We discuss the conductivity of two-dimensional media with coated neutral inclusions of finite conductivity. Such an inclusion, when inserted in a matrix, does not disturb the uniform external field. We are looking for shapes of the core and coating in terms of the conformal mapping ω(z) of the unit disc onto coated inclusions. The considered inverse problem is reduced to an eigenvalue problem for an integral equation containing singular integrals over a closed curve L1 on the transformed complex plane. The conformal mapping ω(z) is constructed via eigenfunctions of the integral equation. For each fixed curve L1, the boundary of the core is given by the curve ω(L1). The boundary of the coating is obtained by the mapping of the unit circle. It is justified that any shaped inclusion with a smooth boundary can be made neutral by surrounding it with an appropriate coating. Shapes of the neutral inclusions are obtained in analytical form when L1 is an ellipse.
Mathematical models of invisibility are of considerable interest in a number of recent publications. One can find the theoretical foundations and results devoted to various approaches of invisibility due to Kerker (1975), Ammari & Kang (2004, 2007), Alu & Engheta (2005), Milton & Nicorovici (2006), Milton et al. (2006), Farhat et al. (2008), Greenleaf et al. (2009), Guenneau et al. (2010), Liu (2010) and Ammari et al. (2011). In this paper, we investigate this problem in the context of conductivity of two-dimensional media with coated neutral inclusions. When the conductivity σ0 of an isotropic matrix is chosen appropriately, one can insert a coated cylinder or sphere, with core conductivity σ1 and coating conductivity σ, into the medium without disturbing the surrounding unidirectional external field. This effect is described in some detail by Milton (2000). Hashin (1962) constructed infinite packings of the plane by such coated cylinders of various sizes, also without disturbing the surrounding field. Hashin & Shtrikman (1962) established that the effective conductivity of such packings satisfy the famous Clausius–Mossoti approximation. These results were extended to coated ellipses and ellipsoids by Milton (1980, 1981). One can find an extended review with corresponding citations devoted to this problem in the book by Milton (2000) and in the paper by Milton & Serkov (2001).
Milton & Serkov (2001) have posed the question of whether there are neutral coated inclusions other than single and multi-coated circles, spheres, ellipses and ellipsoids. By having used the assumption that the coating surrounds a hole or a perfect conductor, Milton & Serkov (2001) solved the problem for two-dimensional geometry. They discovered many possible shapes of the neutral inclusions. This result was obtained by use of the conformal mapping of the doubly connected coated domain onto a circular annulus. As a result, a boundary value problem for analytic functions arises. The latter problem was solved by the use of Laurent’s series. It is worth noting that such problems were discussed by Schiffer (1959) and by Mityushev (1992) as well as Mityushev & Rogosin (1999) in the context of pure mathematical problems.
Our work is motivated by the question posed by Milton & Serkov (2001), which concerns a core of finite conductivity. Milton & Serkov (2001) noted that their analysis does not fit to this case, since they used a conformal mapping of the coating domain. Thus, the core domain cannot be taken into account apart from its boundary effects. Milton & Serkov (2001) tried to solve this problem approximately by small perturbations of the coated circles.
In this paper, the result of Milton & Serkov (2001) is extended to coated inclusions of finite conductivity represented in figure 1, where G1, G, G2 denote core, coating and the exterior domain; curve Γ1 divides G1 and G; curve Γ2 divides G and G2. We also apply the method of conformal mappings, but, contrary to Milton & Serkov (2001), a conformal mapping ζ↦z of the simply connected domain bounded by the exterior coated boundary Γ2 onto the unit disc is used. This conformal mapping yields a boundary value problem that does not have an easier form than the original one. However, the unknown boundary Γ2 transforms onto the known unit circle.
