Research on prediction method of surface roughness in weak magnetorheological shear thickening fluid polishing

A joint prediction method of “mathematical modeling and finite element calculation” is proposed to improve the prediction of machining quality in weak magnetorheological shear thickening fluid polishing of complex surfaces. The study proceeded in several steps. First, based on both impact energy model and material removal model, a numerical prediction model of surface roughness is established. Second, based on the multi-peak fitting method, the field-induced rheological properties of the polishing fluid are characterized and material properties of the flow field medium in the polishing zone are defined. Third, the numerical boundaries of polishing flow velocity and shear stress in the above prediction model are obtained. Fourth, the polishing experiments with parameters consistent with the above simulation model are conducted, and the initial surface roughness values are substituted into the above prediction model. The results show that the joint prediction method can effectively predict the machining quality of workpiece surface. The absolute error of Sa value of surface roughness is up to 10.6 nm, and the maximum relative error is 12.3%.


Introduction
The flexible polishing strategy has become the mainstream in the field of ultra-precision machining, as it is able to meet the requirements of ultra-precision non-destructive machining of hard and brittle materials and complex surfaces [1,2]. Mature flexible processing technologies such as magnetorheological polishing [3], balloon polishing [4], and abrasive flow processing [5] have been developed, moving research in the direction of multi-energy field-assisted processing [6].
Due to the field-induced rheological properties of polishing medium, it is difficult to predict the machining quality, which is needed for the optimization of the machining process. The common research strategy has so far been to apply the theoretical prediction model and the finite element simulation model (both with experimental verification) to predict the processing quality so as to optimize the process of polishing methods [7]. Guo et al. [8] evaluated the influence of polishing flow viscosity, abrasive particle size, extrusion pressure, low temperature, and other parameters on the processed surface and sub-surface quality by modeling the evolution process of abrasive flow machining (AFM) micro-surface topography. Alam et al. [9] proposed the microscopic particle arrangement structure of magnetorheological polishing fluid, analyzed the stress model acted on abrasive particles by the above structure and the wear mechanism between abrasive particles and workpiece surface, and then established an effective surface roughness prediction model. Ghosh et al. [10] established a theoretical prediction model of machined surface roughness for the wheel-based magnetorheological polishing process of oxygen free high conductivity (OFHC) copper, effectively optimizing the process parameters such as polishing wheel speed, machining clearance and feed speed, and finally obtained the ultra-precision machined surface with minimum Sa of 15.5 nm and residual compressive stress of 6.9 MPa. Wang et al. [11] conducted a quantitative analysis of processed surfaces based on mathematical statistics on directional and non-directional texture models, and demonstrated that dual-rotation magnetorheological finishing (DRMRF) can suppress directional texture and effectively improve surface roughness. Arora et al. [12] established a theoretical prediction model on surface roughness of straight bevel gear (SB) machining based on new magnetic bevel gear polishing (BGF), which can effectively complete the surface quality prediction under multiple cycles. Li et al. [13] established the abrasive wear theory based on Brinell hardness value, shear thickening mechanism, and plastic indentation theory to predict the surface roughness value.
Weak magnetorheology shear thickening fluid polishing is a non-Newtonian fluid polishing method based on magnetic field assistance [14,15]. By the synergistic effect of magnetorheological shear thickening, the method has the ability to achieve ultra-precision machining of difficult materials by wrapping the microscopic convex peaks that the abrasive particles collide and act on the rough surface of the workpiece [16]. However, the analysis of the interaction model between abrasive particles and the machined surface is extremely complicated due to the complexity of fieldinduced rheological properties [17][18][19], the complex interaction forces of various particles in the polishing fluid [20,21], and the size effect and other factors [22,23]. To improve the model, researchers have started to employ the strategy of surface roughness prediction and process parameter optimization based on the finite element simulation model. Nguyen [24] used the CFD simulation model to explore the influence of factors such as the workpiece inclination angle, fluid viscosity and polishing flow speed on pressure distribution characteristics, and flow characteristics of the polishing zone, as a way to optimize process parameters. Based on the support vector regression (SVR) technology, Zhang et al. [25] obtained the stress distribution characteristics of the finite element simulation calculation model of the contact area in real time with higher computational accuracy and faster convergence speed, and further optimized the online control of the manipulator. In addition, Fu et al. [26] adopted a research method combining numerical simulation and experiment to obtain the medium flow state and dynamic pressure distribution in the polishing area, thus realizing the design optimization of the processing fixture, and finally obtaining the blade processed surface with high uniformity. However, most studies on this remain at the stage of examining distribution characteristics of magnetic flux, dynamic pressure, or related stress in the material removal process, and research has yet to achieve the direct numerical prediction of machined surface quality.
This study proposes a joint prediction method of "mathematical modeling and finite element calculation" for the numerical prediction of machining quality in the weak magnetorheological shear thickening fluid polishing (WMSTFP) of complex surfaces. The study proceeded as follows. Based on the weak magnetorheological thickening effect and the study of the magnetorheology shear thickening and polishing method [27][28][29][30], a corresponding polishing experimental platform was established and the weak magnetorheological shear thickening polishing fluid (WMSTPF) was prepared. According to the rheological test parameters, a multi-peak fitting method based on Gaussian function was used to establish the constitutive equation, and rheological properties of the medium were compiled and imported by using the function of self-defining material properties. Finally, based on the material removal mechanism of microscopic surface, the WMSTFP machined surface quality prediction of complex workpiece surface was completed by using the joint strategy of medium magnetic flow simulation modeling and surface roughness theory prediction modeling.

