Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 157892, 8 pages

http://dx.doi.org/10.1155/2015/157892

## Modeling Material Flow Behavior during Hot Deformation Based on Metamodeling Methods

^{1}State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China^{2}College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China^{3}Key Laboratory of Nonferrous Materials and New Processing Technology of Ministry of Education of China, Guangxi University, Nanning 530004, China

Received 27 July 2015; Accepted 30 August 2015

Academic Editor: Mohsen Asle Zaeem

Copyright © 2015 Gang Xiao et al. 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

Modeling material flow behavior is an essential step to design and optimize the forming process. In this context, four popular metamodel types Kriging, radial basis function, multivariate polynomial, and artificial neural network are investigated as potential methods for modeling the flow behavior of 6013 aluminum alloy. Based on the experimental data from hot compression tests, the modeling performance of these four methods was tested and subsequently compared from different aspects. It is found that all the methods are capable of constructing models for describing the hot deformation behavior. The merits of Kriging method over other three methods are highlighted when the sample size for modeling is decreased. Furthermore, the applicability of Kriging method is validated while decreasing the sample uniformity with respect to temperature or strain rate. It is proved that Kriging method is competent in modeling the material flow behavior and is the most effective one among the four popular types of metamodeling method.

#### 1. Introduction

Hot forming technology of materials is widely applied in practical manufacturing. In order to achieve the required microstructure and mechanical property of products, the design and optimization of thermomechanical process parameters have to be properly implemented [1, 2]. For this reason, the designers strive to study the effect of the parameters with respect to the kinetics of the metallurgical deformation. The flow behavior of materials is compactly affected by the factors of temperature, strain rate, and strain [3]. For exploring and describing the hot deformation behavior, many research groups have made use of regression method to establish the phenomenological and physically based constitutive models [4–6], which would be committed to provide a complete mathematical description of the flow stress. Lin et al. [5] developed a new phenomenological model for describing high-temperature flow behavior of inconel 718 superalloy. Saadatkia et al. [6] developed a physically based model of low and medium carbon steels to investigate the deformation behaviors. However, the deformation behavior of metal at elevated temperature is always associated with various metallurgical phenomena and thereby complicated in nature [7], such as work hardening, dynamic recovery, dynamic recrystallization, and flow instabilities. The effects of these factors on flow stress are so complex that the relationship among them is highly nonlinear. It can be found that the physical procedure is difficult to be systematically interpreted by the conventional constitutive model [8], which would reduce the prediction accuracy and limit the application range. Moreover, the development of modeling flow behavior in the conventional way is usually time-consuming and relies on amounts of experimental data.

Considering the disadvantages of the abovementioned method, artificial neural network (ANN) method is gradually utilized as an alternative approach for modeling material flow behavior to improve processing scheme [8–10]. In essence, ANN is one type of metamodeling method which can get rid of the constraints from describing physical mechanisms. Metamodeling method is widely accepted as a valuable and efficient technique to model various complex nonlinear relationships. In previous researches [11], some other types of metamodeling method were maturely developed, such as Kriging, radial basis function (RBF), and multivariate polynomial method. Each metamodel type has its associated fitting method and the corresponding characteristics [12]. Due to the complicated nature of the relationships between metamodeling methods and engineering problems, there is no conclusion on which method is definitely superior to the others [11]. In order to investigate the applicability of metamodeling methods and search for the most appropriate one in modeling flow behavior, the four popular methods Kriging, RBF, multivariate polynomial, and ANN were utilized and compared for modeling the flow stress of 6013 aluminum alloy during hot compression.

In this work, the hot plane strain compression tests were carried out on Gleeble-3500 thermomechanical simulator. Considering deformation heating and heat transfer, the measured flow stress data at relatively high strain rates were corrected. Based on the measured and corrected data, the feasibility of these methods is firstly tested in modeling the flow behavior. The most effective modeling method with satisfactory accuracy, stability, and efficiency was then selected on the basis of comparative study and detailed analysis. Finally, the performance of Kriging method that is considered as the best one was further assessed for this modeling process.

#### 2. Metamodeling Methods

##### 2.1. Kriging Method

After being proposed by Krige [13] and improved by others such as Matheron [14], Kriging method was gradually developed to be a popular metamodel type and systematically introduced into the area of computer experiment [15]. In Kriging method, the random output is assumed to be obtained from a linear combination of regression functions plus a random process factor as follows:where is the number of regression functions, is a regression function, is the coefficient for , is the design point, and is the random process function. The correlation function is defined by the following equation:where is the correlation function, and are two design points, and is a structural parameter to be optimized. The correlation function could be defined in several different ways such as Gaussian, exponential, linear, spherical, and cubical [16].

##### 2.2. Radial Basis Function Method

Radial basis function (RBF) method was initially developed as an exact interpolation technique for data in multidimensional space [17]. In RBF method, a series of center points () are chosen first from the design points () based on some criteria. Then the basis functions are constructed by using these center points aswhere is the design point, is the center point for , and is the Euclidian distance between the two points. The function could have different forms such as linear, Gaussian, cubic, thin-plate, spline, and multiquadratic [17]. The relationship between inputs and outputs is constructed as a linear combination of radial basis functions:where is the estimated output value, is a basis function, is a constant, and is the coefficient for .

##### 2.3. Multivariate Polynomial Method

Multivariate polynomial method is one of the most fundamental metamodeling methods used in computer experiment. It is mostly known from response surface method [18], which employs quadratic polynomial in the field of engineering design optimization. In multivariate polynomial method, basis functions are built directly by using input variable components () and their interactions such as

A multivariate polynomial is constructed as the weighted sum of these basis functions:where is the estimated output value, is a basis function, is a constant, and is the coefficient for .

##### 2.4. Artificial Neural Network Method

Artificial neural network (ANN) method is based on a particular set of nonlinear functions to build the relationship between inputs and outputs. Its architecture consists of input layer, output layer, and hidden layer, which are connected by the processing units called neurons [19], as shown in Figure 1. Each neuron in the input layer and output layer represents one independent variable, while the neurons in the hidden layers are only for computation purpose. During training procedure, the known input-output pairs are used to update the weights and biases [20]. The objective is to minimize the errors between the ANN outputs and the targets for the corresponding inputs.