The problem of electromagnetic scattering by composite metallic and dielectric objects is solved using the coupled volume-surface integral equation (VSIE). The method of moments (MoM) based on higher-order hierarchical Legendre basis functions and higher-order curvilinear geometrical elements is applied to transform the VSIE into a system of linear equations. The higher-order MoM provides significant reduction in the number of unknowns in comparison with standard MoM formulations using low-order basis functions, such as RWG functions. Owing to the orthogonal nature of the higher-order Legendre basis functions, the continuity condition at the interface between metal and dielectric can be satisfied explicitly, which further reduces the number of unknowns as well as improves the accuracy of the solution. Numerical results for a metallic sphere with dielectric coating show excellent agreement with the analytical Mie series solution. Scattering by more complex metal-dielectric objects is also considered to compare the presented technique with other numerical methods.
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 The volume-surface integral equation (VSIE) [Sarkar and Arvas, 1989; Lu and Chew, 2000] for the electric field is applied to solve radiation and scattering problems for composite metallic and dielectric objects. The VSIE involves a system of two coupled equations for a dielectric volume and a perfectly electrically conducting (PEC) surface. The VSIE formulation allows full-wave three-dimensional analysis of, e.g., microstrip antennas taking into account the finite-size effects of the dielectric substrate and a ground plane. The VSIE is also suitable for modeling of scattering by metal objects with dielectric coating. Due to its volume integral equation part the VSIE can easily handle inhomogeneous dielectric objects, which are otherwise very inefficient to treat with a surface integral equation (SIE).
 The VSIE has been solved previously with the standard method of moments (MoM) [Sarkar and Arvas, 1989; Lu and Chew, 2000] as well as with its accelerated versions, including the multilevel fast multipole method (MLFMM) [Lu, 2001b], the adaptive integral method (AIM) [Ewe et al., 2004], and the precorrected fast Fourier transform method (P-FFT) [Nie et al., 2005]. All these methods utilize low-order basis functions, e.g., pulse, rooftop, or RWG, to discretize the integral equation. In this paper we solve the VSIE using higher-order basis functions defined on higher-order curvilinear volume and surface elements. Unlike the accelerated solvers, which reduce the complexity of the standard MoM but leave the number of low-order basis functions unchanged, higher order basis functions reduce the number of unknowns, since they generally require much fewer unknowns per wavelength to achieve the same accuracy. Here we utilize higher-order hierarchical Legendre basis functions [Jørgensen et al., 2004b], which have previously been applied to the analysis of metallic objects in free space [Jørgensen et al., 2002] and in layered media [Jørgensen et al., 2004a] by surface integral equations, and to the analysis of dielectric objects by volume integral equations [Kim et al., 2004]. Being near-orthogonal these functions allow a low condition number of the MoM matrix to be achieved. Moreover, we show here that the higher-order Legendre basis functions are able to explicitly enforce the continuity condition for the surface charge density on a PEC surface and the normal component of the electric flux density in the dielectric at those parts where the PEC surface is in contact with the dielectric material. Explicit enforcement of the continuity condition not only reduces the number of unknowns but also improves the convergence of the solution. The exact Mie series expansion for a dielectrically coated PEC sphere is used to validate the presented technique. Numerical results for more complex electrically large scatterers are also given and compared with the results obtained by other numerical methods.
2. Volume-Surface Integral Equation and the Continuity Condition
 The volume-surface integral equation formulation for the electric field comprises two coupled integral equations. The first equation expresses the total electric field E(r) as a sum of the incident electric field Ei(r) and the scattered electric field Es(r) in the dielectric volume V, and the second equation imposes the boundary condition for the electric field on the PEC surface S as
where EVs(r) and ESs(r) are the electric fields scattered by the unknown induced electric volume current density JV(r) and the electric surface current density JS(r), respectively, and is a unit normal vector. We use the electric flux density D(r) as the unknown instead of JV(r) because the normal component of D(r) is continuous across the boundary between two different dielectric materials.
