Finite Element-boundary Integral Method Research Articles

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.

An effective method for electromagnetic scattering from a coated object partially buried in a dielectric rough surface

An effective method combined with finite element method (FEM) and boundary integral method (BIM) is presented in this paper to study the composite electromagnetic scattering from a 2-D coated object partially buried in a 1-D dielectric rough surface. The field within the coated object is modelled by FEM, while the coupling field between the object and rough surface is considered by BIM. The fields inside and outside the object are coupled together via boundary conditions on the boundary of the object. Since the boundary of the object is divided by the rough surface into two parts with different electromagnetic properties, the boundary conditions on the object are more complicated. A detailed derivation of the FEM-BIM formula for the composite model is given in this paper. Our numerical code is validated by the method of moments (MoM), and several numerical simulations are conducted to discuss the scattering characteristics.

Real-time prediction of the deformation of microcapsules using Proper Orthogonal Decomposition

Microcapsules are liquid droplets enclosed by a thin elastic membrane. Being suspended in an external fluid, they undergo large deformations when flowing. Their deformation can be solved numerically, but the resolution of the fluid–structure interactions (FSI) requires extremely long computation times. This is a major constraint for instance when identifying the membrane mechanical properties from experimentations of microcapsules flowing in a microfluidic channel of similar size (Hu et al. PFE 2013). We propose to apply Model Order Reduction (MOR) to predict in real time the steady-state capsule deformed shape, needed to determine the membrane elasticity. A database of the capsule deformed shapes was obtained numerically by solving the three-dimensional (3D) FSI through a finite element-boundary integral method coupling, and by varying systematically the two non-dimensionalized parameters of the problem: the capillary number, ratio of the viscous to the elastic forces, and the capsule-to-tube size ratio. Among the MOR techniques, we chose to apply Proper Orthogonal Decomposition (POD) onto the database, which provides a vector basis of principal components, defining a multi-dimensional vector space. The advantage is that, when all the capsule shapes of the database are mapped into this new vector space, they form a manifold (smooth hypersurface) that represents all the admissible solutions of the problem. We show that POD with a manifold walking technique can be successfully applied to 3D microcapsule data sets, whether they are Lagrangian (e.g. known position vector fields) or Eulerian (i.e. when data is acquired experimentally or numerically using methods like level sets). In both cases, the problem dimensionality is reduced, the predicted capsule shapes are obtained within computation times of milliseconds, and they accurately fit the full FSI simulations. This paves the way to real-time computations for capsules in flow, while retaining all the physical ingredients of the FSI problem.

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.

On capacity computation for symmetric polygonal condensers

Making use of two different analytical–numerical methods for capacity computation, we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers. These two methods are based respectively on the use of the Lauricella function (Bezrodnykh and Vlasov, 2002; Bezrodnykh, 2016 [64,65]) and Riemann theta functions (Bogatyrëv, 2012; Grigoriev, 2013; Bogatyrëv and Grigor’ev, 2017; Bogatyrëv and Grigor’ev, 2018). We apply these results to benchmark the performance of numerical algorithms, which are based on adaptive h p -finite element method (Hakula et al. 2011, 2013, 2018) and boundary integral method (Tsuji, 1959; Jaswon and Symm, 1977; Albrecht and Collatz, 1980).

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.

Efficient hybrid method for electromagnetic scattering from a coated object above a two-layered rough surface

A powerful and efficient hybrid method combining the finite-element boundary-integral method with fast multipole method (FE-BI-FMM) is proposed in this paper to study the electromagnetic scattering characteristics from a coated object above a two-layered dielectric rough surface. The finite-element method (FEM) is applied to model the scattering from the coated object, and multiple interactions between the object and layered rough surface are handled by the FMM-enhanced boundary integral method (BIM). A hybrid solver is adopted to efficiently solve the FEM-BIM matrix equation. Several numerical results are presented, and the influence of several parameters of the composite model on the scattering characteristics is analyzed in detail. Both vertical and horizontal polarizations for the incident waves are considered.

Analysis and numerical solution of transient electromagnetic scattering from two cavities

This paper presents a finite-element boundary integral method to model transient scattering from two cavities embedded in a ground plane. Multiple-cavity scattering in the time domain both increases the computational complexity of the scattering problem and introduces cross-talk between scatterers. By modeling the cavities together, coupling effects are enforced directly and multiple reflections are readily observable. The variational formulation is shown to be well-posed for each time step and a stable numerical implementation for rectangular cavities is demonstrated.

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.

Prediction of metallic nano-optical trapping forces by finite element-boundary integral method.

The hybrid of finite element and boundary integral (FE-BI) method is employed to predict nano-optical trapping forces of arbitrarily shaped metallic nanostructures. A preconditioning strategy is proposed to improve the convergence of the iterative solution. Skeletonization is employed to speed up the design and optimization where iteration has to be repeated for each beam configuration. The radiation pressure force (RPF) is computed by vector flux of the Maxwell's stress tensor. Numerical simulations are performed to validate the developed method in analyzing the plasmonic effects as well as the optical trapping forces. It is shown that the proposed method is capable of predicting the trapping forces of complex metallic nanostructures accurately and efficiently.

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.

Inverse Scattering Approach on Tomography Problem Using Multi-frequency Data

An inverse scattering problem is formulated for reconstructing optical properties of biological tissues. A recursive linearization algorithm is used to solve the inverse scattering problem. We employed the idea of finite element boundary integral method and added suitable boundary conditions on the surface of the domain. The initial guess is obtained by Born approximation based on the fact of weak scattering. The reconstruction is then improved each time by an increment on wave number. Finite element method is used for the interior domain containing inhomogeneity. Nystrom method is used for setting up the boundary conditions and jump conditions. Two numerical examples are presented.

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.

Composite Scattering from an Arbitrary Dielectric Target above the Dielectric Rough Surface with FEM/PML

A numerical approach of the finite element method is extended to study the scattering properties of an arbitrary dielectric target above the dielectric rough surface. For scattering in an open region, the artificial boundaries should be introduced to truncate the infinite computational domain. A perfectly matched layer (PML), as the artificial boundary of the finite-element-method (FEM) region, is employed to absorb the outward wave scattered from the model. The strategies of hybrid FEM/PML are presented with their validity evaluated by finite element/boundary integral method (FE/BIM), and then the scattering properties of the dielectric composite problem with different material permittivity are discussed in detail. Compared with the published works about FEM/PML, we extend the FEM/PML into the simulations of a dielectric target above the dielectric rough surface.

Application of Asymptotic Waveform Evaluation to Hybrid FE-BI-MLFMA for Fast RCS Computation Over a Frequency Band

The asymptotic waveform evaluation (AWE) technique has been widely used in the method of moment (MoM), the finite element method (FEM), and the hybrid finite element-boundary integral method (FE-BI) for fast computation over a wide frequency band. However, how to apply AWE to the multilevel fast multipole algorithm (MLFMA) still remains challenge. This paper proposes an efficient and robust approach to apply AWE to the hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA). The validity of the proposed approach is verified by numerical examples for scattering problems. Numerical experiments show that the application of AWE to FE-BI-MLFMA can greatly improve the efficiency of FE-BI-MLFMA for computation of scattering over a wide frequency band.