An optimum approximation of n-point correlation functions of random heterogeneous material systems

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An optimum approximation of n-point correlation functions of random heterogeneous material systems
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  THE JOURNAL OF CHEMICAL PHYSICS  140 , 074905 (2014) An optimum approximation of n-point correlation functions of randomheterogeneous material systems M. Baniassadi, 1,2,a) M. Safdari, 3 H. Garmestani, 4 S. Ahzi, 2,4 P. H. Geubelle, 3 and Y. Remond 2 1 School of Mechanical Engineering, College of Engineering, University of Tehran,P.O. Box 11155-4563, Tehran, Iran 2 University of Strasbourg, ICube/CNRS, 2 Rue Boussingault, 67000 Strasbourg, France 3  Aerospace Engineering Department, University of Illinois, 104 S Wright St., Urbana, Illinois 61801, USA 4 School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive N.W., Atlanta,Georgia 30332-0245, USA (Received 26 December 2013; accepted 5 February 2014; published online 21 February 2014)An approximate solution for n-point correlation functions is developed in this study. In the approx-imate solution, weight functions are used to connect subsets of (n-1)-point correlation functionsto estimate the full set of n-point correlation functions. In previous related studies, simple weightfunctions were introduced for the approximation of three and four-point correlation functions. Inthis work, the general framework of the weight functions is extended and derived to achieve opti-mum accuracy for approximate n-point correlation functions. Such approximation can be utilizedto construct global n-point correlation functions for a system when there exist limited informationabout these functions in a subset of space. To verify its accuracy, the new formulation is used toapproximate numerically three-point correlation functions from the set of two-point functions di-rectly evaluated from a virtually generated isotropic heterogeneous microstructure representing aparticulate composite system. Similarly, three-point functions are approximated for an anisotropicglass fiber/epoxy composite system and compared to their corresponding reference values calcu-lated from an experimental dataset acquired by computational tomography. Results from both vir-tual and experimental studies confirm the accuracy of the new approximation. The new formulationcan be utilized to attain a more accurate approximation to global n-point correlation functions forheterogeneous material systems with a hierarchy of length scales.  © 2014 AIP Publishing LLC  .[http://dx.doi.org/10.1063/1.4865966] I. INTRODUCTION Statistical description constitutes an essential tool for themathematical representation of heterogeneous materials andtheir reconstruction and homogenization. 1 Statistical contin-uum approaches have been developed to describe the struc-tural morphology through statistical correlation functions  1–3 and experiments or theoretical models have shown that someeffective properties of heterogeneous materials are stronglydependent on their correlation functions. 1,4–6 N-point corre-lation functions can be exploited to characterize the prop-erties of a wide range of heterogeneous systems, includinggalaxies. 1,2,4,5,7,8 For any n-phase random homogenous medium, we candefine the following characteristic function for one of thephases, for instance for phase I, χ (phase i ) ( x )  =  1 if  x inphase i 0 otherwise  ,  (1)where  x  is a position vector within the heterogeneous system.The n-point correlation functions are then defined by 9 C n ( x 1 , x 2 , x 3 , . . . , x n ) = χ α 1 ( x 1 ) χ α 2 ( x 2 ) . . . .χ α n ( x n )  ,  (2) a) Author to whom correspondence should be addressed. Electronicmail: m.baniassadi@ut.ac.ir. Telephone: (9821) 88020-4035. Fax: (9821)8801-3029. where the superscript  α i  shows the desired event for the vectorposition  x i . In general, for a d-dimensional isotropic media,two-point correlation functions can be extracted from an m-dimensional subspace (m  =  1,2,..., d-1). For example, two-point correlation functions can be extracted from a 2d or 1dcut section of the isotropic sample. 