Image-based numerical prediction for effective thermal conductivity of heterogeneous materials: A quadtree based scaled boundary finite element method

International Journal of Heat and Mass Transfer 128 (2019) 335–343

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Image-based numerical prediction for effective thermal conductivity of heterogeneous materials: A quadtree based scaled boundary finite element method Yiqian He, Jie Guo, Haitian Yang ⇑ State Key Lab of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Article history: Received 23 April 2018 Received in revised form 26 July 2018 Accepted 22 August 2018

Keywords: Effective thermal conductivity Heterogeneous materials Quadtree mesh generation Scaled boundary finite element method Inverse heat transfer problem Identification

a b s t r a c t Integrating advantages of the quadtree technique, the SBFEM, the image-based modelling approach, and inverse analysis, a new numerical technique is presented for the evaluation of Effective Thermal Conductivity (ETC) of heterogeneous materials. The quadtree technique provides a convenient way for mesh generation, and facilitates to image-based analysis. The inconvenience of hanging nodes caused in the quadtree mesh generation can be naturally avoided by SBFEM, consequently the temperature solutions of heterogeneous materials can be determined by the combination of quadtree technique and SBFEM, and are partially regarded as ‘experiments values’ for equivalent homogeneous materials. Utilizing a group of such ‘experiments’, the ETC can be obtained by solving a group of inverse heat transfer problems of parameters identification. Numerical examples are provided to demonstrate the effectiveness of the proposed approach, and the impacts of distributions and shapes of inclusions, and volume fractions are taken into account. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Effective thermal conductivity (ETC) is an important parameter for heterogeneous materials [1–4], the evaluation of ETC has been a topic of considerable theoretical and practical interests. There are basically two categories of methods evaluating ETC. The first one is the analytical method, Progelhof et al. [5] provided a detailed review for analytical models, and Carson and co-workers [6,7] developed a unifying equation for five fundamental effective thermal conductivity structural models. The analytical method has good physical basis, however, there does not appear to be any single model equation that is applicable to all types of structures [6]. Another category is the numerical simulations by the computational homogenization approaches mainly based on the high fidelity finite element analysis [8–10]. Due to the complexity of heterogeneous materials, the mesh generation is challenged in numerical simulation, so the structured mesh is suggested [11,12] to decrease the computational cost of the free mesh technique. The quadtree decomposition [13] is a hierarchical-type data structure, in which each parent is recursively divided into four children. As the element solutions for cells of the same pattern ⇑ Corresponding author. E-mail address: [email protected] (H. Yang). https://doi.org/10.1016/j.ijheatmasstransfer.2018.08.099 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

but different sizes are proportional, so this approach is high efficient, because the limited cell patterns generated can be precomputed and quickly extracted when required [14]. The quadtree decomposition is proved to be an effective mean for mesh generation in providing a simple, fast, and efficient way for data storage and retrieval [15]. In addition, the quadtree decomposition provides an effective approach for image-based analysis, which is attractive to perform virtual numerical experiments to evaluate the macro properties of heterogeneous materials [16]. As the quadtree mesh allows a fine representation of the geometry near the interface and a coarser one in the less critical area, it avoids the difficulty in computational cost for the voxel-based approach in which the mesh is built from the conversion of each voxel into a finite element [16]. However, due to the level-mismatches between adjacent elements in quadtree mesh, the ‘hanging nodes’ will produce, and result in a nonconformity across the interfaces and become an obstacle to impede the further application of quadtree mesh in FE analysis, although some remedial techniques have been developed [15,17]. Recently, Saputra et al. [14] presented a quadtree Scaled Boundary Finite Element Method (SBFEM) in stress analysis, where the ‘hanging nodes’ was effectively handled by utilizing the flexibility of polygonal elements in SBFEM [18,19]. It is of interest to integrate advantages of quadtree technique and SBFEM for the direct imagebased numerical simulation for the heterogeneous heat problems.

