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A novel implementation of symmetric boundary condition in harmonic and transient analysis of electromagnetic wave propagation

While doing electromagnetic analysis using FEM, using symmetry, if exists will minimize the computational cost. Through the physical problem definition, it is possible to determine if symmetry exists about one plane for particular electromagnetic problems. The adoption of suitable symmetry boundary conditions can help to minimize the computing domain for these types of problems. But for electromagnetic analysis in potential formulation it is not very straight forward to implement the symmetric boundary condition. In the present work, a novel approach of implementation of symmetry boundary condition in potential formulation within nodal framework has been demonstrated. Also, it is implemented to various electromagnetic harmonic and transient problems. These problems include various domains such as conducting and dielectric sphere, cube with conducting walls and ellipsoid. A significant reduction in computational cost is achieved using the proposed method.


Introduction
In the field of computational electromagnetics, the importance of application of the finite element method (FEM) is universally well established. This field of research has a wide range of applications in antenna radiations, wave guide transmissions, cavity resonant problems, etc [45,16]. In the conventional approach, while solving electromagnetic radiation and scattering problems using finite element method, the infinite region exterior to the scatterer/radiator must be truncated to reduce the size of the computational domain. Then, to absorb the outgoing wave radiation, some appropriate absorbing boundary condition (ABC) is imposed on the truncation surface [32]. This ABC is determined based on Sommerfeld radiation condition [8]. In [8,25,43,46], certain second-and higher-order ABCs are given. In these works, authors have modeled the computational domain with both standard finite element method and Boundary Integral method (BI). Here, the interior domain is discretized with edge elements whereas, BI to the truncation surface. Even though this technique is more accurate than traditional ABCs, a key disadvantage is the problem of generation of densely populated matrix by BI elements which results in higher memory needs and calculation time. Other efficient method is the use of infinite element formulations. In this method, truncation surface which is surrounded by radiator/scatterer are meshed with conventional elements, whereas domain outside the surface is discretized with infinite elements. In [32], A Nandy et al. (2016) proposed an amplitude formulation for the exterior domain, harmonic, electromagnetic analysis for radiation and scattering problems in the framework of nodal finite element method. In this formulation, same element can be used in the complete domain as long as the origin is not part of the domain [32]. Various problems like scattering by conducting and dielectric sphere, conducting cylinder, conducting cube, conducting L shaped domain are solved and the results have been verified either with analytical solutions or with edge element results from HFSS [7]. At higher frequency Amplitude formulation is better than HFSS and conventional nodal formulation totally fails. The basic idea behind the use of the amplitude formulation is to a-priori separate the highly oscillatory − section ( being the radial co-ordinate and being the wavenumber) from the interpolation function so that the finite element requires to interpolate a more gentle function, which it does rather effectively even at higher frequencies. The Wilcox asymptotic expansion for electric field [30] from the radiator/scatter is used in this amplitude formulation. With nodal finite element, presence of spurious eigen values are well known limitations in literature. This can be partly eliminated using a penalty or regularization term [26,34,35,38] and partly by using potential formulations [1]. But these cannot predict singular eigen values for domains with sharp corners and edges. When the non-convex or inhomogeneous domain is involved, the situation becomes much worse. However, mixed finite formulation [18], predicted correct eigenvalues for both convex and non-convex domains except for the curved three-dimensional objects. To tackle this problem, the edge element employs the basis function associated with edge [4,5,6,21,22,23,19,24,20]. Edge elements enable tangential field continuity at element edges and properly represent the curl operator's null space. These elements enforces tangential continuity and normal discontinuity of the field variable across inter-element boundaries.
A wide range of numerical methods are available to solve Maxwell's electromagnetic wave equation in time domain ( [45]). Among the two familiar methods such as Finite Difference Time Domain (FDTD) method and Time Domain FEM (TDFEM) method, former method gained popularity due to its simplicity in numerical integration. In [36], Yee introduced FDTD method to solve Maxwell's equations in isotropic media. Although this method gained popularity, TDFEM method reported in [9,14,27,29,31,37,44,48] has major advantages in dealing with complex geometries. In TDFEM, two approaches are followed to solve for electromagnetic field variables. In first approach [14,27,44], second order electromagnetic wave equation is solved for either electric or magnetic fields. Then from post processing other fields are obtained. In the other approach [9,31,48], two coupled first order Maxwell's equations are solved for both field variables simultaneously. In [11,14,28,37], another method, namely, Finite Element Boundary Integral (FEBI) method is applied to simulate the infinite domains by dividing into exterior and interior domains. These two domains are separated by an artificial boundary. Exterior domains are modeled with boundary integral method and interior domain with finite elements. In transient electromagnetic analysis, different time stepping strategies such as leapfrog strategy [39,40,41], central difference method [10,14,28] are used to discretize the time. Stability of the algorithm is an important factor in simulating electric and magnetic fields in transient analysis. In [33], authors have used the time stepping strategy in nodal element frame work for electromagnetic analysis, which is unconditionally stable from an energy perspective. It is well known, in numerical analysis, if symmetric boundary conditions can be implemented, it can reduce the computational cost. In the current research work, an attempt is made to introduce a novel method of application of the symmetric boundary condition for various harmonic and transient electromagnetic wave problems. We tried to solve the different harmonic analysis problems from [32] and as well as transient problems from [33] where we can implement symmetry boundary condition.
The remaining article is organized as follows: In section 1.1, a brief discussion about the identification of symmetry from the physics of the problem is presented. Then, in section 2, we have briefly presented the mathematical formulation along with finite element implementation for both harmonic and transient analysis. We have presented the implementation of symmetry boundary condition in potential formulation in section 3. Thereafter, in section 4 and section 5, we have solved various numerical examples having symmetry in harmonic and transient analysis respectively with the application of the proposed method. The achieved computational advantage in the proposed formulation has also been presented elaborately.

