International Journal of Engineering Mathematics

Volume 2016, Article ID 9608584, 15 pages

http://dx.doi.org/10.1155/2016/9608584

## Sensitivity Studies and Parameters Identification for Noisy 3D Moving AWJM Model

Laboratoire de Mathématiques J. A. Dieudonné, Université de Nice Sophia Antipolis, Nice, France

Received 10 June 2016; Accepted 3 August 2016

Academic Editor: Z. X. Guo

Copyright © 2016 Didier Auroux and Vladimir Groza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 self-generated 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.

#### 1. Introduction

The abrasive waterjet (AWJ) machining is a nonconventional low-cost process [1] that was developed to give the opportunity to manufacture complex shapes with difficulty in cutting and milling materials regardless of their properties [2, 3]. This machining technique embraces low cutting forces [4] on the target workpiece (reducing the possible risks of damaging the sample) and does not have heat affected zone. The Abrasive Waterjet Milling (AWJM) process involves the high-speed waterjet produced by the water pump with a small nozzle and abrasive particles included in the jet flow. This forms the circular high-energy jet plume of water and abrasives that impacts on the target surface and erodes the material. The behavior of the material removing and the intensity of the etching rate can be changed and controlled by different machine parameters such as the feed speed of jet movement, pump pressure, and mass flow rate of the abrasive particles. All these parameters can be mathematically characterized by etching rate function which is inaccessible from the experiments and plays key role in the modeling and prediction of the trench surface.

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 well-known methods based on the statistical approaches [5] and finite element methods [6–8] have been used and reported. For solving the direct problem, which particularly involves a nonlinear PDE, some information about the model parameters such as coefficients or energy sources is required, but often most of them are unavailable or unknown in advance and need to be identified.

Even if the direct problems are linear under some considerations, however, the inverse problems of the model parameters identification are usually ill-posed [9, 10] and measurement noise and model errors impose regularization needs. The regularization techniques [11–13] in the identification process can assist in performing and coping with such aspects. The posed minimization problem for a cost function (i.e., a mismatch between experimental measurements and modeled estimation) underlies the identification problem. Some common approaches and techniques for various general and particular problems have been used and reported in [14–17].

In this paper, we extend the work presented in [18] about the mathematical method to identify required unknown parameters of the generic Abrasive Waterjet Milling (AWJM) model. This model was previously developed and reported in [19–22] according to the industrial needs for microwaterjet footprints prediction.

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 [23].

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 [24]. Numerical optimization method based on the limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm [25–27], realized in the minimization package N2QN1 from the INRIA MODULOPT library [28], is used to implement the minimization problem. In terms of the adjoint problem, the minimization problem can be represented as an optimal control problem based on the Lagrangian multiplier of the model equations [29, 30].

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 2 consists of representation of the mathematical model describing the AWJM process and explanation of the adjoint approach, which is used to obtain the gradient of the cost function. Further in this section we give a short description of gradient descent algorithms, which are used for the minimization problem. In Section 3 we present the model parameters identification and actual numerical results for the moving waterjet with fixed and varying feed speeds. Results based on artificial and experimental observations are provided. Section 4 presents the sensitivity study of the given AWJM model, where the influence of the various measurement noises is studied and several approaches to improving the accuracy in the surface prediction are demonstrated. This paper finalizes by Section 5 with some conclusions and outcomes.

#### 2. AWJM Direct and Adjoint Model

##### 2.1. Proposed AWJM Model

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 and denote by a bounded domain of where the process takes place.

The proposed Abrasive Waterjet Milling model, coming from previous works [19–21] and already partially studied in [18], is presented aswith initial and boundary conditions:

The given AWJM model in (1) describes the trench creation by impact of an abrasive waterjet of radius with the forces caused by etching rate function on the primarily flat surface.

The final form of the trench, which is described here as a function , depends on the different physical parameters such as pump pressure, abrasives mass flow, velocity, and waterjet nozzle diameter. According to the announced mathematical model, these machine settings could be described and represented as set of model parameters , defining the intensity of the jet impact. The vertical position of the jet is fixed during the process relative to the zero level of the workspace, and the intensity of the jet impact continuously depends on the trench depth.

The schematic representation of the problem is illustrated in Figure 1.