A numericalanalytical coupling computational method for homogenization of effective thermal conductivity of periodic composites
 Quy Dong To^{1}Email author and
 Guy Bonnet^{1}
https://doi.org/10.1186/2196116615
© To and Bonnet; licensee Springer. 2014
Received: 6 October 2013
Accepted: 6 December 2013
Published: 29 April 2014
Abstract
Background
In the framework of periodic homogenization, the conduction problem can be formulated as an integral equation whose solution can be represented by a Neumann series. From the theory, many efficient numerical computation methods and analytical estimations have been proposed to compute the effective conductivity of composites.
Methods
We combine a Fast Fourier Transform (FFT) numerical method based on the Neumann series and analytical estimation based on the integral equation to solve the problem. Specifically, the analytical approximation is used to estimate the remainder of the series.
Results
From some numerical examples, the coupling method have shown to improve significantly the original FFT iteration scheme and results are also superior to the analytical estimation.
Conclusions
We have proposed a new efficient computation method to determine the effective conductivity of composites. This method combines the advantages of the FFT numerical methods and the analytical estimation based on integral equation.
Keywords
Background
Composite materials can exist in nature or be fabricated by purpose. Due to their technological importance, micromechanical approaches are developed to determine the overall behavior of composites from the properties of their constituents. The general procedure comprises two steps: the construction of a representative model, containing information on heterogeneities (morphologies, inclusion shape, volume fraction, local physical properties, etc.), and the analysis of the model by some mathematical methods. Analytical methods are often based on a simplification of inclusion shapes and potential theory, spherical harmonic functions, etc. Many exact and approximate closedform solutions have been derived by such methods for materials having a linear behavior [1–7]. However, if the microstructure is known in all its complexity, numerical methods must be used. Among the numerical methods, finite element method (FEM) and boundary element method (BEM) are widely used for homogenization problems. These methods have been reported in numerous works [8–12]. A more recent method, introduced in the 1990s and described thereafter, uses extensively the Fourier transform and the introduction of a ‘reference material’.
From a theoretical point of view, the introduction of a reference material allows formulation of the localization problem in the form of a LippmanSchwingerDysontype integral equation, whose solution can be represented by a Neumann series. The most convenient way to solve this integral equation is its formulation using Fourier transform of the equations governing the localization problem [13–19]. Some notable variants and improvements of this method can be found in the literature [18, 20–23]. However, it is known that fast Fourier transform (FFT) iterative schemes are very sensitive to the contrast ratio between the phases and may not converge for infinite contrasts. Therefore, an important step when using FFT iterative schemes is to estimate the remainder of the Neumann series whose sum is computed up to a finite number of terms. This paper is devoted to this fundamental question.
In this paper, the remainder of the Neumann series is estimated by a combination between FFT schemes and the NematNasserIwakumaHejazi (NIH) [24, 25] estimation of the effective properties. For twophase systems with spherical inclusions, the NIH estimation in thermal problems leads to closedform solutions which agree with numerical results for a large range of volume fractions. However, the NIH estimation departs from the sum of the Neumann series at high concentrations of inclusions. Since both NIH approximation and FFT schemes are based on integral equations, we use the former to estimate analytically the remainder of the Neumann series and derive the improved effective properties.
The present paper contains four parts. After a brief introduction of the paper’s context, the ‘Methods’ Section is dedicated to the computational methods. The problem statement, FFT methods, and the FFTNIH coupling are also presented in this section. Implementations of the coupling are discussed in the ‘Results and discussion’ Section. Finally, concluding remarks are given in the ‘Conclusions’ Section.
Methods
Problem statement and integral equation formulation
Method of resolution
Although the FFTbased methods produce e(x) at convergence, the main concern is the convergence rate at high contrast ratio. The basic iterative scheme described in (12) and (14) is called the primal iterative scheme (PIS). In the literature, there have been numerous works to improve the convergence of the basic method such as dual iterative scheme (DIS) [20, 28], polarizationbased iterative scheme (PBIS) [21, 22], accelerated scheme (AS) [18], augmented Lagrangian scheme (ALS) [16], etc.
