Linear Embedding Via Green's Operators Research Articles

An efficient approach to the local optimization of finite electromagnetic band-gap structures

We propose a methodology based on linear embedding via Green's operators (LEGO) and the eigencurrent expansion method (EEM) to eciently deal with and locally optimize 2-D electrically large electromagnetic band-gap (EBG) structures. In LEGO terminology, the composite EBG structure is broken up (diakopted) into constitutive elements called \bricks that we characterize through scattering operators by invoking Love's equivalence principle, while, at the same time, the electromagnetic interaction among the bricks is captured by transfer operators. The resulting electromagnetic problem is then succinctly formulated through an integral equation involving the total inverse scattering operator S 1 of the structure. To perform local optimization, the formulation of the problem allows for variations of the electromagnetic properties and the shape of a set of objects in the EBG structure with respect to those of the others, thereby allowing us to tune a compact designated domain within a large one. Finally, the method of moments and the EEM are applied to achieve a considerable reduction in memory use for the overall problem.

CONVERGENCE PROPERTIES OF A DIAKOPTICS METHOD FOR ELECTROMAGNETIC SCATTERING FROM 3-D COMPLEX STRUCTURES

Linear embedding via Green's operators (LEGO) is a diakoptics method that employs electromagnetic \bricks to formulate problems of wave scattering from complex structures (e.g., penetrable bodies with inclusions). In its latest version the LEGO integral equations are solved through the Method of Moments combined with adaptive generation of Arnoldi basis functions (ABF) to compress the resulting algebraic system. In this paper we review and discuss the convergence properties of the numerical solution in relation to the number of ABFs. Besides, we address the issue of setting the threshold for stopping the generation of ABFs in conjunction with the adaptive Arnoldi algorithm.

Wave scattering from random sets of closely spaced objects through linear embedding via Green's operators

In this paper we present the application of linear embedding via Green's operators (LEGO) to the solution of the electromagnetic scattering from clusters of arbitrary (both conducting and penetrable) bodies randomly placed in a homogeneous background medium. In the LEGO method the objects are enclosed within simple-shaped bricks described in turn via scattering operators of equivalent surface current densities. Such operators have to be computed only once for a given frequency, and hence they can be re-used to perform the study of many distributions comprising the same objects located in different positions. The surface integral equations of LEGO are solved via the Moments Method combined with Adaptive Cross Approximation (to save memory) and Arnoldi basis functions (to compress the system). By means of purposefully selected numerical experiments we discuss the time requirements with respect to the geometry of a given distribution. Besides, we derive an approximate relationship between the (near-field) accuracy of the computed solution and the number of Arnoldi basis functions used to obtain it. This result endows LEGO with a handy practical criterion for both estimating the error and keeping it in check.

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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> of the structure: We call this procedure linear embedding via Green's 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (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.

SENSITIVITY ANALYSIS OF 3-D COMPOSITE STRUCTURES THROUGH LINEAR EMBEDDING VIA GREEN'S OPERATORS

We propose a methodology | based on linear embedding via Green's operators (LEGO) and the eigencurrent expansion method (EEM) | for solving electromagnetic problems involving large 3-D structures comprised of ND ? 1 bodies. In particular, we address the circumstance when the electromagnetic properties or the shape of one body difier from those of the others. In real-life structures such a situation may be either the result of a thoughtful design process or the unwanted outcome of fabrication tolerances. In order to assess the sensitivity of physical observables to localized deviations from the \ideal structure, we follow a deterministic approach, i.e., we allow for a flnite number of difierent realizations of one of the bodies. Then, for each realization we formulate the problem with LEGO and we employ the EEM to determine the contribution of the ND i 1 \flxed bodies. Since the latter has to be computed only once, the overall procedure is indeed e-cient. As an example of application, we investigate the sensitivity of a 2-layer array of split-ring resonators with respect to the shape and the ofiset of one element in the array.

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 Green's 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.

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 Green's operators (LEGO) procedure for efficiently dealing with 3-D electromagnetic composite structures. In LEGO's 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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">total inverse scattering operator</i> of the structure, which can be written analytically in terms of scattering operators of the bricks and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">transfer operators</i> 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.