# EM modelling of arbitrary shaped anisotropic dielectric objects using an efficient 3D leapfrog scheme on unstructured meshes

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## Abstract

The standard Yee algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of the classical Yee scheme to 3D unstructured meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This allows the problem of accuracy losses, which are normally associated with the use of the standard Yee scheme and a staircased representation of curved material interfaces, to be circumvented. The 3D dual mesh leapfrog-scheme which is presented has the ability to model both electric and magnetic anisotropic lossy materials. This approach enables the modelling of problems, of current practical interest, involving structured composites and metamaterials.

## Keywords

Anisotropic Co-volume Finite differences Unstructured mesh## 1 Introduction

In anisotropic materials, the electromagnetic material parameters, such as permittivity, permeability and conductivity, may vary in the different crystal directions, so that they must be treated as tensors. It is assumed that already 1000 years ago, before the invention of magnetic compasses, Vikings used crystals, a naturally occuring anisotropic material, in Norse sagas referred to as sunstones to navigate on open water on cloudy days. In accordance to researchers these sunstones could have been calcite crystals where anisotropy leads to the phenomenom of birefrigence (crystalline materials with different indices of refraction with different crystallographic directions). Their sunstone came within 1 % of the true location of the sun [1]. Nowadays anisotropic materials offer many new and interesting perspectives in engineering. A thin anisotropic coating may, for example, significantly change the radar cross section of an aircraft. Composites, anisotropic materials with applications initially limited to stealth bombers, satellites and space shuttles become part of our everyday life. Due to their advantages with respect to mechanical strength and weight compared to metals they are now used in civil aircrafts, trains, automobiles, trucks, sports equipment and so on. Especially in plane and cars electromagnetic compatibility is an issue which can be dealt with using numerical simulations. Other applications are the design of patch antennas where anisotropy can be used as design parameter [2]. Furthermore anisotropy is the basis of metamaterials, which are new materials with electromagnetic properties which cannot be found in nature, e.g. a material may have a negative index of refraction, which could be employed in the design of invisible cloaking devices [3].

Analytical solutions to wave propagation problems in electromagnetics are mainly restricted to problems involving simple geometrical shapes and diagonal, uniaxial or biaxial, tensors [4, 5]. Numerical techniques are required for the solution of the majority of problems, which involve arbitrary shaped objects. The second order accurate standard Yee algorithm implemented on a pair of staggered orthogonal Cartesian meshes is often the favored computational solution technique because of it’s low operation count and its low storage requirements. Unfortunately in the case of a curved boundary where the physical boundary doesn’t conform to the orthogonal mesh a very fine mesh is required due to errors induced by the stair stepping edges. In earlier work [6], we demonstrated the capability of a generalized Yee algorithm adapted to unstructured meshes to accurately model the radar cross section (RCS) of arbitrarily shaped lossy dielectric objects. For isotropic cases our method shows significant savings with respect to memory and time with respect to the standard FDTD scheme due to the unstructured mesh we employ. Here, we describe the extension of the method to deal with anisotropic materials, such as composites.

There are generally two methods to deal with anisotropic materials. Firstly you use the constitutive equation to replace the displacement field in Maxwells equations by the electric field [7]. Another possibility is to obtain the displacement field and afterwards use it in the constitutive equations. We adopt the latter method which has been proposed by [2]. This approach was originally presented within the context of a total field formulation, but the unstructured mesh extension adopted here employs a scattered field formulation.

## 2 Problem formulation

*A*, \(\mathrm {d}\mathbf A \) is an element of surface area, directed normal to the surface, \(\mathrm {d}\mathbf l \) is an element of contour length, in the direction of the tangent to the curve,

*t*denotes time and \(\bar{\bar{I}}\) is the unit matrix. In addition, \({\bar{\bar{\varepsilon }}}\) is the electric permittivity tensor, \(\bar{\bar{\mu }}\) is the magnetic permeability tensor, \(\bar{\bar{\sigma }}\) and \(\bar{\bar{\sigma }}_m\) are the electric and magnetic conductivity tensors respectively and \(\varepsilon _0\) and \(\mu _0\) denote the electric permittivity and magnetic permeability of free space respectively.The subscripts \((.)_{inc}\) and \((.)_{scat}\) refer to incident and scattered field components, with the total fields regarded as being formed as the sum of the corresponding incident and scattered fields. The vectors \(\mathbf D \), \(\mathbf E \), \(\mathbf B \) and \(\mathbf H \) represent the electric flux density, or displacement field, the electric field, the magnetic flux and the magnetic field respectively. The constitutive equations, relating the electric field intensity \(\mathbf E \) to the electric flux density \(\mathbf D \) and the magnetic field intensity \(\mathbf H \) to the magnetic flux density \(\mathbf B \), may be expressed as

