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Модель регрессии со сложной функцией (цепное правило) для оценки производительности производственного процесса
на примере сварки трением с перемешиванием
1 12 1 F. Ramezankhani , R. Noorossana , M.R.M. Aliha
1 Научно-технологический университет Ирана, Тегеран, 16846-13114, Иран 2 Университет Центральной Оклахомы, Эдмонд, Оклахома, 73034, США
Сварка трением с перемешиванием является относительно новым способом соединения твердых материалов с использованием неплавящегося инструмента для сварки деталей без плавления материалов. Данная технология широко применяется в различных отраслях промышленности, включая автомобилестроение, судостроение, авиакосмическую отрасль. Разрушающие испытания являются неотъемлемой, но очень затратной частью прикладных технических исследований. В связи с этим актуальной становится задача сокращения количества разрушающих испытаний за счет использования численных методов для контроля качества сварных соединений. С другой стороны, благодаря достижениям в области компьютерных технологий и встроенных сенсорных систем накоплен огромный объем данных, что привело к необходимости их эффективного использования. Функциональные данные позволяют моделировать и анализировать данные высокой размерности. В настоящей статье предложена полнофункциональная модель линейной регрессии для количественной оценки и прогнозирования качества результатов процесса за счет сокращения количества разрушающих испытаний и использования модели обнаружения точек изменения, чтобы избежать использования модели, которая учитывает изменение в одном из компонентов процесса. В представленной модели учитываются автокорреляция и корреляция. Функциональные переменные модели определяются путем полиномиального разложения базисной функции. Результаты экспериментальных испытаний показывают, что предлагаемый метод эффективен для обнаружения неконтролируемых условий и определения местоположения точки изменения. Подтверждением служат полученные значения коэффициента множественной корреляции 0.98 и соответствующее значение величины F, равное 652.95.
Ключевые слова: функциональная регрессия, моделирование процесса, функциональный анализ данных, обнаружение точек изменения, сварка трением с перемешиванием
DOI 10.55652/1683-805X_2024_27_3_169-172
A function-on-function regression model for monitoring the manufacturing process performance with application in friction stir welding
F. Ramezankhani1, R. Noorossana1,2, and M.R. Mohammad Aliha1
1 Industrial Engineering Department, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran 2 Information Systems and Operations Management Department, College of Business, University of Central Oklahoma, Edmond, Oklahoma, 73034, USA
Friction stir welding is a relatively new way to join solid materials without melting using a nonconsumable tool, which has many applications in different industries including automotive, shipbuilding, and aerospace. Destructive testing is an integral part of engineering science, which costs a lot. Reducing the number of destructive tests via numerical calculations to determine the quality of welded parts is valuable. On the other hand, advances in computer technology and embedded sensing systems in different domains have made it possible to collect a variety of data in huge volume at an unbelievable velocity, which provides an opportunity and at the same time a challenge to engineers and practitioners to utilize this rich source of information efficiently. Functional data as a rich form of structured data allows for high dimensionality modeling and analysis of the data. In this paper, we develop a fully functional linear regression model to quantify and predict the quality of the process outputs by reducing the number of destructive tests and presenting a change-point detection model to avoid using the model when a change has occurred in one of the components of the process. Important issues such as autocorrelation and correlation are taken into account in the presented model. The functional variables of the model are solved by polynomial basis function expansions. The results of the experimental tests indicate that the proposed method performs well in detecting out-of-control conditions as well as estimating the change-point location. The obtained value of the multiple correlation coefficient 0.98 and the corresponding F-value equal to 652.95 support these results.
Keywords: functional regression, process modeling, functional data analysis, change-point detection, friction stir welding
© Ramezankhani F., Noorossana R., Aliha M.R.M., 2024
1. Introduction
Friction stir welding involves a multi-physics phenomenon, including viscoplasticity, material flow, metallurgical transformation, heat generation, thermal straining, and structural interaction. Heat is generated by friction between the tool and the joint line, which leads to a softened region near the FSW tool. While the tool is traversed along the objects, it mixes the two objects of metal, and forges the hot and softened metal by mechanical pressure. The schematic view of the FSW process is shown in Fig. 1.
