Equivalent-inclusion approach for estimating the effective elastic moduli of matrix composites with arbitrary inclusion shapes using artificial neural networks

In this paper, we focus specially on the effective elastic moduli of the heterogeneous composites with arbitrary inclusion shapes. The main idea of this paper is to replace those inhomogeneities by simple equivalent circular (spherical) isotropic inclusions with modified elastic moduli. Available simple approximations for the equivalent circular (spherical) inclusion media then can be used to estimate the effective properties of the original medium. The data driven technique is employed to estimate the properties of equivalent inclusions and the Extended Finite Element Method is introduced to modeling complex inclusion shapes. Robustness of the proposed approach is demonstrated through numerical examples with arbitrary inclusion shapes.

Journal of Science and Technology in Civil Engineering NUCE 2020 14 (1): 15–27 EQUIVALENT-INCLUSION APPROACH FOR ESTIMATING THE EFFECTIVE ELASTIC MODULI OF MATRIX COMPOSITES WITH ARBITRARY INCLUSION SHAPES USING ARTIFICIAL NEURAL NETWORKS Nguyen Thi Hai Nhua , Tran Anh Binha,∗, Ha Manh Hungb a Faculty of Information Technology, National University of Civil Engineering, 55 Giai Phong road, Hai Ba Trung district, Hanoi, Vietnam b Faculty of Building and Industrial Construction, National University of Civil Engineering, 55 Giai Phong road, Hai Ba Trung district, Hanoi, Vietnam Article history: Received 03/12/2019, Revised 07/01/2020, Accepted 07/01/2020 Abstract The most rigorous effective medium approximations for elastic moduli are elaborated for matrix composites made from an isotropic continuous matrix and isotropic inclusions associated with simple shapes such as circles or spheres In this paper, we focus specially on the effective elastic moduli of the heterogeneous composites with arbitrary inclusion shapes The main idea of this paper is to replace those inhomogeneities by simple equivalent circular (spherical) isotropic inclusions with modified elastic moduli Available simple approximations for the equivalent circular (spherical) inclusion media then can be used to estimate the effective properties of the original medium The data driven technique is employed to estimate the properties of equivalent inclusions and the Extended Finite Element Method is introduced to modeling complex inclusion shapes Robustness of the proposed approach is demonstrated through numerical examples with arbitrary inclusion shapes Keywords: data driven approach; equivalent inclusion, effective elastic moduli; heterogeneous media; artificial neural network https://doi.org/10.31814/stce.nuce2020-14(1)-02 c 2020 National University of Civil Engineering Introduction Composite materials often have complex microstructures with arbitrary inclusion shapes and a high-volume fraction of inclusion Predicting their effective properties from a microscopic description represents a considerable industrial interest Analytical results are limited due to the complexity of microstructure Upper and lower bounds on the possible values of the effective properties [1–4] show a large deviation in the case of high contrast matrix-inclusion properties Numerical homogenization techniques [5–8] determining the effective properties give reliable results but challenge engineers by computational costs, especially in the case of complex three-dimensional microstructure Engineers prefer practical formulas due to its simplicity [9–13] but practical ones are built from isotropic inclusions of certain simple shapes such as circular or spherical inclusions In our previous works [14–16] ∗ Corresponding author E-mail address: anh-binh.tran@nuce.edu.vn (Binh, T A.) 15 Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering proposed an equivalent-inclusion approach that permits to substitute elliptic inhomogeneities by circular inclusions with equivalent properties Aiming to reduce the cost of computational homogenization, various methods such as reducedorder models [17], hyper reduction [18], self-consistent clustering analysis [19] have been proposed in the literature Apart from the mentioned methods, surrogate models have been shown their productivity in many studies such as response surface methodology (RSM) [20] or Kriging [21] In recent years, data sciences have grown exponentially in the context of artificial intelligence, machine learning, image recognition among many others Application to mechanical modeling is more recent Initial applications of the machine learning technique for modeling material can be traced back to the 1990s in the work of [22] It has pointed out in [22] that the feed-forward artificial neural network can be used to replace a mechanical constitutive model Various studies have utilized fitting techniques including the artificial neural network (ANN) to build material laws, such as in [23, 24] In this work, we first attempt to build a model to estimate the effective stiffness matrix of materials for some types of inclusion whose analytical formula maybe not available in the literature, with a small volume fraction using ANNs Then, we try to define a model to estimate the elastic properties of equivalent circle inclusion The data in this work is generated by the unit cell method using Extended Finite Element Method (XFEM) which is flexible for the case of complex geometry inclusions The organization of this paper is as follows Section briefly reviews the periodic unit cell problem Section presents the construction of ANN models Numerical examples are presented in Section and the conclusion is in Section Periodic unit cell problem In this section, we briefly summarize the unit cell method to estimate the effective elastic moduli of a homogeneous medium with a Representative Volume Element (RVE) The inside domain and its boundary are denoted sequentially as Ω and ∂Ω The problem defined on the unit cell is as follows: find the displacement field u(x) in Ω (with no dynamics and body forces) such that: ∇ · σ (u(x)) = ∀x in Ω (1) σ=C:ε (2) ε = ∇ · u + ∇ · uT (3) ε = ε¯ (4) where and verifying which means that macroscale field equals to the average strain field of the heterogeneous medium Eq (1) defines the mechanical equilibrium while Eq (2) is the Hooke’s law Two cases of boundary condition can be applied to solve Eq (1) satisfying the equation Eq (4), which are called as kinematic uniform boundary conditions and periodic boundary condition The periodic boundary condition, which can generate a converge result with one unit cell, will be used in this work The boundary conditions can be written as: u(x) = εx ¯ + u˜ (5) where the fluctuation u˜ is periodic on Ω 16 Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering The effective elastic tensor is computed according to Ce f f = C(x) : A(x) (6) where