One of the main results of this paper is that the neutral inclusion problem is reduced to an eigenvalue -linear problem on the auxiliary plane z (see §2b). In §3, the integral equation (3.12) corresponding to the eigenvalue problem is deduced. The solution of this problem depends on the curve L1 dividing the transformed coated and core domains on the plane of variable z. It is worth noting that this eigenvalue problem has solutions for any Hölder continuous curve L1. This observation implies that any shaped inclusion with a smooth boundary can be made neutral by surrounding it with an appropriate coating. Shapes of the neutral inclusions are obtained in analytical form in §§4 and 5 when L1 is an ellipse.
2. Eigenvalue -linear problem
In this section, the problem of neutral inclusion is stated as an inverse boundary value problem for analytic functions with an unknown curve. This problem is reduced to an eigenvalue problem for an integral equation.
(a) Statement of the problem and reduction to -linear problem
Let the plane of complex variable ζ=x+iy be divided onto domains G1, G, G2 (figure 1) by simple closed smooth curves Γ1 and Γ2 oriented in counter-clockwise directions.
For definiteness, it is assumed that the origin belongs to G1 and the domain G2 contains the infinite point. The domains G1 and G2 are simply connected, G doubly connected; G1, G and G2 are, respectively, the core, coated and exterior domains occupied by materials with scalar conductivities σ1, σ and σ0. The potentials u1(x,y), u(x,y) satisfy Laplace’s equation in G1, G and continuously differentiable in the closures of these domains. It is assumed that the potential u2(x,y) in the matrix G2 is a linear function of the form 2.1 where e0=(e1,e2) is the uniform field outside the coated inclusion. Since u1(x,y) satisfies Laplace’s equation in the simply connected domain G1, it is represented as the real part of a function analytic in G1 2.2 where the coefficient in (2.2) is introduced for convenience. The potential in the doubly connected domain G is represented in the form 2.3 where the function is analytic and single-valued in G, and A is a real constant. It will be shown later that A=0. Physically, this comes about because there is no net charge in the inclusion. The potential (2.1) can be also written in the complex form 2.4 where the vector e0 is represented as the complex number e0=e1+ie2; the bar denotes the complex conjugation.
Let n be the unit outward normal vector to the curve Γ2, and let ∂/∂n be the outward normal derivative. Then the perfect contact between materials along the curve Γ2 is described by equations 2.5 Along similar lines 2.6 Following Mityushev & Rogosin (1999), we transform the real conditions (2.5) and (2.6) into complex. Using (2.3) and (2.4), we rewrite the first relation (2.5) in the form 2.7 Let s be the natural parameter of the curve Γ2 and ∂/∂s denote the corresponding derivative along Γ2. Using the Cauchy–Riemann equation ∂u2/∂n=∂v2/∂s, we rewrite the second equation (2.5) in the form of σ∂v/∂s=σ0∂v2/∂s on Γ2 and integrate it along Γ2 on s 2.8 The integration constant in (2.8) is taken to be zero, since the imaginary part of the complex potential is defined up to an arbitrary additive constant. It follows from (2.4) that the imaginary part . Then (2.8) yields 2.9 The two real equations (2.7) and (2.9) can be written in the form of one complex relation: 2.10 The same arguments applied to (2.6) yield 2.11 where the contrast Bergman parameter is introduced as follows: 2.12 This parameter satisfies inequality |ϱ|<1 for finite positive σ1 and σ.
In order to prove that A=0 in the representation (2.3), we calculate the increment [ϕ]Γ1 of the function ϕ(ζ) along the closed curve Γ1. It follows from (2.3) that [ϕ]Γ1=2πiA. On the other hand, [ϕ]Γ1 calculated by (2.11) yields zero. Hence, A=0 in the representation (2.3) and is a single valued function analytic in the doubly connected domain G.
Thus, we arrive at the boundary value problem (2.10)–(2.11) with unknown curve Γ2 with respect to the complex potentials ϕ(ζ), ϕ1(ζ) analytic in G, G1 and continuously differentiable in the closures of the domains considered.
(b) Reduction to eigenvalue problem
The simply connected domain G1∪Γ1∪G can be conformally mapped onto the unit disc U. Let 2.13 denote the inverse conformal mapping of U onto G1∪Γ1∪G. The coefficient in (2.13) is introduced for convenience. Let this conformal mapping be normalized by the condition ω(0)=0. The unit circle L2 transforms, as the boundary of U, onto the curve Γ2, a simple closed curve L1⊂U onto Γ1 (figure 1).