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Because quadtree SBFEM has been demonstrated to be robust, automatic and simple, the results acquired via the simulation with various kinds of distributions and shapes of inclusions can be used as values of ‘artificial measurements’ for equivalent homogeneous materials, and utilized to predict ETC. In this paper, a new numerical technique by combining the quadtree SBFEM and inverse analysis is presented to evaluate the ETC of heterogeneous materials. The technique mainly consists of two parts (1) Reading images of heterogeneous material, generating quadtree meshes, and acquiring SBFEM based solutions with various kinds of distributions and shapes of inclusions. (2) The results acquired via the quadtree SBFEM simulation are partially used as of ‘artificial measurement values’ for equivalent homogeneous materials. Utilizing a group of such ‘artificial experiments’ with different distributions and shapes of inclusions, the ETC can be numerically estimated by solving a group of inverse heat transfer problems of parameters identification. The paper is structured as follows. Section 2 outlines the process of quadtree mesh generation from an image; Section 3 gives a description of building a SBFE element for steady-state heat conduction problems; Section 4 presents an approach to estimate ETC by solving a group of inverse heat conduction problems of parameter identification; and Section 5 verifies the proposed approach via numerical examples. Finally, conclusions are summarized in Section 6.

(3) During the quadtree decomposition, a balanced decomposition (2:1 rule) is used for producing only 16 possible cell patterns. Because coefficient matrix and other matrices of the cells with the same pattern but of different sizes are either identical or proportional [14], it is sufficient to treat only one cell of the same pattern and replicate the results to other cells when needed. As shown in Fig. 1(b), the ‘hanging nodes’ are likely produced in the match of adjacent elements in the quadtree mesh generation. An easy treatment of these hanging nodes by SBFEM is regarding them as nodes of a new polygon element, instead of regular ones. 3. SBFEM based finite polygon element for heat transfer problems Each cell generated in the quadtree decomposition is regarded as a SBFE element, as shown in Fig. 2. For heat transfer problems, the construction of coefficient matrix of a SBFE element is briefly described via several key equations, instead of detailed derivations which have been given by Wolf and Song [20]. Consider a two-dimensional heat transfer problem defined on a RUC (Representative Unit Cell) at steady-state in absence with heat generation. The governing equation is given by

r2 T ¼ 0 in RUC

ð1Þ

with boundary conditions

T ¼ T


on C1

ð2Þ

2. The quadtree mesh generation


 @T ¼ qn k @n

According to the previous work of Saputra et al. [14], this section outlines the key process of quadtree mesh generation based on the color intensity of the pixels (2D) of an image. The key process of quadtree decomposition mainly includes

C1 þ C2 ¼ C stands for the boundary of RUC, T
and qn are the pre-

(1) Each pixel of an image is to be represented as a square domain, as shown in Fig. 1(a), and all the colors of the pixels are stored in an image color matrix. (2) If the difference between the maximum and minimum color intensity in a cell is larger than the color threshold, the cell is recursively divided further into four equal-sized cells, until all the cells satisfy the criterion of homogeneity or reaches the minimum edge length.

on C2

ð3Þ

where T represents the temperature, k is the thermal conductivity;

scribed temperature and heat flux, respectively. In a SBFE element, the solution of temperature is described under a scaled boundary coordinate system [20], as shown in Fig. 2 where the normalized radial coordinate n runs from the scaling center toward the boundary, and the circumferential coordinate s specifies a distance around the boundary from an origin on the boundary. The relationship between x-y and scaled boundary coordinate system is defined by

x ¼ x0 þ nxs ðsÞ

Fig. 1. The quadtree mesh generated from an image with an inclusion (a) Image (b) Quadtree.

ð4Þ

Y. He et al. / International Journal of Heat and Mass Transfer 128 (2019) 335–343

337

4. Estimation of ETC by solving inverse heat conduction problems Using the quadtree based SBFEM, the distribution of temperature of a RUC of heterogeneous material can be determined via Eq. (14), and the results of direct thermal numerical simulation will be partially regarded as a kind of measurement values of an equivalent homogenous material. The evaluation of ETC can be realized by solving an inverse heat transfer problem defined by Seek for unknown ETC represented by / with

Tðn; sÞ ¼ NðsÞTh ðnÞ

ð6Þ

where NðsÞ is the shape function on the boundary n ¼ 1. The unknown vector Th ðnÞ is a set of functions analytical in n. The temperature gradient is given by @T @x @T @y

)

1 ¼ B1 ðsÞTðnÞ;n þ B2 ðsÞTðnÞ n

ð7Þ

where B1 ðsÞ ¼ b1 ðsÞNðsÞ and B2 ðsÞ ¼ b2 ðsÞNðsÞ;s . Th ðnÞ needs to satisfy [20]

h i n2 E0 Th ðnÞ;nn þ E0 þ ET1  E1 nTh ðnÞ;n  E2 Th ðnÞ ¼ 0

ð8Þ

P ¼ E0 Th;n þ E1 Th

ð9Þ

where P is the vector of nodal thermal loads along the boundary n ¼ 1.

