Modified Phase-Extracted Basis Functions for Efficient Analysis of Scattering From Electrically Large Targets

This paper presents a novel Galerkin-type method of moments solution of the volume surface integral equation (VSIE), which is developed for the analysis of electromagnetic scattering and radiation from composite metallic and dielectric structures. Our new scheme utilizes curved tetrahedral elements and curved triangular patches for geometric modeling, and the associated higher order hierarchical vector (HOHV) basis functions for volume/surface current modeling. In contrast to the previous studies on higher order VSIE, our scheme, which is reported for the first time in VSIE, is much more flexible. In comparison with the conventional low-order scheme, our scheme requires much less memory and computational time for the same accuracy level. Numerical results are provided to demonstrate the accuracy, efficiency, and flexibility of our scheme.

In this article, a nonconformal and nonoverlapping domain decomposition method (DDM) based on surface integral equation (SIE) is proposed for the electromagnetic (EM) scattering or radiation of multiscale metallic targets. To reduce the unknown amount of the conventional SIE for multiscale EM simulation, we apply curved triangular elements and higher order hierarchical vector (HOHV) basis functions to the SIE. Next, to increase the flexibility of geometrical modeling and accelerate the convergence of the presented SIE system for electrically large and multiscale metallic targets, a DDM scheme is further developed to employ a discontinuous Galerkin (DG) approach to glue conformal/nonconformal surface grids between adjacent subdomains. In addition, an interior penalty term is introduced to stabilize the DDM solution, and half edge-based HOHV basis functions are introduced to model the current associated with nonconformal surface elements. Meanwhile, the basis expansion and recombination (BER) technique is introduced to significantly accelerate the matrix-filling and improve the efficiency. The flexibility of basis order selection is further enhanced by the hierarchical characteristic of HOHV bases. Finally, several numerical results are provided to demonstrate the accuracy, efficiency, and flexibility of the proposed HO-DG-DDM.

Efficient analysis of electromagnetic-wave responses from electrically large complex objects is a crucial but difficult topic in many applications of electromagnetic engineering. Among various numerical methods, the Method of Moments (MoM) solves for surface- or volume-distributed equivalent currents, which are expanded in terms of basis functions. In this paper, the traveling-wave phase variation has been incorporated into the basis functions to enhance their capability of describing the equivalent surface currents induced over surfaces of perfect electric conductors (PECs). A rigorous derivation is first given as physical insight, to show that the induced surface current is composed of a traveling-wave term and several standing-wave terms. A phase-extracted basis function (PEBF) is then proposed to describe the traveling current wave, and is applied in solving the electromagnetic scattering from three-dimensional (3D) PEC objects with smooth surfaces. By hybridizing the phase-extracted basis function with higher-order hierarchical basis functions, a moving standing-wave (MSW) basis function is further introduced to describe both the traveling-wave and the standing-wave terms in the induced surface current. It is shown from several numerical examples that the moving standing-wave basis function has excellent performance for solving electromagnetic scattering from three-dimensional PEC objects with both smooth and non-smooth surfaces, as well as for three-dimensional PEC cavities. After that, the unique properties of the phase-extracted basis function are utilized in the “sparsification” of the MoM matrix, and in the fast calculation of wideband responses. With the aid of the phase-extracted basis function and the moving standing-wave basis functions, the total memory requirements and the computational time can be significantly reduced, while numerical solutions can be obtained with good accuracy. The advantages of the phase-extracted basis functions and the moving standing-wave basis functions in predicting electromagnetic scattering from electrically large complex objects are summarized before the conclusions are drawn.

The application of p- or hp-refinement techniques on an adaptive finite element procedure requires the use of hierarchical basis functions in order to facilitate the tangential continuity of the solution on the interface between two elements with different order. Interpolatory and hierarchical are nowadays well-established families of basis functions. Hierarchical basis functions have been used already in antenna analysis. In this paper, this kind of basis functions is employed in the full-wave analysis of closed waveguiding structures, which requires scalar/vector interpolation and is compared with interpolatory basis functions, mainly in terms of their convergence.

In this paper, the time domain integral equation is solved by marching-on-in-degree method with near-orthogonal higher order hierarchical Legendre basis as spatial basis functions and causal weighted Laguerre polynomials as temporal basis functions. In the traditional marching-on-in-degree solver of time domain integral equation, RWG basis functions are used as spatial basis functions. The memory requirement and time consuming is very large, which becomes a bottleneck of marching-on-in-degree method. In order to solve this problem, the object is meshed with second-order nine-node curved quadrilateral elements and near-orthogonal higher order hierarchical Legendre basis functions are adopted as spatial basis functions. Numerical results show that this method can greatly reduce the unknowns of the problem, by which it can save memory and CPU time.

