A high-order tangential basis algorithm for electromagnetic scattering by curved surfaces
Introduction
Understanding of many physical phenomena and processes in atmospheric science, climatology, and astronomy can be enhanced through simulation of electromagnetic waves scattered by non-convex particles such as atmospheric aerosols, dust in planetary rings, ice crystals, or interstellar dust [15], [30], [37]. Scattering simulations also play an important role in medical sciences because scattering is an important tool, for example, in classifying bi-concave blood cells [22], [40] and in image-guided neurosurgery using medical imaging, for example, of tumor models [12], [21], [41], [42]. These obstacles are in general dielectric. However, developing algorithms to simulate scattering by such perfectly conducting bodies is a crucial first step towards development of spectrally accurate algorithms for electromagnetic scattering by three-dimensional dielectric bodies.
In this work we construct tangential vector basis functions on the surfaces of these particles, and present a spectrally accurate high-order algorithm to simulate the interaction of electromagnetic waves with a perfectly conducting three-dimensional obstacle D situated in a homogeneous medium with vanishing conductivity, the free space permittivity and permeability H/m, where is the speed of light. We develop the algorithm using a class of surface parametrization (described in Section 2.2) that includes a major class of model deterministic and stochastic surfaces used in the literature.
The electromagnetic waves, with angular frequency ω (rad/s) and wavelength (m), scattered by D are described by the electric field intensity with units and the magnetic field intensity with units [34, Section 1.8]. In this work we consider time harmonic electromagnetic waves, which can be represented asThe complex vector fields , , which describe the magnitude, direction and phase of the electric and magnetic field intensities, satisfy the time harmonic Maxwell equations [9, p. 154]with wave number , the Silver–Müller radiation conditionand the perfect conductor boundary conditionwhere denotes the unit outward normal at on the surface of D and is the incident electric field. In our experiments the incident field , is the plane wavewhere is the direction of the plane wave and its polarization.
Of particular interest is the radar cross section (RCS) [24] of the obstacle D, measured by a receiver with polarization situated in the directionwhere denotes the unit sphere, which is defined as is the electric far field. Two types of RCS are of particular interest: (i) The RCS for all directions , with a fixed incident direction ; (ii) The RCS for all directions , with varying incident directions . These are the bistatic and monostatic RCS respectively [24]. Co-location of the transmitter and receiver in the monostatic RCS requires, unlike in the bistatic case, solution of the Maxwell equations with thousands of distinct boundary conditions of the form (1.4).
The Mie-series solution is useful for checking the accuracy of our algorithm on spherical scatterers. For checking the accuracy of our algorithm on non-spherical scatterers, we also consider the boundary conditionswhere the boundary data is induced by the electric and magnetic dipole solutions and of the Maxwell equations, which describe radiation from a point source with polarization located at [9, (6.20), (6.21), p. 163]:andwhereis the fundamental solution of the Helmholtz equation.
Although scattering problems have a long pedigree, there is substantial interest in establishing reliable, high-order and fast algorithms for their solution. Difficulties arise in this practically important problem due to the shape of curved scattering surfaces and the frequency of the incident wave (or more precisely the electromagnetic size of the scatterers). This is because many established electromagnetic scattering algorithms for small to medium-sized obstacles (say one to one hundred times the incident wavelength) require setting up and solving complex dense linear systems with several thousand to several million unknowns. For example, to compute the bistatic radar cross section of a perfectly conducting sphere of diameter 48 times the incident wavelength at 7.2 GHz, the industrial standard boundary element/method of moments based FISC (Fast Illinois Solver Code) algorithm requires 2,408,448 unknowns to achieve 0.33 dB relative error, measured in a root mean square (RMS) norm [33, p. 29]. Furthermore, as discussed in the preface of [29, p. vi], the effects of the approximation of smooth boundaries is not well understood for Maxwell’s equations.
For high-frequency problems (with obstacle size tens of thousands of times the incident wavelength) one may use appropriate asymptotics (physical optics or Kirchhoff approximation [27]) for the illuminated region and special approximations for the shadow and transition zones to reduce the number of unknowns and the number of integrals required for the discretization, obtaining accuracy that increases as the wave number k increases (specifically, accuracy for ). Such an approach has recently been implemented and analyzed in various ways for high-frequency acoustic scattering (using the Helmholtz equation) by single and multiple two dimensional convex obstacles in [3], [5], [10], [13], [20], [25] and related references therein. However, for high-frequency acoustic scattering by three-dimensional convex obstacles, one encounters substantial difficulties in using such approximations and computations, for example in finding stationary points of various types, and efficient approximations in shadow and transition regions. For recent but limited progress in this area for acoustic scattering by three-dimensional convex obstacles, we refer to [4], [19] and references therein. However, development and realization of a similar asymptotic approach for simulation of scattering by three-dimensional non-convex obstacles is an open problem.
In this work, we use direct (as opposed to asymptotic) methods to develop a spectrally accurate algorithm to simulate the interaction of electromagnetic waves with small to medium electromagnetic-sized deterministic and stochastic obstacles. Spectrally accurate algorithms to approximate the analytic electric and magnetic fields, which are naturally tangential on associated scattering surfaces, have recently been developed in [17], [18] and were mathematically proven to yield spectral accuracy. A large set of simulation results in [17], [18] demonstrate that the algorithms are more efficient than industrial standard electromagnetic scattering algorithms such as FastScat [7], FISC [33], FE-IE and CARLOS [1], and FERM and CICERO [39], for a class of benchmark obstacles. The methods in [17], [18] are based on finite dimensional ansatz spaces that include non-tangential basis functions, leading to the dimension of the ansatz space being three times that required for the acoustic counterpart methods in [16] (due to one normal and two tangential components). The algorithms in [17], [18] are based on a modification of the standard electric field surface integral reformulation of the Maxwell equations that admits non-tangential approximations to the tangential surface electric field.
