Radio Science

Second-kind integral solvers for TE and TM problems of diffraction by open arcs

Authors

  • Oscar P. Bruno,

    Corresponding author
    1. Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California, USA
      Corresponding author: O. P. Bruno, Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA. (obruno@caltech.edu)
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  • Stéphane K. Lintner

    1. Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California, USA
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Corresponding author: O. P. Bruno, Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA. (obruno@caltech.edu)

Abstract

[1] We present a novel approach for the numerical solution of problems of diffraction by open arcs in two dimensional space. Our methodology relies on composition of weighted versions of the classical integral operators associated with the Dirichlet and Neumann problems (TE and TM polarizations, respectively) together with a generalization to the open-arc case of the well known closed-surface Calderón formulae. When used in conjunction with spectrally accurate discretization rules and Krylov-subspace linear algebra solvers such as GMRES, the new second-kind TE and TM formulations for open arcs produce results of high accuracy in small numbers of iterations—for low and high frequencies alike.

1. Introduction

[2] Problems of diffraction by infinitely thin open surfaces play central roles in a wide range of problems in science and engineering, with important applications to antenna and radar design, electronics, optics, etc. Owing to difficulties inherent in rigorous mathematical treatment of open surface scatterers, such problems are often treated by means of physically approximate models [Keller, 1962; Karam and Fung, 1983; Lu and Ando, 2012] whose accuracy is limited by the physical approximations made. Like other wave scattering problems, on the other hand, open surface problems can be rigorously treated—either by means of numerical methods that rely on approximation of Maxwell's Equations over volumetric domains (on the basis of, e.g., finite difference or finite element methods) or by methods based on boundary integral equations. As a result of the singular character of the electromagnetic fields in the vicinity of open edges, open-surface scattering configurations present major difficulties for both volumetric and boundary integral methods. Boundary integral approaches, which require discretization of domains of lower dimensionality than those involved in volumetric methods, can generally be treated efficiently, even for high-frequencies, by means of accelerated iterative scattering solvers [Bleszynski et al., 1996; Bruno and Kunyansky, 2001; Rokhlin, 1993]. Unfortunately, integral methods for open surfaces do not give rise, at least in their classical formulations, to Fredholm integral operators of the second-kind, and can therefore prove to be computationally expensive—as the eigenvalues of the resulting equations accumulate at zero and/or infinity and, thus, iterative solution of these equations often requires large numbers of iterations.

[3] This paper presents new Fredholm integral equations of second-kind and associated numerical algorithms for problems of scattering by two-dimensional open arcs Γ (i.e., infinite cylinders of cross-section Γ) under either transverse electric (TE) or transverse magnetic (TM) incident fields. The new second-kind Fredholm equations (which result from composition of appropriately modified versions Sω and Nω of the classical single-layer and hypersingular integral operators S and N) provide, for the first time, a generalization of the classical closed-surface Calderón formulas to the open-arc case. In particular, the new formulations possess highly favorable spectral properties: their eigenvalues are highly clustered, and they remain bounded away from zero and infinity, even for problems of very high frequency. When used in conjunction with spectrally accurate discretization rules and Krylov-subspace linear algebra solvers such as GMRES, the new open-arc formulations produce results of high accuracy in small numbers of iterations—for low and high frequencies alike.

[4] The new second-kind formulation for the TM problem is particularly beneficial, as it gives rise to order-of-magnitude improvements in computing times over the corresponding weighted hypersingular formulation. Such gains do not occur in the TE case: although the new second-kind TE equation requires fewer iterations than the corresponding weighted first kind formulation, the total computational cost of the second-kind equation is generally higher in the TE case—since the application of the first-kind operator can be significantly less expensive than the application of the composite second-kind operator.

[5] The difficulties that arise as integral equations are used to treat open surface scattering problems are of course well known, and many contributions have been devoted to their treatment; early work in these regards can be found for instance in Meixner [1949], Maue [1949], and Mittra et al. [1973]. Like the present work, Povzner and Suharevskiĭ [1960] and Christiansen and Nédélec [2000] seek to tackle these problems by means of generalizations of the classical Calderón relations to the case of open surfaces. The first of these contributions establishes that the combination NS can be expressed in the form I + TK, where the kernel K(x, y) of the operator TK has local singularity of at most O inline image. This early result however does not take into account the singular edge behavior; the resulting operator TK is not compact (in fact it gives rise to extreme singularities at the edge (S. Lintner and O. Bruno, A generalized Calderón formula for open-arc diffraction problems: Theoretical considerations, submitted to Proceedings of the Royal Society of Edinburgh, 2012; available at http://arxiv.org/abs/1204.3699)) and I + TK is therefore not a second-kind operator in any numerically meaningful functional space. When used in conjunction with boundary elements that vanish on the edges (by means of well-chosen projections) however, the combination NS does give rise to reduction of iteration numbers, as demonstrated in Christiansen and Nédélec [2000] through numerical examples for low frequency problems. This contribution does not include details on accuracy, and it does not utilize integral weights to resolve the solution's edge singularity. A related but different method was introduced in Antoine et al. [2005] which exhibits, once again, low iteration numbers for low-frequency problems, but which does not resolve the singular edge behavior and for which no accuracy studies have been presented. Finally, high-order integration rules for the single-layer and hypersingular operators adapted to open arcs were introduced in a Galerkin framework in Stephan and Wendland [1984], Hsiao et al. [1991], and Stephan and Tran [1998]. These methods have thus far only been applied for simple geometries and at low frequencies, and limited information is available on the actual convergence properties and performance of their computational implementations.

