Combined Field Integral Equation Research Articles

Augmented combined field integral equation for low frequency scattering analysis

In this study, a new augmented combined field integral equation formulation has been proposed to address the low-frequency breakdown problem. The proposed formulation is developed by adding the magnetic field integral operator to the conventional augmented electric field integral equation (AEFIE). In the meanwhile, a saddle point constraint preconditioning is also introduced to improve the convergence property of the system matrix. In some cases, the conventional AEFIE method cannot achieve the accurate solutions when the frequency is extremely low and a perturbation method is needed to tackle this accuracy problem. However, the new method is able to get the accurate solutions without any other strategy in the same cases mentioned above. Some numerical examples have been presented to demonstrate the validity of the proposed method.

An Efficient High-Order Marching-on-in-Degree Solver for Conducting and Dielectric Bodies of Revolution

The transient electromagnetic (EM) scattering from bodies of revolution (BoRs) is analyzed by the time-domain integral equation with the Silvester–Lagrange polynomials. The high-order spatial basis functions are used along the generatrix, while the weighted Laguerre polynomials are served as the temporal basis functions. In this way, the late time stability can be guaranteed by using the marching-on-in-degree scheme. In this communication, the conducting targets are simulated by the time-domain combined field integral equation while the dielectric ones are formulated in terms of the magnetic current combined field integral equation. Moreover, the multilevel adaptive cross approximation algorithm is adopted to accelerate the calculations of the well-separated groups. The numerical results demonstrate that the proposed solver is efficient to analyze the transient EM scattering from BoRs.

A formulation for elastodynamic scattering based on boundary algebraic equations

The solution of the elastodynamic problem arises in many scientific fields such as the wave propagation in the ground, non-destructive testing, the vibration design of buildings or vibroacoustics in general. We present here an integral formulation based on the boundary algebraic equations. It leads to a boundary numerical method and has the important advantage that no contour (2D) or surface (3D) integrals need to be computed. This is very helpful in order to use combined field integral equations (numerical damping of fictitious eigenfrequencies) without the problems caused by the evaluation of hypersingular integrals. The key aspects are: (i) the approach deals with discrete equations from the very beginning; (ii) discrete (instead of continuous) tensor Green’s functions are considered (the methodology to evaluate them is shown); (iii) the boundary must be described by means of a regular square grid. In order to overcome the drawback of this third condition the boundary integral is coupled, if needed, with a thin layer of finite elements. This improves the description of curved geometries and reduces numerical errors. The method is applied to the scattering of waves by objects and holes in an unbounded elastic medium and to the solution of interior elastic problems

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non-smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

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.

Analysis of scattering from composite conductor and dielectric objects using single integral equation method and FG-FFT

A hybrid method of single integral equation and the electric/magnetic current combined field integral equation (JMCFIE) is proposed, abbreviated as SJMCFIE, for analyzing scattering from composite conductor and dielectric objects. Fitting Green’s function with fast Fourier transform (FG-FFT) is adopted to combine SJMCFIE to lastly formulate SJMCFIE-FG-FFT for reducing computation complexity of resultant matrix-vector multiplications further. Next, scattering from the composite hemisphere is computed to verify SJMCFIE-FG-FFT on memory storage, computation efficiency compared to SJMCFIE-MLFMA. Simultaneously, comparison of FG-FFT and the other FFT-based algorithms being combined with SJMCFIE shows that SJMCFIE-FG-FFT is insensitive to grid spacing, easy implementation, and high accuracy. Besides, SJMCFIE-FG-FFT also can be used for analyzing scattering from the coated structure.

EFIE-PMCHWT-Based Domain Decomposition Method for Solving Electromagnetic Scattering From Complex Dielectric/Metallic Composite Objects

In this letter, an electric field integral equation (EFIE)-PMCHWT-based domain decomposition method (DDM) is proposed to solve the scattering from complex multilayer dielectric/metallic composite objects. Existing integral equation-based DDMs take combined field integral equations as subdomain governing equations. Hence, they are not suitable for modeling composite multilayer dielectric with open surfaces because of its poor accuracy. To overcome this limitation, the proposed DDM takes EFIE-PMCHWT as the governing equation for subdomains. The transmission conditions are enforced on the touching-face of neighboring subdomains to ensure the continuity of fields. Compared to the connecting-region-model method, the proposed DDM has a better convergence property because of its locally diagonal-dominant system matrix.

EFIE-Tuned Testing Functions for MFIE and CFIE

A recently developed numerical technique for improving the accuracy of the magnetic-field integral equation and the combined-field integral equation with low-order discretizations using the Rao–Wilton–Glisson functions is demonstrated on iterative solutions of large-scale complex problems, in order to prove the effectiveness of the proposed strategy as an alternative way for accurate and efficient analysis of multifrequency applications.