 If the object contains interfaces between dielectric and PEC surfaces, an additional boundary condition enforcing the continuity between the surface current density on the PEC surface and the normal component of the electric flux density in the dielectric can be introduced as
where D1(r) and D2(r) are the electric flux densities on both sides of the PEC surface, 1 are 2 are unit vector normals to the PEC-dielectric boundary pointing into corresponding media, and ω is the angular frequency. The time factor ejωt is assumed and suppressed throughout the paper. We can distinguish three cases:
 1. The dielectric object is located only on one side of an open PEC surface. This can be for instance a patch printed on a dielectric substrate. Then, with D2(r) being the unknown electric flux density in the dielectric object and D1(r) being the total field on other side of the PEC surface, the continuity condition (2) can be rewritten as
where ɛb is the permittivity of the background medium. The combination of (3) and the VSIE (1) leads to an overdetermined system, and none of the unknowns can be eliminated. In this case, it is more convenient to solve the VSIE (1) alone without enforcing (2).
 2. Parts of the dielectric object are located on both sides of an open PEC surface. In this case, unknowns associated with either D1(r) or D2(r) can be eliminated due to (2), as will be shown in Section 3 below.
 3. If the dielectric object is in contact with a closed PEC surface, the continuity condition (2) simplifies as
Then, as will be shown below, the unknowns associated with D(r) can also be eliminated.
3. Higher-Order Hierarchical Legendre Basis Functions in the VSIE Formulation
 To discretize the integral equation the geometry of the object is first represented by higher-order curvilinear elements: hexahedra for dielectric volumes and quadrilaterals for PEC surfaces. In each element a unique mapping between the local curvilinear coordinate system (u, v, w) ((u, v) for quadrilaterals) and the physical space coordinates (x, y, z) is established using Lagrange interpolation. Then, each contravariant component of the unknown function, D(r) or JS(r), is expanded in the local curvilinear coordinate system, (u, v, w) or (u, v), in terms of the higher-order hierarchical Legendre basis functions [Jørgensen et al., 2004b; Kim et al., 2004]
where (ξ, ζ, η) is (u, v, w), (v, w, u), or (w, u, v) for hexahedrals, and (ξ, ζ) is (u, v) or (v, u) for quadrilaterals, with −1 ≤ u, v, w ≤ 1. In (5),
��(ξ, ζ, η) and ��S(ξ, ζ) are the Jacobians of the parametric transformation for hexahedra and quadrilaterals respectively, Pm(ξ) are Legendre polynomials, amnqξ and bmnξ are unknown coefficients, and Mξ, Nζ, and Qη denote the expansion orders along the parametric directions. Galerkin's testing procedure is applied to transform the integral equation into a system of linear equations. Selection of the expansion orders, numerical treatment of integrals, and singularities arising in the integral equations (1) are discussed in [Jørgensen et al., 2004a; Kim et al., 2004]. Here, we show how the continuity condition can be satisfied explicitly for cases 2 and 3, as defined in the previous section, using the favorable orthogonality properties of the higher-order hierarchical Legendre basis functions.
 Although the expansions of D(r) and JS(r) in (5) are not strictly orthogonal, they yield orthogonal expansions of the corresponding charge densities. For instance, the surface charge density ρSξ(ξ, ζ) associated with the surface current component JSξ(ξ, ζ) is represented by a series of orthogonal Legendre polynomials Pm(ξ) as
Similarly, from (5) and (6) the normal component of the electric flux density D(r) at the face ξ = 1 of a hexahedral element can be written as
where cξ is the covariant unitary vector.
 Consider now case 2 in which two dielectric hexahedral elements are separated by a PEC quadrilateral (see Figure 1). By substituting the expansions (7) and (8) into (2) we can explicitly express the coefficients a1,mnq of the electric flux density over the face of the first hexahedral element in terms of the coefficients a2,mnq for the second hexahedral element and the coefficients bmn associated with the electric currents on the PEC quadrilateral. For instance, for the face w = 1 of the first hexahedral element and the face w = −1 of the second hexahedral element we can write
From this equation, the result for a closed PEC surface (case 3) follows immediately by setting a2,0nqw = 0. Thus, the number of unknowns in the MoM system can be reduced.