The domains D1 and D are transformed by (2.13) onto G1 and G, respectively, the point z=0 belongs to D1. Introduce the complex potentials analytic in the considered domains D1 and D 2.14 where the variables z and ζ are related by (2.13). The conjugation conditions (2.10) and (2.11) become 2.15 and 2.16 where 2.17 One can see that λ−1 is equal to the second contrast parameter and |λ|>1.
It is convenient to introduce the function for |z|≥1 analytic in |z|>1 and continuously differentiable in |z|≥1. The function φ2(z) vanishes at infinity, since ω(0)=0. Then (2.15)–(2.16) takes the form of the standard -linear condition (Mityushev & Rogosin 1999) 2.18 and 2.19 It is necessary to find a curve L1 for which the problem (2.18)–(2.19) has a non-trivial solution. Then the non-zero function 2.20 can be a candidate for the conformal mapping. The function (2.20) besides being analytic must be a one-to-one map of U onto G1∪L1∪G.
Following Milton & Serkov (2001), it is more convenient to treat the above problem as an eigenvalue problem, with respect to the unknown constant λ and a fixed curve L1 as follows: to find non-trivial functions φ(z), φ1(z) and φ2(z) analytic in D, D1 and D2, respectively, and continuously differentiable in the closures of the domains considered with the -linear conjugation conditions (2.18) and (2.19). The constant λ has to be determined. Moreover, φ2(z) vanishes at infinity. The contrast parameter ϱ is supposed to be fixed in such a way that |ϱ|<1.
3. Integral equation
In the present section, the -linear problem (2.18) and (2.19) is reduced to an integral equation. Let D2 be the exterior of the unit circle. Let the curve L1 and the unit circle L2 be oriented in the counter-clockwise direction. Then the boundaries of the open domains D1, D and D2 are expressed by the relations ∂D1=L1, ∂D2=−L2 and ∂D=L2∪(−L1). Let Ψ+(z) and Ψ−(z) be functions analytic in the domains D and D−=D1∪D2, respectively, and Hölder continuous in their closures. Then, Cauchy’s integral formula implies the following relations described by Gakhov (1966): 3.1a 3.1b and 3.1c
Let the function μ(t) be Hölder continuous on ∂D. The Cauchy-type integral 3.2 represents a function analytic in the domains D, D− and it satisfies the jump condition (Gakhov 1966) 3.3 where Ψ±(z) can be considered as the restriction of Ψ(z) to D∪∂D and to D−∪∂D or as the limit values of Ψ(z) from the different sides of the curves ∂D. One can also consider formula (3.2) as the unique solution to the jump problem (3.3) with due to Gakhov (1966).
The relations (2.18) and (2.19) with 3.4 can be written in the form of the jump condition (3.3) with 3.5 Then (3.2) becomes the system of integral equations 3.6 and 3.7 The relation is fulfilled in the unit circle. The function is analytically continued into the unit disc, since φ2(z) is analytic in |z|>1. Then the first integral in (3.6) can be calculated by Cauchy’s formula: 3.8 Substitution of (3.8) into (3.6) and (3.7) yields 3.9 and 3.10 Substitute instead of z in (3.10) and take the complex conjugation 3.11 The function φ2(z) can be eliminated from the system (3.6)–(3.7), which is reduced to the integral equation 3.12 This integral equation has to be stated as the following eigenvalue problem. To find a non-zero function φ1(z) and a constant λ which satisfies equation (3.12). The function φ1(z) has to be analytic in D1 and Hölder continuous in D1∪L1.
