Advanced discontinuous integral-equation schemes for the versatile electromagnetic analysis of complex structures

Premi Extraordinari de Doctorat, promoció 2018-2019. Àmbit de les TICLes Equacions Integrals superficials més importants són l'Equació de Camp Elèctric (EFIE), per a l'anàlisi de la dispersió electromagnètica d'objectes conductors perfectes (PEC), i la formulació Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT), orientada a l'anàlisi d'objectes homogenis penetrables. Ambdues són normalment discretitzades, amb el Mètode dels Moments (MoM), amb funcions base div-conformes, dependents de les arestes del mallat. Les discretitzacions div-conformes de les formulacions EFIE i PMCHWT representen esquemes conformes; és a dir, amb solucions convergents a dins de l'espai físic de corrents. Tanmateix, les implementations MoM div-conformes requereixen que el mallat sigui conforme geomètricament, amb cada parell de triangles adjacents compartint només una aresta. El desenvolupament d'esquemes div-conformes per a objectes compostos amb línies al llarg de les quals tres o més regions hi intersecten, esdevé molt incòmoda perquè cal definir condicions de continuïtat especials en aquestes línies d'intersecció. A més, els mallats que resulten de la juxtaposició de subdominis independentment mallats són típicament no-conformes geomètricament i per tant no aptes per a l'anàlisi div-conforme convencional en Mètode dels Moments. En aquesta Tesi, es tracta l'anàlisi robusta, precisa i versàtil de la dispersió electromagnètica d'objectes conductors o penetrables amb forma arbitrària i d'objectes compostos amb línies d'intersecció entre differents regions, ja sigui amb mallats conformes com no-conformes. Amb aquest objectiu, fem ús de la formulació d'equació integral EFIE–PMCHWT, la qual resulta de l'aplicació de les formulacions EFIE o PMCHWTal llarg de superfícies de contorn, respectivament, incloent regions conductores o separant regions penetrables. Els esquemes proposats en aquesta Tesi es basen en el desenvolupament dels corrents amb conjunts de funcions base discontínues a través de les arestes del mallat i dependents només dels triangles del mallat. Aquesta estratègia dóna lloc a integrals de contorn amb Kernels hypersingulars, que maneguem mitjançant el testeig de les equacions amb funcions de testeig especialment dissenyades, definides fora de les triangulacions de la superfície de contorn, a dins de la regió a on els camps són zero d'acord amb al Teorema d'Equivalència superficial. Les nostres implementacions de la formulació EFIE-PMCHWT, dependents només de triangles, mostren millor precisió respecte dels esquemes continus convencionals en l'anàlisi d'objectes angulosos a on el modelatge acurat del comportament dels camps singulars és d'importància cabdal. A més, els nostres esquemes mostren en general una gran flexibilitat en l'anàlisi d'objectes compostos amb línies d'intersecció entre regions ja que no hi cal el modelatge especial dels corrents. Finalment, les implementacions proposades poden abordar l'anàlisi d'objectes amb forma arbitrària, totalment homogenis o homogenis a trossos, i amb mallats geomètricament no-conformes.The most prominent surface integral equations, the electric field integral equation (EFIE) used for the scattering analysis of perfectly electrically conducting (PEC) targets and the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) formulation commonly utilized for the analysis of homogeneous penetrable objects, are usually discretized, in the context of method of moments (MoM), with edge-based divergence-conforming basis functions. Divergence-conforming discretizations of the EFIE and PMCHWT formulations excel asconforming schemes, hence with converging solutions in the physical space of currents. However, the divergence-conforming MoM implementations require the underlying mesh to be geometrically conformal, with pairs of adjacent facets sharing a single edge. Thedevelopment of divergence-conforming schemes for composite objects with junctions, viz.boundary lines where more than two regions intersect, becomes somewhat awkward because of the definition of special continuity conditions at junctions. Moreover, the meshes arising from the juxtaposition of independently meshed subdomains in the modular design of complex objects are typically nonconformal and thus not suitable for conventional divergence-conforming MoM schemes. In this thesis, we address the robust, accurate and versatile scattering analysis of PEC and penetrable objects with arbitrary shape and composite objects with junctions meshed with conformal or nonconformal meshes. For this purpose, we employ the EFIE–PMCHWT integral-equation formulation, which follows from the application of the EFIE or PMCHWT formulations over boundary surfaces, respectively, enclosing PEC regions or separating penetrable regions. The proposed schemes rely on the expansion of the corrents with the facet-based, discontinuous-across-edges basis functions. This choice gives rise to boundary integrals with hypersingular kernels, which we handle by testing the equations with well-suited testing functions defined off the boundary tessellation, inside the region where, in light of the surface equivalence principle, the fields must be zero. Our facet-based EFIE-PMCHWT implementations exhibit improved accuracy when compared with the conventional continuous schemes in the analysis of sharp-edged targets where the accurate modelling of singular fields is of great importance. Moreover, our schemes manifest in general great flexibility in the analysis of composite objects with junctions as the special modelling of currents at junctions is not required. Finally, the proposed implementations can handle geometrically nonconformal meshes when applied to piecewise (or fully) homogeneous arbitrarily shaped objects.Postprint (published version

