Abstract: The principal objective of this dissertation is to develop and test a robust method based on the method of moments (MoM) surface integral equation (SIE) formulation for electromagnetic analysis of dielectric and magnetic scatterers and antennas in the frequency domain using double higher order (DHO) mesh discretization. It is well known that by using higher order basis functions for current/field modeling in computational electromagnetics (CEM), significant reductions in the number of unknowns, as well as faster system matrix computation/solution, can be achieved when compared to the traditional low order modeling. Tightly coupled with using higher order basis functions is higher order geometry modeling and together they lay foundation to double higher order (DHO) modeling. Double (geometrical and current) higher order modeling enables using large curved patches, which can greatly reduce the number of unknowns for a given problem and enhance the accuracy and efficiency of the computation. Element orders in the model can also be low both in terms of basis function order or geometrical order, so the low-order modeling approach is actually included in the DHO modeling. So, a whole range of element sizes and shapes, geometrical orders, and current approximation orders can be used at the same time in a single simulation model of a complex structure using the high order (more precisely, low-to-high order) CEM technique. The two major issues arising in the application of the MoM-SIE numerical methods when solving large and computationally expensive electromagnetic problems are: 1) fast and accurate calculation of the system matrix entrances arising in the MoM- SIE formulation and 2) overall computational and memory storage complexity of the method. The goal of this dissertation is to propose and validate a solution for both of the major method’s bottlenecks. The accurate and fast computation of the system matrix includes advanced methods for numerical computation of singular and near-singular integrals defined on the surface mesh elements. When the method is aimed at analysis of both metallic and dielectric/magnetic structures, the singularity of the integral kernel increases, and requires special treatment of highly singular integrals. Finally, this problem is even more pronounced when higher order basis functions are used for the approximation of electric and magnetic equivalent surface currents defined on curved patches. This dissertation presents a novel method for numerical computation of near- singular (potential) and near- hypersingular (field) integrals defined on Lagrange-type generalized curved parametric quadrilateral surface elements of arbitrary geometrical orders with polynomial basis functions of arbitrary current-approximation orders. The integrals are evaluated using a method based on the singularity extraction, which consists of analytical integration of a principal singular part of the integrand over a (generally not rectangular) parallelogram whose surface is close to the surface of the generalized quadrilateral near the singular point and numerical integration of the rest. The majority of the existing extraction techniques have been developed for planar triangular patches involving low-order basis functions. Few of those have been extended to curved patches but without really taking into account the curvature of the surface. The presented integration technique considers the curvature of the patch by extracting multiple terms in the evaluation of the principal singular part. Further, the theory behind the extraction technique has been extended to consider integrals with higher order basis functions. Overall computational complexity and memory requirements of the traditional MoM-SIE method are of the O(N3) and O(N2), respectively, where N is the number of unknowns. Even though DHO modelling can reduce number of unknowns by the order of 20, the order of computational complexity remains the same. As the part of this dissertation, a novel fast scalable DHO parallel algorithm on the DHO MoM-SIE in conjunction with a direct solver for dense linear systems with hierarchically semiseparable structures (HSS) is proposed. We are developing asymptotically fast higher order direct algorithms for MoM-SIE solutions which, in a nutshell, are an algebraic generalization to fast multipole methods. In addition to being fast, they offer a promise of being memory- and communication-efficient and amenable to extreme-scale parallel computing. The main advantage of the HSS algorithm is in the linear- complexity ULV-type factorizations (compared to the conventional LU decomposition that has cubic complexity). Our work uses the recently developed new, state-of-the- art, algorithms for solving dense and sparse linear systems of equations based on the HSS method. In addition, rank revealing QR (RRQR) decomposition for the matrix (memory) compression. Its adaptive nature comes from the ability to use the stopping criteria, i.e., relative tolerance value/minimal rank, which allows for the method to store only the low-rank approximation of the original matrix that satisfies predefined accuracy. The standard and most accurate technique for constructing the HSS representation of a dense matrix implies explicit calculation of all matrix elements, and then compression of appropriate blocks using the RRQR decomposition, with an O(rN2) asymptotic cost. Once the HSS construction is done, the other steps are cheaper, with O(r2N) time complexity for ULV factorization and O(rN) for solution, respectively, where N was previously defined and r is the maximum numerical rank. In order to enhance the HSS compression and parallelization i.e. scalability of the method, an algorithm for geometrical preprocessing of the geometrical mesh based on the cobblestone distance sorting technique is utilized. Hence, the MoM unknowns having spatial locality, also exhibit the data locality in the matrix system of equations. To sum up, method is validated and great performance is achieved. Even more, the simulation results show great scalability of the method on more than 1000 processes. Besides developing a fast, parallel and robust method based on the MoM-SIE, in order to extend applicability of the method to the analysis that involves inhomogeneous anisotropic dielectric and magnetic materials, new symmetric hybridization of the finite element method (FEM) and the MoM was developed. The FEM is one of the general numerical tools for solving closed-region (e.g., waveguide/cavity) problems in electromagnetics. It has been especially effectively used in three-dimensional (3-D) frequency- domain modeling and analysis of electromagnetic structures that contain geometrical and material complexities. In addition, as the part of the work included in this dissertation the DHO FEM method was implemented primarily to support analysis of both inhomogeneous and anisotropic materials. Further, numerical computation is accelerated by applying Diakoptic Domain Decomposition approach to divide the original problem of interest into smaller subsystems, analyze subsystems independently, and then connect them back together through the surface equivalence theorem. Finally, all numerical methods described above are validated on a variety of numerical examples and tested across several high performance supercomputing platforms.

