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Numericalestimationofeffectiveelectromagneticpropertiesfordesignofparticulatecomposites

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DOI:10.1016/j.matdes.2016.01.015

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Materials and Design 94 (2016) 546–553

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Materials and Design

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Numerical estimation of effective electromagnetic properties for designof particulate composites

Bhavesh Patel ⁎, Tarek I. ZohdiDepartment of Mechanical Engineering, University of California, Berkeley, CA 94720, United States

⁎ Corresponding author.E-mail addresses: b.patel@berkeley.edu (B. Patel), zod

http://dx.doi.org/10.1016/j.matdes.2016.01.0150264-1275/© 2016 Elsevier Ltd. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 25 October 2015Received in revised form 26 October 2015Accepted 5 January 2016Available online 12 January 2016

Most modern electromagnetic devices consist of dielectric and magnetic particulate composites. Predicting theeffective electric permittivity and effective magnetic permeability of the envisioned composite is of great impor-tance in validating the design for such applications. In this work, we propose a numerical method based on Yee'sscheme and statistically generated representative volume element to estimate these effective electromagneticproperties for linear isotropic composites made with ellipsoidal particles. By considering particle geometry andcomposite microstructure precisely, it provides a more accurate tool for their design than available analyticalbounds. Several numerical examples of composite microstructures are presented to demonstrate the capabilityof the proposedmethod. Comparisonwith analytical bounds and experimental results from literature is conduct-ed to show validity.

© 2016 Elsevier Ltd. All rights reserved.

Keywords:Electrical propertiesMagnetic propertiesComputational modelingComposites design

1. Introduction

Dielectric andmagnetic particulate composites have become thepri-mary choice for manufacturing electromagnetic devices [1,2] due to thepotential they offer in tailoring a material with desired properties [3,4].Dielectric insulators for capacitors [5,6] and magnetic cores for induc-tors [7] are two exampleswhere interest in such composites has recent-ly risen, driven by the requirement for higher performance to achievefurther miniaturization. Particles with characteristic length rangingfrom nm to μm are generally sought for preparing a composite materialin the μm to mm range.

Average properties obtained by homogenization, called effectiveproperties [8], have been introduced for better characterization of themacroscopic response of such composites. For the electromagnetic(EM) applications mentioned previously, estimating the effective elec-tric permittivity ϵ⁎ (in F/m) and the effective magnetic permeability μ⁎(in H/m), is of prime importance to predict the behavior of the compos-ite and assess the success of its design for a desired application. For a lin-ear isotropic composite, these effective properties are defined as follows[9]

Dh iΩ ¼ ϵ� Eh iΩ ð1Þ

Bh iΩ ¼ μ� Hh iΩ ð2Þ

hi@berkeley.edu (T.I. Zohdi).

whereΩ refers to the volume of the compositematerial, ⟨D⟩Ω representsthe average electric flux density over Ω (in C/m2), ⟨E⟩Ω is the averageelectric field intensity over Ω (in V/m), ⟨B⟩Ω gives the average magneticflux density over Ω (in T), and ⟨H⟩Ω refers to the average magnetic fieldintensity over Ω (in A/m). The averaging operator ⟨∙⟩Ω applies to eachcomponent of a vector field, and is defined for the ith component ofany vector G as hGiiΩ ¼ 1

jΩj ∫Ω GidΩ. Taking the dot product of the left

and right hand sides in Eqs. (1) and (2) leads to the explicit formulation

ϵ� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDh iΩ: Dh iΩEh iΩ: Eh iΩ

sð3Þ

μ� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBh iΩ: Bh iΩHh iΩ: Hh iΩ

sð4Þ

Relative effective properties ϵ�r ¼ ϵ�ϵ0

and μ�r ¼ μ�

μ0are commonly

used, where ϵ0 and μ0 are the permittivity and the permeability con-stants of free space, respectively. Direct evaluations of ϵ⁎ and μ⁎ usingformulations 3 and 4 are not possible due to the complexities in-volved in evaluating the average EM fields over a composite. Severalanalytical formulas have been derived in the past, that provide upperand lower bounds to these effective properties values given the EMproperties and volume fractions of the constituents in the composite.We can cite the widely used Wiener bounds [10] and Hashin–Shtrikman bounds [11] — the lower Hashin bound is popularlyknown as the Maxwell–Garnett approximation [12]. They are com-monly used for first hand estimation during design, before

