Build an accurate 3D geometrical model of a soft knife profile of abrasive water jet

Virtual machining simulation is becoming an essential and vital tool currently in traditional machining process. With it, a series of trial machining can be avoided. However, the situation is different for high-energy-beam cutting. Till today, there is no accurate virtual cutting simulation tool that can accurately simulate the physical cutting process. As a soft knife, whose shape is changing dynamically, it is a great challenge to accurately define the tool shape and build an accurate 3D geometrical model to be used as a virtual model for simulation. Aiming at abrasive water jet (AWJ) machining, a new method to build a soft knife profile based on energy distribution analysis has been explored in this paper. Through 3D point cloud data of kerf profile, a 3D model of the abrasive water jet profile has been built under different working conditions. To evaluate the effectiveness of the self-defined tool shape, the compensation strategy based on 3D AWJ profile has been carried out in the actual cutting process. The results show that 3D AWJ profile is beneficial to improve the machining accuracy. And this method can be generalized to other high-energy-beam cutting tools such as laser and plasma.


Introduction
In recent years, digital twin has become a hot topic and an important development direction of smart manufacturing.It is a technology established on the basis of modeling and simulation technology as a core [1].In numerical control machining, a digital twin system should reflect the actual machining process in a virtual environment [2].The virtual cutting simulation is conducted to verify the effectiveness of numerical control strategies, further to avoid various abnormal situations that may occur in the actual cutting process and improve the processing and production efficiency and quality [3].The core of such applications lies in physical modeling and virtual modeling.Physical modeling refers to abstracting, describing, and defining the key features of a physical entity, while virtual modeling refers to building a virtual representation of a physical entity.The virtual model should be a mirror reflection of the physical model and have the same features and behaviors in the virtual environment [4].
For traditional machining processes such as turning and milling, many researchers have built physical models in the actual cutting process and corresponding virtual models to realize virtual cutting simulations [5,6].It has been proven that virtual cutting simulation can save a lot of experimental work and avoid wastage caused by trial cutting in traditional machining processes [7].However, the situation is different in high-energy-beam cutting process.Till today, there is no accurate virtual cutting simulation tool that can accurately simulate the physical cutting process.As a result, high-cost trial cutting has to be carried out in many cases where virtual cutting should be used to play an important role.
In fact, as a non-traditional machining tool, high-energybeam cutting tool has some special advantages, such as high efficiency and small cutting kerf.And with the development of this technology, it has been used in free surface geometry machining which is difficult to be machined by traditional milling.Compared with traditional machining, high-energy beams have "soft knife" features with time-varying nature, as shown in Fig. 1.The energy of the high-energy-beam decreases as the cutting depth of the material increases, leading to the dynamic change of the tool shape and kerf characteristics during the cutting process.Obviously, to use this tool in virtual cutting, a big challenge is to accurately define the tool shape and build a 3D soft knife profile as a virtual model for simulation.For this reason, there is no accurate virtual cutting simulation tool till now.
