Finite Element Analysis with Iterated Multiscale Analysis for Mechanical Parameters of Composite Materials with Multiscale Random Grains

1 School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Hunan 410004, China 2 State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China 3 Division of Mathematical Sciences, School of Physical and Mathematical Science (SPMS), Nanyang Technological University, 21 Nanyang Link, Singapore 637371


Introduction
Predicting the mechanical parameters of a composite material with the multiscale random grains is a very difficult problem because there are too many random grains in the composite material and the range of the scale of the grains is very large in the material field shown in Figure 1.Many studies on predicting physical and mechanical properties of composite materials with random grains have been done: the law of mixtures 1 , the Hashin-Shtrikman bounds 2 , the self-consistent method 3 , the Eshelbys equivalent inclusion method 4 and the Mori-Tanaka method 5 , microanalysis method 6 , and so forth.These methods effectively promoted the development of composite materials, but they simplified the microstructure of real materials in order to reduce the computational complexity.The composite materials with large numbers of grains can be divided into two classes according to the basic configuration: the composite materials with periodic configurations, such as braided composites, and the composite materials with random distribution, such as concrete, foamed plastics.Some physical methods and mathematical methods 7-13 have been proposed to solve these problems.However, most of these techniques and methods are based on empirical, semiempirical models or based on the homogenization methods in the periodical structure.Due to the difference of basic configuration, it is difficult for them to handle the composite material with large numbers of multiscale random grains.Hence, in order to evaluate the physical and mechanical performance of the composite material with random grains, it is necessary to make use of the different advanced numerical methods.
In the recent decades, for the problems with the stationary random distribution, Jikov et al. 14 developed the homogenization method and proved the existence of the homogenization coefficients and the homogenization solution, however, not provided with the numerical techniques to carry out the methods for the stationary random distribution.In addition, their method only deals with the point randomly distributed, not with random grains.
For the porous medias with the random distribution, Hou and Wu 15 developed the the multiscale finite element base function method to compute these problems of the porous medias; this method is much valid to the problems with random coefficients and problems with continuous scale.As for the multiscale systems with stochastic effective, Vanden-Eijnden gave the specific step to carry the multiscale method out 16 ; their methods are very effective to mainly solve the problem with time process.For the perforated domain with small holes, Wang et al. gave an effective macroscopic model for a stochastic microscopic system, and in theory, mainly proved that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution as the size of holes diminishes to zero 17 .However, the above methods are not for the equivalent performance of the composite material with large numbers of the multiscale random grains.
Duschlbauer et al. developed the homogenization method with the Mori-Tanaka scheme averaging microfields extracted for individual fibers and the finite element analysis to estimate the linear thermoelastic and thermophysical behavior of a short fiber reinforced composite material with planar random fiber arrangement 18 .Kari et al. developed a representative volume element RVE approach that was used to calculate effective material properties of randomly distributed short fibre composites and analyzed the properties for the volume of random short fibres 19 .Recently, Kalamkarov et al. gave an asymptotic homogenization model for the 3D grid-reinforced composite structures with the orthotropic reinforcements 20, 21 , and Wang and Pan obtained the elastic property for the multiphase composites with the random microstructures 22, 23 .Their methods are the effective homogenous methods for the equivalent performance of the composites with the random grains.However, in fact, in the engineering fields, for the composite materials with a large number of multiscale random grains, such as concrete, asphalt mixture, rock mass, and foam plastics, owing to the random complexity in configuration and for that the grain scale range is very large from 10 −1 m to 10 −6 m 24 , shown in Figure 1, the above methods find it difficult to analyze the mechanical and physical performance.Hence, in order to deal with the composite materials with multiscale random grains, authors proposed a kind of statistically multiscale analysis SMSA method to predict the effective mechanical parameters of the composite materials with a large number of random multiscale grains 25-27 .In previous papers 25-27 , we proposed an expression for predicting the equivalent mechanical parameters of a composite material with multiscale grains.This method cannot only show the macrocharacteristics and random configurations of a composite material, but also show the contribution of the small-scale grains.In addition, this method can greatly decrease the computing time for the required numerical result.
In these previous papers 25-27 , we gave the multiscale formula to compute the composite material and proved that the expected displacement field is convergent to the equivalent displacement field based on the multiscale methods in the mathematical theory.However, we did not discuss the convergence, the error result, and the symmetric, positive definite property of the equivalent mechanical parameter tensor of the composite material with random grains by the SMSA-FE method.Therefore, in this paper, the convergence and the error result based on statistical multiscale analysis SMSA shall be presented, and the symmetric, positive definite property of the equivalent parameters tensor the random parameters subjected to the uniform distribution shall be proved.
The next section reviews a representation of a composite material with multiscale random grains, some results, and the SMSA-FE procedure 27 .Section 3 is devoted to proving the convergence of the equivalent mechanical parameters computed by the SMSA-FE algorithm.Section 4 obtains the error results of the SMSA-FE algorithm on iterated multiscale analysis.In Section 5, the symmetric, positive definite property of the equivalent mechanics parameters tensor computed by the SMSA-FE algorithm is proved.In Section 6, the numerical results are presented to demonstrate the validity, the convergence, and the proved results of the SMSA-FE algorithm.Finally, we discuss the future work for the inhomogeneous composites with multiscale random grains.