547B. Patel, T.I. Zohdi / Materials and Design 94 (2016) 546–553

conducting experimental validation. The accuracy of these bounds ishowever limited because they provide a range that increases signif-icantly as the differences in the value of the properties between theconstituents increase. Numerical methods were initially developedto estimate effective mechanical properties [13,14]. Extensions foreffective EM properties were introduced in [9] for sphericalparticle-reinforced composites.

In the present study, we develop a numerical framework to esti-mate the effective EM properties ϵ⁎ and μ⁎ for any given linear isotro-pic particulate composite. Following [9,14], it consists of building anumerical sample of the composite material of interest, and applyingEM field at its interface. Then, Maxwell's equations [15] are solvednumerically over the sample to obtain its EM response and subse-quently compute the effective EM properties. The novelty residesin the use of Yee's scheme [16], a Finite Difference Time Domain(FDTD) scheme specifically designed for Maxwell's equations. More-over, this method is designed to handle composites with multiplephases consisting of a matrix material and multiple types of ellipsoi-dal particles. Illustration of such composite is presented in Fig. 1. Ad-ditionally, parallel implementation of the numerical solver is donefor high computational performance, allowing material designer toconduct simulation on a single computer without requiring excep-tional computational power. Precise consideration of the particlesgeometry in 3D allows for a more accurate estimation than the ana-lytical bounds mentioned previously. Also, consideration of dynamicMaxwell's equations provides enough flexibility to further envisioneffect of external dynamic phenomena. For instance temperaturerise could be significant in the applications of interest due to Joule'seffect, and it has been shown to have considerable effect on the EMproperties of certain thermo-sensible materials [17,18,19]. Thiscould be taken into account by solving energy conservation equationsimultaneously with Maxwell's equations, using for instance a stag-gered scheme for coupled physics as in [20]. Thus, this method isintended to set a basis for the multiphysics design of particulatecomposite intended for EM applications. A possibility that is not of-fered by the analytical bounds.

The organization of this paper is as follows. In Section 2, the pro-posed numerical method to compute the effective EM properties ofparticulate composites is presented. Application to model problemsis realized in Section 3 to validate the method using analyticalbounds and experimental results found in literature. Concluding re-marks are finally drawn in Section 4. Throughout the study, we ne-glect thermal, stress, strain, and chemical effects. Constituents andresulting composites are assumed to be linear, isotropic, and non-dispersive. We consider the particles to be non-overlapping and hav-ing hard shell interface.

Fig. 1. Illustration of the homogenization process for a 4 phases compositemadewith Q=3 types of particles.

2. Numerical method

2.1. Representative volume element

Solving Maxwell's equations using any numerical technique wouldrequire meshing of the composite of interest. To capture the particlesinto the numerical scheme, and take the microstructure of the compos-ite properly into account, a fine enoughmeshwould be required. Usual-ly, a mesh size of a tenth of the characteristic length of a particle isprescribed — this is verified during application to model problems inthe next section. Due to the scale difference between the overall com-posite size and the particle size in our problem, this would result in anenormous amount of unknowns to solve for. For instance, for a cubiccomposite of side 1mm filled with particles of diameter 1μm, a meshof 0.1μm would be adequate, resulting in 104 nodes in each directionand a total number of 1012 nodes. This would obviously be too heavyfor efficient computations — a reasonable amount for computationover a single computer is 106 nodes in total. To overcome this problem,commonly encountered while computing effective properties numeri-cally, the notion of a representative volume element (RVE) has been in-troduced [21,22]. We declare L as the characteristic length of the actualcomposite, l as the characteristic length of the RVE, and d as the charac-teristic length of the particles. A schematic of the different scales in-volved is shown in Fig. 2.

