LEGO : linear embedding via Green's operators

Reduction of lead time has long been an important target in product development. Owing to the advance of computer power product optimization has been moved from the production stage to the preceding design stage. In particular, the full electromagnetic behavior of the final product can now be predicted through numerical methods. However, for the tuning of device parameters in the optimization stage, commercial software packages often rely on brute-force parameter sweeps. Further, for each set of parameter values a full recomputation of the entire configuration is usually required. In case of stringent product specifications or large complex structures, the computational burden may become severe. Recently, "marching on in anything" has been introduced to accelerate parameter sweeps. Nevertheless, it remains necessary to further reduce the computational costs of electromagnetic device design. This is the main goal in this thesis. As an alternative to existing electromagnetic modeling methods, we propose a modular modeling technique called linear embedding via Green’s operators (LEGO). It is a so-called diakoptic method based on the Huygens principle, involving equivalent boundary current sources by which simply connected scattering domains of arbitrary shape may fully be characterized. Mathematically this may be achieved using either Love’s or Schelkunoff’s equivalence principles, LEP or SEP, respectively. LEGO may be considered as the electromagnetic generalization of decomposing an electric circuit into a system of multi-port subsystems. We have captured the pertaining equivalent current distributions in terms of a lucid Green’s operator formalism. For instance, our scattering operator expresses the equivalent sources that would produce the scattered field exterior to a scattering domain in terms of the equivalent sources that would produce the incident field inside that domain. The enclosed scattering objects may be of arbitrary shape and composition. The scattering domains together with their scattering operators constitute the LEGO building blocks. We have employed various alternative electromagnetic solution methods to construct the scattering operators. In its most elementary form, LEGO is a generalization of an embedding procedure introduced in inverse scattering to describe multiple scattering be tween adjacent blocks, by considering one of the blocks as the environment of the other and vice versa. To establish an interaction between current distributions on disjoint domain boundaries we define a source transfer operator. Through such transfer operators we obtain a closed loop that connects the scattering operators of both domains, which describes the total field including the multiple scattering. Subsequently, a combined scattering block is composed by merging the separate scattering operators via transfer operators, and removing common boundaries. We have validated the LEGO approach for both 2D and 3D configurations. In the field of electromagnetic bandgap (EBG) structures we have demonstrated that a cascade of embedding steps can be employed to form electromagnetically large complex composite blocks. LEGO is a modular method, in that previously combined blocks may be stored in a database for possible reuse in subsequent LEGO building step. Besides scattering operators that account for the exterior scattered field, we also use interior field operators by which the field may be reproduced within (sub)domains that have been combined at an earlier stage. Only the subdomains of interest are stored and updated to account for the presence of additional domains added in subsequent steps. We have also shown how the scattering operator can be utilized to compute the band diagram of EBG structures. Two alternative methods have been proposed to solve the pertaining eigenvalue problem. We have validated the results via a comparison with results from a plane-wave method for 2D EBG structures. In addition, we have demonstrated that our method also applies to unit cells containing scattering objects that are perfectly conducting or extend across the boundary of the unit cell. The optimization stage of a design process often involves tuning local medium properties. In LEGO we accommodated for this through a transfer of the equivalent sources on the boundary of a large scattering operator to the boundary of a relatively small designated domain in which local structure variations are to be tested. As a result, subsequent LEGO steps can be carried out with great efficiency. As demonstrators, we have locally tuned the transmission properties at the Y-junction of both a power splitter and a mode splitter in EBG waveguide technology. In these design examples the computational advantageous of the LEGO approach become clearly manifest, as computation times reduce from hours to minutes. This efficient optimization stage of the LEGO method may also be integrated with existing software packages as an additional design tool. In addition to the acceleration of the computations, the reusability of the composite building constitute an important advantage. The Green’s operators are expressed in terms of equivalent boundary currents. These operators have been obtained using integral equations. In the numerical implementation of the LEGO method we have discretized the operators via the method of moments with a flat-facetted mesh using local test and expansion functions for the fields and currents, respectively. In the 2D case we have investigated the influence of using piecewise constant and piecewise linear functions. For the 3D implementation, we have applied the Rao-Wilton-Glisson (RWG) functions in combination with rotated RWG functions. After discretization, operators and operator compositions are matrices and matrix multiplications, respectively. Since the matrix multiplications in a LEGO step dominate the computational costs, we aim at a maximum accuracy of the field for a minimum mesh density. For LEGO with SEP, we have determined the unknown currents through inverse field propagators, whereas with LEP, the currents are directly obtained from the tangential field components via inverse Gram matrices. After a careful assessment of the computational costs of the LEGO method, it turns out that owing to the removal of common boundaries and the reusability of scattering domains, the most efficient application of LEGO involves a closely-packed configuration of identical blocks. In terms of the number of array elements, N, the complexity of a sequence of LEGO steps for 2D and 3D applications increases as O(N1.5) and O(N2), respectively. We have discussed possible improvements that can be expected from "marching on in anything" or multi-level fast-multipole algorithms. From an evaluation of the resulting scattered field, it turns out that LEGO with SEP is more accurate than with LEP. However, the spurious interior resonance effect common to SEP in the construction of composite building blocks can not simply be avoided through a combined field integral equation. By contrast, LEGO based on LEP is robust. Further, we have demonstrated that additional errors due to the choice of domain shape or building sequence, or the accumulation of errors due to long LEGO sequences are negligible. Further, we have investigated integral equations for the scattering from 2D and 3D perfectly conducting and dielectric objects. The discretized integral operators directly apply to the LEGO method. For scattering objects that are not canonical, these integral equations are used in the construction of the elementary LEGO blocks. Since we aim at a maximum accuracy of the field for a minimum mesh density, the regular test and expansion integral parts are primarily determined through adaptive quadrature rules, while analytic expressions are used for the singular integral parts. It turns out that the convergence of the scattered field is a direct measure for the accuracy of the scattered field computed with LEGO based on SEP or LEP. As an alternative to the PMCHW and the M¨uller integral equations, we have proposed an new integral equation formulation, which leads to cubic convergence in the 2D case, irrespective of the mesh density and object shape. In case of scattering object with a regular boundary domain scaling may be used to improve the convergence rate of the scattered field.