An efficient higher-order method of moments (MoM) solution of volume integral equations is presented. The higher-order MoM solution is based on higher-order hierarchical Legendre basis functions and higher-order geometry modeling. An unstructured mesh composed of 8-node trilinear and/or curved 27-node hexahedral elements is used for accurate representation of the scattering dielectric object. The permittivity of the object is allowed to vary continuously as a function of position inside each element. It is shown that the condition number of the resulting MoM matrix is reduced by several orders of magnitude in comparison to existing higher-order hierarchical basis functions. Consequently, an iterative solver can be applied even for high expansion orders. Numerical results demonstrate excellent agreement with the analytical Mie series solution for a dielectric sphere as well as with results obtained by other numerical methods.
 The integral equation approach is one of the methods traditionally employed for solving scattering and radiation problems involving dielectric materials [Peterson et al., 1998]. While application of a surface integral equation to this kind of problems is restricted to objects composed by multiple homogeneous dielectric regions [Goggans et al., 1994], the volume integral equation (VIE) can handle general inhomogeneous objects, for instance dielectric lenses. Owing to the three-dimensional nature of the problem, the number of unknowns N in the method of moment (MoM) solution of the VIE grows very fast with the size of the object. For the classical solution the required memory for the MoM matrix storage is O(N2), the solution complexity is O(N3) for a direct matrix solver, and O(N2) for an iterative one. One way to reduce the complexity of the problem is to apply fast iterative solvers, such as fast multiple methods (FMM/MLFMM) [e.g., Chen and Jin, 1999; Sertel and Volakis, 2001; Lu, 2001] or conjugate-gradient algorithms combined with the fast Fourier transform (CG-FFT/BCG-FFT) [e.g., Catedra et al., 1989; Zhang et al., 2003; Millard and Liu, 2003], which do not compute and store the whole MoM matrix but only a matrix-vector product. These methods can reach the solution complexity and memory requirements of O(NlogN), but they usually require a very fine discretization of the object and low-order basis functions (pulse [e.g., Su, 1993], rooftop [e.g., Gan and Chew, 1995], or volume RWG [e.g., Lu, 2003]) in order to be efficient.
 Another way to reduce the excessive CPU and memory requirements of the MoM solution for the VIE is to reduce the number of unknowns N in comparison to low-order methods. A considerable reduction of N can be accomplished by applying higher-order methods involving higher-order basis functions and higher-order geometry modeling. There are two classes of higher-order basis functions, interpolatory [Graglia et al., 1997] and hierarchical [Kolundžija and Popović, 1993]. Hierarchical functions provide greater flexibility, enabling different expansion orders in different elements in the same mesh. The main bottleneck of existing hierarchical basis functions is the ill-conditioning of the MoM matrix which requires a direct solver to be employed [Notaroš et al., 2001]. To apply an iterative solver, which is much more effective, the condition number of the MoM matrix must be improved. Recently, a new type of higher-order hierarchical basis functions was proposed by Jørgensen et al. . Calculated on the basis of appropriately modified Legendre polynomials, the new basis functions are near-orthogonal and therefore provide a low condition number of the MoM matrix. These functions have been successfully applied to surface integral equations for the analysis of metallic objects in free space [Jørgensen et al., 2002] and in layered media [Kim et al., 2002].
 In this paper, we apply the higher-order hierarchical Legendre basis functions by Jørgensen et al.  to solve the electric field VIE. Numerical examples for scattering by a homogeneous dielectric sphere demonstrate excellent agreement between the results of our higher-order hierarchical MoM and the analytical Mie series solution. A very low condition number of the MoM matrix allows the use of an iterative matrix solver even for high expansion orders. A homogeneous dielectric cube serves as an example of a scatterer with corner singularities. The result with a very fine discretization and first-order basis functions is used as a reference. The memory requirements of the higher-order hierarchical MoM are compared with those of an existing multilevel fast multipole method (MLFMM) implementation with low-order basis functions [Sertel and Volakis, 2001]. Results for the scattering by a Luneburg lens are also presented.