Basically a RVE is a cubic piece, “taken” from the actual compositematerial, that is small enough (l≪L) so that efficient computationcould be carried out and estimated properties could be considered asmaterial point properties, but big enough to properly represent the mi-crostructure (l≫d). Additionally, the volume fraction of all constituentsmust be similar to the actual composite for adequate representation.Also, the particles must be randomly distributed and oriented in orderto simulate randomness and isotropy of the actual composite. In thiswork, the simple RVE generation algorithm known as Random Sequen-tial Adsorption (RSA) [23], is employed. Based on the specified RVE sizel, the number of particle Np

q required to satisfy the desired volumefraction vp

q is computed for each type of particles q. Then, all particlesare added to the RVE cube one by one by generating random position(x1q , r, x2q , r,x3q , r) for the center of gravity and random orientation(α1

q , r,α2q , r,α3

q , r) for the principle axis of the rth particle of q type.For each new particle, a check is performed to determine if overlap-ping occurs with any of the previously added particles using themethod from [24], that basically consist of verifying if two ellipsoidsshare any common volume. If overlapping occurs, the position andorientation are rejected, and the process is repeated. The completeprocess is described in Algorithm 1, and the parameters used are il-lustrated in Fig. 3. We notice that this simple RVE generating algo-rithm limit the total volume fraction of particles to about 0.2 that issufficient for most common composites. To incorporate studies ofcomposites with higher particle volume fraction, advanced RVE gen-erator algorithm could be used, as the ones described in [25]. In suchalgorithm, the particles are not placed randomly in the entire RVEvolume but rather into restricted volume, or cell, in order to use

Fig. 2. Illustration of the different scales involved in RVE size estimation.

Fig. 3. Illustration of the parameters used to describe the rth ellipsoidal particle of type q.

Fig. 4. Domain decomposition of a RVE.

548 B. Patel, T.I. Zohdi / Materials and Design 94 (2016) 546–553

available volume optimally and incorporate larger amount of non-overlapping particles.

The conditions mentioned previously provide only vague indica-tions on the size of the RVE, without giving quantitative dimensions.The usual technique is to start with an arbitrary initial size l that sat-isfies previously mentioned qualitative requirements, estimate ef-fective properties over it, then increase the size by a controlledvalue Δl and repeat until variation of results is less than a specifiedtolerance tolS. For each RVE size l during the size estimation process,computation of effective properties over several RVEs is necessary toobtain statistically significative results. The average effective proper-ties over all RVEs are then considered for size convergence analysis.No actual condition is prescribed on the number M of RVEs. Thegreater it is, the better randomness of the particle repartition in theactual composite would be taken into account. However, it has tobe limited based on available computational power to ensure effi-ciency of the method. Evaluation over each RVE is an independentprocess that is parallelized in the proposed method for improvedperformance. Computation over up toMc number of RVEs is conduct-ed simultaneously, where Mc would be set based on the number ofcores and processors available.

Algorithm 1. RVE generator.

2.2. Maxwell's equations

Let Ω be the total volume of the RVE that is considered, Ωm the vol-umeof thematrixmaterial, andΩp the total volume of all of the particles

in the RVE such that Ω=Ωm∪Ωp (if there are Q types of particles,Ωp ¼

∑Q

q¼1Nq

pΩqp). Also, let's designate by Γ any particle/matrix interface. This is

illustrated in Fig. 4. To compute the electromagnetic response of a RVE,the following system of local Maxwell's equations for linear isotropicmaterials [15,26,27] is to be solved over it to get the E and H fields inΩu(u=morp):

∂ ϵuEð Þ∂t

þ σuE ¼ ∇� H ð5Þ

∂ μuHð Þ∂t

¼ �∇� E ð6Þ

on Γ:

n� Em � Ep� � ¼ 0 ð7Þ

n� Hm �Hp� � ¼ 0 ð8Þ

ϵEð Þm � ϵEð Þp� �

� n ¼ 0 ð9Þ

μHð Þm � μHð Þp� �

� n ¼ 0 ð10Þ

whereσdesignates the electric conductivity (in S/m, it would be close to0S/m for the applications of interest aswe are focusing on dielectricma-terials but it is left here in the formulation for generalization), and theunit normal n at Γ is oriented from Ωp to Ωm (particle to matrix).Then, D and B fields are obtained using the constitutive laws for linearisotropic materials [28]

D ¼ ϵuE ð11Þ

B ¼ μuH ð12Þ

where u=p if we are located inside a particle, or u=m otherwise.

