Liliana Borcea, and George C. Papanicolaou y. October 2, Abstract. We show that the eective complex impedance of materials with conductivity and

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1 Network Approximation for Transport Properties of High Contrast Materials Liliana Borcea, and George C. Papanicolaou y October, 996 Abstract We show that the eective complex impedance of materials with conductivity and dielectric permittivity that have high contrast can be calculated approximately by solving a suitable resistor-capacitor network. We use extensively variational principles for the analysis and we assess the accuracy of the network approximation by numerical computations. Contents Introduction Quasistatic Electromagnetics and Associated Variational Principles. The Quasistatic Equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :. The Eective oroverall Impedance : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.3 The Variational Principles : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.4 The High Contrast Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 3 Asymptotic Analysis of High Contrast Problems 7 3. Review of Eective Conductivity Calculations for Some Special Two Component Composites : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3. Local Analysis of Static Transport Properties of a High Contrast Continuum : : : : 3.3 The Eective Quasistatic Impedance of a High Contrast Continuum : : : : : : : : : 4 4 High Contrast Analysis Based on Variational Principles 0 4. The High Contrast Approximation in the Static Case with One Channel : : : : : : : 4. High Contrast Approximation in the Static Case with Many Channels : : : : : : : : The Eective Quasistatic Impedance of a High Contrast Continuum. Case A: Series Connection : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The Eective Quasistatic Impedance of a High Contrast Continuum. Case B: Parallel Connection : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : High Contrast Approximation in the Quasistatic Case with Many Channels : : : : : 35 Scientic Computing and Computational Mathematics, Stanford University, Stanford, CA 94305, borcea@sccm.stanford.edu y Dept of Mathematics, Stanford University, Stanford, CA 94305, papanico@georgep.stanford.edu 0

2 5 Numerical Computations Static Fields and Resistive Circuits : : : : : : : : : : : : : : : : : : : : : : : : : : : : Quasistatic Fields and Resistor-Capacitor Circuits : : : : : : : : : : : : : : : : : : : 39 6 Summary and Conclusions 43 Acknowledgements 44 Introduction The overall electric or other transport properties of materials whose conductivity, dielectric permittivity, etc undergo large variations over short distances are dicult to calculate both analytically and numerically. In this paper we show that in many situations the overall or eective transport properties can be approximated by those of a resistor-capacitor network. High contrast electromagnetic or hydraulic transport problems arise very frequently in geophysical applications and they are a serious obstacle in solving inverse problems [3], [], [3]. This is because most imaging methods use some form of linearization of the unknown medium parameters about a uniform background (Born approximation), which cannot be done in high contrast situations. Understanding how high contrast in material properties aects overall transport behavior is an essential rst step in improving imaging techniques [8]. One of the earliest high contrast problems that was solved analytically, by Keller [8], is the eective conductivity of a periodic array of perfectly conducting cylinders or spheres in a poorly conducting background. Keller's results were extended by Batchelor and O'Brien [4] to random arrangements of highly conducting grains in, or nearly in, contact with each other and immersed in a connected uniform matrix. Conductivity problems with checkerboard geometry were analyzed by Keller in [9]. Network or discrete circuit approximations have been used extensively in the past as a modeling tool for high contrast problems [3,4,33,36,37,] but the relationship to the underlying continuum problem is not made precise. Kozlov [34] rst formulated and analyzed high contrast conductivity problems using variational methods when the approximating network consists of one resistor only. He formulated the calculation of the eective conductivity as a cell problem in homogenization of periodic structures [5,7]. High contrast continuum problems are especially important in connection with imaging [8] where the material properties are not known and it is convenient to assume that the high contrast behavior arises in a simple, generic manner such as in a continuum. Variational principles are used extensively to get bounds for the eective properties of heterogeneous and composite materials [35] (and the references cited there). However, these bounds deteriorate rapidly with high contrast because they do not contain accurate enough geometrical information about the approximating network. In the case of complex coecients variational principles for the eective properties were obtained recently [,6] and they were used for bounds in [9]. They were

3 used for high contrast ow problems (large Peclet number eective diusivity) and elsewhere in [6,7,8]. In this paper we consider general network approximations using again variational principles, including the ones for quasistatic transport problems where the equations have complex coecients. We formulate the eective transport problem as in [34] both in the static (real coecients) and quasistatic (complex coecients) cases. We identify the approximating network of resistors in general, both in the static and in the quasistatic case, when there are many regions of ow concentration due to high contrast. We also present the results of extensive numerical simulations that assess the accuracy of the network approximation as a function of the contrast. We discuss our results and related issues in the Summary and Conclusions at the end of the paper. In section we formulate the calculation of the the eective conductivity and dielectric permittivity as a cell problem in the context of homogenization. We also introduce the variational principles which we use extensively in the analysis. In section 3 we review results for the eective conductivity of some two component periodic structures in order to motivate the results for the continuum problem, which are introduced in a heuristic, physical way in this section. In section 4 we give a detailed analysis of the network approximation based on the variational principles. In section 5 we present numerical computations that help establish the range of validity of the network approximation as a function of the contrast. Quasistatic Electromagnetics and Associated Variational Principles. The Quasistatic Equations We consider the system of Maxwell's equations in a domain that contains no free charges or current sources, with electric and magnetic elds that have the form E(x;t) = real(^e(x)e,i!t ); (:) H(x;t) = real( ^H(x)e,i!t ); (:) where! is the frequency of oscillation. The complex valued eld amplitudes ^E and ^H satisfy the system r ^H = ^E, i!"^e r^e = i! ^H (.3) r("^e) = 0 r( ^H) = 0; where (x) is the electric conductivity, "(x) is the dielectric permittivity and (x) is the magnetic permeability.

