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2 Alied Mathematics and Comutation 27 (2) Contents lists available at ScienceDirect Alied Mathematics and Comutation journal homeage: Stieltjes reresentation of the 3D Bruggeman effective medium and Padé aroximation Dali Zhang a,, Elena Cherkaev b, Michael P. Lamoureux a a University of Calgary, Deartment of Mathematics and Statistics, 25 University Drive NW, Calgary, Alberta, Canada T2N N b University of Utah, Deartment of Mathematics, 55 South East, JWB 233, Salt Lake City, UT 82, USA article info abstract Keywords: Bruggeman self-consistent effective medium aroximation Sectral density function Stieltjes reresentation Padé aroximation Effective comlex ermittivity Two-hase comosite The aer deals with Bruggeman effective medium aroximation (EMA) which is often used to model effective comlex ermittivity of a two-hase comosite. We derive the Stieltjes integral reresentation of the 3D Bruggeman effective medium and use constrained Padé aroximation method introduced in [39] to numerically reconstruct the sectral density function in this reresentation from the effective comlex ermittivity known in a range of frequencies. The roblem of reconstruction of the Stieltjes integral reresentation arises in inverse homogenization roblem where information about the sectral function recovered from the effective roerties of the comosite, is used to characterize its geometric structure. We resent two different roofs of the Stieltjes analytical reresentation for the effective comlex ermittivity in the 3D Bruggeman effective medium model: one roof is based on direct calculation, the other one is the derivation of the reresentation using Stieltjes inversion formula. We show that the continuous sectral density in the integral reresentation for the Bruggeman EMA model can be efficiently aroximated by a rational function. A rational aroximation of the sectral density is obtained from the solution of a constrained minimization roblem followed by the artial fractions decomosition. We show results of numerical rational aroximation of Bruggeman continuous sectral density and use these results for estimation of fractions of comonents in a comosite from simulated effective ermittivity of the medium. The volume fractions of the constituents in the comosite calculated from the recovered sectral function show good agreement between theoretical and redicted values. Ó 2 Elsevier Inc. All rights reserved.. Introduction The Bruggeman effective-medium aroximation (EMA) [6,22] is one of the mixing rules that is widely used in electromagnetics for modeling the effective resonse of heterogeneous comosite materials. Indeed, many hysical henomena in heterogeneous media can be described by the effective medium theory which gives the effective comlex ermittivity of a mixture of two materials with comlex ermittivity and 2. The EMA model is based on an assumtion that the inhomogeneous material is comosed of two tyes of aroximately sherical grains with comlex-valued ermittivity and 2 mixed in the volume fractions f and f. It was shown that the Bruggeman model of the effective ermittivity corresonds to a hierarchical medium with inclusions of very different sizes, so that any two sherical inclusions of similar size are so well searated that the whole assemblage can be viewed as a dilute comosite [27,3]. The derivation of the effective ermittivity is based on the assumtion of self-consistency which allows to relace the material surrounding the inclusions Corresonding author. addresses: dlzhang@math.ucalgary.ca (D. Zhang), elena@math.utah.edu (E. Cherkaev), mikel@math.ucalgary.ca (M.P. Lamoureux) /$ - see front matter Ó 2 Elsevier Inc. All rights reserved. doi:.6/j.amc.2..2

3 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) by a homogeneous matrix material with the same effective ermittivity (f,, 2 ), and use first order aroximation for dilute comosite to calculate the resulting effective roerty. This model is also called self-consistent effective medium aroximation [23,28,3]. Though the model behind the Bruggeman effective medium formula is relatively simle, this aroximation gives a descrition of ercolation henomenon [8], and this is one of the reasons for its wide use in hysics. Indeed, the alications exloiting Bruggeman effective medium formula and sectral reresentation range from characterization of otical and electrical roerties of cermets [2] and modeling linear and nonlinear comosites [3,36] to calculation of the effective resonse of comosite olycrystalline and metamaterials [2,35], dielectric roerties of biological cells [9], and critical behavior of self-similar brine-filled orous rocks [6]. Sectral reresentation of the effective comlex ermittivity of a two-hase comosite was introduced in [3] and used to bound the effective ermittivity of comosites formed from two materials with ermittivity and 2 [,7,26]. It resents the effective ermittivity as a Stieltjes function with secial analytical roerties: FðsÞ ¼ 2 ¼ dlðxþ s x ; s ¼ : ðþ = 2 Here the ositive measure l is the sectral measure of a self-adjoint oerator Cv, with v being the characteristic function of the domain occuied by the first material in the comosite, and C = r( D) (r), where ( D) is the Lalacian oerator. The function F(s) is analytic outside the [,]-interval in the comlex s-lane. The reresentation is valid for other hysical roerties such as electrical conductivity, thermal conductivity, diffusivity and elastic roerties. The secific feature of the sectral reresentation () is that it searates the deendence of the effective ermittivity on the roerties of the comonents from the deendence on the micro-geometry through the comlex variable s. This feature of the analytic reresentation was used to infer information about the microgeometry of the comosite [,2,2,25]. The inverse homogenization method [] allows to estimate the arameters of the microstructure