Iterated Multiscale Analysis Model and Algorithm
In the previous papers 27 , the author had given the algorithm to compute the equivalent mechanical parameters in detail.In order to prove the finite element error and the convergence of the iterated multiscale computed model, we shall review the model and the algorithm in the brief.

Iterated Multiscale Analysis Model
For the brief, all of the grains are assumed as the ellipsoids.Set a domain Ω to represent a composite with multiscale random grains shown in Figure 1 a .Set Ω l to be a set of cube cells of the size ε l shown in Figure 1 b .Based on 27 , the iterated multiscale analysis model can be represented as follows.
1 Obtain the statistical data of the composites and specify the distributions P of the ellipsoid's parameters.
2 Set N to denote the number of the lth scale ellipsoids in the cell ε l Q s ; we can describe the lth scale cube cell ε l Q s as follows: where x 1 0 , x 2 0 , x 3 0 is the center point, a is the long axis, b is the middle axis, c is the short axis, and θ a x 1 x 2 , θ a x 1 , θ b x 1 x 2 and θ b x 1 , are the directions for the axis a and b of the ellipsoids, respectively.One sample ω s l is shown in Figure 1 c . 3 Set the domain Ω l to be logically composed of ε l -sized samples: Ω l ω s ,t∈Z ε l Q s t shown in Figure 1 b .It can be defined as ω l {ω s l , x ∈ εQ s l ⊂ Ω l }.By the SMSA-FE algorithm 27 , the equivalent mechanical parameters can be predicted.Thus, the equivalent material with lth scale random grains in Ω l can be formed.
4 Set m to be the scale number in the composites Ω with multiscale random grains.
For l m − 1, . . ., 1, using the above representation from step 2 to step 3 , recursively and successively, the multiscale random model of Ω can be described.
For example, the asphalt concrete 24 can be considered as the composites with multiscale random grains, respectively.Set ε 2 0.01 m; and ε 1 0.1 m, their configuration can be shown in Figure 1.

SMSA Algorithm Based on Finite Element Method
In the previous section, we introduced the equivalent composites Ω l , l m, m − 1, . . ., 1 with the lth scale grains.In this section, we shall review the mathematical theory that predicts the equivalent mechanical parameters of these composites with random grains by the statistical multiscale analysis SMSA 27 .
For the domain Ω l , l m − 1, m − 2, . . ., 0 shown in Figure 1 b , their corresponding elasticity equation system and the essential boundary condition can be shown as follows: where i, j, h, k 1, 2, . . ., n, ω ω s for x ∈ Ω 1 , ω s ∈ P , P is the probability space, Ω l s∈p,t∈Z n ε l Q s t shown in Figure 1, u ε h x, ω are the displacement field, f i x are the loads, and u x is the boundary displacement vector.
In the paper 25-27 , we had given the SMSA method 25 and established the finite element method 27 to compute the equivalent mechanical parameters.If the FE space V h 0 Q s n can be established and ξ x/ε l , the equivalent mechanical parameters can be computed.