Formulation and Solution of Electromagnetic Integral Equations Using Constraint-Based Helmholtz Decompositions

This dissertation develops surface integral equations using constraint-based Helmholtz decompositions for electromagnetic modeling. This new approach is applied to the electric field integral equation (EFIE), and it incorporates a Helmholtz decomposition (HD) of the current. For this reason, the new formulation is referred to as the EFIE-hd. The HD of the current is accomplished herein via appropriate surface integral constraints, and leads to a stable linear system. This strategy provides accurate solutions for the electric and magnetic fields at both high and low frequencies, it allows for the use of a locally corrected Nyström (LCN) discretization method for the resulting formulation, it is compatible with the local global solution framework, and it can be used with non-conformal meshes. To address large-scale and complex electromagnetic problems, an overlapped localizing local-global (OL-LOGOS) factorization is used to factorize the system matrix obtained from an LCN discretization of the augmented EFIE (AEFIE). The OL-LOGOS algorithm provides good asymptotic performance and error control when used with the AEFIE. This application is used to demonstrate the importance of using a well-conditioned formulation to obtain efficient performance from the factorization algorithm

The Buffered Block Forward Backward technique for solving electromagnetic wave scattering problems

This work focuses on efficient numerical techniques for solving electromagnetic wave scattering problems. The research is focused on three main areas: scattering from perfect electric conductors, 2D dielectric scatterers and 3D dielectric scattering objects. The problem of fields scattered from perfect electric conductors is formulated using the Electric Field Integral Equation. The Coupled Field Integral Equation is used when a 2D homogeneous dielectric object is considered. The Combined Field Integral Equation describes the problem of scattering from 3D homogeneous dielectric objects. Discretising the Integral Equation Formulation using the Method of Moments creates the matrix equation that is to be solved. Due to the large number of discretisations necessary the resulting matrices are of significant size and therefore the matrix equations cannot be solved by direct inversion and iterative methods are employed instead. Various iterative techniques for solving the matrix equation are presented including stationary methods such as the ”forwardbackward” technique, as well its matrix-block version. A novel iterative solver referred to as Buffered Block Forward Backward (BBFB) method is then described and investigated. It is shown that the incorporation of buffer regions dampens spurious diffraction effects and increases the computational efficiency of the algorithm. The BBFB is applied to both perfect electric conductors and homogeneous dielectric objects. The convergence of the BBFB method is compared to that of other techniques and it is shown that, depending on the grouping and buffering used, it can be more effective than classical methods based on Krylov subspaces for example. A possible application of the BBFB, namely the design of 2D dielectric photonic band-gap TeraHertz waveguides is investigated. i

Iso-geometric Integral Equation Solvers and their Compression via Manifold Harmonics

