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

Authors: Youyun Li; Yuqing Pan; Jianlong Zheng; Chiqing Zhou; Desheng Wang
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

Based on the iterated statistically multiscale analysis (SMSA), we present the convergence of the equivalent mechanical parameters (effective moduli), obtain the error result, and prove the symmetric, positive and definite property of the equivalent mechanical parameters tensor computed by the finite element method. The numerical results show the proved results and illustrate that the SMSA-FE algorithm is a rational method for predicting the equivalent mechanical parameters of the composite material with multiscale random grains. In conclusion, we discuss the future work for the inhomogeneous composite material with multiscale random grains.


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## Body

## 1. 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 Figure1. 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.(a)Ω with multiscale grains, (b) equivalent matrix with random grains, (c) ε2 cell.
(a)(b)(c)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.


## 2. 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.


### 2.1. Iterated Multiscale Analysis Model

For the brief, all of the grains are assumed as the ellipsoids. Set a domainΩ to represent a composite with multiscale random grains shown in Figure 1(a). Set Ωl to be a set of cube cells of the size εl 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 distributionsP of the ellipsoid’s parameters.(2)
SetN to denote the number of the lth scale ellipsoids in the cell εlQs; we can describe the lth scale cube cell εlQs as follows:
(2.1)ωls=(a1s,b1s,c1s,θax1x21s,θax11s,θbx1x21s,θbx11s,x101s,x201s,x301s…,θbx1Ns,x10Ns,x20Ns,x30Ns),
where (x10,x20,x30) is the center point, a is the long axis, b is the middle axis, c is the short axis, and θax1x2, θax1, θbx1x2 and θbx1,are the directions for the axis a and b of the ellipsoids, respectively. One sample ωls is shown in Figure 1(c).(3)
Set the domainΩl to be logically composed of εl-sized samples: Ωl=⋃(ωs,t∈Z)εl(Qs+t) shown in Figure 1(b). It can be defined as ωl={ωls,x∈εQls⊂Ωl}. By the SMSA-FE algorithm [27], the equivalent mechanical parameters can be predicted. Thus, the equivalent material with lth scale random grains in Ωl can be formed.(4)
Setm 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 ε2=0.01m; and ε1=0.1m, their configuration can be shown in Figure 1.


