Rooftop Basis Functions Research Articles

NUFFT-accelerated spectral-domain MoM with asymmetric rooftop basis functions for multilayered structures with periodic metallizations

NUFFT-accelerated spectral-domain MoM with asymmetric rooftop basis functions for multilayered structures with periodic metallizations

On the use of combined surface integral equations for the analysis of high contrast penetrable objects

AbstractAccurate solutions of electromagnetic scattering problems involving objects made of materials with large permittivity contrasts are considered. Problems are formulated with different commonly used combined surface integral equations (SIEs). All studied formulations are discretised through the method of moments with rooftop basis functions over flat quadrilaterals represented as bilinear surfaces, with razor‐blade functions being used for the testing procedure. The accuracy of the results is first investigated in detail for several frequencies and permittivity values using different numerical measures. It is shown that numerical instabilities may appear at frequencies corresponding to the physical resonances of the object, in particular in the near field and for large material parameter contrasts. The example of a dielectric resonator (DR) with cubic geometry is considered for the purpose of analysis, especially since to achieve smaller DR type antennas, it is necessary to use higher contrast materials. The accuracy of the combined surface integral equations to determine the natural resonant modes of the DR is investigated. It is found that the resonance modes can be accurately determined by exploring the radar cross section (RCS) of the DR in free space only if a proper combination of the electric and magnetic fields equations is applied.

Low frequency scattering simulation of homogeneous objects in layered medium by Muller formed SIE

ABSTRACT This paper presents a novel approach to accurately simulate the low frequency electromagnetic scattering of a homogeneous dielectric object in layered medium. The N-Muller surface integral equation (SIE) is utilized for accurate electromagnetic modeling at low frequency with a properly designed coefficient to cancel frequency singularity. Matrix-friendly layered medium Green’s functions are adopted and integrated into SIE to account for the effects of the inhomogeneous background. By using Galerkin method and rooftop basis functions, a well-conditioned impedance matrix is obtained and a rapidly converging iterative solution is achieved for low frequencies. Numerical results are provided to validate the proposed method and investigate its properties.

Efficient Analysis of Planar, Arbitrarily Shaped, and (Bi)-Anisotropic Metasurface Antennas

An efficient method of moments (MoM) scheme is presented for the analysis of planar, possibly cascaded metasurfaces (MTSs) implementing high-impedance surfaces. The MTS can be designed on an arbitrarily shaped domain and is accounted for as a set of cascaded electric sheet impedances. In the proposed method, the shaped MTS is included in a larger rectangular domain meshed with rooftop basis functions, resulting in a Toeplitz–block Toeplitz structure for the substrate impedance matrix, which enables fast Fourier transform (FFT)-based acceleration of matrix–vector products. The basis functions lying in the rectangular domain but outside the MTS are disabled by including zeros in the impedance matrix. The obtained linear system of equations is well conditioned, does not require any specific near-field treatment, and can be solved iteratively using the generalized minimal residual (GMRES) method. It has been demonstrated that the algorithm is able to handle efficiently any MTS antenna shape in single-layer or multilayer configuration with almost linear complexity. It is shown that the high-impedance sheets improve the convergence rate. MTSs with more than 40 wavelengths size, meshed with more than 1 million basis functions, can be analyzed on a conventional computer in less than 15 min. A very good agreement is shown through comparison with results from other software packages.

BICGSTAB-FFT Method of Moments with NURBS for Analysis of Planar Generic Layouts Embedded in Large Multilayer Structures

BICGSTAB-FFT method of moment (MM) scheme is proposed to analyze several levels of planar generic layouts embedded in large multilayer structures when the layout geometries are modeled by NURBS surfaces. In this scheme, efficient computation of normalized error defined in iterative bi-conjugate gradient stabilized (BICGSTAB) method for large multilayer structure analysis problems is implemented. The efficient computation is based on pulse expansion with dense equi-spaced mesh of generalized rooftop basis functions (BFs) defined on NURBS surfaces and equivalent periodic problem (EPP) in order to apply fast Fourier transforms (FFT). Moreover, efficient computation of Green’s functions for multilayer structure is implemented for near and far field regions. Experimental and numerical validations of whole printed reflect array antennas of electrical size between 8 and 16 times the vacuum wavelengths are shown. In these validations, CPU time consumptions of the proposed method are obtained with results between few minutes and half an hour using a conventional laptop.

