techniques for numerically efficient analysis of multi ... · multi-scale problems in numerical...

Techniques for Numerically Efficient Analysis of Multi-Scale Problems in Computational ElectromagneticsPresented by: Kapil Sharma

Advisor:Professor Raj Mittra

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Copyright© The use of this work is restricted solely for academicpurposes. The author of this work owns the copyright andno reproduction in any form is permitted without writtenpermission by the author.

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AbstractMulti-scale problems in numerical electromagnetics are becoming increasingly common with the advent and widespread usageof compact mobile phones, body area networks, small and nano antennas, sensors, high-speed interconnects, integrated circuitsand complex electronic packaging structures, to name just a few commercial applications. Numerical electromagnetic modelingand simulation of structures with multi-scale features is highly challenging due to the fact that electrically small as well as largefeatures are simultaneously present in the model which demands for discretization of the computational domain such that thenumber of degrees of freedom is very large, thus, levying a heavy burden on computational resources. The multi-scale nature ofa given problem also exacerbates the challenge of generating very fine meshes which do not introduce instabilities or ill-conditioned behaviors.In this work we introduce a hybrid technique, which combines frequency domain and time domain techniques in a manner suchthat the fine features (electrically small) of the object being modeled are handled by the Method of Moments (MoM) techniquewhile the electrically large parts of the structure are dealt with by using the Finite-Difference Time-Domain (FDTD) technique inorder to reduce the computational burden. Recently, structures with multi-scale features have been simulated by using the dipolemoment (DM) approach combined with the FDTD technique to handle fine features in a multi-scale geometry. However, whenthe size of the scatterer becomes larger in terms of the wavelength and the quasi-static assumption becomes invalid, extensivemodifications of the DM/FDTD hybrid approach are needed resulting in a high computational cost.The research proposes a novel hybrid FDTD technique, which combines the Method of Moments and the Finite-Difference Time-Domain techniques directly in the time domain circumventing the need to carry out frequency transform calculations as requiredin the DM approach when the object size is not small (size>λ/20). The proposed technique utilizes piecewise sinusoidal basisfunctions to represent the currents on arbitrarily shaped wires with fine features, and modified RWG basis function for surfaces.The fields scattered by the object with fine features in MoM region are computed in the time domain on a planar interface. Thetime domain fields obtained at the planar interface are then combined with the FDTD update equations. In contrast to theexisting techniques used to handle this type of problems, the proposed technique is both efficient as well as stable.

Index Terms: MoM, FDTD, DM, RWG, DoF, TDIE, EFIE, MFIE, CFIE, Matrix, RHS, Finite Difference, Lagrange QuadraticInterpolation, Modified RWG

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BiographyKapil Sharma received his B.E. degree in Electronics and CommunicationEngineering from University of Delhi, India in 2008. He worked as a scientificresearcher at Electronics and Radar Development Establishment, India from2008 to 2011, where he was involved with research projects related to EMCand signal integrity analysis of high-speed devices, as well as design of ultrawideband antennas. He received M.S. and Ph.D. in Electrical Engineering fromThe Pennsylvania State University in 2013 and 2017 respectively. He worked atAmphenol FCI as a signal integrity research & development engineer from2015 to 2016, and as a postdoctoral scholar at Penn State University, where heworked on MRI projects to improve signal to noise ratio in MRI scans usinghigh dielectric constant materials. Currently, he is involved with signal integrityrelated research projects at MathWorks Inc. His research interests arenumerical electromagnetics, signal integrity and EMC analysis of high-speedinterconnects, electromagnetic characterization of materials, ultra widebandantenna design, optimization algorithms, and parallel computing.

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Presentation Outline• Introduction• Research motivation• Multi-scale problem definition• Novel techniques to handle multi-scale problems

• Straight wire problem• Arbitrary shaped wire problem• Wire and surface problem• Capacitance matrix computation for coated multi-conductors

• Observations and Conclusions• Future research• References

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External antennaInternal

antenna

FDTD region MoM region

Introduction: Modeling Challenges in Current Technologies

• Ubiquity of multi-scale problems in numerical electromagnetics.