Identification of existence of symmetry from physics of problem
To apply the symmetry boundary conditions, geometry, external loading as well as boundary conditions should be symmetric about the plane. If the problem is symmetric then we can solve using half of the domain with proper implementation of symmetric boundary condition which results in significant reduction of the computational cost. Detailed numerical analysis has been done and electric fields are compared for both full and half domains along with benchmark values. In each and every case, results of both the domains are almost matching with analytical/benchmark values. Moreover, it is also observed that in all the cases, with adoption of the present method, electric fields are simulated efficiently with less percentage of unknowns than full domain. Thus, the proposed method can be computationally very efficient in electromagnetic analysis.
For example, consider an electromagnetic plane wave in the form = − ̂ impinges on a conducting sphere whose center is located at the origin as shown in Fig. 1. From the physical observation, we can observe the geometry of the sphere is symmetry about the origin. The incident wave have electric field components distributed alongdirection, and the wave is traveling along -direction. So, the incident electromagnetic wave vector is symmetric about -plane. Then the resultant scattered electromagnetic field due to the presence of the conducting sphere will also be symmetric about -plane. Hence, we can solve considering half of the domain i.e. one hemisphere with the flat surface along -plane. On that symmetry plane we have to impose zero normal electric field as the symmetry boundary condition. So, on that plane it will be = 0.

Mathematical Formulation
We have Maxwell's electromagnetic wave equation in terms of electric field as stated in [13] where = 1∕ √ 0 0 is the speed of light and is the current density. Relative permittivity, and Relative permeability are given as = 0 and = 0 , where 0 and 0 are the permittivity and permeability for vacuum, and and are the electric permittivity and magnetic permeability of medium, respectively.