### 2.2. SMSA Algorithm Based on Finite Element Method

In the previous section, we introduced the equivalent compositesΩl,(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 domainΩl,(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:
(2.2)∂∂xj[aijhkε(xε,ω)12(∂uhε(x,ω)∂xk+∂ukε(x,ω)∂xh)]=fi(x),x∈Ωl,uε(x,ω)=u¯(x),x∈∂Ω,
where i,j,h,k=1,2,…,n, ω=ωs for x∈Ω1, ωs∈P, P is the probability space, Ωl=⋃s∈p,t∈Znεl(Qs+t) shown in Figure 1, uhε(x,ω) are the displacement field, fi(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 (Vh0(Qs))n can be established and ξ=x/εl, the equivalent mechanical parameters can be computed.Theorem 2.1.
If a composite material with random grains is subjected to the probability distributionP, the equivalent mechanical parameters of a composite material can be approximated as
(2.3)âijhkh0=∑s=1Mâijhkh0(ωs)M,
where ωs∈P(s=1,2,…,M) and âijhkh0(ωs) is computed as
(2.4)âijhkh0(ωs)=1|Qs|∫Qs(aijhk(ξ,ωs)+aiphqεpq(Nkjh0(ξ,ωs)))dξ,(2.5)Nαh0(ξ,ωs)=(Nα1h0,Nα2h0,…,Nαnh0)=(Nα11h0(ξ,ωs)⋯Nα1nh0(ξ,ωs)⋮⋮Nαn1h0(ξ,ωs)⋯Nαnnh0(ξ,ωs).).Nαh0(ξ,ωs) are the FE solutions of (2.6) on unit cell Qs.
(2.6)∂∂ξj(aijhk(ξ,ωs)εhk(Nαm(ξ,ωs)))=-∂aαilm(ξ,ωs)∂ξl,ξ∈Qs,Nα(ξ,ωs)=0,ξ∈∂Qs.Therefore, the equivalent mechanical parameters can be determined by the following SMSA-FE algorithm.SMSA-FE Algorithm  (1)
Specify the scale numberm of random grains in the composites and set the iterative number r=m.(2)
Modelr-scale random grains in Qrs and generate meshes of the sample domain according to the algorithm in [27].(3)
Ifr=m, aijhk(x/εr,ωs) in Qrs can be indicated as follows:
(2.7)aijhk(xεr,ωs)={aijhk1,x∈εrQ̂s,aijhk2,x∈εrQ̃s,⋯aijhkl,x∈εrQ̂s,
where Q̂s⋃Q̃s⋃⋯⋃Q̂s=Qrs and Q̂s⋂Q̃s⋯⋂Q̂s=ϕ, εrQrs⊂Ωdenotes the domain of the basic configuration, εrQ̂s denotes the domain of the matrix, εrQ̃s denotes the domain of the random grains in εrQrs, and εa̅Q̂s denotes the interface domain between the grains and the matrix. Go to step (5).(4)
Ifr<m, aijhk(x/εr,ωs) in Qrs can be indicated as follows:
(2.8)aijhk(xεr,ωs)={âijhkh0(xε(r+1),ωs),x∈εrQ̂s,aijhk1,x∈εrQ̃s,
where Q̂s⋃Q̃s=Qrs and Q̂s⋂Q̃s=ϕ, εrQrs⊂Ω denotes the domain of the basic configuration, εrQ̂s denotes the domain of the equivalent matrix, and εrQ̃s denotes the domain of random grains. Go to step (5).(5)
Compute the FE approximationNαmh0(ξr,ωs) according to (2.6), obtain the FE approximation of sample âijhkh0(εr,ωs) on the rth scale according to (2.4), and compute the FE approximation of the equivalent mechanical parameter tensor {âijhkh0(εr)} on the rth scale using (2.3).(6)
Setâijhk(εr) equal to âijhkh0(εr) on the rth scale and r=r-1. If r>1, go to step (2). Otherwise, the equivalent mechanical parameter tensor âijhk(εr)(ε) is the equivalent mechanical parameter tensor of the composite material with multiscale random grains in Ω.


## 2.1. Iterated Multiscale Analysis Model

For the brief, all of the grains are assumed as the ellipsoids. Set a domainΩ to represent a composite with multiscale random grains shown in Figure 1(a). Set Ωl to be a set of cube cells of the size εl 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 distributionsP of the ellipsoid’s parameters.(2)
SetN to denote the number of the lth scale ellipsoids in the cell εlQs; we can describe the lth scale cube cell εlQs as follows:
(2.1)ωls=(a1s,b1s,c1s,θax1x21s,θax11s,θbx1x21s,θbx11s,x101s,x201s,x301s…,θbx1Ns,x10Ns,x20Ns,x30Ns),
where (x10,x20,x30) is the center point, a is the long axis, b is the middle axis, c is the short axis, and θax1x2, θax1, θbx1x2 and θbx1,are the directions for the axis a and b of the ellipsoids, respectively. One sample ωls is shown in Figure 1(c).(3)
Set the domainΩl to be logically composed of εl-sized samples: Ωl=⋃(ωs,t∈Z)εl(Qs+t) shown in Figure 1(b). It can be defined as ωl={ωls,x∈εQls⊂Ωl}. By the SMSA-FE algorithm [27], the equivalent mechanical parameters can be predicted. Thus, the equivalent material with lth scale random grains in Ωl can be formed.(4)
Setm 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 ε2=0.01m; and ε1=0.1m, their configuration can be shown in Figure 1.