4 The system of equations (.3) takes into account the conducting, dielectric and magnetic properties of the material. In conducting materials at low frequencies we have!" and the dielectric term in (.3) can be neglected. In this case only the magnetic and conducting properties are important. However, if the material is a poor conductor (e.g. glass, semiconductors, etc.) we cannot neglect the dielectric term. In fact, at low frequencies the magnetic properties of such materials are negligible. For example, if the size of the domain is km and the relative dielectric permittivity and relative magnetic permeability are of order one, the magnetic properties of the material will be negligible for frequencies up to hundreds of khz (see for example [7]). Thus, the quasistatic approximation in dielectric media leads to the simplied system of equations which we consider throughout the rest of the paper ^E = ( + ic)r ^H r ^E = 0 (.4) r("^e) = 0 r( ^H) = 0: The rst equation in (.4) involves the complex impedance + ic which consists of a resistance (real part) and a capacitance or capacitive reactance (imaginary part), where (x;!)= +! " ; (:5)!" C(x;!)= +! " : (:6) By combining the rst two equations in (.4) we obtain r[( + ic)r ^H] =0: (:7) In two dimensions, the curl operator r is equivalent to the perpendicular gradient operator r? =(,@ y ;@ x ) and since r ^H is a current density we use the notation r ^H = j(x). Equation (.7) becomes therefore r? [( + ic)j] = 0 rj =0: (.8) Equations (.7) or (.8) can be used for composite materials with ne scale structure. In this context the microscopic elds can be averaged leading to an eective medium describing the large scale behavior of the material. In materials with periodic structure [5] the eective impedance of the composite can be obtained by solving (.8) over a period cell when the average current over the cell is equal to a unit vector. Even though the method we describe in this paper is quite general, we will present it in the context of the periodic homogenization problem in two dimensions. Therefore we take < j >= e ; (:9) 3

5 y r? w(x; ) = r? w(x; 0) < r? w>= e ; r? w(;y)=r? w(0;y) 0 Figure.: Computational domain x where < j > stands for the integral of j over the domain, normalized by the area of and e j ;j=; are unit vectors in the coordinate directions. According to (.8) the eld j is divergence free and so it can be expressed as the perpendicular gradient of a complex scalar function w(x), j(x) =r? w(x). The scalar function w(x) is the component of the magnetic eld perpendicular to the plane of the domain (i.e. H(x) =w(x)e 3 ). With this notation the system of equations (.8) becomes r? [( + ic)r? w] = 0 < r? w> = e ; (.0) where (x);c(x), and r? w(x) are periodic functions in the domain as shown in Fig... Equations (.0) dene the direct cell problem. Independently, by applying the divergence operator to the rst equation in (.4), and recalling that ^E is curl free so that ^E =,r is the gradient of the potential, we obtain the dual cell problem: r[ + ic r(x)] = 0 < r(x) > = e : (.) We will concentrate on the direct problem and rewrite the complex system of equations (.0) as a system of real equations r? [r? u, Cr? v] = 0 r? [Cr? u + r? v] = 0 (.) < r? u> = e < r? v> = 0; 4

6 where we separated the function w into its real and imaginary parts w(x) = u(x)+ iv(x). It can be shown that if and C are bounded functions and either one of them is strictly positive, the system of equations has a unique solution up to an additive constant. When is strictly positive, it can easily be shown that (.) is elliptic and the Lax-Milgram lemma applies. However, we can rotate the system by 90 degrees in the complex plane (u; v) and notice that C > 0 is also a sucient condition for the ellipticity of (.). The positivity of has in fact a physical meaning as shown in []. The resistive part () of the microscopic impedance should always be greater than or equal to zero so that the dissipation energy averaged over a period of oscillations is positive E dissip = (j real(^e) j + j imag(^e) j ) 0: (:3) In our analysis, both and C are greater than or equal to zero, with the constraint that they cannot vanish at the same location in.. The Eective or Overall Impedance The eective impedance associated with the direct problem is the average ux in the direction of the driving =< ( + ic)(r? u + ir? v) e >: (:4) From equations (.) we conclude that the imaginary function v is periodic in, whereas the real function u can be written as u(x) =,y + (x), where (x) isa periodic function in. The identity r? (a) =a r? + r? a; valid for any vector a and scalar function, and the periodicity of; C; v, and give Consequently, < ( + ic)(r? u + ir? v) (r? + ir? v) > = < r? [( + ic)(r? u + ir? v)( + iv)] >, (.5) < ( + iv)r? [( + ic)(r? u + ir? v)] > = 0: =< ( + ic)(r? u + ir? v) (r? u + ir? v) >: (:6) From (.6) we obtain expressions for the eective resistance and capacitive reactance R = real() =<jr? u j >, <Cr? u r? v>, <jr? v j >; (:7) C = imag() =<Cjr? u j > + <r? u r? v>, <Cjr? v j >: (:8).3 The Variational Principles The real and imaginary part of the eective impedance can also be obtained by using the following saddle-point variational principles R = min <r? u>=e max <r? v>=0 (< jr? u j >, <Cr? u r? v>, 5