Harmonic analysis of Maxwell's equations
For harmonic excitation, Eq. 1 can be expressed as follows: where = √ −1. If is the excitation frequency then the wave number of vacuum is given as 0 = ∕ . And also, the wave number of the medium is related as = 0 √ . For no charge condition, the electric field is subjected to following constraint For perfectly conducting boundary ( ) × = 0 whereas for other part ( ℎ ) × is prescribed. Assuming no surface currents exist at material discontinuity, both × and × must be continuous across the material interface [16]. In Potential formulation electric field is replaced as + . So, the governing differential equation given by Eq. (2) and constraint equation Eq. (3) are modified as The variational statements related to Eqs. (4a) and (4b) are given as equations 8a and 8b in [32] in which a regularization term relates to the coulomb gauge condition ⋅ = constant, and the first-order absorbing boundary condition on truncated boundary ( ∞ ) is incorporated. Field variables , , , in equations 8a and 8b are discretized using the equation 9 in [32] as, The finite element equation has been obtained in [32] as where the expressions for the terms , , , , , and are given by equation 10 in [32]. From the expansion theorem of Wilcox [47,30], can be presented as where | | is the distance of a point from the origin, or, alternatively, the radial coordinate in the spherical coordinate system ( , , ). As in [2,3,17,12], this expansion is used to separate out the rapidly-varying part − | | ∕| |. Therefore, the finite element shape functions only have to capture a gently-varying part [32]. From the amplitude formulation proposed in [32], ( , ) and their variations ( , ) are expressed as equation 12, electric field and magnetic field are given as equation 13 from [32]. And also, the variational statements stated as equations 8a and 8b in [32] are modified to equations 15 and 16 in [32] respectively for the amplitude formulation. When origin is part of the domain, the entire domain is divided in two parts -the inner part is sphere containing the origin where conventional formulation is implemented and the outer part is hollow sphere where amplitude formulation is implemented. In order to have smooth transitions from the conventional towards the amplitude formulation, all the discretizations (̄ ,̄ ,̄ ,̄ ) are multiplied with 1 1 where 1 is the radius of the interfacial spherical surface between inner and outer domain. For problems like radiation or scattering from conducting bodies, where the origin is not part of the computational domain, AMP elements are used. In the section 2.2.2 in [32] field variables and their variations are discretized as follows: In [32], resulting finite element equation is given as (see equation 17 in [32]) where from [32] we can see the expressions for the terms , , , , , and .

Transient analysis of Maxwell's equations
From the potential formulation proposed in [33], electric and magnetic fields are expressed in terms of ∶= ∕ and ∶= ( )∕ as follows: In potential formulation, Eq. 1 and Eq. 3 are expressed as The variational statements are presented in equations 2.19 and 2.20 in [33] which are transformed in equations 2.23 and 2.24 after implementing suitable time strategies as expressed in equations 2.21 and 2.22 in [33]. Choice of time stepping strategy is very crucial to conserve certain physical quantities with a view to make our numerical formulation unconditionally stable. As discussed in section 2.3 in [33], field variables and their variations are discretized as follows: We can derive various differential quantities of these variables as presented in [33] and the finite element equation can be written as Different submatrices and subvectors , , , , , and have been elaborately presented after equation 2.28 in [33]. In this section, we explain our novel method of implementation of symmetric boundary condition using a thin patch on the symmetry surfaces. Let us continue our discussion with the same example ( Fig. 1) picked in section 1.1. In order to apply the symmetry boundary condition = 0 on the symmetry boundary i.e., plane, a hollow cylindrical patch of very small thickness is added as shown in Fig. 2. Here, the patch thickness should be very small as compared to the other dimensions of the domain. The symmetric boundary condition ( ⋅ = 0) is applied in accordance with previously discussed potential formulation. On the outer green surface (see Fig. 2) of the cylindrical patch, the normal component of i.e., are made to zero for all the nodes explicitly, to impose the condition ( ⋅ = 0). And also, to satisfy the condition ⋅ = 0, for all the nodes inside the volume of the cylindrical patch, = 0 is applied explicitly. As the patch is very thin, it will result in ⋅ = 0 only on the green surface without disturbing the scattered field in the remaining domain. In the subsequent sections, we can see the efficacy of this method for various harmonic and transient problems.

Numerical examples with electromagnetic harmonic analysis
In this section, we apply the proposed method to implement symmetry boundary condition for both conventional potential formulation and Amplitude formulation for harmonic analysis with hexahedral 27 node brick (B27) and 18 node wedge (W18) elements. In our numerical analysis we have compared the simulation results of both full domain and half domains with proper modeling of symmetric boundary.

