A high-order method of moment solution with quadrature-point-based sampling is presented for the solution of the volume electric field integral equation for the scattering of inhomogeneous dielectric bodies. The proposed scheme efficiently allows for the material profile to be inhomogeneous within a curvilinear cell. It is demonstrated that the method leads to exponential convergence in both the radar cross section (RCS) and the volume current density. It is also demonstrated that the method can be more efficient than surface field integral equation formulations for thin-material scattering.
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 The analysis of the electromagnetic interaction with penetrable inhomogeneous materials is a problem with direct impact to a number of areas of research including medicine, geo-remote sensing, and stealth technology. Due to the necessity of volume discretization, the analysis of such problems has predominately been solved via partial differential equation based solvers that result in sparse linear systems such as the finite element method [Jin, 1993; Volakis et al., 1998] or the finite difference time domain [Taflove and Hagness, 2000]. With the advent of fast solvers such as the fast multipole method [Coifman et al., 1993], integral equation based solvers are becoming a competitive alternative [Lu and Chew, 2000]. In fact, when analyzing the electromagnetic interaction of thin material coatings or shells, volume integral equation methods combined with fast solution algorithms can be more efficient than finite-methods or surface integral equation approaches [Lu and Chew, 2000; Lu, 2001].
 The focus of this paper is the development of a high-order solution of the volume integral equation for the electromagnetic scattering of penetrable inhomogeneous dielectric objects. The proposed scheme leads to exponential convergence in both the radar cross section (RCS) and the volume current density. Previously, a high-order solution method based on the Locally Corrected Nsytröm (LCN) method [Canino et al., 1998] has been proposed for the scattering of two-dimensional inhomogeneous material volumes [Liu and Gedney, 2000, 2001]. More recently, a method of moment analogy to the LCN method via a quadrature-point-based discretization scheme has been proposed [Gedney, 2001, 2003] for the scattering by PEC surfaces. This method is extended herein to the high-order solution of the volume integral equation method.
 Point-based discretization schemes have been previously proposed for volume integral equation methods [e.g., Jin et al., 1989; Papagiannakis, 1997; De Doncker, 2001]. The scheme presented by De Doncker  is based on a potential formulation. The method employs parametric elements for accurate modeling of curved surfaces, arbitrary order basis functions chosen to be Lagrange interpolation polynomials, and a point-based sampling at the interpolation points. In contrast, the proposed scheme is based on the solution of the electric field integration equation (EFIE), and the point matching is done at the appropriate quadrature points. The quadrature point sampling is equivalent to a Galerkin scheme with a fixed-point integration of the outer integral, thus leading to exponential convergence [Gedney, 2001, 2003]. In the proposed formulation, inhomogeneous dielectrics with continuous profiles are also conveniently and efficiently modeled within a cell. This simplifies discretization, and provides a robust solution method for inhomogeneous materials. Finally, it is shown that when analyzing thin materials the volume integral equation (VIE) approach can be more efficient than a surface integral equation (SIE) solution approach [Poggio and Miller, 1973; Mautz and Harrington, 1979; Sheng et al., 1998]. Namely, fewer unknowns can be required by a VIE approach than a SIE approach for the same level of accuracy.
2. Volume Field Integral Equation
 An inhomogeneous and isotropic dielectric object with relative permittivity contained within a volume V and situated in a homogeneous free space is illuminated by an incident electromagnetic field with an ejωt time dependence. The scattered field resulting from the impinging wave can be computed through the use of equivalent volume currents that are distributed within V and that satisfy the electric field integral equation (EFIE) [Livesay and Chen, 1974; Schaubert et al., 1984; Peterson et al., 1998]:
where is the incident electric field in the absence of the dielectric scatterer ko is the free space wave number, ηo is the free space wave impedance, is the dyadic Green's function:
is the free-space Green's function, and is the equivalent polarization current density distributed over V which is defined as:
where is the total electric field in V.