Z

k  B1 ðsÞT B1 ðsÞjJjds

E0 ¼

ð10Þ

S

T

k  B2 ðsÞ B1 ðsÞjJjds Z

T

k  B2 ðsÞ B2 ðsÞjJjds

ð12Þ

S

The Eqs. (8) and (9) can be transformed into an eigenvalue problem defined by

"

T E1 0 E1 1 T E1 E0 E1 

E1 0 E2

E1 E1 0

#

u p



 ¼k

u p

 ð13Þ

By solving the above eigenvalue problem with Eq. (9), the relationship between the nodal temperature and thermal loads for a SBFE element can be obtained

P ¼ QC ¼ Q U1 Th ¼ KTh

ð14Þ

The relationship between Th ðnÞ and u etc., and the definitions of Q , C and U have been given in [20,21] Assembling all cells leads to a FE system equation over the whole domain of RUC

KG TG ¼ PG

ð15Þ

where KG stands for the global coefficient matrix, TG and PG are the vectors of global nodal temperature and thermal loads, respectively.

on C2

ð18Þ

Eqs. (15)–(17) can be formulated in a FE form 0

KT¼P

ð19Þ 0

where K refers to a global coefficient matrix of an equivalent homogeneous material on RUC, T and P are the vectors of global nodal temperature and thermal loads, respectively. Let T refer to a vector of measurement values given by the direct thermal numerical simulation. / can be determined by minimizing a functional defined by

Jð/Þ ¼

1 T ðLT  T Þ ðLT  T Þ 2

ð20Þ

with a constraint of Eq. (19) and L in Eq. (20) refers to a matrix mapping T to T . There are a number of well developed skills to tackle with the above optimization problem [22,23], in this paper, the minimization is carried out by the Gauss-Newton method via an iterative process

(

/ðnþ1Þ ¼ /ðnÞ þ D/ 1

D/ ¼ ðGT GÞ GT ðT  T Þ

ð21Þ

where

ð11Þ

S

E2 ¼

ð17Þ

on C1

G ¼ K0 ð/Þ1

Z E1 ¼

T ¼ T


ð5Þ

The temperature at any point in the domain of a SBFE element is approximated by

(

ð16Þ


 @T / ¼ qn @n

Fig. 2. A SBFE element.

y ¼ y0 þ nys ðsÞ

r2 T ¼ 0 in RUC

@K0 ð/Þ T @/

ð22Þ

When / is determined, an error estimation is conducted via

   n ~ 1X T i  T i  EM ¼   n i¼1  T i 

ð23Þ

and

rM

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   !2 u T~  T   n u 1 X  i t i ¼   EM  n  1 i¼1  T i 

where EM and

ð24Þ

rM stand for a Mean of Error and Variance of Error,

respectively, T i and T~ i refer to the temperature at same ith point in the heterogeneous and equivalent homogeneous materials, respectively. For a RUC, when the volume fraction is given, a group of /i can be gained with different inclusions distributions produced ran
of / is taken as ETC of the heterogeneous domly. The mean value / i material. As illustrated in Fig. 3, the major steps of ETC evaluation includes Step 1: For a prescribed inclusions shape and volume fraction, read N RUC images of with random distribution of inclusions,

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Fig. 3. The process of evaluating ETC.

the image can be acquired by real or virtual experiment. Set i ¼ 1. Step 2: For ith image, generate a quadtree mesh. Step 3: Conduct a Direct Numerical Thermal Simulation (DNTS) using the quadtree based SBFEM on the above mesh. Step 4: Determine /i by utilizing the result of DNTS. Step 5: Set i ¼ i þ 1 if i 6 N goto Step 2 Then verification. Step 6: Evaluate mean value, Mean of Error and Variance of Error
for /. 5. Numerical verification Following assumptions are adopted in the computation (1) Inclusions are randomly distributed. (2) There is no overlap between inclusions. (3) The shape of inclusions is circular or elliptical, with radius R and axial ratio r ¼ a=b, respectively, as illustrated in Fig. 4.