2.3. Yee's scheme

Yee's scheme [16] is implemented to solve the system of Eqs. (5) to(10) numerically. A meticulous explanation of this method and generalcomputational electrodynamics discussions are available in [29,30]. Thisfinite difference scheme uses a particular discretization where alterna-tion in space and time occurs between the components of E and H inorder to provide second order accuracy for both temporal and spatialcomputations. Additionally, it is fully explicit providing thus an efficientcomputational performance. The spatial discretization in 3D is madeusing the so-called Yee cell that is presented in Fig. 5. We call Δx1, Δx2,and Δx3 the spatial discretization step size in direction x1, x2, and x3, re-spectively. Let Δt be the temporal discretization size. We introduce forany function f(x1,x2,x3, t) the notation fi , j ,k

n = f(iΔx1, jΔx2,kΔx3,nΔt).The following centered difference discretization of Eqs. (5) and (6) are

Fig. 5. Schematic of a Yee cell.

549B. Patel, T.I. Zohdi / Materials and Design 94 (2016) 546–553

used in Yee's scheme

H1jnþ12

i; jþ12;kþ1

2¼ H1jni; jþ1

2;kþ12þ Δtμ i; jþ1

2;kþ12

E2jni; jþ12;kþ1 � E2jni; jþ1

2;k

Δx3:

�E3jni; jþ1;kþ1

2� E3jni; j;kþ1

2

Δx2

!

ð13Þ

Fig. 6. Overall flowchart of the proposed numerical method to estimate the effective

H2jnþ12

iþ12; j;kþ1

2¼ H2jniþ1

2; j;kþ12þ Δtμ iþ1

2; j;kþ12

E3jniþ1; j;kþ12� E3jni; j;kþ1

2

Δx1

�E1jniþ1

2; j;kþ1 � E1jniþ12; j;k

Δx3

!

ð14Þ

H3jnþ12

iþ12; jþ1

2;k¼ H3jniþ1

2; jþ12;k

þ Δtμ iþ1

2; jþ12;k

E1jniþ12; jþ1;k � E1jniþ1

2; j;k

Δx2�E2jniþ1; jþ1

2;k� E2jni; jþ1

2;k

Δx1

!

ð15Þ

E1jnþ1iþ1

2; j;k¼

2ϵiþ12; j;k

� σ iþ12; j;k

Δt

2ϵiþ12; j;k

þ σ iþ12; j;k

ΔtE1jniþ1

2; j;kþ 2Δt2ϵiþ1

2; j;kþ σ iþ1

2; j;kΔt

�H3jnþ

12

iþ12; jþ1

2;k� H3jnþ

12

iþ12; j�1

2;k

Δx2�H2jnþ

12

iþ12; j;kþ1

2� H2jnþ

12

iþ12; j;k�1

2

Δx3

0B@

1CA

ð16Þ

E2jnþ1i; jþ1

2;k¼

2ϵi; jþ12;k

� σ i; jþ12;kΔt

2ϵi; jþ12;k

þ σ i; jþ12;kΔt

E2jni; jþ12;k

þ 2Δt2ϵi; jþ1

2;kþ σ i; jþ1

2;kΔt

�H1jnþ

12

i; jþ12;kþ1

2� H1jnþ

12

i; jþ12;k�1

2

Δx3�H3jnþ

12

iþ12; jþ1

2;k� H3jnþ

12

i�12; jþ1

2;k

Δx1

0B@

1CA

ð17Þ

E3jnþ1i; j;kþ1

2¼

2ϵi; j;kþ12� σ i; j;kþ1

2Δt

2ϵi; j;kþ12þ σ i; j;kþ1

2Δt

E3jni; j;kþ12þ 2Δt2ϵi; j;kþ1

2þ σ i; j;kþ1

2Δt

�H2jnþ

12

iþ12; j;kþ1

2� H2jnþ

12

i�12; j;kþ1

2

Δx1�H1jnþ

12

i; jþ12;kþ1

2� H1jnþ

12

i; j�12;kþ1

2

Δx2

0B@

1CA

ð18Þ

EM properties for a composite made with non-overlapping ellipsoidal particles.

