V. Lancellotti

An Eigencurrent Approach to the Analysis of Electrically Large 3-D Structures Using Linear Embedding via Green's Operators

We present an extension of the linear embedding via Greens operators (LEGO) procedure for efficiently dealing with 3-D electromagnetic composite structures. In LEGOs notion, we enclose the objects forming a structure within arbitrarily shaped domains (bricks), which (by invoking the equivalence principle) we characterize through scattering operators. In the 2-D instance, we then combined the bricks numerically, in a cascade of successive embedding steps, to build increasingly larger domains and obtain the scattering operator of the whole aggregate of objects. In the 3-D case, however, this process becomes quite soon impracticable, in that the resulting scattering matrices are too big to be stored and handled on most computers. To circumvent this hurdle, we propose a novel formulation of the electromagnetic problem based on an integral equation involving the total inverse scattering operator of the structure, which can be written analytically in terms of scattering operators of the bricks and transfer operators among them. We then solve this equation by the method of moments combined with the eigencurrent expansion method, which allows for a considerable reduction in size of the system matrix and thereby enables us to study very large structures.

Scattering from large 3-D piecewise homogeneous bodies through linear embedding via green's operators and arnoldi basis functions

We apply the linear embedding via Greens operators (LEGO) method to the scattering by large flnite dielectric bodies which contain metallic or penetrable inclusions. After modelling the body by means of LEGO bricks, we formulate the problem via an integral equation for the total incident currents over the boundaries of the bricks. This equation is turned into a weak form by means of the Method of Moments (MoM) and sub-domain basis functions. Then, to handle possibly large MoM matrices, we employ an order-reduction strategy based on: i) compression of the ofi-diagonal sub-blocks of the system matrix by the adaptive cross approximation algorithm and ii) subsequent compression of the whole matrix by using a basis of orthonormal entire-domain functions generated through the Arnoldi iteration algorithm. The latter leads to a comparatively small upper Hessenberg matrix easily inverted by direct solvers. We validate our approach and discuss the properties of the Arnoldi basis functions through selected numerical examples.

On the Convergence of the Eigencurrent Expansion Method Applied to Linear Embedding via Green's Operators (LEGO)

The scattering from a large complex structure comprised of many objects may be efficiently tackled by embedding each object within a bounded domain (brick) which is described through a scattering operator. Upon electromagnetically combining the scattering operators we arrive at an equation which involves the total inverse scattering operator S-1 of the structure: We call this procedure linear embedding via Greens operators (LEGO). To solve the relevant equation we then employ the eigencurrent expansion method (EEM)-essentially the method of moments with a set of basis and test functions that are approximations to the eigenfunctions of S-1 (termed eigencurrents). We have investigated the convergence of the EEM applied to LEGO in cases when all the bricks are identical. Our findings lead us to formulate a simple and practical criterion for controlling the error of the computed solution a priori.

Electromagnetic modelling of large complex 3-D structures with LEGO and the eigencurrent expansion method

Linear embedding via Greens operators (LEGO) is a computational method in which the multiple scattering between adjacent objects — forming a large composite structure — is determined through the interaction of simple-shaped building domains, whose electromagnetic (EM) behavior is accounted for by means of scattering operators. This method has been successfully demonstrated for 2-D electromagnetic band-gaps (EBG) and other structures [1], and for very simple 3-D configurations [2]. In this communication we briefly report on the full extension of LEGO to large complex 3-D structures, which may be EBG-based but may also include finite antenna arrays as well as frequency selective surfaces, to name but a few applications. In particular, we shall outline two modifications that were crucial for scaling up the LEGO method, namely, the introduction of a total inverse scattering operator S−1 and the eigencurrent expansion method (EEM) [3].

Extended linear embedding via Green's operators for analyzing wave scattering from anisotropic bodies

Linear embedding via Green’s operators (LEGO) is a domain decomposition method particularly well suited for the solution of scattering and radiation problems comprised of many objects. The latter are enclosed in simple-shaped subdomains (electromagnetic bricks) which are in turn described by means of scattering operators. In this paper we outline the extension of the LEGO approach to the case of penetrable objects with dyadic permittivity or permeability. Since a volume integral equation is only required to solve the scattering problem inside a brick and the scattering operators are inherently surface operators, the LEGO procedure per se can afford a reduction of the number of unknowns in the numerical solution with the Method of Moments and subsectional basis functions. Further substantial reduction is achieved with the eigencurrents expansion method (EEM) which employs the eigenvectors of the scattering operator as local entire-domain basis functions over a brick’s surface. Through a few selected numerical examples we discuss the validation and the efficiency of the LEGO-EEM technique applied to clusters of anisotropic bodies.