(4) Inclusions are not divided by the boundary of RCU [8]. (5) In a RUC, the shape and size of all inclusions are same. (6) All examples are mainly designed for numerical verification, therefore there is no specific consideration on materials properties. The process of images creation of inclusions in a RUC basically follows the work of Bessa et al. [24], the main steps include (1) Determine the number of inclusions nc ¼ pL2c =pab where Lc and p denote the length and volume fraction of a RUC. (2) Generate nc random positions and rotation angles (3) Generate a basic inclusion i at a random position in RCU, as plotted in the dash line in Fig. 4. (4) Copy the basic inclusion i to other random positions without rotation till all nc inclusions are generated. (5) Rotate all the inclusions with its own random angles till all nc rotations finished. (6) Guarantee all the assumptions are satisfied. All computation task are fulfilled using a PC with i7-7600U 2.8 GHz CPU and 16.0 GB RAM. 5.1. Verification of quadtree based SBFEM This example provides a comparison of solutions given by quadtree SBFEM and ANAYS with four node quadrilateral element. Consider a RUC with circular inclusions as shown in Fig. 5
¼ 1 W=m2 . The where a ¼ b ¼ 1 m, R ¼ 0:0195 m, T
¼ 2  C and q thermal conductivities of the inclusion and matrix are kp ¼ 10 W=m K and km ¼ 100 W=m K, respectively. Three kinds of volume fractions are taken into computation, and some detailed computing messages are listed in Table 1. As shown in Table 2, two solutions are well matched, the maximum average relative errors on some selected points at random is less than 0.0078%. It implies an ability of quadtree SBFEF to provide a reliable temperature solution for heterogeneous materials. 5.2. Estimation of ETC for circular inclusions

Fig. 4. Image creation of inclusions in a RUC.

The comparison of ETC given by the proposed method and reference solutions is provided for circular inclusions in this example. Two kinds of volume fractions are considered as shown in Fig. 6. For each of them, 30 solutions are obtained for 30 images with

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(a) p = 0.27

(b) p = 0.077

(c)

p = 0.043

Fig. 5. Geometry and boundary conditions of RUC with different volume fractions.

Table 1 The numbers of nodes and elements. Quadtree based SBFEM

Number of Nodes Number of elements

ANSYS

p ¼ 0:27

p ¼ 0:077

p ¼ 0:043

p ¼ 0:27

p ¼ 0:077

p ¼ 0:043

139,588 111,769

44,577 35,392

25,341 20,068

18,984 18,741

13,368 13,113

15,299 15,044

random-distributed circular inclusions, and are taken as 30 artificial experiment results for inverse analysis. The computing parameters are given by R ¼ 0:0195 m, kp ¼ 10 W=m K, and km ¼ 100 W=m K. The solutions given by the Maxwell model and the Effective Medium Theory (EMT) are selected as Ref. [6]. Both of them take randomness of inclusion distribution, volume fraction, and conductivities of components into account. Maxwell model assumes inclusions are circular, but there is no specific consideration on the shape of inclusion in EMT [6].

A comparison of ETC is provide in the Table 3 where the prediction provided by the proposed method agrees well with those given by the Maxwell model, and exhibits a 3.2% relative difference with EMT at p ¼ 0:1. Such a difference may be caused by the regardless of the inclusion shape in EMT model. Table 4 provides a comparison of EM and rM given by the proposed method and reference solutions, a good agreement can be observed. In the reference solutions, the Maxwell model provides an analytical / to compute two indicators defined in Eqs. (22) and (23). Because volume fraction, shape and random distribution

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Table 2 A comparison of temperatures with reference solutions. Positions

Values of temperatures by Quadtree based SBFEM with different volume fractions p ¼ 0:27

(0.125, 0.125) (0.125, 0.500) (0.125, 0.875) (0.500, 0.125) (0.500, 0.500) (0.500, 0.875) (0.875, 0.125) (0.875, 0.500) (0.875, 0.875) Average error

p ¼ 0:077

p ¼ 0:043

Quadtree based SBFEM

ANSYS

Error%

Quadtree based SBFEM

ANSYS

Error%

Quadtree based SBFEM

ANSYS

Error%

2.001822 2.007836 2.013816 2.001808 2.007848 2.013941 2.001822 2.007833 2.013807

2.001795 2.007677 2.013527 2.001782 2.007690 2.013646 2.001795 2.007679 2.013528

1.34e-03 7.94e-03 1.43e-02 1.27e-03 7.85e-03 1.46e-02 1.35e-03 7.68e-03 1.38e-02 7.80e-03