On the Use of Continuity Condition in the Fast Solution of Volume-Surface Integral Equation

The validity of the use of continuity condition (CC) is studied when it is combined with the volume-surface integral equation (VSIE) to solve the electromagnetic scattering/radiation of composite conductor-dielectric objects. It is found that the explicit enforcement of CC in VSIE may be inaccurate if the involved conductor is infinitesimally thin, but it works well for the composite structures that only contain conductors with closed surfaces. A new combined field integral equation-volume integral equation with enforced CC (CC-CFIE-VIE) is then proposed, and a convenient way to embed it into the context of the multilevel fast multipole algorithm is also provided. Numerical results show that CC-CFIE-VIE not only reduces the overall memory usage and the CPU time of a single iteration, but also decreases the number of iterations to reach desired accuracy.

A Design of Vertical Polarized Conformal Antenna and Its Array Based on UAV Structure

The designs of conformal antenna and its array usually need to combine with different carrier structures. As for cone or cylindrical carrier, through using of invasive weed optimization (IWO), the conformal array should be the optimized design as the antenna installation space requirements of cylindrical carrier; through adopting the antenna unit miniaturization method, it designs out a conformal antenna and makes it easier to be installed and the omnidirectional circular polarized beam is also realized. As for unmanned aircraft carrier, it designs out the low profile of conformal vertical polarization antenna unit; through using the electric field integral equation (EFIE) and electric/magnetic current combined-field integral equation (JMCFIE) and multilevel fast multipole algorithm (MLFMA), it analyzes the two different characteristics of unmanned aircraft carrier on the influence of antenna performance. The results obtained through experiments and JMCFIE analysis are in good agreement; according to unmanned aircraft carrier, it also designed out the antenna in X/K-band spectrum; through IWO, the antenna array is optimized; the test results show that the array scanning angle can reach the corresponding beam direction.

A Novel JMCFIE-DDM for Analysis of EM Scattering and Radiation by Composite Objects

In this letter, a new integral equation domain decomposition method (IE-DDM) using electric and magnetic current combined-field integral equation (JMCFIE) is proposed for efficient electromagnetic analysis of composite objects. This method provides a flexible way to decompose a complex composite domain into several easily solvable nonoverlapping subdomains. In order to obtain a good approximation of the global solution, both electric and magnetic field interactions are considered at the subdomain interfaces, in addition to the traditional Robin-type transmission conditions. In this manner, the unphysical reflections at subdomain interfaces can be further suppressed. It is shown that the condition of the new JMCFIE-DDM system is much improved in this new implementation. Accuracy and efficiency of the proposed method are demonstrated through several numerical experiments.

Generalized Hybrid Surface Integral Equations for Finite Periodic Perfectly Conducting Objects

Hybrid formulations that are based on simultaneous applications of diversely weighted electric-field integral equation (EFIE) and magnetic-field integral equation (MFIE) on periodic but finite structures involving perfectly conducting surfaces are presented. Formulations are particularly designed for closed conductors by considering the unit cells of periodic structures as sample problems for optimizing EFIE and MFIE weights in selected regions. Three-region hybrid formulations, which are designed by genetic optimizations, are demonstrated. Comparisons to the conventional combined-field integral equation show the benefits of the generalized formulations in terms of accuracy and efficiency.

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.

IE-DDM with a novel multiple-grid p-FFT for analyzing multiscale structures in half space

We present an integral equation domain decomposition method accelerated by a novel multiple-grid precorrected fast Fourier transform (MG-p-FFT) for the efficient analysis of multiscale structures in a half space. Based on the philosophy of DDM, the original computational domain is partitioned into several non-overlapping sub-domains. By employing non-conformal discretizations to each domain boundaries, combined field integral equation with half-space dyadic Green’s function is proposed for each individual sub-domain. Subsequently, the MG-p-FFT with auxiliary Cartesian grids with different size, order, location, and spacing, is adopted in each sub-domain independently to account for the self-interactions. Here, the proposed MG-p-FFT scheme outperforms the existing single-grid p-FFT scheme for multiscale problems by reducing the computational time and memory consumption. The proposed method can also be viewed as an effective preconditioning scheme for multiscale problems in a half space. The validity and advantages of the proposed method are illustrated by several representative numerical examples.