of a comosite using measured effective roerties of the medium. The method is based on the reconstruction of the sectral measure in the analytic Stieltjes integral reresentation (), the sectral measure contains all information about the microgeometry. It was shown that the sectral measure can be uniquely recovered from the measurements of the effective roerty over a range of frequencies [], but the roblem of reconstruction is ill-osed and requires regularization. The inverse homogenization was extended to viscoelastic comosite media in [7]. Regularized method of constrained rational aroximation of the sectral function was develoed in [39] and used in [38] for evaluation of microstructural arameters of the comosite and in [] for modeling of wave roagation in viscoelastic medium. The aim of this aer is to resent a derivation of the analytic Stieltjes integral reresentation for the 3D Bruggeman EMA model which has a continuous sectral density function. Particular cases of the sectral reresentation for Bruggeman medium or the whole formula have aeared already in several ublications [5,2,23,29,32,36]. For instance, Stieltjes reresentation formula for the case of low volume fraction of the first material (less than threshold value equal /3), was used in [5,2,29] to describe results of exeriments. The whole formula has aeared in (9 52) of Ref. [23], in (2) and (3) of Ref. [36], and in Table of [32]. However these references did not rovide the detailed roof of the sectral reresentation formula. Here we resent two detailed derivations of the Stieltjes integral reresentation, one roof is based on direct calculation and the second one (in Aendix) is based on alication of the Stieltjes inversion formula [37]. In the second art of the aer, we use the derived Stieltjes integral reresentation to investigate the inverse homogenization roblem for Bruggeman comosite and to numerically study the efficiency of the constrained Padé aroximation method for the case of continuous sectral density in the Bruggeman model. Assuming that the values of the effective comlex ermittivity are available in an interval of frequencies (which rovides necessary and sufficient data to uniquely recover the sectral density function []), we use a recently develoed constrained Padé aroximation method [39] to construct a low order Padé aroximation to the continuous sectral function. Padé aroximation is constructed solving constrained minimization roblem followed by artial fraction decomosition. Constrained minimization algorithm rovides regularization of the roblem. The resulting aroximation consists of small number of oles accurately aroximating continuous sectral density. The develoed algorithm is alied to the roblem of estimation of volume fractions of the constituents in a metal insulator comosite material. The roblem of estimation of concentrations or fractions of constituents in the mixture is of significant interest in nondestructive testing of comosite materials, it has various biomedical, geohysical, engineering, and material science alications [8,9,2,2,25,38,39]. We assume that the effective comlex ermittivity is given by the 3D Bruggeman EMA analytical model, construct Padé aroximation of the sectral function, and calculate the volume fraction of one of the constituents as zero moment of the sectral function. The results show good agreement between theoretical and redicted values and demonstrate the efficiency of the resented method. The method can be used for constructing an accurate aroximation to the 3D Bruggeman effective medium in numerical alications dealing with sectral reresentation. In articular, it can be used for extraction of the sectral function from effective measurements to use it in rediction of other effective roerties of the same comosite [,] or in rediction of effective roerties of a different comosite with similar toology [23]. 2. Sectral reresentation of 3D Bruggeman EMA model This section resents the derivation of an analytic Stieltjes integral reresentation for the effective ermittivity of the three-dimensional (3D) Bruggeman self-consistent effective medium analytic model which has a continuous sectral density

4 79 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) function. In 3D Bruggeman s symmetric EMA model, the self-consistency condition results in the equation relating the effective ermittivity of a two-hase mixture with ermittivity and fractions of the comonents: f 2 ð f Þ ¼ : Here f is the volume fraction of the first hase with ermittivity, and f is the volume fraction of the second hase with ermittivity 2. Solving Eq. (2) for in terms of comlex ermittivities, 2 and the volume fraction f of the first hase gives ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c ; where c ¼ð3f Þ ð2 3fÞ 2 : ð3þ Here, c can be exressed as a convex combination of ermittivities and 2 as: c = f(2 2 )+( f)(2 2 ). The arameter c in (3) can also be reresented in a symmetric form as c =(3f ) +(3f 2 ) 2, where f and f 2 are fractions of the comonents in the comosite, which shows that the Bruggeman mixing formula is symmetric with resect to the interchanging the inclusion and matrix materials. We choose the ositive sign of the square root in (3) because for real ositive and 2, the effective ermittivity should lie between the values of and 2. When 2 goes to zero, the solution of Eq. (2) can have different behavior deending on the value of the volume fraction f of the hase. In this case the reresentation in (3) becomes =(c + jcj)/ where c =(3f ). The effective ermittivity is zero if the volume fraction f is less than some threshold volume fraction f c, and is strictly ositive if the volume fraction f is bigger than the threshold value f c : ¼ 3 ðf f c Þ=2 if f c < f < ; if < f 6 f c : This critical volume fraction value f c, f c = /3, is the critical filling factor or the ercolation threshold for the hase. The function F(s) for the 3D Bruggeman effective