6

Research Motivation• Simulation of multi-scale problems with fine features is highly

challenging.

7

Fine feature of object captured by conventional FDTD mesh.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Distance along Z in λ

Am

plitu

de in

V/m

Amplitude Variation of Scattered Ey

DM AppraochHFSS

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.6

0.8

1

1.2

1.4

1.6

Distance along Y in λ

Am

plitu

de in

V/m

Amplitude Comparison of Ez from New Conformal FDTD

New Conformal FDTDGEMSFEKOHFSS

Spurious Ripples!

Dielectric Cuboid

Plasmonic Nano-spheres

λ @ highest frequency of interest and highest dielectric constant.

Extremely fine mesh results in high computational cost.

Utilization of fine mesh results in instability and does not guarantee accurate results using different type of solvers.

Thickness is a fraction of length

Scenario 1: Utilization of FDTD method

Scenario 2: Utilization of FEM method

Scenario 3: Utilization of FEM and MoM method

Novel hybrid technique to handle multi-scale problems

8

The novel hybrid FDTD technique combinesthe MoM and FDTD techniques directly inthe time domain.

Fine features are handled by the time-domain compatible MoM code

Time-domain compatible basis functions are used to represent current on geometry with fine

features. These basis functions are used to obtain scattered field at an observation point directly in the time-domain without using Inverse Fourier

Transform

Time-domain scattered fields are sampled at a planar interface

Sampled time-domain scattered fields are then used as source for the FDTD domain

Time domain compatible MoM code is different than the TDIE formulation, and is stable.

Time step of the time-domain compatible MoM technique is chosen such that it is equal to the time-step of the FDTD technique.

Efficiency factor for the hybrid method is large compared to the previous approach where scattered fields at interface nodes were computed in frequency domain, and inverse transform was used to obtain fields in the time-domain.

Scattered fields from antenna in MoM domain are obtained directly in time-domain.

Different types of problems investigated are discussed in the following sections.

Straight wiresTime-domain compatible basis functions

9

Time-domain compatible MoM formulation is different as compared to the conventional TDIE formulation, and is stable.

The basis function enables us to obtain scattered field directly in the time-domain at interface nodes without using the Inverse Fourier Transform, this is in contrast to the previous approach.

The number of operations required to obtain time-domain fields at interface nodes is significantly reduced compared to past approach.

1 2

1 2

30 2cosj R j R j r

z me e eE j I H

R R r

β β β

β− − −

= − + −

( )1 230 2cosj R j R j rmj IH e e H e

yβ β β

φ βη

− − −= + −

1 2

1 2

2 cos30j R j R j r

y mz H e z H e z H eE j I

y R y R y r

β β ββ− − − − += + −

cosβ0H approximation in the frequency domain makes scattered field expressions time-domain compatible by avoiding causality violation in the time-domain scattered field expressions.

o

o

o

Time-domain scattered field expressions:

Frequency-domain scattered field expressions:

𝐼𝐼 =sin(𝑘𝑘(𝑥𝑥 − 𝑥𝑥𝑛𝑛))

sin(𝑘𝑘∆)

NUMERICAL RESULTS Hybrid FDTD method for problems involving straight wire antennas

10

Validation of Time-Domain compatible MoM Code

X

Z

λ @ highest frequency of interest: 4 GHz

Observation Line A straight wire is directed along x.Length of the straight wire is 0.75 cm.

Straight wire antenna of length 0.75 cm with a delta gap voltage source

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Wire antenna radius = 0.005*length

(Cosine modulated Gaussian pulse)

The radiated fields are computed in the time-domain along the observation line, and transformed to the frequency domain, so as to compare results at a desired frequency with commercial MoM

solver to show results in time-domain using time-compatible MoM code are accurate.

