Hybrid Finite Element-boundary Integral Research Articles

A Hybrid Loop-Tree FEBI Method for Low-Frequency Well Logging of 3-D Structures in Layered Media

Electromagnetic simulation plays an essential role in formation conductivity determination by electromagnetic well-logging tools, especially when the nonvertical borehole and invasion zones are present in a layered earth. A hybrid finite-element boundary integral (FEBI) method is suitable for such well-logging simulations. The usage of layered medium Green's functions allows the simulation background to be a planar stratified medium to characterize the earth formation. However, conventional FEBI using the Rao-Wilton-Glisson (RWG) basis function produces inaccurate results due to the low-frequency breakdown of the boundary integral equation solution. The new contribution of this work is to apply the loop-tree (LT) basis functions in the FEBI method to reduce the numerical error at low frequencies. Such an LT-FEBI method can handle all specific requirements of well-logging simulations at low frequencies. A direct solver is applied to make it more efficient to obtain logging curves with multiple source locations. The LT-FEBI simulation results are validated in several numerical examples, including a challenging well-logging model with a deviated borehole and invasion zones in the Oklahoma formation having 28 layers.

Broadband single-layer reflectarray antenna loaded with meander-delay-lines for X-band applications

In this work, delay-lines loaded broadband reflectarray antenna is proposed for X-band applications. The array cells contain slotted square patch and quad meander delay-lines. The slotted square patch elements are combined with delay-lines to increase the reflection phase range. A smooth linear phase variation of 650° is realized through varying the length of delay-lines. The reflectarray unit element resonant behavior is observed through an equivalent circuit model approach. The reflectarray antenna is simulated and measured on a low-cost FR-4 substrate. A 23 × 23 elements reflectarray antenna is designed and simulated using a hybrid finite element boundary integral (FEBI) method. The proposed reflectarray antenna offers a measured gain of 25 dBi and 52.8% aperture efficiency for 10 GHz frequency band. The cross-polarization levels and side-lobe-levels and in E and H-plane are less than −35 dB and −22 dB respectively. Moreover, a measured 3-dB gain bandwidth of 29% and 1-dB gain of 19.5% is realized for the proposed reflectarray antenna.

An Effective $\mathcal{H}$-LU-Based Preconditioner for the FE-BI-MLFMA for 3-D Scattering Problems

A flexible and efficient Hierarchical LU ( $\mathcal{H}$ -LU)-based preconditioner ( $\mathcal{H}$ -LU-P) is presented for the hybrid finite element-boundary integral (FE-BI) multilevel fast multipole algorithm for solving three-dimensional scattering by inhomogeneous objects. The formulation of FE-BI is first approximated by using locally approximated integral operators for the BI part to construct a finite element method-absorbing boundary condition (FEM-ABC) based preconditioning matrix. Then, the preconditioning matrix equation is solved by the nested dissection accelerated $\mathcal{H}$ -LU-based fast direct solver. A semi-geometric clustering strategy is proposed to simplify the cluster tree-building procedure in representing the preconditioning matrix in the $\mathcal{H}$ -matrix form. Numerical results demonstrate that the $\mathcal{H}$ -LU-P outperforms other alternative preconditioners, based on multifrontal approach. The scattering by a large and complex aircraft model with a radome on its nose section is calculated, showing the capability and efficiency of the preconditioner.

A Cost-Effective Preconditioner for Complete Domain Decomposition Method of Hybrid Finite Element-Boundary Integral

A complete domain decomposition method (CDDM) of hybrid finite element-boundary integral (FE-BI) has been proposed for accurately analyzing multiscale and heterogeneous electromagnetic problem recently. In this communication, a cost-effective additive Schwarz preconditioner is proposed to further improve the capability of the CDDM-FEBI for electrically large scattering and radiation problems. This preconditioner is obtained by rarefying the FE-BI submatrix based on the characteristics of boundary integral equation. Numerical experiments demonstrate that the consumption of CPU time and memory for constructing the proposed preconditioner is greatly reduced comparing with the previous one under the premise of ensuring fast convergence.