Table 1Simulation parameters for two-phase examples.

Parameters Values

Relative electric permittivity (ϵr ,m,ϵr ,p) (1,10)Relative magnetic permeability(μr ,m,μr ,p)

(1,5)

Electric conductivity (σm,σp) (0,0) Sm

Spherical particles radius rp 1μmEllipsoidal particles semi axis (ap,bp,cp) (0.8rp,0.9rp,rp)Initial RVE size l= l0 10rpSize increment Δl 0.1l0Total number of RVE M 32Simultaneous computation Mc 8Tolerance on size variation tolS 3%Tolerance on properties variation tolμ ,ϵ 3%Time step Δt=ΔtCFLElectric field E initial condition (0,0,0) V

m

Electric field E boundary conditions linearly growing up to (1000,1000,1000) Vm

from t=0 to t=1000ΔtMagnetic field H initial condition 0 A/m at t=0

Fig. 7. Representation of RVEs with spherical and ellipsoidal particles.

550 B. Patel, T.I. Zohdi / Materials and Design 94 (2016) 546–553

Because Yee cell, a cubic grid, cannot capture exactly ellipsoidal parti-cle shape, errors known as staircase errors [31] will be introduced at par-ticle/matrix interface. To deal with this problem, we follow [9] andassume that there is no hard jump in properties in the actual compositebut rather a smooth variation due to built up material at the interfacethat is amix ofmatrix and particlematerial. This is taken into account nu-merically by using Laplacian property smoothing [9] at the interface,where any property P is replaced by a smooth spatial representation PS as

PSi; j;k ¼

16

Piþ1; j;k þ Pi�1; j;k þ Pi; jþ1;k þ Pi; j�1;k þ Pi; j;kþ1 þ Pi; j;k�1� � ð19Þ

This omits the presence of hard jump inmaterial properties atmate-rial interface, and remove the requirement to take into account for in-terface conditions. It is also important to mention that Yee's schemeconverges to the actual solution only when the Courant–Friedrichs–Lewy (CFL) condition is met, which restricts the time step such thatΔt≤

ΔtCFL ¼ 1ffiffi3

p mins¼1;3

ðΔxsÞ minu¼m;p

ðμuϵuÞ . Finally, Yee's scheme requires the

specification of boundary conditions for three of the six total compo-nents of E and H combined. In the present problem, the boundary con-ditions must satisfy certain criteria in order for the results computedover a RVE to be valid on the actual macroscale composite. This is de-rived by imposing energy conservation between the two scales as ex-plained originally in [32] for mechanics, which is also extended toelectromagnetism in [20]. It is shown that Dirichlet boundary conditionsgrowing linearly in time satisfy this condition and are used here.We im-pose boundary conditions on the components of E by locating them onthe outer side of the Yee cells on RVE boundary. Boundary values are in-creased up to a constant value at a slow linear rate in order to avoid theintroduction of oscillating fields. Note that themagnitude of the bound-ary conditions should not influence the final estimation of the effectiveproperties that are supposed to be constants of the composite. Thefieldsare averaged numerically by summing over all nodes and dividing bythe total number of nodes. Time stepping over a RVE is stopped oncethe variation of computed effective properties over two successivetime step fall below a specified tolerance tolϵ ,μ.

2.4. Overall description

An overall flowchart of the method is given in Fig. 6.

Algorithm 2. Computing effective EM properties over a RVE.