Theorem 2.1. If a composite material with random grains is subjected to the probability distribution P , the equivalent mechanical parameters of a composite material can be approximated as
where ω s ∈ P s 1, 2, . . ., M and a h 0 ijhk ω s is computed as 4 Therefore, the equivalent mechanical parameters can be determined by the following SMSA-FE algorithm.

SMSA-FE Algorithm
1 Specify the scale number m of random grains in the composites and set the iterative number r m.
2 Model r-scale random grains in Q s r and generate meshes of the sample domain according to the algorithm in 27 .
3 If r m, a ijhk x/ε r , ω s in Q s r can be indicated as follows: where Ω denotes the domain of the basic configuration, ε r Q s denotes the domain of the matrix, ε r Q s denotes the domain of the random grains in ε r Q s r , and ε a Q s denotes the interface domain between the grains and the matrix.Go to step 5 .4 If r < m, a ijhk x/ε r , ω s in Q s r can be indicated as follows: where Ω denotes the domain of the basic configuration, ε r Q s denotes the domain of the equivalent matrix, and ε r Q s denotes the domain of random grains.Go to step 5 .
5 Compute the FE approximation N h 0 αm ξ r , ω s according to 2.6 , obtain the FE approximation of sample a h 0 ijhk ε r , ω s on the rth scale according to 2.4 , and compute the FE approximation of the equivalent mechanical parameter tensor { a h 0 ijhk ε r } on the rth scale using 2.3 .
6 Set a ijhk ε r equal to a h 0 ijhk ε r on the rth scale and r r − 1.If r > 1, go to step 2 .Otherwise, the equivalent mechanical parameter tensor a ijhk ε r ε is the equivalent mechanical parameter tensor of the composite material with multiscale random grains in Ω.

Convergence of SMSA-FE Algorithm
Lemma 3.1.If N h 0 αm ξ, ω and a h 0 ijhk ω are the finite element approximations of the random variables N αm ξ, ω and a ijhk ω , respectively, then there exists a constant Proof.If |a ijhk ξ, ω s | < M for any sample ω s ∈ P , 2.6 has one unique finite element solution where C and M 1 are constants and independent of ξ and ω s .
From 2.4 and 3.1 , one can obtain where M 2 CM 1 M 2 1 and M 1 is independent of both the random variables ω s and the local coordinate ξ.Therefore, for the random variable If ω is a random variable and a h 0 ijhk ω are defined as above, then one unique expected value of the equivalent mechanical parameters tensor E ω a h 0 ijhk ω exists in the probability space Proof.From the definition of the long axis a, the middle axis b, the short axis c, the directions of the long axis and the middle axis, θ a x 1 x 2 , θ a x 1 , and θ b x 1 x 2 , θ b x 1 , and the coordinates of the central points of the ellipsoids x 1 0 , x 2 0 , x 3 0 , their probability density functions are denoted by x , and f x 3 0 x , respectively.The united random variable ω a