 The electric field VIE is written in terms of the vector potential A(r) and the scalar potential Φ(r) for a generally inhomogeneous isotropic dielectric object as [Notaroš and Popović, 1996] (the time dependence ejωt is assumed and suppressed)
where Ei(r) is the incident electric field, D(r) is the electric flux density, G(r, r′) is the free space Green's function, J(r) is the unknown induced current density in the volume V, k0 is the wave number of the free space, and S is the surface enclosing the volume V. The VIE (equation (1)) can be solved with respect to the induced current density J(r). However, it is more convenient to apply the electric flux density D(r) as the unknown because its normal component is continuous across the boundary between two different dielectric materials. Hence the substitution
is applied. The discretization of the VIE (equation (1)) with higher-order MoM involves three steps. First, the dielectric object is accurately represented by trilinear and/or curved hexahedral elements which are defined by 8 and 27 nodes, respectively. Lagrange interpolation is employed to define a parametric transformation from a cube in a curvilinear coordinate system (u, v, w) to the hexahedron in the physical (x, y, z) space, see Figure 1. It is noted that the permittivity inside each element is not necessarily constant but may vary continuously as a function of position. Second, the unknown electric flux density D(u, v, w) in each local curvilinear (u, v, w) coordinate system is expanded as
where au, av, and aw are the covariant unitary vectors. Each of the contravariant components (Du, Dv, Dw) of D(r) are expanded in terms of the higher-order hierarchical Legendre basis functions as [Jørgensen et al., 2004b]
where (ξ, η, ζ) are (u, v, w), (v, w, u), or (w, u, v). In the expansion (4), ��(ξ, η, ζ) is the Jacobian of the parametric transformation, Pm(ξ) are Legendre polynomials, and bmnqξ are unknown coefficients. Mξ, Nη, and Qζ denote the expansion orders along the three parametric directions. The scaling factors Cmξ, Cnη, and Cqζ are chosen such that the Euclidean norm of each function is unity for a unit cubic element. The basis functions are divergence-conforming, i.e., they satisfy continuity of the normal component of the electric flux density at a face shared by two adjacent elements. Despite the necessary modifications to enforce the continuity, the expansion (4) maintains almost perfect orthogonality of the basis functions. Owing to the hierarchical property of the basis functions the expansion order can be selected separately in each hexahedron depending on the electrical size of the element. This allows large smooth objects to be represented by rather large curvilinear hexahedra with high expansion orders while fine parts of the geometry are precisely modeled by small hexahedral elements with low expansion orders. In this way, the number of unknowns can be minimized as opposed to low-order discretization schemes (such as volume RWG or rooftops) that require small elements throughout the mesh. Furthermore, independent selection of the expansion order along the direction of the electric flux density and along the transverse directions provides additional flexibility to the discretization technique. Consequently, even geometries represented by elements that are far from cubic in shape, for instance thin dielectric shells or antenna radomes, can be treated in an efficient manner without introducing unnecessary unknowns. The third and final step of the MoM procedure employs Galerkin's testing to transform the VIE (equation (1)) into a system of linear equations.
 The singularities arising in the integrals in equation (1) when the integration point r′ approaches the observation point r are integrable. However, when integrated numerically these integrals need a special treatment. For the self-term evaluation we use a numerical technique based on the Duffy transform [Duffy, 1982], which was adapted to volumetric hexahedral elements by Sertel and Volakis . The integration domain is transformed into eight subdomains (four subdomains in the surface integral) with the singularity at the origin. Owing to the change of integration variables, the singularity is annihilated. Thus the obtained integrands are well-behaved and present no difficulties for numerical integration. This numerical procedure is very flexible and independent of the basis functions applied.
 Special attention should be given to the selection of the expansion orders Mξ, Nη, and Qζ. Although generally, all nine expansion orders can be chosen independently, we apply the Nedelec constraint [Nedelec, 1980] imposing Nξ = Qξ = Mξ − 1. Thus only three parameters Mu, Mv, and Mw are selected independently. A more detailed discussion about the selection of the expansion orders can be found in the work of Jørgensen et al. [2004a].
 Each element of the MoM matrix requires a number of double volume, volume-surface, and surface-surface integrals to be evaluated numerically [Sertel and Volakis, 2002]. This task is performed using the Gauss-Legendre quadrature integration scheme. The order of the Gaussian integration rule is not fixed for a particular geometrical element; it depends on the electrical size of the geometrical element as well as the distance between geometrical elements for which the integrals are evaluated.
3. Numerical Results
 The first example involves scattering by a homogeneous dielectric sphere with relative permittivity ɛr = 4.0. The sphere is centered at the origin of a rectangular xyz coordinate system and illuminated by an x-polarized plane wave propagating in the z direction. The electrical size of the sphere is k0a = 2.0 where a is the radius. The sphere is represented by 32 hexahedral elements, both trilinear and curved. The number of unknowns, the RMS error, and the 2-norm condition number of the MoM matrix for expansion orders varied from Mξ = 2 to Mξ = 5 are shown in Table 1. The RMS error is computed for the bistatic radar cross section (RCS) for 181 angles in both E and H planes relative to the exact analytical Mie series solution [Wiscombe, 1980]. As it can be seen from Table 1, the result of the higher-order MoM converges to the exact solution as the expansion order Mξ (and corresponding number of unknowns) increases. It is also observed that an RMS error less than one percent is achieved with Mξ = 3.
Table 1. Results for a Homogeneous Sphere (k0a = 2.0; ɛr = 4.0)
 In Table 1 two sets of the 2-norm conditions numbers are presented. We calculated them for the MoM matrices produced respectively by the higher-order hierarchical Legendre basis functions of this paper and the existing higher-order power basis functions by Notaroš et al. . If a direct solver is applied, the accuracy of the solution is exactly (to the machine precision) the same in both cases for the same number of unknowns because the two sets of basis functions span the same polynomial space. However, the higher-order basis functions presented here provide a very well-conditioned MoM matrix system for all expansion orders, while the condition numbers for the power basis functions by Notaroš et al.  increase rapidly, approximately two orders of magnitude for each expansion order. This considerable advantage was achieved due to improved orthogonality of the higher-order Legendre basis functions (as discussed in the work of Jørgensen et al. [2004b]) as well as by careful selection of the scaling factors Cmξ, Cnη, and Cqζ.