7 C = min <r? u>=e max <r? v>=0 <jr? v j >); (.9) (< Cjr? u j > + <r? u r? v>, <Cjr? v j >) (.0) This is because the Euler equations for (.9) and (.0) coincide with equations (.). Variational principles of this and related forms were introduced by Cherkaev and Gibiansky [] and by Fannjiang and Papanicolaou [6]. Besides these variational principles of saddle point type, this problem also has variational principles of Dirichlet type (min min or max max). In general variational principles of Dirichlet type are preferred to saddle point ones in obtaining bounds on the eective parameters and especially in numerical computations. However, the Euler equations associated with them are dicult to use because they are in terms of both current and potential gradients and the coecients are nonlinear functions of of and C. We chose to work with the saddle point variational principles (.9) and (.0) because they are quite suitable to our particular problem of estimating the eective impedance of high contrast media. This is because an asymptotic analysis performed on the simpler system of Euler equations (.) can provide a good guess of the trial elds j R and j I. For example, in order to get an upper bound on the eective resistance R from (.9) we can use such a guess for the real part of the current. Then we are left with a single maximization problem. In general this problem will be dicult to solve but in our case the maximization is equivalent to the minimization occurring in the computation of the eective real conductivity of isotropic media for static elds. The later is a problem we can solve asymptotically (see x4) and therefore we can obtain an upper bound on R. A similar strategy is used in obtaining a lower bound on R. This procedure is used in the proofs we present and works well for high contrast media..4 The High Contrast Model Our aim in this paper is to study the behavior of the eective resistance and capacitance when the microscopic resistance and capacitance have logarithmic high contrast behavior. We assume that the high contrast arises in a simple, generic manner as a continuum and propose the model (x) = 0 e, S(x) ; C(x) = C 0 e, P (x) (.) where S(x) and P (x) are smooth, periodic functions dened on, is a small parameter and 0 ;C 0 are constants which can be viewed as the microscopic resistance and capacitive reactance of a homogeneous reference medium. Model conductivities of form (.) were used by Kozlov [34] and Golden and Kozlov [] for continuum percolation models of media with large variations that are exactly solvable in the 6

8 asymptotic limit of innitely high contrast. Model conductivities of the form (.) were also used in imaging high contrast conductivities from d.c. electric boundary measurements [8]. The high contrast asymptotic analysis is carried out for random resistance and capacitance in the form (.) in [9]. Connections with percolation theory are used in [,0], as well as in [34,6,8]. In the following sections we analyze the behavior of the eective impedance and the current ow in when and C have the form (.) and is small. We can easily see from (.) that when the contrast is high ( ), the resistance will dominate the capacitance when its scaled logarithm S(x) is less than P (x). In fact, and C are comparable in magnitude only when S(x) equals P (x). However, in general one does not expect S(x) to be equal to P (x) over large regions in the domain so we model the medium by alternating regions of dominant resistance and capacitance. Because both and C are assumed to be general continuous functions, the regions are separated by interfaces along which is equal to C. 3 Asymptotic Analysis of High Contrast Problems To understand better the behavior of the eective impedance in high-contrast composites we rst analyze some simpler problems. This will also explain the physical meaning of the results. 3. Review of Eective Conductivity Calculations for Some Special Two Component Composites In the static case! =0wehave from (.5) and (.6) that (x) = ;C = 0 and the medium is purely resistive. The direct cell problem (.0) reduces then to r? (j) = 0 rj = 0 (3.) < j > = e : The current owing in the domain is real and the eective resistance is R =< j j >: (3:) The dual of this problem for the electric potential has from (.) the form r(r) =0 < r >= e ; (3.3) with the eective conductivity given by It is well known (see for example [7]) that = R. =< r r>: (3:4) 7

9 y x Figure 3.: Flow channeling in a high contrast, two component medium r b a Figure 3.: Current ow in a channel Before attempting to solve this problem in the continuous high contrast case we study it rst for some discontinuous two component composites. We consider problems like the one shown in Fig. 3. where the resistance of the material in the period cell can take only two values (for the black region) and (for the white region). Since we are interested in high contrast we assume that. Between the left and right boundaries of the domain we apply a potential dierence r, in the x direction, and study the ow of electric current through the cell. We expect that the current will tend to pass through regions of small resistance. Since we assumed the resistance in the black region ( )tobevery high, the only way the current can pass through the barrier shown in Fig. 3. is by concentrating in the channel near the point. In the rest of the domain the current will be diuse in the white region and will be much smaller than its value near the point. It will be negligible in the black region. The eective resistance of this composite is determined by the resistance (R) of the channel. We expect R to decrease if the 8

10 y c 0 x, c r Figure 3.3: Insulating cylinders in a conducting background width of the channel increases or the radius of curvature of the isolating regions decreases. If the width of the channel is b and the radius of curvature of the isolating regions is a we therefore expect a result of the form R f( a ); where b f is a monotone increasing function. At the point we can take the curvature in the y direction to be k = and the curvature in the b x direction to be k = and a obtain R f( a b )= f( k ): (3:5) k We do not know the exact form of the function f() but we expect it to depend on the geometry of the problem. To illustrate (3.5) we consider some simple geometries. We start with the case shown in gure 3.. We have a conducting channel of resistance embedded in an isolating background ( ). When we apply a unit potential dierence r between the ends of the channel all the current will ow through it. The electric current density isj =, e and its average over the domain is < j >=, be. The eective resistance is computed from (3.) R b= (b, ),b= a, a (, ) dxdy = a b : (3:6) This result agrees with the heuristic form (3.5) if we take the function f(x) =x. The next problem was analyzed by Keller [8]. We consider a periodic array of nonconducting cylinders ( = 0)embedded in a conducting uniform background 0 (see Fig. 3.3). The cylinders have radius a and their axes are at distance c apart. We assume that an electric eld is applied in the x direction and compute the eective resistance of the composite. When the cylinders touch each other it 9