A(x) is the fourth order localization tensor relating micro and macroscopic strains such that: Ai jkl = εkl i j (x) (7) where εkl i j (x) is the strain solution obtained by solving the elastic problem (1) when prescribing a macroscopic strain ε using the boundary conditions with ε¯ = ei ⊗ e j + e j ⊗ ei (8) In 2D problem, to solve this problem, we solve (1) by prescribing strain as in the following: The computation of effective properties and equivalent inclusion coefficients using ANN 0 1/2 (9) ; ε¯ 22 = ; ε¯ 12 = ε¯ 11 = 1/2 0 Artificial Neural Networks have been inspired from human brain structure In such model, each neuron is defined as a simple mathematical function Though some The computation effective andmodern equivalent inclusion coefficients using concepts have appeared of earlier, the properties origin of the neural network traces back to ANN the workArtificial of Warren McCulloch Walter [25]from whohuman have shown that theoretically, Neural Networksand have been Pitts inspired brain structure In such model, each neuron is defined asany a simple mathematical function Though some concepts have appeared ANN can reproduce arithmetic and logical function The idea to determine the earlier, the origin of the modern neural network traces back to the work of Warren McCulloch and Walter equivalent circle inclusions in this work can be seen in Fig Pitts [25] who have shown that theoretically, ANN can reproduce any arithmetic and logical function The idea to determine the equivalent circle inclusions in this work can be seen in Fig lM µM lI µI Network Network eff C ij Generate data from Non-circular inclusions lequ µequ Generate data from circular inclusions Figure equivalent inclusion usingusing ANNANN Fig 1.Computation Computationofof equivalent inclusion Note that, the two networks in Fig are utilized for the same volume fraction of inclusion The Notedetails that, of thethetwo networks ofin the Fig are utilized the same involume fractionThe of first step, construction two1 networks will for be discussed the following inclusion The details the construction of theare twospecified networksFollow will be discussed in the the input fields and of output fields of a network [11], by mapping two formula of an unit cell with a very small volume fraction of inclusion, we first attempt to build an ANN following surrogate based on a square unit cell whose inclusion has a volume fraction (f) of 1% to 5% To The first step, the input fields and output fields of a network are specified Follow [11], simplify problem, in this work, we keep a constant small f which is arbitrary chosen In the two by mapping two formula of an unit cell with a very small volume fraction of inclusion, we first attempt to build an ANN surrogate based17 on a square unit cell whose inclusion has a volume fraction (f ) of 1% to 5% To simplify problem, in this work, we keep a constant small f which is arbitrary chosen In the two cases, an ellipse-inclusion (I2) unit cell or a flower-inclusion unit cell (I3), we attempt to extract two components the Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering cases, an ellipse-inclusion (I2) unit cell or a flower-inclusion unit cell (I3), we attempt to extract two ef f ef f components the effective stiffness matrix including C11 and C33 by the ANN model from the Lamé constants of the matrix λ M , µ M and those of inclusions µI , λI (see ANN2 and ANN4 in Table 1) For the purpose of finding equivalent parameters, with the circle - inclusion unit cell (I1), the outputs of network are Lamé constants of the inclusion while the input are those of the matrix and the expected ef f ef f C11 and C33 of the stiffness matrix (see ANN1 and ANN3 in Table 1) Table Information of ANN model Case Volume fraction f ANN1 ANN2 ANN3 ANN4 I1 I2 I1 I3 Input ef f ef f λ M , µ M , C11 , C33 λ M , µ M , λI , µI ef f ef f λ M , µ M , C11 , C33 λ M , µ M , λI , µI 0.0346 0.0346 0.0409 0.0409 Output Hidden layers MSE λI , µI ef f ef f ef f C11 , C12 C33 15-15 15-15 15-15 10-10 2.2E-3 1.0E-6 3.3E-3 1.0E-6 λI , µI ef f ef f ef f C11 ,C21 ,C33 The second step aims to collect data The calculations are carried out on the unit cell using XFEM The geometry of these inclusions is described thanks to the following level-set function [26], written as 2p 2p x − xc y − yc φ= + (10) rx ry where r x = ry = r0 + a cos(bθ); x = xc + r x cos(bθ); y = yc + ry cos(θ) For inclusion I3 in Fig 2(c)), we fixed r0 = 0.1, p = 6, a = 8, b = For each case, 5000 data sets were generated using quasi random distribution (Halton-set) The data is divided into parts including 70% for training, 15% for validation and 15% for validating Note that, the surrogate model just works for interpolation problem, so the input must be in a range of value In this work, the bound is selected randomly The upper bound of inputs (see Fig 1) are [20.4984 2.0000 50.4937 20.4975] and the lower bound of and ANN3 inANN3 Table in1) Table 1) 0.5011] and ANN3 in0.0001 Table 1) inputs are and [0.5017 0.5027 (a)inclusion I1 inclusion a) I1a)inclusion I1 a) I1 inclusion (b) inclusion b) I2b) inclusion I2I2 b)inclusion I2 inclusion (c)I3 I3 inclusion c) c)inclusion I3c)inclusion I3 inclusion Fig Fig Three types of unit cell Three types of unit cell cell Fig Three types of unit Figure Three types of unit cell The The second step aims to aims collect The calculations are carried on cellunit second step aims to collect data The calculations are carried out the on the cell cell The second step to data collect data The calculations are out carried outunit on unit the The third stepXFEM works on thegeometry architecture of the surrogate Thisthanks step includes the using XFEM The geometry of these is described thanks to the using XFEM The geometry of these inclusions is model described to following the following using The ofinclusions these inclusions is described thanks to determining the following number of layers and neurons, the activation function, the lost function In the following, we employ level-set function [26],[26], written as level-set function written as as level-set function [26], written the Mean square error (MSE) as the lost function For the activation function, tang-sigmoid, which is 2p 2p 2p 2p 2p 2p (10)(10)(10) ưỉyyc -ư yc ưwill be utilized: popular and effective for ỉmany ưỉ xyc -ưỉyproblems, ư-ỉxxc yc x -ỉxxc regression f = ỗf = ỗf =ữ ỗ +ữỗ +ữỗỗ +ữỗ , ữữ , ữ , ỗ ữ ố rxố ứrxố ứrốx rứyố ứrỗốy ứrey x ữứ e x (11) f (x) = x x += arobcos( qx )=;bxqxc)= qcos( ) inclusion where For inI3Fig rx where = rryx = roxy += aroycos( q+)a;bcos( where I3 inI3Fig qy + )=;beyqyc )=+; ryyyc cos( =+ ryycqcos( +) r.yFor q ) inclusion For inclusion in Fig +; rxxxc cos( =+ rxxcbcos( +q r)x;ebcos( 18 we fixed r0.1, p =a 6, a 6, =b 8, b 8, =For each case, 5000 data sets were generated 2c), 2c), we fixed r0 = p0 = 0.1, 6, each case, 5000 data setsdata