Geometric multipolar expansion and its application to neutral inclusion of arbitrary shapeThis work is supported by the Korean Ministry of Science, ICT and Future Planning through NRF grant No. 2016R1A2B4014530.

# Geometric multipolar expansion and its application to neutral inclusion of arbitrary shape††thanks: This work is supported by the Korean Ministry of Science, ICT and Future Planning through NRF grant No. 2016R1A2B4014530.

Doo Sung Choi Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea (7john@kaist.ac.kr, kjb2474@kaist.ac.kr, mklim@kaist.ac.kr).    Junbeom Kim 22footnotemark: 2    Mikyoung Lim22footnotemark: 2
July 24, 2019
###### Abstract

The field perturbation induced by an elastic or electrical inclusion admits the multipolar expansion in terms of the outgoing potential functions. In the classical expansion, the basis functions are defined independently of the inclusion. In this paper, we introduce the new concept of the geometric multipolar expansion for the two dimensional conductivity (or, equivalently, anti-plane elasticity) of which the basis functions are associated with the inclusion’s geometry. We use the series expansion for the complex logarithm by the Faber polynomials that are associated with the exterior conformal mapping of the inclusion. The virtue of this method is that one can express the field perturbation for an inclusion of arbitrary shape in a simple series expansion. Regarding the computation of the exterior conformal mapping, one can use the integral formula for the conformal mapping coefficients obtained in [18]. As an application, we construct multi-coated neutral inclusions of arbitrary shape that do not perturb low-order polynomial loadings. These inclusions are layered structures composed of level curves of one exterior conformal mapping; material parameters in each layer are determined for the low-order terms in the geometric multipolar expansion to vanish. We provide numerical examples to validate the results.

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Mathematics Subject Classification. 35J05; 42C10; 78A46

Key words. Multipolar expansion; Conformal mapping; Multilayer cloaking structure; Shape derivative

## 1 Introduction

Elastic or electrical inclusions induce the field perturbation in an external background field. Analytic and numerical solution methods have been developed and widely applied in various areas, such as imaging, invisibility cloaking, and nano-photonics [3, 10, 22, 23]. In this present paper, we consider the transmission problem of the two dimensional conductivity (or, equivalently, anti-plane elasticity):

 ⎧⎨⎩∇⋅(kχΩ+χR2∖¯¯¯Ω)∇u=0in R2,u(x)−H(x)=O(|x|−1)as |x|→∞, (1.1)

where is a simply connected bounded domain with Lipschitz boundary, is an entire harmonic function, and indicates the characteristic function for a region . We will introduce the new concept of the geometric multipolar expansion of which the basis functions are associated with the inclusion’s geometry.

The transmission problem (1.1) can be expressed by the single layer potential, where the density function is given by the solution to an integral equation involving the so-called the Neumann-Poincaré operator (see [21]). From the Taylor series expansion for the fundamental solution to the Laplacian and the background potential, the field perturbation admits the multipolar expansion with the basis functions independent of . Contracted forms of multipolar expansion employ separation of variables solutions, in polar coordinates for the two dimensions and in spherical coordinates for the three dimensions, as its basis functions. They provide simple solutions to the transmission problem when the inclusion is a disk or a ball.

We use the complex function theory in the derivation. The complex analysis technique has been used in various transmission problems in the dimensions for the conductivity and the linear elasticity problem [8, 9]. Especially in [18], new series representation of single layer potential and Neumann-Poincaré operator using exterior conformal mappaing was derived.

Let us state the results of the paper. We identify in with in . From the Riemann mapping theorem there exists uniquely a conformal mapping, say , that maps the exterior of a disk to . As a univalent function, defines the so-called Faber polynomials ’s, which are complex monomials and form a basis for complex analytic functions in ; see [11]. The complex logarithm admits the following expansion (see [11, 14, 18]): for and ,

 log(z−~z)=logζ−∞∑m=11mFm(~z)ζ−m(in modulo 2πi). (1.2)

This formula sheds new light to the transmission problem (1.1) (see [18]). The fundamental solution to the Laplacian, the real logarithm function, admits the similar expression to (1.2), based on which we define the new concept of the Faber polynomials Polarization Tensors (FPTs) denoted by and (see Definition 2 in section 3). The following theorem is one of main results of this paper. The virtue of the proposed method is that one can express the field perturbation for an arbitrary shape inclusion, which includes a family of layered structure defined by one exterior conformal mapping, in a simple series expansion.