Relevant theory preparation
In the main polishing area, the polishing flow direction is opposite to the rotation speed of the workpiece. Abrasive particles in the polishing flow gather on the surface of the workpiece under the action of coercion. The force of the aggregation behavior is derived from the weak magnetorheological thickening effect [27], which results in the high viscosity of the polishing flow in the polishing region and the shear stress of the fluid at a place on the workpiece surface τ(x), as shown in Fig. 1a. Under the above effects, the thickened phase particles and the free magnetic particle chains form composite clusters under the synergistic effect of shear disturbance and the weak magnetic field, which carry abrasive particles and continuously impact the convex peaks on the rough surface at the inlet velocity u n , leading to the fracture of the material bond (microscopic) and material removal (macroscopic). In the microscopic collision process, abrasive particles and the convex peak act in opposite directions to the shear stress of the polishing flow and the shear yield stress of the material respectively, as shown in Fig. 1b.
As the surface profile of the workpiece is hemispherical in this study, surface roughness evaluation parameter Sa is selected to evaluate the height and numerical distribution of the microscopic surface topography. The equation describing statistical evaluation parameter Sa is as follows: where A is the effective material removal area and h(x,y) is the height of a microscopic bulge or depression.
The abrasive particles bound by the polishing fluid hit the micro-bumps on the rough surface of the workpiece. According to the law of conservation of energy, the kinetic energy of abrasive particles is partly converted into the internal energy of polishing fluid, partly into the internal energy of workpiece, and partly into the kinetic energy of the removed material. Among them, the part converted into the internal energy of the workpiece leads to the local fracture (dislocation, crack) or the overall fracture (material removal) of the crystal structure of the material [31]. Mathematical modeling of the above energy conversion process is extremely complex, so it needs to be analyzed and simplified with statistics. The abrasive collision results are divided into two cases. Firstly, the abrasive particle collides with the convex peak of the rough surface, but no overall fracture is caused, and the kinetic energy is mostly converted into the internal energy of polishing fluid and workpiece. Secondly, the above collision results in the overall fracture of the convex peak, and the kinetic energy of the abrasive particle is mainly converted into the internal energy of the workpiece and the kinetic energy of the workpiece material (fragment). According to Eq. (1), Sa is a statistical evaluation parameter for the fluctuation of the microscopic surface topography. Therefore, it can be approximated that the greater the variation of the Sa parameter value of the microscopic surface before and after machining, the greater the proportion of the overall fracture of the crystal structure of the workpiece surface material in this area. The ratio of before and after machining surface's Sa is constructed by power function to obtain the proportion coefficient of the whole fractured crystal bond, which can be used to describe the probability of the whole fractured crystal bond on the rough surface of the workpiece during the collision process. According to the above analysis, Eq. (2) describing the total energy E TFE of the overall fracture of the crystal bond structure is as follows: where E TFE is the total energy of fracture in the effective material removal region, Sa n i /Sa 0n i is the proportion of surface roughness statistical parameter before and after processing (where n is the experimental group number and i is the test point number), E FE is the fracture energy of single crystal bond of the workpiece material (757.8 kJ/mol of Zr-O bond), N c is the number of fracture crystal bond in the target region (mol), and k 1 is the statistical influence coefficient of surface roughness change ratio. The number of broken bond N c cannot be calculated and solved directly. The larger the material removal rate in the region, the greater the number of crystal bonds broken. Therefore, the volume of material removal and the numerical value of crystal bond fracture can be approximated as a power function [32] (based on the relationship between removal depth and area, k 3 is roughly estimated to be about 2). Therefore, according to the mathematical model of material removal rate established in the preceding research [27], the expression of N c can be obtained: wherein K is the correlation coefficient of material removal rate, m is the velocity index, k 3 is the fracture correlation index, and k 4 is the fracture correlation coefficient.