## 3 Discrete equations

*i*th Delaunay edge corresponds to the projection, \((D_{scat,i},E_{scat,i})\), of the scattered electric field onto the direction of the edge, as illustrated in Fig. 1a. The unknown at the centre of the

*j*th Voronoi edge corresponds to the projection, \((B_{scat,j},H_{scat,j})\), of the scattered magnetic field onto the direction of the edge, as illustrated in Fig. 1b. In the scattered field formulation, the incident field is a known function, while the scattered field is unknown. At the interface boundaries, the material parameters in the Eqs. (1) and (2) are not constant. In this case, we average the values of \({\bar{\bar{\varepsilon }}}\), \(\bar{\bar{\mu }}\), \(\bar{\bar{\sigma }}\) and \(\bar{\bar{\sigma }}_{m}\) at a dielectric interface, leading to the values \({\bar{\bar{\varepsilon }}}_{av}\), \(\bar{\bar{\mu }}_{av}\), \(\bar{\bar{\sigma }}_{av}\) and \(\bar{\bar{\sigma }}_{m_{av}}\). The method for determining these averaged values is detailed in Sect. 5. Direct discretization of Ampère’s Law and Faraday’s Law then leads to the equations

*n*denotes an evaluation at time level \(t=n\triangle t\), \(l_{i}^{D}\) represents the length of the

*i*th Delaunay edge and \(A_{i}^{V}\) corresponds to the area of the Voronoi face spanned by the Voronoi edges surrounding Delaunay edge

*i*. Similarly, \(l_{j}^{V}\) represents the length of the

*j*th Voronoi edge and \(A_{j}^{D}\) corresponds to the area of the Delaunay face spanned by the Delaunay edges surrounding Voronoi edge

*j*. The numbers \(j_{i,k}\), \(k=1,\ldots ,M_{i}^{V}\) refer to the \(M_{i}^{V}\) edges of the Voronoi face corresponding to the

*i*th Delaunay edge, while the numbers \(i_{j,k}\), \(k=1,\ldots ,M_{j}^{D}\) refer to the \(M_{j}^{D}\) edges of the Delaunay face corresponding to the

*j*th Voronoi edge. In addition, \(\left\langle \mathbf F ,{\hat{\mathbf{e }}_{i}} \right\rangle \) denotes the dot product (projection) of any field vector \(\mathbf F \) along the \(i^{th}\) edge. We will use both \(\left\langle \mathbf F ,{\hat{\mathbf{e }}_{i}} \right\rangle \) and \(\mathbf F \cdot {\hat{\mathbf{e }}_{i}}\) to represent the scalar product between a field and a unit edge vector. The projection of the scattered electric field vector onto Delaunay edge

*i*is denoted by \(E_{scat,i}\), while \(\mathbf E _{scat}|_i\) denotes the scattered electric field vector at the location of the

*i*th Delaunay edge. Defining the quantities

*i*th Delaunay edge and the

*j*th Voronoi edge respectively. These field values have to be determined from their corresponding stored projections and this calculation, which is not direct, is described in detail in Sect. 6. In contrast to the isotropic case, these equations cannot be updated in one step, as vector-matrix multiplications are involved. These staggered equations are used to advance the solution in a leapfrog manner. The magnetic field is updated over the dual mesh at the half time step, using Eq. (11), and the electric field is updated over the primal mesh at the full time step, using Eq. (12).

The scheme is based upon the projections of the field unknowns at the edge centres. This allows us to use unstructured meshes and to model electromagnetic scattering, even for objects of arbitrary shape. Full details of the mesh generating process that is employed, to ensure Delaunay and Voronoi meshes with the correct properties, may be found elsewhere [8, 9, 10]. Here, the additional challenge that is faced, is that we have no direct access to the full field vectors \(\mathbf B \), \(\mathbf H \) \(\mathbf D \) and \(\mathbf E \) at a given location. Nevertheless, we are able to get accurate values for the approximated field vectors, that are needed for the matrix-vector multiplication of the updating equations, and the results are then projected to the corresponding Delaunay edge \({\hat{\mathbf{e }}}_{i}\) or Voronoi edge \({\hat{\mathbf{e }}}_{j}\). This updating process is explained in detail in Sect. 8.

