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Green's kernels for transmission problems in bodies with small inclusions

The uniform asymptotic approximation of Green's kernel for the transmission problem of antiplane shear is obtained for domains with small inclusions. The remainder estimates are provided. Numerical simulations are presented to illustrate the effectiveness of the approach.


Introduction
Our goal is to obtain a uniform asymptotic approximation of Green's function for a transmission problem of antiplane shear in a domain with small inclusions.
Exact solutions to singularly perturbed problems corresponding to bodies with defects are often unavailable. For complicated geometries involved in problems of this kind, i.e. domains with multiple small perforations, numerical algorithms may become incapable of reaching the required accuracy. Also, when the right-hand sides of such boundary value problems are singular, numerical procedures can suffer from the same deficiencies. In this case, asymptotic solutions to these problems are desirable.
The approximation of Green's kernels for regularly perturbed problems, for the Laplace operator and the biharmonic operator, was first studied by Hadamard in [4]. More recently, the question of uniformity for the approximations of Green's kernels for boundary value problems in domains with singularly and regularly perturbed boundaries, was addressed in [9]. The uniform approximations in [9] were derived using the method of compound asymptotic expansions.
The paper [7] contains the rigorous proofs and remainder estimates for uniform approximations of Green's kernels, given in [9], for −∆ in an n-dimensional domain (n ≥ 2), with a single small rigid inclusion. Uniform asymptotic formulae of Green's functions for mixed problems of antiplane shear in domains with a small hole or a crack, are discussed in detail in [8]. In [10], an extension of the theory developed in [7] is made to the case of Green's tensors of vector elasticity, for an elastic body with a small inclusion. This was followed by [11], where uniform asymptotics of Green's kernels for planar and three-dimensional elasticity in bodies with multiple rigid inclusions are given. The paper [11] also includes analysis of Green's kernels and numerical simulations for antiplane shear and plane strain.
In the present paper, the new feature of the problem tackled is that on the small inclusions we prescribe transmission conditions (the continuity of tractions and displacements). The inclusions are assumed to be occupied by materials which are different from that of the ambient medium. Compared to previous expositions into the uniform approximation of Green's kernels in [7,8,10], where the kernels are approximated in the bodies containing small holes we also must approximate the Green's kernel inside the inclusions. The analysis also brings additional boundary layers when the point force is placed inside the inclusion.
Below, we illustrate one of the main results in this article, for the case when the domain has a single inclusion. Let ω ε be a small planar inclusion, occupied by a material of shear modulus µ I , containing the origin O and which is perfectly bonded to the rest of the matrix Ω ε ⊂ R 2 whose shear modulus is µ O . Here, ε is a small positive parameter characterising the normalized size of the inclusion. Consider the antiplane shear Green's function N ε for the transmission problem inside the perturbed domain Ω ε ∪ω ε . We also use N (Ω) , as the Neumann function in the unperturbed domain (without the inclusion) and R (Ω) as its regular part. We denote by N the Green's function for the transmission problem, in the unbounded domain with the scaled inclusion containing the origin. Also let us define the vector function D = (D 1 , D 2 ) T , where the components D j , j = 1, 2 are the dipole fields for the scaled inclusion in the unbounded domain.
As one of the results we state Theorem The approximation of Green's function for the transmission problem of antiplane shear in Ω ε ∪ ω ε ⊂ R 2 , is given by uniformly for x, y ∈ Ω ε ∪ ω ε .
The structure of the article is as follows. In Section 2, we introduce the main notations that will be adopted throughout the text and define Green's function for the domain containing several inclusions. Section 3 contains the description of model fields used to construct the uniform asymptotic approximations of the Green's function. In Section 4, we state and prove an estimate for solutions to model transmission problems in an unbounded domain with an inclusion.
Solutions to transmission problems in a domain with several small inclusions are studied in Section 5. Results of these sections will aid us in deriving the remainder estimate present in the generalization of approximation (1.1) to the case of multiple inclusions. Asymptotic properties of the boundary layer fields, involving the regular part of Green's function for the transmission problem, in the unbounded domain with an inclusion are investigated in Sections 6. Then, we consider the uniform approximation of Green's function for the transmission problem in the domain with small inclusions in Section 7, and give the formal algorithm together with the remainder estimates. Finally, in Section 8, we demonstrate the effectiveness of our approach and present the numerical simulations comparing the asymptotic formula in Section 7 with the finite element calculations in COMSOL.
The asymptotic formulae obtained in the sequel are readily applicable to numerical simulations. As an example, Figure 1 shows the comparison between an asymptotic approximation and COMSOL computation for the modulus of the gradient of the regular part of Green's function for the transmission problem in a domain with a circular inclusion, for the case when the point force is applied outside the inclusion in a planar body. This Figure represents the regular part of the displacement field produced by a point force inside a Cast Iron disk containing an Aluminum inclusion: Fig. 1a, shows the computations obtained through the formula (1.1) when y ∈ Ω ε , while Here, we consider a circular cylinder of radius 30m containing a circular inclusion of radius 7m, the shear modulus of the inclusion is µ I = 2.6316 × 10 10 Nm −2 (an Aluminum inclusion), where the shear modulus for the rest of the matrix is µ O = 5.6×10 10 Nm −2 . The position of the unit point force is y = (10m, 10m), which is quite close to the boundary of the inclusion.