###### Theorem 1.1 (Geometric multipolar expansion).

Assume that is a simply connected bounded domain in and is the exterior conformal mapping associated with . Then, for a harmonic function given by with complex coefficients ’s and ’s, the solution to (1.1) satisfies that for ,

 u(z)=H(z)−∞∑k=1∞∑m=114πk[(αmF(1)mk+βm¯¯¯¯¯¯¯¯¯¯F(2)mk)ζ−k+(αmF(2)mk+βm¯¯¯¯¯¯¯¯¯¯F(1)mk)¯¯¯¯¯¯¯¯ζ−k].

As an application of the geometric multipolar expansion, we construct neutral inclusions of arbitrary shape that does not perturb low-order polynomial loadings. The FPTs are linear combinations of the generalized polarization tensors (GPTs), which are coefficients in the classical multipolar expansion. As the GPTs contain the information on the geometry and material parameter of the inclusion, so do the FPTs.

The GPT-vanishing structure has been used in effective medium theory and the invisibility cloaking. The concept of GPT-vanishing structure is first introduced in [5], which cancels the GPTs of low orders using multi-coated sturcture. The GPT-vanishing structure makes near-cloaking for some background field with higher orders. In [5], the multi-coated inclusion with circular layers was introduced. This idea was extended to 3-dimension [26]. Recently, the shape derivative was used to make the coating on an inclusion with general shape [15]. Also, vanishing the first order GPT by giving imperfect interface instead of multi-coating was introduced [19].

In this paper, by use of the exterior conformal mapping and the concept of the FPTs we construct neutral inclusions of arbitrary shape. The enhanced neutral inclusion is a layered structure made of level curves of the exterior conformal mapping. The material parameters are determined such that the resulting FPTs to vanish for low-order terms, by using the Fourier series expansion for the potential function in terms of the curvilinear coordinates associated with the exterior conformal mapping.

The remainder of this paper is organized as follows. Section 2 is devoted to review the classical multipolar expansion. We define the FPTs and derive the geometric multipolar expansion in section 3. We explain the numerical scheme to design the FPT-vanishing structure with arbitrary shape in section 4 and 5. We then conclude with some discussion.

## 2 Classical multipolar expansion

Let be a simply connected domain in with Lipschitz boundary.

For we define the single layer potential and the Neumann-Poincaré operator associated with as

 S∂Ω[φ](x)=∫∂DΓ(x−~x)φ(~x)dσ(~x),x∈Rd, K∗∂Ω[φ](x)=p.v.12π∫∂Ω⟨x−~x,νx⟩|x−~x|2φ(~x)dσ(~x),x∈∂Ω,

where is the outward unit normal vector to , stands for the Cauchy principal value, and is the fundamental solution to the Laplacian, i.e., . On the interface the following jump relations holds:

 S∂Ω[φ]∣∣+(x) =S∂Ω[φ]∣∣−(x)        a.e. x∈∂Ω, ∂∂νS∂Ω[φ]∣∣±(x) =(±12I+K∗∂Ω)[φ](x)        a.e. x∈∂Ω. (2.1)

We also set for and .

The solution to (1.1) satisfies the relations and on . Hence, one can express as

 u(x)=H(x)+S∂Ω[φ](x),x∈R2, (2.2)

where

 φ=(λI−K∗∂Ω)−1[ν⋅∇H]with λ=k+12(k−1). (2.3)

As shown in [12, 20, 25] the boundary integral problem is invertible on for . The stability of the transmission solution have been established; see for example [12, 13]. We recommend the reader to see [3, 4] for more properties of the NP operator. The boundary integral equation and the spectrum of the NP operator can be numerically solved with high precision by the Nyström discretization method [16, 17].

The boundary integral formulation (2.2) implies the multipolar expansion for the transmission problem as follows. We use the conventional multi-index notation: , . We remind the reader the Taylor series expansions

 Γ(x−y) =∞∑|α|=0(−1)|α|α!∂αΓ(x)yα, (2.4) H(y) =∞∑|β|=01β!∂βH(0)yβ (2.5)

for and with sufficiently large magnitude. The so-called generalized polarization tensors (GPTs), associated with the domain and the parameter , are defined as

 Mαβ(Ω,k)=∫∂Ωyα(λI−K∗∂Ω)−1[ν⋅∇yβ](y)dσ(y) (2.6)

for the multi-indices , . By putting the series expansion (2.4) and (2.5) to the equations (2.2) and (2.3), it follows the multipole expansion for the solution to (1.1) (see [4]):

 u(x)=H(x)+∞∑|α|,|β|=1(−1)|α|α!β!∂αΓ(x)Mαβ∂βH(0),|x|≫1, (2.7)

The main properties of GPTs are introduced in [4, 1]. Recently the expansion for was generalized for a smooth domain. In [7], was expanded in terms of the eigenfunction of the symmetrized for a -smooth domain with .

