An Evolutionary Algorithm for the Texture Analysis of Cubic System Materials Derived by the Maximum Entropy Principle

Abstract

Based on the principle of maximum entropy method (MEM) for quantitative texture analysis, the differential evolution (DE) algorithm was effectively introduced. Using a DE-optimized algorithm with a faster but more stable convergence rate of iteration, more reliable complete orientation distribution functions (C-ODF) have been obtained for deep-drawn IF steel sheets and the recrystallized aluminum foils after cold-rolling, which are both designated as the macroscopic cubic-orthogonal symmetry. With special reference to the data processing, no more other assumptions are required for DE-optimized MEM except that the system entropy approaches the maximum.

1. Introduction

With regard to the quantitative analysis of textured materials, when the representation method for orientation distribution function (ODF) appeared primarily in an approximation, it was defined as the series of a reduced ODF (R-ODF) for crystalline grain clusters [1–5]; furthermore, the complete orientation distribution (C-ODF) also evolved with the development of quantitative texture analysis through several specific mathematical treatment procedures [6–9]. Among the different methods currently utilized, the maximum entropy method (MEM) established in keeping with the principle of maximum entropy has become the most accessible technique for the resolving of C-ODF based on a nonlinear parametric system, which was proposed independently by Wang et al. and Schaeben in the late 1980s [8–11]. However, during the calculation process, a massive system of nonlinear equations must be solved to reach the genuine solution. Moreover, for all the conventional algorithms for solving the nonlinear equations based on the system of analytical mathematics, a set of specific initial solutions should be selected or defined. If the designation on initial solution is not appropriate, it usually results in the iterative failure of the whole calculation process. That’s exactly what makes MEM a difficult approach to the genuine solution by means of conventional iterative algorithms [12–16].

In recent years, the maximum entropy principle has been used in a wider range of applications for dealing with complex systems consisting of nonlinear variables [17–21]. It is crucial for such an analytical method to solve the fuctional relevancy among multivariate nonlinear variables by means of the most accessible variables that are linearly independent of each other [22–29]. In order to improve the efficiency for solving this problem, a DE algorithm was introduced for the solution procedure. The DE algorithm is a newly developed meta-heuristic approach that mainly has three advantages: detecting the true global minimum regardless of the initial parameter values, fast convergence, and the use of few control parameters. The DE algorithm is a population-based algorithm like genetic algorithms using similar operators: crossover, mutation and selection [30–35]. It works to prevent the solving procedure from obscure selection of the initial solution.

2. General Principle Description

2.1. ODF Representation Derived by MEM

As with the implementation of MEM, sub-orientation space without symmetry is presented in response to the crystalline structure and texture symmetry of different materials, and then, divided into equivalent orientation units whose total number is J. V_j is defined as the volume percentage of crystalline orientation that falls coincidently in orientation unit j of the detected sample; ω (θ_j, ψ_j, φ_j) denotes the relative orientation density of unit j identified (θ_j, ψ_j, φ_j are the central Euler angles of j):

$$\begin{array}{cc}{V}_j=\omega ({\theta }_j,{\psi }_j,{\phi }_j)\sin {\theta }_j\mathrm{\Delta }\theta \mathrm{\Delta }\psi \mathrm{\Delta }\phi ,& j=1,2,\dots \dots \end{array},J$$

(1)

$${\displaystyle \sum _{j=1}^J{V}_j}=1$$

(2)

In the R-ODF series, the coefficient of the lmn th item, W_lmn, is substituted by W_r in subscript label, then, r = 1, 2, …, R; the value of R is determined by lmn where the expanded series is truncated. According to series expansion method [2,3], there should exist:

$$\begin{array}{cc}{W}_r={\displaystyle \sum _{j=1}^J{V}_j{R}_{rj}},& r=1,2,\cdots \cdots ,R\end{array}$$

(3)