According to the energy conservation law in the collision process, the energy of the material removal process on the microscopic surface, or the fracture behavior of the crystal bond structure at the convex peak, comes from the kinetic energy of abrasive particles flowing with the polishing fluid [31,32]. However, in the process of collision, the kinetic energy of abrasive particles is not only converted into the fracture energy of the crystal bond but also consumed in the composite cluster containing abrasive particles due to the reaction force. When the shear stress of weak magnetorheological shear thickening fluid containing abrasive particles is large, the kinetic energy of abrasive particles can be more effectively converted into fracture energy needed for material removal Fig. 1 a Simplified model of material removal mechanism during machining; b particle flow thickening removal mechanism on workpiece surface. Therefore, the ratio of shear stress of polishing flow to shear yield stress of workpiece material can reflect the energy conversion efficiency in the material removal process. Based on the above analysis, E TFE based on energy conservation can be described as follows: where N a is the number of abrasive particles in the flow channel section corresponding to entering the polishing area, m a is the weight of abrasive particles, u(x) is the velocity distribution function of polishing fluid on the workpiece surface, τ(x) is the shear stress distribution on the workpiece surface, τ w is the shear yield stress of the workpiece, and k 2 is the effective collision coefficient.
Assuming that abrasive particles are spherical with uniform texture, the expression of the abrasive mass is as follows: where ρ a is the density of abrasive grains and d a is the average diameter of abrasive grains.
Assuming that the abrasive particles in the polishing flow are evenly distributed, the number of abrasive particles entering the polishing zone can be expressed as: where u n is the flow velocity of polishing fluid inlet, S is the flow channel cross section area of polishing area, ρ F is the polishing fluid density, and w a is the mass fraction of abrasive particles. Simultaneous Eqs. (2) and (4), and substituted into Eqs. (3) and (6), simplification can be obtained: In particular, due to the synergy of weak magnetorheological effect and shear thickening effect, when the polishing fluid flows through the hemispherical surface of the workpiece and the clearance of the polishing pool wall, as shown in Fig. 2a, it presents a flow variant state; that is, the fluid shear stress is not evenly distributed in space. This is because when the polishing fluid with field-induced rheological properties flows through the gap between the workpiece and the polishing pool wall, the flow section shape becomes sharply narrow, resulting in a sharp increase in shear rate, and the numerical distribution is directly related to the section shape [33], as shown in Fig. 2b.
It is difficult to calculate the flow field on the workpiece surface because of the complex cross section shape of the flow channel in the polishing area. In the aforementioned study [27], simplified mathematical modeling was established for the shear rate and polishing flow velocity of workpiece surface in the polishing zone.
The above model is an approximate representation of the actual situation. In addition, according to the numerical distribution function of shear rate and the constitutive equation of polishing fluid, the shear stress of the corresponding position of polishing flow should be calculated. In order to avoid the complicated mathematical modeling process, the numerical distribution of velocity (u(x)) and shear stress (τ(x)) in the polishing area are calculated by using the finite element method.