 The solution to the EFIE in (1) will be obtained via the high-order method of moment scheme proposed by Gedney . To this end, V is discretized with volumetric curvilinear cells that represent the surface defining V to high order. Within each cell, the current is expanded using a set of smooth vector basis functions that are complete to polynomial order p − 1. A Gauss-quadrature rule of order 2p − 1 with abscissas and weights is introduced over each volume cell. At each quadrature point three-independent test vectors are introduced . A point-matched formulation is obtained by sampling (1) at the quadrature points, leading to:
It is shown by Gedney  that the formulation in (4) is equivalent to that employing test functions complete to polynomial order p-1 and evaluating the inner product with a fixed point quadrature rule. It is also shown by Gedney  to be equivalent to a locally corrected Nyström (LCN) formulation [Canino et al., 1998].
 When integrating over a volume cell Vn for which , the volume integral in (4) has a leading singularity of 1/R3 arising from the ∇∇ term in (2). The singularity can be reduced through the identity:
where . Since the Gaussian-quadrature point lies within V and not on S, the closed surface integral on the right-hand side of (5) is integrable. The volume integral now contains a 1/R2 singularity which is numerically tractable using a scheme such as the Duffy transformation [Duffy, 1982]. It was found that reducing the order of singularity further can accelerate the numerical computation of this integral. This is can be done by extracting the singular term as:
and where is the Jacobian, and χm is a constant defined as:
Exploiting the fact that , (6) can be rewritten as:
In this form, the volume integral has a 1/R singularity that is efficiently computed numerically using the Duffy transformation [Duffy, 1982].
 The volume V defined by the dielectric scatterer is assumed to be discretized by curvilinear cells. In this work, the cells are chosen to be curvilinear hexahedron, though the proposed scheme is not limited to this choice. Each cell is uniquely described by a local curvilinear coordinate system (u1, u2, u3). The three independent unitary vectors defined for this space are described as [Stratton, 1941]
where . The Jacobian for the volume integration is
 The current within each hexahedron is expanded via a set of vector basis functions weighted by constant coefficients bn as:
Each basis function is defined as the product of one-dimensional Legendre polynomials with support limited to the hexahedron weighted and directed along the unitary axis. Thus, a set of basis functions complete to polynomial order p − 1 are defined for an individual cell as:
where, Pl (u) is the lth order Legendre polynomial, Thus, there are a total of 3p3 basis functions per cell. (For simplicity, the same order is assumed for each one-dimensional Legendre polynomial in (14). However, this is not a limitation of the algorithm as each dimension can assume arbitrary order.) The test vectors are also chosen as the unitary vectors.
 A quadrature rule is also defined for each cell. For hexahedron, the rule is conveniently defined via a product rule of one-dimensional Gauss-Legendre quadratures. While optimal cubature rules exist with fewer points [Stroud, 1971; Cools and Rabinowitz, 1993], this choice of rule is the most appealing since it leads to p3 quadrature points. At each quadrature point the three unitary vectors are used for the test vectors. This leads to a total of 3p3 constraints per cell and consequently a square linear system results from (4).
4. Nyström Formulation
 The formulation presented in the previous two sections can be projected into a point-based discretization as defined by the Nyström method [Canino et al., 1998; Liu and Gedney, 2000, 2001]. This is done by mapping the current as represented by basis functions to the current at the quadrature points. To this end, (1) can be expressed in discrete form as:
where [D] is a block diagonal matrix arising from the leading term in (4) and [Z] is a dense matrix arising from the convolutional operator. The matrices [D] and [Z] can be broken up into nine blocks corresponding to the reaction of the test vector at with the basis vectors of the nth basis function.
 Consider the region where a cell c is sufficiently far from the observation point such that the kernel is smooth and the convolutional integrals can be approximated to sufficient accuracy by a fixed-point quadrature rule. Assuming a p3-point Gauss-quadrature rule, the volume integral in (4) over a cell c can be approximated as:
where are the abscissa points and ωq are the weights of the three-dimensional quadrature rule defined over the source cell, and , which is the discrete dyadic Green's function. From (16), the contribution to the mth row of the impedance matrix can be represented in vector form as:
where is the vector of constant coefficients weighting the basis functions defined on cell c, the ith row of the matrix [L]T represents the basis functions evaluated at the quadrature point , and the qth entry of the row vector . A change of variable is introduced such that:
where is simply the vector of currents evaluated at the quadrature points in cell c. Consequently, the contribution to the mth row of the impedance matrix can also be expressed in vector form as:
where, jq is a constant coefficient representing the current at quadrature point .