[6] A second class of methods include those proposed in Atkinson and Sloan [1991], Mönch [1996], and Jiang and Rokhlin [2004]. The contribution by Atkinson and Sloan [1991], some aspects of which are incorporated in our method, treats the Dirichlet problem for Laplace's equation by means of second kind equations. The basis of this approach lies in the observation that the cosine basis has the dual positive effect of diagonalizing the logarithmic potential for a straight arc and removing the singular edge behavior—so that the inverse of the logarithmic potential can be easily computed and used as a preconditioner to produce a second kind operator for a general arc. The approach by Mönch [1996], which also uses a cosine basis, treats the Neumann problem for the non-zero frequency Helmholtz equation with spectral accuracy by means of first kind equations. The contribution by Jiang and Rokhlin [2004], finally, treats, just like Atkinson and Sloan [1991], the Laplace problem by means of second-kind equations resulting from inversion of the straight arc logarithmic potential; like Mönch [1996], further, it produces spectral accuracy through use of the cosine transforms. The second-kind integral approach developed in Atkinson and Sloan [1991] and later revisited in Jiang and Rokhlin [2004] seems essentially limited to the specific problem for which it was proposed: neither an extension to the Neumann problem nor to the full three dimensional problem seem straightforward. And, more importantly, this approach does not lead to adequately preconditioned equations for non-zero frequencies: a simple experiment conducted in section 3.4 shows that a direct generalization of the algorithm Atkinson and Sloan [1991] to the Helmholtz problem generally requires significantly more linear algebra iterations than are necessary if the operator Sω alone is used.

[7] The remainder of this paper is organized as follows: after recalling in section 2 the classical boundary integral formulations for the TE and TM open-arc scattering problems, in sections 3.1 and 3.2 we present our new weighted operators Sω and Nω as well as certain periodized counterparts inline image and inline image (which are obtained by considering sinusoidal changes of variables for source and observation points). Our main result, the generalized Calderón formula, is presented in section 3.3. Section 3.4 then presents results of numerical evaluation of eigenvalues for a non-trivial open arc problem, illustrating the spectral properties of previous open-arc operators as well as the second kind operators introduced in this paper. Theoretical considerations concerning the new second-kind equations are presented in section 4, including a succinct but complete proof of the open-arc Calderón formulae; a more detailed theoretical discussion, including full mathematical technicalities, can be found in Lintner and Bruno (submitted manuscript, 2012). The high-order quadrature rules we use for evaluation of the new integral operators are described in section 5. Numerical results, finally, are presented in section 6 for a wide range of frequencies and for various geometries (including a brief study of resonant open cavities) demonstrating the uniformly well conditioned character of the integral formulations proposed in this paper.

2. Preliminaries

[8] Let Γ denote a smooth open arc in the plane. The TE and TM problems of scattering by the open arc Γ amount to two-dimensional problems for the Helmholtz equation with Dirichlet and Neumann boundary conditions on Γ, respectively:

display math

Here n denotes the normal to Γ, and f and g are given in terms of the incident electric excitation uinc: f = −uinc and g = −∂uinc/∂n on Γ.

2.1. Classical Open-Arc Equations for TE and TM Problems

[9] The (unique) solutions u and inline image of the TE and TM problems above can be expressed, for r ∉ Γ, in terms of single- and double-layer potentials of the form u(r) = ∫Γ Gk(r, r′)μ(r′)d inline image and inline image(r) = ∫Γ inline imageν(r′)d inline image [see, e.g., Stephan and Wendland, 1984; Wendland and Stephan, 1990; Saranen and Vainikko, 2002]. Here inline image is a unit vector normal to Γ at the point r′ ∈ Γ (we assume, as we may, that inline image is a smooth function of r′ ∈ Γ), and, calling H01 the first Hankel function of order zero,

display math

and inline image = inline image · inline image Gk(r, r′). Denoting by S and N the single-layer and hypersingular operators

display math

and

display math

further, the densities μ and ν are the unique solutions of the integral equations

display math

[10] It is known that the eigenvalues of the integral operators in equations (5) accumulate at zero and infinity, respectively (see, e.g., Figure 1). As a result (and as illustrated in section 6), the solution of these equations by means of Krylov-subspace iterative solvers such as GMRES generally requires large numbers of iterations. In addition, as discussed in section 2.3, the solutions μ and ν of equations (5) are not smooth at the end-points of Γ, and, thus, they give rise to low order convergence (and require high discretization of the densities for a given accuracy) if standard quadrature methods are used. Details in these regards are provided in what follows.