Scattering from a conducting sphere
The first example is taken from section 5.4 in [32]. Here, we want to solve for scattering electric fields from a conducting sphere. Let us consider a conducting sphere of radius on which an electromagnetic plane wave inc = 0 − is incident. To implement FEM, our computational domain domain (air surrounding the conducting sphere) is hollow sphere which is truncated at radius ∞ . On this truncated surface, we apply Sommerfeld absorbing boundary condition. Conducting boundary conditions is imposed on the inner spherical surface at radius . Our interest is to find the scattered electric field in the hollow spherical air domain having inner and outer radii of and ∞ respectively. From the discussion in section 1.1, we can treat the problem is symmetric about -plane. Therefore, we consider half of the hollow sphere as our computational domain. As discussed in the section 3 we apply symmetric boundary conditions explicitly on the symmetry face ( ). We discretize the truncated annular hemisphere with mesh size of 16 × 12 × 12 B27 and W18 elements whereas for the thin cylindrical patch we used 16 × 12 × 1 ( × × ). In Table 1, mesh details along with total number of equations are presented for both full sphere and half sphere with thin patch. We have considered 0 = 1, 0 ∞ = 5 with 0 = 1. At two different locations: (a) at near field ( 0 = 1.15) and (b) at far field ( 0 = 4.75), for = 4 real and imaginary scattered electric field components are plotted along in Fig. 3 and Fig. 4. During numerical analysis for half of the domain, we directly solve for 0 ≤ ≤ , then we have ( − ) = ( ) because of symmetry of the problem for the remaining half domain. We can notice from both the plots (see Fig. 3 and Fig. 4) that the results from both full domain, half domain are closely matching with the analytical benchmark given as equation 33 in [32]. But the total numbers of equations are 25% less for the half domain (see Table 1) as compared to the full domain. As computational cost is directly co-related with the total no. of equations, hence proposed method is computationally very efficient.

Scattering from a conducting ellipsoid
This example is chosen to show that the proposed implementation of symmetry boundary condition also works for non spherical geometry. A wave of the form − 0 impinges on the ellipsoid having diameters 2a, 2a, and 2c along , and axes, respectively. The domain is truncated at a spherical surface given by 0 ∞ = 24. We present our results for 0 = 12 and = ∕4. In this problem, the symmetry boundary condition ⋅ = is implemented as described in section 3. The half annular ellipsoidal domain is meshed with 12 × 20 × 8 ( × × ) B27 and W18 elements and an additional cylindrical patch has 12 × 40 × 1 ( × × ) B27 elements. Then we have to solve 73,949 equations. Table 2 summarizes the mesh details of half domain along with the full domain. We have taken 0 = 1, 0 = 12, 0 ∞ = 24. The scattered electric field components are compared in Fig. 5 along with 0 = 24 and = 4 . We have compared the results of half domain with symmetry (HDWS) in both conventional (Con) and amplitude formulation (OWF) with the converged HFSS [7] result obtained using a very fine mesh (1,49,530 unknowns i.e., unsuppressed degree of freedoms). In Fig. 5, we have also presented the respective full domain result for both Con and OWF formulations. In the current problem we directly solve for 0 ≤ ≤ , because of symmetry of the problem we have (2 − ) = ( ) for the remaining half domain. We obtain equivalent accuracy with half domain (thin patch) using a system of equations having 40% less unknowns than that of full domain.

Scattering from a dielectric sphere
Consider an incident wave of the form inc = 0 − impinges on a dielectric sphere. Let's assume the relative permeability and relative permittivity of the dielectric sphere and the surrounding medium be denoted by ( 1 , 1 ) and ( 2 , 2 ), respectively. The analytical solution has been presented by Stratton [42]. In our numerical analysis, we assume the dielectric sphere to be in vacuum i.e., 2 = 2 = 1 and it has material properties 1 = 1 and 1 = 1.5. We choose = 8 and ∞ = 24 and the incident wave is of the form − 0 . For both formulations i.e., conventional and amplitude formulation, 1000 B27 and W18 elements are used to mesh the half domain with symmetry plane, out of which 750 elements constitute the hemisphere and the remaining 250 B27 elements model the cylindrical patch of thickness = 0.01. Mesh and equation details are given in Table 3. Since the origin is part of the domain, to mesh the dielectric sphere we use conventional nodal elements. Whereas, AMP elements are used to mesh the vacuum part of the domain.  Fig. 6 shows the results for the scattered electric field inside the dielectric sphere at 0 = 6.4, at a point just outside the dielectric sphere ( 0 = 9.6), and at a point in the far-field ( 0 = 16) respectively. In all the plots, we have compared the results of both full domain (FD) and half domain with symmetry (HDWS) with analytical benchmark results as given by equation 34 in [32]. In Fig. 6, results from both conventional and OWF formulations have been presented. In order to obtain same level of accuracy, for HDWS we have to solve around 36% less no. of equations as compared to the full domain which clearly depict the computational efficacy of the proposed method.