ermittivity model has the following form as a function of arameter s = 2 /( 2 ) on the comlex s-lane: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FðsÞ ¼ ¼ c c 2 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 3 3f 8 2 3f s s s sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 3 3f 9s 2 6ðf Þs ð3f Þ 2 : ð5þ s s 2 We can check that as exected, F(s)? ass?. We assume here that 2, otherwise we consider a symmetric analytic reresentation on a comlex lane of variable t = /( 2 ). Theorem. Let l(x) reresent the sectral measure function. The sectral function F(s) in (5) has an analytic Stieltjes integral reresentation on the comlex lane of variable s, s = x + iy: FðsÞ ¼ A HðmÞ s dlðxþ s x ¼ A HðmÞ s mðxþdx s x ; where the function H(m) is the Heaviside ste function, the arameters m and A are m ¼ f f c ; A ¼ 3m=2 ð7þ and the sectral density function m(x) is ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mðxþ ¼ x 9x 2 6ðf Þx ð3f Þ 2 if x < x < x 2 ; if x 6 x or x P x 2 ; where x ;2 ¼ f ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8f ð f Þ ; 6 x < x 2 < : ð9þ 3 To rove Theorem, we first derive the sectral density formula (8) using Stieltjes inversion formula ð2þ ðþ ð6þ ð8þ mðxþ ¼ lim ImFðx iyþ: ðþ y! In fact, for s and a fixed volume fraction f of one comonent in a two-hase comosite ( < f < ), the quadratic inside the square root of (5) is zero at the oints x ;2 ¼ f ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8f ð f Þ ; 6 x < x 2 < : ðþ 3

5 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) which can be verified using the quadratic formula. We see that for real values of s, the quadratic under the square root of (5) is negative for x 2 (x,x 2 ) and ositive outside. In articular, we can define the square root (and hence the function F(s)) to be analytic on the whole comlex lane, with only a finite cut removing the interval (x,x 2 ), and the oint zero, from the domain of analyticity. To ensure that F(s)? ass?, we have to choose the sign of the square root in (5) to be ositive for real values for s > x 2. By analyticity, as we rotate 8 degrees from real s > x 2 to real s < x 2 through the uer half lane, the square root rotates through 9 degrees, giving a ure imaginary root, with ositive imaginary art. Thus, by continuity we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fðx iþ¼ 3 3f i j9x 2 6ðf Þx ð3f Þ 2 j ; 8x 2ðx x x 2 ; x 2 Þ: ð2þ Noting that only the square root term in (2) gives an imaginary art, we see immediately that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j9x 2 6ðf Þx ð3f Þ 2 j ; 8x 2ðx ; x 2 Þ: ð3þ lim Im Fðx iyþ ¼ y! x 2 Since the term in the absolute value is negative when x 2 (x,x 2 ), (3) can be simlified as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim Im Fðx iyþ ¼ 9x y! x 2 6ðf Þx ð3f Þ 2 ; 8x 2ðx ; x 2 Þ: ðþ On the other hand, for real values of x outside of (x,x 2 ), the square root in (5) is a real number, so that F(x + + i) is real, its imaginary art is zero, and thus lim y! Im Fðx iyþ ¼; 8x R ðx ; x 2 Þ: ð5þ From the Stietjes inversion formula (), the limits () and (5) show that ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mðxþ ¼ x 9x 2 6ðf Þx ð3f Þ 2 ; x 2ðx ; x 2 Þ; otherwise; ð6þ where x, x 2 are given in (9). To comlete the roof of Theorem, the following lemmas will be used to directly derive the Stieltjes integral reresentation formula (6). Lemma. For any comlex number a with jaj >, the integral identity ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi dw ¼ða a w a 2 Þ holds. ð7þ Proof. This is a standard integral identity on the real line, for real a >, as can be verified by Mathematica or a table of integrals. The integrand on the left is analytic in variable a and hence the arameterized integral can be extended to an analytic function on most of the comlex lane, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi only excluding the real interval [,], which is the interval of integration. Similarly, we extend the real root a 2 to the comlex-valued function vðaþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2, choosing a branch cut so ffiffiffiffiffiffiffiffiffiffiffiffiffiffi that it is continuous and analytic on most of the comlex lane, only excluding the interval [,]. ffiffiffiffiffiffiffiffiffiffiffiffiffiffi We choose signs so that a 2 is ositive for real a >, and as a consequence of the choice of branch cut we will have a 2 negative for real a <. In fact, we may note that rffiffiffiffiffiffiffiffiffiffiffiffiffi vðaþ ¼a for all jaj > ; ð8þ a 2 where we use the usual ower series exansion for the square root on the unit disk centered at. Therefore, we have the signs correct and the two sides of Eq. (7) agree for jaj > in the comlex lane. By analyticity, they agree on the entire region on which they are analytic, namely the comlex lane with the real interval [,] removed. This comletes the roof of lemma. h Lemma 2. Let a and b be comlex numbers, a b, satisfying jaj > and jbj >. The integral identity ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 ðw aþðw bþ dw ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi b a a b b 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 holds. ð9þ