Figure 1.

Eigenvalue distribution for the spiral-shaped arc displayed in Figure 2, with frequency inline image = 100, for the various operators under consideration. (top left) inline image, (top right) inline image, (bottom left) inline image inline image, and (bottom right) inline image. Note the differences of scale that occur in these graphs.

2.2. Calderón Relation for Closed and Open Arcs

[11] As is well known Colton and Kress [1983], Nédélec [2001], and Colton and Kress [1997] second-kind formulations for closed surfaces (arcs) result from: (1) The classical jump relations associated with singular integral operators, as well as (2) application of the Calderón formula

display math

which shows that the composition of the closed-surface hypersingular and single-layer operators Nc and Sc equals a compact perturbation Kc of the identity.

[12] In the open-surface context however, jump relations cannot be exploited since, according to the boundary conditions (1), the same limits must be reached on both sides of the open surface Γ. Further, use of the combined operator NS does not lead to well-posed equations: as shown in Lintner and Bruno (manuscript in preparation, 2012), the image under NS of the constant function 1 is highly singular, with edge asymptotics NS[1](r) = O inline image, r ∈ Γ, where d(r) denotes the Euclidian distance to the edge. The combination NS is also problematic in the functional framework set forth in Stephan and Wendland [1984], Wendland and Stephan [1990], and Stephan [1987], since the image under S of the Sobolev space inline image(Γ) (as defined in Stephan [1987]) is larger than the domain of definition of inline image(Γ) (as defined in Stephan [1987]) of the operator N.

2.3. Regularity and Singular Behavior at the Edge

[13] The singular character of the solutions of equations (5) is well documented [Stephan, 1987; Costabel et al., 2003]: μ and ν can be expressed in the forms

display math

where d denotes the distance to the edge, χ1 and χ2 are smooth cut-off functions, and where the functions η and ζ are somewhat smoother than μ and ν. More recently furthermore, it was established in Costabel et al. [2003] that expressions such as (7) can be viewed as part of an expansion in powers of d1/2 which can be carried out as long as the degree of smoothness of the surface and the right hand side of the equations allow it. For example, if the curve itself and the functions f and g in (5) are infinitely differentiable, it follows that

display math

where α and β are infinitely differentiable functions throughout Γ, up to and including the endpoints [Costabel et al., 2003]. Thus the singular character of these solutions is fully characterized by the factors d1/2 and d−1/2 in equation (8).

3. Well-Posed Second-Kind Integral Equation Formulations

3.1. Weighted Operators

[14] In view of the regularity results (8), we introduce a positive integral weight ω(r) > 0 with asymptotic behavior ωd1/2 at the edge (by which it is implied that the quotient ω/d1/2 is infinitely differentiable up to the edge), and we define the weighted operators

display math

so that, in view of the discussion of section 2.3, for smooth functions f and g, the solutions of the equations

display math

are also smooth.

[15] Without loss of generality, we choose a smooth parameterization r(t) = (x(t), y(t)) of Γ defined in the interval [−1, 1], and we define the weight ω canonically as ω(r(t)) = inline image. As a result, the operators Sω and Nω give rise to the parameter-space operators

display math

and

display math

defined on functions φ and ψ of the variable t, −1 ≤ t ≤ 1; in these equations we have set τ(t) = inline image. Clearly, for φ(t) = α(r(t)) and φ(t) = β(r(t)) we have Sω[α](r(t)) = Sω[φ](t) and Nω[β](r(t)) = Nω[ψ](t).

3.2. Periodized Spaces and Regularity

[16] Introducing the changes of variables t = cos θ and t′ = cos θ′, and defining nθ = nr(cosθ), we obtain the periodic weighted single-layer and hypersingular operators

display math

and

display math

Clearly then, the solutions to the periodic equations

display math

where inline image(θ) = f(r(cos θ)) and inline image(θ) = f(r(cos θ)), are related to the solutions of equations (11) and (12) by the relations

display math

In view of (15), the solutions to equations (14) are smooth and periodic, and it is therefore natural to study the properties of inline image and inline image in the Sobolev spaces Hes(2π) of 2π periodic and even functions [cf. Yan and Sloan, 1988; Bruno and Haslam, 2007].

[17] Definition 1.

[18] Let s ∈ ℝ. The Sobolev space Hes(2π) is defined as the set of all 2π-periodic functions of the form inline image(θ) = inline imagea0 + inline imageam cos() for which the s-norm ∥ inline images2 = |a0|2 + 2 inline imagem2s |am|2 is finite.

[19] Clearly the set {cos(): n ∈ ℕ} is a basis of the Hilbert space Hes(2π) for all s.