Instead of finding the full field solution of (1), the mean value of e^{∗} and the effective thermal conductivity K^{eff} can be estimated from (7) with NIH approximation [24, 25]. Such an approximation has been shown to predict very well the overall elastic and thermal properties of twophase composites for a large range of volume fractions of inclusions [25, 29]. However, it fails at higher concentrations. Generally, the estimation of K^{eff} requires only the computation of a lattice sum which admits closedform expressions in many cases, as seen thereafter.
Residual integral equation and estimation of the remainder of the Neumann series
The main scope of this paper is to combine the advantages of the analytical approximation and FFT numerical methods to improve the prediction of the effective properties. The material under consideration is a twophase matrixinclusion composite with conductivity of both phases being K^{ M } (matrix) and K^{ I } (inclusion). The volume fraction of the inclusions is f, and the distribution of the inclusions in the unit cell is taken to be general at this stage.
The method presented in this paper can be used in coupling with any FFTbased iterative scheme. An algorithm presenting the implementation with the basic scheme (PIS) is presented in Algorithm 1 and used later in this work. In the following, this scheme will be stopped before convergence, in view to evaluate the performance of the estimation of the remainder of the series.
Algorithm 1 Algorithm of the iterative scheme PIS coupled with NIH approximation
It is clear that, from (26), the effective conductivity is obtained in the same form as in the previous work [25].
Coupled method in special cases
The coupled algorithm is significantly accelerated if the shape functions I(ξ) or P(ξ) are determined from closedform expressions, for example, in the case of ellipsoidal inclusions. Firstly, it is no longer necessary to compute numerically the Fourier transform of the characteristic function [28]. Secondly, the lattice sum $\sum _{\mathit{\xi}\ne \mathbf{0}}P\left(\mathit{\xi}\right){\hat{\mathbf{\Gamma}}}^{M}\left(\mathit{\xi}\right)$ can also be estimated by a closedform expression.
The parameter ξ_{ c } defines the number of initial terms of the sum that we keep in the approximation formula. The final analytical expression is given in the following:

Simple cubic system$\begin{array}{l}P\left(\mathit{\xi}\right)=P\left(\xi \right)=\frac{9f{(\eta cos\eta sin\eta )}^{2}}{{\eta}^{6}},\phantom{\rule{2em}{0ex}}\\ \sum _{\mathit{\xi}\ne \mathbf{0}}P\left(\xi \right)\simeq \sum _{0<\left\mathit{\xi}\right<{\xi}_{c}}P\left(\xi \right)+\frac{3cos2{\eta}_{c}}{\pi {\eta}_{c}}+\frac{2\stackrel{2}{sin}{\eta}_{c}}{\pi {\eta}_{c}^{3}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{46.0pt}{0ex}}\frac{2sin2{\eta}_{c}}{\pi {\eta}_{c}^{2}}\frac{2}{\pi}\text{Si}\left(2{\eta}_{c}\right)+1,\phantom{\rule{1em}{0ex}}\eta =\mathrm{\xi R},\phantom{\rule{1em}{0ex}}{\eta}_{c}={\xi}_{c}R,\phantom{\rule{2em}{0ex}}\end{array}$(33)

with Si(η) being the sine integral$\text{Si}\left(\eta \right)={\int}_{0}^{\eta}\frac{sin{\eta}^{\prime}}{{\eta}^{\prime}}d{\eta}^{\prime}.$(34)

Bodycentered cubic system$\begin{array}{l}P\left(\mathit{\xi}\right)=\frac{9f}{4}\frac{{[\eta cos\eta sin\eta ]}^{2}}{{\eta}^{6}}{[1+cos\pi ({n}_{1}+{n}_{2}+{n}_{3}\left)\right]}^{2},\phantom{\rule{2em}{0ex}}\\ \sum _{\mathit{\xi}\ne \mathbf{0}}P\left(\xi \right)\simeq \sum _{0<\left\mathit{\xi}\right<{\xi}_{c}}P\left(\xi \right)+\frac{3cos2{\eta}_{c}}{\pi {\eta}_{c}}+\frac{2\stackrel{2}{sin}{\eta}_{c}}{\pi {\eta}_{c}^{3}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{46.0pt}{0ex}}\frac{2sin2{\eta}_{c}}{\pi {\eta}_{c}^{2}}\frac{2}{\pi}\text{Si}\left(2{\eta}_{c}\right)+1.\phantom{\rule{2em}{0ex}}\end{array}$(35)

