A Higher Order Hierarchical Generalized VolumeSurface Integral Equation using Parametric Finite Elem

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A Higher Order Hierarchical Generalized
Volume-Surface Integral Equation using Parametric
Finite Elements and a novel Petrov-Galerkin
Testing Scheme

• Brian C. Usner, Kubi Sertel PhD., John L. Volakis
  PhD.
• The ElectroScience Laboratory, Department of
  Electrical Engineering
• The Ohio State University

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Background
Surface Integral Equation
Volume Integral Equation

• Can handle PEC structures
• Can handle homogeneous materials
• Cant handle anisotropic materials
• Can handle high contrast materials
• Can be generalized to handle arbitrary piecewise
  homogeneous structures having arbitrary boundary
  conditions

• Cant handle PEC structures
• Can handle inhomogeneous materials
• Can handle anisotropic materials
• Computational difficulties in handling high
  contrast materials
• System matrix is well conditioned

Can we find construct Unified Theory for Integral
Equations ???
Formulation
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Whats Been Done??
The Volume Surface Integral Equation

• Use Volume Integral Equation for Materials
• Use Surface Integral Equation for PEC Surfaces

Free-Space Greens Function
These formulations are just subsets of a more
general formulation!!!!!!
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Pertinent Vector Wave Equations
Material Factorizations
Homogeneous Term
Modulation Term
Inhomogeneous Term
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Partitioning of the EM Structure
Some Notations
Each Volume is characterized by the parameter set
with
Enumerate the Surfaces
such that
Associated with each Surface
is two volumes
Then for convenience define the function
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Equivalent Surface Currents
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The General VSIE
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Integral Equation Discretization
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Higher Order Basis Functions
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Petrov-Galerkin Volume Testing
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Junction Resolution
Thanks to discussions with Dr. Carr
Step by Step
Step 1. Pair all the unknowns that radiate into
the same volume Step 2. Eliminate all unknowns
that radiate into an enclosed PEC or PMC volume,
as well as unknowns defined over PMC/PEC
surface Step 3. If more than one unknown remains,
enforce all dielectric boundary conditions Step
4. If more than one unknown remains, then for
global currents radiating into the same volume,
only keep enough so that all surfaces contain a
current
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The GenVSIE System
VIE
MFIE
EFIE
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Validations Junction Resolution
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Validations Junction Resolution
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Validations Junction Resolution
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Validations Junction Resolution
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Validations Junction Resolution
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Validations Junction Resolution
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Validations Junction Resolution
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VSIE Validations Dielectric Sphere
Scattering from Relatively Large High Contrast
Sphere
Total Unknowns 9,540
Typical VIE would require 19,494
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VSIE Validations Coated Ogive
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Antenna Feed Modeling
If the jth unknown in the MoM system is
associated with impressed current basis function,
then the jth column becomes the excitation vector
after the jth row has been removed
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Validation Tapered Slot Antenna
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Validations High Order
First-order solution requires 15,000 unknowns
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Validations High Order
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Infinite Doubly Periodic Structures
Frequency Selective Volumes (FSVs)
Main Advantage

• Only need to evaluate the periodic Greens
  function for currents radiating into free-space

Simply Replace Free-Space Greens Function with
Periodic Greens Function
Need a Modified Junction Resolution Algorithm
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Periodic Boundary Conditions
From Floquets Theorem
Unit Cell For Joined FSV
Aperture-PBC Junction
PBC-PBC Junction
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Modified Junction Resolution
PBC-PBC Junction Resolution
Super Junctions
Aperture-PBC Junction Resolution
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Slot Array FSV
Unit Cell
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Dielectric Slab
Unit Cell
Incident Angle Scan at 300MHz
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Dielectric Slab
Unit Cell
Frequency Scan for Normal Incidence
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Embedded Material FSV
Unit Cell