11 r 0 y a, c c a x Figure 3.4: Conducting cylinders in an insulating background is clear that the eective conductivity is zero so we are interested in the case of a narrow channel (i.e., b = c, a small). The electric current concentrates in the channel and the eective resistance is given by R, 0 r a b ; (3:7) where b = c, a is the width of the channel. This result agrees with (3.5) with f(x) = p x which is dierent from the previous case. Another conguration, also analyzed by Keller [8], is shown in Fig We have a periodic array of perfectly conducting cylinders (!)embedded in a uniform insulating background ( 0 ). The cylinders have their axes a distance c apart and their radii are equal to a. When we apply an electric eld in the x direction we have currentowing through the gap between the cylinders. When the cylinders touch each other the eective conductivity is innite so we are interested in narrow gaps. The eective resistance of this composite is R, 0, s c, a a =, 0, s b a ; (3:8) which is the reciprocal of (3.7). These results can be easily extended to cylindrical inclusions of dierent radii. If the conducting cylinders in gure 3.4 have radii a and a, with c the intercenter distance and b = c, (a + a ) the gap, then the eective resistance is given by R, 0, s (a + a )b a a ; (3:9) which reduces to (3.8) when the radii are equal a = a = a. All the cases considered so far conrm the expression (3.5), with dierent forms for the function f(), the common feature being current channeling in regions of high contrast. 0

12 Figure 3.5: The eective resistor network for high-conductivity cylindrical inclusions. If there are several conducting cylinders in the region (Fig. 3.5) then the overall conductivity can be calculated by solving the resistor network, shown in Fig. 3.5, in which each resistance is given by a formula like (3.9). We consider now another two component conguration that is dierent from the previous ones because there are no channels. It is the checkerboard geometry shown in Fig. 3.6 and analyzed by Keller in [9]. The rectangles have conductivities and with. The eective conductivity tensor has its principal axes in the x and y directions and is diagonal. Because, the current ows from one conducting region to another across the corner so the eective conductivity of the composite is controlled by the conductivity of the corner. In fact, the eective resistance is R c h =h, where the conductivity of the corner is c = p. This result is dierent from (3.5). While the factor h =h suggests a geometry dependent function f(x) =x, the absence of a channel leads to an eective resistance depending on the geometric average of the component resistances rather than on the resistance of only one component. Similar asymptotic techniques can be used in the computation of eective conductivity of high contrast discontinuous materials in three dimensions. Keller [8] studied such a problem for a dense cubic periodic array of perfectly conducting identical spheres immersed in a material of small uniform conductivity 0. For a dense packing, Keller shows that most of the current ows across narrow gaps separating adjacent spheres and that the leading term in the asymptotic approximation of the eective conductivity is 0 log(a ); (3:0) h where a is the radius of the spheres and h is the width of the gap. The asymptotic approximation (3.0) agrees with the heuristic result (3.5) stated in the two-

13 y h h x Figure 3.6: Rectangular checkerboard dimensional case and shows that when the spheres touch each other the eective conductivity has a logarithmic singularity. We consider next continuous media with logarithmic high contrast properties. 3. Local Analysis of Static Transport Properties of a High Contrast Continuum We restrict attention rst to a purely resistive problem (equations (3.), (3.)) for a continuous medium with the microscopic resistance (x) = 0 e, S(x). The geometry of the composite is sketched in gure 3.7. The microscopic resistance (x) has two minima at x m and x m,two maxima at x M and x M, and a saddle point x s.atthis point the function (x) is maximum in the x direction and minimum in the y direction. In a small neighborhood of x s the function S has the approximate expression S(x) S(x s )+ k, (x, x s ), k+ (y, y s ) : (3:) The geometry we consider here is the continuous equivalent of the discrete case we presented before and, therefore, we expect the following ow behavior. The eective resistance is given by R (x s ) the driving direction around the saddle point x s. q k + k, with the ow concentrated in We give a heuristic derivation of this result motivated by the previous examples and return in x4 to give a proof based on variational principles. Just like the previous cases the current will be small in regions with high resistance and in particular in

14 y x M x m x s x m x M 0 Figure 3.7: Resistance (x) with a single saddle point in the domain x the vicinity of the maxima x M and x M. Around the minima x m and x m, the bowl shaped resistance function (x) will cause the ow to stall. The only way for the current to pass through the highly resistive region created by the two maxima of (x) isby concentrating in a small neighborhood of the saddle point x s. The small neighborhood of the saddle-point acts like a bottleneck for the current ow, so it is sucient to solve the cell problem (3.) in the vicinity ofx s and to look for a current in the x direction. The current is given by j = r? u, where u can be viewed as the magnetic eld in the direction perpendicular to. In order for the current to be in the x direction, u must be a function of y only. In a small neighborhood of x s, where S(x) has the expression (3.), the direct cell problem (3.) simplies to d (y, y s ) dy [exp(k+ ) du ]=0 (3:) dy with the normalization condition (, du )dy = (3:3) jy,ysj dy for some >0 xed. Equation (3.) gives u 0 (y) =,D exp(, k+ (y,y s) ), where D is a constant obtained from (3.3): D q k + =. The solution of the direct problem (3.) is then j q (y, y s ) )e exp(,k+ (3:4) k + 3

