algorithm seems to be appropriate [
27] for this work. In addition, the influence of algorithm parameters such as the influence of weight, the number of initial points to train the surrogate model, and different sets of bounds are investigated in this work, which contributes to additional novelty to the paper.
The Machine Learning algorithm combined with automation makes an efficient Artificial Intelligence (AI) platform to identify the parameter’s value of the models. The proposed AI platform is significant in identifying the best parameter’s values for the JC constitutive and Coulomb’s friction model to predict the observables with less deviation from experiments when compared with the best parameter set stated in the literature by Ducobu et al. [
28]. In addition, the optimization procedure reduces the total computation time to a maximum of 8 days without the need for a parallel computing domain.
The paper is organized as follows: In
Section 2, Materials and Methods, the constitutive and friction models of the surrogate-guided optimization theoretical background and the EGO algorithm used within this work are outlined, followed by the experimental reference. The Arbitrary Lagrangian–Eulerian Finite Element model implemented for the simulation of orthogonal cutting is given in
Section 3. In
Section 4, the FE simulation-based optimization procedure is discussed in detail, followed by the framing of the optimization problem and the methodology to investigate the algorithm. The numerical results are also presented and briefed in
Section 4. The identified parameters’ values are analyzed and their applicability for other cutting conditions is critically discussed in
Section 5. The conclusions are drawn based on the significance of the optimization procedure and the identified parameter sets in
Section 6.
3. Finite Element Orthogonal Cutting Model
In FE modeling, the Eulerian and the Lagrangian formulations [
1,
5] are usually considered. In the Eulerian approach, the computational mesh is fixed, and the material moves with respect to the grid, which allows handling large distortions. Prior information on the chip geometry is required to model the machining simulations with the Eulerian formulation [
1], and it is adopted only for steady-state chip formation. In Lagrangian formulation, the nodes of the mesh are attached to the material and follow the material’s deformation. It may induce large distortions in the domain, and frequent remeshing operations may be necessary to adapt large deformations. In addition, without remeshing, the Lagrangian formulation needs chip separation criteria [
5].
To overcome the drawbacks of the purely Eulerian and purely Lagrangian formulations, two other formulations which combine the merits of Lagrangian and Eulerian have been developed. They are the Arbitrary Lagrangian–Eulerian (ALE) and the Coupled Eulerian–Lagrangian (CEL) formulations. In the ALE formulation, the material flows through the mesh like in the Eulerian formulation. Because of this freedom in movement of the mesh, the ALE description can accommodate high distortions with more resolution [
40]. In the CEL formulation, a Lagrangian part is modeled within a Eulerian domain, and the efficiency of the model depends on the Eulerian mesh definition; no mesh distortion occurs [
6].
In this work, an explicit ALE finite element formulation was adopted to simulate the orthogonal cutting process of Ti6Al4V. This ALE formulation combines the advantage of Lagrangian and Eulerian formulations, which allows one to take into account the large deformations during the material flow around the cutting edge of the tool without using a chip separation criterion. A two-dimensional (2D) plane strain model with orthogonal cutting assumption was considered for this work. The finite element software Abaqus was used to model the thermo-mechanical chip formation process.
In this FE model, the tool is fixed, and the workpiece moves at the prescribed cutting speed. The tool and the workpiece are meshed with quadrilateral elements with reduced integration for a coupled temperature–displacement calculation (CPE4RT). The length of the workpiece is 3
h, where
h is the uncut chip thickness. To achieve a better trade-off between the element size and computation time, the area near the cutting zone (near the tool-tip) was modeled with a finer mesh of size 5 µm. In this approach, the initial geometry of the chip must be predefined with respect to the uncut chip thickness (
h). The workpiece inflow and outflow surfaces, as well as the chip top surface, were modeled as Eulerian surfaces, and adaptive constraints were applied. For the tool, tungsten carbide was considered, and the linear elastic law was imposed [
41]. The chemical composition of Ti6Al4V is given in
Table 3.
The material properties of Ti6Al4V considered for this work are given in
Table 4.
The tool geometry and the cutting conditions are given in
Table 5.