3.3. Generalized Calderón Formula

[20] Given the smoothness of the solutions of the equations arising from the weighted periodic operators inline image and inline image, and in light of equation (6), it is reasonable to consider the composition inline image as a possible basis for solution of open-arc problems. As stated in the following theorem and demonstrated below, this combination does indeed give rise to a second-kind integral formulation. The statement of the theorem makes use of the concept of bicontinuity: an operator L between two Hilbert spaces is said to be bicontinuous if and only if L is invertible and both L and L−1 are continuous operators.

[21] Theorem 1.

[22] (i) For all s > 0 the operators inline image and inline image define bicontinuous mappings inline image: Hes(2π) → Hes+1(2π) and inline image: Hes+1(2π) → Hes(2π). Thus, the composite operator inline image is bicontinuous from Hes(2π) into Hes(2π) for all s > 0.

[23] (ii) The generalized Calderón formula

display math

holds, where inline image: Hes(2π) → Hes(2π) is a compact operator, and where inline image: Hes(2π) → Hes(2π) is a bicontinuous operator independent of the frequency k.

[24] (iii) The set σs of eigenvalues of inline image: Hes(2π) → Hes(2π) equals the union of the discrete set Λ = {λ0 = − inline image, λn = − inline imageinline image: n > 0} and a certain open set Λs which is bounded away from zero and infinity. The sets Λs are nested, they form a decreasing sequence, (Λsinline image, s > s′) and they satisfy inline image, where inline image denotes the closure of Λs.

[25] It follows that the open-arc TE and TM scattering problems (1) can be solved by means of the second-kind integral equations

display math
display math

respectively. The smooth and periodic solutions inline image and inline image of these equations are related to the singular solutions μ and ν of equations (5) via μ(r(cos θ)) = inline image(θ)/sin θ and ν(r(cos θ)) = ( inline image[ inline image](θ)) · sin θ. We thus see that, through introduction of the weight ω and use of spaces of even and 2π-periodic functions, a picture emerges for the open-surface case that resembles closely the one found for closed-surface configurations: the generalized Calderón relation (16) is analogous to the Calderón formula (6), and mapping properties in terms of the complete range of Sobolev spaces are recovered for inline image and inline image, in a close analogy to the framework presented in Kress [1999] and Nédélec [2001] for the operators Nc and Sc.

3.4. Eigenvalue Distributions

[26] In order to gain additional insights on the character of the various k ≠ 0 open-arc operators under consideration (namely, inline image, inline image, inline image as well as a generalization to non-zero frequencies of the operator introduced in Atkinson and Sloan [1991]), we consider their corresponding eigenvalue distributions. In Figure 1 we thus display the eigenvalues associated with these operators, for the spiral-shaped arc displayed in Figure 2 and described in section 6, as they were produced by means of the quadrature rules presented in section 5 and subsequent evaluation of matrix eigenvalues. The frequency was chosen to ensure a size to wavelength ratio inline image = 100, where L is the length of the arc and λ the wavelength of the incident wave. As expected, the eigenvalues of inline image tend slowly to zero, the eigenvalues of inline image are large, while the eigenvalues of inline image are bounded away from zero and infinity, and they accumulate at − inline image.

Figure 2.

TE-diffraction by a spiral-shaped arc of size L = 200 λ, for incidence angles of (top) 135° and (bottom) 45° from the positive x-axis. Figure 2 (top) shows internal reflections that enable the field to penetrate to the center of the spiral, giving rise to an interesting array of caustics. The relative error ϵr in both numerical solutions is no larger than 10−5.

[27] The k > 0 generalization of the second-kind equation introduced in Atkinson and Sloan [1991], whose operator eigenvalues are displayed in Figure 1 (bottom left), is obtained from right-multiplication of the single layer operator in equation (10) by the inverse of the flat-arc zero-frequency single-layer operator inline image (defined in equation (39) below); the resulting equation is given by

display math

This equation, which can be re-expressed in the form inline image, is a second-kind Fredholm integral equation: the operator inline image−1: Hes(2π) → Hes(2π) is compact. Unfortunately, the spectrum of the operator in equation (19) is highly unfavorable at high frequencies, as illustrated in Figure 1 (bottom left). Such poor spectral distributions translate into dramatic increases, demonstrated in Table 6, in the number of iterations required to solve (19) by means of Krylov subspace solvers as the frequency grows. In fact, a direct comparison with Table 3 shows that the second-kind integral equation (19) may require many more iterations at non-zero frequencies than the original first-kind equation (14).

4. Theoretical Considerations

[28] This section presents a proof of Theorem 1. The argument proceeds by consideration of the flat-arc zero-frequency case (sections 4.1 through 4.3) and subsequent extension to the general case (sections 4.4, 4.5 and 4.6). A more detailed discussion, including full mathematical technicalities, can be found in Lintner and Bruno (submitted manuscript, 2012).

