This work focuses on the identification of optimal model parameters related to Abrasive Waterjet Milling (AWJM) process. The evenly movement as well as variations of the jet feed speed was taken into account and studied in terms of 3D time dependent AWJM model. This gives us the opportunity to predict the shape of the milled trench surfaces. The required trench profile could be obtained with high precision in lack of knowledge about the model parameters and based only on the experimental measurements. We use the adjoint approach to identify the AWJM model parameters. The complexity of inverse problem paired with significant amount of unknowns makes it reasonable to use automatic differentiation software to obtain the adjoint statement. The interest in investigating this problem is caused by needs of industrial milling applications to predict the behavior of the process. This study proposes the possibility of identifying the AWJM model parameters with sufficiently high accuracy and predicting the shapes formation relying on selfgenerated data or on experimental measurements for both evenly jets movement and arbitrary changes of feed speed. We provide the results acceptable in the production and estimate the suitable parameters taking into account different types of model and measurement errors.
The abrasive waterjet (AWJ) machining is a nonconventional lowcost process [
One of the most challenging and crucial questions among the industrial and manufacturing problems which can be interpreted with partial differentiation equations (PDEs) is the identification of the optimal model parameters. The goal is to reproduce the required shapes and processes relying only on the available experimental measurements related to the real systems. In the conditions of complexity and nonlinearity of the problem, the determination process becomes one of the critical questions and leads to involvement of various techniques and approaches.
There were several reported studies in consideration of the direct problem when it is necessary to predict the trench surface with a given model and its set of parameters. Some wellknown methods based on the statistical approaches [
Even if the direct problems are linear under some considerations, however, the inverse problems of the model parameters identification are usually illposed [
In this paper, we extend the work presented in [
The inverse problem consists of the identification of AWJM model parameters from the experimental observations. These results have to be further used to simulate the required surface profile. Recent research of linear AWJM inverse problems focused on the identification of the beam path has been previously reported in [
In our work, the gradient vector of the cost function, which is required for all the family of the gradient descent algorithms, is found numerically by use of the automatic differentiation software TAPENADE [
The parameter identification problem significantly depends on input measurements and is very unstable, which is shown by inclusion of noise in the generated data. In addition, we indicate the influence of using Tikhonov regularization on accuracy of the surface prediction and improvement of the AWJM model parameters identification in case of noisy data.
The paper is organized as follows. Section
The milling process perpetrated by abrasive waterjet machine is represented as a nonlinear partial differential equation with initial and boundary conditions. This model characterizes the process of the trench surface formation by the jet impact on the workpiece and is suitable for various jet feed speeds independently of the target material properties. To define the problem, we suppose the time interval of the continuous milling process
The proposed Abrasive Waterjet Milling model, coming from previous works [
The given AWJM model in (
The final form of the trench, which is described here as a function
The schematic representation of the problem is illustrated in Figure
Schematic of the AWJM process and jet footprint.
Crosssection
Top view
In order to identify the optimal AWJM model parameters
In expression (
To obtain the solution of minimization problem (
To solve the minimization problem and find an optimal solution
It is necessary to consider not the continuous but the discrete system to figure out numerically the optimal problem and to find its solution. Actually we have to minimize the discrete cost function which requires the gradient of the discrete cost function. We involve the automatic differentiation software (i.e., TAPENADE) to obtain the gradient of the discretized cost function. Once the gradient is computed we can solve minimization problem (
Mainly most minimization techniques, which are used to evaluate approximate gradients for constrained problems and to find local minimum of cost functions, are iterative gradient descent algorithms. The quasiNewton type methods to compute the approximate gradient and descent step of the minimization process have been applied due to complexity and high costs to compute the Hessian on each iteration. To perform the minimization process, we use the N2QN1 minimization package for constrained optimization problems from “MODULOPT” library [
The LBFGS [
In case of evenly moving abrasive waterjet, we assume the proposed AWJM model in (
In this subsection we study two different cases, when the input data are considered selfgenerated surfaces or averaged experimental observations.
For the numerical implementation we define a domain
In order to obtain a smooth solution (
The value of Tikhonov regularization multiplier
The “pseudoexperimental” surface was generated with arbitrary values of model parameters
Based on the demonstrated correctness of the identification process reported in the previous work [
Comparison of the identified etching rate function with the original one, used for the direct simulation.
From the results shown in Figure
Comparison of the crosssections of obtained solution and selfgenerated surface.
Further, we base our search of the AWJM model unknowns on the trench surface obtained from the real experimental measurements by extending the average crosssection in the direction of the jet movement (Figure
Extended average crosssection profile to the trench surface.
Unlike the previous occasion, we have no estimation of the etching rate function that was used in the produced experiment; thus we start the determination from the zero assumption
Here, we provide the results of the identification of AWJM model parameters, inaccessible from the experiments, which should be used in the direct simulation to reproduce the required workpiece shape. The identified etching rate function (Figure
Identification of the etching rate function and comparison of the crosssections of obtained solution with experimental measurements for averaged trench surface.