The initial geometry and the boundary conditions are illustrated in
Figure 3. The thermal properties are adopted from the literature [
44,
45]. The initial temperature for tool and work piece is set to 298 K. The mass scaling was considered to artificially increase the critical time increment in the simulations. A mass scaling factor of 1000 was considered, as it shows a significant decrease in the computational time without affecting the results [
28]. This approach is crucial to reach the steady state (it is enough in these cutting conditions) for force calculations with less computation time (42 min with 6 cores Intel
® Core
™ i7-10700 CPU @2.90 GHz with the memory of 16 GB). The specifications of the PC machine used for the FE simulations are given in
Table 6.
The reaction forces and the chip characteristics from the FE simulation were processed to calculate the cutting force, feed force, and the chip thickness
5. Discussion
The four optimization models (
) considered in this work perform well in determining the model parameters value with a maximum computation time of 8.5 days. Indeed, the total deviation of the simulated results from the experimental results lies in the range of 19% to 29%. The identified parameter sets from the optimization for the JC model along with the friction coefficient can predict the forces and the chip thickness 62% more efficiently than the best parameter sets determined by Ducobu et al. [
32] (Their method was a critical investigation of the available parameter sets in the literature for the machining of Ti6Al4V alloys, and friction was not included).
In the machining process, knowledge about the forces is of utmost importance in optimizing the cutting process. Taking that into account, more interest is put on the forces’ prediction with the identified parameter sets. The difference (
) of simulated results with the experimental results with respect to the forces is given in
Table 14.
The parameters set found by the optimization models
and
predict the cutting force and feed force with less deviation from the experimental results when compared with models
and
. Indeed, the cutting force predicted by the parameter sets identified by
and
is in the range of 9% to 10%, which is slightly more when compared with the reference parameter sets considered in this study [
1,
46]. However, the feed force predicted by the identified parameter sets by the optimization models
and
significantly reduces the deviation in the numerical results from the experimental results. The parameter set from
predicts the cutting force near the experimental result, as the optimization model is defined with a greater weight coefficient on the cutting force than the feed force, which is reflected in the parameter identification process of the algorithm. Even though the parameter set from
predicts the total forces with a deviation of around 10%, the cutting force deviation could be improved. This is due to the definition in the equal weight coefficients of the observables. The model
predicts the forces with a deviation of 18% from the experimental results. Indeed, the identified parameter sets can predict the same chip thickness as the experimental reference but fail to predict the forces with less deviation from the experiments. The model
predicts the forces with a deviation of 15%; this can be explained by the increase in the range of the
.
The JC parameter sets and the Coulomb’s friction coefficient identified by the optimization models
and
are selected for further analysis, as they provide a good trade-off between the predicted forces and the chip thickness. Two other values of uncut chip thickness,
h = 0.04 mm and
h = 0.06 mm, are introduced to further analyze and validate the identified parameters from the optimization. The results from these FE simulations are compared with the experimental results. The RMS values of the forces and the chip thickness for uncut chip thicknesses of
h = 0.04 mm and 0.06 mm are given in
Table 15.
The identified parameter sets for the JC and Coulomb’s friction coefficients by the and optimization models can accurately predict the cutting force (deviation within 4%) for uncut chip thicknesses of h = 0.04 mm and 0.06 mm. Meanwhile, the feed force is overestimated (16% to 33%). The chip thickness prediction shows some improvement when compared with the reference model, even though they are overestimated (14% to 19%) when compared with the experimental measurements.
The inversely identified parameters values of the JC model and Coulomb’s friction coefficient value qualitatively predict the observables for the cutting condition considered for optimization. The accuracy is lower for the other cutting conditions but still quite good, particularly for the cutting force. However, this analysis on cutting conditions is different than the one conducted for the optimization, showing the limits of an optimization performed for a single cutting condition. Nevertheless, the prediction of the cutting forces with the identified parameter sets are in close agreement (<10%) with the industrial trends, where the cutting force prediction has fundamental importance for the optimization of cutting conditions, tool design, and also for tool wear/life prediction [
1]. Indeed, the identified parameter sets can predict the cutting forces within a deviation of 6% for all the cutting conditions considered in this study, while Arrazola et al. [
1] state that the experimental measurement trials and the variations in the material properties of the same material can cause variations of around 10% in forces.