A total inverse scattering operator formulation for solving large structures with LEGO

We propose a methodology — based on Linear Embedding via Greens Operators (LEGO) and the eigencurrent expansion method — for solving large 3-D structures comprised of N<inf>D</inf> ≫ 1 objects. The problem is formulated through an integral equation involving the total inverse scattering operator S⁻¹ of the structure. Upon analyzing an electromagnetic band-gap open cavity, as an example of application, we show that hundreds of thousands of unknowns can be effortlessly handled and that the CPU times scale just linearly with N<inf>d</inf>.

Computational aspects of 2D-quasi-periodic-green-function computations for scattering by dielectric objects via surface integral eEquations

We describe a surface integral-equation (SIE) method suitable for computation of electromagnetic fields scattered by 2D-periodic high-permittivity and plasmonic scatterers. The method makes use of fast evaluation of the 2D-quasi-periodic Green function (2D-QPGF) and its gradient using a tabulation technique in combination with tri-linear interpolation. In particular we present a very efficient technique to create the look-up tables for the 2D-QPGF and its gradient where we use to our advantage that it is very effective to simultaneously compute the QPGF and its gradient, and to simultaneously compute these values for the case in which the role of source and observation point are interchanged. We use the Ewald representation of the 2D-QPGF and its gradient to construct the tables with pre-computed values. Usually the expressions for the Ewald representation of the 2D-QPGF and its gradient are presented in terms of the complex complementary error function but here we give the expressions in terms of the Faddeeva function enabling efficient use of the dedicated algorithms to compute the Faddeeva function. Expressions are given for both lossy and lossless medium parameters and it is shown that the expression for the lossless case can be evaluated twice as fast as the expression for the lossy case. Two case studies are presented to validate the proposed method and to show that the time required for computing the method of moments (MoM) integrals that require evaluation of the 2D-QPGF becomes comparable to the time required for computing the MoM integrals that require evaluation of the aperiodic Green function.

SIE approach to scattered field computation for 2D periodic diffraction gratings in 3D space consisting of high permittivity dielectric materials and plasmonic scatterers

We describe a surface integral-equation (SIE) method suitable for reliable computation of electromagnetic fields scattered by 2D periodic gratings in homogeneous 3D space in which the gratings may consist of high permittivity dielectric materials and metals. More in particular we brie y describe the formulation, the discretization and efficient evaluation of the Quasi Periodic Green Function (QPGF) and its gradient using Ewalds method. We present a case study to illustrate the methods capability of handling high permittivity dielectric materials and a second case study to demonstrate the effectiveness and indispensability of interpolating the QPGF and its gradient using tables with precomputed values.

Scattering from inhomogeneous anisotropic bodies with 3-D linear embedding via Green's operators

We describe the application of the linear embedding via Greens operator (LEGO) technique to the solution of scattering problems that involve objects with anisotropic electromagnetic properties. Thanks to this approach, the usage of volume integral equations is restricted to the calculation of the scattering operators of a small number of LEGO sub-domains. Since the scattering operators are surface operators, they constitute a means to reduce the computational burden. Further reduction is accomplished by applying the eigencurrent expansion method. We elaborate on the overall procedure by discussing a fewnumerical examples.

A priori error estimate and control in the eigencurrent expansion method applied to linear embedding via Green's operators (LEGO)

Linear embedding via Greens operators (LEGO) [1, 2] is a domain decomposition method in which the electromagnetic scattering by an aggregate of N<inf>D</inf> bodies (immersed in a homogeneous background medium) is tackled by enclosing each object within an arbitrarily-shaped bounded domain D<inf>k</inf> (brick), k = 1, …, N<inf>D</inf> (e.g., see Fig. 1). The bricks are characterized electromagnetically by means of scattering operators S<inf>kk</inf>, which are subsequently combined to form the total inverse scattering operator S⁻¹ of the structure [1]. Finally, we use the eigencurrent expansion method (EEM) [1,3] to solve the relevant equation involving S⁻¹, viz.