As one of the high-energy-beam cutting tools, abrasive water jet (AWJ) is a complex and hard-to-study tool which is formed by the high-energy mixture of water, air, and abrasive particles.One of the unique advantages of AWJ is that the cutting process is a cold process, which means it does not produce a heat-affected zone or cause thermal distortion [8].Additionally, AWJ offers environmental benefits and is capable of producing high-quality surface finishes [9].Due to its versatility in cutting almost any material, AWJ technology has been adopted in over 30 industries [10].
Same as other high-energy-beam tools, the tool shape of AWJ changes dynamically.Therefore, to realize the virtual cutting simulation of AWJ, it is necessary to accurately define the tool shape of the abrasive water jet profile during the actual cutting process.In order to achieve this, the physical cutting process of AWJ needs to be analyzed.
In the past 40 years, many researchers have focused on how the material of workpiece responds to AWJ in terms of behavior [8] [11].For example, Zeng et al. proposed a mathematical model, separation speed, which determines the velocity that can just cut through the target material [12].Hashish measured the width profile and jetlag geometry and proposed dynamic angle compensation [13], while Hlaváˇc established a new model to calculate the jet exit angle [14].Karthik et al. addressed the influence of input process parameters on material removal rate and kerf top width [15].Wang et al. established a kerf taper prediction model and a cutting front profile model through theoretical and experimental analysis [16].Llanto et al. investigated the impacts of input parameters, including water pressure, abrasive mass flow rate, traversing speed, and material thickness on material removal rate and surface roughness [17].Hlaváˇc et al. used a waterjet-force measuring device to describe the cutting process and study the respective force signals [18].In addition, many researchers established different models through theoretical and experimental methods [19].
While these studies provide insight into single kerf characteristics, they are insufficient to building an accurate virtual model for simulation.Therefore, the accurate definition of the tool shape of the AWJ profile during the actual cutting process is crucial for realizing the virtual cutting simulation of AWJ.
Actually, several virtual cutting tools of AWJ have been developed in the last few years.However, the time-varying nature of AWJ is often ignored in current CAM solutions [21].As shown in Fig. 2, AWJ is considered as a fixed tool in some NC machining simulation software, such as VERICUT and NCSIMUL.Moreover, it is simply modeled as a cylinder without any soft knife features [20].It should be noted that the tool shape cannot change with the variation of process parameters.Obviously, these solutions cannot be used to simulate the actual cutting process.
As the demand for precision in AWJ machining increases, virtual cutting simulation must take into account the timevarying nature of the soft knife profile.By utilizing digital twin technology, a 3D soft knife profile for virtual cutting can be constructed based on actual process parameters, and these parameters can be fed back in real-time.Since the soft knife profile can be used to adjust the compensation strategy, a better definition of the tool shape can further improve precision.Therefore, building an accurate 3D geometrical model of a soft knife profile becomes the most important module of AWJ virtual cutting simulation.The more accurate the 3D model is, the smaller the error between the virtual cutting process and the actual cutting process will be.This paper focuses on building a 3D soft knife profile to define the tool shape in the AWJ actual cutting process.Based on the theoretical analysis of the jet profile energy distribution and experimental data abstraction of the kerf profile, a novel method to build a 3D soft knife profile is proposed.This approach provides a potential solution for virtual cutting simulation.With it, a digital twin system for AWJ is expected.