In the Equation (3), the coefficient of ODF series W_r can be calculated from experimental data. The coefficient A_rj is an identified value in the case of complete macroscopic symmetry. The deduction process is elucidated in the Equation (4) as follows:

$$\begin{array}{c}{A}_{rj}=\frac{1}{3}\{[Z_{\ell mn}(\cos {\theta }_j)\cos (\mathrm{m}{\psi }_j+\mathrm{n}{\phi }_j)+{(-1)}^{\ell +m}{Z}_{\ell m\overline{n}}(\cos {\theta }_j)\cos (\mathrm{m}{\psi }_j-\mathrm{n}{\phi }_j)]\\ +[Z_{\ell mn}(\cos {\theta }_2)\cos (\mathrm{m}{\psi }_2+\mathrm{n}{\phi }_2)+{(-1)}^{\ell +m}{Z}_{\ell m\overline{n}}(\cos {\theta }_2)\cos (\mathrm{m}{\psi }_2-\mathrm{n}{\phi }_2)]\\ +[Z_{\ell mn}(\cos {\theta }_3)\cos (\mathrm{m}{\psi }_3+\mathrm{n}{\phi }_3)+{(-1)}^{\ell +m}{Z}_{\ell m\overline{n}}(\cos {\theta }_3)\cos (\mathrm{m}{\psi }_3-\mathrm{n}{\phi }_3)]\}\end{array}$$

(4)

In Equation (4)Z_lmn(θ) denotes the augmented Jacobi polynomials; and two sets of Euler angles with the subscripts of 2 or 3 are derived from {θ_j, ψ_j, ϕ_j} due to the tri-fold symmetry in crystalline structure, whose values can be determined according to the reference [13,16].

In accordance with the principle of entropy maximum, there should be:

$$V_j=e^{(-1-x_0-{\displaystyle \sum _{r=1}^R{x}_r{A}_{rj}})};r=1,2,\dots ,R;j=1,2,\dots ,J$$

(5)

In Equation (5), x (x₀, x₁, x₂, …, x_R) are the undetermined Lagrange multipliers. When Equation (5) is substituted into both Equations (2) and (3), a set of nonlinear equations on solving different x whose total number is (R + 1) are obtained:

$$\{\begin{array}{c}{\displaystyle \sum _{j=1}^J{e}^{(-1-x_0-{\displaystyle \sum _{r=1}^R{x}_r{A}_{rj}})}}=1\\ {\displaystyle \sum _{j=1}^J{e}^{(-1-x_0-{\displaystyle \sum _{r=1}^R{x}_r{A}_{rj}})}\cdot A_{rj}=W_r}\end{array}$$

(6)

Based on the theorem that the number of unknown variables equals the amount of equations, Equation (6) is. However, it holds a set of nonlinear equations with the unknown variables sitting on their perch among the exponential terms, so it is practically infeasible to solve such a system of equations by the regular mathematical method. For this purpose, the objective function S is established for the DE algorithm.

The objective function of S, which is composed by the nonlinear equation set, can be expressed as:

$$MinS={(1-{\displaystyle \sum _{j=1}^J{e}^{(-1-x_0-{\displaystyle \sum _{r=1}^R{x}_r{A}_{rj})}}})}^2+{\displaystyle \sum _{r=1}^R{f}_r{(W_r-{\displaystyle \sum _{j=1}^J{A}_{rj}{e}^{(-1-x_0-{\displaystyle \sum _{r=1}^R{x}_r{A}_{\mathrm{rj}}})}})}^2}$$

(7)

where f_r ∊(0,1] is a positive parameter for controlling the effect of W_r on S, whose value should be assumed as less than 1. When Equations (6) and (7) are both substituted into DE, the values of x₀, x₁, x₂, …, x_R can be solved. Then, after being substituted back to Equations (5) and (1), the complete ODF (C-ODF) can be obtained.

2.2. Differential Evolution Algorithm

Differential evolution (DE) is an algorithm based upon swarm evolution, which is characterized by memorizing the individual optima and sharing information in the swarm. It performs according to the mechanism of cooperation and competition among the individuals to achieve the solution of an optimization problem.

As a classic formulation, DE begins with a randomly initiated swarm population of N_P in the whole accessible solution space, which is designated as $X^0=\{{\mathrm{x}}_1^0,{\mathrm{x}}_2^0,\cdots \cdots ,{\mathrm{x}}_{N_p}^0\}$, N_p denotes the swarm population; furthermore, an individual with D-dimensional parameter vectors represents the initial solution of the specified problem, which is designated as ${\mathrm{x}}_i^0=[x_{i1}^0,x_{i2}^0,\cdot \cdot \cdot \cdot \cdot \cdot ,x_{iD}^0]$, i = 1, 2, …, N_p. D is the dimensionality of the optimization problem.