6 796 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) Proof. For any comlex numbers a, b with a b, jaj > and jbj >, a artial fraction exansion of the integrand on the left hand side of (9) shows that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 ðw aþðw bþ dw ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z w 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w dw 2 b a w a b a w b dw ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi a a b a 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi b a b b 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi b a a b b 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 ð2þ by Lemma. This comletes the roof of lemma. h Now we are ready to comlete the roof of Theorem. Proof of Theorem. The sectral density function m(x) in(8) can be rewritten as mðxþ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 6ðx f Þ 2 3ðx f Þ 2 ; for x < x < x 2 ; ð2þ x where the values of x, x 2 are given in (9). We now intend to directly evaluate the integral qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 6ðx f Þ 2 3ðx f Þ 2 GðsÞ ¼ mðxþ s x dx ¼ Z x2 x xðs xþ We introduce the variable u = x ( + f)/3, in terms of which the function G(s) can be rewritten as GðsÞ ¼ Z u2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8f ð f Þ 9u 2 u ðu ðfþ=3þðs u ðfþ=3þ du; ð23þ where u = x ( + f)/3 and u 2 = x 2 ( + f)/3. Making a second change of variable with w ¼ 3u= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8f ð f Þ, we obtain GðsÞ ¼ 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 dw: ð2þ w ffiffiffiffiffiffiffiffiffiffiffi f w ffiffiffiffiffiffiffiffiffiffiffi f ffiffiffiffiffiffiffiffiffiffiffi 3s 8f ð f Þ 8f ð f Þ 8f ð f Þ The integral on the right hand side of (2) has the same form of Eq. (9) as in Lemma 2, with f f 3s a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and b ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for < f < : ð25þ 8f ð f Þ 8f ð f Þ Note that a is real, and greater than, since ð f Þ P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8f ð f Þ, and b can be assumed large by suitable choice of the comlex variable s. Analytic continuation will imly agreement between F(s) and the integral reresentation (6) for all s in the domain of analyticity. Hence we can aly Lemma 2 and solve for G(s) as GðsÞ ¼ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffi b a a b b 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 ¼ 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8f ð f 3s ð f 3sÞ 2 ð f Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3s 8f ð f Þ 8f ð f Þ 8f ð f Þ A ¼ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð f 3sÞ 2 8f ð f Þ ð f Þ 2 8f ð f Þ s ¼ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9s s 2 6ð f Þs ð 3fÞ 2 ð 3f Þ 2 ¼ 3 j 3f j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9s s s 2 6ð f Þs ð 3fÞ 2 : ð26þ In order to move the variable s into the square root in the last term on the right hand side of (26), we rewrite this term by denoting the comlex variable s = jsjarg (s) with n =, as in the following vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u s s 2 9 6ð f Þ ð 3f Þ2 t ¼! u s s jsjeðargðsþinþ 9 6ð f Þ ð 3f Þ2 t : ð27þ s s 2 Since lim s? G(s)=in(22), we must choose the branch cut with n = on the right hand side of (27), so that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðsÞ ¼ 3 j 3f j 9 6ð f Þ ð 3f Þ2 ¼ 3 j 3f j 9s 2 6ðf Þs ð3f Þ 2 : ð28þ s s s 2 s s 2 dx: s 2 ð22þ It should be noted that ð 3f Þ¼j 3f j2ð 3f ÞHð3f Þ; ð29þ

7 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) where H is the Heaviside ste function. The function F(s) in(5) can be calculated directly in terms of G(s) in(28) as FðsÞ ¼ 3f Hðf =3ÞGðsÞ: 2s Therefore, we obtain the Stieltjes integral reresentation FðsÞ ¼ 3f Hðf f c Þ 2s Z x2 x mðxþdx s x Here the arameters m and A are given by ¼ A HðmÞ s ð3þ mðxþ dx s x : ð3þ m ¼ f f c ; A ¼ 3m=2; f c ¼ =3: ð32þ Next we will examine the ole at s = for the Stieltjes integral reresentation of F(s) in(3). For a small s, from (5) and (26), we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3f FðsÞ ð3f Þ 2 ð3f Þj3f j ¼ ; ð33þ s s s 2 which is zero when (3f ) is negative (i.e. F(s) has no ole at s = in this case), and is just (3f )/2s when (3f ) is ositive. Thus, in the case where f 6 f c = /3, the Stieltjes integral reresentation FðsÞ ¼ Z x2 x mðxþdx s x holds because F(s) has no ole at the origin. In the case where f > f c = /3, from (33), we have to insert a ole at s = with residue (3f )/2 and obtain FðsÞ ¼ 3f 2s mðxþ s x dx; which agrees with the integral reresentation (3). Therefore, from (3), (3) and (35), we conclude that 8 Z A s mðxþdx >< if f s x c < f < ; FðsÞ ¼ mðxþdx >: if < f 6 f c ; s x where A =(3f )/2, m(x) is given in (8) and (9). This comletes the roof of theorem. h ð3þ ð35þ ð36þ Remark. An alternative roof of Theorem based on the Stieltjes inversion formula () is given in Aendix A. The sectral density function m(x) satisfies the sum rule: Z x2 x mðxþdx ¼ Z x2 x dlðxþ ¼f : ð37þ The sum rule roerty (37) gives the zero moment l of the sectral measure l which can be used to calculate the volume fraction f of one of the constituents from known effective comlex ermittivity of the comosite. 3. Rational (Padé) aroximation In this section we discuss Padé aroximation of the sectral function which allows us to construct accurate aroximation to the continuous sectral density in the Bruggeman effective medium model using a small number of oles. Based on the theory of reconstruction of the sectral measure from known effective ermittivity develoed in [] and the aroach to reconstruction of the sectral measure using rational function aroximation in [38,39], the sectral function l(x) in(6) can be aroximated by a ste function with a finite number of stes, so that the sectral density function m(x) in(6) has the form: mðxþ ^mðxþ ¼ Xq n¼ A n dðx s n Þ; x 2½; Þ; ð38þ where d(x) is the Dirac delta function. It was shown in [39], that the sectral roerties of the related oerator imose constraints on the amlitudes A n and location of the oles s,s 2,...,s q of the delta functions in this sum: 6 A n <, 6 s < s 2 < < s q <. In this case, the function l(x) satisfying (6) can be aroximated by lðxþ ¼ Z x dlðtþ Z x d^lðtþ ¼ Xq n¼ Z x A n dðt s n Þdt; ð39þ