weresets generated 2c), we fixed p= =8, a= =8 b =For For each case, 5000 were generated = r0.1, using quasi random distribution (Halton-set) The data is divided into parts including using quasi random distribution (Halton-set) The dataThe is divided into parts using quasi random distribution (Halton-set) data is divided intoincluding parts including for training, for validation 15% for validating that, the surrogate 70%70% for70% training, 15%15% for15% validation and and 15%and for15% validating NoteNote that, thethat, surrogate for training, for validation for validating Note the surrogate works here in were trained by the popular Lavenberg-Marquardt algorithm fifth step is to train the network: use the constructed data to fit the diffe Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering meters and weighting functions in the ANN Various factors can affect the trai The input data was then normalized using Max-min-scaler, written as: which can be defined by the trainer In case the expected performance is obtai x − xmin −1 (12) x=2 xmin + xmax raining process is stopped, and the result will be employed In contrast, when The fourth step selectsthe a training algorithm Various algorithm is availableprocess in literature,may however, ormancethedoes not reach expectation, another training be condu most effective one is unknown before the training process is conducted Some are available in Matlab Bayesian Algorithm may combinegradient a change in are theLavenberg-Marquardt, parameters (e.g theRegularization, number ofGenetic echoes, theOne minimum several algorithms to obtain the expected model Evaluating each algorithm or network architecture is ning rateoutinof gradient-based training )by the popular Lavenberg-Marquardt scope of this work All ANN networksalgorithm here in were trained algorithm fifth step is to train the network: use the constructed data to fit the different parameters and r the sixthThe step, which aims to analyze the performance, we use the network N weighting functions in the ANN Various factors can affect the training time which can be defined by the trainer.of In case the expected obtained, training process which is stopped, has and thebeen ch the application network isperformance limited isby the the input range result will be employed In contrast, when the performance does not reach the expectation, another re training training process may be conducted with a change in the parameters (e.g the number of echoes, the minimum gradient, the learning rate in gradient-based training algorithm ) After the sixth step, which aims to analyze the performance, we use the network Note that the umerical results application of network is limited by the input range which has been chosen before training Numerical Computation of results the effective stiffness matrix Ceff using surrogate models 4.1.cell Computation of the effective stiffness matrix C e f f using surrogate models for periodic unit cell odic unit problem problem Figure A multilayer perceptron The details for each ANN models are depicted in Table g 2: A multilayer perceptron The details each ANN models This section shows some information of the for trained networks which will be are used depicted for the prob- in Table lem in Section 4.2 and 4.3 We compare the results generated by trained ANNs and XFEM method Specifically, we used ANN2 and ANN4 for I2 and I3, respectively As discussed in Section 3.4, we fix Table Information of ANN model f and vary the elastic constant The agreement of ANN models and the unit cell method using XFEM is depicted in Fig and Fig 5, which show that the surrogate models are reliable Note that, we don’t Case N1 I1 Volume fraction f 0.0346 N2 I2 0.0346 Input Output 19 eff lM, µM, C11eff , C33 lM, µM, lI, µI lI, µI eff C11eff C12eff C33 Hidden layers 15-15 M 2.2 15-15 Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering attempt to use any type of realistic materials and the problem is plain strain In the relation with the two Lamé constants, the material stiffness matrix is written as:    λ + 2µ 2λ    2µ  C =  2λ (13)   0 µ 18 16 18 14 16 12 14 10 12 10 84 14 10 128 106 84 62 62 40 40 10 10 8 12 10 14 12 16 12 1410 14 18 lM lM 1612 16 1814 16 18 18 0.4 0.4 ef f 11 11 12 10 119 128 106 84 108 97 6 62 10 lM 10 lM 11 13 10 11 12 lM 14 ef f C11 (c) 9λ M −10 11 l M 11 12 13 14 0.4 0.8 0.6 0.6 0.4 0.8 0.6 0.6 µM 16 14 16 12 14 10 13 96 84 62 16 XFEM XFEM 14 12 Neural network results Neural network 13 results 106 0.81.2 0.8 1.2 1.4 µM 1.2 1.4 1.2 1.4 µM ef f 11 eff µ - C eff b) µ M - Cb) M 11 11 XFEM Neural network results Neural 15 network results 1410 128 M 15 XFEM16 14 86 85 1814 XFEM XFEM 12 Neural16 network results Neural network results eff µ - C eff b) b) µ(b)M µ- −CC11 M 11 M M XFEM XFEM16 Neural network results Neural network results µM l eff l a) lM - C11effa) l(a)M λ- −CC11 eff a) l16M - C11effa) lM - C11 M 18 18 XFEM XFEM 16 Neural network results Neural network results 14 XFEM18 XFEM Neural network results 16 12 Neural network results 12 13 14 40 0.35 14 16 12 14 10 12 10 0.42 0.35 0.45 12 13 14 0 0.35 0.4 XFEM XFEM Neural network results Neural network results XFEM XFEM Neural network results Neural network results 0.40.5 µM 0.45 0.55µM 0.50.6 0.55 ef f 0.35(d) µ 0.4 0.45 0.5 0.55 M − C 11 µM 0.45 0.5 0.55 0.6 0.6 µM eff µ - C µ - C11 c) lM - C11effc) l - C C11eff λM decreases from 16 to while µM decrease d) µ M - Cto11eff0.4870 c) lM -4(b): In Figs from eff 4(a) and eff 1.3870 d) µ0.5023); c) lsimonteneously and respectively, (λI , µI ) are constant at (0.5058, M - C11 M - CIn 11 Figs 4(c) and 4(d): ef f eff Figure Comparison of results (C11 components) of ANN2 and XFEM eff d) M 11 M d)I2 M (periodic unit cell problem) for case 11 λ M decreases from 14 to while µ M increase from 0.3971 to 0.5771 (λI , µI ) are fixed at (44.1500, eff Fig 3: 14.9600) Comparison ( C11eff components) XFEM unit (periodic for all theof omparison of results (cases components) of ANN2 of andANN2 XFEMand (periodic cell un C11results In Figs 5(a) and 5(b): λ M increases eff from 17.3918 to 8.3918 while µ M increases from 1.4670 to Fig 3: Comparison of results ( components) of16λANN2 and XFEM (periodic unit C problem) for case I2 In (a), (b): l decreases while µ decrease 11 M M 1.2870 respectively.from In Figs and 5(d): from 16from to while ν Mfrom Mto or case I2 In simonteneously (a), (b): leff decreases 165(c) tofrom while µdecreases 1.3870 to 1.38 Mand M decrease mparison of results ( C11 components) of ANN2 and XFEM (periodic unit cell 20 0.4870 simonteneously (lI, µfrom constant at (0.5058, ; 1.387 In (c problem) for case I2 In and (a), respectively, (b):(llI,Mµdecreases to while µM decrease from I) are16 onteneously and respectively, ; 0.5023) In (c), (d): I) are constant at (0.5058, 0.5023) I2.simonteneously In (a), (b):14 l todecreases 16I, to while µM decrease