3.3
Therefore, there exists one unique expected value of the equivalent mechanical parameters tensor E ω a h 0 ijhk ω i, j, h, k 1, 2, . . ., n in the probability space P .
Lemma 3.3.If a h 0 ijhk ω s i, j, h, k 1, 2, . . ., n. s 1, 2, . . . .have the expected value E ω a h 0 ijhk ω in the probability space and ω is the random variable, one obtain Proof.Because { a h 0 ijhk ω s , s 1} are the independent and identical distribution random variables and Set a 1 E ω a h 0 ijhk ω ; from Kolmogorov's classical strong law of large numbers, one obtains Proof.Set r m; define It is easy to see that a ijhk x/ε r , ω s i, j 1, 2, 3 are bounded and measurable random variables.Based on Lemmas 3.1, 3.2, and 3.3, there exist the expected values E ω a r,h 0 ijhk ω of the equivalent mechanical parameter tensor a r,h 0 ijhk ω of the material with only the rth random grains.Therefore, the equivalent mechanical parameter tensor of the composites with the rth random grains can exist as follows: Mathematical Problems in Engineering 9 Set r r − 1 and a r,h 0 ijhk x/ε r 1 , ω s E ω a r,h 0 ijhk ω r , the mechanical parameters of the equivalent matrix material and the grain material in the equivalent composite material with the rth random grains can be obtained.
That is, It is easy to see that a ijhk x/ε r , ω s i, j 1, 2, 3 are bounded and measurable random variables.By the iterated loop proof for r as above, based on Lemmas 3.2 and 3.3, the convergence of the equivalent mechanical parameter tensor E ω a r,h 0 ijhk , r m, m − 1, . . ., 1 can be obtained.

Error Analysis for Equivalent Mechanical Parameter Tensor Computed by SMSA-FE Algorithm
Based on the SMSA-FE algorithm, if the equivalent mechanical parameter tensor is computed, three kinds of errors are considered: the homogenization error, the random error based on Monte Carlo simulation method, and the finite element computation error.For the homogenization error, the composite materials with multiscale random grains are the special cases of the random coefficient problems whose convergence was proved in 14 .For the random error, we have obtained the convergence of the equivalent mechanical parameter tensor of the composite material with multiscale random grains as above.Therefore, in the following section, we will devote to analyzing the finite element error based on SMSA.Firstly, we give the finite element error estimation of the statistical two-scale analysis.Then we give the error estimation of the SMSA-FE algorithm.Lemma 4.1.If N αm ξ, ω s , α 1, 2, . . .n, is the variational solution of 2.6 and N h 0 αm ξ, ω s is the corresponding finite element solution of 2.6 and N h 0 αm ξ, ω s ∈ H 2 Q n , then we have where the constant C > 0 is independent of the size h 0 of mesh.
Proof.Since E s αm N αm ξ, ω s − N h 0 αm ξ, ω s ∈ H 1 0 Q , based on Korn inequality and the interpolation theorem, we have that Mathematical Problems in Engineering

4.5
In fact, based on the idea and the method in 28 , it is easy to prove Lemma 4.2.
Lemma 4.3.Let a ijhk be the equivalent mechanical parameter tensor matrix based on STSA and a h 0 ijhk its finite element approximation as above, if there exists one constant N such that for all ω s ∈ P, N αpm ξ, ω s H 2 Q ≤ N. Then one has Proof.From the above algorithm, the following equation is held.
where R ijhk is defined by where Since R ∞ denote the maximum norm of matrix R ijhk n×n , applying Lemma 4.2 to the above 4.8 , we deduce Since there exists one constant N such that the function matrices N α ξ, ω s ≤ N, the above inequality 4.9 yields Then inequality 4.6 follows from the above inequality 4.10 .
Based on the SMSA-FE algorithm, the equivalent mechanical parameter tensor of the equivalent material with the mth random grains can be obtained.Therefore, the matrix material with m − 1 th random grains can be considered as the equivalent matrix material.Using the loop proof by Lemmas 4.2 and 4.3, it is easy to obtain the following theorem on the equivalent mechanical parameter tensor of a composite material with m-scale random grains.
Theorem 4.4.Let a r ijhk r m, m − 1, . . ., 1 be the equivalent mechanical parameters tensor of the composite material with r th scale random grains and a r,h 0 ijhk r m, m − 1, . . ., 1 its finite element approximation; set the size of the last mesh of the finite element in Q s to be h 0 ; based on the SMSA-FE algorithm, if there exists one constant N such that for all, ω s ∈ P, N αpm ξ r , ω s where C is one constant that is independent of h 0 but dependent on the sizes of the other finite element mesh h 0r r m, m − 1, . . ., 2 in the cell Q r r m, m − 1, . . ., 2 with rth random grains.
From Theorem 4.4, it is easy to see the error of the equivalent mechanical parameter tensor of the composite material with the biggest grains being the main error by the SMSA-FE algorithm.Hence, we only need to consider the error in the composite material with the biggest random grains.