Polishing liquid
To simulate more accurately how the field condition of the workpiece surface is subjected to fluid action, it is necessary to detect and characterize the rheological characteristics of polishing flow. The preparation scheme and information about the manufacturer is shown in Table 1. The particles selected for the test sample, the proportion of preparation, and the preparation method are all used in the above study [28,29], which will not be described here. The aforementioned rheological properties of WMSTPF at different magnetic field strengths are tested, and the results are shown in Fig. 3. It can be seen that the WMSTPF exhibits the best thickening performance when the strength of the applied magnetic field is around 100 mT. Therefore, the magnetic field strength of the surface polishing area of the workpiece in the experiment is designed to be this parameter, that is, 100 mT. In addition, in the flow field simulation analysis, the rheological properties under this parameter are chosen as the fluid material properties.

Polishing experiment
Establish the processing platform as shown in Fig. 4. The workpiece is set as hemispherical zirconia ceramics, whose mechanical properties are shown in Table 2. The rotating motion of workpiece fixture is provided by the vertical milling machine (working speed range); The rotary motion of the polishing groove is provided by the lower belt drive motor (rated power, working speed range). The uniform magnetic field near the polishing area of the workpiece is provided by the rectangular block of strong magnetic material (NdFeB), and its magnetic energy parameters are shown in Table 3.
Based on the weak magnetorheological strengthening thickening effect and rheological properties of the polishing fluid, it is necessary to ensure the weak-magnetorheological  To obtain a more uniform weak magnetic field in the polishing area, the excitation device was improved using an electromagnetic simulation software (Ansoft Maxwell). The simulation results in Fig. 5 show that the scheme of placing permanent magnets at an appropriate distance inside and outside the workpiece can produce a more uniform strong and weak magnetic field. At the same time, it can be seen from the rheological property parameters that the weakmagnetorheological strengthening and thickening effect can be obtained under the condition of weak magnetic field of about 70-100 mT [27], with higher thickening performance. Therefore, based on the numerical distribution results of magnetic field intensity attenuation from the wall to the workpiece surface, the space position of the workpiece is set at 10 mm from both the side wall surface (D z ) and the bottom wall surface (D y ). Furthermore, according to Newton's law of internal friction, as shown in Eq. (8), and rheological properties of polishing flow, the wall velocity corresponding to the shear rate range of fluid with high viscosity can be calculated. In addition, the polishing experiment set four different speeds of the polishing pool (N 1-4 ), which were respectively classified into four experimental groups G1-4. These parameters and others are shown in Table 4. In particular, a test point is set every 15° along the surface of the workpiece, and the numbers are 0-6 in sequence. The points 3-6 in the main polishing area are set as a-d (as shown in the subgraph of Fig. 4). Then, the naming rule of experimental data number is "experimental group number + test point number," such as G1-a.   where τ is the shear stress, η is the viscosity of the fluid, and dv/dy is the rate of change of flow velocity distribution in the wall clearance, i.e., velocity gradient. Meanwhile, the experimental parameter of the aforementioned material removal rate prediction model [27], as shown in Table 4, was used in the surface roughness prediction method of this study.