2.001418 2.005671 2.009925 2.001417 2.005671 2.009924 2.001417 2.005671 2.009923

2.001410 2.005641 2.009871 2.001410 2.005640 2.009871 2.001410 2.005641 2.009871

3.56e-04 1.48e-03 2.66e-03 3.61e-04 1.52e-03 2.63-03 3.53e-04 1.48e-03 2.57e-03 1.49e-03

2.001369 2.005369 2.009376 2.001365 2.005366 2.009355 2.001369 2.005369 2.009374

2.001365 2.005353 2.009348 2.001360 2.005349 2.009329 2.001365 2.005353 2.009348

2.67e-04 8.04e-04 1.38e-03 2.44e-04 7.95e-04 1.30e-03 2.56e-04 7.93e-04 1.33e-03 7.98e-04

(a) p = 0.1

(b) p = 0.4 Fig. 6. RCU with circular inclusions.

Table 3 A comparison of ETC identified with analytical models (W=m K) p

ETC given by the proposed method

ETC given by Maxwell model

ETC given by EMT model

0.1 0.4

84.4332 50.0348

84.7803 49.9952

87.1470 49.9941

of inclusions are taken into account for Maxwell model and proposed approach, it is reasonable that the results obtained from them are well matched. The slight difference in Table 4 may be caused by size effect fo RUC that was not considered in Maxwell model. The size effect of RUC is an issue concerned in the effective evaluation of heterogeneous material [25], Fig. 7 provides three different sizes of RUC which are taken into account for the impact of RUC
with same volume fraction. EM and rM in Table 5 indisize on / cates the size of RUC has not significant impact on the results of estimation of ETC.

Table 4 A comparison of EM and p

0.1 0.4

Table 5 also provides a comparison of computing time for different RUC sizes. As the increase of RUC size, the scale of quadtree SBFE analysis increases rapidly, and results in more computational expense in the direct simulation). While in the homogeneous inverse analysis, there is no significant increase of problem scale, therefore only slight change can be observed in the numerical efforts with different RUC sizes. The results of EM and rM in Table 6 indicate that ETC evaluated by proposed method is well matched with the Maxwell model; and neither the number nor the locations of measurement points has significant impact on the accuracy of estimation. Three selections of measurement points given in Fig. 8 exhibit small impact on the estimated ETC.

5.3. Estimation of ETC for elliptical inclusions The comparison of ETC given by the proposed method and reference solutions is provided for elliptical inclusions in this example.

rM with Maxwell model. Quadtree based SBFEM

Maxwell model

EM %

rM %

EM %

rM %

1.87 1.93

0.048 0.0589

2.05 1.92

0.036 0.064

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(a) 1*1

(b) 2*2

(c) 4*4 Fig. 7. RUC with different sizes.

Table 5 The identified ETC and CPU time with different RUC sizes. RUC size

ETC (W=m K)

EM %

rM %

Degree of freedom (DOF)

1m1m 2m2m 4m4m

84.8594 84.0203 82.5620

1.69 1.87 1.32

0.022 0.0048 0.023

14,183 55,062 221,431

CPU time Direct simulation (s)

Inverse analysis (s)

Total (s)

1.56 6.75 21.00

1.38 1.66 0.73

2.94 8.41 21.73

Table 6 The impact of number of measurement points on ETC. Number of measurement points

25 50 100

RUC size 1 m  1 m

RUC size 4 m  4 m

ETC (W=m K)

EM %

rM %

ETC (W=m K)

EM %

rM %

84.8594 86.0749 86.1423

1.69 1.21 1.30

0.022 0.016 0.015

82.5620 82.8221 82.8275

1.32 0.83 0.81

0.023 0.068 0.006

(a) 25 measurement points

(b) 50 measurement points Fig. 8. Locations of measurement points.

(c) 100 measurement points

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(a) p = 0.1

(b) p = 0.15

(d) p = 0.25

(c) p = 0.2

(e) p = 0.3

Fig. 9. RCU with elliptical inclusions.

Fig. 10. The comparison of variation of ETC with p for a fixed r.

