Extended Linear Embedding via Green's Operators for Analyzing Wave Scattering from Anisotropic Bodies

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.

Electromagnetic problems with inhomogeneous or anisotropic penetrable media require using volume integral equations (VIEs) to describe in the integral equation approach. The VIEs are usually solved by the method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function, but the implementation may not be convenient because conforming meshes are required to define the basis function. In this paper, we propose a novel meshless method to discretize the VIEs. The method uses the Green-Gauss theorem to transform a volume integral into a boundary integral so that discretizing volume domains can be removed, resulting in a truly meshless method. For electrically large problems, we incorporate the method with multilevel fast multipole algorithm (MLFMA) to accelerate the solving process. A numerical example is presented to demonstrate the approach and its effectiveness and robustness have been validated.

Reconstruction of unknown objects requires efficient inversion for measured electromagnetic scattering data. In frequency-domain integral equation method for reconstructing dielectric objects, the volume integral equations (VIEs) are involved because the imaging domain with both true unknown objects and part of background is inhomogeneous. When solving the forward scattering integral equation (FSIE), the Nyström method is used since the traditional method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function may not be convenient due to the inhomogeneity of the imaging domain. The benefits of the Nyström method include the simple implementation without using any basis and testing functions and lower requirement on geometrical discretization, so it is very suitable for inhomogeneous problems. When solving the inverse scattering integral equation (ISIE), the Gauss-Newton minimization approach (GNMA) with a multiplicative regularization method (MRM) is employed. Numerical examples for reconstructing three-dimensional dielectric objects are presented to illustrate the inversion approach.

The improved hybrid discretisation of volume integral equation (VIE) combined with the characteristic basis function method is proposed to analyse the scattering from complex dielectric objects in this study. To mitigate the excessive number of discrete elements which arise from the conformal discretisation on the complex dielectric objects, the hybrid discretisation is developed. For the conventional hybrid discretisation, the conformal Schaubert–Wilton–Glisson (SWG) basis functions are used to discretise the homogeneous regions, meanwhile, the non‐conformal half‐SWG basis functions are employed to discretise the boundary regions which separate different materials or multiscale structures. The hybrid discretisation takes advantage of the merits of both conformal and non‐conformal basis functions and improves the computational efficiency for solving the VIE. In our study, a kind of edge basis function defined within boundary tetrahedral elements substitutes for the SWG basis functions to model the boundary regions, leading to the further reduction on the number of basis functions. Moreover, the characteristic basis functions extracted from the low‐level basis functions are employed to construct the reduced matrix of which the size is considerably smaller than that of the system matrix derived from the VIE. Several numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Linear embedding via Green's operators (LEGO) is a domain decomposition method for solving electromagnetic problems which involve composite 3-D structures. By combining the standard Method of Moments with a set of macro basis functions obtained through the Arnoldi iteration, the algebraic system of LEGO can be effectively compressed. However, under general circumstances it is not easy to choose the minimum number of macro basis functions for a given level of accuracy, since many physical and geometrical factors come into play. To shed light on this topic, we discuss a few examples with a focus on the convergence of the numerical solution.

A novel equivalence principle algorithm (EPA) domain decomposition method previously for the surface integral equation is explored for volume integral equation in this work. The whole object is decomposed into several subdomains, the equivalence spheres are automatically produced to enclose each subdomain. The Rao–Wilton–Glisson (RWG) and body of revolution (BoR) basis functions are defined on the equivalence sphere surface. The equivalence currents defined on the RWG basis functions are evaluated with multilevel fast multipole algorithm, and then the currents defined on the RWGs can be projected onto the BoR basis functions. Compared with surface integral equation EPA-BoR algorithm, the ratio of the number of average original unknowns in each subdomain with respect to the equivalence unknowns is relatively larger, which leads to significant reduction of computations. Numerical simulations validate the accuracy and efficiency of the proposed method.

Electromagnetic wave scattering by three dimensional inhomogeneous dielectric sphere is analyzed using Volume Integral Equation (VIE) and Method of Moments (MoM). The VIE/MoM technique deals with the formulation of an integral equation and then finding its solution by converting it into a system of linear equations with the help of set of pulse basis and delta testing functions. In this paper a simple object like dielectric sphere is considered and its scattering characteristics are determined through solving the linear equations by iteration based numerical procedure called Generalized Minimum Residual (GMRES) method. The bistatic Radar Cross Section (RCS) results of this implementation are verified with the theory based Mie series and commercially available HFSS results, and they are found to be in good agreement.