Adviser: Prof. Branislav Notaros

Co-Adviser: N/A

Non-ECE Member: Prof. Iuliana Oprea

Member 3: Prof. Steven Reising

Addional Members: Prof. Sourajeet Roy, Prof. Milan Ilic

Publications:

JOURNAL PAPERS

1) S. V. Savic, A. B. Manic, M. M. Ilic, and B. M. Notaros, “Efficient Higher Order Full-Wave Numerical Analysis of 3-D Cloaking Structures,” Plasmonics, 2012 (published online: 8 July, 2012), 10.1007/s11468-012-9410-0.

2) A. B. Manic, S. B. Manic, M. M. Ilic, and B. M. Notaros, “Large Anisotropic Inhomogeneous Higher Order Hierarchical Generalized Hexahedral Finite Elements for 3-D Electromagnetic Modeling of Scattering and Waveguide Structures,” Microwave and Optical Technology Letters, vol. 54, No. 7, July 2012, pp. 1644-1649.

3) A. B. Manic, M. Djordjevic, and B. M. Notaros, “Duffy Method for Evaluation of Weakly Singular SIE Potential Integrals over Curved Quadrilaterals with Higher Order Basis Functions,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 6, June 2014,

4) A. B. Manić, D. I. Olćan, M. M. Ilić, and B. M. Notaroš, “Diakoptic approach combining finite-element method and method of moments in analysis of inhomogeneous anisotropic dielectric and magnetic scatterers,” Electromagnetics, vol. 34, no. 3–4, pp. 222–238, 2014.

5) M. Thurai, V. N. Bringi, A. B. Manić, N. J. Šekeljić, and B. M. Notaroš, “Investigating rain drop shapes, oscillation modes, and implications for radiowave propagation,” Radio Science, vol. 49, no. 10, pp. 921-932, October 2104.

6) E. Chobanyan, N. J. Šekeljić, A. B. Manić, M. M. Ilić, V. N. Bringi, and B. M. Notaroš, “Efficient and Accurate Computational Electromagnetics Approach to Precipitation Particle Scattering Analysis Based on Higher Order Method of Moments Integral-Equation Modeling”, Journal of Atmospheric and Oceanic Technology, accepted.

7) Ana B. Manić, François-Henry Rouet, Xiaoye Sherry Li, and Branislav M. Notaroš, “Efficient Scalable Parallel Higher Order Direct MoM-SIE Method with Hierarchically Semiseparable Structures for 3D Scattering, ” submitted to IEEE Transactions on Antennas and Propagation.

8) Ana B. Manić, and Branislav M. Notaroš, “Fast Computation of Near-Singular and Near-Hypersingular Integrals in Higher Order Method of Moments Using Curved Quadrilateral Patches, ” IEEE Transactions on Antennas and Propagation, manuscript completed, to be submitted.

BOOK CHAPTER

1) B. M. Notaros, M. M. Ilic, S. V. Savic, and A. B. Manic, “Construction, Modeling, and Analysis of Transformation-Based Metamaterial Invisibility Cloaks,” accepted for publication in The Annual Reviews in Plasmonics, 2015.