## 2.2. SMSA Algorithm Based on Finite Element Method

In the previous section, we introduced the equivalent compositesΩl,(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 domainΩl,(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:
(2.2)∂∂xj[aijhkε(xε,ω)12(∂uhε(x,ω)∂xk+∂ukε(x,ω)∂xh)]=fi(x),x∈Ωl,uε(x,ω)=u¯(x),x∈∂Ω,
where i,j,h,k=1,2,…,n, ω=ωs for x∈Ω1, ωs∈P, P is the probability space, Ωl=⋃s∈p,t∈Znεl(Qs+t) shown in Figure 1, uhε(x,ω) are the displacement field, fi(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 (Vh0(Qs))n can be established and ξ=x/εl, the equivalent mechanical parameters can be computed.Theorem 2.1.
If a composite material with random grains is subjected to the probability distributionP, the equivalent mechanical parameters of a composite material can be approximated as
(2.3)âijhkh0=∑s=1Mâijhkh0(ωs)M,
where ωs∈P(s=1,2,…,M) and âijhkh0(ωs) is computed as
(2.4)âijhkh0(ωs)=1|Qs|∫Qs(aijhk(ξ,ωs)+aiphqεpq(Nkjh0(ξ,ωs)))dξ,(2.5)Nαh0(ξ,ωs)=(Nα1h0,Nα2h0,…,Nαnh0)=(Nα11h0(ξ,ωs)⋯Nα1nh0(ξ,ωs)⋮⋮Nαn1h0(ξ,ωs)⋯Nαnnh0(ξ,ωs).).Nαh0(ξ,ωs) are the FE solutions of (2.6) on unit cell Qs.
(2.6)∂∂ξj(aijhk(ξ,ωs)εhk(Nαm(ξ,ωs)))=-∂aαilm(ξ,ωs)∂ξl,ξ∈Qs,Nα(ξ,ωs)=0,ξ∈∂Qs.Therefore, the equivalent mechanical parameters can be determined by the following SMSA-FE algorithm.SMSA-FE Algorithm  (1)
Specify the scale numberm of random grains in the composites and set the iterative number r=m.(2)
Modelr-scale random grains in Qrs and generate meshes of the sample domain according to the algorithm in [27].(3)
Ifr=m, aijhk(x/εr,ωs) in Qrs can be indicated as follows:
(2.7)aijhk(xεr,ωs)={aijhk1,x∈εrQ̂s,aijhk2,x∈εrQ̃s,⋯aijhkl,x∈εrQ̂s,
where Q̂s⋃Q̃s⋃⋯⋃Q̂s=Qrs and Q̂s⋂Q̃s⋯⋂Q̂s=ϕ, εrQrs⊂Ωdenotes the domain of the basic configuration, εrQ̂s denotes the domain of the matrix, εrQ̃s denotes the domain of the random grains in εrQrs, and εa̅Q̂s denotes the interface domain between the grains and the matrix. Go to step (5).(4)
Ifr<m, aijhk(x/εr,ωs) in Qrs can be indicated as follows:
(2.8)aijhk(xεr,ωs)={âijhkh0(xε(r+1),ωs),x∈εrQ̂s,aijhk1,x∈εrQ̃s,
where Q̂s⋃Q̃s=Qrs and Q̂s⋂Q̃s=ϕ, εrQrs⊂Ω denotes the domain of the basic configuration, εrQ̂s denotes the domain of the equivalent matrix, and εrQ̃s denotes the domain of random grains. Go to step (5).(5)
Compute the FE approximationNαmh0(ξr,ωs) according to (2.6), obtain the FE approximation of sample âijhkh0(εr,ωs) on the rth scale according to (2.4), and compute the FE approximation of the equivalent mechanical parameter tensor {âijhkh0(εr)} on the rth scale using (2.3).(6)
Setâijhk(εr) equal to âijhkh0(εr) on the rth scale and r=r-1. If r>1, go to step (2). Otherwise, the equivalent mechanical parameter tensor âijhk(εr)(ε) is the equivalent mechanical parameter tensor of the composite material with multiscale random grains in Ω.