Basic shape and trend of abrasive water jet profile based on jet energy distribution theory
When machining hard materials using AWJ, the removal process mainly depends on the erosion caused by abrasive particles rather than water.If the jet beam is considered as a tool, the tool shape is mainly determined by the distribution of abrasive particle during the cutting process.As mentioned above, thousands of abrasive particles work on the material simultaneously.Each abrasive particle will impact on the material with different velocities and impact angles and some abrasive particles may impact material multiple times.Since the positions, velocities, and impact angles of each particle are different, it is difficult to describe the entire removal process from the perspective of a single abrasive particle.However, the basic shape and trend of AWJ profile can be analyzed from the energy distribution of the whole jet beam.And then the accurate 3D profile can be built with the kerf profile of experiments.In this paper, the energy distribution of AWJ and the profile data of cutting experiments are used together to build the profile of this soft knife.

Energy distribution of the abrasive water jet beam in non-cutting process
The kinetic energy of a single particle is proportional to the particle mass multiplied by the square of the particle velocity.Due to the uneven distribution of abrasive particle quantity and velocity, the energy density of this AWJ beam is also uneven.Only when the energy density within a unit is higher than the critical destruction energy of the workpiece material, the material in this unit can be removed.In this case, the profile is a set of such units with sufficiently high energy density.Therefore, the quantity and velocity distribution of the abrasive particles in non-cutting process must first be analyzed.When the abrasive water jet is ejected from the focusing tube, the jet flow detaches from the constraint and forms an unsubmerged jet in the air, as shown in Fig. 3.In the initial segment, the velocity in the core area of the jet flow can still maintain the initial velocity V 0 .On the one hand, the jet diffuses outward, introducing more air into the boundary.On the other hand, the boundary area of the jet flow extends inward and the jet flow in boundary area exchanges momentum and mass with the jet flow in core area.The core area will gradually decrease.In the main segment, the velocity of the jet center will not be able to maintain the initial velocity.Also in main segment, the velocity gradually decreases with the increasing distance, as given by Eq. (1).
where V m is the velocity of the jet center in this section, d is the distance from this section to the outlet of the focusing (1) The simulation tools of AWJ in VERICUT [20] tube, d 0 is the initial segment length which depends on factors such as tube structure, size, roughness, etc.
In the main segment, the velocity distribution changes as the distance increases.The further away from the outlet of the focusing tube, the flatter the distribution.But at different sections, if the radius on each section is kept the same, the ratio of the velocity at that position to the velocity at the center remains unchanged.The velocities at each cross section of the turbulent jet change according to the one seventh power law [22], as given by Eq. ( 2).
where V is the velocity at a certain position on this section, R is the position relative to the jet center, and R max is the maximum radius of this section.
Based on the above analysis, the velocity distribution and trend can be obtained.However, since the velocity gradient is different on the whole horizontal section, the abrasive particle quantity distribution is not uniform for all abrasive particles passing through the section in a specified time.As a result, under the effect of Magnus force and Saffman force [23], more abrasive particles intend to move from edge to the center, as shown in Fig. 4. Chen et al. used Laser Doppler Anemometry technique to analyze the abrasive particle distribution in the jet beam [24].They found that the peak of the quantity distribution occurred at about 1/3 to 2/3 of the radius of the jet.Balz et al. presented a method to determine the velocity and position of abrasive particles by using laserinduced fluorescence technology [25].The results showed that the quantity distribution of abrasive particles can be well fitted to a Gaussian curve.Ke et al. designed a testing device to test the abrasive distribution on the section of the jet beam [26].They found that the abrasive quantity distribution obeys the normal distribution and the distribution range has a first-order linear relationship with the stand-off distance.Although different studies show that the abrasive distribution law is not completely consistent, the general trend can be confirmed as the closer the units to the jet center and to the nozzle tip, the bigger the amount of the abrasive particles in those units.
The trend of the energy distribution on a certain section can be derived from the quantity and velocity distribution, as shown in Fig. 5.It can be observed that the energy is higher near the center of the jet and lower near the edges.Moreover, as the cutting depth increases and the particles participate in more material removal, their velocity decreases, resulting in a decrease in the energy density in the vertical direction.
Based on the analysis above, the material in a unit can only be removed when the energy density reaches a certain value.According to the energy distribution trend of AWJ, less material is removed from the bottom compared to the top, as shown in Fig. 6.Point O represents the jet center position.The gray circle represents the ideal jet profile boundary in the air, while the blue circle represents the actual jet profile boundary according to the energy distribution.The abrasive feed rate is generally fixed during a cutting process, meaning the quantity of abrasive particles in a unit mainly depends on the traversing speed of the cutting head.Once the traversing Fig. 3 The velocity distribution of abrasive water jet in noncutting process Fig. 4 The effect of Magnus force and Saffman force speed decreases sufficiently, the blue circle will approach the gray circle.