In this paper, S(X) is defined as the objective function to be optimized for solving the MEM nonlinear equation set, and X (x₀, x₁, x₂, …, x_R) is the decision vector consisting of R + 1 variables and designated as components for the equation set to be solved, i.e., X is the solution vector and in addition termed as an individual that consists of an R + 1 dimensional parameter vector. DE algorithm executes to evolve a population including N_P individuals in number, i.e., X_ik =(x_i0,x_i1,x_i2⋯x_iR) ∊ S(X), i=1,2⋯N_p. So, the amount of candidate solutions is N_P. $x_{i,k}^G$ signifies the kth parameter of the ith solution vector. G denotes the Gth generation of evolution. N_P doesn’t change during the evolution process once determined.

The essential feature for DE is an iterative algorithm for generating trial vectors. Mutation and crossover operators are used to generate trial vectors, and the selection operator then determines which of the vectors will survive into the next generation [31–36]. In this paper, we adopt the fundamental scheme whose notation is described as “DE/rand/1/bin” strategy for solving ODF utilizing a set of experimental information [2–6].

2.2.1. Initialization

DE operates in a straight parallel search method. An initial population begins randomly with a uniform distribution in the whole accessible search space to activate the iterative scheme of DE. When G = 0, it starts as a population of N_P parameter vectors in the R + 1 dimensionality, i.e., $X(0)=\left\{{\mathrm{x}}_1^0,{\mathrm{x}}_2^0,\mathrm{...},{\mathrm{x}}_{N_P}^0\right\}$, and ${\mathrm{x}}_i^0=[x_{i,0}^0,x_{i,1}^0,\dots ,x_{i,R}^0]$.

2.2.2. Mutation

DE mutates and recombines the population to produce a population of N_P mutant vectors, $V_i^G=({\nu }_{i,0}^G,{\nu }_{i,1}^G,{\nu }_{i,2}^G,\dots ,{\nu }_{i,R}^G)$. For each target vector $x_{i,k}^G$, a mutant vector ${\nu }_{i,k}^G$ is generated according to the following:

$${\nu }_{i,k}^G=x_{r_1,k}^G+F\cdot \left(x_{r_2,k}^G-x_{r_3,k}^G\right),k=0,1,2,\dots ,R$$

(8)

In Equation (8)F is commonly known as a scale factor and is a positive real number. It controls the amplification of the different vector $\left(x_{r_2,k}^G-x_{r_3,k}^G\right)$. According to [10], the range of F varies within [0,2]; Three individuals, $x_{r_1,k}^G$, $x_{r_2,k}^G$ and $x_{r_3,k}^G$, are selected randomly from the current population, where the indices r₁, r₂ and r₃ represent the random and mutually different integers generated within the range of [1,N_P] so that r₁,r₂,r₃∊{1,2,…N_p}, and furthermore different from index i such that r₁ ≠ r₂ ≠ r₃ ≠ i. It should be noted that these indices must be different from each other and out of the proceeding index labeled with i so that N_p should be ≥ 4. If one component of the mutant vector overflows the search space, then a new substitute value is generated by activating initialization.

2.2.3. Crossover

The target vector is mixed with the mutated vector, utilizing that the parent vector is discretely recombined with the mutated vector to yield a trial vector, $U_i^G=(u_{i,0}^G,u_{i,1}^G,u_{i,2}^G,\dots ,u_{i,R}^G)$ so as to balance the differential mutation search strategy:

$$u_{i,k}^G=\{\begin{array}{l}{v}_{i,k}^G,rand(i,k)\le C_{r}ork=k_{rand},\\ x_{i,k}^G,rand(i,k)>C_{r}ork\ne k_{rand}\end{array}$$

(9)

In Equation (9), rand(i, k) ∊ [0,1] is a uniformly generated random number ranging within [0,1]. i=1,2,…,N_p and k = 0,1,2,…,R refers to the kth parameter of the ith vector, which performs to establish a $u_{i,k}^G$ through discrete recombination. $k_{rand}\in \left\{0,1,2,\mathrm{...},R\right\}$ denotes a randomly triggered integer index, which ensures that $u_{i,k}^G$ acquires at least one mutant vector different from $v_{i,k}^G$; otherwise, no new parent individual would be generated and the population should still remain unchanged. C_r ∊ [0,1] is the crossover probability constant, which is empirically calibrated by the user.