## 3. Convergence of SMSA-FE Algorithm

Lemma 3.1.
IfNαmh0(ξ,ω) and âijhkh0(ω) are the finite element approximations of the random variables Nαm(ξ,ω) and âijhk(ω), respectively, then there exists a constant M2>0 such that |âijhkh0(ω)|<M2.Proof.
If|aijhk(ξ,ωs)|<M for any sample ωs∈P, (2.6) has one unique finite element solution Nαmh0(ξ,ωs)∈(H1(Qs))n(n=2,3) such that
(3.1)‖Nαmh0(ξ,ωs)‖(H1(Qs))n<C|aijhk(ξ,ωs)|L∞(Qs)<CM1,
where C and M1 are constants and independent of ξ and ωs.
From (2.4) and (3.1), one can obtain
(3.2)|âijhkh0(ωs)|=|1|mes(Qs)|∫(Qs)(aijhk(ξ,ωs)+aiphq(ξ,ωs)12(∂Nkpjh0(ξ,ωs)∂ξq+∂Nkqjh0(ξ,ωs)∂ξp))dξ|<1|mes(Qs)|(‖aijhk(ξ,ωs)‖L∞(Qs)+‖aiphq(ξ,ωs)‖L∞(Qs)‖Nkj(ξ,ωs)‖(H1(Qs))n)≤1|mes(Qs)||M1+M1*CM1||mes(Qs)|=CM1+M12=M2,
where M2=CM1+M12 and M1 is independent of both the random variables ωs and the local coordinate ξ. Therefore, for the random variable ω in Section 2, |âijhkh0(ω)|<M2.Lemma 3.2.
Ifω is a random variable and âijhkh0(ω) are defined as above, then one unique expected value of the equivalent mechanical parameters tensor Eωâijhkh0(ω) exists in the probability space P=(Pa)N×(Pb)N×(Pb)N×(Pθax1x2)N×⋯×(Px0)N×(Py0)N×(Pz0)N.Proof.
From the definition of the long axisa, the middle axis b, the short axis c, the directions of the long axis and the middle axis, θax1x2, θax1, and θbx1x2, θbx1, and the coordinates of the central points of the ellipsoids (x10,x20,x30), their probability density functions are denoted by fa(x), fb(x), fc(x), fθax1x2(x), fθax1(x), fθbx1x2(x), fθbx1(x), fx10(x), fx20(x), and fx30(x), respectively. The united random variable ω=(a1,b1,c1,θax1x21,θax11,θbx1x21,θbx11,x101,x201,x301…,θbx1N,x10N,x20N,x30N) has the the united probability density function fa1,b1,c1,…,x0N,y0N,z0N(ω)=faN(ω)·fbN(ω)·fcN(ω)⋯fx0N(ω)·fy0N(ω)·fz0N(ω). From Lemma 3.1, one can obtain
(3.3)Eωâijhkh0(ω)=∫Pâijhkh0(ω)dP=∫-∞+∞⋯∫-∞+∞âijhkh0(ω)fa1,b1,c1,…,x0N,y0N,z0N(ω)dω<M2∫-∞+∞⋯∫-∞+∞fa1,b1,c1,…,x0N,y0N,z0N(ω)dω<M2∫-∞+∞⋯∫-∞+∞faN(ω)dω⋯∫-∞+∞⋯∫-∞+∞fz0N(ω)dω<M2.
Therefore, there exists one unique expected value of the equivalent mechanical parameters tensor Eωâijhkh0(ω)(i,j,h,k=1,2,…,n) in the probability space P.Lemma 3.3.
Ifâijhkh0(ωs)(i,j,h,k=1,2,…,n.s=1,2,….)have the expected value Eωâijhkh0(ω) in the probability space and ω is the random variable, one obtain
(3.4)∑s=1Mâijhkh0(ωs)M→a⋅eEωâijhkh0(ωs)(M⟶∞).Proof.
Because{âijhkh0(ωs),s⩾1} are the independent and identical distribution random variables and
(3.5)SM=∑s=1Mâijhkh0(ωs)(M=1,2,…),
from Lemma 3.2, |Eωâijhkh0(ωs)|<∞. Set a1=Eωâijhkh0(ω); from Kolmogorov’s classical strong law of large numbers, one obtains
(3.6)SMM-a1→a⋅e0(M⟶∞).
Therefore, we have SM/M→a·eEωâijhkh0(ωs)(M→∞), that is,
(3.7)∑s=1Mâijhkh0(ωs)M→a⋅eEωâijhkh0(ω)(M⟶∞).Theorem 3.4.
Ifâijhkr,h0(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 Eωâijhkr,h0(ω) exist in the probability space P=(Par)N×(Pbr)N×(Pcr)N×⋯×(Px0r)N×(Py0r)N×(Pz0r)N.Proof.
Setr=m; define
(3.8)aijhk(xεr,ωs)={aijhk2,x∈εQ̂s,aijhk1,x∈εQ̃s,
because both aijhk1 and aijhk2 are constants satisfying |max{aijhk1,aijhk2}|<M1. It is easy to see that aijhk(x/εr,ωs)(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 Eωâijhkr,h0(ω) of the equivalent mechanical parameter tensor âijhkr,h0(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:
(3.9)∑s=1Mâijhkr,h0(ωs)M→a⋅eEωâijhkr,h0(ωs)(M⟶∞).
Set r=r-1 and âijhkr,h0(x/ε(r+1),ωs)=Eωâijhkr,h0(ωr), 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,(3.10)aijhk(xεr,ωs)={âijhkr,h0(xε(r+1),ωs),x∈εQ̂s,aijhk1,x∈εQ̃s,|aijhk(x/εr,ωs)|<M1. It is easy to see that aijhk(x/εr,ωs)(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 Eωâijhkr,h0,(r=m,m-1,…,1) can be obtained.


