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Using the numerical discrete technique with unstructured grids, conformal perfectly matched layer (PML) absorbing boundary in the discontinuous Galerkin time-domain (DGTD) can be set flexibly so as to save lots of computing resources. Based on the DGTD equations in an orthogonal curvilinear coordinate system, the processes of parameter transformation for 2-D UPML between the coordinate systems of elliptical and Cartesian are given; and the expressions of transition matrix are derived. The calculation scheme of conductivity distribution in elliptic cylinder absorbing layer is given, and the calculation coefficient of DGTD in elliptic UPML is calculated. Furthermore, the 2-D iterative formulas of DGTD and that of auxiliary equation in the elliptical cylinder UPML are derived; the conformal UPML calculation in DGTD is realized. Numerical results show that very good accuracy and computational efficiency are achieved by using the method in this paper. Compared to the rectangular computational region, both the memory and computation time of conformal UPML absorbing boundary are reduced by more than 20%.

Truncated boundary condition is the key to ensure the accuracy for many electromagnetic numerical methods. In the past decades, different kinds of absorbing boundary conditions (ABC) have been proposed and successfully applied to the finite difference time-domain (FDTD) method and other methods. The perfectly matched layer (PML) ABC is presented by Berenger in 1994 [

The first-order Silver-Muller (SM) ABC is widely used owing to its easy realization [

In order to apply the UPML technique to DGTD in two-dimensional cases, the wave equations of UPML in an orthogonal curvilinear coordinate system are considered with the theory of parameter transformation, the processes of parameters transformation for 2-D UPML, and the specific expressions of transition matrix between the coordinate systems of elliptical and Cartesian are derived. The penalty flux is used for field exchange between units. The 2-D DGTD iterative formula and auxiliary equation in elliptical cylindrical UPML are derived. Numerical results show that very good effectiveness is achieved by using the proposed algorithm.

Suppose that the interface from free space to PML is the isosurface

Discrete unit of PML in orthogonal curvilinear coordinate system.

From the literature [

When the interface is the isosurface

UPML equations in an orthogonal curvilinear coordinate system are as follows:

In practical applications, (

Taking the example of tensor

It is noted that after the transformation, the

Since the expression of the transformation matrix depends on the specific coordinate system, the process of calculational parameters in the elliptical cylinder UPML can be described as follows.

Let the ellipse cylindrical axis be

Elliptical cylinder coordinate systems.

From [

The slope of the tangent line

The

Then, the direction cosine and direction sine between normal unit vectors and

Thus, the transformation matrix in (

Substituting (

UPML are distributed between the regions

Supposing that

Thus, the formulas of semimajor axis

Substituting (

Inserting (

In the case of two-dimensional TM wave,

The auxiliary equations in a two-dimensional case are

There are nondiagonal elements in (

After using the penalty flux [

After (

The iterative formulas of auxiliary equations are

In the elliptical cylindrical computing region, the semimajor and semishort axes of the outer PML boundary are 4 m and 3 m, respectively. And the inner PML boundary has a semimajor axis of 3.5 m. Both the inner and outer PML boundaries have the same focus length of 2.29 m. The region is divided into 46,122 triangular units and 23,298 nodes with the discrete scale of 0.05 m. The time discrete interval is taken as

Radiant field and reflection error.

Time-domain waveform and spectrum

Reflection error

The PEC elliptic column has a semimajor axis and a semishort axis of 1.5 mm and 0.5 mm, respectively. Incident plane wave has a wavelength of 2 mm. The computational region is truncated with conformal PML layer. The result of bistatic scattering for metal elliptical column is shown in Figure

Bistatic RCS of PEC elliptic column.

Two nested dielectric elliptical columns target truncated by conformal PML. Sparse discrete conformal model is shown in Figure

Bistatic scattering for target.

Sparse discrete model

Bistatic RCS for two nested elliptical columns

In order to verify the benefits of conformal UPML, the numbers of unit and node in two different regions by three discrete scales of 0.1 mm, 0.08 mm, and 0.04 mm are given in Table

Comparison of the number of unit and node in different calculated regions.

Shape | Discrete scale | |||||||
---|---|---|---|---|---|---|---|---|

0.1 mm | 0.04 mm | 0.08 mm | ||||||

Number of unit | Number of node | Number of unit | Number of node | Number of unit | Number of node | Storages (MB) | Time (mins) | |

Nonconformal | 77,670 | 39,184 | 489,534 | 245,638 | 121,474 | 61,175 | 116.5 | 134.7 |

Conformal | 61,088 | 30,821 | 383,690 | 192,534 | 94,634 | 47,662 | 92.8 | 101.6 |

According to the DGTD equations and the parameter transformation theory in orthogonal curvilinear coordinate systems, the processes of parameter transformation for 2-D UPML between the coordinate systems of elliptical and Cartesian are presented, and expressions of transition matrix are derived. The 2-D iterative formulas of DGTD and that of auxiliary equation in elliptical cylinder UPML are derived, and the conformal UPML calculation in DGTD are realized. Numerical results show that good absorption for the outgoing wave can be realized by using conformal UPML. Because of the discrete characteristics of the unstructured grids, the absorption layer can be set up flexibly according to the shape of the target. Compared with the computational cost in rectangular PML layer, the memory and computation time for conformal structure decrease by 20.3% and 24.5%, respectively. Thus, the method presented in this paper will be very beneficial to the engineering computation since lots of computing resources can be saved.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported in part by the National Natural Science Foundation of China under Grant nos. 61571348, 61231003, and 61401344.

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