15 y I II III L L 0 I L II x x x 3 x x C C C C C x Figure 3.8: Medium layered across the driving direction Figure 3.9: Enlarged view of the interface between two layers and by substituting it in (3.) we obtain the eective resistance s R (x s ) k ys+, k+ exp( k+ (y, y s ), k+ (y, y s ) )dy y s, s s s (x s ) k k, k+ k = (x + s) + k : (3.5), Since the curvatures k + ;k, have dimensions of =L, this result for the eective resistance of the composite agrees with (3.5) with f(x) = x. In fact the result obtained here is more typical because the saddle points are just continuous versions of the channels, gaps or corners occurring in the discontinuous problems. 3.3 The Eective Quasistatic Impedance of a High Contrast Continuum Case A: Series Connection We consider the direct problem (.) with the coecients and C given by (.), dened over a region with the geometry shown in Fig The domain is divided in three regions by curves L and L 0. In regions I and III the resistance dominates and has the saddle points x region I and x 3 region III oriented in the x direction. In region II the capacitance is dominant and has a saddle point x oriented in the x direction. We choose this orientation of the saddles for simplicity. However, the results derived in this section apply to more general orientations, as well. The functions (x) and C(x) are smooth and periodic and they equal each other along L and L 0. 4

16 We apply a driving potential dierence in the x direction and study the ow through the cell. From (.) we see that in each one of the regions of dominant capacitance or resistance the problem reduces to the one studied in x3:. Thus, the current ow is concentrated at the saddle points of and C and is oriented in the driving direction x. The question that remains is how the three regions connect with each other. From Fig. 3.8 we see that it is sucient to study the connection between regions I and II. The connection between regions II and III is similar. Thus, we concentrate attention to the domain shown in Fig From the periodicity iny of the current and the lack of concentration of the vertical ow in both regions I and II we have < j > I =< j > II = e : (3:6) Thus, in region I, where the resistance is dominant, the problem reduces to r? [(j R + ij I )] = 0 rj R = rj I =0 (3.7) < j R > = e ; < j I >= 0; where j R ; j I are the real and imaginary current densities. It is easy to see that the imaginary current is zero, and that the problem reduces to the one studied in x3:. A similar conclusion applies to region II where the capacitive reactance is dominant. Consequently, the ow behavior in regions I and II is exactly the one studied in section x3:. In the vicinity of the curve L the function (x) (C(x)) is continuously decreasing (increasing) with x. Because the ow is driven in the x direction, it must cross the curve L. The current will be diuse in regions of this neighborhood where and C are fairly at, and it will concentrate close to the possible common minima of and C along L. This current concentration is weaker than the one at saddle points because it occurs only along one direction (i.e. along L). There is no concentration in the direction perpendicular to the boundary interface and this leads to a lower order term in the eective impedance. In the asymptotic limit of innitely high contrast the saddle points of and C will dominate and the contribution of other features like the minima along L are negligible. This conclusion is supported by numerical experiments (see x5:) and it is proved in x4 using variational principles. We should point out that the same kind of features can occur in the purely resistive problem and exactly as we concluded above they are negligible in the computation of the eective resistance for very high contrast. The validity of this is again given by variational principles (see x4:). When the contrast is not very high, features other than the saddle points become important in the computation of as is seen in [0]. Thus, for computing the eective impedance we need only consider small neighborhoods of the saddle points of (x) and C(x) in regions I, and II. Near these 5

17 R C Figure 3.0: Equivalent Series Network points we have (x) (x )exp( k+ (y, y ), k, (x, x ) C(x) C(x )exp( k+ (y, y ) ); j x, x j< (3:8), k, (x, x ) ); j x, x j< (3:9) In these small regions we can solve our cell problem by separation of variables, exactly as in x3:: j R j R r k + r j I 0: k + exp(, k+ (y, y ) )e ; j x, x j< exp(, k+ (y, y ) )e ; j x, x j< (3.0) We use the expression (.6) for the eective impedance and obtain (x ) y + x + y, x, y + x + ic(x ) y, x, vu u (x ) t k+ k, k + + ic(x ) k + vu u t k+ k, exp(, k+ (y, y ), k, (x, x ) )dxdy + exp(, k+ (y, y ), k, (x, x ) )dxdy (3.) : This result, together with the observation that the current has the same normalized average over regions I or II as over the entire cell, tells us that our problem is 6

18 y C C C L 0 L x 3 x x III II I II I L C x x C x Figure 3.: Medium layered along the driving direction Figure 3.: Enlarged view of the interface between twolayers asymptotically equivalent to the discrete series circuit r shown in Fig We have k only two network elements: one resistor R = (x ) + placed at x k,, and a capacitor r k C = C(x ) + placed at x. The current passing through R is < j R + ij I > I = e, k, and the current passing through C is < j R + ij I > II = e. Case B: Parallel Connection We consider the direct problem (.) with the coecients and C given by (.), dened over a region with the geometry shown in Fig. 3.. The domain is divided in three regions by curves L and L 0. In regions I and III the resistance dominates and has the saddle points x region I and x 3 region III oriented in the x direction. In region II the capacitance is dominant and has a saddle point x oriented in the x direction, as well. The functions (x) and C(x) are smooth and periodic and they equal each other along L and L 0. We apply a driving potential dierence in the x direction and study the ow through the cell. From (.) we obtain that r? [(j R + ij I )] = 0 for x region I or III r? [C(j R + ij I )] = 0 for x region II; (3.) where j R and j I are the real and imaginary current densities. The ow is driven in the x direction so we have < j R + i j I >= Ke, where K is a complex constant ux. We can rewrite the current densities as j R = real(k)e + r? u; j I = imag(k)e + r? v; where u and v can be viewed as the real and imaginary parts of the magnetic eld. 7