PEER-REVIEWED CONFERENCE PAPERS AND ABSTRACTS

1) A. B. Manic, M. Djordjevic, E. Smith, and B. M. Notaros, “Numerical Computation of Singular Integrals in Higher Order Method of Moments Using Curved Quadrilateral Patches,” Proc. 2013 USNC-URSI National Radio Science Meeting, January 9-12, 2013, Boulder, Colorado.

2) A. B. Manic, M. M. Ilic, and B. M. Notaros, “Symmetric Coupling of Finite Element Method and Method of Moments Using Higher Order Elements,” 2012 IEEE Antennas and Propagation Society International Symposium Digest, July 8-14, 2012, Chicago, Illinois.

3) A. B. Manic, D. I. Olcan, M. M. Ilic, and B. M. Notaros, “FEM-MoM-Diakoptic Analysis of Scatterers with Anisotropic Inhomogeneities Using Hierarchical Vector Bases on Large Curved Elements,” invited paper, Special Session “Advances in Vector Bases for CEM,” 11th International Workshop on Finite Elements for Microwave Engineering – FEM2012, June 4-6, 2012, Estes Park, Colorado.

4) A. B. Manic, M. M. Ilic, and B. M. Notaros, “Symmetric and Nonsymmetric FEM-MoM Techniques Using Higher Order Hierarchical Vector Basis Functions and Curved Parametric Elements,” invited paper, Special Session “Advances in Hybrid Methods and Multiphysics Problems,” 11th International Workshop on Finite Elements for Microwave Engineering –FEM2012, June 4-6, 2012, Estes Park, Colorado.

5) A. B. Manic, S. B. Manic, S. V. Savic, M. M. Ilic, and B. M. Notaros, “Efficient Electromagnetic Analysis Using Electrically Large Curved p-Refined Hierarchical Anisotropic Inhomogeneous Finite Elements,” Proc. 2012 USNC-URSI National Radio Science Meeting, January 4-7, 2012, Boulder, Colorado.

6) A. B. Manic, D. I. Olcan, M. M. Ilic, and B. M. Notaros, “Diakoptic FEM-MoM Analysis Using Explicit Connection between Field and Current Bases,” 2013 IEEE Antennas and Propagation Society International Symposium Digest, July 7-12, 2013, Lake Buena Vista, Florida.

7) E. Chobanyan, N. J. Sekeljic, A. B. Manic, M. M. Ilic, and B. M. Notaros, “Atmospheric Particle Scattering Computation Using Higher Order MoM-SIE Method,” 2013 IEEE Antennas and Propagation Society International Symposium Digest, July 7-12, 2013, Lake Buena Vista, Florida.

8) N. J. Sekeljic, A. B. Manic, M. M. Ilic, and B. M. Notaros, “Transient Analysis of 3D Waveguides Using Double-Higher Time-Domain Finite Element Method,” 2013 IEEE Antennas and Propagation Society International Symposium Digest, July 7-12, 2013, Lake Buena Vista, Florida.

9) M. Thurai, V. N. Bringi, A. B. Manic, and B. M. Notaros, “Ongoing Investigations of Rain Drop Shapes and Oscillation Modes,” Proc. URSI Commission F Triennial Open Symposium on Radiowave Propagation & Remote Sensing, April 30-May 3, 2013, Ottawa, Canada.

10) Ana Manic, Elene Chobanyan, Milan Ilic, and Branislav Notaros, “Parallelization of Double Higher Order FEM and MoM Techniques,” 2014 IEEE Antennas and Propagation Society International Symposium Digest, July 6-11, 2014, Memphis, Tennessee.

11) Nada Sekeljic, Ana Manic, Elene Chobanyan, Merhala Thurai, V. N. Bringi, and Branislav Notaros, “Electromagnetic Scattering by Oscillating Rain Drops of Asymmetric Shapes,” 2014 IEEE Antennas and Propagation Society International Symposium Digest, July 6-11, 2014, Memphis, Tennessee.

12) B. M. Notaros, M. M. Ilic, D. I. Olcan, M. Djordjevic, A. B. Manic, and E. Chobanyan, “Hybrid Higher Order Num

Program of Study:

ECE 536

ECE 512

ECE 742

MATH 652

ECE 540

ECE 742

ECE 641

ECE 642