Simplified model and grid division
The polishing area of the experimental platform is approximated as a rectangular DC channel model centered on the position of the workpiece, and the geometrical size of the channel is 200 × 50 × 50 mm. Local mesh encryption was performed on the surface of the hemispherical workpiece, and the mesh was divided. The number of grids was 531,037, as shown in Fig. 6. The x-direction is the flow direction of the fluid, the negative direction is the inlet, and the positive direction is the outlet. The remaining boundary is the wall.

User-defined material properties
Based on the above rheological test results, the material properties (constitutive equation) of the polishing fluid were characterized. The power law constitutive model is built into the simulation software, which can be used for fluid materials with traditional power law characteristic curves. However, for the fluid used in this study, the characterization accuracy is extremely low, as shown in Fig. 7a. In order to characterize the viscosity function of polishing flow in the non-Newtonian fluid polishing process, rheological properties at the shear rate range under working conditions were characterized based on the power-law constitutive equation  and the multi-mode model for the characterization of complex rheological properties [34,35]. Further, according to this method, the polishing fluid used in this study was segmented, as shown in Fig. 7b. However, this method has some shortcomings in fluid characterization and cannot deal with the simulation calculation at the inflection point of fluid viscosity change. The multi-peak fitting method based on the previous study [28] can effectively achieve the fitting characterization of the rheological properties of the whole region through Gaussian function (Formula 9), and the fitting accuracy is as high as R-square 0.993, as shown in Fig. 7c. Further, based on the above fitting method and constitutive parameters, fluid material properties are defined in the C language compilation environment ( Table 5).

Simulation parameter setting
On the basis of the above simplified mathematical modeling and finite element mesh division, the flow field finite element simulation software (WORKBENCH 2021 R1) and simulation parameters are set and solved. Firstly, the general setting was carried out: gravitational acceleration and centrifugal acceleration parameters were introduced by Formula (10), and the centrifugal acceleration values were set as − 5.48, − 14.03, − 26.52, and − 42.96 m/s 2 , respectively, according to the experimental process parameters (as shown in Table 4; N 1−4 = 50, 80, 110, 140 rpm; R = 200 mm).
where ω is the angular velocity of rotation of the polishing pool, R is the radius of the polishing pool, and N is the rotational speed of the polishing pool. The model to be solved is set as the steady-state solver and the pressure boundary conditions. The Eulerian multinomial model was used, the main phase was set as the base carrier liquid (WMSTF) of the continuous phase, and the secondary phase was set as the abrasive dispersed phase (SiC), and the related physical characteristics parameters were set. In addition, due to the high viscosity and shear thickening characteristics of the polishing fluid, the laminar viscosity model and dispersed phase model were turned on and relevant parameters were set [35,36]. The material properties of continuous and dispersed phases were respectively set: the density of the continuous phase was 1530 kg/m 3 (measured by experiment), and the material properties of the dispersed phase were directly called from the material library. In addition, viscosity parameters of the continuous phase are set by introducing custom material properties based on the above constitutive equations and their empirical parameters, as shown in Table 4 and Eq. (9). Then, the boundary conditions of import are set. According to the process parameters of the polishing experiment and calculation (Formula (11)), the inlet flow rates of polishing fluid were set as 1.05, 1.68, 2.30, and 2.93 m/s, respectively. Finally, the basic initialization of the calculation is completed and the calculation can be committed.

Flow field characteristics of WMSTP polishing area
According to the above CFD simulation method in Section 2.4, the corresponding energy field and velocity field distribution of the flow field can be obtained. Figure 8 shows the numerical distribution of dynamic pressure in the flow field. The peak value of dynamic pressure appears on the negative side of the workpiece near the z-axis, which is the main polishing area of fluid polishing. A low dynamic pressure area appeared on both sides of the workpiece along the x-axis, which was caused by the lateral vortices caused by the workpiece rotation, forming a local low pressure. On the side near the positive direction of z-axis, the dynamic pressure is higher. This is because the workpiece rotation is consistent with the polishing flow direction, which leads to the increase in both velocity on this section and dynamic pressure.