Similar problems were already discussed by Bojarski (1960) and by Schiffer (1959), who considered the eigenvalue problem (3.6)–(3.7) when ϱ=λ. Schiffer (1959) proved that the eigenfunctions of that problem generate a complete orthogonal basis in a Hilbert space; all eigenvalues are real, generate a countable set and their moduli are greater than unity. This result does not fit to the considered case with fixed ϱ satisfying the inequality |ϱ|<1. However, it is possible to extend Schiffer’s result to the problem (3.6)–(3.7) or to the equivalent problem (3.12). This result will be presented in a separate paper with necessary mathematical description of the functional space and singular integral operators.
In order to study the properties of the coatings and its dependence on the core, one can numerically solve the integral equation (3.12) for various L1 and ϱ. Applying polynomial approximations for the function φ1(z), consider some examples of a numerical solution to (3.12). All the computations are performed for the normalized conductivity of the matrix σ0=1. Let the shape of the core of conductivity σ1=3.36 be given by the curve ζ=0.6eiθ+0.31e2iθ+0.07e3iθ+0.013e4iθ (0≤θ<2π). Three neutral coatings with different conductivity σ are presented in figure 2. It is interesting to note that the non-convex of the core have convex and non-convex exterior boundaries of the coatings.
Another example with the boundary curve of core ζ=0.5eiθ+0.35e2iθ+0.12e3iθ+0.06e4iθ (0≤θ<2π) is presented in figure 3. Here, we are looking for similar shapes of the coatings for different σ1. All the found shapes differ locally near the point z=−0.5, where the boundary of coatings approaches to the core with increasing σ1.
4. Functional equation
As noted at the end of §3, it is convenient to consider (3.12) as an eigenvalue problem with a fixed contour L1. Each fixed contour L1 on the plane z produces a neutral inclusion on the plane ζ. In order to constructively solve the integral equation (3.12), one can take an algebraic curve L1 and reduce (3.12) to a functional equation. Any smooth curve can be approximated by an algebraic one. The simplest algebraic curve L1=∂D1 is a circle. One can check that the circle L1 yields the circular annulus on the physical plane ζ described by Milton (2000). In the present section, D1 is taken as an ellipse. This gives other non-trivial shapes of the neutral coated domain.
Let semi-axis of the ellipse L1 be denoted by r(1+α) and r(1−α) (0<α<1, r>0). The Joukowsky conformal mapping 4.1 transforms the annulus onto D1\Γ, where Γ is the slit along the x-axis (figure 4). The mapping (4.1) transforms D′ onto D and D2′ onto D2. The curve L2′ bounds a Bell’s domain (Bell et al. 2009). The inverse mapping to (4.1) has the form 4.2 where the branch of the root is chosen in such a way that for
Let τ run over the unit circle. Then t=r(τ+α/τ) runs over L1. Introduce the function 4.3 analytic in , (z∈D1\Γ) and Hölder continuous in . It satisfies the condition 4.4 Equation (4.4) implies that the function Φ1(w) can be written in the form 4.5 where Φ(w) is analytic in |w|<1.
Let z belong to D1. The integral from (3.9) becomes 4.6 where z=r(w+α/w), |w|<1. The integral (4.6) can be calculated by residues. First, consider the integral 4.7 where the function is analytically continued into |τ|>1. It follows from inequality that all roots of the denominator of (4.7) belong to the unit disc except at infinity. Hence, 4.8 The second part of the integral (4.6) can be calculated by residues at the points τ=0; w; α/w lying in the unit disc: 4.9 since the function is analytic in |τ|<1. The integral (4.6) is equal to J1+J2. Substitution of this result into (3.9) and the use of (4.3) and (4.5) yields 4.10 where .
Now we transform the integral (3.10) written in the form 4.11 where z=r(w+α/w) with |z|>1, which implies that |w|>1. Calculate the first part of the integral (4.11) by residues in |w|>1 4.12 The second part of the integral (4.11) is calculated by residues in |w|<1 4.13 Substitution of (4.12) and (4.13) into (4.11) yields 4.14 for z and w related by (4.1) and (4.2) when |z|>1 and w∈D′2. Introduce the function Φ2(w) analytic in D′2 4.15 The relation (4.15) can be written on the plane z 4.16 It follows from (4.16) that where Therefore, equations (3.9) and (3.10) take the form 4.17 Substitute Φ2 from the second equation of (4.17) to the first one 4.18 where 4.19 This functional equation (4.18) can also be obtained from equation (3.12).