## 4 Mesh generation

Structured meshes that are employed for wave propagation problems are generally constructed to have a uniform element size, \(\delta \), that is related to the characteristic wavelength, \(\lambda \), of the problem. A value of \(\delta \) in the range \(\lambda /30\) to \(\lambda /10\) is typical for practical applications of the Yee scheme on regular cartesian grids [11]. Consider now the problem of generating a body fitted mesh of this form for use with the co-volume algorithm outlined above. The computational domain surrounding a general scattering obstacle is discretised employing a hybrid mesh, which is generated in four stages. In the first stage, an unstructured triangulation of the surface of the scatterer is produced [12] and the triangulation is then placed inside a hexahedral box. The region inside this box is discretised using a regular cartesian mesh of cubes, of a prescribed edge length \(\delta \). Cubes within a prescribed distance of the scatterer, or lying internal to the scatterer, are removed in a second stage, to create a staircase shaped surface that completely encloses the scatterer. In the third stage, a point distribution is specified to completely cover the unmeshed region, with those points from the distribution that lie either inside the scatterer or outside the staircase surface being removed. The fourth stage consists of using the points that remain to generate an unstructured tetrahedral mesh in the region between the surface triangulation of the scatterer and the staircase surface. The problem of fitting the hexahedral and tetrahedral meshes is overcome by placing a pyramidal element on each exposed square face, leading naturally to a consistent mesh. The unstructured mesh is optimised to ensure that both the primal and the dual mesh are of the highest possible quality. The approach adopted is to relax the requirement that a dual edge must be a bisector of the corresponding Delaunay edge. At the same time, the corresponding dual mesh vertex is moved to a point which still ensures orthogonality between the two grids and which lies inside the corresponding primal element. Primal elements with a common circumcentre, and hence a corresponding Voronoï edge length of zero, will automatically be merged during the solution process, creating general polyhedral cells.

## 5 Boundary conditions

### 5.1 PEC boundary conditions

### 5.2 Material interface boundary conditions

When the boundary is an interface between two different media, the update Eqs. (1) and (2) require integration across the interface. These integrals are evaluated by assigning a weighted average value to the material parameters, based upon the mesh structure.

*q*,

*l*) of the material property tensors is evaluated, using weighted average formulae, as

*q*and

*l*can take the values 1, 2 or 3, corresponding to the

*x*,

*y*or

*z*directions respectively, while \(M_i^V\) refers to the number of Voronoi edges surrounding a given Delaunay Edge

*i*. As there are two sub-volumes associated to each Voronoi edge, we have to sum over \(2M_i^V\) Voronoi edges. These account for the contribution of the material parameter assigned to each of the cells surrounding Delaunay edge \((.)_{Del,i}\), weighted by a coefficient \(w_{k}\) that corresponds to the volume spanned by the two endpoints of the Delaunay edge, the intersection point of the Voronoi edge with the Delaunay face and the position of the circumcentre of the cell. For example, to obtain \(\left. {\varepsilon _{av}}_{q,l}\right| _{Del,i}\), in Fig. 3a, \(w_{1}\) would be the volume spanned by the points

*N*1,

*N*2,

*C*1 and

*Fi*1, while \(\left. {\varepsilon _{q,l}}\right| _{Cell,1}\) would correspond to the permittivity component of the element

*Cell*1. The points labelled

*N*belong to the Delaunay mesh, the points labelled

*C*are the circumcentres of the corresponding cells and

*Fi*refers to the intersection point of the Voronoi edge with a face spanned by Delaunay edges.

*g*1 and

*g*2, inside

*cell*1 and

*cell*2 are the distances between the intersection point

*Fi*1 of the Voronoi edge \((.)_{Vor,j}\) with the Delaunay face and the circumcentre of the cell. For example, in Fig. 3b, the distance between the points

*C*1 and

*Fi*1 is \(g_{1}\).