3. Applications and discussion

3.1. Validation with analytical bounds

The proposed numerical method has been implemented in Fortranlanguage. It is first applied to problems where two-phase composites

are considered. The numerical results are validated by verifying thatthey fall between the corresponding Hashin–Shtrikman bounds [11] —the tightest known bounds for effective electromagnetic properties oflinear isotropic two-phase composites — that are expressed as

P1 þ v21

P2 � P1þ 1� v2

3P1

≤P�≤P2 þ 1� v21

P1 � P2þ v23P2

ð20Þ

where P=ϵ or μ, and for each property the subscript 2 refers to thema-terialwith the highest property value (in our examples it is the particles'property).We consider two types of microstructure, with one being theparticular case of spherical particles. A sample RVE for each one isshown in Fig. 7. A list of all the parameters used is given in Table 1. Ma-terial properties are intentionally set close to obtain very narrow rangefor the bounds and validate the method properly (in an actual design,the composite would probably be made with polymer matrix withϵr ,m and μr ,m close to 1, and dielectric or magnetic particles with ϵr ,pand μr ,p around 1000). Mesh refinement tests have been conducted be-fore hand to estimate the required mesh size to ensure convergence ofthe numerically computed E and H fields in Yee's scheme. It was

found that convergence occur for mesh size of around minðap ;bp ;cpÞ5 . This

value is thus used here. Evolution of relative effective electromagneticproperties for various volume fractions of particles vp is presented inFig. 8. We observe that numerical results successfully lie within thebounds defined by Hashin and Shtrikman.

To illustrate the validity of the method for composites made withseveral types of particles, it is also applied to a three-phase compositecontaining simultaneously the two types of particles used in the previ-ous examples. A sample RVE is shown in Fig. 9. The best known bounds

Fig. 8. Evolution of numerically estimated εr⁎ and μr⁎ for different volume fraction of particles, along with analytical Hashin and Shtrikman bounds.

551B. Patel, T.I. Zohdi / Materials and Design 94 (2016) 546–553

for these types of microstructure are Wiener bounds [10], expressed as

vmPm

þ vspPspþ vepPep

!�1

≤P�≤vmPm þ vspPsp þ vepP

ep ð21Þ

where P=ϵ or μ, and superscripts s and e indicates properties of spher-ical and ellipsoidal particles, respectively. Material properties used arelisted in Table 2, and all other parameters are chosen similar to theone listed in Table 1. The volume fraction of ellipsoidal particles isfixed to vp

e=0.05. We show the evolution of relative effective electro-magnetic properties for various volume fractions vps of spherical parti-cles in Fig. 10. The results again pass the analytical bounds testsuccessfully.

3.2. Validation with experimental data

Finally, numerical results are compared to experimental data from[33] where the effective electric permittivity of a composite made bydoping polyethylene (PE)with BariumTitanium (BaTiO3) spherical par-ticles is measured. The permittivity of matrix and particles as well asparticles size are listed in [33]. The properties values not given in [33]were taken from various other sources. This is presented in Table 3.All other numerical parameters are chosen similar to the one listed inTable 1. Comparison between experimental and numerical results isshown in Fig. 11. We can observe that better estimation than analyticalbounds is provided by the proposedmethod. However, differences withexperimental data are noticeable, especially with increasing volumefraction. This could be due to several factors. Approximation of interfaceby smooth variation of properties rather than hard jump could reduceaccuracy. The higher the volume fraction of particles, the higher is thenumber of interface, resulting in a higher overall error in effective prop-erties estimation. This could be remedied by using alternate scheme for

Fig. 9. Representation of a RVE with both spherical and ellipsoidal particles.

Maxwell's equations, able to capturemore accurately particle geometry,such as the one described in [34] and [35]. Due to the complexity ofthese methods, we would however loose the simplicity and efficiencythat motivated the choice of Yee's scheme. Moreover, it has beenshown that interfacial layers that appear around particles for certainmaterials play an important role in macroscopic physical properties[36]. This is not taken into account in the numerical method and couldlead to increasing error as particle volume fraction increases. Finally, ex-ternal phenomena, such as temperature rise or material deformation,could play a significant role. However, they are not currently takeninto account in the numerical modeling.