The basic shape and trend of abrasive water jet profile in cutting process
In the above analysis, the jet beam is assumed to be ejected in the air instead of to be ejected on the target material.A circle could be used as the boundary for abrasive particle distribution on each section of the jet beam.In actual cutting process, the shape of the jet would change as the target material kicks back to the jet beam.That is why jet will deflect in the opposite direction of the cutting head's traversing direction, and this is related to the energy loss and deflection of the AWJ [27].An empirical formula can be used to build a self-defined parabolic curve-like profile.The 3D model proposed by Henning reflects such features of a soft knife [21], and this geometrical model is used as the virtual cutting model in simulations.Furthermore, Chen proposed a self-defined virtual cutting method with a soft knife that uses a circle as the basic shape [28].
As mentioned previously, the cutting process of AWJ is very complicated as some abrasive particles might impact the target material many times.To illustrate the complex process, two representative cases have been presented as examples, as shown in Fig. 7. Generally, a particle may have multiple impacts on the target material until it leaves the bottom of material [29].In case 1, the abrasive particles would give several shots to the material since the wall on the opposite side would kick them back each time.Undoubtedly, a portion of abrasive particles would follow this kind of trajectory.Another representative trajectory of abrasive particle is presented as case 2. As the AWJ cutting head moves along the traversing direction, since the material behind the jet has been removed, a portion of the abrasive particles would Fig. 5 The energy distribution at one horizontal cross section of jet beam Fig. 6 The horizontal cross section of jet beam at the top and bottom based on energy distribution trend Fig. 7 The different case of particle trajectories escape since no wall on the back side would kick them back.In both cases, those abrasive particles which deviate their original trajectories are named as "escaped particles" in this paper.As the impact times of particle increases, the escape probability also increases.
In previous jet profile building, the impact of the escaped particles was not taken into account, resulting in inaccurate circle-based definitions of the basic shape of the jet profile.The escaped particles leave the ideal jet profile boundary before they leave the bottom of material.But the energy of these escaped particles is still sufficient to remove material on the sidewall behind the jet.So the actual jet profile boundary on the back side will approach and, in some cases, exceed the ideal jet profile boundary.
If the escaped particles are considered, the basic shape of AWJ profile should be an ellipse or quasi-ellipse, as shown in Fig. 8.And the section shape of the front wall coincides with the front half of the jet profile in AWJ cutting process.
The shape of the cutting edge in AWJ cutting process is affected by the escaped particles, which varies with the cutting depth.As the cutting depth increases, the energy distribution becomes more dispersed.And more escaped particles appear at the larger cutting depth.This results in a larger ratio of the kerf width removed by escaped particles to the kerf width removed by core energy, resulting in a more elliptical cutting-edge shape.This change and analysis are confirmed by the comparison of the cuttingedge on the top surface and the bottom surface in Fig. 9.
Due to the multiple impacts of abrasive particles with different impact angles and velocities in the cutting process, it is difficult to calculate specific AWJ profile theoretically.However, based on the above energy distribution trend, the basis for AWJ profile building can be obtained.The cutting-edge shape at different cutting depth can be fitted from the data by experiments.And then a 3D model of AWJ profile that conforms to both theory and experience can be built.

Building a 3D geometrical model of abrasive water jet profile by combining mechanism analysis and experimental results
To fit the cutting-edge shape, it is necessary to collect complete data on the kerf profile.Since the previous methods only focused on the 2D features and 2D models of AWJ kerf profiles, the kerf will be damaged and some profile data will be lost.So a new method is needed to obtain the complete kerf profile and build the jet profile.
Fig. 8 The AWJ profile boundary considering the impact of escaped particle Fig. 9 The cutting-edge shape difference on the top surface and the bottom surface

A method to obtain a complete kerf profile
In this method, two blocks are integrated to replace one block sample.As shown in Fig. 10, the experiment consists of four steps.Firstly, two blocks are fabricated as required and integrated together.The roughness of the joint face should be less than 0.8 um, and the flatness of the joint face should be less than 0.01 mm.Secondly, the sample is clamped to the workbench of the AWJ machine.Thirdly, cutting is performed along the joint face.Finally, the machine is operated to stop abrasive supply and jet flow abruptly during the cutting process.After these four steps, a sample with a complete kerf profile can be obtained.Through comparison, it is found that the 2D features obtained from the integrated block sample and the whole-piece sample under the same process parameters are basically consistent.The effectiveness of this method in obtaining a complete kerf profile has been verified in the previous study [30].Subsequently, a high-precision 3D laser scanner is used to generate complete point cloud data, as shown in Fig. 11.The sample is separated to facilitate scanning the kerf profile on each block.The two point cloud data are then merged together based on the relationship between the two blocks.The laser scanner type used in this paper is HOLON380, and the scanning resolution is set to 0.05 mm to balance the scanning efficiency and accuracy.With the point cloud data of the kerf profile, the cutting-edge shape can be further abstracted.

Segmentation of kerf profile
First, a rough segmentation is performed by plane segmentation, keeping only the point cloud data near the kerf, as Fig. 10 Experimental method for integrated block sample Fig. 11 The complete point cloud data of kerf profile from one sample as example shown in Fig. 12a.This step is taken to remove most of the irrelevant point cloud data.Next, the point cloud beyond the kerf profile is further removed by fitting three reference planes from the point clouds-the joint surface of the blocks, the top surface, and the bottom surface.Three rectangular bounding boxes based on the reference planes are used for selection and segmentation, with the bounding box thickness set to 0.1 mm, slightly greater than the scan resolution.The color of the point cloud segmented by the bounding box is white, as shown in Fig. 12b.Finally, unwanted point clouds are deleted, leaving only the point cloud data of the kerf profile, as shown in Fig. 9c.