A Geometry-Aware Domain Decomposition Preconditioning for Hybrid Finite Element-Boundary Integral Method

Fast, scalable, and robust solution of the hybrid finite element-boundary integral (FE-BI) linear system of equations is traditionally considered a challenge due to multifaceted technical difficulties. This paper proposes a nonoverlapping geometry-aware domain decomposition (DD) preconditioning technique for iteratively solving hybrid FE-BI equation. The technique ingredients include a volume-based Schwarz FE DD method and a surface-based interior penalty BI DD method. Compared with previous algorithms, the work has two major benefits: 1) it results in a robust and cost-effective preconditioning technique for the solution of the FE-BI linear system of equations and 2) it provides a flexible and natural way to set up the mathematical models, to create the problem geometries, and to discretize the computational domain. The capability and performance of the computational algorithms are illustrated and validated through numerical experiments.

Application of AIM and MBPE Techniques to Accelerate Modeling of 3-D Doubly Periodic Structures with Nonorthogonal Lattices Composed of Bianisotropic Media

An efficient methodology is introduced for rapid analysis and design of three-dimensional (3-D) doubly periodic structures over a wide frequency range based on hybrid finite element boundary integral (FEBI) methods. The 3-D doubly periodic structures can be represented as nonorthogonal lattices composed of general inhomogeneous bianisotropic media with arbitrarily-shaped metallic patches. Based on Floquet theory and periodic boundary conditions, the original stated problem that involves infinite periodic structures can be converted into a single unit cell. Using the equivalence principle, the derived BI equation formulation is applied to the top and bottom surfaces of the unit cell, which results in a perfectly reflectionless boundary condition for the FE-based approach. Then, the unit cell was meshed using triangular prismatic volume elements, which provide a great deal of flexibility in modeling complex planar geometries with arbitrary shapes in the transverse direction. The adaptive integral method (AIM) was employed to accelerate the calculation of the matrix-vector product for the BI portion within the iterative solver. Furthermore, a model-based parameter estimation (MBPE) technique was proposed for the wide-band interpolation of the required impedance matrix elements in the BI part for near field components that were used in the AIM procedure. The accuracy and efficiency of the proposed hybrid algorithms are demonstrated by the presented numerical results (e.g., in comparison with analytical solutions). Several simulation results are presented to illustrate the flexibility of the proposed methods for analysis of frequency selective surfaces with arbitrarily-shaped metallic patches, bianisotropic materials, and nonorthogonal lattice configurations.

Investigation on scattering from a plasma-coated target over a rough sea surface using a multi-hybrid method

An iterative strategy combining the Kirchhoff approximation (KA) with the hybrid finite element–boundary integral (FE-BI) method is presented to consider electromagnetic scattering from a coated target above a rough sea surface. The multi-hybrid method considers the multiple-scattering interactions between target and underlying surface based on equivalent principle and currents updating scheme. The hybrid FE-BI-KA method, which is an improved and generalized version of a previous KA–Method of Moments (MoM) technique, can deal with an inhomogeneous target. Numerical results are given to evaluate the validity of the multi-hybrid technique, and then the hybrid method is employed to investigate electromagnetic scattering from a plasma-coated airfoil above a rough sea surface, including the effects of several key parameters on stealth performance, such as plasma angular frequency and electron collision frequency.