## 4. 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.
IfNαm(ξ,ωs),α=1,2,…n, is the variational solution of (2.6) and Nαmh0(ξ,ωs) is the corresponding finite element solution of (2.6) and Nαmh0(ξ,ωs)∈(H2(Q))n, then we have
(4.1)∥Nαm(ξ,ω⁮s)-Nαmh0(ξ,ωs)∥(H01(Q))n≤Ch0∥Nαm(ξ,ωs)∥(H2(Q))n,∥Nαm(ξ,ωs)-Nαmh0(ξ,ωs)∥(L2(Q))n≤Ch02∥Nαm(ξ,ωs)∥(H2(Q))n,
where the constant C>0 is independent of the size h0 of mesh.Proof.
SinceEαms=Nαm(ξ,ωs)-Nαmh0(ξ,ωs)∈H01(Q), based on Korn inequality and the interpolation theorem, we have that
(4.2)∥Eαms∥(H1(Q))n2≤Ca(Eαms,Eαms)=Ca(Nαm(ξ,ωs)-Nαmh0(ξ,ωs),Nαm(ξ,ωs)-γh0Nαmh0(ξ,ωs))≤Ch0∥Nαm(ξ,ωs)∥(H2(Q))n∥Eαms∥(H1(Q))n.
That is,
(4.3)∥Nαm(ξ,ωs)-Nαmh0(ξ,ωs)∥(H1(Q))n≤Ch0∥Nαm(ξ,ωs)∥(H2(Q))n.
Aubin-Nitsche lemma [28] yields
(4.4)∥Nαm(ξ,ωs)-Nαmh0(ξ,ωs)∥(L2(Q))n≤Ch02∥Nαm(ξ,ωs)∥(H2(Q))n.Lemma 4.2.
LetNαpm(ξ,ωs),α1=1,2,…,n, be the variational solution of the (2.6) and Nαpmh0(ξ,ωs) the finite element solution such that Nαpm(ξ,ωs)∈W2,∞(Q), and then one has
(4.5)‖∇ξ(Nαpm(ξ,ωs)-Nαpmh0(ξ,ωs))‖≤Ch0|ln⁡h0|∥∇ξ2Nαpm∥L∞(Q),‖(Nαpm(ξ,ωs)-Nαpmh0(ξ,ωs))‖≤Ch02|ln⁡h0|∥∇ξ2Nαpm∥L∞(Q).In fact, based on the idea and the method in [28], it is easy to prove Lemma 4.2.Lemma 4.3.
Letâijhk be the equivalent mechanical parameter tensor matrix based on STSA and âijhkh0 its finite element approximation as above, if there exists one constant N̂ such that for allωs∈P,∥Nαpm(ξ,ωs)∥H2(Q)≤N̂. Then one has
(4.6)‖âijhk-âijhkh0‖L∞≤Ch0|ln⁡h0|N̂.Proof.
From the above algorithm, the following equation is held.(4.7)âijhk=âijhkh0+Rijhk,
where Rijhk is defined by
(4.8)Rijhk=∑s=1M∫Qaiphq(ξ,ωs)(1/2)(H/∂ξp+A/∂ξq)M,
where ℋ denotes ∂(Nkqj(ξ,ωs)-Nkqjh0(ξ,ωs)) and𝒜 denotes ∂(Nkpj(ξ,ωs)-Nkpjh0(ξ,ωs))
Since∥R∥∞ denote the maximum norm of matrix (Rijhk)n×n, applying Lemma 4.2 to the above (4.8), we deduce
(4.9)‖R‖∞=‖∑s=1M∫Qaiphq(ξ,ωs)(1/2)(H/∂ξp+A/∂ξq)M‖∞≤1M∑s=1M‖∫Qaiphq(ξ,ωs)12(H∂ξp+A∂ξq)‖∞≤C1Mh0|ln⁡h0|∑s=1M‖∇ξ2Nk(ξ,ωs)‖L∞,
Since there exists one constantN̂ such that the function matrices ∥Nα(ξ,ωs)∥≤N̂, the above inequality (4.9) yields
(4.10)∥R∥∞≤Ch0|ln⁡h0∥N̂|.
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 themth 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.
Letâijhkr(r=m,m-1,…,1) be the equivalent mechanical parameters tensor of the composite material with r th scale random grains and âijhkr,h0(r=m,m-1,…,1) its finite element approximation; set the size of the last mesh of the finite element in Qs to be h0; based on the SMSA-FE algorithm, if there exists one constant N̂ such that for all, ωs∈P,∥Nαpm(ξr,ωs)∥H2(Q)≤N̂,(r=m,m-1,…,1), then one has
(4.11)‖âijhkr-âijhkr,h0‖≤Ch0|ln⁡h0|N̂,
where C is one constant that is independent of h0 but dependent on the sizes of the other finite element mesh h0r(r=m,m-1,…,2) in the cell Qr(r=m,m-1,…,2) with rth random grains.From Theorem4.4, 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.