Main Notations
Let Ω be a subset of R 2 , with smooth boundary ∂Ω and compact closureΩ. Also let ω (j) ⊂ R 2 , have smooth boundary ∂ω (j) and compact closureω (j) , whose complement in the infinite plane is Cω (j) = R 2 \ω (j) , j = 1, . . . , M . The sets ω (j) , j = 1, . . . , M , are assumed to contain the origin O, and the maximum distance between O and ∂ω (j) is 1. Let ω (j) ε , be a subset of Ω, with centre O (j) , 1 ≤ j ≤ M . We assume that the minimum distance between O (j) and O (k) , k = j, k = 1, . . . , M and the minimum distance between O (j) and ∂Ω is 1. We relate the domain ω The perturbed geometry is defined by Ω ε = Ω\ j ω (j) ε , and we say that this domain is occupied by a material with shear modulus µ O and the domain ω (j) ε is occupied by a material with shear modulus µ Ij , where µ O , µ Ij > 0, 1 ≤ j ≤ M . In the subsequent sections, along with x and y we will also use the scaled variables By χ T we mean the characteristic function of the set T , that is Our goal is to obtain uniform asymptotics for the Green's function N ε , of the transmission problem in j ω The normal derivative of N ε on the exterior boundary satisfies where |∂Ω| is the one-dimensional measure of the set ∂Ω; ∂/∂n x = n · ∇ x is the normal derivative, where n is the unit outward normal.
The proof of symmetry of N ε , for this problem, is addressed below.
The symmetry of the Green's function N ε Proposition 1 Green's function N ε for the transmission problem in (2.7) We apply Green's formula for the functions N where the right-hand side is zero as a result of the transmission conditions (2.4), the exterior boundary condition (2.3) and the normalization (2.5). Thus , j = 1, . . . , M . When x, y ∈ Ω ε , (2.8) gives Similarly, for x ∈ ω (j) ε , y ∈ Ω ε , we deduce The above relations together with (2.7) lead to (2.6).