10 The two-point correlations of eigen-microstructures arecalculated using Fast Fourier Transform. 11,12 In fact, thesquare of the amplitude of the Fast Fourier Transform (FFT)of the characteristic function is equal to the FFT of any au-tocorrelation of respective microstructure function. Higher-order correlation functions can be computed using higher-order spectra. 8 In the present work, our previously developed approxi-mate solution for n-point correlation functions 13 is modifiedto derive a comprehensive equation of the N-point correla-tion functions of heterogeneous materials. The main objectiveof this work is to propose a new set of weight functions forthe previously developed formulation. 13 However, our currentwork is not limited to three-and four-point correlation func-tions, as in our previous work, but is extended to higher-ordercorrelation functions. This approximation can be optimizedfor each microstructure (or heterogeneous system) using neu-ral network or other optimization techniques when thereis limited information available about these functions. To 0021-9606/2014/140(7)/074905/6/$30.00 © 2014 AIP Publishing LLC 140 , 074905-1  074905-2 Baniassadi  et al.  J. Chem. Phys.  140 , 074905 (2014) illustrate this point, the optimum approximation is imple-mented in the current study to approximate three-point cor-relation functions from two-point function in two differentrandom heterogeneous systems; the first one comprising ahard-core platelets/polymer matrix system generated virtu-ally,andthesecond one composed ofaglassfiber/epoxy com-posite system processed and assessed experimentally. II. APPROXIMATION OF N-POINT CORRELATIONFUNCTIONS For non-eigen microstructures, using n-point correlations(n  >  2) of the structure can reveal more morphological de-tails of the features and their distribution. Theoretically, non-eigen structures can be uniquely defined by exploiting aninfinite-order correlation functions ( n  → ∞ ). In statisticalcontinuum theory, approximating higher-order statistical cor-relation functions provides a good precision and efficiencyfor the homogenization and reconstruction of heterogeneousmicrostructures.In our previous research work, 13 n-point correlation func-tions have been approximated with the aid of m-point corre-lation functions, where 0  <  m  <  n . In that approximation,n-point correlation functions were divided into n subsets of (n-1)-point correlation functions. The key idea of this approx-imation was that the correlation functions can be approxi-mated by a weighted sum of the products of n subsets of (n-1)-point correlation functions, which are linked using theconditional probabilities. In addition, the weight functionswere calculated using the boundary limiting conditions. Thegeneral formulation of the approximation of n-point correla-tion functions ( n  >  3) was derived as 13 C n ( x 1 ,x 2 ,x 3 ,....,x n ) = n  i = 1  W  ni   n − 1 n − 2  l = 1 C ( n − 1) ( x i ,....,x ( n − 1) )   n − 1 n − 3  l = 1 C ( n − 2) ( x i ,....,x ( n − 2) ) .   n − 1 n − 4  l = 1 C ( n − 3) ( x i ,....,x ( n − 3) )   n − 1 n − 5  l = 1 C ( n − 4) ( x i ,....,x ( n − 4) ) ...  ,  (3)where  W  ni  are the dependency weight functions. In the formulation above, (  x  m ,...  x  m  ...,  x   p ) is defined as the subset of   (n − 1 ) points that include  x  i  as a member of the subset. The weight functions  W  nm  can be calculated using the boundary limits. The firstboundary condition is given for each  i  aslim x ( i ) →∞ C n ( x 1 ,x 2 ,x 3 ,... ,x ( i − 1) ,x ( i ) ,x ( i + 1)  ....,x n ) = C 1 ( x i ) C n − 1 ( x 1 ,x 2 ,x 3 ,... ,x ( i − 1) ,x ( i + 1)  ....,x n ) .  (4)Here,  C  1  represents the one-point correlation function. This boundary limit condition can be written aslim x ( i ) →∞ n  i = 1  W  ni   n − 1 n − 2  l = 1 C ( n − 1) ( x i ,....,x ( n − 1) )   n − 1 n − 3  l = 1 C ( n − 2) ( x i ,....,x ( n − 2) )   n − 1 n − 4  l = 1 C ( n − 3) ( x i ,....,x ( n − 3) )   n − 1 n − 5  l = 1 C ( n − 4) ( x i ,....,x ( n − 4) )  ...  = C 1 ( x i ) C n − 1 ( x 1 ,x 2 ,x 3 ,...,x ( i − 1) ,x ( i + 1)  ....,x n ) ..  (5)Applying this boundary limit, we get x i  →∞ :  W  ni  = 0 ,W  nj   = 0 for  j   = i.  (6)The second boundary condition is given bylim  x 1 →∞ ... x n →∞ C n ( x 1 ,x 2 ,x 3 ,...,x n ) = n  i = 1 C 1 ( x i ) .  (7)This equality condition leads to x i  →∞  ( i  = 1 ,...,n ) , n  i = 1 W  ni  = 1 .  (8)The third boundary limit is expressed aslim x ( i ) → x ( j  ) C n ( x 1 ,x 2 ,x 3 ,...,x ( i ) ,...,x ( j  ) ,....,x n ) = C n − 1 ( x 1 ,x 2 ,x 3 ,...,x ( j  ) ,....,x ( n − 1) ) .  (9)From thisboundary condition for compatible events and usingEq. (3), we get x i  → x  j  (for  i  = j  ) , W  nk  = 0 ,k  = i  and  k  = j.  (10)Therefore, the necessary conditions for the weight functionsare summarized as follows: x i  →∞  W  ni  = 0 , W  nj   = 0 for  j   = i,  (11) x i  →∞  ( i  = 1 ,...,n ) , n  i = 1 W  ni  = 1 ,  (12) x i  → x j   (for  i  = j  ) , W  nk  = 0 , k  = i  and  k  = j. (13)In the proposed approximation, a unique solution does notexist for the weight functions. Therefore, any chosen set of the weight functions that satisfy the necessary boundary limitconditions is useful for this approximation. For example, for  074905-3 Baniassadi  et al.  J. Chem. Phys.  140 , 074905 (2014) the approximation of three-point correlation functions, a sim-ple choice for the weight functions has been proposed in ourprevious work  13 as W  3 m  =| x k x l || x 1 x 2 |+| x 1 x 3 |+| x 2 x 3 |  m  =  lm  =  k ,  (14)where m , k  ,and l areequalto1,2,or3.Inthecurrentproposedwork, we extend the approximation for the n-point correlationfunctions by finding the best weight functions for each systemstructure. First, we show that weight functions for the approx-imation of 3-point, 4-point, and 5-point correlation functionscan be expressed using Cayley-Menger 14 determinant as W  nm  =   ( − 1) n − 1 2 n − 2 (( n − 2)!) 2  n − 1 ( {| x 1 x 2 | , | x 1 x 3 | ,..., | x ( n − 2) x ( n − 1) |} m | n  =  m )  αm 2 n  k = 1   ( − 1) n − 1 2 n − 2 (( n − 2)!) 2  n − 1 ( {| x 1 x 2 | , | x 1 x 3 | ,..., | x ( n − 2) x ( n − 1) |} k | n  =  k )  αk 2 | n <  6 ,  (15)where  x  i  are the position vectors and  x  i  x   j  are the correlationvectors (Figure 1). In Eq. (15), {} ς   represents the subset of vector lengths of correlations which do not include  x  ς  , theexponents  α m and  α k  are optimization parameters, and   n − 1  isexpressed using Cayley-Menger determinant 14 as  n − 1 ( {| x 1 x 2 | , | x 1 x 3 | , | ,..., | x ( n − 2) x ( n − 1) |} ) =  0 1  ... ...  1 sym  0  | x 1 x 2 | 2 | x 1 x 3 | 2 | x 1 x 4 | 2 ...  sym  0  | x 2 x 3 | 2 | x 2 x 4 | 2 ......  sym ...............  sym ... sym sym sym sym  0  (16)We note that   k   gives a formula for (k-1)-dimensional vol-ume of convex hull of the points (  x  i ) in terms of the Euclideandistances which are defined using the magnitude of vectorlengths for n-point correlation functions with n  <  6 (seeFigure 1). For example, the second polynomial (or   3 ) yieldsthe well-known Heron’s formula for the area A of a trian-gle with known edge lengths. These proposed weight func-tions, using Cayley-Menger determinant, are no longer phys-ically meaningful for k   >  4 (or n  >  5) because the Cayley-Menger determinant in the Euclidean 3D space becomesequal to zero. 15 We also note that the proposed weight func-tion in the work of Baniassadi  et al. 13 , for n  <  5, can also be FIG. 1. Schematic of correlation vectors of the n-point correlation functions. used to derive the above Cayley-Menger determinant-basedapproximation.Given the limitation of the above proposed approxima-tion for the weight functions, we propose a new generalizedapproximation based on simple weight functions that satisfyall necessary conditions in Eq. (11)–(13) and which are valid for n-point correlation functions with n  >  2. These are givenby W  nm  = (( {| x 1 x 2 || x 1 x 3 | ... | x ( n − 2) x ( n − 1) |} m | n  =  m )) α m n  k = 1 ( {| x 1 x 2 || x 1 x 3 | ... | x ( n − 2) x ( n − 1) |} k | n  =  k ) α k .  (17)These weight functions are derived by simply consideringmultiplier correlation lengths of the subsets. Unlike the pre-vious approximation (Eq. (15)), the new generalized approxi-mation is a mathematical description with no particular phys-ical meaning, particularly for n  >  3. In fact, we note that, forn = 3, the previous (Eq. (15)) and new (Eq. (17)) approxima- tions yield the same result, and that Eq. (17) reduces toW 3 m  =| x k  x l | α | x k  x l | α +| x m x k  | β +| x m x l | γ   m  =  lm  =  k,  (18)where  α ,  β , and  γ   are non-zero positive real numbers and  l , m , and  k   are equal to 1, 2, and 3. FIG. 2. Two-point correlation functions (TPCF) via correlation length  r   /  d  ,where  d   is the diameter of the platelets, for the two-phase heterogeneoussystems (shown in inset image).  074905-4 Baniassadi  et al.  J. Chem. Phys.  140 , 074905 (2014) FIG. 3. Dispersion contour of the error for optimization parameters entering Eq. (18): (a)  α  or X-direction, (b)  β or Y-direction, and (c)  γ   or Z-direction. III. COMPUTATIONAL VERIFICATION To show the validity of the proposed weight functionsapproximation, we use a virtual system based on random mi-crostructures and consider the 3-point correlation functionsonly. The virtual microstructure consists of a representativevolume element (RVE) filled with 4.5% inclusions. Isotropicrealizations of randomly distributed hard-core platelet inclu-sions, with three different aspect ratios, are generated andused to calculate the optimum three-point correlation func-tions of the generated virtual microstructures. In this study, asa first step, three-point correlation functions are approximatedusing Eq. (18). In the next step, optimum values of the opti-mization parameters ( α ,  β ,and γ  ) are calculated using neuralnetworks. To calculate the two-point correlation functions, weuse Monte-Carlo simulations 2 for the virtual non-eigen sam-ples. The obtained results for the two-point correlation func-tions are then used to get the approximate solution for thethree-point correlation functions. Then, we optimize the ap-proximation using Monte Carlo results for the 3-point corre-lation points. For this purpose, we use a three layer neuralnetwork with one input and one output. The network has twohidden layers with 32 and 10 neurons and one output layerwith one neuron and uses Levenberg-Marquardt back propa-gation algorithm for training.The platelet geometries of inclusions are defined by thecorresponding radius center and the normal vector for eachinclusion surface. The size of the RVE is chosen large enoughto produce convergence for the two-point correlation func-tions. In Figure 2, we show the calculated two-point corre- lation functions (TPCF) of the three realized microstructureswith different inclusion aspect ratios.Three-point correlation functions (platelet/platelet/ platelet) are calculated using Monte Carlo simulation, and theresults are used to optimize the weight functions in Eq. (18). TABLE I. Optimum values of optimization parameters.Aspect ratio  α β γ   Average of minimum error10 2.2 2.2 2.2 0.1215 2.1 2.1 2.1 0.1320 2 2 2 0.16 In the simulation, more than 1000 three-point correla-tion points with different magnitude of vector lengths havebeen computed and used to optimize the three-point correla-tion functions (THPCF) given by Eq. (3). In this optimization process, we define an error function asError =| THPCF( r ) simulation − THPCF( r ) approximation | THPCF( r ) simulation .  (19)The average error contours are reported in Figure 3 via theoptimization parameters ( α ,  β ,and γ  ) of the approximation,showing a large dispersion of the error. However, we can findthe optimum values of these parameters to have the best ap-proximation of the THPCF for the considered microstructure(see Table I). IV. EXPERIMENTAL VALIDATION To scrutinize the accuracy of the proposed approxima-tion, a composite specimen composed of 52 vol. % unidi-rectional glass fibers loaded into an epoxy matrix is ana-lyzed. The internal microstructure of the specimen has beenobtained using a high-resolution 3D X-ray imaging system FIG. 4. (a) Sample x-ray projection image used for reconstruction; (b) 2Dcross-section view of binary label matrix; and (c) 3D volume rendering of the arrangement of the glass fibers in the unidirectional composite specimen.  074905-5 Baniassadi  et al.  