Velocity distribution in the polishing zone
In fluid polishing, the distribution of polishing flow velocity on the workpiece surface has a direct effect on the machining quality. Therefore, the velocity field of the polishing zone under boundary conditions of different inlet polishing flow velocity was calculated with the finite element method, as shown in Fig. 9. It can be seen from the simulation results that the maximum velocity in the polishing area increases with the inlet velocity of polishing fluid, which peaks at 15.9 m/s. Based on the Preston equation, the tangential velocity of the polishing flow acting on the workpiece surface is positively correlated with the material removal rate. However, due to the complex field-induced non-Newtonian rheological properties of the polishing fluid in this study, it is difficult to simply map the flow rate variation to the microscopic material removal process. Nevertheless, this result is based on the self-defined material properties of the polishing fluid, so it can establish the boundaries of numerical velocity distribution on the workpiece surface for the subsequent prediction model. It can be clearly seen that the area near the workpiece has a high flow rate. This phenomenon conforms to the classical hydrodynamics law, under the assumption of an ideal incompressible fluid; the flow rate is inversely proportional to the cross-sectional area.

Shear stress distribution in the polishing zone
From the above microscopic material removal mechanism, we infer that in the fluid polishing process, the main force causing the removal of microscopic convex surface of the workpiece is the shear stress of microscopic clusters (or fluid layer) carrying abrasive particles. The aforementioned research on the mathematical model of material removal rate also supports our inference. Therefore, it is necessary to further investigate the shear stress distribution on the workpiece surface under the dynamic pressure of polishing fluid. As shown in Fig. 10a-d, high shear stress regions appeared in the workpiece area on one side of the polishing area in all experimental groups. In group G1, the shear stress on the side of the main polishing area is mainly distributed between 9.89 × 10 4 and 1.96 × 10 5 Pa. Subsequently, the upper and lower limits of the shear stress distribution in G2 and G3 groups gradually increase. Finally, the peak shear stress of group G4 in the main polishing area reaches 3.25 × 10 5 Pa.
Combined with the distribution characteristics of polishing fluid velocity in the above polishing area, it can be confirmed that the area near the outside of the workpiece (the negative side of the x-axis) is the main area causing material removal and polishing, namely, the main polishing area. In this section, the numerical distribution characteristics of flow field on and around the workpiece surface are studied, and the main areas causing surface polishing of the rotating workpiece are revealed. In order to further explore the numerical prediction model of the surface roughness of the machined workpiece, it is necessary to establish the numerical matrix of the polishing fluid flow rate and shear stress in the corresponding position of the region, and verify the numerical prediction model of the surface roughness with relevant empirical parameters.

Machined surface quality
In order to verify the effectiveness of the above prediction model and polishing method, it is necessary to detect and compare the surface roughness of the workpiece before and after machining. The value of surface roughness Sa in the main polishing zone on the workpiece surface under different polishing parameters is shown in Fig. 11. The initial surface roughness Sa before machining ranges from 164 to 512.5 nm, with an average value of 319 nm. After processing, this range was reduced to 27.5-60.2 nm, with an average of 39.2 nm. The width of the numerical distribution decreased by 90.6% and the mean by 87.7%. In addition, we used the surface roughness improvement rate (SRIR), as shown in Eq. (12), to describe the ability of WMSTP to improve the quality of the machinated surface, of which each group ranged from 63.3 to 93.1%, with an average of 85.8%. Therefore, the polishing technology adopted in this study can effectively improve the surface quality of each point on the surface, greatly reduce the surface roughness value, and remarkably improve the uneven distribution of surface quality. Therefore, it contributes an ultra-precision fluid polishing method with adaptive polishing ability. where Sa before represents the surface roughness value before machining, and Sa after represents the surface roughness value after machining at the same detection point.