 In the near region, the convolutional integral in (4) must be performed via an adaptive quadrature to desired precision (using (10) for self-cells). In this region, the contribution to the impedance matrix is expressed in vector form as:
where the nth term of the row vector is
which is evaluated by adaptive quadrature to sufficient precision. Next, using the transformation of variables in (18), and (21) is rewritten as:
Equation (23) then represents the contribution to the mth row of the impedance matrix from near cells.
 Finally, after applying this transformation to the block diagonal matrix, [D] has entries:
 The proposed method has been validated through the analysis of the electromagnetic scattering by a number of canonical structures. For each case, the error in the calculated RCS and/or the total electric field is presented. The error in the RCS is calculated as the mean of the error relative to a known value over the range of discrete angles over a sphere. The known reference RCS is derived analytically for spheres or from numerical simulation from a fine discretization for nonseparable geometries. The mean absolute error in the total electric field is computed at the quadrature points as compared to a known solution. It is noted that the absolute error as opposed to relative error is presented for the electric field so that large relative errors at discrete nulls in the field do not dominate the global error estimation.
 Initially, consider the scattering by a piecewise homogeneous spherical shell with relative permittivity defined as:
Curvilinear hexahedrons that represent the spherical surfaces exactly were used to model the volume. Symmetry was not enforced in the simulation. The shell is excited by both vertical and horizontal polarized plane waves with λo = 1 m. To model the step discontinuity in the material profile, two layers of cells along the radial dimension were necessary. Figure 1 illustrates the mean error in the RCS relative to the exact Mie-series solution as a function of the total number of unknowns (N) for increasing order when 48 curvilinear cells and 108 curvilinear cells were used to model the geometry. The basis function order used in each cell is indicated next to each data point (e.g., 3 × 3 × 2). Note that the first two numbers represent the transverse coordinate orders and the third the radial coordinate. The RCS is converging to high order. It is noted that a solution with two digits of accuracy is obtained with 48 cells and order 3 × 3 × 2. The mean absolute error in the electric field in the sphere is presented in Figure 2. It is noted that the maximum Etot in the sphere is approximately 4.2 V/m. Comparing Figures 1 and 2, it is seen that the field converges with roughly the same order, but to a lower level of accuracy, as expected.
 A second example is the scattering by a spherical shell with a continuous radial inhomogeneity. The relative permittivity of the sphere is defined as:
where a = 0.3 m, and b = 0.5 m. The relative permittivity thus varies from 2.0 at the inner radius to 9.0 at the outer radius with a quadratic profile. The shell was modeled with only a single layer of curvilinear hexahedral cells (a total of 24 cells) that exactly represented the inner and outer surfaces. The volume current density and RCS were computed using the proposed high-order scheme. The bistatic RCS of the shell resulting from axially incident vertically and horizontally polarized waves is illustrated in Figure 3 for two different basis orders and is compared against the exact solution. For a basis order of 3 × 3 × 3, there is noticeable error in the RCS (about 1 dB on average). For a basis order of 4 × 4 × 4, the predicted RCS is indistinguishable from the exact solution at the given resolution of the graph.
 The mean relative error in the RCS as a function of unknowns for a single layer of 24 hexahedral cells and various orders is illustrated in Figure 4. The error is computed relative to the known analytical solution. Superimposed in this graph is the mean absolute error in the total electric field, again referenced to the analytical solution. Note that the maximum electric field magnitude in the shell was about 3 V/m. Exponential convergence is again realized for this case by the high-order method.