Multi-hybrid method for investigation of EM scattering from inhomogeneous object above a dielectric rough surface

An iterative strategy combining Kirchhoff approximation^(KA) with the hybrid finite element-boundary integral (FE-BI) method is presented in this paper to study the interactions between the inhomogeneous object and the underlying rough surface. KA is applied to study scattering from underlying rough surfaces, whereas FE-BI deals with scattering from the above target. Both two methods use updated excitation sources. Huygens equivalence principle and an iterative strategy are employed to consider the multi-scattering effects. This hybrid FE-BI-KA scheme is an improved and generalized version of previous hybrid Kirchhoff approximation-method of moments (KA-MoM). This newly presented hybrid method has the following advantages: (1) the feasibility of modeling multi-scale scattering problems (large scale underlying surface and small scale target); (2) low memory requirement as in hybrid KA-MoM; (3) the ability to deal with scattering from inhomogeneous (including coated or layered) scatterers above rough surfaces. The numerical results are given to evaluate the accuracy of the multi-hybrid technique; the computing time and memory requirements consumed in specific numerical simulation of FE-BI-KA are compared with those of MoM. The convergence performance is analyzed by studying the iteration number variation caused by related parameters. Then bistatic scattering from inhomogeneous object of different configurations above dielectric Gaussian rough surface is calculated and the influences of dielectric compositions and surface roughness on the scattering pattern are discussed.

Simulation of electromagnetic scattering by random discrete particles using a hybrid FE-BI-CBFM technique

A hybrid finite element–boundary integral–characteristic basis function method (FE-BI-CBFM) is proposed for an efficient simulation of electromagnetic scattering by random discrete particles. Specifically, the finite element method (FEM) is used to obtain the solution of the vector wave equation inside each particle and the boundary integral equation (BIE) using Green's functions is applied on the surfaces of all the particles as a global boundary condition. The coupling system of equations is solved by employing the characteristic basis function method (CBFM) based on the use of macro-basis functions constructed according to the Foldy–Lax multiple scattering equations. Due to the flexibility of FEM, the proposed hybrid technique can easily deal with the problems of multiple scattering by randomly distributed inhomogeneous particles that are often beyond the scope of traditional numerical methods. Some numerical examples are presented to demonstrate the validity and capability of the proposed method.

Hybrid FE-BI-KA method in analysing scattering from dielectric object above sea surface

A new and more powerful algorithm combining Kirchhoff approximation (KA) with the hybrid finite element-boundary integral (FE-BI) method is presented to study the interactions between an inhomogeneous object and the underlying sea surface. This hybrid FE-BI-KA scheme is an improved and generalised version of the previous KA-MoM. A reduced model composed of a PEC object above the conducting sea surface is simulated to evaluate the validity of the multi-hybrid technique by comparing with MoM. Then the bistatic scattering pattern of a more complex layered object above the dielectric sea surface is calculated at different wind speeds.

Directly Connected Hybrid Finite Element–Boundary Integral–Ray-Optical Models

The hybrid finite element–boundary integral–uniform geometrical theory of diffraction method with acceleration by the multi-level fast multi-pole method is extended in a way that allows the application of the hybrid method to problems, where the finite element–boundary integral–multi-level fast multi-pole method parts of the model can be directly connected to uniform geometrical theory of diffraction regions. This becomes possible by an appropriate utilization of Huygens' principle, together with the fact that the regions exterior to the introduced Huygens surface is free of any fields and can thus be filled with arbitrary materials. As such, the appropriate materials are chosen in order to obtain simple enough bodies, which can be treated by the uniform geometrical theory of diffraction. Since some of the boundary integral basis functions typically touch flat parts of these bodies, the image principle is employed for the accurate evaluation of the corresponding boundary integrals. Numerical results for radiation and scattering problems are presented and compared to reference numerical data. The obtained results prove the feasibility of the procedure and considerable computation time and memory savings.