Introduce the operator P+, which transforms Laurent’s series to its regular part. More precisely, let , Then Application of P+ to equation (4.18) yields 4.20 Thus, we arrive at the following eigenvalue problem. To find Φ(w) analytic in |w|<1 and continuous in |w|≤1 with a constant λ satisfying equation (4.20).
5. Solution to functional equation
In this section, we solve the functional equation (4.20) using the polynomial approximation 5.1 with a fixed number M. The operator from the right-hand part of (4.20) is compact in the Banach space endowed with the norm (see Mityushev 1984). Therefore, approximate solutions of (4.20) can be found by replacement of (4.20) with equations of finite rank operators. Substitution of (5.1) into (4.20) and selection of the coefficients on wm (m=0,1,…,M) yields such an equation. Having used these arguments, Mityushev (1984) showed that the polynomial from (5.1) tends to Φ(w) as M tends to infinity.
We also use approximations on r2 (r<1). More precisely, all symbolic computations are performed with the accuracy O(r2(2N−1)). For definiteness, the power 2(2N−1) is taken with an odd number 2N−1. Substitute (5.1) and (4.19) in the functional equation (4.20), and then select the terms with the same powers wm for m=0,1,…,M. As a result, we obtain a system of homogeneous linear algebraic equations with respect to the coefficients αm and the spectral parameter λ. This eigenvalue problem can be solved by a standard scheme. Substitution of the rational approximation from (4.19) into (4.20) instead of f(w) is justified by the compactness of the operator containing f(w). In order to illustrate the algorithm, we take M=4 and N=8. Then the system becomes 5.2 Denote λj, (j=1,…,4) the eigenvalues of the system (5.2). For brevity, only the first eigenvalue λ is explicitly written 5.3 Each λj yields a non-zero solution of the system (5.2). If λ=λ1, then α1 is an arbitrary number, α3=k1α1 and α2=α4=0, where 5.4
If λ=λ2, then α2 is arbitrary, α3=k2α1 and α1=α3=0. If λ=λ3, then α3 is arbitrary, α3=k3α1 and α2=α4=0. If λ=λ4, then α4 is arbitrary, α2=k4α4 and α1=α3=0. Coefficient kj tends to zero as . This means that |kj| is small for sufficiently small r. We do not write them all, because of their long form. The normalized eigenfunctions corresponding to the eigenvalues (5.3) have the form 5.5 The map is a one-to-one if and only if φ2(z) is a one-to-one map. The function φ2(z) is related to Φ(w) by (4.14). Therefore, φ2(z) is one-to-one in |z|>1 if and only if the function F(w)=Φ(α2w)−Φ(w) is one-to-one in the domain (D2′)*, the image of D2′ under the inversion with respect to unit circle (figure 4). The function Φ(w) is one for the functions (5.5). Consider the first function Φ(1)(w). Then F(w)=−w−k1w3+wα+k1w3α3. Let w0 belong to . Then equation F(w0)=F(w) with respect to w0 has the following three roots: 5.6 For sufficiently small r, the roots w2 and w3 lie outside the unit disc. This implies that F(w) is one-to-one mapping in (D2′)*. Similar calculations can be done for F(w) with Φ(j)(w) (j=2,3,4) given by (5.5). In all these cases, F(w) is not a one-to-one mapping.
For instance, α1=1, α3=k11, α4=k12, α2=α4=α6=0 if λ=λ1. The coefficients kij are not written here because of their long form. We only note that |kij| are sufficiently small for small r. This implies that the eigenfunction Φ(1)(w)≈w+k11w3+k12w5 is a one-to-one mapping.
