Finite Element-boundary Integral Research Articles

An analytical study of curved frequency selective surfaces for shielding applications

AbstractThis paper demonstrates an analytical study of different nonplanar frequency selective surfaces (FSSs) for X‐band (8–12 GHz) electromagnetic (EM) shielding applications. A well‐known metallic square loop unit cell is used for the analysis, which is arranged in a periodic pattern on a three‐dimensional (3‐D) printer compatible dielectric substrate. The geometry is wrapped on various kinds of curved surfaces (cylindrical, paraboloid, and hemispherical surfaces) and investigated using the finite element boundary integral (FEBI) technique. The structure exhibits a band stop characteristic with a fractional bandwidth of 62.74% (6.68–12.66 GHz, having transmission coefficient below −10 dB), resonating around 9.53 GHz under different angles of incident EM wave. Finally, the 3‐D printing method is used to fabricate the prototypes for performing laboratory measurements. The simulated and measured results follow each other closely, thereby establishing the proposed technique in realizing nonplanar FSSs.

Surface Integral Equation Solutions for Electromagnetic Scattering From a Special Class of Anisotropic Media

We present a surface integral equation (SIE) solution for a special class of anisotropic media, whose permittivity and permeability are both symmetric full tensors with the same or multiple values. This kind of media is generated from the invisible cloaking or absorbing material of a perfectly matched layer (PML) design. A new derivation based on the potential functions is presented to obtain the closed-form Green’s function. Detailed and rigorous proofs of some vector identities are given for the first time. Special treatments are detailed to address the singularity of the anisotropic Green’s function for discretizing the SIE. Numerical experiments validate the accuracy, stability, and high efficiency of the proposed solution compared with the analytical solution or the finite element-boundary integral (FE-BI) method.

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.

Electromagnetic waves scattering from an infinite elliptical metal cylinder covered by a plasma layer in presence of external magnetic field

In this paper, scattering of the electromagnetic waves from a metal elliptical cylinder covered by a plasma layer is investigated by the Finite Element-Boundary Integral (FE-BI) method. The under study object that can be supposed as an elliptic complex antenna is located in a constant magnetic field. The inner cylinder can be rotated to plasma cover so that they are nonconfocal. Generally, elliptic antennas have azimuthal asymmetry and their response to electromagnetic waves is sensitive to incident angle of landing wave. This effect is more severe in under study antenna due to its geometrical configuration that depends on the rotating angle of the conducting core related to plasma cover. Furthermore, presence of constant magnetic field along the antenna axis causes permittivity tensor of the plasma as an influential factor in radar cross section, becomes anisotropic. Here, backscattering radar cross section (BRCS) of the antenna is calculated when irradiated by TE and TM waves under different incident angles. It will be shown how one can control BCRS of the antenna by varying its configuration and the external magnetic field intensity. Validation of numerical computations applied in this work is demonstrated by comparing it with an analytical method for a special case.

A Feasibility Study of 2-D Microwave Thorax Imaging Based on the Supervised Descent Method

In this paper, the application of the supervised descent method (SDM) for 2-D microwave thorax imaging is studied. The forward modeling problem is solved by the finite element-boundary integral (FE-BI) method. According to the prior information of human thorax, a 3-ellipse training set is generated offline. Then, the average descent direction between an initial background model and the training models is calculated. Finally, the reconstruction of the testing thorax model is achieved based on the average descent directions online. The feasibility using One-Step SDM for thorax imaging is studied. Numerical results indicate that the structural information of thorax can be reconstructed. It has potential for real-time imaging in future clinical diagnosis.

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.

Direct Solution of Finite Element-Boundary Integral System for Electromagnetic Analysis in Half-Space

An efficient direct solution scheme is presented to solve the finite element-boundary integral (FEBI) system for the analysis of open-region electromagnetic problems in half-space. Data-sparse representation based on hierarichal (H-) matrix is employed to construct the direct solution framework. The FEBI system matrix is first represented to be an H-matrix form, in which the FE submatrices are accurately represented by H-matrices, while the H-matrix approximations of the BI submatrices are constructed by the adaptive cross approximation algorithm combined with singular value decomposition recompression (ACA-SVD). Based on the data-sparse formatted algorithm, the lower and upper (LU) triangular factors of the FEBI system matrix can be computed and stores as H-matrix forms with low computational costs. The resulting method, refered to as the direct FEBI method (D-FEBI), is free of convergence problem and well-suited for multiple right-hand side (RHS) computations. The performance of the D-FEBI is examined by two examples of monostatic electromagnetic scattering above a lossy half-space.

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.