Special solutions in model domains
The asymptotic algorithm uses special fields defined in model domains including the unperturbed set and the exterior of a scaled inclusion.
1. The regular part R (Ω) of the Neumann function N (Ω) in Ω is defined as a solution of where y ∈ Ω.
To guarantee the symmetry of R (Ω) we impose the orthogonality condition The function N (Ω) is related to R (Ω) by 2. The next model field is Green's function for the transmission problem in the domain Cω (j) ∪ ω (j) , j = 1, . . . , M . This function is denoted by N (j) and, for η ∈ Cω (j) ∪ ω (j) , is subject to where the transmission conditions across the interface of the inclusion are given as and at infinity we will prescribe the condition and then it can be shown that The proof is analogous to the one of Proposition 1.
We set and Assume η ∈ Cω (j) , and let B R = {ξ : |ξ| < R} be a disk, with sufficiently large radius R, so that η ∈ B R \ω (j) . By applying Green's formula to Taking the limit as R → ∞ and employing (3.4), we deduce which is equivalent to . In a similar way, it can be shown that and the condition (3.5) implies that the above right-hand sides are zero. Thus N (j) is symmetric.
Regular part of N (j) . Let the function N (j) have the form The symmetry of N (j) , then implies the conditions 3. We also make use of model solutions known as the dipole fields D The transmission conditions on the boundary of the inclusion ω (j) are where in (3.9), n (j) is the unit normal to ω (j) . At infinity the vector function ∞ is a constant.

An estimate for solutions to transmission problems for antiplane shear in unbounded domains
The next result plays an important role in the asymptotic algorithm. It will allow us to obtain estimates for the boundary layer fields and derive the estimate for a solution of the transmission problem in a domain with multiple inclusions.
Proof. We note that transmission problems are studied in detail in [1] and [2], in the context of boundary integral equations and their solvability.
Let us first represent the solution U (j) by two functions U (j,O) and U (j,I) , harmonic in the domains Cω (j) and ω (j) , respectively. These functions satisfy For the function U (j,O) , the condition also holds. Applying Green's formula to the functions N (j) (see Problem 2, Section 3), U (j,O) and U (j,I) , one obtains for ξ ∈ Cω (j) , and for ξ ∈ ω (j) . First, let |ξ| ≥ 2, then using the asymptotics for the function N (j) at infinity, and the condition (4.1), we deduce Also by the Cauchy-Schwarz inequality and (4.4) Similarly the integral representation (4.5) gives (4.8) The combination of (4.6), (4.7) and (4.8), leads to (4.2).
As an immediate corollary of Lemma 1, we have an estimate for the dipole fields associated with the scaled inclusion ω (j) , j = 1, . . . , M : holds, where the constant in the right-hand side can depend on µ O , µ Ii and ω (i) .
5 An estimate for the maximum modulus of solutions to transmission problems for antiplane shear in a domain with several small inclusions Here we obtain an estimate for solutions to transmission problems for antiplane shear in domains with small inclusions. The next lemma will be used in Section 7 incorporating the remainder estimates produced by the approximations of Green's function in a perforated domain.
To provide uniqueness we also assume ∂Ω u(x)dS x = 0 .
Then there exists a positive constant A, independent of ε and such that Proof. a) The inverse operators to model problems in Ω and Cω (j) , j = 1, . . . , M . Let us introduce the operators which are the inverse operators of the problems where ψ ∈ L ∞ (∂Ω), ϕ ∈ L ∞ (∂ω (j) ), j = 1, . . . , M , also b) An estimate for solutions to the model Neumann problem in Ω. Let N (Ω) denote the Neumann function (3.2) in Ω.
Then an application of Green's formula to N (Ω) (x, y) and w(y) yields the representation for w The solution w of the Neumann problem in Ω is subject to the orthogonality condition in problem (5.4), and hence the last term on the right-hand side of (5.6) is zero, so that From this we obtain the estimate c) The case of the homogeneous boundary condition on ∂Ω. When the right-hand side of the Neumann condition on ∂Ω in (5.1) is zero we look for a solution of the form ε is an unknown function defined on ∂ω ε ∪ Ω ε , and is continuous across the boundaries of the small inclusions. On ∂Ω, u 1 satisfies Computing the jump in tractions of u 1 on ∂ω where g ε (x) = (g (1) (x), . . . , g (M ) (x)) T and Then, from the local estimate for harmonic functions, we can also assert the inequality where in moving to the second inequality we used estimate (5.7), and then Lemma 1 brings us to the last inequality. Then, the preceding estimate and (5.9), (5.10) lead to Hence, from the smallness of S (m) ε , we can write where g ε (x) = (g  In this situation we look for the solution u 2 in the form