One can rewrite (2.7) in a simpler form by use of the complex formulation. As before we identify in with in .

###### Definition 1 (The complex contracted GPTs).

For , we define

 N(1)mk(Ω,λ)=∫∂Ωzk(λI−K∗∂Ω)−1(∂zm∂ν)dσ(z), N(2)mk(Ω,λ)=∫∂Ωzk(λI−K∗∂Ω)−1(∂¯¯¯¯¯¯zm∂ν)dσ(z).

We call and the complex contracted GPTs (See [1]).

The complex contracted GPTs are linear combinations of ’s with the coefficients of the Taylor series expansion of and .

###### Lemma 2.1 ([1]).

For a harmonic function given by with complex constants ’s and ’s, the solution to (1.1) satisfies that for ,

 u(z)=H(z)−∞∑k=1∞∑m=114πk[(αmN(1)mk+βm¯¯¯¯¯¯¯¯¯¯N(2)mk)z−k+(αmN(2)mk+βm¯¯¯¯¯¯¯¯¯¯N(1)mk)¯¯¯¯¯¯¯¯z−k].
###### Proof.

We use the Taylor series of the complex logarithm to deduce

 log(z−~z)=logz+log(1−~zz)=logz−∞∑k=11k~zkz−kfor |z|>|~z|.

Hence, the fundamental solution admits the expansion in complex variables as follows:

 Γ(x−~x) =12πln|z−~z| =14π(log(z−~z)+¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯log(z−~z)) =12πln|z|−∞∑k=114πk(~zkz−k+¯¯¯¯¯~zk¯¯¯¯¯¯¯¯z−k).

By modifying the expansion (2.7) we prove the lemma.

The contracted GPTs were used in making a near-cloaking structure [5, 6] and also used as the shape descriptor [2]. We refer the reader to [1] and references therein for more applications of the (contracted) GPTs.

Since decays as as , one can reduce the perturbation in potential for large by designing of such that

 Mijmn=0for all m,n≤N

for a given integer . We call such a conductivity distribution a GPT-vanishing structure or coating of order .

## 3 Geometric multipolar expansion

We assume further that is with some .

### 3.1 The geometric series solution method

From the Riemann mapping theorem there exist uniquely and the conformal mapping from onto such that

 Ψ(w)=w+a0+a1w+a2w2+⋯. (3.1)

One can numerically compute the ’s by solving a boundary integral equation; see [18].

The Faber polynomials, first introduced by G. Faber in [14], have been extensively studied in various areas. The Faber polynomials associated with are defined such that

 ζΨ′(ζ)Ψ(ζ)−z=∞∑m=0Fm(z)ζm,z∈Ω, |ζ|>γ. (3.2)

For example, the first three polynomials are

 F0(z)=1,F1(z)=z−a0,F2(z)=z2−2a0z+(a20−2a1).

The relation (3.2) impies

 log(Ψ(ζ)−Ψ(w))=logζ−∞∑n=11mFm(Ψ(w))ζ−m, (3.3)

and it can be further extended to (1.2). The Faber polynomial is an -th order monic polynomial satisfying

 Fm(Ψ(w))=wm+∞∑k=1cm,kw−k. (3.4)

Here, ’s are called the Grunsky coefficients. Recursive relations for the Faber polynomial coefficients and the Grusnky coefficients are well-known. We recommend the reader to see [11] for further details on the Faber polynomials and the Grunsky coefficients. The expansion (1.2) sheds new light to better understanding for the solution to the transmission problem (1.1) and the related boundary integral operators; see [18].

Let us introduce the orthogonal curvilinear coordinates:

 z=Ψ(eρ+iθ)for ρ>ρ0=lnγ, θ∈[0,2π).

We denote the scale factors as . One can easily see that on and for a function ,

 ∂v∂ν∣∣+∂Ω(z)=1h(ρ0,θ)∂∂ρv(Ψ(eρ+iθ))∣∣ρ→ρ+0. (3.5)

For , we set the density basis functions for as

 ψm(z) =ψm(θ)=eimθh(ρ0,θ), ζm(z) =√mψm(θ), ηm(z) =1√meimθ.
###### Lemma 3.1 ([18]).