Iterative FEBI-PO Hybrid Method for Analyzing Scattering Problems of Complex Structures Having Junctions With Large-Scale Platform

An iterative hybrid of finite element boundary integral (FEBI) method and physical optics (PO) is presented for scattering problems of complex structures having junctions with electrically large-scale platform. The treatment of basis functions on the junctions between complex structures and platform is discussed in detail. Moreover, the multilevel fast multipole algorithm and the adaptive cross approximation method are employed to accelerate the solution of FEBI and the computation of interaction between FEBI and PO regions, respectively. Compared with FEBI method, the hybrid method can efficiently reduce the number of unknowns and the computational complexity while preserving acceptable accuracy. In addition, the alternative implementation of FEBI and PO can also maintain good independence of the two methods. Some numerical results are discussed to demonstrate the efficiency and accuracy of the approach.

Nonlinear Reconstruction of Multilayer Media in Scanning Microwave Microscopy

Scanning microwave microscopy (SMM) is a nanoscale characterization technique widely used to image electrical or magnetic properties of materials at microwave frequencies. Nevertheless, it is challenging to reconstruct the properties of materials from measured SMM signals, and it is even difficult to retrieve multilayer media directly from measured signals. This paper proposes a nonlinear inversion scheme to quantitatively reconstruct multilayer media. In the forward solver, hybrid boundary integral-finite element method is used to reduce the computational cost. In the inverse problems, capacitance or conductance derivative signals at different probe–sample distances are used to retrieve multilayer media. Reconstructed results show that the proposed approach is able to reconstruct both permittivity and conductivity distributions under noisy environment in 3-D samples. Most importantly, it is found that the resolution has been significantly improved in the retrieved images compared with capacitance or conductance signals. To the best of our knowledge, it is the first time that the multilayer media in 3-D media is reconstructed in SMM.

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 hybrid numerical treatment of nonlinear reaction-diffusion equations with memory: A prototypical Fisher-Kolmogorov-Petrovskii-Piskunov equation

In reaction-diffusion systems with non-standard diffusion, the memory of the transport process causes a coupling of reaction and diffusion. A generalisation of the Fick's law has been suggested to account for this coupling. Furthermore, the resultant effects of the interplay of transport, memory and reaction lend themselves to some interesting physics which is still not well understood because the governing equation as well as the accompanying memory integral and nonlinear reaction terms are not always amenable to tidy analytic or numerical expressions. Hence, the derivation of a suitable governing integro-differential equation as well as the approximate solution demands a considerable level of attention'. The main focus of this work can be seen as a contribution towards this objective. In this report, we develop and apply a hybrid boundary integral-finite element - finite difference numerical procedure to investigate an integro-differential-Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) - type kinetics. We also focus on scalar evolution for cases where the reaction coefficient takes on relatively large values. Although we are still far from a rigorous mathematical analysis, it has been found that the numerical results obtained compared favourably with existing benchmark solutions. This not only validates the current numerical formulation but also justifies the physics of the resulting front propagation.

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.

A Multi-Solver Scheme Based on Combined Field Integral Equations for Electromagnetic Modeling of Highly Complex Objects

A combined field integral equation (CFIE)-based multisolver scheme is presented for electromagnetic modeling of objects with complex structures and materials. In this scheme, an object is decomposed into multiple bodies based on its material property and geometry. To model bodies with complicated materials, the finite element-boundary integral (FE-BI) method is applied. To model bodies with homogeneous or conducting materials, the method of moments is employed. Specifically, three solvers are integrated in this multi-solver scheme: the FE-BI(CFIE) for inhomogeneous objects, the CFIE for dielectric objects, and the CFIE for conducting objects. A mixed testing scheme that utilizes both the Rao–Wilton–Glisson and the Buffa–Christiansen functions is adopted to obtain a good accuracy. In the iterative solution of the combined system, the multilevel fast multipole algorithm is applied to accelerate computation and reduce memory costs, and a preconditioner based on an absorbing boundary condition is employed to speed up the convergence. In the numerical examples, the individual solvers are first demonstrated to be well conditioned and highly accurate. Then, the validity of the proposed multisolver scheme is demonstrated and its capability is shown by solving scattering problems of electrically large missilelike objects.

A Multisolver Scheme Based on Robin Transmission Conditions for Electromagnetic Modeling of Highly Complex Objects

A multisolver scheme based on Robin transmission conditions (RTCs) is proposed for electromagnetic modeling of highly complex objects. Different from the traditional finite element–boundary integral (FE-BI) method that applies BI equations to truncate the FE domain, the proposed multisolver scheme employs both FE and BI equations to model an object along with its background. To be specific, the entire computational domain consisting of the object and its background is first decomposed into multiple nonoverlapping subdomains with each modeled by either an FE or BI equation. The equations in the subdomains are then coupled into a multisolver system by enforcing the RTC at the subdomain interfaces. Finally, the combined system is solved iteratively with the application of an extended preconditioner based on an absorbing boundary condition and the multilevel fast multipole algorithm. To obtain an accurate solution, both the Rao–Wilton–Glisson and the Buffa–Christiansen functions are employed as the testing functions to discretize the BI equations. This scheme is applied to a variety of benchmark problems and the scattering from an aircraft with a launched missile to demonstrate its accuracy, versatility, and capability. The proposed scheme is compared with the multisolver scheme based on combined field integral equations to illustrate the differences between the two schemes.