Some scattering results computed by surface-integral-equation and hybrid finite-element - boundary-integral techniques, accelerated by the multilevel fast multipole method

Method-of-moments (MoM) solutions of surface integral equations are especially well suited for scattering computations involving metallic objects. Improved modeling flexibility for dielectric (possibly lossy) and mixed dielectric/metallic bodies is obtained by combining a surface-integral-equation formulation, involving electric and magnetic equivalent surface-current densities, with a volumetric finite-element (FE) model of the dielectric regions. This results in the well-known hybrid FEBI (finite-element-boundary-integral) technique. For many years, hybrid FEBI techniques, as well as stand-alone Bl (surface-integral equation, often just termed MoM) techniques, were restricted to relatively small (with respect to a wavelength) geometries. However, with the development of powerful multilevel fast multipole methods/algorithms (MLFMM/MLFMA), it has become possible to compute a larger variety of practical scattering and radiation problems with the hybrid FEBI-MLFMM technique. In this contribution, we give a short review of our hybrid FEBI-MLFMM approach, with a focus on mixed dielectric/metallic geometries and multiple Bl domains. We then present a variety of scattering results for metallic and mixed dielectric/metallic objects, together with comparisons with measured RCS (radar cross section) data. Broadband computations are used to derive high-resolution range (HRR) profiles of several configurations.

Analysis of a Rectangular Waveguide, Edge Slot Array With Finite Wall Thickness

A method to analyze a one-dimensional array comprising tilted edge slots cut in the narrow wall of a rectangular waveguide is presented. The fields in the slots are calculated from a hybrid finite element-boundary integral (FE-BI) equation method. The boundary surface integrals on both the interior and exterior surface of the slots are computed from a spectrum of two-dimensional solutions (S2DS), which allows the influence of both the interior and exterior structure of the waveguide to be included rigorously. For initial array design, the mutual coupling between slots in the exterior region is treated approximately with an infinite array assumption modeled by two slots surrounded by periodic boundary walls with an imposed linear phase progression. A short, finite slot array is designed based on the infinite array assumption and fabricated. Results from full-wave analysis of the array and measurements are presented

The Implicit, Element Residual Method for a Posteriori Error Estimation in FE-BI Analysis

Hybrid finite element-boundary integral (FE-BI) formulations are widely used for electromagnetic analysis. In such analysis, estimation of the mesh-wide error distribution and global solution quality may be used to drive adaptive refinements. In this paper, an a posteriori error estimator based on the implicit, element residual method is constructed for the cavity FE-BI formulation, with dominant-mode coaxial port excitation. It is based on solving elemental variational boundary value problems of the error, using approximate, localized boundary conditions. Results show that the scheme can be applied to practical problems, yielding significant reductions in computational cost.

A hybrid FEBI-MLFMM-UTD method for numerical solutions of electromagnetic problems including arbitrarily shaped and electrically large objects

Numerical solutions of electromagnetic scattering and radiation problems including arbitrarily shaped objects are obtained by solving integral equations with the method of moments (MoM). Fast and efficient solution of the integral equation with low computation and memory complexity is provided by the multilevel fast multipole method (MLFMM). The presence of electrically large conducting objects leads to hybrid MoM techniques with high-frequency methods. For ray-based high-frequency methods no discretization of the electrically large objects is needed, resulting into a more efficient numerical treatment of the problem. However, in order to retain low computation and memory complexity, the high-frequency fields must be taken into account in the matrix-vector product computations in the various levels of the MLFMM. In this contribution, a ray-based hybridization of the MLFMM with the uniform geometrical theory of diffraction (UTD) is proposed within a hybrid finite element-boundary integral (FEBI) technique, using the combined field integral equation (CFIE), resulting into a hybrid FEBI-MLFMM-UTD method. The hybridization is performed at the translation procedure on the various levels of the MLFMM, using a far-field approximation of the appropriate translation operator to obtain the high-frequency incident fields at the critical points of the UTD. The formulation of this new hybrid technique is presented and numerical results are shown.