2.2.4. Selection

A greedy criterion is adopted for DE to decide whether or not the trial vector $u_{i,k}^G$ should become a member generation of G + 1 after it is compared to the target vector $x_{i,k}^G$. Only when the $u_{i,k}^G$ yields a better fitness function value than the $x_{i,k}^G$ does can it be set to $x_i^{G+1}$. Otherwise, the previous target vector $x_{i,k}^G$ becomes retained. The selection scheme is described as follows for an issue on minimization:

$$x_{i,k}^{G+1}=\{\begin{array}{l}{u}_{i,k}^G,S(u_{i,k}^G)<S(x_{i,k}^G),\\ x_{i,k}^G,S(u_{i,k}^G)\ge S(x_{i,k}^G).\end{array}$$

(10)

A typical pseudo-code of DE algorithm according to the “DE/rand/1/bin” mutation scheme [11–13] is described in Algorithm 1 as follows:

Algorithm 1. Pseudo-code flowchart of a typical DE algorithm with “DE/rand/1/bin” mutation scheme

Algorithm 1. Pseudo-code flowchart of a typical DE algorithm with “DE/rand/1/bin” mutation scheme

The number of control parameters in DE is very small. The classic DE has three parameters that need to be adjusted: (a) the population size N_P; (b) the mutation scale factor F∊ [0,2]; (c) the crossover rate C_r∊[0,1]. How these parameters affect the performance of DE algorithm is well discussed in the references [30–34]. As for the termination conditions, either a maximum iteration number corresponding to the objective function for evaluation or a fixed tolerance on precision error of the desired solution can be proposed in advance. It can be shown that this algorithm is favorable due to its simplicity in formulation, easiness to be operated and suitability for calculation by computers.

In DE algorithm, the population size is set as N_P = 300, which is determined according to the principle: 3R ≤ N_P ≤ 9R, suggested by Storn and Price et al. [33–36] due to the large value of R = 58, when l equals to 16; cross rate, C_r = 0.2, which is confirmed by the 59 linearly independent vectors of X_ik and also referred to the suggestion by Storn and Price et al. [32,33]; the scaling factor is referred to as F = randreal_j (0.2,05); the tolerance on precision error in equality is set as δ=5×10⁻⁴ according to the references proposed in [31–33]. The algorithm is quite robust with respect to the algorithmic parameters F and C_r. Setting F = 0.35 and C_r = 0.2 will give generally good convergence on a wide range of problems according to references [34–40].

The calculated results should be adjusted in terms of normalization for comparison with each other. For this purpose, in the whole programming procedure of solving genuine C-ODF, the parameter settings are fixed as the followings unless a change is mentioned: In MEM, for simplicity, all of f_r (r = 1, 2, 3, …, R) in the proposed objective function S(X) are suggested to perform as a constant of 1/16 experientially since that the truncation point was set to l = 16. Since that the DE optimized MEM is designated for the quantitative texture analysis on the cubic materials with macro symmetric texture, therefore, the total amount of orientation units corresponding to sub-space is set to 1296 (72 × 18), i.e., J = 1296.

3. Experimental Results and Discussion

The test samples were the recrystallized aluminum foils after cold-rolling (fcc), and the deep-drawn IF steel sheets (bcc), respectively. More than two samples for each group in comparison were examined systematically in this work. The measurements were carried out using an X'Pert diffractometer from PANalytical (Almelo, The Netherlands) equipped with a Cobalt target (λ_kα = 0.179 nm). Three experimental pole figures of (111), (200), and (220) corresponding to the fcc structure, and (110), (200), and (211) to the bcc structure have been measured (α = 0°~70°, β = 0°~360°, △α = △β = 5°), respectively.

3.1. ODF Representation on Aluminum Foils

The results of ODF representation derived by different algorithms are shown in comparison as Figure 1. It can be obviously observed that the ODF representation of the fcc aluminum foil sample consists of exclusively intensive {100} <001> cubic texture. By adopting the traditional Two-step Method [6], its maximum orientation density is ω_max = 167.51, and there still exist “ghost peaks” together with negative zones in divergence shown as Figure 1a, whose negative value reaches as low as −60.16. The C-ODF derived by a damped Newtown method based on MEM is shown in Figure 1b, ω_max = 260.05. There exist no “ghost peaks” or negative areas any more. Moreover, the C-ODF derived by DE optimized MEM is shown as Figure 1c, ω_max = 265.97. It is verified that the positions of the texture components, i.e., the {ψ,θ,φ} presented in the ODF configuration are completely identical to each other except that their intensities differ more or less.

Although the value of the primary {100} <001> cubic texture component in Figure 1b appears almost identical to that in Figure 1c, it has been achieved through 4000 iterations on condition that an initial solution selection must be selected appropriately. By comparison, its magnitude is ten times larger than that derived by DE. The results calculated for the Aluminum foils corresponding to the iteration number, G_max, are list in details in

Table 1. The effect of iteration number on the calculation results of Al foils using DE optimized MEM.