Calderón Preconditioned Spectral-Element Spectral-Integral Method for Doubly Periodic Structures in Layered Media

An efficient and accurate spectral-element spectral-integral method is developed for rigorously modeling arbitrary inhomogeneous objects with doubly periodicity embedded in a layered medium background. It is an improvement over the traditional finite-element boundary-integral (BI) method. In the proposed approach, the domain decomposition between the interior spectral-element domain and the outer surface BI domain enables the BI domain to be solved by a far more efficient spectral integral method using a uniform mesh; without any loss of generality, it leaves the interior spectral-element domain with flexibility to model any inhomogeneous objects. For the BI domain, with a uniform mesh and rooftop basis functions, we can exploit the Toeplitz structure of the BI matrices with the fast Fourier transform algorithm and apply the Calderón preconditioner to achieve a high iteration convergence rate that will not slow down as the problem gets larger. With the periodic layered medium Green's functions, the layers without objects can be treated as background and the discretization is avoided. We show that this method is very promising for large-scale problems, because the CPU time and memory required with respect to the number of unknowns grow slowly. We validate this method with a commercial finite element solver and show its ability by solving a large-scale optical lithography problem.

Method of Moments Based on Equivalent Periodic Problem and FFT with NURBS Surfaces for Analysis of Multilayer Periodic Structures

In this paper, an efficient technique of computation of method of moments (MM) matrix entries for multilayer periodic structures with NURBS surface and Bézier patches modelling is proposed. An approximation in terms of constant pulses of generalized rooftop basis functions (BFs) defined on Bézier patches is proposed. This approximation leads discrete convolutions instead of usual continuous convolution between Green’s functions and BFs obtained by the direct mixed potential integral equation (MPIE) approach. An equivalent periodic problem (EPP) which contains the original problem is proposed to transform the discrete convolutions in discrete cyclic convolutions. The resultant discrete cyclic convolutions are computed by efficiently using the Fast Fourier Transform (FFT) procedure. The performance of the proposed method and direct computation of the MM entries are compared for phases of reflection coefficient. The proposed method is between 9 and 50 times faster than the direct computation for phase errors less than 1 deg. The proposed method exhibits a behaviour of CPU time consumption of O(NbLog10Nb) as the number Nb of BFs increases. This behaviour provides significant CPU time savings with respect to the expected behaviour of O(Nb2) provided by the direct computation of the MM matrix entries.

Fully Analytical Evaluation of Singular Integrals With RWG and Rooftop Basis Functions

When the method of moments is used to solve surface integral equations, it is inevitable to encounter the problem of singularity because of Green's function in integral kernel. In this article, by expanding Green's function with Taylor series and applying appropriate coordinate transformation, closed-form expression of each term is derived. After proper linear combination, analytical solutions of singular integrals on triangular and rectangular domains with Rao–Wilton–Glisson and rooftop basis function can be achieved. Compared with numerical solutions, this approach is more efficient and can considerably decrease the computational cost, for a given precision.

Simulation of Electromagnetic Scattering of 3-D Inhomogeneous Biaxial Anisotropic Magnetodielectric Objects Embedded in Uniaxial Anisotropic Media by the Mixed-Order BCGS-FFT Method

This paper presents a volume integral equation (VIE) solver for the forward electromagnetic scattering of 3-D inhomogeneous biaxial anisotropic objects embedded in uniaxial anisotropic media. The optical axes of the objects can be rotated with arbitrary angles. The mixed-order basis functions are employed to discretize the VIE, i.e., the flux densities $(D, B)$ are expanded by the volumetric rooftop basis functions and the vector potentials $(A, F)$ are expanded by the second-order curl conforming basis functions. The weak form of the VIE is formulated by testing it using the same volumetric rooftop basis function and solved by the biconjugate gradient stabilized fast Fourier transform (BCGS-FFT) method. Several numerical simulations of different shapes anisotropic objects are performed and the results are compared with commercial software simulations to validate the accuracy and efficiency of the proposed solver based on different discretization schemes. The major new contribution of this paper is that not only the scatterer but also the background medium is magnetodielectrically anisotropic. Therefore, the dyadic Green’s function for the uniaxial anisotropic background medium is evaluated before solving the VIE.

Efficient Numerical Model to Analyze the Conformal Frequency-Selective Surfaces

A modified model is presented in this letter to efficiently analyze the electromagnetic properties of the conformal capacitive frequency-selective surfaces (FSSs). By employing a dual set of quasi-orthogonal rooftop basis functions for the surface current distribution over the FSS conducting surface, the conditioning of the system matrix is significantly improved without decreasing the solution accuracy or increasing the computational costs. As a result, the iteration steps and CPU time are dramatically reduced in comparison with the conventional rooftop-expanded scheme. Besides, without subdividing the narrow side of the conducting cells of the FSS in the space discretization, the proposed model is also geometrical-adaptive and engineering-accordant, which results in a significant reduction of the number of unknowns compared to the conventional RWG-expanded scheme. Moreover, the multilevel fast multipole algorithm is employed to further improve the computational efficiency. Numerical experiments demonstrate the satisfactory accuracy and much higher efficiency of the proposed scheme.