The three times computational penalty in [17], [18] is substantial for medium frequency scattering and for monostatic RCS computations, for which linear systems are to be set up and solved with thousands of different right hand sides, arising due to co-location of the transmitter and receiver. In [18], the penalty is avoided if the scatterer is a sphere, for which spectrally accurate tangential vector basis functions are well known and widely used for the analytic Mie-series solution [9], [29]. Algorithms based on ansatz spaces spanned by just surface tangential vector fields lead to only a two times computational penalty compared with their acoustic counterparts [16]. Construction of ansatz spaces spanned by vector basis functions that are tangential on general curved surfaces and development of algorithms utilizing these spaces, in which tangential electric fields can be approximated with spectral accuracy, is a significant challenge.
In this work we describe such ansatz spaces, and subsequently develop a new algorithm that gives high-order spectral accuracy with two-thirds of the unknowns required by the algorithms in [17], [18]. We demonstrate this important advancement of the spectrally accurate algorithms in [17], [18] with numerical experiments using various deterministic and stochastic surfaces. The reduction in unknowns (and hence reduced memory usage) allows us to simulate electromagnetic waves with the excellent accuracy achieved in [17], [18] at frequencies up to one hundred percent higher, using a small cluster computing environment similar to that used in [17], [18].
Our ansatz space is based upon tangential vector spherical harmonics that arise via the surface-gradient and surface-curl of spherical harmonics. Such vector spherical harmonics are widely used in Mie series [36, Chapter 9] and T-matrix [28] computations for electromagnetic scattering by spheres and non-spherical particles respectively, where they arise as components of fundamental solutions of the Maxwell equations that are defined throughout the exterior of the scatterer. In our boundary integral equation (BIE) algorithm, transformed vector spherical harmonics (with a tangential-like property on a given non-spherical surface) are used to approximate a vector potential on a spherical reference surface, which yields the exterior field via a surface integral representation. In contrast, in the Mie and T-matrix computations vector spherical harmonics are used to approximate the exterior field throughout the exterior of the scatterer.
The truncated T-matrix is often computed using the null-field method [28]. Null-field method based T-matrix computations are numerically unstable, and round-off errors become significant when the dimension of the truncated T-matrix is large. Consequently, such computations can become divergent for large and/or highly non-spherical particles [28, p. 543]. Such instability problems are avoided by, for example, using slow extended precision arithmetic to minimize the effect of round-off errors [28, p. 544].
It is known [26, Section 7.9.4] that numerical instability involved in the null-field method can be avoided by using a BIE to compute the T-matrix. The only disadvantage of using a BIE to compute the T-matrix is that this approach requires solving a large number of (boundary integral) linear systems with a fixed scattering matrix (obtained by discretizing the associated boundary integral operator) but different right hand sides (corresponding to each wave function used in expanding the incident field). Consequently, it is crucial to develop a spectrally accurate scattering algorithm that requires fewer unknowns, allowing utilization of the LU-factorization of the scattering matrix. In this work, we develop such an algorithm, and application of the algorithm to compute the T-matrix in a stable way for various non-spherical obstacles will be considered in a future work.
Section snippets
Surface integral equation reformulations
In this section, we describe analytical reformulations of the electromagnetic scattering problem involving surface integral operators, and a class of parametrization of the scatterer’s surface. Let and denote the space of all continuous and uniformly Hölder continuous tangential vector fields on with the supremum and Hölder norm respectively. Let denote the subspace of consisting of all r-times continuously differentiable functions on with norm .
Ansatz spaces
The crucial component of our algorithm is the development of a suitable sequence of ansatz spaces for the Galerkin scheme to approximate electromagnetic fields that are tangential on .
We have the following requirements from each ansatz space: (i) spectrally accurate approximation of tangential functions on the parametrized surface (that is, for any the ansatz space should contain a function with , where N represents the dimension of the ansatz space); (ii) is
Exterior field and far field computations
Using (2.5), (3.23), the far field corresponding to is given bywhereHence the representation (3.19) yieldsWe compute a spectrally accurate approximation to using the approximation to described in [18, Section 4.1], which we denote . Our approximation is thenThe
Numerical experiments
We compute approximations to the surface current, exterior field, and monostatic and bistatic RCS by solving the electromagnetic scattering problem (1.2), (1.3), (1.4) at various frequencies. The boundary condition (1.4) is induced by the plane wave (1.5) with incident unit vector direction , given by (1.6) and with polarization that is either horizontal (H, where ) or vertical (V, where ) [35, pp. 8–9].
In the case of spherical
Conclusions
In this work we have utilized a new tangential basis that gives spectrally accurate approximation to simulate electromagnetic scattering by perfect conductors using only about twice as many unknowns as related acoustic scattering algorithms, and considerably fewer unknowns than industrial standard algorithms. We have demonstrated that this reduction in unknowns allows solution of problems at up to one hundred percent higher frequency than was possible using previous spectral Galerkin methods
Acknowledgments
Support of the Australian Research Council is gratefully acknowledged.
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