8 798 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) so that lðxþ ^lðxþ ¼ Xq n¼ A n Hðx s n Þ; x 2½; Þ; ðþ where H(x) is the Heaviside ste function. The function l(x) defined for x 2 [, ) is a non-decreasing, non-negative function corresonding to the Stieltjes function F(s). Thus, the aroximation b FðsÞ of the function F(s) is given by FðsÞ b FðsÞ ¼ Xq with constraints n¼ A n s s n ðþ 6 s n < ; 6 A n < ; < X A n < : ð2þ Here s n is the n-th simle ole on the unit interval with ositive residue A n, q is the total number of oles. Note that the effective ermittivity of a comosite material could be frequency deendent; it follows from () that the aroximation of the frequency-deendent effective ermittivity is given by ðxþ ¼ 2 ðxþ½ FðsÞŠ 2 ðxþ Xq " # A n ; s ¼ s s n n¼ 2 ðxþ 2 ðxþ ðxþ : ð3þ The real arameters A n and s n in the reresentation (3) deend urely on the microgeometry of the comosite. The right hand side of () can be viewed as rational function aroximation: bfðsþ ¼ X n A n ¼ /ðsþ s s n wðsþ ; ðþ where the degree of the olynomial /(s) is lower than the degree of the olynomial w(s). Let and q be the orders of olynomials /(s) and w(s) with 6 q, resectively. The rational [, q]-padé aroximation of F(s) can be written as (see []) bfðsþ ¼F ½;qŠ ðsþ ¼ /ðsþ wðsþ ¼ a a s a 2 s 2 a s b b s b 2 s 2 b q s q ; ð5þ where a l (l =,,...,) and b k (k =,,...,q) are the coefficients of real olynomials /(s) and w(s), resectively. Since (38) rovides an aroximation to a function in the unit interval, all the zeroes of the denominator w(s) should be simle and all oles s n of the function F(s) lie in the interval [,). We normalize the olynomial coefficient b = in the denominator w(s); this allows us to model the hysically realizable zero ole of F(s). We further assume that the measurements (x j ) of the effective ermittivity are available at samle frequencies x j (j =,2,...,N). The measured data airs (x j, (x j )) can be transformed to data airs (z j,f j ) in the comlex s-lane: f j ¼ ðx j Þ 2 ðx j Þ ; z j ¼ 2 ðx j Þ ; ðj ¼ ; 2;...; NÞ: ð6þ 2 ðx j Þ ðx j Þ Here f j is the measured value of the function F(s) at the samle oint z j, f j = F(z j ) with N being the total number of data oints. To formulate the otimization roblem for the coefficients a l s and b k s in (5), we require that the constructed aroximation b FðsÞ agreed with the measured values of F(s) at the oints z j. Then Eq. (5) can be written as /ðz j Þ wðz j Þ ¼ a a z j a 2 z 2 j a z b z j b 2 z 2 j b q z q ¼ f j ; j j ð7þ where a l (l =,...,), b k (k =,...,q,k ) are required unknown coefficients. Eq. (7) is equivalent to following system a a z j a z j b f j b 2 f j z 2 j b q f j z q j ¼ f j z j ; ðj ¼ ; 2;...; NÞ: ð8þ Therefore, the system (8) for the unknown real coefficients a l s and b k s of the rational aroximation /(s)/w(s) can be further exressed as the following system Sc ¼ d; ð9þ where z z 2... z f f z 2 f z 3... f z q z 2 z z 2 f 2 f 2 z 2 2 f 2 z f 2 z q 2 S ¼ B A : z N z 2 N... z N f N f N z 2 N f N z 3 N... f N z q N

9 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) c ¼½a ; a ;...; a ; b ; b 2 ; b 3 ;...; b q Š > ; d ¼½f z ; f 2 z 2 ;...; f N z N Š > ð5þ and the symbol [ ] > indicates a transosed matrix. It is clear that in order for the Padé coefficients a l s and b k s to be uniquely determined, the total number of the measurements must be greater or equal to the number of coefficients, i.e., N > + q +. The reconstruction roblem of determining the column vector c =[a,a,...,a,b,b 2,...,b q ] > of real coefficients in (9), (5) is an inverse roblem. It is ill-osed and requires regularization to develo a stable numerical algorithm. In the resent work we solve a constrained minimization roblem described below with constraints given in (2). Let comlex matrix S be S = S R +is I and the vector of data oints d be d = d R +id I where subindices R and I stand for the real and imaginary arts. To construct a real solution vector c of [, q]-padé coefficients for the roblem (9), (5), we introduce a minimization functional T k ðc; d R ; d I Þ with a enalization term which constrains the set of minimizers. The inequalities (2) for the residues and oles of the function F(s) are used to imose constraints for the set of minimizers of the roblem. We reformulate the roblem as a constrained minimization using Tikhonov regularization [33] with the regularization arameter k ( < k < ) to be chosen roerly: min c T k ðc; d R ; d I Þ¼minfkS R c d R k 2 ks I c d I k 2 k 2 kck 2 g c s:t: 6 A n < ; 6 s n < ; < X A n < ; n ¼ ; 2;...; q: ð5þ Here kk denotes the usual Euclidean norm and arameters s n, A n are oles and residues of the artial fractions decomosition () of the reconstructed [, q]-padé aroximation of the sectral function. To find the minimizer of the roblem, we solve the corresonding Euler equation; its solution is given by c ¼ S > R S R S > I S I ki q S > R d R S > I d I : ð52þ Here I +q+ denotes the ( + q +) ( + q + ) identity matrix. After reconstruction of the vector c of the coefficients of rational function aroximation of b FðsÞ, its decomosition into artial fractions (), gives [,q]-padé aroximation of the sectral function F(s). In the next section, we show results of numerical simulations and alication of the technique to the roblem of estimation of volume fractions of constituents in the mixture, which is a articular case of the roblem of extraction of information about structural arameters of comosites. Indeed, it was shown in [5] for a case of discrete sectral density function in analytical reresentation, that the volume fraction f of subdomains occuied by the first material in comosite is a sum of all residues in the analytical integral reresentation: X An ¼ f : ð53þ The reconstructed Padé aroximation, function b FðsÞ, can be used to calculate the volume fraction of the first material in the comosite using formula (53) which gives an aroximation to the sum rule (37).. Numerical examles This section describes numerical exeriments using the inversion technique in Section 3 alied to the roblem of reconstruction of the sectral density of the 3D Bruggeman EMA model derived in Section 2. To demonstrate the effectiveness of the method, we aly it to the roblem of recovering the volume fractions of the constituents from the effective ermittivity of the Bruggeman comosite. 