Multimode Coherent Pattern in Bistatic Scattering From Randomly Corrugated Surfaces With Irregular Grooves at L-Band

This paper investigates the bistatic scattering from randomly corrugated surfaces with irregular grooves. For a given incident wave, the surface fields and the scattered field were computed by the method of moments in which the rooftop basis function was used to account for fast phase changes due to steep surface slopes. The total scattered field is decomposed into coherent and incoherent components to analyze their respective contributions. We found that, for randomly corrugated surfaces with irregular ridges or grooves, the coherent scattering is profound at several scattering angles with strong main lobes, whose beamwidths are strongly correlated with the ridge density. The numerical simulation has shown that the lobe angular shift away from the specular direction is quasi-linearly dependent on the ridge density, suggesting that the coherent scattering pattern substantially contains the surface geometric information. We expect this paper to offer deeper understanding of coherent imaging of rough surface and to help in designing a novel imaging system.

The Mixed-Order BCGS-FFT Method for the Scattering of Three-Dimensional Inhomogeneous Anisotropic Magnetodielectric Objects

This paper presents the first FFT-based fast volume integral equation solver for inhomogeneous anisotropic magnetodielectric objects. The volume integral equations are formulated by employing mixed-order basis functions that expand the flux densities and vector potentials in the coupled field integral equations in terms of different sets of basis functions with different orders. Volumetric roof-top basis functions are used for flux densities whereas second-order curl conforming basis functions are used for vector potentials. A fast volume integral equations solver namely the BCGS-FFT method is then applied to accelerate the solution of this mixed-order weak-form formulation. Examples show that the mixed-order BCGS-FFT method has high accuracy compared to both analytical and numerical solutions. Examples also show the mixed-order BCGS-FFT method has high computational efficiency compared to commercial software.

Modeling Electromagnetic Field Coupling Through Apertures for Radio-Frequency Interference Applications

Electromagnetic field coupling between a digital platform and a radio-frequency antenna through an aperture represents one of the typical coupling paths responsible for radio-frequency interference (RFI) issues in mixed-signal systems. Conventional approaches of modeling radiation through an aperture and evaluating the shielding effectiveness of a metallic enclosure with apertures for classic electromagnetic interference (EMI) applications are not sufficient to handle modern RFI challenges. This paper discusses a segmentation method, together with impedance network representation, employed to divide the enclosure EMI problem into exterior and interior problems. With a port definition based on the vector roof-top basis function, the impedance network for the exterior domain was rigorously formulated using a method of moments solution for mixed potential integral equations. The impedance network for the interior domain was computed using the cavity method based on the analytical formula of 3-D Green's functions in a rectangular cavity. Connecting a pair of impedance networks yielded the overall impedance relationship between the interior port and the exterior port. Ewald's transform was used to accelerate the computation of Green's functions in a cavity with metallic walls, which yielded fast convergence of the series summation in Green's functions. Good agreement with a commercial full-wave tool validated the proposed approach.

Electromagnetic modelling of surfaces using method of moments with calculated phase mesh

An alternative method of moments (MoM) solution for the radiation and scattering of surfaces is presented. One limit of MoM is the high number of basis functions needed, for the segmentation of electrically large surfaces. In this study, a conventional MoM solution is defined with quadrilateral surface patches and at least eight rooftop basis functions per wavelength. Galerkin testing is applied. This conventional MoM is extended with a new approach for the surface segmentation, called the calculated phase mesh (CPM). Efficient basis functions are chosen, and their positions on the surface are calculated with the incident field. In the theory part, the number of unknowns for scatterers with electrically large dimensions is reduced by 50%, compared with conventional MoM meshes with eight functions per wavelength. The incident field on surfaces is calculated numerically. This allows the application of CPM on scattering bodies with a variety of field sources, for example, both the near and the far field of antennas. The valid frequency range for the meshes is described. Three numerical examples are presented. In one example, the number of basis functions was reduced to 69.3%, compared with conventional MoM. The third example shows the application on three-dimensional scattering bodies.