A coupled FE-BIE model for the static analysis of Timoshenko beams bonded to an orthotropic elastic half-plane

Interface displacements, surface tractions and stresses of a flexible beam bonded to an elastic orthotropic half-plane are analysed by means of a Finite Element-Boundary Integral Equation (FE-BIE) method. Numerical results are obtained by using locking-free shear deformable beams and piecewise constant interfacial reactions. Making use of the generalised Green's function for the half-plane, the mechanical behaviour of fully bonded or detached beams subjected to force, couple or thermal load is investigated. The special cases of a beam in bilateral frictionless contact with the half-plane and a beam having a vanishing bending rigidity (thin film) are considered also. In particular, the maximum bending moment of beams subjected to a vertical point force are compared with some closed-form solutions of the contact problem of a rigid indenter and with the solution of an infinite Euler–Bernoulli beam in bilateral frictionless contact with an isotropic substrate.

FE-BI Formulations for Characteristic Modes

Two types of finite-element boundary-integral (FE-BI) formulations are developed to determine the characteristic modes (CMs) of general composite material objects. The electromagnetic (EM) fields inside the general objects with inhomogeneous anisotropic material are flexibly formulated using the finite-element (FE) formulations, while the EM fields outside the general material objects are accurately described using the boundary-integral (BI) equations defined on the general objects’ boundary. The developed FE-BI formulations result in two generalized Hermitian eigenvalue (GHE) equations for the solution of CMs. These two novel FE-BI based formulations provide clear physical explanation for the analysis of the CMs of the general objects. It is found that their corresponding eigenvalues indicate the imaginary part of the complex power of CMs, which are zero at resonance. Simulated results for typical dielectric resonators are presented to clearly demonstrate the accuracy of the proposed FE-BI formulations.

An Accurate and Efficient Finite Element-Boundary Integral Method With GPU Acceleration for 3-D Electromagnetic Analysis

An accurate and efficient finite element-boundary integral (FE-BI) method with graphics processing unit (GPU) acceleration is presented for solving electromagnetic problems with complex structures and materials. A mixed testing scheme, in which the Rao-Wilton-Glisson and the Buffa-Christiansen functions are both employed as the testing functions, is first presented to improve the accuracy of the FE-BI method. An efficient absorbing boundary condition (ABC)-based preconditioner is then proposed to accelerate the convergence of the iterative solution. To further improve the efficiency of the total computation, a GPU-accelerated multilevel fast multipole algorithm (MLFMA) is applied to the iterative solution. The radar cross sections (RCS) of several benchmark objects are calculated to demonstrate the numerical accuracy of the solution and also to show that the proposed method not only is free of interior resonance corruption, but also has a better convergence than the conventional FE-BI methods. The capability and efficiency of the proposed method are analyzed through several numerical examples, including a large dielectric coated sphere, a partial human body, and a coated missile-like object. Compared with the 8-threaded CPU-based algorithm, the GPU-accelerated FE-BI-MLFMA algorithm can achieve a total speedup up to 25.5 times.

Analytical finite element matrix elements and global matrix assembly for hierarchical 3-D vector basis functions within the hybrid finite element boundary integral method

Abstract. A hybrid higher-order finite element boundary integral (FE-BI) technique is discussed where the higher-order FE matrix elements are computed by a fully analytical procedure and where the gobal matrix assembly is organized by a self-identifying procedure of the local to global transformation. This assembly procedure applys to both, the FE part as well as the BI part of the algorithm. The geometry is meshed into three-dimensional tetrahedra as finite elements and nearly orthogonal hierarchical basis functions are employed. The boundary conditions are implemented in a strong sense such that the boundary values of the volume basis functions are directly utilized within the BI, either for the tangential electric and magnetic fields or for the asssociated equivalent surface current densities by applying a cross product with the unit surface normals. The self-identified method for the global matrix assembly automatically discerns the global order of the basis functions for generating the matrix elements. Higher order basis functions do need more unknowns for each single FE, however, fewer FEs are needed to achieve the same satisfiable accuracy. This improvement provides a lot more flexibility for meshing and allows the mesh size to raise up to λ/3. The performance of the implemented system is evaluated in terms of computation time, accuracy and memory occupation, where excellent results with respect to precision and computation times of large scale simulations are found.

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.