An improved generalized single-source tangential equivalence principle algorithm (GSST-EPA) is proposed for analyzing array structures with connected elements. In order to use the advantages of GSST-EPA, the connected array elements are decomposed and computed by a contact-region modeling (CRM) method, which makes that each element has the same meshes. The unknowns of elements can be transferred onto the equivalence surfaces by GSST-EPA. The scattering matrix in GSST-EPA needs to be solved and stored only once due to the same meshes for each element. The shift invariant of translation matrices is also used to reduce the computation of near-field interaction. Furthermore, the multilevel fast multipole algorithm (MLFMA) is used to accelerate the matrix-vector multiplication in the GSST-EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

Surface integral equation (SIE) methods have been widely used in the simulation of complex electromagnetic phenomena in practical engineering. Application of SIE methods always leads to a dense matrix which limits its applicability. Many fast solvers have been developed to reduce the memory requirement and CPU time, such as fast multipole methods [

In recent years, domain decomposition (DD) methods have been attracting much attention and regarded as effective techniques to solve multiscale systems. The DD methods rely on decomposing the problem into several smaller subdomains and solving each subdomain independently. In this manner, the dimension of the matrix can be reduced and the condition number of the matrix can be improved. Due to these advantages, the DD methods have been combined with finite element (FE) methods [

In this paper, the improved generalized single-source tangential equivalence principle algorithm (GSST-EPA) is proposed to address the array structures with connected elements. Although the GSST-EPA has been developed to solve the connected structures [

Without loss of generality, consider the scattering problem of a three-dimensional dielectric domain with the media

(a) The scattering problem of a three-dimensional dielectric object. (b) The object decomposed into two subdomains using the contact-region modeling method.

By using the extinction theorem, the total fields on the inner boundary of subdomain

PV stands for the Cauchy principle value integration. For the residue term

For the interior equivalence situation, the total fields are zero on the outer boundary of

Combining (

The H-field equation can be obtained by combining (

Similarly, the PMCHWT E-field and H-field equations for surface

Finally, the matrix form of these four PMCHWT equations is written as

It is worth mentioning that the additional transmission conditions are unnecessary for the contact surface, because the continuity of the currents at the contact surface is contained in the above PMCHWT equations automatically. To prove this fact, a simple model is considered as shown in Figure

The model for the continuity of the currents at the contact surface.

Because the surfaces

The definition of the

Since

It can be seen that the continuity of the electric currents at the contact surface is included in the MFIE. Similarly, the continuity of the magnetic currents at the contact surface

Consider the scattering of a dielectric object in free space. As shown in Figure

The dielectric object decomposed into subdomains and each subdomain enclosed by an equivalence surface.

The scattering operator can be derived from a dielectric object enclosed by an equivalence surface with three steps as shown in Figure

A dielectric object enclosed by an equivalence surface with three steps.

The coupling between objects via the equivalence surfaces can be described by a translation operator. The translation operator has two different forms: one is suitable for nonadjacent cases, and the other is suitable for adjacent cases. For the two nonadjacent subdomains, the translation operator is quite simple. As shown in Figure

The translation operator of two nonadjacent subdomains.

Therefore, the equivalence scattered current

The expression of the translation operator

For the two adjacent subdomains, the translation operator can be derived by the source reconstruction method (SRM) [

The translation operator of two adjacent subdomains.

Thirdly, the equivalent current produced by the incident fields

Finally, by combing (

Since the scattering operator and translation operator have been derived, the GSST-EPA equations for the object in Figure

In this paper, the periodic array structures will be solved by the GSST-EPA. If the array elements are connected with each other, then the CRM method can be used to decompose the elements. Next, each element is enclosed by an equivalence surface. Due to the same geometry and meshes of each element and equivalence surface, the scattering operator

Since the array structures have a lot of elements, the MLFMA is used to accelerate the matrix-vector multiplication in the GSST-EPA. It is notable that the translation operators have two different forms. The

Eight kinds of near-interaction mechanism between nine elements.

In this section, the performance of the proposed method is studied via several numerical experiments. All of the simulations are performed on PC with Intel Core i7-2760 2.4 GHz CPU and 32.0 GB memory.

Firstly, the accuracy of the CRM is investigated. The scattering of a dielectric sphere with

(a) The model of a dielectric sphere. (b) Subdomain partitions of the dielectric sphere.

The bistatic RCS of a dielectric sphere at 300 MHz, HH polarization.

The second example is used to investigate the accuracy and convergence of the GSST-EPA with CRM. As shown in Figure ^{−6} is used to solve the final equations. With the increase in subdomains, the iteration number only grows from 20 to 25, which shows the convergence stability of the method, while the PMCHWT equations converge to 10^{−6} with 1219 steps.

The model of a dielectric slab.

The dielectric slab decomposed into subdomains and each subdomain enclosed by an equivalence surface: (a)

The bistatic RCS of a dielectric slab at 300 MHz, HH polarization.

The convergence of GSST-EPA for the dielectric slab.

The third example is a

(a) The dimension of a bowtie antenna element and (b)

The bistatic RCS of a

The computational results of two methods.

Method | Unknowns | Memory (MB) | Iteration number (0.001) | Total time (s) |
---|---|---|---|---|

EFIE-PMCHWT | 13,404 | 1747.9 | 1177 | 894.5 |

GSST-EPA | 1674 | 53.3 | 10 | 442.6 |

Finally, to investigate the capability of GSST-EPA with CRM to solve large array structures, the scattering of the ^{−3} using 74 iteration steps. It can be seen that GSST-EPA still exhibits fast convergence as the increment of the element number.

The bistatic RCS of a

In this paper, GSST-EPA with the CRM method is introduced to solve the array structures with connected elements. Firstly, the array structures are decomposed into subdomains by CRM. Then, the unknowns on the subdomains are transferred to the unknowns on the equivalence surfaces by GSST-EPA. Moreover, the meshes of subdomains can be the same with each other, which makes the scattering matrix solved and stored only once. The shift invariant of translation matrices is also used according to the characteristic of array structures. Using this strategy, the number of unknowns can be reduced and the conditioning of the final matrix equation is significantly better than the traditional PMCHWT and EFIE-PMCHWT. Therefore, the improved GSST-EPA can be viewed as an efficient and robust solver to solve array structures.

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

This work is supported by National Natural Science Foundation of China (Grant nos. 61501267, 61631012, 61671257, and 61425010), Natural Science Foundation of Zhejiang Province (Grant no. LQ16F010003), and Ningbo University Discipline Project (Grant nos. XKL14D2058 and XYL15008) and is partially sponsored by a K.C. Wong Magna Fund in Ningbo University.