Table 7 The impact of different volume fraction on EM and p

Axial ratio r ¼ 0:4

Axial ratio r ¼ 0:6

Proposed method

0.10 0.15 0.20 0.25 0.30

rM with different axial ratios. Maxwell model

Proposed method

Maxwell model

EM %

rM %

EM %

rM %

EM %

rM %

EM %

rM %

2.44 3.69 2.80 2.65 3.45

0.039 0.12 0.077 0.078 0.11

2.37 3.14 4.53 5.09 4.67

0.016 0.065 0.045 0.068 0.058

2.52 3.98 3.40 2.89 3.62

0.046 0.23 0.21 0.076 0.13

2.83 3.85 4.16 3.78 4.60

0.031 0.15 0.12 0.047 0.075

Y. He et al. / International Journal of Heat and Mass Transfer 128 (2019) 335–343

Two kinds of axial ratios, i.e. r ¼ 0:4 and r ¼ 0:6 are considered with five volume fractions, as shown in Fig. 9. Computing parameters are given by a ¼ 0:0156 m, kp ¼ 10 W=m K and km ¼ 100 W=m K, the size of RUC is 1 m  1 m. Utilizing a fitting skill given in [8], Fig. 10 gives a comparison of variation of ETC with a fixed r, the blue and purple lines refer to r ¼ 0:4 and r ¼ 0:6, respectively. The variation with r ¼ 0:6 is closer to Maxwell model than r ¼ 0:4, it may be illustrated as r ¼ 0:6 is closer to a circle than r ¼ 0:4. The comparisons of EM and rM in Table 7 indicate that ETC evaluated by proposed method is well matched with the Maxwell model; and neither the volume fraction nor the axial ratio has significant impact on the accuracy of estimation.

5.4. Computing remarks Numerical comparisons indicate (1) In comparison with ANSYS, the results acquired in Example 1 indicates the solutions provided by quadtree SBFEM are reliable, and can be taken as ‘measurements values’ for ETC evaluation. (2) In comparison with Maxwell model and EMT, the results acquired in Examples 2 and 3 indicates that the proposed method is available for the ETC evaluation of heterogeneous materials with circular or elliptical inclusions for different selections of number and location of measurement points, RUC sizes, volume fractions, and axial ratios as well.

6. Conclusion A new numerical technique is presented for the prediction of ETC of heterogeneous materials. The major steps of implementation of the technique include (1) Acquire or generate a group of RUC images of heterogeneous materials with inclusions distributed randomly. (2) Generate a quadtree mesh for each of RUC images. (3) Conduct a Direct Numerical Thermal Simulation (DNTS) using SBFEM on each RUC mesh with a prescribed boundary conditions. (4) Utilizing the result gained by DNTS, predict ETC by solving a number of inverse heat transfer problems on RUC with same prescribed boundary conditions The major merits of the new technique include: (1) It integrates the advantages of the quadtree mesh generation, SBFEM, the image-based modelling approach, and the inverse analysis as well. (2) It provides a way to predict ETC of heterogeneous materials via numerical experiments, the volume fractions, shape and randomness of distribution of inclusions can be conveniently taken into account when geometric images of heterogeneous materials are given, either by practical or virtual experiments, an artificial numerical experiment based prediction of ETC can be implemented. (3) Numerical verifications are provided to demonstrate the effectiveness of the proposed method with different RUC sizes, volume fractions, and distribution of inclusions. (4) The presented work is devoted to the 2-D problem with circular or elliptical inclusions, the achievement in this paper encourages authors’ further investigation on the 3-D problem and more complex inclusions.