3.3. Benefits of parallelization

All the results shown previously were obtained on a single laptopwith 2.6 GHz Intel Core i7 processor containing a total of 8 cores. Allcores were used for best performance during computation of effec-tive properties in the previous examples. To assess improvementprovided by parallelization, we provide in Table 4 the central pro-cessing unit's (CPU) processing time required to compute effectiveproperties using one core (no parallelization) to all 8 cores over 32different RVEs of same size l0 for the two-phase spherical particlecomposite case studied previously. We see that the computationtime is significantly decreased with increasing the number of cores,justifying the parallel implementation.

4. Conclusions

Driven by the requirement to design particulate composites for elec-tromagnetic applications, a numerical method to estimate the effectiveelectromagnetic properties has been presented in this study. The overallmethod is designed toworkwith linear isotropic compositesmadewithlinear isotropic non-overlapping ellipsoidal particles. Multi-phase com-positeswith several types of particles, varying in geometry andmaterialnature, can be taken into account. Parallelization of a portion of the nu-merical solver has also been achieved decreasing the computationaltime significantly. The overall numerical implementation is realizedsuch that the computation could be conducted on a single computer.The method has been tested on several composite microstructures,and the numerical results showed great agreement with the corre-sponding analytical bounds, ensuring validity of the method.

Table 2Material properties used for three-phase composite simulation.

Material parameters Values

Relative electric permittivity (ϵr ,m ,ϵr ,ps ,ϵr ,pe ) (1,10,7)Relative magnetic permeability (μr ,m ,μr ,ps ,μr ,pe ) (1,5,2)Electric conductivity (σr ,m ,σr ,p

s ,σr ,pe ) (0,0,0) S/m

Table 4CPU time required to compute effective electromagnetic properties over 32 different RVEsof same size.

Number of coresused

CPU time(s)

Time reduction compared tonon-parallelized computation (%)

1 5438 02 2801 484 2132 618 1685 69

Table 3Parameters used for validation with experimental data.

Parameters Values

Relative electric permittivity (ϵr ,m,ϵr ,p) [33] (1,72.5)Relative magnetic permeability (μr ,m,μr ,p) [37] (1.2,3)Particle radius rp [33] ≈50μm

Fig. 10. Evolution of numerically estimated εr⁎ and μr⁎ for different volume fractions of spherical particles vps , along with analytical Wiener bounds.

552 B. Patel, T.I. Zohdi / Materials and Design 94 (2016) 546–553

Comparisonwith experimental data from literature showedmore accu-rate results with the proposedmethod compared to the same analyticalbounds.

Thismethod is setting a basis uponwhichmodifications could be re-alized to successively improve accuracy, expand to larger type of com-posites, and consider effect of external physical phenomena, in orderto build the ideal design tool. An alternate scheme for Maxwell's equa-tions, able to capture more accurately particle geometry can be imple-mented to potentially improve the accuracy of the results, especiallyat higher volume fraction. Factors such as temperature variation or in-terfacial layer at particles surface have been shown to influence the ef-fective properties significantly, and taking them into account wouldfurther improve accuracy of the numerical estimations. Because dynam-ic Maxwell's equations have been considered here, these dynamicphenomena could be taken into account simply by adding relevantgoverning equations along with adequate numerical method to solvethem. Extension of the method to larger type of composite could berealized. For instance, particle shape other than ellipsoidal could be con-sidered by incorporating adequate RVE generating and non-overlapping

Fig. 11. Evolution of the numerically estimated effective electric permittivity εr⁎ alongwithanalytical bounds and experimental results from [33].

check methods. Moreover, adaption to anisotropic materials could beenvisioned by considering properties to be represented as tensor quan-tities instead of scalars. Each effective property would then have nineunknowns that could be estimated using nine independent numericaltests similar to the one described in this work. Finally, combination ofthis method with an optimization technique, to determine optimal pa-rameters for a composite to achieve desired effective EM properties,would provide a complete design tool. This is all currently under consid-eration by the authors.

Acknowledgments

The authors gratefully acknowledge the financial support of KingAbdullah University of Science and Technology (KAUST) during this re-search. The authors also thank Aashish Ahuja, Zeyad Zaky, and RishiGaneriwala for their assistance in the revision of the paper.

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