The density functions for a basis of and for a basis for . Let be arbitrary natural number. The single layer potential satisfies

 S∂Ω[ζm](z) =−12√mγmFm(z)in ¯¯¯¯Ω, (3.6) S∂Ω[ζm](z) =−(12I+K∂Ω)[ηm](z)on ∂Ω. (3.7)

The NP-operator and its -adjoint satisfy the following:

 K∗∂Ω[ψ0]=12ψ0,K∂Ω[1]=12

and

 K∗∂Ω[ζm]=12∞∑k=1√m√kck,mγm+k¯¯¯¯¯ζk,K∗∂Ω[¯¯¯¯¯¯ζm]=12∞∑k=1√m√k¯¯¯¯¯¯¯¯¯ck,mγm+kζk, (3.8) K∂Ω[ηm]=12∞∑k=1√m√kck,mγm+k¯¯¯¯¯ηk,K∂Ω[¯¯¯¯¯¯ηm]=12∞∑k=1√m√k¯¯¯¯¯¯¯¯¯ck,mγm+kηk. (3.9)

One can approximate the solution to (2.3) by the finite section method to the relations (3.8) and (3.9); see [18].

### 3.2 The Faber polarization tensors (FPTs)

As a concept corresponding to the contracted GPTs we define the following. For a disk centered at , the FPTs coincide with the contracted GPTs since the corresponding Faber polynomials are simply monomials .

###### Definition 2 (FPTs).

For , we define

 F(1)mk(Ω,λ)=∫∂ΩFm(z)(λI−K∗∂Ω)−1(∂Fk∂ν)dσ(z), (3.10) F(2)mk(Ω,λ)=∫∂ΩFm(z)(λI−K∗∂Ω)−1(∂¯¯¯¯¯¯Fk∂ν)dσ(z). (3.11)

We call and the Faber polynomial Polarization Tensors (FPTs).

Proof of Theorem 1.1 From (1.2), we have

 12πln|z−~z| =14π(log(z−~z)+¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯log(z−~z)) =12πln|z|−∞∑k=114πk(Fk(~z)ζ−k+¯¯¯¯¯¯¯¯¯¯¯¯¯Fk(~z)¯¯¯¯¯¯¯¯ζ−k).

Applying this relation to the equations (2.4) and (2.5), we prove the theorem.

Let us define the coefficients and ’s as

 (λI−K∗∂Ω)−1[ζm]=:12∞∑m=1[√m√kak,mγm+kζk+√m√kbk,mγm+k¯¯¯¯¯ζk]. (3.12)
###### Lemma 3.2.

The FPTs satisfy

 F(1)mk(Ω,λ) =4πkcm,k+4πk(14−λ2)bm,k, F(2)mk(Ω,λ) =8πkλγm+kδmk+4πk(14−λ2)¯¯¯¯¯¯¯¯¯¯am,k.
###### Proof.

From the interior jump relation for the single layer potential and (3.6), we have

 (−12I+K∗∂Ω)[ζk]=−12√kγk∂Fk∂ν∣∣∂Ω. (3.13)

By use of (3.6), (3.7) and (3.13), we compute that

 F(1)mk =∫∂Ω(−2√mγm)S∂Ω[ζm](λI−K∗∂Ω)−1(−2√kγk)(−12I+K∗∂Ω)[ζk]dσ =4√m√kγm+k∫∂Ω(12+K∂Ω)[ηm](λI−K∗∂Ω)−1(12I−K∗∂Ω)[ζk]dσ =4√m√kγm+k∫∂Ωηm(12+K∗∂Ω)(λI−K∗∂Ω)−1(12I−K∗∂Ω)[ζk]dσ

and, similarly,

 F(1)mk=2√m√kγm+k∫∂Ωηm(12+K∗∂Ω)(λI−K∗∂Ω)−1(12I−K∗∂Ω)[¯¯¯¯¯ζk]dσ.

One can easily see that

 (12+K∗∂Ω)(λI−K∗∂Ω)−1(12I−K∗∂Ω)=(λI+K∗∂Ω)+(14−λ2)(λI−K∗∂Ω)−1.

Note that and . Applying these orthogonality relations to (3.8) and (3.12) we prove the lemma.

One can easily see that

 F(1)11 =M(1)11=M11−M22+i(2M12), F(2)11 =M(2)11=M11+M22,

where the matrix

 M=(M11M12M12M22)

denote the polarization tensor (PT) associated with .

For , the FPTs show simple relations with the Grunsky coefficients:

 F(1)mk(Ω,±1/2) =4kπcm,k, F(2)mk(Ω,±1/2) =±δm,k4kπγm+k.

If , then we have

 F(1)11(Ω,1/2) =4πc11=4πa1, F(2)11(Ω,1/2) =4πγ2.

Then, the corresponding PT is

 M=2π(γ2+R{a1}I{a1}I{a1}γ2−R{a1}).