거친 도체 표면 후방 산란 계산을 위한 모서리 효과 저감 기법

Abstract An improved numerical scattering model with the 2-dimensional moment method including roof-top basis and a modified window- function to reduce edge-effect is presented in this study. The roof-top basis function is used to depict randomly positioned surface cu-rrents and increase an efficiency of the moment method. To reduce the edge-effect which occurs at the end of numerically generated surfaces, an enhanced window-function which is weighted by incident angle variable is proposed. To validate an proposed 2-dimensional scattering model and numerical analysis techniques for randomly rough surfaces, computational results are compared and analyzed to SPM(Small Perturbation Model) as well. Key words: Rough Surface, Moment Method, Roof-Top Basis, Edge-Effect Reduction, Small Perturbation Model _`a22013bYc(^def c)%g!CE hg I%i!+jk lKmn(No. 2013-005336).o pqre^:sYtuvwe(Department of Electronic Information and Communication Engineering, Hongik University)Manuscript received December 20, 2013 ; Revised February 20, 2014 ; Accepted February 24, 2014. (ID No. 20131220-129)Corresponding Author: Yisok Oh (e-mail: yisokoh@hongik.ac.kr)

Characterization of Superstrate-Loaded Resistive Rectangular Patch Antenna

The scattering radar cross section (RCS) of a superstrate loaded resistive rectangular microstrip patch which is printed on isotropic or uniaxial anisotropic substrate are investigated, where an accurate design based on the moment method technique in the spectral domain is developed. Entire domain sinusoid basis functions without edge condition and roof top sub-domain basis functions are introduced to expand the unknown current on the metal patches. The integral equation includes a superstrate resistive boundary condition on the surface of the patch and the effects of anisotropic substrate are developed. The necessary terms for representing the surface resistance on the patch were derived and were included in the equation in the form of a resistance matrix. Comparative study between our results and those available in the literature is done and showed a very good agreement.

Calculating the surface currents in electromagnetic scattering by screens of complex geometry

The vector problem of scattering an external electromagnetic field by a perfectly conducting thin bounded screen is considered. The approximation property of rooftop basis functions is proven. Numerical solutions of the problem for a screen of canonical shape are obtained using the Galerkin method. Numerical solutions of the problem for screens of complex geometry are obtained by a subhierarchic method.

HIGHER ORDER METHOD OF MOMENTS FOR BISTATIC SCATTERING FROM 2D PEC ROUGH SURFACE WITH GEOMETRIC MODELING BY NURBS SURFACE

The higher order method of moments (HMOM) has been proposed to calculate the bistatic scattering from two-dimensional (2D) perfectly electric conducting (PEC) rough surface in this paper. The electric fleld integral equation (EFIE) is solved through the HMOM with the hierarchical higher order basis functions which are the modifled Legendre polynomials. The non-uniform rational B-spline (NURBS) surface is applied to model the plane surface related to the rough surface. Validity of this approach is shown by comparing the bistatic scattering coe-cient (BSC) to that of lower order MOM (LMOM) with the Rao-Wilton-Glisson (RWG) or rooftop basis function. This approach has fewer segments in the parametric directions than the LMOM with rooftop basis, and is more e-cient for the fewer unknowns and requires less memory than the LMOM with RWG basis. Properties of bistatic scattering from a 2D Gaussian rough surface are also exhibited and analyzed.

Large Overlapping Subdomain Method of Moments for the Analysis of Frequency Selective Surfaces

A new set of basis functions is presented for the analysis of frequency selective surfaces (FSSs) by the method of moments (MoM). Each of the separate patches in the unit cell of an FSS is covered with some large overlapping sub-patches. Each sub-patch has a shape that is simple enough for obtaining an appropriate set of basis functions analytically from an eigenvalue problem. This technique is called the large overlapping subdomain MoM. It is shown that the proposed method has a better convergence than the standard MoM with rooftop basis functions in terms of both required basis functions and Fourier modes. Furthermore, this approach offers a much easier way to model patches with curved boundaries. The extraction of the proposed basis functions involves mostly analytical procedures. Therefore, this kind of moment method is computationally more efficient than versions with entire domain basis functions, in which the basis is obtained from the boundary integral resonant mode expansion technique. Nonetheless, this is accompanied by a small decay in the convergence rate, i.e., the large overlapping subdomain is clearly placed between the standard subdomain and entire domain versions of the MoM. It is shown that the developed method is advantageous for usual FSSs and unit cell configurations with several patches.

Effective medium theory applied to frequency selective surfaces on periodic substrates

We apply the effective medium theory combined with the conventional periodic method of moments (MoM) to analyze frequency selective surfaces (FSSs) on periodic and anisotropic substrates. Based on the effective medium theory, even periodic and anisotropic substrates can be considered homogeneous; thus, the Green's function can be obtained. The resulting integral equation can then be solved by the MoM using rooftop basis functions and Galerkin testing functions. We analyze an example using this technique, and the numerical results agree with Fallahi's full-vector semi-analytical method, showing an increasing difference between the results as the frequencies increase. These results show that the proposed method is effective for analyzing FSSs on periodic and anisotropic substrates.