Electromagnetic potential-based volume integral equations (VIEs) for the efficient analysis of large-scale problems are presented. Different from the conventional VIEs the proposed approach is based on scalar and vector potential unknowns. A VIE and the Lorentz gauge equations constitute a complete required set. The introduced potential VIEs (PVIEs) allow using scalar interpolatory nodal basis functions, do not require any surface integrals (both for homogeneous or inhomogeneous domains), and do not involve any hyper-singular integrals. This method works from very low- to high-frequency regimes, does not have high-density mesh breakdowns, requires a significantly reduced number of quadrature nodes as compared to methods based on vector basis functions, and is easy to implement for low- and high-order basis functions. The method is coupled with fast interpolation based techniques implemented on Graphics Processing Units (GPUs) to allow for the rapid analysis of complex structures.

Finite arrays of waveguides and horn antennas are widely used both as direct radiators as well as feed clusters for reflector antennas. Accurate and efficient numerical simulation techniques represent a powerful tool for the design of such antennas. Many approaches have been proposed in the literature that employ hybrid techniques to efficiently analyze both the interior waveguide problem and as well as their coupling to the outer free space [1]–[3]. The mode matching (MM) is used to compute the generalized scattering matrix (GSM) for the inner waveguide problem, whereas the Method of Moments (MoM) evaluates the GSM for the outer free space coupling. When the array is electrically large in terms of number of elements or radiating apertures dimensions, the applicability of this approach is limited because the associated MoM matrix is large. Some novel approaches have been proposed in the literature to reduce the matrix size [4]. Recently the Characteristic Basis Function method (CBFM) [5] has been proposed to solve electrically large problems by using a set of high-level basis functions, called CBFs, that are physics-based and tailored to the problem.

The Adaptive Cross Approximation (ACA) algorithm has been used to compress the rank- deficient sub-blocks of the matrices that arise in the numerical solution of integral equations (IEs) with the Method of Moments. In the context of the linear embedding via Green's operator (LEGO) method — a domain decomposition technique based on IEs — an electromagnetic problem is modelled by combining bricks in turn described by scattering operators which, in many situations, are singular. As a result, macro basis functions defined on the boundary of a brick can be generated by applying the ACA to a scattering operator. Said functions allow compressing the weak form of the LEGO functional equations which then use up less computer memory and are faster to invert.

A technique for estimating a plasma drift velocity distribution in the ionosphere is presented. This technique is based on a framework for representing a global vector field on a sphere by using a set of localized basis functions which is newly derived as a variant of the spherical elementary current system (SECS). A vector field on a sphere can be divided into its divergence-free (DF) component and curl-free (CF) component. The DF and CF components can then be represented by weighted sums of the DF and CF vector-valued basis functions, respectively. While the SECS basis functions have a singular point, the new basis functions do not diverge over a sphere. This property of the new basis function allows us to achieve robust prediction of the drift velocity at any point in the ionosphere. Assuming that the ionospheric plasma drift velocity has no divergence, its distribution can be represented by a weighted sum of the DF basis functions. The proposed technique estimates the ionospheric plasma drift velocity distribution from the SuperDARN data by using the DF basis functions. Since there are some wide gaps in the spatial coverage of the SuperDARN, an empirical convection model is combined with the framework based on the new basis functions. It is demonstrated that the proposed technique is useful for the estimation and modeling of the ionospheric plasma velocity distribution.

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).

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.

Natural convection in a pentagonal house with a protruding cooler for summer is numerically investigated. The sloping roof is subjected to constant heat flux, and the cooler maintains at a uniform cold temperature. The remaining walls of the house are insulated. Local multi-quadratic radial basis function (LMQRBF) method is applied in solution of Navier–Stokes equations in stream function-vorticity form. The working fluid in the house is air with Pr = 0.71. The effects of Rayleigh number (103 ≤ Ra ≤ 106), heat flux of the roof (0.1 ≤ q ≤ 2.0), and the cooler’s vertical position (0.65 ≤ H/L ≤ 0.85) are explored. Streamlines, isotherms, velocity profiles, and the distribution of Nusselt number are displayed. Numerical results indicate that for the top and right wall of the cooler, the heat transfer is stronger than that for the bottom wall. It reveals that all the governing parameters efficiently impact the generation and evolution of the secondary and tertiary vortex in the gable roof. It is also observed that the variation of average Nusselt number on the right wall with Rayleigh number is under the combined influence of the heat flux and the cooler’s vertical position.

Boundary element methods (BEM) have been successfully applied towards solving a broad array of complicated electromagnetic problems. Most BEM approaches rely on flat triangular discretizations and discretization via the Method of Moments (MoM) and low-order basis functions. Although more complicated from an implementation standpoint, it has been shown that high-order methods based on curvilinear patch mesh discretizations can significantly outperform low-order MoM in both accuracy and computational efficiency. In this work, we review a new high-order Nystrom method based on using Chebyshev basis functions with curvilinear elements that we have recently developed, present a few scattering examples, and discuss related on-going and future work.