from 1.3870 to lcase decreases while from µMfrom increase from (l fix 0.4870 (l µI)7are constant atto(0.5058, 0.5023) In (c), M from I , µI;) are es 14 tofrom whileMand µM5respectively, increase 0.3971 to 0.3971 0.5771 (l0.5771 I , µI) are fixed at nteneously and respectively, (l constant at (0.5058, InI ,(c), 14.9600) for all 5thewhile cases I, µµ I)Mare l(44.1500, 14 to increase from 0.3971 to0.5023) 0.5771.; (l µI) (d): are fixe M decreases 14.9600) for allfrom the cases simonteneously and respectively, (lI, µI) are(lconstant (0.5058, In (c), (d): 0.4870 simonteneously and respectively, constant at 0.5023) (0.5058,; 0.5023) ; In I, µI) are at reases from 14 to while increase 0.3971 0.5771.to(l0.5771 at I , µI) are lM decreases from 14 toµ5M while µM from increase fromto0.3971 (lI fixed , µI) are 0, (44.1500, 14.9600) for all the for cases 14.9600) all the cases Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering 25 15 10 10 XFEM 1.8 XFEM Neural network results Neural network results 1.6 1.6 20 1.8 XFEM XFEM Neural network results Neural network results 10 14 11 15 12 16 13 17 14 18 15 118 129 13 lM lM 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 1.25 1.3 1.251.35 1.3 1.4 1.351.45 1.4 1.5 1.45 16 17 18 (b) µµ − C- C b)33eff µ M - C33eff b) M ef f 11 M 20 18 XFEM XFEM Neural network results Neural network results 16 1.8 1.6 1.8 XFEM XFEM 1.6network results Neural network results Neural 1.4 1.4 12 1.2 1.2 10 1 8 0.8 0.8 6 0.6 0.6 4 0.4 0.4 2 0.2 0.2 0 10 lM 10 15 lM (c) λ − C eff eff c) lM - Cc) 11 lM - C11 M ef f 11 15 0.4 ef f 33 14 µM µM eff eff (a) λ − C a) lM - Ca) 11 lM - C11 M 0.6 0.4 0.8 0.6 µM 0.8 1.2 µM 1.4 1.2 eff d) µ M - Cd)33eff µ M - C33 ef f (d) µ M − C33 ef f ef f Figure Comparison of results (C11 and C33 components) of ANN4 and XFEM for case I3 eff eff Fig 4: Comparison and C33eff components) ANN4 andforXFEM forInca Comparison of results of ( Cresults of ANN4of and XFEM case I3 C11eff 11 and( C 33 components) decreases from 1.3870 to 0.4870 simonteneously and respectively In both all the cases, (λI , µI ) are at (0.5058, 0.5023) (a)land (b) lM increases from to 17.3918 8.3918 µM increases fromto1.4670 (b) increases from 17.3918 8.3918towhile µM while increases from 1.4670 1.2870to M fixed d 4.2 Computation C equivalent of l I2M (ellipse simonteneously andof respectively and (d)inclusion) lM decreases 16 toµ7M while µM d eneously and respectively In (c) inclusion andIn(d)(c) decreases from 16 tofrom while decreases We aim to0.4870 find λequsimonteneously , µequ of the equivalent inclusion (I1),all which has the volume fromto1.3870 tosimonteneously and respectively Inthe both all same the 3870 0.4870 andcircle respectively In both cases, (lI,cases, µI) are(lfixed I, µI) fraction with other type of inclusion (case I2, I3 in this work) To compute these coefficients, we at 0.5023) (0.5058, 058, combine0.5023) two networks as shown in Fig 1: ANN1 for Network1 and the ANN2 for Network Three tests will be computed to validate the surrogate models: In Test (Fig 6), the sample has the size of × 1mm2 and contains halves of an ellipse inclusion; in Test (Fig 7), the sample has 4.2 Computation of C equivalent of I2inclusion) (ellipse inclusion) omputation of C equivalent inclusioninclusion of I2 (ellipse the size of × 1.73 mm2 in which inclusions distribute hexagonally and Test (Fig 8) which contains 100 random inclusions these consider two of data Assuming that λ M , µ M ,inclusion λ(I1), known, wehas choose a I , µI are which aimInlto lofequ , µequ ofsetsthe circle equivalent (I1), which m We to find , tests, µequwe the circle equivalent inclusion the equfind small volume fraction and using ANN1 to generate the input for ANN2 Two data sets are examined: has th same volume with fraction with other type of inclusion I2, I3 in thisTowork) To compu e fraction other type of inclusion (case I2, (case I3 in this work) compute these 21 coefficients, we combine two networks Fig 1: for ANN1 for Netwo cients, we combine two networks as shownasinshown Fig 1:in ANN1 Network1 and thefor ANN2 for Network NN2 Network Three will be to surrogate validate the surrogate models: In Test (Fig 5), ree tests will be tests computed to computed validate the models: In Test (Fig 5), the sample has1 the size2 of x 1mm and contains halves inclusion; of an ellipse in Tes mple has the size of x 1mm and1 contains halves of an4 ellipse in inclusion; Test 2 (Fig.6),has thethe sample the size ofin1x1.73mm in which inclusions distribute hexagona ig.6), the sample size ofhas 1x1.73mm which inclusions distribute hexagonally and 7) Test (Fig 7) which contains 100andrandom inclusions d Test (Fig which contains 100 random inclusions Nhu, N T H., et al / Journal of Science Technology in Civil Engineering (a) A sample with halves of ellipse inclusions (b) The equivalent medium of the sample in Fig 6(a) A 4sample halves of(b) ellipse (b) The equivalent (a) A sample(a) with halves with of ellipse The equivalent medium of themedium sample of in the sample inclusions Figof5a/b (a)= 1.5 inclusions (a) radius Figure Test 1: The sample in (a) has the size of × mm2 and theFig ratio5between Test 1:inThe size2 of x 1mm the ratio between radius g Test 1:Fig The5.sample (a) sample has the in size(a)ofhas xthe 1mm and1 the ratio2 and between radius of a/b = 1.5 (a) (a)AAsample samplewith with44and and4x1/2 4x1/2 ellipse ellipse a/b = 1.5 (b) (b)The Theequivalent equivalentmedium mediumof ofthe thesample samplein in inclusions Fig inclusions Fig66(a) (a) (b) The equivalent medium 22 of the sample in Fig 7(a) Fig Test 2:2:awith sample has size of 1x1.73mm (a) its equivalent Fig 6.A Test a rectangular rectangular sample hasthe the size ofequivalent 1x1.73mm (a)and and its equivalent (a)(a) A sample and 4x1/2 ellipse (b) The medium of of the sample in in sample with and 4x1/2 ellipse (b) The equivalent medium the sample its (a) medium Figure Test 2: ainclusions rectangular size of 1(b) × 1.73 mm (a)Fig and equivalent medium (b) inclusionssample has the Fig (a) medium (b) 2 Fig 6.6 Test 2:2: a arectangular sample has thethe size ofof 1x1.73mm andand its its equivalent Fig Test rectangular sample has size 1x1.73mm(a)(a) equivalent medium (b)(b) medium (a) A sample with and × 1/2 ellipse inclusions (a) 100 ellipse inclusions (a)A(a) Asample sample with100 100 ellipse inclusions A samplewith with ellipse inclusions (b) medium the in (b) The equivalent medium of thesample sample in (b)The The equivalent equivalent medium of theof sample in Fig 8(a) Fig Fig66(a) (a) (a)(a) AAsample with ellipse (b)(b) The equivalent medium of of thethe sample in in sample with 100 ellipse inclusions The equivalent medium sample Figure Test 3: A100 sample withinclusions 100 random ellipse inclusions (a) and its equivalent medium Fig (a) with 100 ellipse circular inclusions 6equivalent (a) Fig (a) its medium Fig.7.7.Test