Ana Manic

Ph.D. FinalAug 19, 2015, 10:00 am - 1:00 pm

ENGRG B4

Fast and Accurate Double-Higher-Order Method of Moments Accelerated by Diakoptic Domain Decomposition and Memory Efficient Parallelization for High Performance Computing Systems

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 methodâ€™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.

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 methodâ€™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

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, â€œEfficient Higher Order Full-Wave Numerical Analysis of 3-D Cloaking Structures,â€ 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, â€œLarge Anisotropic Inhomogeneous Higher Order Hierarchical Generalized Hexahedral Finite Elements for 3-D Electromagnetic Modeling of Scattering and Waveguide Structures,â€ Microwave and Optical Technology Letters, vol. 54, No. 7, July 2012, pp. 1644-1649.

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

4) A. B. ManiÄ‡, D. I. OlÄ‡an, M. M. IliÄ‡, and B. M. NotaroÅ¡, â€œDiakoptic approach combining finite-element method and method of moments in analysis of inhomogeneous anisotropic dielectric and magnetic scatterers,â€ Electromagnetics, vol. 34, no. 3â€“4, pp. 222â€“238, 2014.

5) M. Thurai, V. N. Bringi, A. B. ManiÄ‡, N. J. Å ekeljiÄ‡, and B. M. NotaroÅ¡, â€œInvestigating rain drop shapes, oscillation modes, and implications for radiowave propagation,â€ Radio Science, vol. 49, no. 10, pp. 921-932, October 2104.

6) E. Chobanyan, N. J. Å ekeljiÄ‡, A. B. ManiÄ‡, M. M. IliÄ‡, V. N. Bringi, and B. M. NotaroÅ¡, â€œEfficient and Accurate Computational Electromagnetics Approach to Precipitation Particle Scattering Analysis Based on Higher Order Method of Moments Integral-Equation Modelingâ€, Journal of Atmospheric and Oceanic Technology, accepted.

7) Ana B. ManiÄ‡, FranÃ§ois-Henry Rouet, Xiaoye Sherry Li, and Branislav M. NotaroÅ¡, â€œEfficient Scalable Parallel Higher Order Direct MoM-SIE Method with Hierarchically Semiseparable Structures for 3D Scattering, â€ submitted to IEEE Transactions on Antennas and Propagation.

8) Ana B. ManiÄ‡, and Branislav M. NotaroÅ¡, â€œFast Computation of Near-Singular and Near-Hypersingular Integrals in Higher Order Method of Moments Using Curved Quadrilateral Patches, â€ 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, â€œConstruction, Modeling, and Analysis of Transformation-Based Metamaterial Invisibility Cloaks,â€ 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, â€œNumerical Computation of Singular Integrals in Higher Order Method of Moments Using Curved Quadrilateral Patches,â€ 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, â€œSymmetric Coupling of Finite Element Method and Method of Moments Using Higher Order Elements,â€ 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, â€œFEM-MoM-Diakoptic Analysis of Scatterers with Anisotropic Inhomogeneities Using Hierarchical Vector Bases on Large Curved Elements,â€ invited paper, Special Session â€œAdvances in Vector Bases for CEM,â€ 11th International Workshop on Finite Elements for Microwave Engineering â€“ FEM2012, June 4-6, 2012, Estes Park, Colorado.

4) A. B. Manic, M. M. Ilic, and B. M. Notaros, â€œSymmetric and Nonsymmetric FEM-MoM Techniques Using Higher Order Hierarchical Vector Basis Functions and Curved Parametric Elements,â€ invited paper, Special Session â€œAdvances in Hybrid Methods and Multiphysics Problems,â€ 11th International Workshop on Finite Elements for Microwave Engineering â€“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, â€œEfficient Electromagnetic Analysis Using Electrically Large Curved p-Refined Hierarchical Anisotropic Inhomogeneous Finite Elements,â€ 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, â€œDiakoptic FEM-MoM Analysis Using Explicit Connection between Field and Current Bases,â€ 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, â€œAtmospheric Particle Scattering Computation Using Higher Order MoM-SIE Method,â€ 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, â€œTransient Analysis of 3D Waveguides Using Double-Higher Time-Domain Finite Element Method,â€ 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, â€œOngoing Investigations of Rain Drop Shapes and Oscillation Modes,â€ 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, â€œParallelization of Double Higher Order FEM and MoM Techniques,â€ 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, â€œElectromagnetic Scattering by Oscillating Rain Drops of Asymmetric Shapes,â€ 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, â€œHybrid Higher Order Num

JOURNAL PAPERS

1) S. V. Savic, A. B. Manic, M. M. Ilic, and B. M. Notaros, â€œEfficient Higher Order Full-Wave Numerical Analysis of 3-D Cloaking Structures,â€ 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, â€œLarge Anisotropic Inhomogeneous Higher Order Hierarchical Generalized Hexahedral Finite Elements for 3-D Electromagnetic Modeling of Scattering and Waveguide Structures,â€ Microwave and Optical Technology Letters, vol. 54, No. 7, July 2012, pp. 1644-1649.

