Modeling of impact toughness of cold formed material by genetic programming

Abstract

In the paper, an approach completely different from the conventional methods for determination of accurate models for the change of properties of cold formed material, is presented. This approach is genetic programming (GP) method which is based on imitation of natural evolution of living organisms. The main characteristic of GP is its non-deterministic way of computing. 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. First, copper alloy rods were cold drawn under different conditions and then impact toughness of cold drawn specimens was determined by Charpy tests. The values of independent variables (effective strain, coefficient of friction) influence the value of the dependent variable, impact toughness. On the basis of training data, different prediction models for impact toughness were developed by GP. Only the best models, gained by genetic programming were presented in the paper. Accuracy of the best models was proved with the testing data set. The comparison between deviation of genetic model results and regression model results concerning the experimental results has showed that genetic models are more precise and more varied then regression models.

Introduction

In order to reach high quality of the metal forming process and full functionality of the product, the properties of the material which the future product will be made of, have to be determined as precisely as possible. One of very important properties of material is impact toughness, which is defined as the ability of a material to resist fracture under the effect of shock loading. It is defined by the energy required to break a piece of metal of standard shape and with a cross-sectional area of 1 cm² [1]. A test called the Charpy test is used for evaluation of impact toughness (CVN) of a variety of mass produced materials and is of great value for the selection of materials and for quality control [2].

To get precise data and to reduce the cost of the experiment, several modeling methods predicting the values of the dependent output variables (i.e. system behavior) have been used so far [3], [4], [5]. Traditional methods often employed to solve complex real problems tend to inhibit elaborate explorations of the search space. They can be expensive and often results in sub-optimal solutions. In most traditional modeling methods, such as multiple regression analysis, a prediction model is determined in advance. Merely a set of coefficients has to be found by the deterministic procedure. Because of the pre-specified shape and size of the model, the latter is often not capable of capturing the complex relation among influencing parameters and the model would not be precise enough for industrial applicability.

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 limitations encountered in traditional methods. EC harnesses the power of natural selection to turn computers into optimization tools. It is very applicable to different problems in manufacturing industry [6], [7]. One of the core methods of evolutionary computing is GP method, which use the genetic algorithm paradigm to derive computer expressions to solve a given problem. The aim is often to build models capable of predicting output values from input values. It is very important that the influence of the independent input variables on the dependent output variables and, consequently, on the product quality can be examined in the early stage of process planning. GP method is most often used for complex system modeling [8], [9]. But with proper selection of several genetic parameters, GP can also be effectively used for modeling of relatively simple system, such as system described in our paper, since it has a great advantage of being more accurate than conventional methods (e.g. regression analysis). In addition, when the complexity of the environment is increasing (e.g., more system parameters, more experiments), the adaptation of GP run on such new environment is relative easily. In GP, prevention against quick fall in local optimum (i.e. genetic model, which does not provide a suitable system solution) can be assured by different measures:

• With sufficient number of input variables (i.e. terminal genes T), influencing the output variable of the GP process.

• With proper selection of function genes F for adequate description of the relation between system variables.

• With adequate mutation probability (assures new genetic material in the population).

In the paper, genetic programming (GP) approach for modeling the system (i.e. to describe the change of impact toughness in cold formed copper alloy) is proposed. Experimental data obtained during cold drawing processes at different conditions have served as an environment which, during simulated evolution, the models for the impact toughness have to be adapted to. Different values of effective strain and coefficients of friction were used as independent input variables (parameters), while impact toughness was a dependent output variable. No assumptions about the form and size of impact toughness expressions were made in advance, but they were left to the self organization and intelligence of evolutionary process. Finally, prediction accuracy of different generated models was proved with the testing data set.

The paper is organized as follows. A short description of GP method is given in Section 2. A description of experimental work and experimental results is given in Section 3. Section 4 shows the fitness measure, genetic operations and values of genetic parameters used for GP method. Results, discussion and comparison between the best genetically developed model results, regression model results and experimental results are given in Section 5. Finally, some concluding remarks are given in Section 6.