 To validate the higher-order MoM for objects with corners and edges we consider a dielectric cube of dimension 0.5λ0 × 0.5λ0 × 0.5λ0 with ɛr = 4.0. The parameters of the incident plane wave are the same as in the above example, and the edges of the cube are aligned along the axes of a rectangular xyz coordinate system. Figure 2 shows the bistatic RCS computed with 2 × 2 × 2 = 8 cells and the expansion order Mξ = 3, which corresponds to 756 unknowns. For the reference RCS the rooftop basis functions (Mξ = 1) on a finely discretized mesh (33 × 33 × 33 = 35937 cells) are employed. Figure 3 displays the RMS error as a function of the number of unknowns for several mesh densities. Two sets of curves are presented. The first set shows how the result converges for a given mesh size as the expansion order (and hence, the number of unknowns) increases. For curves in the second set the expansion orders are kept constant while the mesh is refined. It is observed that the curve for Mξ = 2 has higher slope, e.g., the error decays much faster when the second-order expansion is used instead of first-order (rooftop) expansion. For instance, to achieve the RMS error of one percent the second-order solution requires approximately ten times less unknowns than the first-order solution. The solution with Mξ = 3 converges even faster.
 In the next example, a spherical shell with an outer radius a = 2λ0 and thickness 0.2λ0 is considered. The relative permittivity of the shell is ɛr = 2.75 − j0.3. In contrast to the previous examples, in which the expansion order was fixed for all hexahedra in all directions, the expansion order is here allowed to vary depending on the electrical size of each element. Thus the expansion orders are chosen to be Mξ = 4 for the components tangential to the shell surface and Mξ = 2 for the radial components. 96 curvilinear hexahedral elements are used in total to represent the geometry of the shell. The solution with 10752 unknowns requires 884 Mb of memory. The bistatic RCS is plotted in Figure 4. Excellent agreement with the exact Mie series result is observed.
 For this example, the MLFMM solution with first-order basis functions [Sertel and Volakis, 2001] requires 24642 hexahedra, 73962 unknowns, and 2GBytes of memory to achieve the same accuracy. Thus, despite the higher memory requirements of MoM (O(N2)) in comparison to MLFMM (O(NlogN)), the large reduction in the number of unknowns provided by the higher-order technique in comparison with the low-order technique results in a lower memory consumption.
 In the last example, scattering by a Luneburg lens is considered in order to validate the higher-order MoM for inhomogeneous dielectric object with continuous variation of the permittivity. The Luneburg lens possesses the property of focusing parallel rays to a point, or, conversely, transforming rays from a point source into a beam of parallel rays. The relative permittivity of a Luneburg lens varies as function of the radial coordinate r as [Luneburg, 1944]
 The bistatic RCS, normalized by πa2, in the E and H plane for two Luneburg lenses of radii a = λ0 and a = 2λ0 are presented in Figure 5. Owing to the symmetry of the problem, only one half of the RCS plot in each of the planes is shown on the same polar plot. Both lenses are represented by 575 hexahedral elements, and the maximum expansion order for the first lens is Mξ = 2 and Mξ = 4 for the second. Reference results also shown in Figure 5 are obtained by other numerical methods. For the first lens the reference result is computed using a finely stratified sphere model by Kai and Massoli ; the second reference result was obtained by Greenwood and Jin  by the finite-element method for bodies of revolution (FEM-BOR). A small discrepancy in the backlobe level in Figure 5b occurs at a low level of −30dB relative to the maximum. The discrepancy is most likely caused by different accuracies of the two different numerical results. In general, a good agreement between the results of the higher-order MoM and the reference results is observed.
 A new higher-order method of moment technique for volume integral equations is presented. The technique is based on higher-order hierarchical Legendre basis functions and higher-order geometry modeling. The improved orthogonality of the basis functions allows the condition number of the resulting MoM matrix to be reduced by several orders of magnitude in comparison to existing higher-order hierarchical basis functions and consequently, an iterative solver can be applied even for high expansion orders. 8-node trilinear and/or 27-node curved hexahedral elements are used for accurate representation of dielectric objects. Numerical examples for scattering by a dielectric sphere and a shell demonstrate excellent agreement between the results of our higher-order hierarchical MoM and the analytical Mie series solution. The higher-order convergence is demonstrated in the examples involving a dielectric sphere and a cube. While for extremely large scattering problem the low-order MLFMM will still be more memory efficient, it is shown that for fairly large scattering problems the higher-order hierarchical MoM requires much fewer unknowns and less computer memory than the low-order MLFMM.
 The Danish Technical Research Council and the Danish Center for Scientific Computing are acknowledged for supporting this work.