Symmetry and Positive Definite Property for Equivalent Mechanical Parameter Tensor
From 7, 26 , if the parameters of the ellipsoids are subjected to the uniform probability P , the equivalent mechanical parameter tensor a ijhk shall satisfy the following conditions: where for any symmetry matrix η η ij n×n , μ 1 ≥ 0, and μ 2 ≥ 0. Therefore, if the parameters of the ellipsoids are subjected to the uniform probability P , it is important to keep the symmetric, positive definite property of mechanical parameter tensor computed by the finite element method.So we shall give the following theorem to illustrate it.Lemma 5.1.Let { a ijhk } be the equivalent mechanical parameter tensor based on STSA [26] and a h 0 ijhk its finite element approximation; if there exists one constant N such that for all ω s ∈ P, N αpm ξ, ω s H 2 Q ≤ N, the matrix a h 0 ijhk satisfies the following property: for any symmetric matrix η η ih n×n , where K 1 , K 2 are positive constants.
Proof.Taking into consideration the fact that a ijhk ω s a jikh 1 s a hjik ω s , based on the concept of N h 0 α ξ, ω s , taking into account the idea in 7 , we deduce that

5.4
So we have proved a h 0 jikh ω s a h 0 hjik ω s .Let us establish the first equality in 5.2 , which is equivalent to prove A h 0 hk * A h 0 kh , where A * denotes the transpose of the matrix A and A h 0 hk is the matrix a h 0 ijhk n×n .
From the integral identity for solution of problem 2.6 , for any matrix A pk ξ, ω s M ξ ∂ξ p dξ.

5.5
Based on the relations A pq ξ * A qp ξ and AB * B * A * for matrices A, B, it is easy to obtain N α from 5.5 that If we set M N h 0 h , the following equations are obtained.
Taking into account the idea in 7 , 5.7 , and the concept of N h 0 * h ξ, ω s , the following equations can be obtained.

5.8
In the second equation of the above equations, the property of δ ij is used; in the third equation, 5.7 is applied.It follows that the equivalent mechanical parameter matrices can be written in the following form by the relationship of A jq A qj :

Mathematical Problems in Engineering
Thus Hence, we see that A h 0 hk ω s A h 0 kh ω s * .Thus we obtain the following equations: hjik ω s .

5.11
Making use of the relations 5.11 , one has hjik ω s .

5.12
That is 5.2 is proved.
In the sequel, we shall prove 5.3 .From the inequality 4.6 and Theorem 4.4, there exists one h 0 ≥ 0 that is small enough such that

5.13
We have

5.14
Setting K 1 μ 1 /2 and K 2 μ 2 μ 1 /2 yields the inequality 5.3 .By the iterated multiscale analysis and Lemma 5.1, the finite element approximation of the equivalent mechanical parameter tensor of the composite material with multiscale random grains satisfies the following property.Theorem 5.2.Let { a r ijhk } be the equivalent mechanics parameter tensor of the composite material with r-scale random grains based on the SMSA algorithm and a r,h 0 ijhk their finite element approximation by 2.3 .Set the size of the last mesh of the finite element in Q s to be h 0 ; if there exists the constant N α 1 such that for all, ω s ∈ P, N r αpm ξ, ω s H 2 Q ≤ N αpm , then a r,h 0 ijhk satisfy the conditions a r,h 0 ijhk a r,h 0 jikh a r,h 0 hjik ,

5.15
where η η ih n×n is the symmetry matrix and K 1 , K 2 are the positive constants that are independent of h 0 but dependent on the sizes of the other finite element meshes h 0r , r m, m − 1, . . ., 2 in the rth scale cells Q r r m, m − 1, . . ., 2, 1 .