Test3:3:AAsample samplewith with100 100random random ellipseinclusions inclusions(b) (a)and andFig itsequivalent medium with with 100 circular inclusions (b) 100 circular inclusions (b) Fig 7.7 Test 3: AA sample with 100 random ellipse inclusions (a)(a) and itsits equivalent medium with Fig Test 3: sample with 100 random ellipse inclusions and equivalent medium with 2 Dataset 1: λ = 17.3918 N/mm ; λ = 0.5058 N/mm , µ = 1.4870 N/mm , µ = 0.5023 M I M I 100 circular inclusions (b) 100 circular inclusions (b) N/mm2 , and λequ = 0.3822 N/mm2 , µequ = 1.4528 N/mm2 -these Dataset 2: λ M = 18.7749 N/mm ;sets λI =of 40.2908 N/mm , µ M that =that 0.4822 = 16.4163N/mm InInthese tests, we consider two data llMM, ,µN/mm known, tests, we consider two2sets of data.Assuming Assuming µMM, ,llI,I2,,µµI IIare are known,we we , 2 and λthese =aa39.9912 N/mm ,fraction µequtwo =two 16.2965 N/mm Assuming equ InIn tests, consider sets data Assuming that lMl, Mµthe , Mlinput are known, weTwo Mµ I,lµ choose small volume and ANN1 totogenerate ANN2 choose small volume fraction and using ANN1 generate ANN2 Two these tests,we we consider setsofusing of data that ,the ,input are known, we I,I µfor Ifor Figs 9–11 compare the effective properties of the two media in Test 1, Test 2, Test respectively choose volume and using ANN1 to to generate thethe input forfor ANN2 Two data sets examined: data setsaare are examined: choose asmall small volumefraction fraction and using ANN1 generate input ANN2 Two We can see that with the equivalent properties of inclusions, equivalent media reflect very well it data are data sets areexamined: •• sets Dataset 1:examined: Dataset 1:llMM==17.3918 17.3918N/mm N/mm22; ;llI I=0.5058 =0.5058N/mm N/mm22, ,µµMM==1.4870 1.4870N/mm N/mm22,, µµI I== referenced media 2 2 2 • •0.5023N/mm Dataset 1:1: lMl2,2M =,and 17.3918 N/mm ; l;N/mm =0.5058 1.4870 , µ, I µ =I = 22 N/mm 22N/mm Il , µ,Mµ= Dataset = 17.3918 N/mm =N/mm 1.4870 I =0.5058 M N/mm llequ 0.3822 , ,µµequ ==1.4528 N/mm 0.5023N/mm and 0.3822 N/mm 1.4528 equ==N/mm equ 2 2 2 0.5023N/mm , and = 0.3822 , µ,equ = 1.4528 22 N/mm 22N/mm 0.5023N/mm , 18.7749 andlequ lequ = 0.3822 àequ =N/mm 1.4528 N/mm 22 ãã Dataset 2:2:llMM==18.7749 N/mm ; ;llN/mm N/mm Dataset N/mm =40.2908 0.4822N/mm N/mm22,, µµI I== I I=40.2908 , ,µµ MM==0.4822 2 • • Dataset 2:2: lMlM = 18.7749 N/mm ;2l;I l=40.2908 , 2µ, I µ =I = M = 0.4822 N/mm ,µ 22 = 22 N/mm 2N/mm Dataset 18.7749 N/mm =40.2908 N/mm IN/mm , µM = 0.4822 16.4163N/mm , and l , µ N/mm 16.4163N/mm , and l 39.9912 N/mm , µ 16.2965 N/mm equ equ equ==39.9912 equ==16.2965 2 16.4163N/mm , 2and lequ = 39.9912 N/mm , µ equ = 16.2965 N/mm 16.4163N/mm , and lequ = properties 39.9912 N/mm ,the µequtwo = 16.2965 1, Figs.8-10 8-10 comparethe the effective properties ofthe two mediaN/mm in Test Test 1, Test Test 2, 2, Test Test 33 Figs compare effective of media in Figs 8-10 compare the effective properties of the two media in Test 1, Test 2, 2, Test 3 Figs 8-10 compare the effective properties of the two media in Test 1, Test Test respectively.We Wecan can see see that that with with the the equivalent equivalent properties properties of of inclusions, inclusions, equivalent equivalent respectively Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering a) a) a)a) a)a) a) a) c) c) c)c) c) c)c) c) 25 2525 25 XFEM -ref 2525 XFEM -ref 25 XFEM- -ref XFEM equXFEM -ref 20 XFEM - -ref equ XFEM - equ XFEM 20 XFEM XFEM-ref - equ XFEM -ref 20 2025 XFEM - equ XFEM - equ XFEM - equ 2020 20 XFEM -ref 15 1515 XFEM - equ 1520 1515 15 10 1010 1015 1010 10 55 10 5 0 0.05 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 05 0.2 0.2 0.05 0.1 0.15 0.25 0.3 0.35 0.4 0.4 0.05 0.1 0.3 0.20.15 0.050 00.1 0.15 0.25 0.30.25 0.35 0.40.35 0 0.2 0.05 0.1 0.15 0.25 0.3 0.35 0.4 f0.10.15 0.20.25 0.05 0.050.1 0.150.2 0.250.3 0.30.35 0.350.40.4 ff f 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 f eff f f 25 (a) C11 2525 25 f 2525 25 20 2020 2025 XFEM -ref 20 2020 XFEM-ref -ref XFEM XFEM- -ref XFEM equ 15 XFEM- -ref - equ XFEM XFEM XFEM equ XFEM-ref - equ XFEM -ref 1515 1520 XFEM - equ XFEM - equ 15 1515 XFEM -refXFEM - equ 10 XFEM - equ 1010 1015 10 1010 10 55 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 05 00 0.05 0.15 0.2 0.25 0.3 0.35 0.4 0.4 0.050 00.1 0.15 0.2 0.25 0.30.25 0.35 0.40.35 0.05 0.10.1 0.15 0.2 0.3 0 0.0500 0.1 0.150.1 0.2 0.3 0.35 0.3 0.4 0.05 f 0.25 0.05 0.10.15 0.150.20.20.25 0.25 0.30.35 0.350.40.4 f f 0.3 0.35 0.4 0.05 0.1 0.15 f 0.2 0.25 b) b) b) b) d) d) d) d) f 1.82 1.8 1.6 1.8 1.6 1.42 1.6 1.4 1.8 1.2 1.4 1.2 1.6 1.21 1.4 0.8 11.2 0.8 0.6 0.8 0.61 0.4 0.6 0.4 0.8 0.2 0.4 0.2 0.6 0.2 00 0.4 0 0.2 0 0.91 0.9 0.8 0.9 0.81 0.7 0.8 0.9 0.7 0.6 0.7 0.8 0.6 0.5 0.6 0.7 0.5 0.4 0.5 0.6 0.4 0.3 0.4 0.5 0.3 0.2 0.3 0.4 0.2 0.1 0.2 0.3 0.1 0 0.1 0.2 0 0.1 0 b)b) b)b) d)d) d)d) f f 22 2 1.8 1.8 1.8 1.6 1.8 1.6 XFEM -ref XFEM -ref XFEM- -ref XFEM equXFEM -ref XFEM ref XFEM -equ equ XFEM XFEM XFEM-ref - equ XFEM -ref XFEM - equ XFEM - equ XFEM - equ XFEM -ref XFEM - equ 1.6 1.4 1.4 1.6 1.4 1.2 1.2 1.4 1.2 111.2 1 0.8 0.8 0.8 0.6 0.6 0.8 0.6 0.4 0.4 0.6 0.4 0.2 0.2 0.4 0.05 0.2 00.1 0.15 0.2 0.25 0.3 0.35 0.4 00.2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4 0.050000.10.05 0.150.1 0.20.15 0.250.2 0.30.25 0.350.3 0.4 0.35 0 0.3 0.4 0.4 0.05 0.10 0.05 0.15 0.2 0.25 0.3 0.35 0.4 0.050.1 0.150.2 0.20.25 0.25 0.30.35 0.35 f 0.10.15 0.05 0.1 11 0.15 f0.2 ef f (b) fC33 1 0.9 0.9 0.9 0.8 0.8 0.9 f f f 0.25f 0.3 0.35 0.4 f 0.8 0.7 0.7 0.8 0.7 0.6 0.6 0.7 0.6 0.5 0.5 0.6 XFEM -ref XFEM-ref -ref XFEM XFEM- equ -refXFEM XFEM ref -equ equ XFEM XFEM XFEM XFEM-ref - equ XFEM -ref XFEM - equ XFEM - equ XFEM -ref XFEM - equ XFEM - equ 0.5 0.4 0.4 0.5 0.4 0.3 0.3 0.4 0.3 0.2 0.2 0.3 0.2 0.1 0.1 0.2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 00.1 0.2 00.10.05 0.05 0.1 0.15 0.25 0.3 0.35 0.4 0.4 0.20.15 0.050 0.150.1 0.250.2 0.3 0.25 0.350.3 0.4 0.35 0 0.2 0.25 0.2 0.05 00.1 0.05 0.15 0.1 0.3 0.35 0.3 0.4 0.4 0.4 0.05 f 0.10.15 0.15 0.20.25 0.25 0.30.35 0.35 0.05 0.1 0.15 f0.2 f 0.25ff 0.3 0.35 0.4 f f ef f ef f Fig 8: Comparison of C11effeff andf (c) in Test (Fig 5): using Data set (a, Data set (c, (d)b)Cfand C effeffC11 effeff33 effeff in Test (Fig 5): using Data set 133(a, b) and Data set (c, Fig Comparison and C Fig 8: Comparison of Ceff11ofof and in Test (Fig 5): using Data set (a, b) and (c, C Fig 8:8:Comparison and in Test (Fig 5): using Data set (a, b)Data andset Data set (c, CC C 11 33 33f 11 33 effeeff eff e f f f Fig Comparison of C11 of and in 1inTest (Fig using Data set (a,setb) Data setData 2Data (c,set effin d) 8:Fig Ceff 8:8:9 Comparison and 15): (Fig 5):5): using Data 1and (a, b) and 22(c, C11 CTest 33 Figure Comparison of and C in Test (Fig 6): set (a, b) and Data set 33 Fig Comparison of and Test (Fig using Data (a, b) and set 2(c, (c,d) C C eff eff 11 33 33 (Fig 5): using