19 The functions u(x) and v(x) are periodic in the domain and satisfy the conditions I < r? u>=< r? v>= 0: From the periodicity inx of both u and v it follows that in each region there is no net ux in the y direction. For the ow inthex direction we have j e dx + j e dx + j e dx = Ce (3:3) which is equivalent to i i II III j R e dx = i e ;i= I;II; or III j I e dx = i e (3.4) X i X i i = real(k); i = imag(k): Equations (3.) with the driving conditions (3.4) are very similar to the problem studied in x3: and give that the ow ineach region of the domain is concentrated at the saddle points of and C, respectively. The remaining question is how the regions connect with each other. We notice that the connection between regions I and II is similar to the connection between regions II and III. Thus, it is sucient to study the ow in the domain shown in Fig. 3.. For simplicity we assume that the average ow over regions I and II is jdx + jdx = e : I II A dierent driving condition with both real and imaginary net uxes does not aect the expression of the eective impedance, it just modies the uxes of current in the channels (saddles). Consequently, we have r? [(j R + ij I )] = 0 < j R > = e < j I > = e (3.5) rj R = rj I =0 x region I; r? [ic(j R + ij I )] = 0 < j R > = e < j I > = e (3.6) rj R = rj I =0 x region II: 8

20 In the vicinity of the curve L the function (x) (C(x)) is continuously decreasing (increasing) with y. In this region the current is diuse and small where and C are fairly at. If for example andchave two maxima along L and a minimum in between, because we have no driving force in the y direction the current avoids the two maxima and is relatively small at the minimum as well. Consequently, the behavior of the ow for this problem is dictated by what happens in regions I, and II, for which we already know the result. This case is a little more complicated than case A because the current is divided between the two regions of our domain. Since the average current over the domain shown in Fig. 3. is e,wemust have + = + = 0: (3.7) We already know that in order to obtain the eective impedance of the regions I and II it is sucient to look only in small neighborhoods of saddle points of and C. In the vicinity of these points we have (x) (x )exp( k+ (y, y ), k, (x, x ) ); j x, x j<; (3:8) C(x) C(x )exp( k+ (y, y ), k, (x, x ) ); j x, x j< (3:9) and by using separation of variables in (3.5) and (3.6) we obtain j R + ij I r k + j R + ij I r k + exp(, k+ (y, y ) )+ir k + exp(, k+ (y, y ) )+ir k + exp(, k+ (y, y ) ); j x, x j< exp(, k+ (y, y ) ); j x, x j<: (3.30) We can use these results in the formula for the eective impedance (.6) and, after performing straightforward computations, we obtain =[R(, ), C ]+i[c(, )+R ]; (3:3) r r k where R (x ) + k and C C(x ) +. In order to determine the uxes i ; i ; k, i =; we look at the potential dierence r(x) which isgiven by k, r(x) =( + ic)(j R + ij I ): (3:3) We multiply equation (3.3) by e and average over the domain to obtain < r e >=U =< ( + ic)(j R + ij I ) e >= : (3:33) 9

21 We can do the same thing for regions I and II. Because r is periodic over the cell and regions I and II have the same boundaries as the cell in the x direction, the average potential dierence for the entire cell equals the average potential dierence over regions I and II. Consequently we have U = = R( + i )=ic( + i ): (3:34) Relations (3.34) and (3.7) form a linear system of four equations and four unknowns with solution = C R + C ; = RC R ; + C (3.35) =, RC R + C ; = R R + C : Using (3.35) in (3.3) we obtain the following expression for the eective impedance = RC R + C + i R C R + C =( R + ic ), : (3:36) This result suggests that the continuous problem is asymptotically equivalent to the discrete network shown in Fig. 3.3, where a resistor R placed at x is connected in parallel with a capacitor C placed at x. The current passing through R is < j R + ij I > I = e + i e, and the current passing through C is < j R + ij I > II = e + i e. 4 High Contrast Analysis Based on Variational Principles In this section we prove that in the asymptotic limit of innitely high contrast the computation of the eective impedance can be reduced to the solution of an equivalent network of resistors and capacitors. The eective impedance is given by where (x) = 0 e, S(x), C(x) =C 0 e, P (x) satises =< ( + ic)j j > (4:) and j is the complex current density which r? [( + ic)j] =0 rj =0 (4.) < j >= e ; 0

22 C R Figure 3.3: Equivalent Parallel Network over the periodic cell. The eective impedance can also be obtained from variational principles of saddle point type (.9) and (.0). The proofs presented in this section assume a logarithmic high contrast of the coecients and C, as! The High Contrast Approximation in the Static Case with One Channel We return to the problem considered in x3: in the static, purely resistive circuit, with (x) = 0 e, S(x). The function S(x) is smooth and periodic. The geometry of the composite is sketched in gure 3.7. The resistance (x) has two minima at x m and x m,two maxima at x M and x M, and a saddle point x s. Near the saddle point wehave S(x) S(x s )+ k, (x, x s ), k+ (y, y s ) : (4:3) Under these assumptions we state the following theorem: Theorem. The eective resistance is given by s k + R (x s ) k ; (4:4), with the ow in the driving direction and concentrated around the saddle point x s. The proof of this theorem is based on the variational principles and it was rst given by Kozlov [34]. We dene the direct cell problem r? (j) = 0 < j > = e (4.5) rj = 0

23 with the associated eective resistance and the dual cell problem with the eective conductivity R =< j j >; (4:6) r( r) = 0 < r > = e (4.7) =< r r>: (4:8) The eective parameters R, and can also be obtained from the Dirichlet-type variational principles R = min <r? u r? u>; (4:9) <r? u>=e = min < r r> (4:0) <r>=e since the Euler equations associated with them are exactly (4.5), and (4.7). Note that in (4.9) we replaced the current density by j = r? u, where u is the magnitude of the magnetic eld which is perpendicular to the plane of. We begin with a lemma which is proved in [7], pp Lemma.The eective parameters R and satisfy the relation R =,. In order to prove Theorem, we will obtain an upper bound for R from (4.9) which we will match with a lower bound for R obtained from the dual variational principle (4.0). If we select from the eligible elds r? u in (4.9), those which are nonzero only in a neighborhood of x s,we obtain the inequality R < min <r? u> =e jx,xsj jr? u j dx; (4:) where < r? u > is the average of r? u in the - neighborhood of the saddle point. From all the possible elds r? u in (4.) we choose those which have the component in the direction e equal to zero, so that u = u(y) only, and therefore we obtain R < xs+ (x s )exp(, k, (x, x s ) ys+ )dx exp( k+ (y, y s ) )( du x s, y s, dy ) dy; (4:) where R y s+ du y s, (, )dy =. We point out that the inequalities above are only asymptotic and depend on the parameter which is xed as! 0. More dy precisely,