There are many studies on various aspects of FSW [1-5] aimed to achieve high-quality welded products [6-9]. Residual stress, like other response variables extracted from the literature and presented in Table 1, can be an indicator of output quality showing whether the final quality of a weld is acceptable. Residual stress that can result from a variety of mechanisms, including structural changes, inelastic deformations or temperature gradients, is a measure of resistance to localized plastic deformation induced by either mechanical abrasion or indentation. The heat generated in the process and the axial tool force are more significant parameters for residual stress. The temperature value is affected by unknown factors such as microstructure of the object, vibration of the device or any other factor that is not controllable.
Many manufacturing processes including FSW involve destructive tests performed to determine the quality of products. Destructive tests are very expensive and time consuming. It is very important to be able to determine the output of the process by reducing the number of destructive tests. Therefore, the main purpose of this study is to develop a model for quantifying and predicting the quality of the process outputs by reducing the number of destructive tests.
On the other hand, significant growth in the volume, variety and acquisition rate of data in various fields including healthcare [10, 11], supply chain [12, 13], internet of things (IOT), social networking, manufacturing [14], geographic information system, and quality management [15] indicate that big data is the environment that cannot be effectively processed by conventional data analysis methods [9].
In manufacturing environments, different software packages and equipment are used to collect valuable data to increase the efficiency and productivity of systems [16]. Enterprise resource planning, coordinate measuring machine, automated storage and retrieval systems, programmable logic controllers, computerized numerical control machines, automated guided vehicles, and cloud manufacturing environ-
ment [17] are just a few examples of such software packages and equipment. However, the lack of effective analysis methods has contributed to inefficient use of such data that has high or ultrahigh dimensionality and structural complexity, which poses novel challenges both theoretically and computationally to classical methods.
Functional data is one of the most important types of complex structured data with high dimensionality [18]. Such data provide information about curves, surfaces, or anything varying over a continuum. Each functional observation should be considered as a single entity rather than as a sequence of discrete observations. In many operational stochastic processes, the generated data have a functional form. However, one could misrepresent functional data inappropriately as multivariate data, which leads to losing important inherent information. Functional data can be defined as multivariate data where the number of variables is infinite.
Functional data analysis (FDA) has many advantages including reduction of the curse of dimensionality, easier data modeling, and recovering the entire function or its derivatives. There are some important challenges in technical aspect of functional data such as autocorrelation, correlation, and changes in the parameters of the underlying distribution. Core theoretical contributions to FDA were made by J.O. Ramsay in the early 1990s [18]. For more information on FDA see [10-12].
Generalized linear model (GLM) is a useful classical regression model with many applications, however, it is not adequate for cases when independent variables are digitized points of a curve [20]. When one or more dependent or independent variables in a regression equation is in the form of a function, functional regression is used. Functional regression analysis that combines FDA and regression is used to describe the relationship between response variables and their predictors when at least one of the random variables is a function. In statistical process monitoring and control, in addition to parameter prediction [21], parameter monitoring [22] and change-point analysis are also of great importance.
In this paper, we develop a fully functional linear regression model in which both the response variable and covariate(s) are functions (function-on-function regression) by utilizing functional data analysis to quantify the relationship between dependent and independent variables. The focus of this paper is on functional analogues of linear regression analysis. The output of the regression model is the estimation
of coefficients to predict the value of the response variable. For change-point estimation, a likelihood ratio-based model is applied to cluster the functions. The remainder of this paper is structured as follows. A literature review on functional regression analysis is presented in Sect. 2. Functional regression model is explained in Sect. 3. Section 4 is dedicated to simulation analysis and a real-world case study. Our concluding remarks and future research topics are presented in the final section.
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Received 01.02.2024, revised 19.04.2024, accepted 19.04.2024
This is an excerpt of the article "A Function-on-Function Regression Model for Monitoring the Manufacturing Process Performance with Application in Friction Stir Welding". Full text of the paper is published in Physical Mesomechanics Journal. DOI: 10.1134/S102995992404009X
Сведения об авторах
Farshad Ramezankhani, Iran University of Science and Technology, Iran, f.ramezankhani@ut.ac.ir
Rassoul Noorossana, Prof., Iran University of Science and Technology, Iran; University of Central Oklahoma, USA, rassoul@iust.ac.ir Mohammad Reza Mohammad Aliha, Prof., Iran University of Science and Technology, Iran, mrm_aliha@iust.ac.ir