V-cycle and non-standard W-cycle versions of a multiplicative Schwarz preconditioner based on hierarchical basis functions are presented. It is used together with a Krylov subspace method to efficiently solve the linear system that results from higher-order finite element (FE) discretisations of time-harmonic Maxwell equations. A recently developed hierarchical basis for H(curl)-conforming (vector-valued and tangentially continuous) FE spaces on tetrahedral meshes is also briefly presented. On this basis, a certain amount of orthogonality between basis functions of different orders is obtained through the requirement that the Nédélec interpolation of higher-order basis functions vanishes in lower-order FE spaces. Numerical experiments are used to show the good performance of the presented schemes. In these experiments, the performance obtained with the presented basis is compared with the performance obtained by several other hierarchical bases found in the literature. For third-order elements, it is observed that most bases give very similar performance and that the V-cycle preconditioner typically requires about 30% more computing time than the W-cycle one. For fourth-order elements, the new basis combined with the non-standard W-cycle preconditioner leads to the best performance. The computing times for the other combinations are about 40% longer, at best.

The fast multipole method (FMM) is developed to analyse electromagnetic scattering using higher-order hierarchical Legendre basis functions, which have been proposed recently and make the impedance matrix, from method of moments, a low condition number. In this method, the point-to-point based-on-sample quadrature is used because the basis functions are defined on large surface patches of scatterers. The numerical implementations of matrix–vector multiply by FMM are particularly presented. These numerical results show that the proposed second- or third-order FMM (high-order FMM) is accurate and efficient.

Printed circuits in bounded media encompass a wide range of practical structures such as discontinuities in waveguides, planar circuits embedded in shielded multilayered media or even two-dimensional printed periodic structures. The Electromagnetic (EM) modeling of printed circuits in layered bounded media is performed via an Integral Equation (IE) technique. Green's functions (GFs) are specially defined to satisfy both the Boundary Conditions (BCs) imposed by the layered media and by the transverse boundary enclosing the entire structure. Finally, a system of IEs on the equivalent sources can be solved numerically by means of the Method of Moments (MoM). Each of the problems enumerated above has already been solved by other authors using IE-MoM techniques. Nevertheless, our formulation introduces a unified approach applicable to all the aforementioned problems. Due to the symmetry presented by a bounded layered media, the GF problem can be reduced into a two-dimensional transverse boundary problem and a one-dimensional transmission line problem in the normal direction. Both problems can be treated independently. This thesis proposes and fully develops an efficient technique that encompasses different laterally bounded multilayered problems with a seamless transition between them. The method is based on a modal representation of the transverse boundary problem and on the expansion of the equivalent surface currents by zero-curl & constant-charge Basis Functions (BFs). It offers a unified and versatile approach that, on one hand eliminates redundancy in the formulation and on the other hand simplifies each particular problem to the evaluation of constant coefficients or basic line integrals. Analytical solutions can be found for the combination of linear subsectional basis functions in rectangular and circular Perfect Electric Conductor (PEC) boundaries as well as for periodic lattices. This thesis then solves the problem of transmission line model in the longitudinal direction by proposing an efficient algorithm that guarantees numerical stability under a variety of known critical conditions where other already known formulations fail. In addition, it introduces alternate equivalent expressions of this formulation that allow new interpretations of the problem. Due to its practical interest, the method is applied for the EM modeling of multilayered boxed printed circuits. This motivated the implementation of a dedicated software tool for the efficient analysis of these topologies including losses. Extensive numerical experiments have been carried out to assess the validity of the aforementioned theory and some properties of test-structures (losses, mesh, etc).