24 R < F ; means R F ; ( + o()) with the small order one term tending to zero as! 0 with >0 xed. After doing the x integration, (4.) becomes s R < ys+ (x s ) k min exp( k+ (y, y s ) )( du, u y s, dy ) dy: (4:3) We choose u in (4.3) to minimize the right side. Then u satises the Euler equation d (y, y s ) dy [exp(k+ ) du dy ] = 0 ys+ y s, (, du )dy = : (4.4) dy Equations (4.4) are exactly the same as (3.) and u is given by (3.4), which substituted in (4.3) gives the upper bound s R < (x s ) k ys+, k+ exp( k+ (y, y s ), k+ (y, y s ) )dy = y s, s s s (x s ) k k, k+ k = (x + s) + k ; (4.5), This bound is gotten under the hypothesis that the current r? u is concentrated around x s, and ows in the driving direction. To obtain a lower bound for R we use the dual formulation with the variational principle (4.0) and Lemma. In (4.0) we select elds r which are concentrated around x s and have nonzero components only in the e direction. The minimum over such elds can only give a larger value of the functional in (4.0) and we obtain xs+ )( d ys+ dx ) dx y s, exp(, k+ (y, y s ) )dy; < min, (x s )exp( k, (x, x s ) x s, (4:6) where R x s+ x ( d s, )dx =: The computation for the minimum in (4.6) is exactly the dx same as the one for the direct problem. By solving the Euler equation associated with (4.6) we obtain d dx = q (x, x s ) exp(,k, ) (4:7) k, and hence s <, (x s ) k xs+ + exp( k, (x, x s ), k, (x, x s ) )dx = k, x s, s s, (x s ) k + : (4.8) k k,, 3

25 Using Lemma and (4.8) we see that R satises s = R, < k,, (x s ) k : (4:9) + Combining (4.5), and (4.9) we get the asymptotic behavior of the eective resistance s k R (x s ) + k : (4:0), Since we used a current r? u in the e direction and concentrated around x s to get (4.0), Theorem is proved. 4. High Contrast Approximation in the Static Case with Many Channels We shall now describe how to analyze (3.)-(3.) and (3.3)-(3.4) when (x) has several, say M, saddle points in. With each saddle point we associate an eective resistance R j ;j=; ;...;M given by the right side of (4.4). This comes from the local analysis around a saddle point that was just carried out in Theorem. To nd the currents owing through the saddle points we must specify how the eective saddle resistances are connected to each other. We do this by identifying the network associated with the resistance function (x) as follows. With each minimum of the resistance (x) we associate a node of a network on the torus T ( which is the region ). The nodes are connected to adjacent nodes if there is a path between two nodes that goes over a single saddle. This path is a branch or a bond that connects two nodes. To avoid complicated terminology and notation we will assume that the network that is associated with the resistance is a rectangular lattice network as in the gure 4.. Recall that the network is periodic. Let the index j N denote the set of nodes and I jk be the current from node j to node k that is a neighbor of j, k N j. Then I jk =,I kj and X kn j I jk =0;j N (4:) which is the discrete analog of rj = 0, Kircho's current law for circuits when there are no sources or sinks of current in the network. We denote the resistance of the (j; k) bond by R jk ( and note that R kj = R jk ). With the above notation and Theorem we now conclude that in the multisaddle case the eective resistance (4.9) is bounded from above by R < X jn X kn j R jk I jk (4:) where the factor = is needed because each bond is counted twice. We now minimize (4.) subject to the condition (4.) and the discrete analog of the driving 4

26 k 0 j k j 0 Figure 4.: Periodic resistor network condition < j >= e. If for example e = e, the unit vector in the x direction, we add to the currents I jk satisfying (4.) a constant current I in the horizontal direction so that the net current crossing any vertical line in equals one while the net current crossing any horizontal line is zero. The equation that the minimizing currents must satisfy is from (4.) X jn X kn j R jk I jk I jk =0 (4:3) for every current perturbation I jk that satises (4.) and has no net ow inthe horizontal or vertical direction. The implications of (4.3) are best analyzed by using the dual network which in the case of a rectangular network has the form shown in Fig. 4. with broken lines. We denote by j 0 N 0 the collection of dual nodes and by Nk 0 the collection of dual 0 nodes adjacent to the dual node k 0. The dual network is, of course, the one that is associated with the minima of, (x), the conductivity of the continuous medium. A distribution of currents I jk that satises (4.) and has no net ow inthex or y direction can be written in the form I jk = H k 0, H j 0 (4:4) in terms of a function H j 0 dened on the dual network. Here (j; k) and (j 0 ;k 0 ) are related to each other, as is shown in the gure, and the relation X I jk =0 kn j 5