The state of art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low frequency and dense discretization breakdown, preconditioning, and so on. Likewise, the community has seen extensive investment in development of methods for higher order analysis, in both geometry and physics. Unfortunately, these standard geometric descriptors are $C^0$ at the boundary between patches with a few exceptions; as a result, one needs to define additional mathematical infrastructure to define physical basis sets for vector problems. In stark contrast, the geometric representation used for design is higher-order differentiable over the entire surface. Geometric descriptions that have $C^{2}$-continuity almost everywhere on the surfaces are common in computer graphics. Using these description for analysis opens the door to several possibilities, and is the area we explore in this paper. Our focus is on Loop subdivision based isogeometric methods. In this paper, our goals are two fold: (i) development of computational infrastructure necessary to effect efficient methods for isogeometric analysis of electrically large simply connected objects, and (ii) to introduce the notion of manifold harmonics transforms and its utility in computational electromagnetics. Several results highlighting the efficacy of these two methods are presented

Advanced discontinuous integral-equation schemes for the versatile electromagnetic analysis of complex structures

Les Equacions Integrals superficials més importants són l'Equació de Camp Elèctric (EFIE), per a l'anàlisi de la dispersió electromagnètica d'objectes conductors perfectes (PEC), i la formulació Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT), orientada a l'anàlisi d'objectes homogenis penetrables. Ambdues són normalment discretitzades, amb el Mètode dels Moments (MoM), amb funcions base div-conformes, dependents de les arestes del mallat. Les discretitzacions div-conformes de les formulacions EFIE i PMCHWT representen esquemes conformes; és a dir, amb solucions convergents a dins de l'espai físic de corrents. Tanmateix, les implementations MoM div-conformes requereixen que el mallat sigui conforme geomètricament, amb cada parell de triangles adjacents compartint només una aresta. El desenvolupament d'esquemes div-conformes per a objectes compostos amb línies al llarg de les quals tres o més regions hi intersecten, esdevé molt incòmoda perquè cal definir condicions de continuïtat especials en aquestes línies d'intersecció. A més, els mallats que resulten de la juxtaposició de subdominis independentment mallats són típicament no-conformes geomètricament i per tant no aptes per a l'anàlisi div-conforme convencional en Mètode dels Moments. En aquesta Tesi, es tracta l'anàlisi robusta, precisa i versàtil de la dispersió electromagnètica d'objectes conductors o penetrables amb forma arbitrària i d'objectes compostos amb línies d'intersecció entre differents regions, ja sigui amb mallats conformes com no-conformes. Amb aquest objectiu, fem ús de la formulació d'equació integral EFIE–PMCHWT, la qual resulta de l'aplicació de les formulacions EFIE o PMCHWTal llarg de superfícies de contorn, respectivament, incloent regions conductores o separant regions penetrables. Els esquemes proposats en aquesta Tesi es basen en el desenvolupament dels corrents amb conjunts de funcions base discontínues a través de les arestes del mallat i dependents només dels triangles del mallat. Aquesta estratègia dóna lloc a integrals de contorn amb Kernels hypersingulars, que maneguem mitjançant el testeig de les equacions amb funcions de testeig especialment dissenyades, definides fora de les triangulacions de la superfície de contorn, a dins de la regió a on els camps són zero d'acord amb al Teorema d'Equivalència superficial. Les nostres implementacions de la formulació EFIE-PMCHWT, dependents només de triangles, mostren millor precisió respecte dels esquemes continus convencionals en l'anàlisi d'objectes angulosos a on el modelatge acurat del comportament dels camps singulars és d'importància cabdal. A més, els nostres esquemes mostren en general una gran flexibilitat en l'anàlisi d'objectes compostos amb línies d'intersecció entre regions ja que no hi cal el modelatge especial dels corrents. Finalment, les implementacions proposades poden abordar l'anàlisi d'objectes amb forma arbitrària, totalment homogenis o homogenis a trossos, i amb mallats geomètricament no-conformes.The most prominent surface integral equations, the electric field integral equation (EFIE) used for the scattering analysis of perfectly electrically conducting (PEC) targets and the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) formulation commonly utilized for the analysis of homogeneous penetrable objects, are usually discretized, in the context of method of moments (MoM), with edge-based divergence-conforming basis functions. Divergence-conforming discretizations of the EFIE and PMCHWT formulations excel asconforming schemes, hence with converging solutions in the physical space of currents. However, the divergence-conforming MoM implementations require the underlying mesh to be geometrically conformal, with pairs of adjacent facets sharing a single edge. Thedevelopment of divergence-conforming schemes for composite objects with junctions, viz.boundary lines where more than two regions intersect, becomes somewhat awkward because of the definition of special continuity conditions at junctions. Moreover, the meshes arising from the juxtaposition of independently meshed subdomains in the modular design of complex objects are typically nonconformal and thus not suitable for conventional divergence-conforming MoM schemes. In this thesis, we address the robust, accurate and versatile scattering analysis of PEC and penetrable objects with arbitrary shape and composite objects with junctions meshed with conformal or nonconformal meshes. For this purpose, we employ the EFIE–PMCHWT integral-equation formulation, which follows from the application of the EFIE or PMCHWT formulations over boundary surfaces, respectively, enclosing PEC regions or separating penetrable regions. The proposed schemes rely on the expansion of the corrents with the facet-based, discontinuous-across-edges basis functions. This choice gives rise to boundary integrals with hypersingular kernels, which we handle by testing the equations with well-suited testing functions defined off the boundary tessellation, inside the region where, in light of the surface equivalence principle, the fields must be zero. Our facet-based EFIE-PMCHWT implementations exhibit improved accuracy when compared with the conventional continuous schemes in the analysis of sharp-edged targets where the accurate modelling of singular fields is of great importance. Moreover, our schemes manifest in general great flexibility in the analysis of composite objects with junctions as the special modelling of currents at junctions is not required. Finally, the proposed implementations can handle geometrically nonconformal meshes when applied to piecewise (or fully) homogeneous arbitrarily shaped objects