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

4) A. B. ManiÄ‡, D. I. OlÄ‡an, M. M. IliÄ‡, and B. M. NotaroÅ¡, â€œDiakoptic approach combining finite-element method and method of moments in analysis of inhomogeneous anisotropic dielectric and magnetic scatterers,â€ Electromagnetics, vol. 34, no. 3â€“4, pp. 222â€“238, 2014.

5) M. Thurai, V. N. Bringi, A. B. ManiÄ‡, N. J. Å ekeljiÄ‡, and B. M. NotaroÅ¡, â€œInvestigating rain drop shapes, oscillation modes, and implications for radiowave propagation,â€ Radio Science, vol. 49, no. 10, pp. 921-932, October 2104.

6) E. Chobanyan, N. J. Å ekeljiÄ‡, A. B. ManiÄ‡, M. M. IliÄ‡, V. N. Bringi, and B. M. NotaroÅ¡, â€œEfficient and Accurate Computational Electromagnetics Approach to Precipitation Particle Scattering Analysis Based on Higher Order Method of Moments Integral-Equation Modelingâ€, Journal of Atmospheric and Oceanic Technology, accepted.

7) Ana B. ManiÄ‡, FranÃ§ois-Henry Rouet, Xiaoye Sherry Li, and Branislav M. NotaroÅ¡, â€œEfficient Scalable Parallel Higher Order Direct MoM-SIE Method with Hierarchically Semiseparable Structures for 3D Scattering, â€ submitted to IEEE Transactions on Antennas and Propagation.

8) Ana B. ManiÄ‡, and Branislav M. NotaroÅ¡, â€œFast Computation of Near-Singular and Near-Hypersingular Integrals in Higher Order Method of Moments Using Curved Quadrilateral Patches, â€ 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, â€œConstruction, Modeling, and Analysis of Transformation-Based Metamaterial Invisibility Cloaks,â€ 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, â€œNumerical Computation of Singular Integrals in Higher Order Method of Moments Using Curved Quadrilateral Patches,â€ 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, â€œSymmetric Coupling of Finite Element Method and Method of Moments Using Higher Order Elements,â€ 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, â€œFEM-MoM-Diakoptic Analysis of Scatterers with Anisotropic Inhomogeneities Using Hierarchical Vector Bases on Large Curved Elements,â€ invited paper, Special Session â€œAdvances in Vector Bases for CEM,â€ 11th International Workshop on Finite Elements for Microwave Engineering â€“ FEM2012, June 4-6, 2012, Estes Park, Colorado.

4) A. B. Manic, M. M. Ilic, and B. M. Notaros, â€œSymmetric and Nonsymmetric FEM-MoM Techniques Using Higher Order Hierarchical Vector Basis Functions and Curved Parametric Elements,â€ invited paper, Special Session â€œAdvances in Hybrid Methods and Multiphysics Problems,â€ 11th International Workshop on Finite Elements for Microwave Engineering â€“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, â€œEfficient Electromagnetic Analysis Using Electrically Large Curved p-Refined Hierarchical Anisotropic Inhomogeneous Finite Elements,â€ 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, â€œDiakoptic FEM-MoM Analysis Using Explicit Connection between Field and Current Bases,â€ 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, â€œAtmospheric Particle Scattering Computation Using Higher Order MoM-SIE Method,â€ 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, â€œTransient Analysis of 3D Waveguides Using Double-Higher Time-Domain Finite Element Method,â€ 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, â€œOngoing Investigations of Rain Drop Shapes and Oscillation Modes,â€ 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, â€œParallelization of Double Higher Order FEM and MoM Techniques,â€ 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, â€œElectromagnetic Scattering by Oscillating Rain Drops of Asymmetric Shapes,â€ 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, â€œHybrid Higher Order Num

Program of Study:

ECE 536

ECE 512

ECE 742

MATH 652

ECE 540

ECE 742

ECE 641

ECE 642

ECE 536

ECE 512

ECE 742

MATH 652

ECE 540

ECE 742

ECE 641

ECE 642