3D Bruggeman EMA model was used to simulate the effective comlex ermittivity of the two-hase comosite material for a range of frequencies, then these simulated values were taken as data for the rational aroximation algorithm which reconstructs oles and residues of the aroximation b FðsÞ. The aroximation ^mðxþ of the sectral density function m(x) at each location of the reconstructed oles s n with corresonding residues A n is calculated using the following formula dlðs n Þ dx ¼ mðs n Þ ^mðs n Þ¼ A n ðs n s n Þ=2 ; ðn ¼ ; 2;...; q Þ ð5þ where s ¼ x ; s q ¼ x 2, the values of x, x 2 are given in (9), and q is the total number of the validly reconstructed oles of sectral function. We model a frequency-deendent metallic articles comosite of magnesium and magnesium fluoride (MgMgF 2 ) using the described 3D Bruggeman EMA model. The frequency-deendent ermittivity of the metallic articles of magnesium (Mg) taken as inclusion material in the comosite, is given by the Drude dielectric model (see [2]): x 2 ¼ metal ðxþ ¼ xðx is Þ : ð55þ Here x is the circular frequency, x is the lasma frequency, s is the relaxation time and i ¼ ffiffiffiffiffiffiffi. The arameter x and the relaxation time s of the dielectric metallic grains of magnesium (Mg) are given in Table. The ermittivity of magnesium fluoride ( MgF 2 ) considered as the background matrix material in the mixture, is taken as a disersionless constant, 2 =.96, it is shown in Table as well. The frequency-deendent values of the effective comlex ermittivity for the

10 7 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) Table Physical arameters of comlex ermittivity of MgMgF 2 comosite. Material s x Material Permittivity 2 Mg s 9. 5 s MgF A n.2 A n.2.. Im(s n ) 2 Re(s ) n Im(s n ) Re(s ) n Fig.. Poles and residues of the sectral function recovered without constraints for 2%Mg-8%MgF 2 mixture with = q = (left) and = q = (right). Sectral density function.5. Comuted (=q=) Comuted (=q=) Analytic Comuted (=q=) Comuted (=q=) Residues Location of oles Fig. 2. Valid oles and residues of the sectral function (left) and the sectral density function m(x) (right) recovered with constraints for 2%Mg-8%MgF 2 mixture ( = q = and = q = ). mixture of MgMgF 2 with magnesium (Mg) inclusions were simulated in a range of frequency: 6 x 6 x = 9. 5 s using described model of microstructure of comosite materials. Two cases of the comosites of magnesium and magnesium fluoride (MgMgF 2 ) with low volume fraction and large volume fraction of the magnesium (Mg) inclusion hase are tested in the numerical exeriments... Results for low volume fraction f ( < f 6 /3 ) The recovered oles and residues of the sectral function for 3D Bruggeman EMA model with volume fraction f =.2 of magnesium (Mg) comonent are shown in the left art of Fig. for the case = q = and in the right art of Fig. for the case = q =. Shown are results of inversion without constraints in the inversion rocedure when no noise was added to the data. It can be seen from the left art of Fig. that there are 9 validly reconstructed oles which are located between.23 and.777 in the unit interval [,) and the other 5 oles located off the unit interval in the comlex s-lane. The left art of Fig. 2 illustrates the recovery of valid oles and residues of the sectral function for 3D Bruggeman EMA model for the case with constraints in the inversion rocess. The volume fraction of the magnesium (Mg) comonent is calculated by the sum of residues corresonding to validly reconstructed oles using formula (53) as f ¼ P A n : with q = 9 for the case when = q = and f ¼ P A n :275 with q = when = q =, resectively. We comared analytically and numerically calculated sectral density functions. The right art of Fig. 2 shows the true and comuted sectral density functions calculated using formula (5). The true and comuted real and imaginary arts of F(s) at 3 data oints are

11 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) A n. A n Im(s n ) 2 2 Re(s ) n 2 Im(s n ) 2 2 Re(s ) n Fig. 3. 3D Bruggeman EMA model of 75%Mg-25%MgF 2 comosite. Poles and residues reconstructed without constraints, Padé aroximations of order = q = 7 (left) and = q = 3 (right). Sectral density function Residues Comuted (=q=7) Comuted (=q=3) Analytic Comuted (=q=7) Comuted (=q=3) Location of oles Fig.. Reconstructed with constraints oles and residues (left) and sectral density function m(x) (right) for 3D Bruggeman EMA model of 75%Mg-25%MgF 2 mixture ( = q = 7 and = q = 3). demonstrated in left art of Fig. 5. The imaginary art of the sectral function F(s) is a non-ositive function in the comlex s- lane..2. Results for large volume fraction f (/3 < f < ) For large volume fraction of the magnesium inclusion hase in the mixture of magnesium and magnesium fluoride (MgMgF 2 ), the sectral function exhibits different behavior comared with the case of low volume fraction of inclusion comonent. There is an additional isolated ole of the sectral function which corresonds to the delta function of the sectral density function at the origin. Fig. 3 shows the recovered oles and residues of the sectral function without constraints in the inversion rocess for different orders = q = 7 (left) and = q = 3 (right) for 75%Mg 25%MgF 2 mixture when there is no noise in the data. The additional isolated ole with amlitude A =.625 is located at s =. There are other 9 validly reconstructed oles of small amlitudes lying in the unit interval between.75 and.992 shown in the left art of Fig.. The right art of Fig. shows the analytical sectral density function and the one numerically reconstructed using Padé aroximation of orders = q = 7 and = q = 3. The volume fraction of the magnesium (Mg) inclusion hase is estimated fairly well using formula (53) calculated as f ¼ P A n : for = q = 7 and f ¼ P A n : for = q = 3. The true effective ermittivity used at data oints and comuted effective ermittivity are illustrated in the right art of Fig Sensitivity analysis The urose of the next series of comutations is to numerically examine the stability of the calculated volume fractions of magnesium (Mg) comonent in the mixture of magnesium and magnesium fluoride (MgMgF 2 ) for the 3D Bruggeman effective medium model. To simulate the noisy data, we used a uniformly distributed random noise calculated as ercentage of the true value of effective ermittivity at each measured oint of samle data in the same range of frequency as

