Numerical simulation of electromagnetic radiation using high-order discontinuous galerkin time domain method

ABSTRACT


INTRODUCTION
To date, electromagnetic (EM) phenomena play a crucial role in any aspect of human life.The modern lifestyle has become a source of omnipresent electromagnetic since the used devices generate electromagnetic fields and produce electromagnetic radiation.Television and mobile phone are the good example devices used daily.Furthermore, there are many instances in the real word, which reflect the electromagnetic (EM) phenomena, such as the radiation of microwave [1], laser [2], lightning [3], etc. Shortly, we cannot leave the EM from our life; therefore, EM simulation has been developed by many scientists to figure out any real-world phenomena.
Currently, many scholars have developed research on the numerical simulation of EM, due to the performance of the digital computer is increased significantly but the price is decreased.Thus, the numerical simulation of EM will be more attractive than both experimental and analytical methods since the cost is reduced.Furthermore, the research is aimed to improve the performance of the method concerning both efficiency problems, primarily when the method should deal with the complex problems [4].It should also be noted that the numerical method is aimed to solve the problem of EM by using Maxwell's equations as the governing equations.
In the beginning, the numerical simulation in EM is performed in the frequency domain [5], [6].The equation, which is established in the frequency domain, is resulted from the transformation of the time domain equation.As a result, the method is simple however the solution is limited on calculation one frequency at a time.So, it can not be used for broadband frequency analysis.Regarding the limitation, Yee [7] proposed finite difference time domain method (FDTD) to solve Maxwell's equations in the time Finite difference (FD) methods [7], [8] are the most popular methods for numerical simulation of wave propagation.While the methods have gained the both of advantages, i.e., it is simple and robust, they have some disadvantages.For example, they are not well suited to problems with complicated problems, and the handling of the boundary condition is not an easy task.Finite volume (FV) [9], [10] and finite element (FE) methods can handle complicated spatial domains easily [11]- [13].Unfortunately, FV methods have only second-order accuracy, and FE method based on Bubnov-Galerkin projection suffers from spurious Gibbs oscillation as well as the overshoot or undershoot at sharp gradient region.Numerous efforts have been conducted to improve the performance of the FE method.
In this paper, we described a high order discontinuous Galerkin (DG) method for simulating twodimensional electromagnetic wave radiation.Discontinuous Galerkin (DG) method is one of the advanced, improved FE methods.The DG method combines the flexibility of finite element methods with the accuracy of spectral methods.The DG method allows unstructured mesh configuration, and inter-element continuity is not required.The basis function is discontinuous across mesh boundaries.With a proper choice of numerical flux at the element boundaries, the spurious Gibbs oscillation can be suppressed, and the DG method only requires communication between mesh that has common faces [14]- [17].

GOVERNING EQUATIONS AND NUMERICAL SCHEME
We use the two-dimensional transverse electric (TE) Maxwell's equations as the governing equations [7].We assumed that there is no field variation in the z-direction, E field is lying in the (x,y) plane and H field is parallel to the z-direction.
where 0  is the dielectric conductivity, 0  is the magnetic susceptibility,  is the electric conductivity and *  is the magnetic resistivity.
To simulate the infinite spatial domain we truncated the domain by adding Berenger's Perfectly Matched Layer (PML) boundary conditions in an outer truncated region [18], [19] where the parameters where The boundary conditions on metalic surfaces are taken as perfect electrical conductor (PEC), so the electrical fields are set to be zero.By applying the Bubnov-Galerkin procedure, i.e., integrating the (2) partially twice and still retained the flux terms in each element D k , we obtain the weak form. where is the Lagrange basis function, * F is the numerical flux, and D k are the faces of an element.In this paper, we used the central difference as the numerical flux for simplicity.
The local solution is approximated by N p is the number of grid points in each element, k h q and n q ˆare nodal and modal expansion coefficients respectively.
k h q and n q ˆare related by using Vandermonde matrix V q q V  ˆ; is modal basis function defined on tetrahedron element and obtained by combined Jacobi polynomials.
is the n th -order polynomial Jacobi.
As the FE method procedure, the physical triangular is mapped as shown in Figure 1   The detail of 2-D derivation of high order Discontinuous Galerkin method is described in [20].The semi-algebraic ( 2) is integrated into time marching by using five stages of fourth order low storage Runge-Kutta scheme as developed by Carpenter & Kennedy [21].

RESULTS AND DISCUSSION
In this section, we present 2 (two) numerical examples to demonstrate the performance of the DG method.The first example illustrated the modeling of 6 cm diameter metal cylindrical scatterer in free space.The spatial domain is divided into 1753 triangular elements, and the thickness of PML is 0.024 m as shown in Figure 2. The hard source excitation is the combination of an exponential and sinusoidal pulse with the carrier frequency of 5 MHz.
We took the time stepping dt=2.43e-12 and polynomial order N=3.The numerical calculations are compared with the results of the FDTD method.The FDTD code is provided by Susan Hagness (https://github.com/cvarin/FDTD/blob/master/Taflove/fdtd2D.m). Figure 3(d) and Figure 4 show the spatial distribution of the magnetic field and Figure 5 shows the comparison of magnetic pulse at position (0.0465, 0.675).The comparisons show excellent agreement.

CONCLUSION
In this paper, we have proposed the simulations of 2-D electromagnetic wave radiation in the time domain using a high order discontinuous Galerkin method and PML boundary condition.The use of unstructured triangular elements makes the methods very attractive and complex geometries can be handled easily.The numerical examples and the comparison with the FDTD method indicate the capability of the proposed approach for electromagnetic wave simulation.

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ISSN: 2088-8708 Int J Elec & Comp Eng, Vol. 9, No. 2, April 2019 : 1267 -1274 1268 domain.Furthermore, the numerical simulation for the Maxwell equation is developed, and the numerical methods are usually based on the finite difference, finite volume, and finite element methods.

x
 and y  represent an anisotropic electric conductivity in x and y directions respectively and * x  * y  represent an anisotropic electric conductivity Int J Elec & Comp Eng ISSN: 2088-8708  Numerical simulation of electromagnetic radiation using high-order discontinuous… (Pranowo) 1269 To simplify matters, let us express Maxwell's equations in conservation form: to standard  ISSN: 2088-8708 Int J Elec & Comp Eng, Vol. 9, No. 2, April 2019 : 1267 -1274 1270 triangular by a relation:

Figure 2 .Figure 3 .
Figure 2. The domain of the first example

Figure 4 .Figure 5 .Figure 6 .Figure 7 .
Figure 4. Hz Pulse propagation att =8.51e-10 s of FDTD method 1273 . The critical part of Berenger's PML definition for the 2D TE case is that the magnetic field