 It is worth noting that for other types of higher-order hierarchical basis functions, which do not possess the favorable feature of orthogonality as (7) and (8) (for instance power basis functions), it is much more difficult to satisfy the continuity condition explicitly. The continuity condition can be enforced explicitly also with the RWG basis functions [Lu and Chew, 2000]. However, in accelerated solvers it is usually satisfied only numerically for the sake of implementation simplicity [Lu, 2001a].
4. Numerical Results
 In the first example the scattering of a plane wave by a dielectrically coated PEC sphere is considered. The exact Mie series solution for this problem is used to validate the technique presented in this paper. The sphere of radius a = 0.5λ0 is coated by a shell made of lossless dielectric with relative permittivity ɛr = 2.56 and thickness d = 0.25λ0. In this case, 54 second-order quadrilaterals and 54 second-order hexahedral elements are utilized to represent the sphere and the shell, respectively. The expansion order Mξ = 3 results in 5346 unknowns after eliminating 486 unknowns due to the enforcement of the continuity condition (4). Figure 2 illustrates an excellent agreement between the computed bistatic radar cross section (RCS) and the Mie series result.
 To study the accuracy in detail, the root-mean-square (RMS) error of the RCS is defined as
where Ns is the number of sampling points (bistatic observation angles), and σref and σVSIE are the reference and computed RCS measured in dB, respectively. The number of unknowns and the RMS error for different current and geometry expansion orders are shown in Table 1. It can be seen that the enforcement of the continuity condition improves the accuracy and the convergence of the solution with respect to the current expansion order. Since the MoM based on higher-order basis functions implies rather large geometry discretization elements, special attention should be paid to the modeling of curvilinear surfaces. As it follows from the results in Table 1, the second-order quadrilaterals and hexahedral elements do not approximate the spherical surfaces accurately enough, which deteriorates the convergence with respect to the current expansion order. In this case, at least the third-order geometrical elements should be utilized.
Table 1. Results for the Dielectrically Coated PEC Sphere Versus the Expansion Order Mξ (a = 0.5λ0; d = 0.25λ0, ɛr = 2.56)
Expansion Order Mξ
Continuity Condition Enforced
RMS error, dB
Continuity Condition Neglected
RMS error, dB
 In the next example the presented technique for the VSIE is investigated for increasing size of a dielectrically coated PEC sphere. The radius a of the PEC sphere is varied from 0.43λ0 to 1.52λ0, while the thickness d = 0.1λ0 of the dielectric coating is kept constant. The relative permittivity of the coating is ɛr = 1.5 − j0.5. As the radius of the sphere increases the mesh is refined so that the size of the geometry discretization elements are about the same in all cases. For each value of the radius, the convergence with respect to the current expansion order is obtained. The generalized minimal residual (GMRES) method with restart after 30 iterations is utilized to solve the MoM system. No preconditioner is applied. The statistics for the geometry and current discretization, the number of unknowns, the 2-norm condition number, the RMS error for the bistatic RCS, and the number of iterations to converge to the relative residual error of 10−3 are summarized in Table 2. As opposite to the previous example the solutions with the continuity condition enforced produce somewhat larger RMS error, which can be attributed to the small thickness of the coating. In Table 2 it is observed that the condition number and the number of iterations weakly depends on the number of unknowns, indicating the good conditioning of the MoM matrices. Although the condition number is nearly the same for the cases with and without the continuity condition enforced, the solution with the continuity condition takes less number of iteration to converge. The eigenvalue decompositions of the corresponding matrices show that the enforcement of the continuity condition changes the distribution of the eigenvalues, while their maximum and minimum values remain unchanged. The distribution of the eigenvalues for the case a = 0.65λ0 is plotted in Figure 3. It is seen that enforcement of the continuity condition groups the eigenvalues spread between 3 and 30 (Figure 3a) into a compact cluster around 10 (Figure 3b). Thus, the convergence of the solution improves.