4.1. Flat Arc at Zero-Frequency

[29] In the particular case where the frequency vanishes (k = 0) and the curve under consideration is the flat strip (Γ = [−1, 1]), the operator inline image reduces to Symm's operator

display math

whose well-known diagonal property in the cosine basis en(θ) = cos , [Yan and Sloan, 1988; Bruno and Haslam, 2007], namely

display math

plays a central role in our analysis. Note that, in particular, equation (21) establishes the bicontinuity of the operator inline image from Hes(2π) into Hes+1(2π). The corresponding property of bicontinuity (from Hes+1(2π) into Hes(2π)) for the weighted flat-strip, zero-frequency hypersingular operator

display math

results, in turn, from our study of the operator inline image in section 4.2.

[30] An integration by parts argument presented in Mönch [1996] gives rise to the factorization inline image where

display math

and

display math

As a result, for the flat-arc at zero-frequency case, the composite operator inline image, is given by

display math

4.2. The Operator inline image

[31] It is easy to evaluate the action of the operator inline image on the cosine basis: in view of equation (24), using (21) and expressing inline image[en] as a linear combination of cosines (which results from simple trigonometric manipulations) we obtain the relation

display math

Clearly then,

display math

where the operator inline image is defined by

display math

[32] The right hand side of equation (26) may appear to bear a direct connection with the classical closed-curve Calderón formula (6). Yet, unlike the operator Kc in equation (6), the operator inline image is not compact–see section 4.3. It is easy, however, to verify that inline image can also be expressed in the form

display math

and is therefore closely related to the Césaro operator inline image[η](x) = inline imageη(u)du. Since inline image is bounded (but not compact) from L2[0, b] into L2[0, b] [Brown et al., 1965] (where L2[0, b] denotes the space of square-integrable functions in the interval [0, b]), it follows that inline image: Hes(2π) → Hes(2π) is a continuous operator for all s > 0, and, therefore, inline image: Hes(2π) → Hes(2π) is also a continuous operator.

[33] Composing the operator inline image with inline image from the right on one hand, and from the left, on the other, and applying the two resulting composite operators to the basis en, shows that Ĩ0 is the inverse of inline image, and that Ĩ0 is a continuous operator from Hes(2π) into Hes(2π). It follows that inline image is a bicontinuous operator. As indicated in Section 4.1, finally, the bicontinuity of inline image0 from Hes+1(2π) into Hes(2π) follows directly from equation (24) and the corresponding bicontinuity properties of inline image and inline image.

4.3. Eigenvalues of inline image

[34] Re-expressing (25) in the form

display math

and making use the well-known expansion [Erdélyi et al., 1981; Mason and Handscomb, 2003]

display math

we obtain

display math

where the diagonal elements λn are given by

display math

In view of (29), the operator inline image can be viewed as an infinite upper-triangular matrix. The diagonal terms λn of this matrix are eigenvalues of inline image; the corresponding eigenvectors inline imagen, in turn, can be expressed in terms of a finite linear combination of the first n cosine basis functions: inline imagen = inline image. This establishes that the set Λ defined in Theorem 1 is contained in the spectrum σs for all s > 0.

[35] As is known, the diagonal elements of an infinite upper-triangular matrix are a subset of all the eigenvalues of the corresponding operator; for instance, the point spectrum of the upper-triangular bounded operator C*[a](n) = inline image, (the adjoint of the discrete Cesàro operator C) is given [Brown et al., 1965, chapter 2] by the entire disc |λ − 1| < 1. A similar situation arises for the operator inline image: searching for sequences {fn, n ≥ 0} such that f = ∑n fnen satisfies J0[f] = λf leads to the relations

display math

and

display math

The spectrum of inline image equals the set of all values λ for which the relations (33) and (34) admit solutions satisfying ∑|fnns|2 < ∞. This set is the union of the discrete set Λ (see point (iii) in Theorem 1), and the open set Λs defined by

display math

It follows in particular that the operator inline image is not compact from Hes(2π) into Hes(2π): if it were, in view of equation (26), the spectrum of inline image would be discrete, in contradiction with (35).

[36] We thus see that, even though inline image is not compact (so that, in particular, the decomposition (26) does not present inline image as the sum of a multiple of the identity and a compact operator), it follows from (32) and (35) that the eigenvalues of inline image are all bounded away from zero and infinity, and that they are tightly clustered around − inline image—in close analogy to the eigenvalue distribution associated with the closed-surface Calderón operator (6).

4.4. Properties of the Operator inline image

[37] The study of inline image in the general case (possibly curved arc, k ≠ 0) hinges on the decomposition

display math

of the kernel Gk, which results from the expression [Colton and Kress, 1997, p. 64]

display math

where J0(z) is the Bessel function and G(z) is analytic. Clearly, the functions A1 and A2 are smooth functions of t and t′, and the singular behavior of Gk thus resides entirely in a logarithmic term. In particular, the continuity of the operator inline image follows directly from the corresponding continuity of the operator inline image.