Numerical Experiment
To test the validity of the error result, the convergence, and the symmetric, positive definite property of the mechanical parameter tensor computed by the SMSA-FE algorithm, two numerical examples are given as follows.
The first example models a composite material.The grains are divided into two classes according to the sizes of their long axis shown in Table 1.We use one statistical window ε 1 0.1 to predict the mechanical parameters of the equivalent matrix with small random grains.In each window, small grains occupy approximate 30% of the volume.Their long axis a, short axis b, and angle θ are subjected to the uniform distributions shown in Table 1.Using the different finite element sizes h 0 , based on the data of Table 2, we obtain the equivalent mechanical parameters tensor { a ijhk } that are given in Table 3.
From Table 3, it is easy to see that convergence of the equivalent mechanical parameter tensor computed by the SMSA-FE algorithm exists.From Table 3, the symmetric, positive definite property of the equivalent mechanical parameter tensor and the convergence of the finite element errors with the different mesh sizes h 0 are proved.
The second example is a concrete named as C30 with three-scale random rock grains whose sizes are from 0.3 mm to 19 mm.Its matrix is made up of the cement and the sand.Their sizes of the three-scale rock grains in the concrete are shown in Table 4. Their elastic parameters are shown in Table 5.If all grains are generated in a large statistical window of   500 mm, the number of grains is approximately 6360.In each window, small grains, middle grains, and the large grains occupy approximately 57.7% of the volume and are subjected to the uniform probability distribution in the range of their sizes.Therefore, we set three kinds of sizes of the statistical windows: ε 3 5 mm, ε 2 50 mm, and ε 1 100 mm to compute the equivalent mechanical parameter tensor with the different scale random grains.Numerical results for the mechanical parameter tensor of a composite with only small, middle, and large rock grains are listed in Tables 6, 7, and 8 by the SMSA-FE algorithm.Tables 6-8 also show that the equivalent parameter tensor computed by the SMSA-FE algorithm possess the symmetrical, positive, and definite properties.The expected values of Young's modulus and Poisson's ratio for the different number of samples with three-scale random grains are shown in Table 9 and in Figure 2. Table 9 and Figure 2 show that the equivalent Young's modulus and Poisson's ratio are convergent.
A comparison of the numerical results for Young's modulus and Poisson's ratio by the SMSA-FE procedure, by the mixed volume method, and by the experiment method in the lab is also shown in Table 10.Table 10 also shows that Young's modulus and Poisson's ratios produced by SMSA-FE procedure are very close to that by the experiment method.It proves the SMSA-FE algorithm to be feasible and valid for predicting the effective modulus of the composites with random grains.

Conclusion
In this paper, we proved that the equivalent mechanical parameter tensor for the composite materials with multiscale random grains is convergent and obtained the error result by the finite element analysis.At the same time, we also prove that the equivalent parameter tensor matrix should satisfy the symmetric, positive, and definite property.

Figure 1 :
Figure 1: a Ω with multiscale grains, b equivalent matrix with random grains, c ε 2 cell.

3 . 7 Theorem 3 . 4 .
If a r,h 0 ijhk ω r m, m − 1, . . ., 1 are computed as the equivalent mechanical parameter tensor of the composite material with r-scale random grains by the SMSA-FE algorithm, then the expected values of the equivalent mechanical parameters tensor E ω a r,h 0 ijhk ω exist in the probability space P

Table 1 :
Probability distributions of grains in composite material.

Table 2 :
Mechanical parameters of the matrix and grains.

Table 3 :
Equivalent mechanical parameters { a ij ω s } for different mesh sizes h 0 .

Table 4 :
Size of rock random grains and the number in a statistical window.

Table 5 :
Elasticity mechanical parameters of the matrix, grains, and joint interface materials.

Table 6 :
Equivalent mechanical parameters of a composites with small rock random grains by the SMSA-FE algorithm kPa .

Table 7 :
Equivalent mechanical parameter tensor for concrete with middle rock random grains by the SMSA-FE algorithm kPa .

Table 8 :
Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm kPa .

Table 9 :
Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm GPa .