## 6 Obtaining approximated field vectors from edge projections

*i*and Voronoi edge

*j*respectively. Unfortunately, we cannot get exact full field vector components from field to edge projections. However, we can approximate the full field components at any location in the mesh. To achieve this, we assume that, in \({\mathbb {R}}^3\), with a set of three orthogonal vectors \(\mathbf v _1,\mathbf v _2,\mathbf v _3\), a general vector \(\mathbf x \) can be reconstructed as

*i*, denoted by the red line in Fig. 4a, we have to construct the set of equations

*i*. The matrix \({\bar{\bar{\mathbf{P }}}}\), based upon the

*x*,

*y*,

*z*components of the vector \({{\hat{\mathbf{e }}}}_i\) and the projections \(\mathbf D _{scat} \cdot {{\hat{\mathbf{e }}}}_i \) are known.

*i*is approximated by considering the sum of the system in Eq. (18) for each Delaunay edge connected to that node, as depicted in Fig. 4b. In this case, we solve the system

*N*is the number of Delaunay edges connected to the node and \(e_{i_1}=e_{i_x}\), \(e_{i_2}=e_{i_y}\) and \(e_{i_3}=e_{i_z}\). We solve this system of equations locally, node by node, until we have an approximated field vector at all nodes of the Delaunay mesh. Finally, we have to link the computed field vector to the corresponding edges. A Delaunay edge is assumed to connect the two nodes \(n_1\) and \(n_2\) and the displacement field vector, associated to the Delaunay edge

*i*, is obtained by averaging the electric field vectors at nodes \(n_1\) and \(n_2\), i.e. \(\mathbf D _{scat,i}=\left( \mathbf D _{scat,n_1}+\mathbf D _{scat,n_2}\right) \)/2. The same procedure is applied for approximating the magnetic flux vectors \(\mathbf B _{scat}\), but using now the Voronoi edges. However, unlike the Cartesian Yee scheme, where we have one update equation for each component of the field vectors, the approximations of the full vector fields obtained using this method are not good enough for time iterating the field components. It is found that error accumulation causes the algorithm to become unstable. In the following sections, we suggest how this difficulty may be circumvented.

## 7 Local orthogonal unit vectors

When using the integral formulation of Maxwell’s equations, it is not the displacement field vector \(\mathbf D _{scat}\) that is updated, but rather its projection, \(D_{scat,i}=\mathbf D _{scat}\cdot \mathbf e _i\), onto a Delaunay edge \(\mathbf e _i\). In the case of an isotropic material, the electric permittivity and the magnetic permeability are scalars, so that updating the fields only involves scalar multiplication between the field projections and the scalar material properties. For an anisotropic material, the integrals in Ampère’s and Faraday’s laws contain matrix-vector multiplications between material tensors \(({\bar{\bar{\varepsilon }}}, \bar{\bar{\mu }}, \bar{\bar{\sigma }}, \bar{\bar{\sigma }}_{m})\) and the fields \((\mathbf D _{scat},\mathbf B _{scat})\). To deal with these matrix-vector multiplications, we first create two linearly independent vectors for each Delaunay and Voronoi edge. Using the stabilized Gram-Schmidt orthonormalization procedure, we generate three orthonormal vectors, as illustrated in Fig. 5. The first vector \(\mathbf e _{1}\), to which the two linearly independent vectors \((\mathbf e ^{\prime }_{2}, (\mathbf e ^{\prime }_{3})\) are added, remains unchanged during the whole process. Each set of three orthogonal vectors represents one local coordinate system, leading to as many local systems as Delaunay and Voronoi edges. Due to the discretisation, we can reconstruct approximated field vector components for each local coordinate system, using the field projections of the surrounding Delaunay or Voronoi edges, as explained in Sect. 6. The field vectors, obtained in this way, are projected onto the two orthogonal vectors forming the local frame. The first vector of each subset, which remains unchanged during the orthonormalization procedure, can be immediately updated using the projection equation employed in the isotropic case. For this projection, no error is induced by the field averaging.