## 5. 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 âijhk shall satisfy the following conditions:
(5.1)âijhk=âjikh=âhjik,μ1ηijηij≤âijhkηijηhk≤μ2ηijηij,
where for any symmetry matrix η=(ηij)n×n, μ1≥0, and μ2≥0.Therefore, if the parameters of the ellipsoids are subjected to the uniform probabilityP, 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{âijhk} be the equivalent mechanical parameter tensor based on STSA [26] and âijhkh0 its finite element approximation; if there exists one constant N̂ such that for all ωs∈P,∥Nαpm(ξ,ωs)∥H2(Q)≤N̂, the matrix âijhkh0 satisfies the following property:
(5.2)âijhkh0=âjikhh0=âhjikh0,(5.3)K̂1ηijηij≤âijhkh0ηijηhk≤K̂2ηijηij
for any symmetric matrix η=(ηih)n×n, where K̂1,K̂2 are positive constants.Proof.
Taking into consideration the fact thatâijhk(ωs)=âjikh(1s)=âhjik(ωs), based on the concept of Nαh0(ξ,ωs), taking into account the idea in [7], we deduce that
(5.4)âjikhh0(ωs)=1|Qs|∫Qs[ajikh(ξ,ωs)+ajpkq(ξ,ωs)12(∂Nhpih0(ξ,ωs)ξq+∂Nhqih0(ξ,ωs)ξp)]dξ=1|Qs|∫Qs[ahjik(ξ,ωs)+ahqip(ξ,ωs)12(∂Nkpjh0(ξ,ωs)ξq+∂Nkqjh0(ξ,ωs)ξp)]dξ=âkjihh0(ωs).
So we have proved âjikhh0(ωs)=âhjikh0(ωs). Let us establish the first equality in (5.2), which is equivalent to prove (Âh0hk)*=Âh0kh, where A* denotes the transpose of the matrix A and Âh0hk is the matrix (âijhkh0)n×n.
From the integral identity for solution of problem (2.6), for any matrix Mh0(ξ)∈(Vh0(Qs))n×n, we have
(5.5)-∫Qs∂(M(ξ)∂ξpApq(ξ,ωs)∂(Nk*(ξ,ωs))∂ξqdξ=∫QsApk(ξ,ωs)(M(ξ))∂ξpdξ.
Based on the relations(Apq(ξ))*=Aqp(ξ) and (AB)*=B*A* for matrices A,B, it is easy to obtain Nα from (5.5) that
(5.6)-∫Qs∂Nkh0(ξ,ωs)∂ξpAqp(ξ,ωs)∂M*(ξ)∂ξqdξ=∫QsAkq(ξ,ωs)∂M*(ξ)∂ξqdξ.
If we set M=Nhh0⋆, the following equations are obtained.
(5.7)-∫Qs∂Nkh0(ξ,ωs)∂ξpApq(ξ,ωs)∂Nhh0(ξ,ωs)∂ξqdξ=∫QsAkq(ξ,ωs)∂Nhh0(ξ,ωs)∂ξqdξ.
Taking into account the idea in [7], (5.7), and the concept of Nhh0*(ξ,ωs), the following equations can be obtained.