[38] To establish the invertibility of the operator inline image and the continuity of its inverse inline image−1, on the other hand, we consider the decomposition

display math

where the operator inline image, defined by

display math

differs from inline image only in the additional multiplicative term inline image[γ](θ) = γ(θ)τ(cosθ). Since the HesHes operator inline image−1 inline image is compact (in view of classical embedding results and the increased regularity of the kernel of the operator inline image) and since the operator inline image is injective (as it follows from results established in Stephan and Wendland [1984], Wendland and Stephan [1990], and Stephan [1987] for the operator (3)), the bicontinuity of inline image follows from the invertibility of inline image (and, thus, of inline image) together with an application of the Fredholm theory to the term in parenthesis in equation (38).

4.5. Properties of the Operator inline image

[39] As is known, for closed curves, the operator Nc (the normal derivative of the double-layer potential), can be expressed in terms of a composition of a single layer potential and derivatives tangential to the curve Γc [Colton and Kress, 1983, p. 57]. While for open surfaces the conversion of normal derivatives into tangential derivatives for the operator N gives rise to highly singular boundary terms (which arise from the integration-by-parts calculation that is part of the conversion process), the presence of the boundary-vanishing weight ω in our Nω operator eliminates all singular terms, and we obtain Nω[φ](t) = Nωg[φ](t) + Nωpv[φ](t) where

display math

and where

display math

[see Mönch, 1996]. Using the changes of variables t = cos θ and t′ = cos θ′ in equations (40) and (41), it follows that

display math

where

display math

and where, defining inline image,

display math

Note for future use that equation (44) can be re-expressed in the form

display math

where inline image[γ] = inline image[γ].

[40] In view of equation (42), the operator inline image equals the sum of inline imageg (which, like inline image, maps Hes(2π) into Hes+1(2π)) and inline imagepv. But, from (45), we see that inline imagepv can be expressed as the sum of inline image and a compact perturbation. Since, like inline image0, the operator inline image maps Hes+1(2π) into Hes(2π), it follows that inline imagepv is a bounded operator from Hes+1(2π) into Hes(2π), and, thus, inline image:Hes+1(2π) → Hes(2π) is a bounded operator as well.

[41] The bicontinuity properties of inline image can now be established by an argument similar to the one applied to inline image in the previous section using, this time, the identity inline image

4.6. Generalized Calderón Formula

[42] The generalized Calderón formula is obtained by expressing the combination inline image in the form

display math

where the operator inline image, given by inline image is compact in view of the classical Sobolev embedding theorems and the (increased) smoothness properties of the kernels of the operators inline image and inline image respectively. In view of the relation inline image inline image and inline image have the same spectrum: (λ, inline image) is an eigenvalue-eigenvector pair for inline image if and only if (λ, inline image[ inline image]) is an eigenvalue-eigenvector pair for inline image. Clearly, inline image has the same mapping properties and regularity as inline image ( inline image is bicontinuous from Hes(2π) into Hes(2π)), and therefore, the decomposition (46) shows that inline image equals the sum of an invertible bicontinuous operator and a compact operator; that is, inline image is a second-kind Fredholm operator.

5. High-Order Numerical Methods

[43] In this section we present spectral quadrature rules for the operators inline image, inline image and inline image which give rise to an efficient and accurate solver for the general open arc diffraction problem (1).

5.1. Spectral Discretization for inline image

[44] Use of the nodes inline image, n = 0, ⋯, N − 1, gives rise [Press et al., 1992, equations (5.8.7) and (5.8.8)] to a spectrally convergent cosine representation for smooth, π-periodic and even functions inline image:

display math

Thus, applying equation (21) to each term of expansion (47), we obtain the well-known spectral quadrature rule for the logarithmic kernel

display math

where

display math

Following Colton and Kress [1997], Martensen [1963], and Kussmaul [1969] we then devise a high-order integration rule for the operator inline image, noting first from (2) and (36) that

display math

where, letting R = |r(cos θ) − r(cos θ′)| we have

display math

and

display math

In view of (37) and the smoothness of the ratio inline image, the functions A1 and A2 are even, smooth (analytic for analytic arcs) and 2π-periodic functions of θ and θ′—and, thus, in view of (48), the expression

display math

provides a spectrally accurate quadrature rule. By making use of trapezoidal integration for the second term in the right-hand side of (50) we therefore obtain the spectrally accurate quadrature approximation of the operator inline image:

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5.2. Efficient Implementation

[45] The right-hand side of (53) can be evaluated directly for all θ in the set of quadrature points {θn, n = 0, …, N − 1} by means of a matrix-vector multiplication involving the matrix S(N) whose elements are defined by

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A direct evaluation of the matrix S(N) on the basis of (49) requires O(N3) operations; as shown in what follows, however, the matrix S(N) can be produced at significantly lower computational cost. Indeed, expressing the product of cosines in (49)) as a sum of cosines of added and subtracted angles, the quantities

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can be expressed in the form

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where the vector R(N) is given by

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Our algorithm evaluates this vector efficiently by means of an FFT, and produces as a result the matrix S(N) at an overall computational cost of O (N2 ln N) operations. This fast spectrally-accurate algorithm could be further accelerated, if necessary, by means of techniques such as those presented in Bleszynski et al. [1996], Bruno and Kunyansky [2001], and Rokhlin [1993].