For arbitrary even M and N (N≥2M), we have M eigenvalues and eigenfunctions. Only the first function realizes one-to-one mapping. It can be approximately represented in the following form: 5.7 where kN,j tends to zero as . The conformal mapping is constructed via Φ by using (2.20) and (4.14): 5.8 where w(z) is given by (4.2). Consider an example based on the approximation (5.5). Substitution of (5.5) into (5.8) yields ω(z)≈zϱ(1+r2z2α)(−1+α2). Examples of computations are presented in figure 5.
The coated neutral inclusion problem for two-dimensional media is formulated as the eigenvalue -linear problem (2.18)–(2.19) on the auxiliary complex plane z. The later problem is reduced to the integral equation (3.12), where the closed curve L1 can be arbitrarily fixed in the unit disc.
Each solution of the discussed eigenvalue problem can produce a solution for the neutral coated problem, if only the function ω(z) from (2.20) is a one-to-one map. Examples given in §5 show that the following conjecture can be posed. There exists at least one eigenfunction of the problem (3.9)–(3.10) or of (3.12) so that the corresponding function constructed by (3.11) is a one-to-one map of the unit disc onto G1∪Γ1∪G2. This conjecture is equivalent to the existence of at least one eigenfunction from an infinite set of all eigenfunctions such that ω(z) in |z|<1 has the unique zero at origin. Integral equation (3.12) gives a constructive algorithm to solve the neutral inclusion problem, since eigenvalues λ and eigenfunctions ϕ1(z) can be computed by standard methods due to Krasnosel’skij et al. (1969). Then ω(z) is constructed by (2.20) and (3.11). Furthermore, zeros of ω(z) in the unit disc have to be investigated (see for instance Kravanja & Van Brel 2000). If z=0 is the unique zero of ω(z) in the unit disc, the shape of the neutral inclusion is given by formula ζ=ω(eiθ), 0≤θ≤2π.
In the earlier mentioned algorithm, ϱ is a fixed number and λ is unknown. Using formulae (2.12) and (2.17), it is easy to state the eigenvalue problem in which conductivity σ has to be found. The spectral parameter λ is a function of ϱ, i.e. λ=λ(ϱ) (see for instance (5.3) when L1 is an ellipse). Then σ can be determined from equation (σ+σ0)/(σ−σ0)=λ((σ1−σ)/(σ1+σ)) with fixed σ0 and σ1.
It is difficult to precisely compare our results with Milton & Serkov (2001) since the -linear problem (2.18)–(2.19) does not coincide with the corresponding problem (3.17)–(3.18) from Milton & Serkov (2001) in the limit cases when σ1=0 or . Moreover, it is difficult to match two results taken from two different infinite sets of all results. We can suggest that fig. 2(a) and (b) from Milton & Serkov (2001) can be obtained by using our approach when L1 is an ellipse.
The results obtained in this paper allow us to make the following conclusion. Any two-dimensional core, i.e. a core of an arbitrary smooth shape and of an arbitrary conductivity, can be coated by such a material that the coated inclusion inserted in a matrix of an arbitrary fixed conductivity does not disturb the uniform field outside the inclusion. This conclusion can be justified by the following arguments. Any simply connected domain G1∪Γ∪G can be conformally mapped onto the unit disc; L1 in figure 4 is the image of Γ1 in figure 1. For any Hölder continuous curve L1, the integral equation (3.12) has non-trivial eigenfunctions. For sufficiently small ratios of the areas |G1|/(|G1|+|G|) (that is equivalent to small r in §§4 and 5), one of the eigenfunctions produces the required conformal mapping ω(z). Therefore, this eigenfunction determines the shape of the coating Γ2 (figure 1), and the corresponding eigenvalue determines the conductivity σ of the coating. It is not yet known whether the area of coating |G| can be small, i.e. |G1|/(|G1|+|G|) is of order 1. This question is related to the condition that ω(z) must be a one-to-one map. The earlier mentioned discussion is restricted to inclusions with smooth boundaries. The physical and geometrical properties of the coating inclusions can be systematically investigated via the integral equation (3.12). Some examples are discussed at the end of §3.
- Received April 12, 2011.
- Accepted October 19, 2011.
- This journal is © 2011 The Royal Society