Now, this and (5.11) give
Then, v is a solution of (5.1) with the boundary conditions This inequality, (5.7) and (5.13), give Finally, we obtain (5.2) through the combination of (5.12) and (5.14).
6 An asymptotic approximation for the regular part of the Green's function N (j) at infinity In this section, we shall prove a result which concerns the asymptotics of h Proof. First we study the asymptotic behaviour of the functions h N is a solution of the transmission problem We introduce one more auxiliary vector function It solves the transmission problem which is consistent with Problem 3 of Section 3. One can also represent Υ (j) , as ii) The asymptotics of h For |ξ| > 2, η ∈ Cω (j) , we have from Lemma 2 in [6], that the function h where the remainder r The vector function is evaluated below. Evaluation of C (j,O) (η). Let B R be a disk centered at the origin with a sufficiently large radius R. We apply Green's formula to h (j) where while combining the integrals over ∂ω (j)+ and ∂ω (j)− we have used the continuity conditions for the functions h for η ∈ Cω (j) . The last integral, owing to (6.7), is equal to where η ∈ Cω (j) . Now, using the jump in tractions for the dipole fields, stated in Problem 3 of Section 3, we can write Using the continuity of the displacements for the dipole fields (see Problem 3, Section 3), we write the above right-hand side as Then, upon applying Green's formula to D , ω (j) , respectively, with the fundamental solution of −∆, and using the definition of Υ (j) , we obtain the above in the form where η ∈ Cω (j) and by (3.10) the integral over ∂B R decays as R → ∞ like O(| log R|/R). Therefore, we have for η ∈ Cω (j) . Equation (6.11), with substitution of the preceding equality, leads to (6.13) We now aim to determine the vector function in (6.9). Taking the limit R → ∞ in (6.13) and using (6.9) (where h Thus, we have derived that which corresponds to the leading order term of the function h iii) The asymptotics of h (j,I,O) at infinity. We have already shown that for In order to deduce the leading order term of h (j,I,O) at infinity, we recall the relations (3.8). First, since h (j,O,O) N is symmetric for ξ, η ∈ Cω (j) we have the above asymptotic representation also holds for h (j,O,O) N (η, ξ). Next we allow η to approach the boundary of the inclusion ω (j) . For η ∈ ∂ω (j) we have Therefore, allowing η ∈ ω (j) , for |ξ| > 2, we arrive at is the remainder term and subject to its smallness the leading order part of (6.2) has been formally deduced.
Remainder estimate. Consider the function which by (6.14) and (6.15) is Let |ξ| > 2 and write the problem for r (j) N with respect to η as follows N (ξ, η) → O, as |η| → ∞ . We have that the right-hand side of (6.16) is self balanced and where η ∈ ∂ω (j) , |η| ≤ 1 and |ξ| > 2. Now an application of Lemma 1, leads to the estimate for r iv) The asymptotics of h . We once again refer to (3.8), for the relation For |ξ| > 2, η ∈ ω (j) , this can be rewritten as By combining this with (6.2) we obtain (6.3).

Uniform approximation of N ε in a domain with multiple inclusions
The aim of the current section is to present the uniform asymptotic approximation for N ε in a domain with several inclusions.