On New Triangle Quadrature Rules for the Locally Corrected Nyström Method Formulated on NURBS-Generated Bézier Surfaces in 3-D

Nonuniform rational b-splines (NURBS) are the most widely used technique in today’s geometric computer-aided design systems for modeling surfaces. Combining the locally corrected Nystrom (LCN) method with NURBS requires formulating LCN on both quadrilateral and triangular Bezier surfaces, as a typical NURBS-generated Bezier mesh includes elements of both the types. While on quadrilateral elements the product of 1-D Gaussian quadrature rules can be applied to LCN effectively, Gaussian integration rules available for triangles cannot efficiently be applied to LCN for two reasons. First, they do not possess the same number of quadrature points as the number of functions in a complete set of polynomial basis at an arbitrary order. Second, they exacerbate the condition number of the resulting matrix equation at higher orders due to an increasing density of quadrature points near the edges and corners of triangles. In this paper, we construct a new set of quadrature rules for Bezier triangles (i.e., degenerate quadrilaterals) based on the Newton–Cotes (equidistant) quadrature rules and apply these rules to the LCN solution of the electric, magnetic, and combined field integral equations. Results show that the new family of quadrature rules overcomes both the aforementioned issues and can be applied to LCN effectively for orders from 0 to 9, inclusively.

Stable Solution of Time-Domain Combined Field Integral Equations for Transient Electromagnetic Scattering by Composite Structures Based on Nyström Scheme and Laguerre Function

Transient electromagnetic scattering by composite structures is formulated by time-domain combined field integral equations (TDCFIEs). Traditionally, the TDCFIEs are solved by combining the method of moments (MoMs) in space domain and march-on-intime (MoT) scheme in time domain. The space-domain MoM requires two basic functions to represent the electric and magnetic current densities on material interfaces, respectively, and the conventional choice of $\hat {n}\times {\mathrm{ RWG}}$ basis function for representing the magnetic current density may not be good in the TDCFIEs. In addition, the MoT scheme has a well-known late-time instability problem, which will aggravate in the surface integral equations with penetrable media. In this communication, the TDCFIEs for composite structures are solved by a different approach in which the Nystrom method is used to discretize the space domain, while the Galerkin method with Laguerre basis and testing functions is employed to discretize the time domain. The proposed approach can fully overcome the drawbacks of the traditional approach as demonstrated by numerical examples.

Dual-Surface Electric Field Integral Equation Solution of Large Complex Problems

The integral equation analysis of large perfect electric conductor bodies is commonly formulated in terms of combined field integral equations (CFIE) that avoid spurious internal resonances. The dual-surface electric field integral equation (DSEFIE) is a less employed alternative approach; we discuss here how to cure its shortcomings and enhance its advantages over the CFIE for large bodies of nontrivial geometrical complexity. We study the convergence and accuracy of the DSEFIE also with respect to the current state of the art in the testing of the magnetic field integral equation. Examples of applications to large and realistic composite structures show the effectiveness of the proposed approach.

Evaluation of Electromagnetic Scattering by Conducting Bodies of Revolution With Discontinuous Currents

In this paper, the electromagnetic scattering by conducting bodies of revolution with discontinuities in the equivalent surface currents is investigated. The simulations are based on the combined field integral equation evaluated by the method of moments. A strategy for implementing an integral equation formulation with appropriate testing functions is presented. The procedure is applied in the analysis of the TE11 mode propagation inside a circular waveguide and the radiation from a circular corrugated horn antenna. The effectiveness of the proposed method is illustrated by comparing numerical results with analytical and experimental data.

Solutions of plasmonic structures using the multilevel fast multipole algorithm

We consider accurate full-wave solutions of plasmonic problems using the multilevel fast multipole algorithm (MLFMA). Metallic structures at optical frequencies are modeled by using the Lorentz-Drude model, formulated with surface integral equations, and analyzed iteratively via MLFMA. Among alternative choices, the electric and magnetic current combined-field integral equation (JMCFIE) and the combined tangential formulation (CTF), which are popular integral-equation formulations for penetrable objects, are discretized with the conventional Rao-Wilton-Glisson functions and used to model plasmonic structures. We discuss electromagnetic interactions in plasmonic media and show how far-field interactions may be omitted for improving the efficiency without sacrificing the accuracy of results. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:335–341, 2016.

On the Hierarchical Preconditioning of the Combined Field Integral Equation

This paper analyzes how hierarchical bases preconditioners constructed for the Electric Field Integral Equation (EFIE) can be effectively applied to the Combined Field Integral Equation (CFIE). For the case where no hierarchical solenoidal basis is available (e.g., on unstructured meshes), a new scheme is proposed: the CFIE is implicitly preconditioned on the solenoidal Helmholtz subspace by using a Helmholtz projector, while a hierarchical non-solenoidal basis is used for the non-solenoidal Helmholtz subspace. This results in a well-conditioned system. Numerical results corroborate the presented theory.