12 72 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) Effective ermittivity True Analytic Comuted (=q=7) Comuted (=q=3) Frequency (Hz) True Re F Re F (=q=) Re F (=q=) True Im F Im F (=q=) Im F (=q=) Im ε Re ε Fig. 5. Left: True and comuted real and imaginary arts of F(s) for 3D Bruggeman EMA model of 2%Mg-8%MgF 2 mixture ( = q = and = q = ). Right: True and comuted real and imaginary arts of for 3D Bruggeman EMA model of 75%Mg-25%MgF 2 comosite ( = q = 7 and = q = 3). Table 2 Volume fractions calculated for the 3D Bruggeman EMA model using Padé aroximation of order = q = 7 at samle data oints with added noise levels of %, 3%, 5%. Noise (%) f =.5 f =.5 f =.25 f =.35 f =.5 f =.55 f =.65 f = described in this section. This sensitivity test was done with a Padé aroximation of order = q = 7. A summary of the sensitivity analysis for the calculated volume fractions of magnesium comonent for MgMgF 2 comosites of various volume fractions of magnesium inclusion hase using data with %, 3%, 5% noise is shown in Table 2. The first row in the table shows the true volume fractions of Mg comonent and the other three rows resent the calculated volume fractions of the Mg hase. The results of comutations show that even with added noise, the recovered volume fractions of magnesium comonent agree with the true values. This demonstrates the stability of the reconstruction algorithm. 5. Conclusion In this aer, we have resented the derivation of an analytic Stieltjes integral reresentation for the 3D Bruggeman effective medium model which has continuous sectral density function. The sectral function containing imortant information about the microgeometry of the medium can be used to find the effective behavior of heterogeneous material, it also can be used to infer information about the comosite s structure. We used constrained rational Padé algorithm [39] to construct a low order discrete aroximation of the continuous sectral density function, such aroximation consists of small number of oles. Using derived Stieltjes integral reresentation, we investigated efficiency of the constrained Padé aroximation method for inverse homogenization for the case of Bruggeman comosite model with continuous sectral density. The erformed numerical exeriments for estimation of the fractions of comonents in a mixture of magnesium and magnesium fluoride demonstrate the effectiveness of the algorithm. Acknowledgments This work was suorted by NSF Grants DMS-589 and DMS/CMG We gratefully acknowledge generous suort from NSERC, MITACS, PIMS, and sonsors of the POTSI and CREWES rojects. The first author is also funded by a Postdoctoral Fellowshi at the University of Calgary. Aendix A. An alternative roof of Theorem based on Stieltjes inversion formula Here we first intend to show that the sectral density function m(x) in(8) is equivalently given by ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mðxþ ¼ x 8f ð f Þ ð3x fþ 2 if x < x < x 2 ; if x 6 x or x P x 2 ; ða:þ

13 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) where h x ;2 ¼ 9 3m 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ð 3mÞð 6mÞ ; 6 x < x 2 < : ða:2þ In fact, for s = x +iy, the comlex sectral function F(s) in(5) can be written exlicitly in terms of x and y as Fðx iyþ ¼ 3 ð 3f Þx ð3f Þy i ðx 2 y 2 Þ ðx 2 y 2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 3 ig ; ða:3þ where and g 3 ¼ ðx2 y 2 ð3f ÞxÞ 2 y 2 ð3f Þ 2 ðx 2 y 2 Þ 2 8g ; ða:þ g ¼ 2yð3f Þðx2 y 2 ð3f ÞxÞ ðx 2 y 2 Þ 2 8yg 2 ða:5þ g ¼ x x 2 y ; g 2 2 ¼ x 2 y ; 2 s ¼ g iyg 2 : ða:6þ Let w denote exression under the square root in (A.3), w = g 3 +ig = jg 3 +ig je ih, where h = arg (w) 2 [,2), so that g 3 g cos h ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sin h ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ða:7þ g 2 3 g2 g 2 3 g2 Thus, the imaginary art of the square root term in (A.3) can be exressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Im g 3 ig ¼ g 2 3 g2 sin h 2k ; k ¼ ; : ða:8þ 2 In the following analysis, it should be noted that the sectral density function m(x) is a real and non-negative function which is defined in the unit interval x 2 [,] by the Stieltjes inversion formula (). In order to obtain the non-negative real-valued function m(x) in the unit interval, the imaginary art of the square root term in (A.3) should be ositive so that we must choose the ositive sine function in (A.8). This requires k = due to the fact that function sin ( + h/2) < when k = in (A.8). Thus, from (A.8), the imaginary art of the comlex function F(x + iy) in(a.3) must have the following form ð3f Þy ImFðx iyþ ¼ ðx 2 y 2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 3 g2 sin h 2 ; ða:9þ where g 3 and g are given in (A.) and (A.5), resectively. Case I: x 2 (,] and volume fraction f 2 (,). For a fixed x, taking limit as y goes to + for the functions g, g 2, g 3 and g in (A.) (A.6), we obtain and lim g y! ¼ ; lim x g y! 2 ¼ x ; lim g 2 y! ¼ ða:þ lim g y! 3 ¼ x ½ð3x ðf 2 ÞÞ2 8f ð f ÞŠ: ða:þ The right hand side function in Eq. (A.) is a concave uward quadratic olynomial in x which has two different roots in the unit interval: x ;2 ¼ h 9 3m 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ð 3mÞð 6mÞ ; 6 x < x 2 < ; ða:2þ where m = f f c, so that and lim g y! 3 < where x < x < x 2 ða:3þ lim g y! 3 > where x < x or x > x 2 : ða:þ We consider searately the two cases given in (A.3) and (A.). It is seen from (A.3) that for x 2 (x,x 2 ), the function g 3 (x,y) < for sufficiently small ositive values of y. Hence, the angle h of the comlex number w must be restricted to the interval h 2 (/2,3/2). Thus, from the fact that