Dual-Polarized Frequency-Tunable Composite Left-Handed Slab

A dual-polarized frequency-tunable composite left-handed slab (CLS) is proposed. The CLS consists of periodic split-ring resonators (SRRs) along with thin wires. Two orthogonal SRR/wire arrays are utilized for dual-polarization purposes. Polarization-selectivity and frequency-tuning are achieved by means of on/off switches placed between the rings of the SRR. By opening all of the switches for one array and closing them for the other, the left-handed behavior can be switched on and off, respectively. Frequency-tuning performance is then achieved by turning on/off the switches in the former array. Transmission characteristics of the composite structure are predicted using a fast periodic array simulator based on the hybrid finite element-boundary integral (FE-BI) method in conjunction with the fast spectral domain algorithm (FSDA). Numerical results demonstrate dual-polarization and frequency-tuning performance of the proposed switching configuration.

Scattering of Three-Dimensional Chiral Objects above a Perfect Conducting Plane by Hybrid Finite Element Method

Electromagnetic scattering from three-dimensional chiral objects above a perfectly electric or magnetic conducting plane (PEC and PMC) is analyzed by the hybrid finite element-boundary integral (FEM-BI) method. The chiral image principle is used to transform the original single chiral target in half space into two chiral targets with the same permittivity and permeability yet conjugate chirality parameters. Numerical examples demonstrate that the FEM-BI method provides a flexible and powerful solution of multiple scattering from inhomogeneous chiral objects. Both near field distribution and radar cross section (RCS) of chiral objects illuminated by plane waves are provided to demonstrate the effectiveness of the hybrid method.

Modeling Conformal Antennas on Metallic Prolate Spheroid Surfaces Using a Hybrid Finite Element Method

In this paper, the hybrid finite element-boundary integral (FE-BI) method appropriate for modeling conformal antennas on doubly curved surfaces is developed. The FE-BI method is extended to model doubly curved, convex surfaces by means of a specially formulated asymptotic dyadic Green's function. The FE-BI method will then be used to examine the effect of curvature variation on the resonant input impedance of a cavity-backed, conformal slot antenna and a conformal patch antenna recessed in a perfectly conducting, electrically large prolate spheroid surface. The prolate spheroid shape provides a canonical representation of a doubly curved mounting surface. The numerical results for conformal slot and patch antennas on the prolate spheroid are compared as a function of curvature and orientation.

A highly effective preconditioner for solving the finite element-boundary integral matrix equation of 3-D scattering

A highly effective preconditioner is presented for solving the system of equations obtained from the application of the hybrid finite element-boundary integral (FE-BI) method to three-dimensional (3-D) electromagnetic scattering problems. Different from widely used algebraic preconditioners, the proposed one is based on a physical approximation and is constructed from the finite element method (FEM) using an absorbing boundary condition (ABC) on the truncation boundary. It is shown that the large eigenvalues of the finite element (FE)-ABC system are similar to those of the FE-BI system. Hence, the preconditioned system has a spectrum distribution clustered around 1 in the complex plane. Consequently, when a Krylov subspace based method is employed to solve the preconditioned system, the convergence can be greatly accelerated. Numerical results show that the proposed preconditioner can improve the convergence of an iterative solution by approximately two orders of magnitude for large problems.

Radiation and scattering by complex conformal antennas on a circular cylinder

A technique for characterizing and designing complex conformal antennas flush-mounted on a singly-curved surface is presented. This approach is based on the hybrid finite element–boundary integral (FE–BI) method. A related method was proposed in the past utilizing cylindrical-shell finite element and roof-top rectangular basis functions for the boundary integral. Although that method proved very powerful for analyzing cylindrical–rectangular patch arrays flush-mounted to a circular cylinder, the requirement for uniform meshing in the aperture ultimately limited its usefulness. In this present formulation, tetrahedral elements are used to expand the volumetric electric fields while similar basis functions are used for the boundary integral. The curvature of the aperture is explicitly included via the use of the circular cylinder dyadic Green's function. After presentation of the formulation and validation using several well-understood examples, an example is presented that illustrates the capabilities of this method for modeling complex conformal antennas heretofore not examined by rigorous methods due to inherent limitations of the various published methods.