27 is equivalent to X j loop H =0 which is the sum of H dierences taken over the dual loop surrounding the node j. We now use (4.4) in (4.3) and note that 0= X jn X R jk I jk I jk = X ( X kn j j 0 N 0 (RI) j0 k 0)H j0: (4:5) k 0 N 0 j 0 This basic identity is the analog of integration by parts in the continuum case and says that the sum over nodes of the network can be replaced by a sum over loops of the voltage dierences (RI) or, what is the same, a sum over the dual nodes. But in (4.5) the H j 0 are arbitrary since (4.4) must hold for all I jk subject to (4.). Thus, X k 0 N 0 j 0 (RI) j 0 k 0 =0;j0 N 0 (4:6) which is Kircho's loop voltage-dierence law for circuits. Equation (4.6), the current law (4.) and the driving condition determine uniquely the minimizing currents I jk in the upper bound (4.). To get the asymptotic equality of the eective resistance R of the original problem to the upper bound we just described we must show that the lower bound is the same as the upper bound. From Theorem and Lemma, we know that we have the bound R, < X j 0 N 0 X R, j 0 k 0V j 0 k (4:7) 0 k 0 N 0 j 0 where R j 0 k is the resistance associated with the dual bond 0 (j0 k 0 ) and equals R jk from formula (4.4) for the saddle is associated with the bond (jk) in the network or (j 0 k 0 ) in the dual network (see Fig. 4.). The potential dierences V j 0 k0 are dened on the dual network and satisfy the relation V j 0 k = 0,V k 0 j0 as well as X k 0 N 0 j 0 V j 0 k 0 =0;j0 N 0 (4:8) which says that the potential dierences around a loop (a dual node) vanish. The driving condition in (4.7) is that the net potential dierence across any vertical line must equal one ( driving in the x direction) while the one across any horizontal line is zero. With these conditions on V j0 k we can now minimize (4.7) and we nd, using 0 the basic identity (4.5) again, Kircho's current law for the minimizing potential dierences X jk V jk =0;jN: (4:9) kn j R, But (4.8) and (4.9) are exactly the same equations as (4.) and (4.6) so, up to scaling by the overall eective resistance, it is clear that the upper bound (4.7) is 6

28 the same as the reciprocal of the upper bound (4.) and this proves the asymptotic form of the eective resistance R as the one associated with the resistances (4.4) in the network formed from the minima (nodes) of the resistance function (x). 4.3 The Eective Quasistatic Impedance of a High Contrast Continuum. Case A: Series Connection In this section we consider the problem dened in x3:3 case A. We study the direct problem (.8) with coecients (x) = 0 e, S(x) and C(x) =C 0 e, P (x) dened over a region with the geometry shown in Fig The functions S(x) and P (x) are smooth, 0 and C 0 are constants and is a small parameter. The domain is divided in two regions by a curve L. In region I the resistance dominates ( C), in region II C and along curve L= C. We assume that the resistance (x) has a single saddle x in region I and that the capacitance C(x) has a single saddle x in region II. Both saddles are oriented in the x direction. The current density is divergence free and can be written as j = r? (u + iv), where u and v are the real and imaginary parts of the magnetic eld which is perpendicular on the plane. In x3:3 case A we stated that when we have a driving in the x direction (i.e. < j >= e ), in the asymptotic limit! 0wehave an equivalent series circuit consisting of a resistance R placed at the saddle x connected with a capacitance ic placed at the saddle x. In this section we use the variational principles (.9) and (.0) to prove this result. In the vicinity of the saddle points x and x wehave for x N(x ): (x) (x )exp(, k, (x, x ) + k+ (y, y ) ) C(x) 0; (4.30) for x N(x ): (x) 0 C(x) C(x )exp(,k, (x, x ) + k+ (y, y ) ): (4.3) In order to prove the series connection result we have to show that real() = R and imag() = C. This is done by obtaining matching lower and upper bounds for the eective impedance. For example, in order to obtain an upper bound for real() wechoose some particular real current eld r? u and then perform the maximization over all the eligible imaginary current elds. From the previous 7

29 results, we have a good clue for what choice of a real current to make. The current averaged over the whole domain should be e, therefore the function v(x; y) is periodic in the domain, whereas the real current can be rewritten with the help of a periodic function (x; y) asr? u = e + r?. We note that our domain has in the vertical direction the same boundaries as its component regions I and II and we use the periodicity iny of and v to show that the currents in the x direction averaged over regions I or II satisfy (,u y )dxdy = (, y )dxdy =; (,v y )dxdy =0; A I A I I A II II (,u y )dxdy = I A II II (, y )dxdy =; A I I A II II (,v y )dxdy =0; (4.3) where A I and A II are the areas of regions I and II. The currents in the y direction can have dierent averages as long as they algebraically add to zero. However in this problem both and C have a single saddle point in regions I and II. From the results presented in x3: and x5: we know that all the current is concentrated around these saddle points. Consider for example x, where the function has a minimum in the y direction and a maximum in the x direction. The current will have x component,u y concentrated around x, but the y component u x should be small. The only place where we could have owinthey direction would be away from the saddle point, where according to Theorem it should be negligible. The same conclusion applies to region II. In order to obtain an upper bound on real(), we choose the following real current r? u = 8 >< >: ( q k + ( q k + exp(, k + (y,y ) ); 0) 8x N(x ) exp(, k + (y,y ) ); 0) 8x N(x ) 0 otherwise which is clearly divergence free, and satises (4:33) < r? u>=< r? u> I =< r? u> II = e ; (4:34) where < > stands for the normalized average over the indicated region. We use (4.33) in the variational principle (.9) to obtain real() < max <r? v>=0 [ N(x) (x )e, k, (x,x ) + k + (y,y ) e, k + C du N(x) dy v ydx, <(v x + v y ) >]: k + (y,y ) dx, (4.35) 8