Numerical examples with electromagnetic transient analysis
In this section we apply the novel implementation of the symmetry boundary condition for various electromagnetic transient problems from [33] to analyze the efficacy of the proposed method in time domain. These problems includes radiation inside a cube with conducting faces, scattering from a conducting and dielectric sphere. Again, we choose B27 and W18 elements to discretize the computational domains and we take = 3 × 10 8 ∕ √ m/s.

Electromagnetic radiation inside a cube with conducting walls
A cube of dimension × × with conducting walls is excited with and with initial values given as equation 3.2 in [33]. We have taken = 3 × 10 8 m/s, and simulated upto 4 × 10 −8 sec with a time step of Δ = 1 × 10 −9 sec.  To implement the symmetric boundary conditions, we truncated the cube from × × to cuboid with 2 × × 2 . On the symmetry faces i.e., on and -planes, we have applied ⋅ = explicitly as explained in section 3, and on the remaining faces of the cuboid, conducting boundary conditions × = are applied. For transient analysis, 36 B27 elements are used to mesh cuboid and the thin patches, whereas full domain is meshed with 64 elements. Table 4 summarizes the numerical analysis details for both the domains. At a location with coordinates ( , , ) = (1.1780, 0, 3926, 0.7853), , , and field components obtained for both full and half domains with symmetry are plotted in Fig. 7 and compared with analytical solutions given as equation 3.1 in [33]. Again, it is evident, by adopting the symmetric boundary conditions, 890 equations are sufficient for the perfect match with analytical benchmark as compared to 1666 unknowns for the full domain.  To know the performance of the proposed method on curved objects, we consider the present example where we simulate the transient scattering electric fields from a conducting sphere. An incident wave given as equation 3.5 in [33] impinges on a conducting sphere, other problem parameters are followed same as section 3.6 in [33]. Computational full domain which is truncated at a radius ∞ = 3.8 m is meshed with 640 B27 and W18 elements which results in 14,946 equations in [33]. In the current symmetry boundary condition implementation, the half domain with cylindrical patch is meshed with 448 B27 and W18 elements resulting in 11,169 equations (Table 5). Boundary conditions such as ⋅ = is implemented on the surface of the thin patch as described in section 3 and × = is applied on the conducting surface. We have taken a time step of 5 × 10 −10 sec in our simulation. From Fig. 8, it is evident, values at a point (0.33, −1.03, −0.45) in full domain and half domain with symmetry are perfectly matching with the standard benchmarks in [14]. Around 25% less unknowns have to be solved using symmetry boundary condition which clearly represent the computational efficacy of the proposed method.

Electromagnetic scattering from a dielectric sphere
This example is chosen from section 3.7 in [33] to test the effectiveness of the proposed method in predicting scattering electric fields from a dielectric sphere. On a dielectric sphere of radius 0.5 m having = 1 and = 6.0, Neumann pulse wave as given in equation 3.5 in [33] is impinged. The computational domain which is truncated at radius ∞ = 2 m is modeled with B27 and W18 elements. Table 6 gives the numerical analysis details for both full and half domains. Proper symmetric boundary condition is applied with thin cylindrical patch as described in section 3.
With a time step of 0.5 ns simulations are performed. We have plotted ⋅ along time in Fig. 9 for full domain, half domain with symmetry and compared it with the benchmark edge element results in [15]

Conclusions
In the present work, an attempt is made to implement symmetric boundary conditions to electromagnetic problems for both harmonic and transient analysis. In nodal framework, potential formulations are followed and it is not very straight forward to implement the symmetry boundary condition in that formulation. In the proposed method, a very thin patch is attached to the symmetry faces of the domain. Then, proper symmetric boundary conditions are applied to the nodes of the thin patch explicitly. We have validated the proposed method solving a wide range of problems from radiation to scattering problems, from spherical to non-spherical domains, from harmonic to transient analysis.
For all the problems we have validated our results against either analytical benchmark results or other benchmark from existing literatures. Furthermore, we have presented the computational efficacy of our proposed method as compared to existing full domain analysis for all the examples.