14 7 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) we get lim y! cos h ¼ ; lim sin h ¼ ; ða:5þ y! lim h ¼ : ða:6þ y! Therefore, taking the limit in (A.9) as y goes to +, it follows from (A.6) that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim ImFðx iyþ ¼ y! lim g 2 y! 3 g2 sin h ¼ 2 lim ffiffiffiffiffiffiffi jg 3 j : ða:7þ y! By substituting (A.7) into (), the sectral density function m(x) is obtained as mðxþ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8f ð f Þ ð3x fþ 2 if x < x < x 2 : x ða:8þ For the second case x < x and x > x 2, (A.) gives that g 3 (x,y) > for sufficiently small ositive values of y, so that the angle h of the comlex number w must be in the interval h 2 (,/2) S (3/2,2). Thus, lim y! and we obtain cos h ¼ ; lim sin h ¼ ða:9þ y! lim h ¼ 2l; l ¼ ; : ða:2þ y! By substituting (A.2) into () and (A.7), we obtain for x < x and x > x 2 (l =,): mðxþ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim Im g 3 ig ¼ y! lim ffiffiffiffiffiffiffi jg 3 j sinðlþ ¼: ða:2þ y! From (A.8) and (A.2), we conclude that (A.) holds where x and x 2 are given in (A.2). It should be noted that when volume fraction f = f c = /3, the sectral density function in (A.) has the following simle form ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xð8 9xÞ if < x < 8 x 9 mðxþ ¼ ; if 8 6 x 6 : ða:22þ 9 (A.22) gives a maximal subinterval (, 8/9) [, ] in which the function m(x) is ositive. Next, we consider asymtotic behavior of the imaginary art of the sectral function F(s), i.e., ImF(x + iy) when x?, to analyze the behavior of the function m(x) at the origin. For sufficiently small ositive x (i.e., x ),we introduce olar coordinates x = rcos/, y = rsin/ in the first quadrant where r > and / < / 6 /2. The real and imaginary arts for the comlex number w = g 3 +ig = jg 3 +ig je ih in (A.), (A.5) where h 2 [,2) can be reresented as g 3 ¼½9r 2 6rð f Þ cos / ð3f Þ 2 cosð2/þš=r 2 ; g ¼½ð82ð3f ÞÞr sin / ð3f Þ 2 sinð2/þš=r 2 : ða:23þ ða:2þ Case II: x and volume fraction f /3. In this case, for sufficiently small r, it is seen from (A.23), (A.2) that we must have h 2 (/2,3/2) because g 3 <, and we also have g lim tan h ¼ lim ¼ tanð2/þ; r! r! g 3 < / 6 2 ; ða:25þ so that h =(l +) 2/ with l =,. It should be noted that if l =, i.e., h = 2/, this results in h 2 [,/2) for / 2 (/,/2] which contradicts h 2 (/2,3/2). Therefore, l should be equal to, so that h =2 2/ 2 [,3/2) [,2) for / (/ < / 6 /2). For sufficiently small r, the leading term O(/r 2 ) of the modulus of w, w = g 3 +ig = jg 3 +ig je ih is obtained from (A.23), (A.2) as jg 3 ig j¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 3 ð3f Þ2 g2 ¼ oðr 2 Þ; ða:26þ r 2 the sine function in (A.9) becomes sinðh=2þ ¼sinð /Þ, and (A.9) becomes ð3f Þy ImFðx iyþ ¼ ðx 2 y 2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 3 g2 sinð /Þoðr 2 Þ: ða:27þ

15 D. Zhang et al. / Alied Mathematics and Comutation 27 (2) Therefore, for small ositive x, the leading order term in (A.27) has the form ð3f Þy ImFðx iyþ ¼ ðx 2 y 2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 3 ð3f Þy j3f j g2 sin / ¼ sin /: ðx 2 y 2 Þ r We note that sin/ = y/r and r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 y 2, so that the Dirac delta function can be written as dðxþ ¼ lim y! y x 2 y 2 ¼ lim ða:28þ sin / : ða:29þ y! r Therefore, by substituting (A.28) into (), and taking the limit as y aroaches to +, the non-negative sectral density function at the origin is obtained as mðxþ ¼ 3f j3f j dðxþ dðxþ; ðx Þ: ða:3þ Case III: x and volume fraction f = f c = /3. In this case, the sectral function F(s) has a simle form of FðsÞ ¼ 3 rffiffiffiffiffiffiffiffiffiffiffiffi 9 8 s and (A.23), (A.2) become g 3 ¼ 9 8 cos / ; g r ¼ 8 sin / > ; r < / 6 2 ða:3þ ða:32þ for sufficiently small r. For such a small r, the angle h of the comlex number w = g 3 +ig must be restricted to the interval (,). (A.32) yields to g lim tan h ¼ lim ¼ tan /; r! r! g 3 < / 6 2 ða:33þ so that h =(l +) / with l =,. It should be noted that l cannot be equal to because if l =, i.e., h =2 /, then h 2 [3/ 2,7/) for / 2 (/,/2] which contradicts h 2 (,). Therefore, l should be equal to, so that h = / (/ < / 6 /2). For small r, the leading term O(/r) of the modulus of w, w = g 3 +ig = jg 3 +ig je ih can be calculated using (A.32) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jg 3 ig j¼ g 2 3 g2 ¼ 8 r oðr Þ ða:3þ and (A.9) becomes ImFðx iyþ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 3 g2 sin / 2 o r 2 : ða:35þ Therefore, for small ositive x and y, the leading order term in (A.35) has the form ImFðx iyþ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 3 g2 cos / rffiffiffi 2 8 oðr 2 Þ¼ cos / r 2 oðr 2 Þ: ða:36þ For small ositive x and r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 y 2, the leading term on the right hand side of (A.36) as y goes to + is a bounded integrable function of x, i.e., m ðxþ ¼ Z x lim ImFðt iyþdt 6 y! Z x t / 2 cos 2 2 ffiffiffi 2 ffiffiffiffiffi / dt ¼ 2x cos 2 6 ffiffiffi 2 ; ða:37þ which imlies that there is no delta function at the origin in the reresentation of the sectral density function. From the analysis of Cases I, II and III, we conclude that (36) holds. This comletes the roof of theorem. h Remark 2. An alternative roof of Case II and Case III is given in Aendices B and C, resectively. Aendix B. An alternative roof of Case II Let x =,f /3 and / = /2 in (A.23) and (A.2), then (A.23) and (A.2) become ð3f Þ2 g 3 ¼ 9 < ; g y 2 ¼ 6ð f Þ > ðb:þ y for sufficiently small ositive y. Thus, the angle h of w = g 3 +ig must be in the second quadrant, i.e., h 2 (/2,). Equations in (B.) imly g lim tan h ¼ lim 6ð f Þy ¼ lim y! y! g 3 y! 9y 2 ð3f Þ ¼ 2 ; ðb:2þ