The trace and eigenvalues, say and , of the PT satisfies

 λ1+λ2 =4πγ2, 1λ1+1λ2 =1πγ2γ4−|a1|2.

Note that

 |D|=πγ2−π∞∑k=1k|ak|2γ2k.

Hence, we have the following lemma.

###### Lemma 3.3 (Perfect conducting case).

Assume that is a perfect conductor. Let and be the eigenvalues of the PT associated with . Then, we have

 |D|(1λ1+1λ2) =(πγ2−π∞∑k=1k|ak|2γ2k)1πγ2γ4−|a1|2≤1, (3.14) 1|D|(λ1+λ2) =(πγ2−π∞∑k=1k|ak|2γ2k)−14πγ2<∞. (3.15)

The equality holds in (3.14) if and only iff for all (or, equivalently is an ellipse). This is in accordance with the Pólya-Szegö conjecture [24].

### 3.3 An ellipse case

We set . As shown in [18], it holds that

 cm,k=δmkak1 (3.16)

and

 K∗∂Ω[ζm](z)=am12γ2m¯¯¯¯¯¯¯¯¯¯¯¯¯ζm(z),K∗∂Ω[¯¯¯¯¯¯ζm](z)=¯¯¯¯¯a1m2γ2mζm(z).

Hence, in the space spanned by and , the operator corresponds to the matrix whose inverse is

 1λ2−|a1|2m4γ4m⎡⎢ ⎢⎣λam12γ2m¯¯¯¯¯a1m2γ2mλ⎤⎥ ⎥⎦,

Hence, it follows from (3.12) that

 am,k=δkm2λγ2mλ2−|a1|2m4γ4m,bm,k=δkm2am1λ2−|a1|2m4γ4m.

The FPTs satisfies

 F(1)mk(Ω,λ) =δmk⎡⎢ ⎢ ⎢⎣4πmam1+4πk(14−λ2)2am1λ2−|a1|2m4γ4m⎤⎥ ⎥ ⎥⎦, (3.17) F(2)mk(Ω,λ) =δmk⎡⎢⎣8πmλγ2m+4πk(14−λ2)2λγ2mλ2−|a1|2m4γ4m⎤⎥⎦. (3.18)

## 4 Multi-coated inclusion of arbitrary shape

### 4.1 FPTs of a multi-coated inclusion

Let be a simply connected and bounded inclusion in . We define a conformal mapping from onto such that

 Ψ(w)=w+a0+a1w+a2w2+⋯, (4.1)

as the expansion in (3.1). For a positive integer , let and define

 Ωj:={Ψ(w):|w|≤eρj},j=1,2,...,N.

Let . We set be the conductivity for each disjoint region, and put . In other words, the conductivity distribution is given by

 σ=σ0χ(Ω0)+χ(ΩN+1)+N∑j=1σjχ(Ωj∖Ωj−1).

To compute and , we look for solution to

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∇⋅σ∇um=0in Cum∣∣+=um∣∣−on ∂Ωj,j=0,1,...,Nσj+1∂um∂ν∣∣+=σj∂um∂ν∣∣−,on ∂Ωj,j=0,1,...,Num(z)−Fm(z)=O(|z|−1)as |x|→∞ (4.2)

of the form

 um(z)=∞∑k=1(a1,jmkζk+b1,jmk¯¯¯¯¯ζk+a2,jmkζ−k+b2,jmk¯¯¯¯¯¯¯¯ζ−k)in Ωj, (4.3)

for , and (4.3) holds on for .

From Theorem 1.1, we have

 um(z) =Fm(z)−∞∑k=114πk(F(1)mkζ−k+F(2)mk¯¯¯¯¯¯¯¯ζ−k) =ζm+∞∑k=1⎛⎝cm,k−F(1)mk4πk⎞⎠ζ−k−∞∑k=1F(2)mk4πk¯¯¯¯¯¯¯¯ζ−kin ΩN+1.

Moreover, we set

 um(z) =∞∑k=1[amkFk(z)+bmk¯¯¯¯¯¯¯¯¯¯¯¯¯Fk(z)] =∞∑k=1[amkζk+bmk¯¯¯¯¯ζk+(∞∑l=1amlcl,k)ζ−k+(∞∑l=1bmlcl,k)¯¯¯¯¯¯¯¯ζ−k]on ∂Ω0.

From the condition for ,

 a1,0mk=amk,b1,0mk=bmk,a2,0mk=∞∑l=1amlcl,k,b2,0mk=∞∑l=1bmlcl,k. (4.4)

From the condition for ,

 a1,N+1mk=δmk,b1,N+1mk=0,a2,N+1mk=cm,k−F(1)mk4πk,