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Conflict of interest We have no conflict of interest to declare. Acknowledgements The research leading to this paper is funded by NSF [11572077], NKBRSF [2015CB057804], and Natural Science Funding of Liaoning Province [2015020141]. References [1] Y. Asakuma, I. Honda, T. Yamamoto, Numerical analysis of effective thermal conductivity with thermal conduction and radiation in packed beds, Int. J. Heat Mass Transf. 114 (2017) 402–406. [2] A.F. Abuserwal, E.M.E. Luna, R. Goodall, R. Woolley, The effective thermal conductivity of open cell replicated aluminium metal sponges, Int. J. Heat Mass Transf. 108 (2017) 1439–1448. [3] D. Sova, M. Porojan, B. Bedelean, G. Huminic, Effective thermal conductivity models applied to wood briquettes, Int. J. Therm. Sci. 124 (2018) 1–12. [4] C. Lee, L. Zhuang, D. Lee, S. Lee, I. Lee, H. Choi, Evaluation of effective thermal conductivity of unsaturated granular materials using random network model, Geothermics 67 (2017) 76–85. [5] R.C. Progelhof, J.L. Throne, R.R. Reutsch, Methods for predicting the thermal conductivity of composite systems: a review, Polym. Eng. Sci. 16 (1976) 615– 625. [6] J. Wang, J.K. Carson, M.F. North, D.J. Cleland, A new approach to modelling the effective thermal conductivity of heterogeneous materials, Int. J. Heat Mass Transf. 49 (17) (2006) 3075–3083. [7] J.K. Carson, S.J. Lovatt, D.J. Tanner, A.C. Cleland, Predicting the effective thermal conductivity of unfrozen, porous foods, J. Food Eng. 75 (2006) 297–307. [8] W. Kaddouri, A. El Moumen, T. Kanit, S. Madani, A. Imad, On the effect of inclusion shape on effective thermal conductivity of heterogeneous materials, Mech. Mater. 92 (2016) 28–41. [9] R.P.A. Rocha, M.E. Cruz, Computation of the effective conductivity of unidirectional fibrous composites with an interfacial thermal resistance, Numer. Heat Transf. Part A 39 (2001) 179–203. [10] A. El Moumen, T. Kanit, A. Imad, H. El Minor, Computational thermal conductivity in porous materials using homogenization techniques: numerical and statistical approaches, Comput. Mater. Sci. 97 (2015) 148–158. [11] N. Lippmann, Th. Steinkopff, S. Schmauder, P. Gumbsch, 3D finite element modelling of microstructures with the method of multiphase elements, Comput. Mater. Sci. 9 (1997) 28–35. [12] T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach, Int. J. Solids Struct. 40 (2003) 3647–3679. [13] M. Yerry, M. Shephard, Automatic three-dimensional mesh generation by the modified-octree technique, Int. J. Numer. Methods Eng. 20 (11) (1984) 1965– 1990. [14] A. Saputra, H. Talebi, D. Tran, C. Birk, C. Song, Automatic image-based stress analysis by the scaled boundary finite element method, Int. J. Numer. Meth. Engng 109 (2017) 697–738. [15] A. Tabarraei, N. Sukumar, Adaptive computations on conforming quadtree meshes, Finite Elem. Anal. Des. 41 (2005) 686–702. [16] G. Legrain, P. Cartraud, I. Perreard, N. Moës, An X-FEM and level set computational approach for image-based modelling: application to homogenization, Int. J. Numer. Meth. Eng. 86 (2011) 915–934. [17] M. Yerry, M. Shephard, A modified-quadtree approach to finite element mesh generation, IEEE Comput. Graph. Appl. 3 (1) (1983) 39–46. [18] J.P. Wolf, C. Song, Finite-Element Modelling of Unbounded Media, John Wiley and Sons, Chichester, 1996. [19] E.T. Ooi, C. Song, F. Tin-Loi, Z.J. Yang, Polygon scaled boundary finite elements for crack propagation modelling, Int. J. Numer. Meth. Eng. 91 (3) (2012) 319– 342. [20] C. Song, J.P. Wolf, The scaled boundary finite element method – Alias consistent infinitesimal finite element cell method for diffusion, Int. J. Numer. Meth. Eng. 45 (1999) 1403–1431. [21] A.J. Deeks, J.P. Wolf, A virtual work derivation of the scaled boundary finiteelement method for elastostatics, Comput. Mech. 28 (2002) 489–504. [22] C.H. Huang, C.Y. Huang, An inverse problem in estimating simultaneously the effective thermal conductivity and volumetric heat capacity of biological tissue, Appl. Math. Model. 31 (2007) 1785–1797. [23] S. Znaidia, F. Mzali, L. Sassi, A. Mhimid, A. Jemni, Nasrallah S. Ben, D. Petit, Inverse problem in a porous medium: estimation of effective thermal properties, Inverse Prob. Sci. Eng. 13 (6) (2005) 581–593. [24] M.A. Bessa, R. Bostanabad, Z. Liu, A. Hu, D.W. Apley, C. Brinson, W. Chen, W.K. Liu, A framework for data-driven analysis of materials under uncertainty: countering the curse of dimensionality, Comput. Methods Appl. Mech. Eng. 320 (2017) 633–667. [25] D. Balzani, L. Scheunemann, D. Brands, J. Schröder, Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations, Comput. Mech. 54 (2014) 1269–1284.