### 7.1 Coordinate transformation

*x*,

*y*,

*z*) to the global coordinate system. Each component of the Jacobian can be interpreted as an amplification factor, describing how one coordinate in a given reference frame stretches, shrinks or rotates with respect to another coordinate in another reference frame. In our case, the Jacobian is pure a rotation matrix \({\bar{\bar{J}}}_R\) which can be directly calculated as

## 8 Time updating scheme

## 9 Validation

*x*direction. In each case, the electrical length of the sphere is \(2\lambda _0\). The mesh employed is illustrated in Fig. 6 and has an average edge length of \(\lambda _0/20\). For each example, the PML region is located at a minimum distance of \(\lambda _0\) from the scatterer and the PML is discretised using 10 layers of hexahedra. The minimum distance between the inner boundary of the PML and the surface of the scatterer is represented by 8 cells. The complete mesh consists of 876, 116 cells, 1, 673, 527 Delaunay edges and 2, 076, 019 Voronoi edges. The distribution of the radar cross section of the cross- and co-polarized scattered waves is computed as

It is worth noting that our unstructured mesh scheme only needs 8 points per wavelength to obtain accurate results in free space [20]. However, inside a dielectric, the wavelength \(\lambda _{Diel}\) is less than the wavelength in free space \(\lambda _0\). For the anisotropic case, it is estimated that a mesh spacing of \(\lambda _0/20\) mesh should be sufficient. As we include a test case with full anisotropic tensors for both electric and magnetic properties, we decided to use the same mesh for all the test cases.

### 9.1 Magnetically uniaxial non-lossy anisotropic sphere

### 9.2 Electrically uniaxial non-lossy anisotropic sphere

In this case the only term that require matrix multiplication in Eq. (6) is \(({\bar{\bar{\varepsilon }}}_{av}-\varepsilon _{0}\bar{\bar{I}}\ )\frac{\partial }{\partial t}\left. \mathbf E _{inc}^{n+0.5}\right| _{j}\), as the other matrices \(\bar{\bar{a}}'_{\varepsilon +}\) and \(\bar{\bar{a}}'_{\varepsilon -}\) reduce to the unit matrix. The update of the electric field from the constitutive Eq. (31) only involves multiplication by the coefficient \(({\bar{\bar{\varepsilon }}}^{'-1}_{av})_{11}\). Figure 8 shows good agreement between the computed RCS distributions and those produced with the DDA method. There seems to be a big difference between our solution and the solution from DDA for the co-polarized RCS distribution in Fig. 8a but this is mainly due to the logarithmic scaling we are using highlighting even small differences between the results.

### 9.3 Electrically lossy anisotropic sphere

### 9.4 Magnetically lossy anisotropic sphere

### 9.5 Computational cost of the anisotropic model

The extra operations, needed for averaging and reconstructing the vectors described in Sect. 6, imply a cost penalty. For each node, we require 18 averaging operations for each field, in addition to \(3 \times 3\) operations of vector reconstruction for each edge. For the 6 tetrahedra lying inside one cube, the number of operations will be \(2 \times 8 \times 18 + 9 \times (19+14)=585\) for all degrees of freedom, taking into account the 19 electric field vectors and 14 magnetic field vectors. For one cell of a Cartesian mesh, 8 operations are required for averaging the two offset components of a field vector, leading to 48 extra operations on one cell node or 144 per Yee’s cell. However, in our case dealing with curved boundaries, to achieve the same level of accuracy, the standard scheme will actually cost \(6^3 \times 48=10368\) operations per same computational volume whereas we only have 585 operations.

### 9.6 Transmission of a narrow band pulse

## 10 Conclusion

A Yee type algorithm has been implemented, on an appropriately generated unstructured mesh, to model electromagnetic wave scattering by bodies consisting of electrically and magnetically anisotropic and conducting dielectric materials. The implementation has been successfully validated by comparison with the results obtained using the discrete dipole approximation. The generalization from isotropic to anisotropic materials allows us now to accurately model anisotropic objects of complex geometry.

In future work, we expect to extend the method to allow for the modelling of anisotropic dispersive materials. Bi-anisotropic dispersive materials, such as metamaterials, which involve coupling electric and magnetic fields in the constitutive equations could also be included. This can be achieved using the Z-Transform [21, 22, 23]. It is also proposed to incorporate a multi-scaling procedure, to allow for the modelling of more complex materials, such as composites.

## Notes

### Acknowledgments

A.Gansen gratefully acknowledges the financial support provided by the National Research Fund, Luxembourg AFR-5977057. Furthermore, I acknowledge the support of EPSRC funding (Grant Number: EP/K502935).

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