5.3. Spectral Discretization for inline image

[46] In order to evaluate inline image we use (42), the first term of which is a single-layer operator which can be evaluated by means of a rule analogous to (53) and a rapidly computable matrix Ng,(N) (similar to S(N)) with elements

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To evaluate the second term in (42), in turn, we make use of the decomposition (44), and we approximate the quantity inline image by means of term per term differentiation of the sine expansion of the function inline image(θ)sin(θ) (which can itself be produced efficiently by means of an FFT). Since inline image is essentially the differentiation operator in the x variable,

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our solver evaluates the quantity inline image by invoking classical FFT-based Chebyshev differentiation rules [Press et al., 1992, p. 195].

6. Numerical Results

[47] The numerical results presented in what follows were obtained by means of a C++ implementation of the quadrature rules introduced in section 5 for numerical evaluation of the operators inline image and inline image (and thus, through composition, inline image), in conjunction with the iterative linear algebra solver GMRES [Saad and Schultz, 1986]. In all cases the errors reported were evaluated by comparisons with highly-resolved numerical solutions. All runs were performed in a single 2.2 GHz Intel processor.

6.1. Spectral Convergence

[48] To demonstrate the high-order character of the algorithm described in previous sections we consider the problems of TE and TM scattering by the exponential spiral x(s) = es cos(5 s), y(s) = es sin(5 s) of size inline image = 800, where L and λ denote the perimeter of the curve and the electromagnetic wavelength, respectively. Table 1 demonstrates the spectral (exponentially-fast) convergence of the TE and TM numerical solutions produced by means of the operators inline image, inline image and inline image for this problem (cf. equations (14), (17), and (18)); note from Figure 2 the manifold caustics and multiple reflections associated with this solution.

Table 1. Scattering by a Spiral-Shaped Arc of Size inline image = 400: Far-Field Errors
NTE Case, inline imageTE Case, inline imageTM Case, inline imageTM Case, inline image
30005.4 × 10−68.2 × 10−61.5 × 10−32.8 × 10−4
31004.5 × 10−84.9 × 10−81.0 × 10−55.5 × 10−6
33508.4 × 10−128.5 × 10−123.7 × 10−103.7 × 10−11

6.2. Limit of Closed Curves

[49] In order to obtain an indication of the manner in which an open arc problem can be viewed as a limit of closed-curve problems (and, in addition, to provide an independent verification of the validity of our solvers) we consider a test case in which the open-arc parabolic scatterer x = 1 − 2y2 is viewed as the limit as a → 1 of the family of closed curves x(s) = (1 − a)cos s + a cos(2 s), y(s) = sin(s) Using the closed-curve Nyström algorithms [Colton and Kress, 1997] we evaluate the TE fields scattered by these closed curves at k = 10 for values of a approaching a = 1. Figure 3 displays the k = 10 far fields corresponding to a = 0.9 and a = 0.99 side-by-side the corresponding far-field pattern for the limiting open parabolic arc as produced by the inline image-based open-arc solver. Clearly the closed-curve and open-arc solutions are quite close to each other. As might be expected, as a approaches 1 an increasingly dense discretization is needed to maintain accuracy in the closed-curve solution: for a = 0.9, 256 points where needed to reach a far field error of 10−4, while for a = 0.99 as many 1024 points were needed to reach the same accuracy—even for the low frequency under consideration. The corresponding open-arc solution, in contrast, was produced with 10−4 accuracy by means of a much coarser, 64 point discretization.

Figure 3.

A sequence of increasingly thin closed curves converging to the open parabolic scatterer x = 1 − 2y2 and corresponding far-field patterns. (top) Closed-curve with a = 0.9 and corresponding far field, (middle) closed-curve with a = 0.99 and corresponding far field, and (bottom) parabolic (open) arc and corresponding far field. Note the convergence of the far-field patterns as the closed scatterers approach the open parabolic scatterer.

6.3. Solver Performance

[50] The TE (Dirichlet) problem can be solved by means of either the left-hand equation in (14) or equation (17), which in what follows are called equations TE inline image and TE inline image, respectively. The TM (Neumann) problem, similarly, can be tackled by means of either the right-hand equation in (14) or equation (18); we call these equations TM( inline image) and TM inline image, respectively. Results for TE and TM problems obtained by the various relevant equations for two representative geometries, a strip [−1, 1] and the exponential spiral mentioned above in this section, are presented in Figures 2 and 4 and Tables 23456. In the tables the abbreviation “It.” denotes the number of iterations required to achieve an ϵr relative maximum error in the far field (calculated as the quotient of the maximum absolute error in the far field by the maximum absolute value of the far-field), “Mat.” is the time needed to build the inline image matrix given by equation (54), as well as (when required) the corresponding matrix for inline imageg which can be constructed in O (N2) operations from the inline image matrix (see section 5), and “Time” is the total time required by the solver to find the solution once the matrix is stored.