Data set (a, b) and Data set (c, d).Comparison d) Fig.d) 8: of C11 and 11C33 in Test 25 d) d).d) 25 2525 1.82 XFEM -ref XFEM -ref 25 d) 25 1.8 20 XFEM equXFEM -ref 1.8 XFEM - equXFEM -ref XFEM- -ref 25 1.6 a) a) a) a) a) a)a) a) c) c) c) c) c) c) c) c) 2025 20 15 20 15 15 10 15 10 10 10 5 50 0 0 30 0 30 30 25 2530 20 25 2025 15 20 1520 10 15 1015 105 510 50 05 0 0 2020 20 20 15 15 15 15 10 10 10 10 5 5 0.05 00.050 00 0.05 00 0.05 30 30 30 30 25 25 25 25 20 20 20 20 15 15 15 15 10 10 10 510 5 0.05 05 00.050 00 0.05 00 0.05 XFEM -ref XFEM- -ref - equ XFEM XFEM-ref - equ XFEM XFEM equ XFEM -ref XFEM equ XFEM - equ XFEM -ref XFEM - equ XFEM - equ b) b) b) b) 1.8 1.62 1.4 1.8 1.6 1.4 1.2 1.6 1.4 1.2 1.4 1.2 0.81 11.2 C0.6 0.8 0.81 0.6 C0.4 0.8 0.6 0.4 0.2 C 0.40.6 0.2 0.40 0.2 0 0.2 0 10 0.9 1 0.9 0.8 0.9 0.8 0.7 0.9 0.8 0.6 0.7 0.8 0.7 0.6 0.5 0.7 0.6 0.4 0.5 0.6 0.5 0.3 0.4 0.5 0.4 0.2 0.3 0.4 0.3 0.1 0.2 0.3 0.2 0.1 0.2 0.1 0 00.1 0 b)b) b)b) 0.1 0.15 0.2 0.25 0.3 0.05 0.15 0.15 0.2 0.3 0.25 0.3 0.1 f 0.1 0.2 0.05 0.1 0.15 0.25 0.2 0.25 0.3 0.1 0.05 0.15 0.1 0.15 0.2 0.3 0.25 0.3 f 0.25 f 0.1 0.2 0.05 0.15 0.2 0.25 0.3 f 0.1 0.2 0.25 0.3 f0.15 f ef f f f (a) C 11 XFEM -ref XFEM equXFEM -ref XFEM- -ref XFEM -ref XFEM XFEM-ref - equ XFEM-ref - equ XFEM XFEM - equ XFEM -ref XFEM -ref XFEM - equ XFEM - equ XFEM - equ XFEM - equ 0.1 0.15 0.2 0.25 d) d) d) d) d)d) d)d) 0.3 f 0.1 0.20.15 0.25 0.05 0.15 0.2 0.3 0.25 0.3 0.1 0.05 0.15 0.1 0.2 0.15 0.25 0.2 0.3 0.25 0.3 0.1 f 0.05 0.1 0.15f 0.2 0.25 0.3 0.1 f0.15 0.250.2 0.30.25 0.05 f0.15 0.1 0.2 0.3 f f (c) XFEM -refXFEM -ref 1.8 XFEM ref -equ equ XFEM 1.8 1.6 XFEM-ref - equ XFEM XFEM 1.6 1.8 XFEM -ref XFEM - equ XFEM - equ XFEM -ref 1.6 1.4 XFEM - equ 1.41.6 XFEM - equ 1.4 1.2 1.21.4 1.21 1.2 0.8 0.8 C 0.8 0.6 C0.6 0.8 C0.6 0.4 C 0.4 0.6 0.2 0.4 0.20.4 0.01 0.20 0.30 0.2 f 0.01 0.20 0.01 0.20 0.30 0.30 00.2 0.01 0.20 0.30 f 0.20 f f 0 0.01 0.01 0.20 0.30 0.30 0.01 0.01 0.20 0.30 f e f f 0.20f 0.30 (b) C33 XFEM f-ref f 0.9 XFEM 0.9 XFEM- equ -ref XFEM -ref XFEM -ref 0.8 0.9 XFEM XFEM -ref - equ XFEM-ref - equ XFEM 0.80.9 XFEM - equ XFEM -ref XFEM - equ XFEM -ref 0.7 0.8 XFEM - equ 0.70.8 XFEM - equ XFEM equ 0.6 0.7 0.60.7 0.5 0.6 0.50.6 0.4 0.5 0.40.5 0.3 0.4 0.30.4 0.2 0.3 0.20.3 0.1 0.2 0.05 0.15 0.2 0.25 0.3 0.2 0.1 0.1 0.10 0.2 0.30.25 0.3 0.10.05 0.15 0.1 f0.1 0.20.15 0.25 00.05 0.05 0.15 0.1 0.20.15 0.250.2 0.30.25 0.3 00 0.05 0.1 00 0.05 0.1 0.15f 0.2 0.25 0.3 f 0.050 0.1 0.050.15 0.1 0.2 0.15 0.3 f 0.25 0.2 0.3 0.25 f f f ef f C11 (d) f f ef f C33 ef f ef f Figure 10 Comparison of C11 and C33 in Test (Fig 7): using Data set (a, b) and Data set (c, d) 23 eff eff (Fig 6): using Data set (a, b) and Data set (c, Fig 9:Fig Comparison of C11effofand in C Test Cand 9: Comparison in Test (Fig 6): using Data set (a, b) and Data set (c, 33 C11effeff 33 eff eff eff Fig 9: Comparison of and in Test (Fig 6): using Data set (a, b) and Data set (c, C C Fig 9: Comparison of and in Test C C 11 33 33 (Fig 6): using Data set (a, b) and Data set (c, 11 d) d) d) d) Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering 10 10 98 87 76 65 a) a) a) a) 54 43 32 21 10.15 0.15 30 30 25 25 20 c) c) 20 15 c) c) 10 10 78 67 b) b) 0.25 0.2 0.25 0.2 0.3 0.25 f 0.3 0.25 f 0.35 0.3 0.35 0.3 f fe f f 0.4 0.35 0.4 0.35 b) 0.5 b) 0.5 0.15 0.15 0.4 0.4 10 5 00.2 0.15 0.2 0.15 0.2 0.15 0.2 0.15 0.3 0.25 0.3 0.25f f 0.35 0.3 0.35 0.3 0.4 0.35 0.4 0.35 ef f f (b) C33 0.35 0.3 0.35 0.3 f f 0.4 0.35 0.40.35 0.4 0.4 XFEM -ref XFEM - equXFEM -ref XFEM -ref XFEM XFEM- -ref equ XFEM - equ XFEM - equ 0.7 0.6 0.6 0.5 0.5 0.6 0.4 0.5 d) 0.3 d) 0.4 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.10.15 0.15 0.4 0.4 0.3 0.25 f 0.25 0.3 0.9 0.8 0.8 0.7 0.2 0.3 0.1 0.2 0.25 0.2 f 0.25 0.2 f 0.25 0.2 0.25 0.2 0.91 0.7 0.8 0.6 0.7 d) d) XFEM -ref XFEM -ref XFEM - equ XFEM- -ref XFEM -ref XFEM equ XFEM - equ XFEM - equ 0.5 0.5 0.9 0.8 0.9 20 20 10 10 1 (a) C11 30 XFEM -ref XFEM -ref 30 XFEM - equ XFEM -refXFEM XFEM- -ref equ 25 XFEM - equXFEM - equ 25 15 15 1.5 1.5 34 23 15 10 0.15 0.15 1.5 56 45 12 00.2 0.15 0.2 0.15 1.5 XFEM -ref XFEM -ref XFEM - equ XFEM- -ref equ XFEM -ref XFEM XFEM - equXFEM - equ 89 0.1 0.2 0.15 0.2 0.15 ef f 0.25 0.2 f 0.25 0.2 f 0.3 0.35 0.25 0.3 0.30.25 0.35 0.3 f f 0.4 0.35 0.40.35 0.4 0.4 ef f C11in Test (Fig 7): using Data set (d) C33and Data set (c, Fig 10: Comparison of C11eff and (c) (a,b) C eff eff 33 eff Fig 10: Comparison and in Test (Fig 7): using Data set (a,b) and Data set (c, C eff of C11 eff e f f e f f 33 eff eff Fig Comparison of C11 of andof in Test (Fig using Data setData (a,b) Data set (c, 10: Comparison and Test 3(Fig using Data set1and 1(a, (a,b) and Data (c, Figure 11 Comparison C33 C33in in Test7): (Fig.7): 8): using set b) and Data setset (c, d) CC C33 d) 10:Fig 11 11 and d) d) d) 4.3 Computation C equivalentinclusion inclusion of of I3 4.3 Computation of Cofequivalent I3 (flower (flowerinclusion) inclusion) 4.3 Computation of C equivalent inclusion of I3 (flower inclusion) 4.3 Computation C equivalent inclusion of I3 inclusion) 4.3to Computation C equivalent inclusion of I3 (flower inclusion) Similar toof the I2, we employ and(flower ANN4 (for Network and Network Similar the case I2, case weofemploy ANN3 ANN3 and ANN4 (for Network and1 Network in2 in Fig 1, Similar to the case I2, we employ ANN3 and ANN4 (for Network and Network in respectively) to generate the equivalent parameter for circle inclusion As the geometry Similar to the case employ ANN3 and ANN4 (for Network and Network 2the in of22 flower Fig 1,Similar respectively) towegenerate the equivalent parameter for inclusion to theI2,case I2, we employ ANN3 and ANN4 (forcircle Network and As Network in inclusion is quite complicated, we reduce the input dimension by exclude the properties of matrix Fig 1, respectively) to generate the equivalent parameter for inclusion circle inclusion As the Fig 1, respectively) to generate the equivalent parameter for circle As the geometry inclusion isforquite complicated, weparameter reduce dimension 2thefor 2by As the Fig of 1, flower respectively) toisgenerate the equivalent circle inclusion Specifically, network the case = 17.3918 N/mm , µ Minput =the 1.4870 The data geometry ofthe flower inclusion is complicated, quiteλ M complicated, we reduce inputN/mm dimension by of geometry of flower inclusion is quite we reduce the input dimension by 2 exclude the properties of N/mm matrix the network isequivalent for the case linput M = 17.3918 geometry flower inclusion quite N/ complicated, reduce the dimension inclusion λI of = 0.5058 , Specifically, µI =is0.5023 mm and thewe inclusion computed by by ANNs exclude the properties of matrix Specifically, the 2network is for the case lM = 17.3918 2 2 exclude the properties of matrix Specifically, the network is for the case l = 17.3918 M includes 0.3872 ,data µequSpecifically, = 