It has been shown recently that the p-version of the finite element method is well suited for the analysis of thin walled three-dimensional continua [2]. However, designing and implementing software for the p-version of FEM is a challenge because of the increased complexity compared to the h-version. In this paper, we present an object-oriented finite element system implemented in Java. It is pointed out how the object-oriented paradigm, suitable design patterns and thorough unit testing can help to develop and maintain a complex engineering application. The basic idea of the software design is to separate generally applicable mathematical concepts, like basis functions and geometrical mappings from concrete element formulations that contain the physics of the actual problem. In the mathematical package, there are interfaces representing the concepts of basis functions (forming an Ansatz space) and functions defined on R 1, R 2 and R 3. Several concrete classes implement these basic interfaces and realize for instance Lagrangian basis functions, hierarchical basis functions, functions constructed by a linear combination of basis functions or functions constructed by the blending function method. The use of NURBS curves hereby allows for the representation of complex geometries like that of the shell structure shown below. The power of this approach lies in the fact that on the element level only the interface types are used. Thus, an element formulation solely contains the physics and is not limited to a certain type of Ansatz space. Also, new Ansatz spaces can be easily incorporated lateron. Generally Java is considered as being slow for numerically intense applications. On the other side there are many advantages that make Java attractive also for simulation software [1]. In this paper, a hybrid approach is presented where the overall program is implemented in Java but numerically intense linear algebra operations are delegated to native code. In the full paper, it is shown that the software can be easily applied to problems involving 75 000 DOFs and more. Open image in new window

A nonconformal scheme for scattering analysis from PEC objects

The main aim of this work was to assess the ability of Bayesian multivariate adaptive regression splines (BMARS) and Bayesian radial basis function (BRBF) techniques for modelling the gas chromatographic retention indices of volatile components of Artemisia species. A diverse set of molecular descriptors was calculated and used as descriptor pool for modelling the retention indices. The ability of BMARS and BRBF techniques was explored for the selection of the most relevant descriptors and proper basis functions for modelling. The results revealed that BRBF technique is more reproducible than BMARS for modelling the retention indices and can be used as a method for variable selection and modelling in quantitative structure–property relationship (QSPR) studies. It is also concluded that the Markov chain Monte Carlo (MCMC) search engine, implemented in BRBF algorithm, is a suitable method for selecting the most important features from a vast number of them. The values of correlation between the calculated retention indices and the experimental ones for the training and prediction sets (0.935 and 0.902, respectively) revealed the prediction power of the BRBF model in estimating the retention index of volatile components of Artemisia species.

2-D local hp adaptive isogeometric analysis based on hierarchical Fup basis functions

This paper presents in detail the extension of the T-Ω formulation for eddy currents based on higher-order hierarchical basis functions so that it can automatically deal with conductors of arbitrary topology. To this aim, we supplement the classical hierarchical basis functions with nonlocal basis functions spanning the first de Rham cohomology group of the insulating region. Such nonlocal basis functions may be efficiently and automatically found in negligible time with the recently introduced Dłotko–Specogna (DS) algorithm. The approach presented in this paper merges techniques together which to some extent already existed in literature but they were never grouped together and tested as a single unit.

Plane wave scattering by a cylindrically conformal periodic finite array (CCPFA) with perfectly conducting (PEC) and resistive patches, including graphene patches are studied. CCPFA represents a dielectric rod (DR) with patches placed in azimuth direction. The problem solution was obtained by the Method of Moments (MoM) applied in the spectral domain using Piece Wise Sinusoidal (PWS) basis functions. The structure geometries in the case when the PEC patches are nearly touching are carefully analyzed. It was found in the discussed 3D case that for a half wavelength resonance the natural frequency tends to zero when the slot is decreasing (gap-coupled case). Also it was shown that it is in a very good agreement with 2D analytical solution until the slot angular size is less than 40°. For CCPFA with resistive, including graphene patches the patch resistivity influence on the resonant frequency is discussed. The current distribution, far fields and the resonant frequency behavior versus frequency are presented.

A method is introduced for reducing the exorbitant dependence on computer storage and solution time in the method of moments (MoM) for electrically large electromagnetic (EM) scattering problems. The unknown surface currents on large, smooth parts of a perfect electrical conductor (PEC) scatterer are expressed by an efficient set of linearly phased surface current basis functions. The phasefront characteristics of the surface currents are numerically extracted from known current samples obtained from a lower-frequency solution of the same configuration. The use of such basis functions for efficiently representing the surface currents that are constructed in terms of linearly phased currents at higher frequencies is justified by considering the form of the surface currents predicted by high-frequency asymptotic ray methods. The procedure for extracting the current phasefronts is purely numerical, obviating computationally expensive and nonrobust operations such as ray-tracing, and thus, is amenable to general purpose scattering codes. The new MoM with linearly phased basis functions is shown to greatly relieve the storage and solution time of the conventional MoM while accurately reproducing the induced surface currents and scattered fields of some chosen targets.