Etching rate function
Crosssections
One can observe the mismatch on the edges of the slopes of the trench, but this aspect was not considered and modeled in the used AWJM model in (
According to the considered mathematical model, we assume the constant movement of the jet in straight direction and in this subsection we base our determination process on the original measurements of the real experiments done with waterjet machining tool (Figure
Original measurements of the part of the trench, milled by microwaterjet machine.
Usually parameter identification problems induce various difficulties caused by model errors and rather measurement noise. To be able to search unknowns from rough and noisy initial measurements, we include in the AWJM model the error term
For the direct simulations, we use the following model:
Here the available experimental measurements differ from the previous ones and correspond to waterjet milling process with the jet feed speed of 3000 mm/min. Due to provided data, we define the squared domain
Results of the determination of the etching rate function and comparison of the reproduced surface with original profile are given in Figures
Identification of the etching rate function and comparison of the crosssections of obtained solution and experimental measurements in case of evenly moving waterjet.
Etching rate function
Crosssections
To extend the possibility of the application of demonstrated identification mechanism in the manufacturing, we include another particular case belonging to the variations of the waterjet feed speed during the milling process. To meet the practical capabilities of waterjet machine, we assume that it accelerates constantly during the movement from the initial position to the final one. For the numerical implementation, it can be described as a change of the time spent by jet beam on each position of the workpiece where we examine the problem.
We define the domain
Averaged experimental measurements of the trench, milled by microwaterjet machine with a feed speed change from 600 to 2000 mm/min.
A Tikhonov regularization factor
Results of the identification of etching rate function
Numerical results of the identification of the etching rate function and comparison of the crosssections of obtained solution and experimental measurements for the feed speed acceleration of the waterjet.
Etching rate function
Crosssections
Sensitivity study plays one of the key roles in the plenty of the parameters identification problems. By this, it is possible to deeply understand the behavior of the model and improve the correctness of the identification process. Observation of the influences and sensitivity of the AWJM model on measurement or model errors provides us with the opportunity to see what the possibilities of reconstructing the required shape of the trench are regardless of the input data.
In this subsection we demonstrate and compare the numerical results of the proposed approach to identify the etching rate function
As explained above, the simulated input data is obtained by adding a Gaussian white noise of various levels of intensity to the initial trench surface (e.g., Figure
Selfgenerated surface measurements with 15% of applied noise and form of the initial etching rate function.
Generated input data
Initial etching rate function
The general purpose is to identify the unknown parameter
Results of the identification of the etching rate function
Results of numerical identification of the etching rate functions for AWJM model and prediction of the surface shapes based on single trench measurement with applied noises of 5%, 15%, and 30%, respectively.
Single trench, 5% of noise
Single trench, 15% of noise
Single trench, 30% of noise
Further we assume that there are several different available experimental measurements of exactly the same trench that can be used to identify the unknown AWJM model parameters and to model the required surface. We generate them identically with the same parameters, but the distribution of the noise is always random, so the difference between them is only the random noise applied to the initial surface. To diversify the study we assume two different cases when there are three and ten available measurement inputs, which are shown in Figures
Diversity of the available input measurements for different cost functions in case of 15% applied noise.
3 input measurements
10 input measurements
Here the identification is based on the minimization of the cost function, which measures the difference between numerical solution and each of the experimental observations. In both cases, our cost function transforms to
The use of several independent trench measurements leads to the following results for the identification of the unknown function
Results of identification of the etching rate functions and reconstruction of the surface shapes based on 10 independent trench measurements with the 15% level of the measurement errors.
The given numerical results of the surface prediction (Figure
Considering several independent trench measurements can be interpreted as the averaging of the surfaces in some sense, but to clarify this aspect we demonstrate another case of our identification problem with the use of several superposed trench measurements. In theory, the use of the average of the noisy trenches will provide less rough and noisy input data and will lead to the identification of model parameters more precisely, which in its turn implies the better reproducing of the required surface. Based on that proposition, we introduce the superposition of the experimental observation, taken from the previous test and introduced in the cost function as follows:
A comparison of the various configurations of the cost function is represented in Figure
Comparison of the proposed constructions of the cost function, based on different amount of the available experimental observations.
Holding the acceptable level of accuracy (less than 10%) in the surface reconstruction in comparison with the experimental measurements, using several inputs or either of their averages in (
The difference between using one and several measurements is not very impressive due to random nature of the noise applied to the input and could be strongly increased by involving hundreds of experimental observations to reduce the influence of the errors and by adjusting the regularization coefficient according to the averaging of the input. The given overview of the identification based on 1, 3, or 10 trenches shows us that, from the other side, we can identify unknowns with reasonable accuracy even with only one trench measurement.
Cost functions (
Comparison of the accuracy in the trench surface prediction, corresponding to different cases of the cost functions and different levels of applied noise.