Fitting of cutting-edge shape
As analyzed above, the shape of the cutting-edge changes with the cutting depth in accordance with the energy distribution trend.Therefore, to fit the cutting-edge shape, point clouds within a specific cutting depth range should be used.As shown in Fig. 13a, the point cloud data for this layer is obtained by abstracting a bounding box.The abstracted results are shown in Fig. 13b using a top-down view.On the same plane, two straight lines can be fitted through the point clouds on both sidewalls to represent the kerf width at that cutting depth.The point cloud data for the cutting-edge can be filtered by evaluating the distance between the points and the lines.The filtered point clouds for the cutting-edge are shown in Fig. 13c.A least squares method is used to fit ellipses to these scattered points [31], and the fitted ellipse is shown in Fig. 13d.The fitted profile is the section shape of the 3D AWJ profile at this layer.

Modeling of jet profile
By abstracting point cloud data from different cutting depths, section shapes of different layers can be fitted, as shown in Fig. 14.Both the point clouds and the fitted profiles show that as the cutting depth increases, the section shapes become more elliptical, and the ratio of the major axis to the minor axis of the fitted ellipse increases accordingly.This result is consistent with the energy distribution trend analyzed above.In a virtual environment, a 3D model can be built by lofting two profiles in different planes.The level of detail in the 3D model of the AWJ profile is related to the number of layers involved in the lofting process, as shown in Fig. 15.To obtain a 3D model of the AWJ profile that closely matches the kerf, the point cloud data could be divided into more closely spaced layers.
According to the energy distribution trend and the basic shape of jet profile, the section shape in AWJ cutting process can be abstracted from the experimental point cloud data.The 3D AWJ profile is built through the section shape at different cutting depths.This model represents the tool shape of AWJ under a certain working condition in actual cutting process.
This method for building the AWJ profile can be applied to construct the 3D geometrical model of soft knife under a variety of working conditions.The new self-defined tool shape will replace the original simulated tool in virtual cutting process.

Verification of the 3D geometrical model
Based on the method of building a soft knife profile, 3D geometrical models can be constructed under different experimental conditions.To verify the effectiveness of the self-defined tool shape, it is necessary to determine whether it can accurately simulate the features of the AWJ beam during virtual cutting processes.These features are dynamically related to the AWJ process parameters.Therefore, the experiment and verification process are divided into two steps.In the first step, the influence of process parameters on the jet profile is studied by comparing the 3D models of AWJ profile.In the second step, the compensation strategy is implemented based on the 3D AWJ profile in the virtual cutting process.This compensation strategy is further applied in the actual cutting process to verify its effectiveness.

3D AWJ profiles under different experimental conditions
Among the AWJ process parameters, the cutting head traversing speed has a significant influence on the kerf characteristics.In industry, the cutting head traversing speed is often divided into five grades based on cutting surface quality.Q1 represents the material separation speed corresponding to the lowest quality level and is often determined by separation speed experiments [32].Q2, Q3, Q4, and Q5 correspond to better quality and can be calculated using the separation speed formula given in Eq. ( 3).
where u is the cutting head traversing speed (mm/s); u 0 is the separation speed (mm/s); h 0 is the material thickness used in separation experiment; h is the target material thickness; and q is the surface quality level.A total of 25 experiments were designed and conducted, with the experimental working conditions listed in Table 1.The experimental parameters, except for the target material thickness and traversing speed, are widely used in industry.All experiments were conducted on the Shanghai Lionstek Waterjets LTJ1613-5A.The block experiment samples are shown in Fig. 16, and a total of 25 profiles were built under 25 different experimental parameter combinations.
Taking the profiles with a material thickness of 100 mm as an example, 3D models at five different traversing speeds are shown in Fig. 17.By comparing the features of these profiles, it can be observed that the AWJ profiles at different traversing speeds become more elliptical as the cutting depth increases.It confirms the above analysis of energy distribution trend again.In addition, as the cutting head traversing speed increases, the curvature of the AWJ profile increases and the width of the AWJ profile decreases.This result is consistent with the conclusions of previous studies by other researchers.
When the 3D model of AWJ profile is used as a simulation tool in the virtual cutting process, it helps in the formation of kerf characteristics.The formation of kerf can be simulated by performing a Boolean subtraction between the 3D model of AWJ profile and the target material in the virtual cutting process [28].The virtual cutting process in simulation is shown in Fig. 18.The jetlag in the simulation corresponds to the curvature of AWJ profile, while the kerf width corresponds to the width of AWJ profile.These features serve as the basis for compensation strategies based on 3D AWJ profile.