Table 2. Results for the Dielectrically Coated PEC Sphere Versus Its Radius a (d = 0.1λ0, ɛr = 1.5 − j0.5)
Radius a, λ0
Number of quadrilaterals
Number of hexahedrals
Continuity Condition Enforced
Number of unknowns
Number of iterations
RMS error, dB
Continuity Condition Neglected
Number of unknowns
Number of iterations
RMS error, dB
 The third example involves a PEC disk of diameter Dd = 0.5λ0 enclosed in a dielectric cylinder (Figure 4) and illuminated by a plane wave propagating along the center line. The diameter of the cylinder is Dc = 0.5λ0 and the height is Hc = 0.6λ0. The relative permittivity of the dielectric cylinder is ɛr = 2.0. This example illustrates case 2, i.e., parts of the dielectric object are located on both sides of the open PEC surface. Thus, the continuity condition (2) is applied. The bistatic RCS computed with 12 third-order quadrilaterals, 48 third-order hexahedral elements, and Mξ = 3 current expansion order is shown in Figure 5 along with the reference result obtained with a SIE approach [Jørgensen et al., 2004b]. The results are in a good agreement. The total number of unknowns in the VSIE is 4344 with 108 unknowns excluded by the continuity condition.
 In the fourth example a cubic PEC cavity without top wall filled with dielectric is considered. The cavity side length is 0.5λ0 and the relative permittivity of the dielectric is ɛr = 2.3. Since the cavity does not form a closed object the continuity condition is not applied here. Figure 6 presents the RMS error for the bistatic RCS as function of the number of unknowns with the current expansion order Mξ as a parameter. The reference result is computed with the VSIE approach with Mξ = 4 current expansion order, 125 first-order quadrilaterals, and 125 first-order hexahedral elements, yielding 29160 unknowns in total. It is seen that better convergence is achieved for the higher current expansion orders. This fact is more clearly illustrated in Figure 7, in which the RMS error is plotted versus the inverse of the discretization element size h. For each current expansion order a line indicating an approximate decrease rate of the error is also drawn.
 The fifth example is a composite metal-dielectric sphere combined of PEC and dielectric hemispheres covered by two dielectric hemispherical shells, as shown in Figure 8. This is a challenging modeling problem involving both flat and curved surfaces, a PEC wedge, and junctions of up to four different materials. The scattering of a plane wave is simulated with the presented VSIE formulation as well as with a SIE approach [Jørgensen et al., 2004b], and the excellent agreement between the two methods is illustrated in Figure 9. The VSIE solution employs 96 quadrilaterals with the current expansion order Mξ ranging from 2 to 4 and 224 hexahedrals with the current expansion order from 2 to 6. This results in 37612 volume unknowns and 2248 surface unknowns. By explicitly enforcing the continuity at the interface between metal and dielectric, 1160 unknowns can be eliminated, yielding 38700 in total.
 The final example involves a 25.4 mm-thick trapezoidal PEC plate covered around the edges by 50.8 mm of lossy dielectric with ɛr = 4.5 − j9.0. The geometry and the coordinate system are shown in Figure 10. The problem is used to validate the presented VSIE formulation for objects with corners and sharp edges. The mesh for the object is created with 422 quadrilaterals and 56 hexahedral, and up to 5th order expansions are utilized in the solution, yielding 10496 unknowns in total. The enforcement of the continuity condition allows 780 unknowns to be eliminated. In Figure 11 the monostatic RCS obtained from the present method at 1 GHz is compared with the result of the MLFMM for SIE given in [Donepudi et al., 2003]. The discrepancies observed at the minimum levels are most likely due to different θ steps utilized in the simulations. Apart from this, the agreement between the results of two different numerical techniques is very good.
 The coupled volume-surface integral equation is solved using the method of moments based on higher-order hierarchical Legendre basis functions and higher-order curvilinear geometrical elements. By using the favorable orthogonality properties of the higher-order Legendre basis functions the continuity condition on the PEC-dielectric interface is explicitly enforced. The given numerical examples demonstrate that the explicitly enforced continuity condition not only reduces the number of unknowns but also improves the accuracy and the convergence of the solution.
 The Danish Technical Research Council and the Danish Center for Scientific Computing are acknowledged for supporting this work. The authors would like to thank Jianming Jin for the reference result for the trapezoidal plate.