0.4547 N/ l mm These N/mm results validated in the two N/mm , µM =λ1.4870 N/mmN/of mm The of inclusion µI then =case 0.5023 exclude the properties matrix the network is for ,are the lM =N/ 17.3918 I =0.5058 equ = 2 2 2 the same 2N/mm N/mm , µ 1.4870 N/mm The data of inclusion l =0.5058 , µ N/ M =which IFigs I = 0.5023 following tests have size of × mm (see 12(a) and 12(b)) 2 N/mm , µ = 1.4870 N/mm The data of inclusion l =0.5058 N/mm , µ = 0.5023 N/ M I includes I mm and the ,equivalent inclusion lequ = N/mm 0.3872 mm , N/mm µM = 1.4870 N/mm computed The databy of ANNs inclusion lI =0.5058 , N/ µI = 0.5023 N/ 2 mm and the equivalent mm 2, inclusion computed by ANNs includes l = 0.3872 N/ equ inclusion mm and the equivalent computed by ANNs includes l = 0.3872 N/ mm , equ µequ =mm 0.4547 mm These results are then validated in the two following tests which andN/the equivalent inclusion computed by ANNs includes l = 0.3872 N/ mm , equ mm22 These µ = 0.4547 N/ results are then validated in the two following tests which equ µhave 0.4547 N/size mmof 1x1 These are in the two following tests which equ = the (see results Fig.then 11are a,validated b) µequsame = 0.4547 N/ mmmm results These then validated in the two following tests which mm22 (see Fig 11 a, b) have the same size of 1x1 have the same size of 1x1 mm (see Fig 11 a, b) have the same size of 1x1 mm (see Fig 11 a, b) (a) Anwith unit cell halves inclusions (b) unit An unitcell with 4040 random I3 inclusions (a) An (a) unitAn cell halves ofofI3I3inclusions (b) An with random I3 inclusions unit cell4 with with halves of I3 inclusions (b) Ancellunit cell with 40 random I3 inclusion 2 12.11: Two unit cells ofcells the ×1x1mm mm Fig Figure 11:Fig Two unit cells of thesize size Two unit of1 the size 1x1mm a) a) 18 18 16 16 14 14 12 12 10 10 b) XFEM -ref XFEM -ref 24 XFEM - equ XFEM - equ b) 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 XFEM -ref XFEM -ref XFEM - equXFEM - equ (a)cell An with unit cell with 4ofhalves of I3 inclusions (b) cell An unit 40 random I3 inclusions (a) An unit halves I3 inclusions (b) An unit withcell 40 with random I3 inclusions (a) cell An unit with 4ofhalves of I3 inclusions (b)cell An with unit cell with 40I3random I3 inclusions (a) An unit withcell halves I3 inclusions (b) An unit 40 random inclusions Engineering Nhu,11: N T.Fig H., et11: al /Two Journal of Science and1x1mm Technology in Civil cells of the size 21x1mm Fig Two unit cellsunit of the size cells of1x1mm the size2 1x1mm2 Fig 11: Fig Two 11: unitTwo cellsunit of the size a) a) a)18 a)1816 18 18 16 XFEM -ref XFEM -ref 16 -ref- equ XFEM XFEM -ref- equXFEM 14 XFEM 14 XFEM - equ XFEM - equ 12 12 10 10 8 6 4 2 0.05 0.0500 00.10.05 0.15 0.10.2 0.15 0.25 0.20.3 0.25 0.30.3 0.05 0.1 0.15 0.10.2 0.15 0.25 0.20.3 0.25 b) b) 16 14 14 12 12 10 10 8 6 4 2 0 0 f f b) b) 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2 1.8 1.8 XFEM -ref XFEM -ref 1.6 XFEM -ref- equ XFEM XFEM -ref - equ 1.6 XFEM 1.4 XFEM - equ XFEM - equ 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.3 0 0.1 0.050.15 0.1 0.2 0.150.25 0.2 0.3 0.25 0.05 0.3 0.05 0.1 0.05 0.15 0.1 0.2 0.15 0.25 0.2 0.3 0.25 f f ef f f f f f ef f (b) C33 (a) C11 eff eff eff and eC Fig 12: Comparison ofand (a) (b) for Test 11 (Fig 11result a): thecomputed result computed using C effe11 eff effeff (a) eff f(a) fC f f33 Fig.12: 12:Fig Comparison (b) Test Test (Fig a): the using using 12: Comparison ofand and (b) (Fig a): the result computed Cfor Fig Comparison ofofCC (a) (b) for Test 4for(Fig 114 a): the11 result computed using C Figure 13 Comparison ofCC (a)33 and C 1111 33 1111 33 33 (b) for Test (Fig 12(a)): the result computed using equivalent equivalent inclusion (XFEM-equ) shows a good match with the reference result (XFEM-ref) inclusion (XFEM-equ) aamatch good match thethe reference result (XFEM-ref) equivalent inclusion (XFEM-equ) shows ashows good with the reference result (XFEM-ref) equivalent inclusion (XFEM-equ) shows good match with reference result (XFEM-ref) equivalent inclusion (XFEM-equ) shows a good match with thewith reference result (XFEM-ref) a) a) a)a) 1818 18 18 1616 16 16 1414 14 14 1212 12 12 1010 10 10 88 66 44 00 b)b) XFEM-equ XFEM-equ XFEM-equ XFEM-equ XFEM-ref XFEM-ref XFEM-ref XFEM-ref b)b) 2 1.5 1.5 1 0.5 0.5 4 0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.15 0.25 0.3 0.25 0.3 0.05 0.10.10.05 0.150.10.20.20.15 f0.250.2 0.3 f f f (a) ef f C11 2 XFEM-equ XFEM-equXFEM-equ XFEM-equ XFEM-ref XFEM-ref XFEM-ref XFEM-ref 1.5 1.5 1 0.5 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.05 0.15 0.15 0.2 0.2 0.25 0.25 0.3 0.3 00.1 0.1 0.05 0.1 0.15 0.25 0.3 f 0.2 f (b) f f ef f C33 eff eff eff eff eff eff Fig.13: 13: Comparison of (a) and (b)for(Fig forTest Test 4(Fig (Fig 11b):b): the result computed using C C eff eff Fig Comparison ofofCC (a) and (b) for 11411 b): the11 result computed using C e11 f (a) f 33 e Test f 33 f (b) of and the result computed C C 11 Fig.13: 13:Fig Comparison (a)of and (b) (Fig b): the result computed using using C 33 Test Figure Comparison 14 Comparison C1111 (a)33 and Cfor 11 33 (b) for Test (Fig 12(b)): the result computed using equivalent equivalent inclusion (XFEM-equ) shows a good match with the reference result (XFEM-ref) equivalent inclusion (XFEM-equ) shows a good match with thewith reference result (XFEM-ref) inclusion (XFEM-equ) aamatch good match thethe reference result (XFEM-ref) equivalent inclusion (XFEM-equ) shows good match with reference result (XFEM-ref) equivalent inclusion (XFEM-equ) shows ashows good with the reference result (XFEM-ref) The resultscompared compared inand Fig.1212and and Fig.13 13again again show a good match between two The compared ininFig again show a good match between the twotwo The results in12 Fig Fig show a match good match between thethe two Theresults results compared Fig.12 andFig Fig.1313 again show a good between the The resultssuggests compared in reliability Figs 13 and 14 again show a approach good match between the two media, which media, which suggests the reliability ofthe theproposed proposed approach media, which suggests the of the proposed approach which the of media, media, which suggests thereliability reliability of the proposed approach suggests the reliability of the proposed approach Conclusion Conclusion 5.5.Conclusion Conclusion Conclusions this paper, wehave havepresented presented anovel novelapproach approach estimating equivalent circular In this paper, we havewe presented a novel approach for estimating the equivalent circular circular InInthis paper, forforestimating thethe equivalent In this paper, we have presented a novela approach for estimating the equivalent circular In this paper, we have presented a novel approach for estimating the equivalent circular inclusion inclusion We’ve shown thecapacity capacity ofthe theANN ANNsurrogate