Facecentered cubic system$\begin{array}{l}P\left(\mathit{\xi}\right)=\frac{9f}{16}\frac{{[\eta cos\eta sin\eta ]}^{2}}{{\eta}^{6}}[cos\pi {n}_{1}+cos\pi {n}_{2}+cos\pi {n}_{3}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{116.0pt}{0ex}}{+cos\pi ({n}_{1}+{n}_{2}+{n}_{3})]}^{2},\phantom{\rule{2em}{0ex}}\\ \sum _{\mathit{\xi}\ne \mathbf{0}}P\left(\xi \right)\simeq \sum _{0<\left\mathit{\xi}\right<{\xi}_{c}}P\left(\xi \right)+\frac{1}{4}\left[\frac{3cos2{\eta}_{c}}{\pi {\eta}_{c}}+\frac{2\stackrel{2}{sin}{\eta}_{c}}{\pi {\eta}_{c}^{3}}\right.\phantom{\rule{2em}{0ex}}\\ \left.\phantom{\rule{120.0pt}{0ex}}\frac{2sin2{\eta}_{c}}{\pi {\eta}_{c}^{2}}\frac{2}{\pi}\text{Si}\left(2{\eta}_{c}\right)+1\right].\phantom{\rule{2em}{0ex}}\end{array}$(36)
Results and discussion
In this section, we study the results coming from the implementation of the coupled method for the case of a simple cubic system. The representative cell is a cube with the spherical inclusion located at its center (the first figure from the left in Figure 1). The periodic problem with prescribed temperature gradient E is solved by three approaches: the NIH approximation (31), the conventional PIS, and the coupled method. The last two methods are based on the same iterative scheme, and in the coupled method, the reevaluation of the effective conductivity after each iteration is done using (30). All results are compared with the results coming from the conventional PIS method at convergence. The analytical expression of $\sum _{\mathit{\xi}\ne \mathbf{0}}P\left(\xi \right)$ described in (33) is used to accelerate the computation and to improve the accuracy. Regarding the iterative scheme, the number of harmonic terms retained in the Fourier series is limited to 128*128*128, i.e., n_{ i } < 128, i = 1, 2, 3, and the precision of the computation ε = 0.001 is adopted. Different contrast ratios k_{ I } / k_{ M } ranging from 0.1 to 50 are considered in this work, and the results of the three approaches are discussed and compared.
Numerical examples at different contrast ra tios k_{ I }/k_{ M } also demonstrate a considerable improvement of the coupling in comparison with the basic FFT method. More particularly, for small k_{ I }/k_{ M } = 0.1, it generates a very good approximation of k_{eff} even at N = 1, 2, where the error of the FFT method is of order 20%. At high k_{ I }/k_{ M } = 50, the coupling performs less well but still reduces the relative error, the effect being important for lower number of iterations.
Conclusions
A coupled method is developed for computing the effective conductivity of periodic composites. The method uses a FFT iterative scheme to solve the localization problem and the NIH approximation to estimate analytically the remainder at any iteration N of the Neumann series. As a result, a new expression for the effective conductivity is derived on the basis of the current flux and temperature gradient field. Numerical tests on various cases have shown that the expression coming from the coupled method improves considerably the results issued from the uncoupled methods for small numbers of iterations. The contribution of the coupling improves the results at any contrast ratio and any volume fraction.
The application domain of the coupled method is large. Although the numerical examples given in this work concern the PIS scheme and spherical inclusions, the method can be applied to any existing FFTbased methods and arbitrary inclusion shapes to improve the accuracy of the predicted properties. The method can be extended to deal with other physical problems such as elasticity, piezoelectricity, etc.
Declarations
Authors’ Affiliations
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.