A LOCALLY CORRECTED NYSTRM METHOD FOR SURFACE INTEGRAL EQUATIONS: AN OBJECT ORIENTED APPROACH

Classically, researchers in Computational Physics and specifically in Computational Electromagnetics have sought to find numerical solutions to complex physical problems. Several techniques have been developed to accomplish such tasks, each of which having advantages over their counterparts. Typically, each solution method has been developed separately despite having numerous commonalities with other methods. This fact motivates a unified software tool to house each solution method to avoid duplicating previous efforts. Subsequently, these solution methods can be used alone or in conjunction with one another in a straightforward manner. The aforementioned goals can be accomplished by using an Object Oriented software approach. Thus, the goal of the presented research was to incorporate a specific solution technique, an Integral Equation Nystrm method, into a general, Object Oriented software framework

Optical Near-Field Enhancement by Micro/Nano Particles for Nanotechnology Applications

Ph.DDOCTOR OF PHILOSOPH

Computational Scattering Models for Elastic and Electromagnetic Waves in Particulate Media

Numerical models were developed to simulate the propagation of elastic and electromagnetic waves in an arbitrary, dense dispersion of spherical particles. The scattering interactions were modeled with vector multipole fields using pure-orbital vector spherical harmonics, and solved using the full vector form of the boundary conditions. Multiple scattering was simulated by translating the scattered wave fields from one particle to another with the use of translational addition theorems, summing the multiple-scattering contributions, and recalculating the scattering in an iterative fashion to a convergent solution. The addition theorems were rederived in this work using an integral method, and were shown to be numerically equivalent to previously published theorems. Both ordered and disordered collections of up to 5,000 spherical particles were used to demonstrate the ability of the scattering models to predict the spatial and frequency distributions of the transmitted waves. The results of the models show they are qualitatively correct for many particle configurations and material properties, displaying predictable phenomena such as refractive focusing, mode conversion, and photonic band gaps. However, the elastic wave models failed to converge for specific frequency regions, possibly due to resonance effects. Additionally, comparison of the multiple-scattering simulations with those using only single-particle scattering showed the multiple-scattering computations are quantitatively inaccurate. The inaccuracies arise from nonconvergence of the translational addition theorems, introducing errors into the translated fields, which minimize the multiple-scattering contributions and bias the field amplitudes towards single-scattering contributions. The addition theorems are shown to converge very slowly, and to exhibit plateaus in convergence behavior that can lead to false indications of convergence. The theory and algorithms developed for the models are broad-based, and can accommodate a variety of structures, compositions, and wave modes. The generality of the approach also lends itself to the modeling of static fields and currents. Suggestions are presented for improving and implementing the models, including extension to nonspherical particles, efficiency improvements for the algorithms, and specific applications in a variety of fields