 The next case studied is a dielectric ogive shell. The geometry of this object is illustrated in Figure 5. The shell has a uniform thickness of 0.05 m and is composed of a homogeneous dielectric with εr = 2.56. The ogive was discretized using a single layer of curvilinear hexahedron (as illustrated in Figure 5). The mesh was generated using a commercial mesh generator and cubic elements were used. The bistatic RCS of the ogive in the θ = 90° plane is illustrated in Figure 6 for two different discretizations and are compared to a reference solution that was finely discretized. For this simulation, the frequency was 299.796 MHz and the incident field was incident on the tip of the ogive (θinc = 90, ϕinc = 0). Both vertical and horizontal polarizations were simulated. The results are indistinguishable for the resolution presented. The mean relative error in the bistatic RCS versus the total number of unknowns is presented in Figure 7. For this data, the error was computed relative to a finely discretized reference solution. For each curve, fixed discretizations were used, 24 cubic hexahedron and 56 cube hexahedron, and increasing order. For each data point, the basis order is provided on the graph (e.g., 4 × 4 × 2). The first two orders are the transverse coordinates and the third is the radial. Overall, the solution exhibits exponential convergence, though there is a stutter in the convergence for the 56-cell discretization.
 The final case presented is the scattering due to a lossy dielectric slab. The slab was rectangular in shape with dimensions 0.6 m × 0.6 m × 0.075 m. The thin slab had a homogeneous profile with εr = 3 − j0.09 and was illuminated by a plane wave with a frequency of 1 GHz. For comparison, the monostatic RCS of the slab was predicted using both the proposed high-order volume field method and a surface field integral equation solution method [Mautz and Harrington, 1979; Sheng et al., 1998; Liu and Gedney, 2003]. The surface field integral equation method used was based on a high-order point-based discretization solution scheme [Liu and Gedney, 2003]. Both the electric and magnetic surface current densities were modeled. A combined-field integral equation based on the Poggio, Miller, Chang, Harrington, Wu, and Tai (PMCHWT) formulation [Mautz and Harrington, 1979; Sheng et al., 1998] is then solved via the high-order method. The monostatic RCS predicted by these schemes is illustrated in Figure 8. For the surface formulation, the top and bottom surfaces were meshed using 25 square quadrilaterals. Each side was meshed with five rectangular quadrilaterals for a contiguous mesh of 70 rectangular cells. A product of 1-D Legendre polynomial basis functions was used to model the electric and magnetic currents on each patch. The basis order for the first curve calculated via the SIE is 5 × 5 for the upper and lower surfaces and 5 × 3 for the side patches (3 along the vertical dimension). This leads to a total of 6200 unknowns. The second set of results for the SIE assumed basis order of 5 × 5 for all patches (a total of 7000 unknowns). For the volume formulation, nine rectilinear hexahedron were used to discretize the slab. Basis function orders of 5 × 5 × 3, 5 × 5 × 5 and 7 × 7 × 3 were used, where the first two numbers indicate the horizontal dimensions and the third the thin vertical dimension. This leads to 2025, 3375, and 3969 unknowns, respectively. Observing Figures 8a and 8b, the SIE with 5 × 3 order on the thin side patches encounters significant errors as compared to the other simulations. For both discretizations, the SIE leads to significant error near grazing incidence (θ = 90°) as compared to the VIE. The mean difference in the VIE simulations is < 0.1 dB.
 A high-order method of moment solution with quadrature-point-based sampling was presented for the solution of the volume electric field integral equation for the scattering of inhomogeneous dielectric bodies. It was demonstrated through examples of the electromagnetic scattering of a number of canonical geometries that the method leads to exponential convergence in both the RCS and the volume current density. The proposed scheme efficiently allows for the material profile to be inhomogeneous within a curvilinear cell. This was also demonstrated through the example of the scattering of a spherical dielectric shell with a quadratic radial profile. It was also demonstrated that the method can be more efficient than surface field integral equation formulations for the scattering of thin-material structures.
 This work was funded by DARPA under grant MDA972-01-1-0022 with the University of Kentucky.