(5.8)∫Qs∂∂ξq(Nkh0*+ξkE)Aqp(ξ,ωs)∂∂ξp(Nhh0+ξhE)dξ=∫Qs(∂Nkh0*∂ξqAqp(ξ,ωs)∂Nhh0∂ξp+δqkAqp(ξ,ωs)∂Nhh0∂ξp+∂Nkh0*∂ξqAqp(ξ,ωs)δphE+δqkAqp(ξ,ωs)δphE∂Nkh0*∂ξq)dξ=∫Qs(∂Nkh0*∂ξqAqp(ξ,ωs)∂Nhh0∂ξp+Akp(ξ,ωs)∂Nhh0∂ξp+∂Nkh0*∂ξqAqh(ξ,ωs)+Akh(ξ,ωs))dξ=∫Qs(∂Nkh0*∂ξqAqh(ξ,ωs)+Akh(ξ,ωs)dξ)=(Ah0hk⋆)(ωs).
In the second equation of the above equations, the property of δij 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 Ajq⋆=Aqj:
(5.9)Âh0hk(ωs)=∫Q㘱∂∂ξq(Nhh0*+ξhE)Apq(ξ,ωs)∂∂ξp(Nkh0+ξkE)dξ,Âh0kh(ωs)=∫Qs∂∂ξq(Nkh0*+ξkE)Aqj(ξ,ωs)∂∂ξj(Nhh0+ξhE)dξ.
Thus
(5.10)(Âh0kh)*(ωs)==∫Qs∂∂ξj(Nhh0*+ξhE)Ajq(ξ,ωs)∂∂ξq(Nkh0+ξkE)dξ=(Âh0hk)(ωs).
Hence, we see thatÂh0hk(ωs)=(Âh0kh(ωs))*. Thus we obtain the following equations:
(5.11)âijhkh0(ωs)=âjikhh0(ωs)=âhjikh0(ωs).
Making use of the relations (5.11), one has
(5.12)1M∑s=1Mâijhkh0(ωs)=1M∑s=1Mâjikhh0(ωs)=1M∑s=1Mâhjikh0(ωs).
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 h0≥0 that is small enough such that
(5.13)Ch0|ln⁡h0||N̂α|L∞<μ12.
We have
(5.14)μ12ηijηij≤âijhkh0ηijηhk≤(μ2+μ12)ηijηij.
Setting K̂1=μ1/2 and K̂2=μ2+μ1/2yields the inequality (5.3).
By the iterated multiscale analysis and Lemma5.1, 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{âijhkr} be the equivalent mechanics parameter tensor of the composite material with r-scale  random grains based on the SMSA algorithm and âijhkr,h0 their finite element approximation by (2.3). Set the size of the last mesh of the finite element in Qs to be h0; if there exists the constant N̂α1 such that for all, ωs∈P,∥Nαpmr(ξ,ωs)∥H2(Q)≤N̂αpm, then âijhkr,h0 satisfy the conditions
(5.15)âijhkr,h0=âjikhr,h0=âhjikr,h0,K1¯ηijηij≤âijhkr,h0ηijηhk≤K2¯ηijηij,
where η=(ηih)n×nis the symmetry matrix and K1¯,K2¯ are the positive constants that are independent of h0 but dependent on the sizes of the other finite element meshes h0r,(r=m,m-1,…,2) in the rth scale cells Qr(r=m,m-1,…,2,1).