Theorem 1
The approximation of Green's function for the transmission problem of antiplane shear in l ω (l) ε ∪ Ω ε ⊂ R 2 , is given by Proof. Formal asymptotic algorithm. First, we give a plausible argument concerning the representation of N ε . We propose the function N ε to be given in the form In the above problem we have that R ε is subject to the orthogonality condition First order approximation for R ε . If we allow y to be located inside one of the M inclusions, then we assume y ∈ ω (m) ε , where m is fixed, 1 ≤ m ≤ M . We first rewrite the boundary conditions for R ε .
The boundary condition for R ε when x ∈ ∂Ω, y ∈ Ω ε ∪ ω (m) ε , is equivalent to Using scaled variables we can also rewrite the transmission conditions (7.6) on ∂ω ε . Finally, the transmission condition (7.6) on the m th inclusion becomes ε . We therefore have the representation The construction of boundary layer terms. The function Z ε , given in (7.9), is harmonic for and is continuous across the boundaries of the small inclusions. The normal derivative of Z ε on the exterior boundary is for x ∈ ∂Ω, and the traction transmission conditions for Z ε on the small inclusions are where 1 ≤ q ≤ M , q = m and y ∈ Ω ε ∪ ω ε . Next we apply Lemma 4, in order to rewrite the above boundary condition on ∂Ω for Z ε as where n (m) is the unit outward normal to the inclusion ω Then, we approximate Z ε by Combined formula for N ε . The substitution of (7.9), (7.10) into (7.2), for ε (x, y) .

and the same lemma also gives
Estimate of the right-hand side of (7.23) on ∂Ω. The definition of N (j,I) in Section 3 and the boundary condition (3.1) for R (Ω) , give us Using the asymptotics of h (j,I) N at infinity contained in Lemma 4b), we obtain Finally, the above estimate for h (j) ε , j = 1, . . . , M and (7.22), combined with (7.12) complete the proof.

Numerical simulations
In the current section, we implement the asymptotic formulae derived in Section 7 for Green's function N ε in numerical simulations. The numerical computations are carried out for the regular part R ε of Green's function in l ω (l) ε ∪ Ω ε for the transmission problem. In other words, let this regular part be given by the formula Then in accordance with the boundary value problem for N ε given in Section 2, the function R ε , which we choose to consider for our numerical schemes, when y ∈ Ω ε is a solution of the problem (7.3)-(7.8).

Asymptotic formulae for R ε
From formula (8.1) and Theorem 1, we can immediately state the asymptotic formulae for the regular part R ε that will be used in the examples below. When y ∈ Ω ε we have from (7.1), that R ε admits the asymptotic representation For the case η j ∈ ω (j) , ξ j ∈ Cω (j) h (j,I) (ξ j , η j ) = 1 2π and for η j , ξ j ∈ ω (j) we have h (j,I) (ξ j , η j ) = 1 2π

The dipole fields for the circular inclusion in the infinite plane
Here, we give the vector function D (j) whose components are the dipole fields for the circular inclusion of radius a j in the infinite plane and We have for ξ j ∈ Cω (j) , and D (j,I) (ξ j ) = µ Ij − µ O µ Ij + µ O ξ j , for ξ j ∈ ω (j) .

Example 1
The case of the force applied outside the inclusions For our first example, we look at the case when y ∈ Ω ε . We therefore base our computations on the asymptotic formula (8.2) when comparing with those of COMSOL. The coordinates of the point force are given as y = (−10m, −80m). We plot the modulus of the gradient of the regular part R ε in Figure 2a) according to the analytical formulae (8.2). Figure 2b) is the same quantity computed using the method of finite elements in COMSOL. Both figures are very similar, the maximum absolute error between these computations is 7.666 × 10 −16 occurring on the exterior boundary.

Example 2
The case of the force positioned inside an inclusion In the second example, we aim to compare the computations produced by formula (8.3) with those generated by COMSOL. Now the point force is assumed to be situated at y = (60m, 0m), in the inclusion ω (5) ε , containing the Steel AISI 4340. Figure 3a), gives the surface plot for the modulus of the gradient of the regular part, provided by formula (8.3). The numerical solution given in COM-SOL is shown in Figure 3b). The maximum absolute error here is 7.98 × 10 −16 , which occurs on the boundary of the inclusion ω (1) ε . We conclude that the asymptotic formulae and numerical computations are in a good agreement with each other.