Figure 4.

Sample diffraction patterns. (top) TE scattering (total field) by an infinitely thin strip of size L = 200λ for horizontal left-to-right incidence. (The TM total field at this incidence equals the incident field.) (bottom) TM solution for a half-circle of size L = 200λ under normal incidence from below. Note the trailing shadow in the wake of the strip, and the caustics inside the circular reflector.

Table 2. Iteration Numbers and Computing Times for the TE (Dirichlet) Problem on the Flat Strip
inline imageNϵrMat.TE( inline image)TE( inline image)
It.TimeIt.Time
50400<10−5<1 s24<1 s8<1 s
2001600<10−54 s331 s82 s
8006400<10−554 s4518 s815 s
Table 3. Iteration Numbers and Computing Times for the TE (Dirichlet) Problem on the Spiral-Shaped Arc
inline imageNϵrMat.TE( inline image)TE( inline image)
It.TimeIt.Time
50400<10−5<1 s64<1 s46<1 s
2001600<10−54 s933 s628 s
8006400<10−555 s13658 s79158 s
Table 4. Iteration Numbers and Computing Times for the TM (Neumann) Problem on the Strip
inline imageNϵrMat.TE( inline image)TE( inline image)
It.TimeIt.Time
50400<10−5<1 s67<1 s9<1 s
2001600<10−54 s16016 s91 s
8006400<10−555 s298415 s917 s
Table 5. Iteration Numbers and Computing Times for the TM (Neumann) Problem on the Spiral-Shaped Arc
inline imageNϵrMat.TE( inline image)TE( inline image)
It.TimeIt.Time
50400<10−5<1 s202<1 s48<1 s
2001600<10−53 s43265 s638 s
8006400<10−555 s8491692 s83160 s
Table 6. Iteration Numbers and Computing Times Required for the Solution of TE Problems (on the Spiral-Shaped Arc Depicted in Figure 2) Using inline image−1 as a Preconditionera
inline imageNϵrIt.Mat. TimeSol. Time
504001.2 × 10−5124<1 s1 s
20016006.3 × 10−62933 s15 s
80064002.2 × 10−567256 s411 s

[51] As can be seen from these tables, the TM equation TM( inline image) requires very large number of iterations as the frequency grows and, thus, the computing times required by the low-iteration second-kind equation TM inline image are significantly lower than those required by TM( inline image). The situation is reversed for the TE problem: although, the corresponding second-kind equation TE inline image requires fewer iterations than TE inline image, the total computational cost of the second-kind equation is generally higher in this case—since the application of the operator in TE inline image is significantly less expensive than the application of operator in TE inline image.

6.4. Resonant Cavities

[52] We have found that interesting resonant electromagnetic behavior arises from diffractive elements constructed as almost-closed open-arcs. As can be seen in Figure 5, for example, circular and rocket-shaped cavities with small openings (a few wavelengths in size), can give rise to interesting and highly energetic field patterns within the open cavity. The number of iterations required for each of these configurations is of course much larger than for simpler geometries, such as the strip. Yet, overall reduction in number of iterations and computing times is observed when the equation TM inline image is used in lieu of TM( inline image). For the TE problem, once again TE inline image gives rise to faster overall numerics than TE inline image, although the latter equation still requires fewer GMRES iterations.

Figure 5.

(top) Field leakage and resulting resonances for a circular cavity of size inline image = 200 with a bottom aperture of size equal to the wavelength λ; TM polarization, 90° incidence. (middle) Same as Figure 5 (top) for a rocket-shaped cavity of perimeter L = 200 λ and a bottom aperture of size 6λ; TE polarization, 45° incidence. (bottom) Same as the Figure 5 (middle) but under 90° incidence—for which a strong resonance develops in the tail of the rocket.

7. Conclusions

[53] We have introduced new second kind equations and associated numerical algorithms for solution of TE and TM scattering problems by open arcs. The new open-arc second-kind formulations are the first ones in the literature that lead to reduced number of GMRES iterations consistently across various geometries and frequency regimes. The new second-kind formulation TM inline image is highly beneficial for the TM problem, giving rise to order-of-magnitude improvements in computing times over the hypersingular formulation TM( inline image). Such gains do not occur in the TE case: although, the second-kind equation TE inline image requires fewer iterations than TE inline image, the total computational cost of the second-kind equation is generally higher in this case—since the application of the operator in TE inline image is significantly less expensive than the application of operator in TE inline image. Generalization of these methods enabling efficient second-kind solution of problems of scattering by open-surfaces in three dimensions will be presented elsewhere (O. Bruno and S. Lintner, A high-order integral solver for scalar problems of diffraction by screens and apertures in three dimensional space, submitted to Journal of Computational Physics, 2012; available at http://arxiv.org/abs/1208.5173).

Acknowledgments

[54] The authors gratefully acknowledge support from AFOSR, NSF, JPL and the Betty and Gordon Moore Foundation.

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