Trenches  1%  2%  5%  10%  15%  20%  30%  40% 

1 trench  3.74 × 10^{−2}  3.92 × 10^{−2}  3.65 × 10^{−2}  4.08 × 10^{−2}  5.46 × 10^{−2}  7.62 × 10^{−2}  0.118329  0.142537 
3 trenches  3.75 × 10^{−2}  3.95 × 10^{−2}  3.61 × 10^{−2}  4.14 × 10^{−2}  6.02 × 10^{−2}  7.73 × 10^{−2}  0.115635  0.138153 
10 trenches  3.75 × 10^{−2}  3.97 × 10^{−2}  3.61 × 10^{−2}  4.22 × 10^{−2}  5.67 × 10^{−2}  7.45 × 10^{−2}  0.108759  0.100952 
Superposition of 10  3.75 × 10^{−2}  3.97 × 10^{−2}  3.54 × 10^{−2}  4.10 × 10^{−2}  5.67 × 10^{−2}  7.45 × 10^{−2}  0.108724  0.115774 
The analysis of the obtained results induces thinking about the particular random distribution of the noise, applied with high level to the original input, which has very high influence on the identification process. The trench surface simulated with the use of the identified etching rate function is quite close to the input (noisy or average of several trenches) in all the cases, which were not aligned and fitted to the original surface due to the fairly random distribution of the noise. It engenders the conclusion that the use of much larger number of measurements can negotiate the noises or make them more uniform, calibrate and fit by this the inputs to the original data, and improve the accuracy of the surface prediction.
Mostly, the use of several trenches (and their average) instead of only one can essentially improve the accuracy in the parameters identification, conducting to reduction of the errors in the surface prediction up to 20% in cases of adverse available inputs. Certainly, it should be noted that sometimes only one measurement is available, and it might be enough to obtain the model parameters required to reconstruct the profile.
The identification of the unknown AWJM model parameters in Section
To ensure first the correctness and possibility of the identification of the existing noise in the input data, we first use the “true” values of
Firstly, it is necessary to check if the minimizer is able to faithfully determine the randomly distributed values of the model parameter. The initial noise applied to the trench with the level of 5% is shown in Figure
Comparison of the original generated noise of the 5% level with the identified noise.
Initial applied noise
Identified noise
Usually there is no information about the behavior and type of the etching rate function
We start with some assumptions about the form of the etching rate function
The results of the identification of the measurement errors for the cases of 5% and 30% are presented in Figure
Comparison of the identified noise form in case of 5 and 30% of applied noise.
Identified measurement errors 5%
Identified measurement errors 30%
Numerical results of the etching rate identification and trench surface reconstruction with identified measurement noise of 5%.
Identified etching rate function
Surface reconstruction
Considering the identification result for all the range of noise levels, we can note that, with the decrease of the measurement errors, their influence on the surface formation decreases as well and becomes less significant. It leads to the modification of the form of identified noise, which can be distinctly seen in Figure
One more interesting aspect of this work is the ability to improve the surface reconstruction by the improvement of the input data. Assume now that we identified quite acceptable and useful values of the measurement errors (e.g., Figure
Numerical results corresponding to the removed noise of 20% from the initial measurements and prediction of the milled trench.
Trench without the identified noise
Surface crosssection reconstruction
After the use of such manipulation, we can perform again the identification of the unknown AWJM model parameter
Accuracy in the trench surface prediction, corresponding to different levels of applied noise. One measurement as input.
Measurement errors  1%  2%  5%  10%  15%  20%  30%  40% 

Removed noise 








Noisy input 






0.118329  0.142537 
Parameter identification is a highly challenging problem in AWJM problem, particularly from the noisy experimental measurements and in case of uneven movement of the waterjet with varied feed speed in 3D case. In this paper we demonstrated the possibility of using the application of inverse problems theory, based on minimization problems, in the real manufacturing problems to estimate the process behavior and forecast the trench shape formation. The general high precision of the AWJM model parameters identification provides good opportunity to predict and simulate the milled trench surfaces regardless of the quality and density of available experimental observations. We showed the capability of the proposed method to cope with different cases independently of type and size of the input data, depth of the milled trench, microwaterjet feed speed, kind of the jet movement, and level of the measurement noise. We gave an overview of how an even minor and insensitive level of noise can affect the accuracy of the results and occasionally leads to considerable errors in surface reconstruction.
We presented the comparison of different approaches of the cost function formulation in accordance with various number of available data, which leads to several particular improvements in cases of high measurement errors. Moreover, we demonstrated the importance of the regularization terms, which have to be considered and carefully adjusted to obtain a more real and precise surface shape. In order to control the surface prediction under noisy conditions, we introduced and implemented a technique to identify the measurement noise independently of the other model parameters and to remove it from the input data of the minimization problem, thereby increasing the quality of the identification. Also, the combined identification of all the possible and not fixed model parameters was presented in this work, to explain how widely this approach can be used.
The authors declare that there are no competing interests regarding the publication of this paper.
The authors would like to acknowledge the funding support of the EUFP7ITN (Grant no. 316560) for the works presented as a part of the STEEP ITN project. The authors thank Mr. Pablo Lozano Torrubia from the University of Nottingham for his help in the experimental work.