surrogate forcell unitcell cell method inclusion We’ve shown the capacity of the ANN surrogate for the unit method to inclusion We’ve shown the of forunit thethe unit method to to inclusion We’ve shown the capacity of the ANN surrogate for the cell method to We’ve shown the capacity of the ANN surrogate for the unit cell method to compute the effective stiffness matrix in some cases of inclusion with a small specified volume fraction Using a second network, which interpolates properties of the circular inclusion from the expected effective stiffness matrix, we have proposed a new approach to deal with the equivalent inclusion problem by combining two ANN models The proposed approach allows us to apply for the case when an analytic formula to compute effective elastic moduli is not available, eg I3 in this work The results in section show a good agreement between the equivalent medium and the referenced medium, which reveals the potential of data driven approach for this problem For future works, we’ll try to improve the quality of the network and apply for various types of inclusions 25 Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.02-2017.309 References [1] Hashin, Z., Shtrikman, S (1962) A variational approach to the theory of the effective magnetic permeability of multiphase materials Journal of Applied Physics, 33(10):3125–3131 [2] Miller, M N (1969) Bounds for effective electrical, thermal, and magnetic properties of heterogeneous materials Journal of Mathematical Physics, 10(11):1988–2004 [3] Phan-Thien, N., Milton, G W (1982) New bounds on the effective thermal conductivity of N-phase materials Proceedings of the Royal Society of London A Mathematical and Physical Sciences, 380 (1779):333–348 [4] Chinh, P D (2011) Bounds on the effective conductivity of statistically isotropic multicomponent materials and random cell polycrystals Journal of the Mechanics and Physics of Solids, 59(3):497–510 [5] Dunant, C F., Bary, B., Giorla, A B., Péniguel, C., Sanahuja, J., Toulemonde, C., Tran, A.-B., Willot, F., Yvonnet, J (2013) A critical comparison of several numerical methods for computing effective properties of highly heterogeneous materials Advances in Engineering Software, 58:1–12 [6] Tran, A B., Yvonnet, J., He, Q.-C., Toulemonde, C., Sanahuja, J (2011) A multiple level set approach to prevent numerical artefacts in complex microstructures with nearby inclusions within XFEM International Journal for Numerical Methods in Engineering, 85(11):1436–1459 [7] Moulinec, H (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites Comptes Rendus Mecanique, 318:1417–1423 [8] Michel, J C., Moulinec, H., Suquet, P (2001) A computational scheme for linear and non-linear composites with arbitrary phase contrast International Journal for Numerical Methods in Engineering, 52 (1-2):139–160 [9] Pham, D C., Tran, N Q., Tran, A B (2017) Polarization approximations for elastic moduli of isotropic multicomponent materials Journal of Mechanics of Materials and Structures, 12(4):391–406 [10] Pham, D C., Tran, A B., Do, Q H (2013) On the effective medium approximations for the properties of isotropic multicomponent matrix-based composites International Journal of Engineering Science, 68: 75–85 [11] Tran, A B., Pham, D C (2015) Polarization approximations for the macroscopic elastic constants of transversely isotropic multicomponent unidirectional fiber composites Journal of Composite Materials, 49(30):3765–3780 [12] Norris, A N., Callegari, A J., Sheng, P (1985) A generalized differential effective medium theory Journal of the Mechanics and Physics of Solids, 33(6):525–543 [13] Milgrom, M., Shtrikman, S (1989) A layered-shell model of isotropic composites and exact expressions for the effective properties Journal of Applied Physics, 66(8):3429–3436 [14] Quyet, T N., Chinh, P D., Binh, T A (2015) Equivalent-inclusion approach for estimating the elastic moduli of matrix composites with non-circular inclusions Vietnam Journal of Mechanics, 37(2):123– 132 [15] Do, Q H., Tran, A B., Pham, D C (2016) Equivalent inclusion approach and effective medium approximations for the effective conductivity of isotropic multicomponent materials Acta Mechanica, 227(2): 387–398 [16] Chinh, P D., Binh, T A (2016) Equivalent-inclusion approach for the conductivity of isotropic matrix composites with anisotropic inclusions Vietnam Journal of Mechanics, 38(4):239–248 [17] Hernández, J A., Oliver, J., Huespe, A E., Caicedo, M A., Cante, J C (2014) High-performance model reduction techniques in computational multiscale homogenization Computer Methods in Applied Mechanics and Engineering, 276:149–189 26 Nhu, N T H., et al / Journal of Science and Technology in Civil Engineering [18] Zahr, M J., Avery, P., Farhat, C (2017) A multilevel projection-based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics International Journal for Numerical Methods in Engineering, 112(8):855–881 [19] Bessa, M A., Bostanabad, R., Liu, Z., Hu, A., Apley, D W., Brinson, C., Chen, W., Liu, W K (2017) A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality Computer Methods in Applied Mechanics and Engineering, 320:633–667 [20] Myers, R H (1999) Response surface methodology—current status and future directions Journal of Quality Technology, 31(1):30–44 [21] Simpson, T W., Mauery, T M., Korte, J J., Mistree, F (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization AIAA Journal, 39(12):2233–2241 [22] Ghaboussi, J., Garrett Jr, J H., Wu, X (1991) Knowledge-based modeling of material behavior with neural networks Journal of Engineering Mechanics, 117(1):132–153 [23] Yvonnet, J., He, Q.-C (2007) The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains Journal of Computational Physics, 223(1):341–368 [24] Le, B A., Yvonnet, J., He, Q.-C (2015) Computational homogenization of nonlinear elastic materials using neural networks International Journal for Numerical Methods in Engineering, 104(12):1061–1084 [25] McCulloch, W S., Pitts, W (1943) A logical calculus of the ideas immanent in nervous activity The Bulletin of Mathematical Biophysics, 5(4):115–133 [26] Yvonnet, J., Quang, H L., He, Q C (2008) An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites Computational Mechanics, 42(1):119–131 27 ... of inclusion The Notedetails that, of thethetwo networks ofin the Fig are utilized the same involume fractionThe of first step, construction two1 networks will for be discussed the following inclusion. .. capacity ofthe theANN ANNsurrogate surrogate forcell unitcell cell method inclusion We’ve shown the capacity of the ANN surrogate for the unit method to inclusion We’ve shown the of forunit thethe unit... architecture of the surrogate Thisthanks step includes the using XFEM The geometry of these is described thanks to the using XFEM The geometry of these inclusions is model described to following the following