Gmax	Maximum iteration number
	50	100	200	600	1000	4000
Objective values, Smin	5.19 × 10−2	2.09 × 10−2	1.76 × 10−3	4.91 × 10−4	1.05 × 10−4	7.62 × 10−5
CPU time, t (s)	3.205	6.795	12.895	29.940	63.245	234.125
ωmax	283.07	281.95	263.50	265.97	264.96	265.95

.

It is indicated from Table 1 that, when k exceeds 600, S_min converges to a convinced value. Hence, $S(u_{i,k}^{600})$ is designated as the genuine solution when G_max equals to 600, and the C-ODF can be resolved consequently.

3.2. ODF Representation on IF-steel Sheets

The samples are 0.8 mm deep-drawn plates of standard IF steel provided by a commercial steel plant. More than two groups of samples were compared in practice. The results of ODF representation derived by different algorithms are compared with each other and shown in Figure 2. In R-ODF, its maximum orientation density is ω_max = 7.07, and the “ghost peaks” and dispersive negative value zones still appear, as shown in Figure 2a, whose negative value grade approaches as low as −0.69. Correspondingly, the magnitude of C-ODF derived by DE optimized MEM, shown in Figure 2c, reaches ω_max = 9.75. No “ghost peaks” or negative areas appear. The result derived by a damped Newtown method is also shown in Figure 2b. It is observed that the positions of the primary γ-fiber (<111>//ND) texture component derived by different methods are completely identical to each other in Figure 2. Therefore, it can be verified that the positions of the texture components, i.e., the {ψ,θ,φ} presented in the ODF configuration are completely identical to each other except that their intensities differ more or less. Although the value of the strongest texture component is nearly identical to that in Figure 2c, ω_max = 9.53, it has been achieved through 5000 iterations with a set of initial solution being designated appropriately. The iteration number also stands ten times greater than that derived by DE. A typical calculation result of the IF steel sheets corresponding to the iteration number, G_max, is list in details in

Table 2. The effect of iteration number on the calculation results of IF steel using DE optimized MEM.

Gmax	Maximum iteration number
	50	100	200	500	1000	5000
φ = 45° section in details
Objective values, Smin	5.19 × 10−2	2.09 × 10−2	3.76 × 10−3	4.97 × 10−4	1.35 × 10−4	3.94 × 10−5
CPU time(s)	3.205	6.795	12.895	29.940	83.765	312.350
ωmax	11.36	11.04	10.42	9.75	9.81	10.03

.

It is indicated from Table 2 that, when G_max exceeds 500, S_min converges to a fixed value. Hence, $S(u_{i,k}^{500})$ is designated as the genuine solution when G_max equals to 500, and the C-ODF can be resolved consequently. The φ = 45° section diagrams in Table 2 are the concise representation on bcc materials, which indicates the {111} γ-fiber texture component parallel to the rolling surface of IF steel. The diagrams illustrate the variation tendency of the primary texture component in IF steel corresponding to different iterations obviously. When G_max stands at 500, the configuration of φ-section is nearly identical to that with 5000 iterations.

It should be noted that DE is a population-based search strategy and performs through the stochastic optimization algorithm, which differs prominently in the sense that distance and direction information from the current population are utilized to guide the search process compared with other evolutionary algorithms (EA). Consequently, the optimum solution for each different calculation under the same initial conditions doesn’t remain constant but rather varies within a very limited range. The variation is rational due to the robustness of global optimum corresponding to the prerequisite that the solution’s behavior changes continuously with the initial conditions. Since that there is no need for an initial preset solution for executing the MEM, it is a well-posed problem.

In conclusion, it is proved that DE is a more efficient algorithm for global optimization and also an evolutionary method for ODF representation of textured materials so as to make the results much more reliable with much less computation time.

4. Conclusions

In this article the differential evolution algorithm has been effectively applied for an improved ODF representation on cubic textured materials by MEM without designating a set of initial solutions.

• The complete ODF derived by differential evolution is entirely feasible due to its simple parameterization scheme, whose primary texture component is endowed with the configuration pattern in more details with much less iteration rounds.
• The differential evolution algorithm is suitable for solving the nonlinear equations of quantitative texture analysis, which is proposed in this article for the ODF representation on the cubic textured materials.
• Differential evolution is characterized by global optima to provide better computation results via much less iterations in comparison with those deduced by conventional damped Newton iteration, which shows a universal optimization mechanism for quantitative texture analysis that follows the maximum entropy principle.