Elsevier

Expert Systems with Applications

Volume 38, Issue 12, November–December 2011, Pages 15014-15019
Expert Systems with Applications

Application of genetic programming for modelling of material characteristics

https://doi.org/10.1016/j.eswa.2011.05.045Get rights and content

Abstract

Genetic programming, which is one of the most general evolutionary computation methods, was used in this paper for the modelling of tensile strength and electrical conductivity in cold formed material. No assumptions about the form and size of expressions were made in advance, but they were left to the self organization and intelligence of evolutionary process. Genetic programming does this by genetically breeding a population of computer programs using the principles of Darwinian’s natural selection and biologically inspired operations. In our research, copper alloy was cold formed by drawing using different process parameters and then tensile strengths and electrical conductivity (dependent variables) of the specimens were determined. The values of independent variables (effective strain, coefficient of friction) influence the value of the dependent variables. Many different genetic models for both dependent variables were developed by genetic programming. The accuracies of the best models were proved by a testing data set. Also, comparison between the genetic and regression models is presented in the paper. The research showed that very accurate genetic models can be obtained by the proposed method.

Highlights

Genetic programming harnesses the power of natural selection to turn computers into optimization tools. ► Many different genetic models for material characteristics were developed by genetic programming. ► Genetic models show much greater accuracy then regression models.

Introduction

For high quality and full functionality of the formed product, the properties of the material, which the future product will be made of, have to be determined as precisely as possible. Several modelling methods for predicting dependent output variables have been developed to reduce the costs of the experiments and computer computations. In most conventional deterministic modelling methods, such as regression analysis, a prediction model is determined in advance. Traditional methods can be expensive and often result in sub-optimal solution. Because of the pre-specified size and shape of the model, the latter is often incapable of capturing complex relationships between influencing parameters. It is very important that the independent input variables influence on the dependent output variables and, consequently, on the product quality has been already examined in the early stages of a metal forming process.

Evolutionary computation (EC) is generating considerable interest for solving real engineering problems. They are proving robust in delivering global optimal solutions and helping to resolve those limitations encountered in traditional methods. EC harnesses the power of natural selection to turn computers into optimization tools. This is very applicable to different problems in the manufacturing industry (Koza, 1992, Paskowicz, 2009). One of most important EC methods is genetic programming (GP) which is, similarly to a genetic algorithm, an evolutionary computation method for imitating biological evolution of living organisms. Several researches have been carried out using a neural network or genetic algorithms for modelling, thus forming process parameters (Fakhrzad and Khademi Zare, 2009, Ganguly et al., 2007, Odugava et al., 2005, Özel and Karpat, 2005, Pierrevall et al., 2003, Tanguy et al., 2005, Zadeh et al., 2005), but only a few dealing with much more general genetic programming method (Baykasoğlu et al., 2008, Brezocnik et al., 2005, Chang et al., 2005, Dimitriu et al., 2009). In the GP method, the structure subject for adaptation is the population of hierarchically-organized computer programs. The GP method is most often used for complex system modelling, but it can also be effectively used for the modelling of a relatively simple system, such as the systems described in our paper.

This paper describes an evolutionary computation method approach for the modelling of tensile strength and electrical conductivity of formed copper alloy. Experimental data obtained during the cold drawing processes under different conditions serves as an environment which, during simulated evolution, models for the tensile strength and electrical conductivity have to be adapted to. Different values for effective strains and coefficients of friction were used as independent input variables, while tensile strength and electrical conductivity were dependent output variables. GP method was used for the evolutionary computation of the models for prediction of both dependent variables, on the basis of a training data set.

Section snippets

Method used

Genetic programming is evolutionary computation methods in which the structures subject to adaptation are those hierarchically organized computer programs whose size and form dynamically change during simulated evolution.

The space for solutions in the GP method is the huge space of all possible computer programs consisting of components describing the problem area studied. Possible solutions in genetic programming are all those possible computer programs that can be composed in a recursive

Experimental work

The aim of the experimental work was to determine the influence of the effective strain εe and coefficient of friction μ during cold drawing on the change of tensile strength and electrical conductivity of cold drawn copper alloy CuCrZr. This special copper alloy has high electrical and thermal conductivity, with excellent mechanical and physical properties at elevated temperatures.

Copper alloy rods were deformed by cold drawing under different conditions. The drawing speed was 20 m/min and the

Genetic programming modelling of tensile strength and electrical conductivity

In the GP method the initial random population P(t) consists of randomly generated organisms which are, in fact, mathematical models. The variable t represents the generation time. Each organism in the initial population consists of the available function genes F and terminal genes T. Terminal genes are in fact independent variables: strain and coefficient of friction. Random floating-point numbers within the range [−10, 10] are added to the set of terminals to increase the genetic diversities

Genetic models – results and discussion

GP modelling of tensile strength was executed by two different genes function sets F = (+, − , *, /) and F = (+, −, *, /, ZEXP). The best (the most accurate) model for tensile strength obtained with genes function set F = (+, −, *, /) is quite complicated and is written in LISP as:(-(+(%(+-6.72502μ)(-(+(-(ε-5.63273)(-εε))((+(+-6.72502μ)μ)(ε(-(-0.168495ε)4.86573))))(+(%(%-5.63273ε)(+-6.72502ε)))))((-7.44391(+7.15074μ))(+μ-9.92379)))(+(-(%(%-5.63273ε)(+-3.00006ε))((+(+-6.72502(+(-μμ)(+-5.92924μ)))μ)(%(ε-

Modelling results obtained by regression analysis

A mathematical model for regression method was chosen according to (Montgomery, Runger, & Hubele, 2001):y(x)=b0+i=1nbixi+i=1n-1j=i+1nbijxij+i=1nj=inbijxij2In Eq. (6.1) y is dependent variable, xi, xij are independent variables, while b0, bi, bij are coefficients to be determined by using regression analysis. In our case, the dependent variables were tensile strength (Rm) and electrical conductivity (a) while effective strain εe and coefficient of friction μ were independent variables.

Conclusion

The genetic development of models took place on the basis of experimental data. The experimental data in this research were in fact the environment to which the population of models had to be adapted as much as possible. The models presented are a result of the self-organization and stochastic processes taking place during simulated evolution, and not of human intelligence. The accuracies of the models developed during the training phase were also confirmed using testing data not included

References (15)

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