## 6. 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 Table1. We use one statistical window ε1=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 h0, based on the data of Table 2, we obtain the equivalent mechanical parameters tensor {âijhk} that are given in Table 3.Table 1
Probability distributions of grains in composite material.
Small grainsLarge grainsθ[0,2π]θ[0,2π]a[0.03,0.08]a[0.1,1]b[0.02,a]b[0.1,a]Table 2
Mechanical parameters of the matrix and grains.
MatrixGrains8.415×1057.041×10502.99×1061.50×10607.041×1058.415×10501.50×1062.99×1060006.870×104007.50×105Table 3
Equivalent mechanical parameters{âij(ωs)} for different mesh sizes h0.
h0.a11a12a21a22a330.041.083×1068.323×1058.323×1051.092×1061.327×1050.021.095×1068.302×1058.302×1051.099×1061.295×1050.011.097×1068.377×1058.373×1051.108×1061.319×1050.0051.098×1068.375×1058.371×1051.112×1061.311×105From Table3, 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 h0 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 Table4. 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: ε3=5mm, ε2=50mm, and ε1=100mm to compute the equivalent mechanical parameter tensor with the different scale random grains.Table 4
Size of rock random grains and the number in a statistical window.
ClassGrain size rangeAverage sizeNumber of rock grainsSizes of statistical windowsLarge grains10–19 mm14.5 mm8ε1=100mmMiddle grains1–10 mm5 mm20ε2=50mmSmall Grains0.3–1 mm0.6 mm28ε2=5mmTable 5
Elasticity mechanical parameters of the matrix, grains, and joint interface materials.
ClassYoung’s modulus (GPa)Poisson's ratioRock grains74.50.15Matrix13.50.25Joint interface50.50.20Numerical results for the mechanical parameter tensor of a composite with only small, middle, and large rock grains are listed in Tables6, 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.Table 6
Equivalent mechanical parameters of a composites with small rock random grains by the SMSA-FE algorithm (kPa).
20341770.66054537.66054645.8−119951.6164516.8−12764.76054537.620128556.86060485.8−106579.519649.2−141765.46054645.86060485.820061719.4−18765.7140779.4−116731.90.0000000.0000000.0000006923128.00.0000000.000000164516.819649.25140779.480.0000006902337.6−81061.0−12764.7−141765.40.00000098172.3−81061.06859146.1Table 7
Equivalent mechanical parameter tensor for concrete with middle rock random grains by the SMSA-FE algorithm (kPa).
25464900.26882577.06881924.2−135341.116667111669.7−2238.36882577.025410941.86916568.5−123921.97075.0−187899.586881924.26916568.525291123.2−17145.989634.5−192916.50.0000000.0000000.0000009071093.30.0000000.000000111669.77075.089634.50.0000009042380.8−82547.1−2238.3-187899.50.00000069773.7−82547.19060481.0Table 8
Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm (kPa).
34510679.48073028.88076036.2−75074.2111127.6−4769.68073025.034121732.28092588.8−36814.05862.0−110837.98076036.88092589.334065284.2−531.7123664.7−79363.60.0000000.0000000.00000012860237.80.0000000.000000111127.65862.0123664.70.00000012862877.8−30079.5−4769.6−110837.90.00000078302.2−30079.512783465.6Table 9
Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm (GPa).
Iterative number51015202530Esmall17.498539216.916038416.915131717.000444817.024519616.9342101Vsmall0.2313750.2346530.2342710.2339510.2339420.233926Emiddle22.591713622.209825622.229875222.130273622.105740822.0321386Vmiddle0.2146310.2154080.2153420.2154430.2155520.215537Elarge30.644806.430.368940830.592396830.681593630.693693430.6862218Vlarge0.1929510.1934190.1930650.1931160.1930660.193082Young's modulus and Poisson's ratio for 5, 10, 15,…, 30 samples with different scale random grains by the SMSA-FE procedure.
(a)(b)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 Table10. 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.Table 10
Equivalent mechanical parameters for a concrete by the different methods (GPa).
ESMSA-FE 57.7%=30.686221VSMSA-FE 57.7%=0.193082Etest 60%=30.700000Vtest 60%=0.198000Eaverage 60%=50.600000Vaverage 60%=0.19000


## 7. 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.Various test examples were solved by the SMSA-FE procedure. The numerical results show that the SMSA-FE procedure is feasible and valid and that these data satisfy the properties of the equivalent mechanical parameter tensor proved in the pervious sections.The procedure can also be extended to other composite materials with random short fibers, random foams, random cavities, and so forth. Although we have given the specific steps, some theory results and numerical examples to carry out the SMSA-FE method and to illustrate the validity, the influence of shape, size, component, orientation, spatial distribution, and volume fraction of inclusions on inhomogeneous macromechanical properties, analyze the calculated results, and capture the information of microbehaviors to the macromechanical properties are also our important future work for the composite materials with random grains.



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*Source: 585624-2011-06-14.xml*