Optimized compensation strategy based on 3D AWJ profile
In the current compensation strategy, a five-axis cutting head eliminates jetlag error by tilting an angle along the traversing direction.Before 3D AWJ profile is built, the angle is calculated by the jetlag of kerf.The jetlag of kerf is the projection distance between the cutting front point of the top and bottom surfaces.As shown in Fig. 19, the original compensation angle θ 1 is the ratio of jetlag to thickness.According to the above analysis of the basic shape of jet profile, the original compensation strategy is not accurate enough.Since the jet center projection of section shape on the top and bottom surfaces are not at the same position after compensation, it will affect the further compensation for jet width.To achieve a better compensation effect, the jetlag of profile is used to calculate the optimized compensation angle θ 2 .The jetlag of AWJ profile is the projection distance between the center points of the top and bottom section shape.By tilting the angle θ 2 , the spatial attitude of AWJ is adjusted to eliminate the jetlag error.The compensation angles of 25 experimental conditions under different strategies were calculated and compared, as illustrated in Fig. 20.It is apparent that the compensation angle of the optimized strategy is generally greater than the compensation angle of the original strategy.The reason is that as the cutting depth increases, the section shape of AWJ profile gradually becomes more elliptical.This results in a larger compensation angle calculated based on 3D AWJ profile.

Verification in both virtual and actual cutting process
To verify the effectiveness of the optimized compensation strategy based on the 3D AWJ profile, both virtual and actual cutting processes were performed.The virtual cutting process was implemented in simulation, while the actual cutting process was carried out by block experiments.The traversing speed grade was Q5, and the other experimental processes were consistent with those listed in Table 1.The optimized compensation angle is first calculated through 3D AWJ profile and then applied to the control strategy in virtual and actual cutting process.As shown in Fig. 21, the virtual and actual results were compared, and the cutting front curve was plotted in a coordinate system.As shown in Fig. 22, the cutting front curves of the experimental results and the simulation results are very close to each other at different thicknesses.The maximum profile error is greatly eliminated after compensation.On the one hand, the results show that the optimized compensation strategy based on 3D AWJ profile is effective in eliminating the jetlag error.On the other hand, it indicates that the virtual cutting simulation can effectively replace the actual cutting of some experimental cuts.This new method of building a 3D soft knife profile is significant for industrial simulation solutions and digital twin systems.It can also be applied to other high-energy beam technologies with soft knife features, such as laser and plasma.

Conclusion
The method to build an accurate 3D geometrical model of a soft knife profile of AWJ presented in this paper leads to the following conclusions: 1.A new method to build a 3D soft knife profile of abrasive water jet is provided in this paper.Using this new method, a self-defined soft knife with a shape is closely resembling the real tool shape under various cutting conditions is obtained.This soft knife could be used to simulate the cutting process accurately in a virtual environment.2. The analysis of the "escaped particles" phenomenon and its incorporation into the jet profile modeling has enabled a more precise assessment of the AWJ cutting process.It has been revealed that the basic shape of the jet profile is elliptical rather than circular.3. The optimized compensation strategy based on the 3D AWJ profile is applied in both virtual and actual cutting process, and its effectiveness is verified.The use of the 3D AWJ profile has been proven beneficial to improve the machining accuracy of AWJ.

Fig. 12
Fig. 12 Point cloud segmentation of kerf profile

Fig. 13 Fig. 14
Fig.13 Point cloud fitting approach of cutting-edge shape

Fig. 21 Fig. 22
Fig.21 Comparison between virtual cutting and actual cutting result under optimized compensation strategy (a sample of 40 mm thickness)

Table 1
Experimental parameters of AWJ cutting process Fig. 16 All experimental samples under different experimental parameter combinations Fig. 17 Comparison of 3D AWJ profile under different traversing speed