Numerical distribution of parameters related to machining surfaces
The shear stress τ(x) and the flow velocity u(x) of the polishing fluid can be obtained based on the above finite element simulation results of the flow field in the polishing zone and on the workpiece surface. As shown in Fig. 12, the shear stress distribution on the workpiece surface along the positive x-axis can be derived from the model calculation results. According to the position coordinates of effective polishing detection points, the numerical distribution results of shear stress in corresponding areas are obtained. Similarly, the flow rate of the polishing fluid in the polishing area can be obtained. Shear stress τ(x) and the polishing fluid flow rate u(x) can be approximated by selecting values near the coordinates of corresponding test points in the effective polishing area, as shown in Tables 6 and 7.

Validation of the numerical prediction model
It is necessary to determine the constant values in the above prediction model under relevant conditions such as polishing fluid and the experimental platform, and determine empirical values of corresponding parameters based on experimental data. For the given volume of WMSTPF samples, the polishing fluid density (ρ F ) of 15 wt% was 1.53 × 10 −3 g/mm 3 . Based on the geometric dimensions of the polishing platform and the position parameters of the workpiece, the section area (S) of the polishing area was determined to be 448.3 mm 2 . In addition, the physical property parameter table of zirconia shows that the bond energy of Zr-O bond is 757.8 kJ/mol. Finally, as is known from the aforementioned study [27], the related parameter K and velocity index m are 2.60 × 10 −6 and 0.28, respectively. Based on the flow rate of the inlet polishing fluid given by the experimental parameters, a pre-experiment was carried out, and the empirical parameters k 1 to k 4 of the prediction model are 162, 34, 2, and 4.15 × 10 7 , respectively, where k 1-3 is a dimensionless number, and k 4 unit is mol. Based on the above parameter setting and the numerical boundary of initial surface roughness Sa, the roughness values of the machined surface under the corresponding process parameters were obtained, as shown in Table 8. Figure 13 shows the experimental values of surface roughness of the workpiece before and after machining and the calculated values of corresponding positions based on the prediction model. It can be seen that the value of surface roughness before machining is large and the distribution threshold is wide. After machining, the surface roughness Sa has obviously improved in terms of both numerical value and numerical distribution threshold. The calculated values

Conclusions
This study established a numerical prediction model of surface roughness for the material removal process of surface polishing with field induced non-Newtonian polishing fluid--WMSTPF. The velocity field and stress field of the polishing area and workpiece surface are calculated based on the material attributes of customized polishing fluid and the Gaussian constitutive model. And this provides numerical boundaries for the above prediction model and effectively predicts surface roughness Sa value of the machined surface.
Below is a summary of key findings of this study.   (1) Based on the finite element analysis of the flow field in the polishing area, the distribution characteristics of dynamic pressure, flow rate, and shear stress on the workpiece surface in the polishing area are obtained, and it is verified that the main polishing area is located on the quarter-arc surface near the outside of the workpiece. (2) By comparing the Sa value of surface roughness of the workpiece before and after machining, the study demonstrates that WMSTFP is an ultra-precision machining method with adaptive polishing characteristics and that it can effectively polish the surface of difficult-tomachining materials. The distribution width of Sa value decreases by 90.6%, and the average value decreases by 87.7%. (3) Based on the hypothesis of abrasive collision and the law of energy conservation, the statistical significance of the surface roughness of the workpiece was introduced into the crystal bond fracture of the material removal on the micro surface, and the above macroscopic and microscopic concepts were unified through the power law function of the material removal process, and finally, the WMSTP surface roughness model was constructed. (4) The study has proposed a joint numerical prediction method of machining surface roughness based on the numerical model and finite element calculation, and used experiments to verify the effectiveness of the prediction method. The absolute error of numerical prediction of average surface roughness is up to 10.6 nm, and the relative error is below 12.3%.