J. Chem. Phys.  140 , 074905 (2014) FIG. 5. In-plane variation of TPCF 01  (a) and TPCF 11  (b), with 1 and 0 re-spectively denoting the glass and epoxy phases; (c) Variation of the four two-point correlation functions for a random in-plane direction, showing a corre-lation length = 10  µ m for all in-plane directions (All dimension are in  µ m). (MicroXCT-400, Xradia). Figure 4(a) shows a sample 2Dprojection generated by the X-rays passing through the speci-men. A number of these X-ray projections have been acquiredfrom the specimen in different angles (from  − 170 ◦ to 170 ◦ around the main axis of the specimen). Using a filtered back projection method, the 3D microstructure of the specimen hasbeenreconstructedfromtheseprojectionimages.Toeliminatenoise and improve quality, a Gaussian smoothing filter hasbeen applied to the raw data. The binary representation of themicrostructure has been segmented from gray-scale data us-ing a threshold filter. A 2D cross-section of the binary matrixis shown in Figure 4(b). Each voxel of the binary matrix (also known as label matrix) represents a cubic chunk of the ma-terial and a non-zero value is assigned to the each voxel cor-responding to the phase occupying the location of the voxel.These operations have been preformed in  MATLAB  using theImage-Processing toolbox. Figure 4(c) also shows a volumet-ric rendering generated from the acquired data revealing theanisotropic arrangement of unidirectional glass fibers in thecomposite specimen.Two-point correlation functions (TPCFs) have been eval-uated directly from the binary label matrix. Global 3D TPCFhave been constructed by systematically evaluating these two-point functions in three orthonormal directions over the corre-lation range. Figures 5(a) and 5(b) show the two-point corre- TABLE II. Optimum values of optimization parameters. α β γ   Maximum error1.56 1.77 1.66 0.15FIG. 6. Error distribution in the approximation of THPCF 111  obtained withproposed method, showing a maximum of 15% error. lationfunctionsTPCF 11  andTPCF 01  with1representingglassphase and 0 the epoxy evaluated along a random in-plane di-rection. Figure 5(c) shows a 2D slice of the global two-pointfunctions.Next the three-point correlation functions (THPCFs) areapproximated from the two-point functions utilizing Eqs. (3)and (18). For comparison, the reference value of the THPCFs are also evaluated directly from binary label matrix and com-pared against their approximate counterparts. An error costfunction is constructed simply by taking absolute differencebetween the approximate and reference values and it is min-imized to find best values for  α ,  β , and  γ  . Unique weightingconsidered for all three optimization parameters in the costfunction. Using a collocation approach, a number of pointshave been selected randomly in the correlation range for thespecimen; i.e., about 10  µ m as shown in Figure 5(c), and the average value for each parameter are reported in Table II.The distribution of error corresponding to the proposed ap-proximation is depicted in Figure 6 showing a maximum of 15% error. V. CONCLUSIONS In the present work, a previously developed approxima-tion for n-point correlation functions based on the condi-tional probability theory has been modified. In this study, aset of weight functions has been proposed to obtain an ac-curate approximation for n-point correlation functions of het-erogeneous materials or systems. The approximation can beadapted to different microstructures. Two examples have beenshown to validate our approach. In both examples, three-pointcorrelation functions are approximated from the set of two-point functions using our proposed methodology for optimiz-ing the accuracy. This methodology can be readily extendedto higher-order correlation functions that are needed for ap-plications such as cosmology, biology, and materials science. 1 S. Torquato,  Random Heterogeneous Materials: Microstructure and  Macroscopic Properties  (Springer, New York, 2002). 2 M. Baniassadi, A. Laachachi, F. Hassouna, F. Addiego, R. Muller, H.Garmestani, S. Ahzi, V. Toniazzo, and D. Ruch, Compos. Sci. Technol. 71 , 1930 (2011).
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