Results along the observation line are compared at 2 GHz.

In the frequency domain, the Gaussian pulse has a range of 50 MHz to 4 GHz

with 50 MHz frequency step

Note: The wire can be tilted or have an arbitrary curved shape

Time-Domain compatible MoM Code vs FEKO at 2 GHz

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The radiated fields are computed in the time-domain along the observation line, and transformed to the frequency domain. Results along the observation line are compared at 2 GHz which is the center frequency of the modulated Gaussian pulse.

Hybrid FDTD Technique Example: Tilted Wire Antenna in Transmit Mode

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X

Z

FDTD domain mesh size: λ/20λ @ highest frequency of interest: 8 GHz

Observation Line

A straight wire antenna is inclined to x axis at an angle=45o in xz plane.

Length of the straight wire antenna is 0.1875 cm.

Straight wire antenna of length 0.1875 cm with a delta gap voltage source

7.5 cm pec plate

Interface (26.25 cm x 26.25 cm)

0.1875 cm

Variable Value

λ 3.75 cmλ0 7.5 cmλ/20 0.1875 cm FDTD domain

MoM domain


MoM domain radiated fields at interface nodes are obtained in the time-domain

pec plate touches FDTD domain pml boundaries along x and y

directions which renders it infinite

x

yCross-sectional view

(Cosine modulated Gaussian pulse)

Tilt angle=45o

λ0 @ center frequency : 4 GHz The modulated Gaussian pulse spectrum has a range of 40 MHz to 8 GHz with 40

MHz frequency step.

Ex Comparison at 4 GHz

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Results for radiated fields along observation line are in close agreement 4 GHz is the center

frequency of the modulated Gaussian pulseEfficiency factor =

Number of operations using I.F.T. method at interfaceNumber of operations using hybrid method > 104

Hybrid FDTD Technique Example: Scattering from a Tilted Wire

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X

Z

Mesh size: λ/20λ @ highest frequency of interest: 8 GHz

Observation Line

0.1875 cm

Interface(26.25 cm x 26.25 cm)

A straight wire is inclined to x axis at an angle=45o in xz plane.

Length of the straight wire is 0.1875 cm.

Straight wire antenna of length 0.1875 cm7.5 cm

pec plate

Variable Value

λ 3.75 cmλ0 7.5 cmλ/20 0.1875 cm

FDTD domainMoM domain

k


Incident modulated gaussian pulse

MoM domain scattered field solution at interface nodes is obtained in the time-domain pec plate touches FDTD domain pml

boundaries along x and y directionswhich renders it infinite

x

yCross-sectional view

Tilt angle=45o

Ei

λ0 @ center frequency : 4 GHz In the frequency domain, the modulated Gaussian pulse has a range of 40 MHz to

8 GHz with 40 MHz frequency step.

Ex Comparison at 4 GHz

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Efficiency factor =Number of operations using I.F.T. method at interface

Number of operations using hybrid method > 104

Scattered field results along observation line are in close agreement 4 GHz is the center

frequency of the modulated Gaussian pulse

Handling arbitrary shaped wires with bends

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𝐸𝐸𝑢𝑢1 = −𝑗𝑗30𝐼𝐼𝑚𝑚 �𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅1𝑅𝑅1

−𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗

𝑟𝑟cos 𝛽𝛽𝛽𝛽 − 𝑗𝑗𝑢𝑢1 sin 𝛽𝛽𝛽𝛽 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗(

1𝑟𝑟2

+1

𝑗𝑗𝛽𝛽𝑟𝑟3

Scattered field expressions from basis function current:

𝐸𝐸𝑣𝑣1 =𝑗𝑗30𝐼𝐼𝑚𝑚𝑣𝑣1

𝑢𝑢1 − 𝛽𝛽)𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅1𝑅𝑅1

− 𝑢𝑢1cos 𝛽𝛽𝛽𝛽𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗

𝑟𝑟−

)𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗(𝛽𝛽𝛽𝛽𝛽𝛽𝑟𝑟3

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗(𝑟𝑟𝛽𝛽𝑢𝑢12 + 𝑗𝑗𝑣𝑣12

𝐸𝐸𝑢𝑢2 = −𝑗𝑗30𝐼𝐼𝑚𝑚 �𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅2𝑅𝑅2


𝑟𝑟cos 𝛽𝛽𝛽𝛽 + 𝑗𝑗𝑢𝑢2 sin 𝛽𝛽𝛽𝛽 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗(

1𝑟𝑟2

+1


𝐸𝐸𝑣𝑣2=𝑗𝑗30𝐼𝐼𝑚𝑚𝑣𝑣2

�𝑢𝑢2 +𝛽𝛽𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅2𝑅𝑅2

− 𝑢𝑢2cos 𝛽𝛽𝛽𝛽𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗

𝑟𝑟+

)𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗(𝛽𝛽𝛽𝛽𝛽𝛽𝑟𝑟3

𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗(𝑟𝑟𝛽𝛽𝑢𝑢22 + 𝑗𝑗𝑣𝑣22

Bent wire and time compatible scattered fields

Bent wire and basis function current on the branches

Branch 1 current distribution along u1

Branch 2 current distribution along u2

𝐼𝐼 = 𝐼𝐼𝑚𝑚𝑗𝑗𝑗𝑗𝑗𝑗𝛽𝛽 𝛽𝛽 − ℎ , ℎ > 0

𝐸𝐸𝑢𝑢1 = −𝑗𝑗30𝐼𝐼𝑚𝑚𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅1𝑅𝑅1


𝑟𝑟 cos 𝛽𝛽𝛽𝛽−

𝐸𝐸𝑣𝑣1 =𝑗𝑗30𝐼𝐼𝑚𝑚𝑣𝑣1

𝑢𝑢1 − 𝛽𝛽− 𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅1𝑅𝑅1

− 𝑢𝑢1cos 𝛽𝛽𝛽𝛽− 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗

𝑟𝑟

Time compatible scattered field expressions from branch 1 using virtual current approach

Virtual current approach

Corner

Branch 1

(a) Current distribution on bent wire (b) Virtual current approach

NUMERICAL RESULTS Hybrid FDTD method for problems involving arbitrary shaped wire antennas

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Problem definition Validation of the MoM code

19

Loop diameter: 5.625 mm. Loop cross section radius = 1

100of the loop circumference.

Scattered fields are computed in the time-domain along the observation lines, and transformed to the frequency domain.

Incident modulated gaussian pulse

0.75 cm

(5.625 mm)

In the frequency domain, the modulated Gaussian pulse has a range of 40 MHz to

8 GHz with 40 MHz frequency step.

MoM code vs FEKO results comparisonObservation line 1

20

Line 1: x = 0.75 cm, z = 0 cm

MoM results agree well with FEKO results.Results along the observation line are compared at 4 GHz which is the center

frequency of the modulated Gaussian pulse.


21

Line 2: x = 0.75 cm, z = -3.75 cm




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Line 3: x = 0.75 cm, z = -7.5 cm



Multi-scale problem: Transmit Case

23

(-7.5 cm, 0 cm, 0 cm)(-1.5 cm, 0 cm, 0 cm)

12 cm

Loop is centered at origin, and has a diameter of 5.625 mm Tilt angle of loop is 45o

Interface has 121 x121 nodesNodes on interface have a spacing of λhigh/20 = 0.1875 cmλhigh @ 8 GHz (highest frequency of interest)λhigh = 3.75 cm

Loop cross section radius = 1100

of the loop circumference

Modulated Gaussian pulse spectrum

The modulated Gaussian pulse spectrum has a range of 40 MHz to 8 GHz with 40 MHz

frequency step

Scattered fields are computed in the time-domain along the observation line, and transformed to the frequency domain.

Hybrid FDTD vs FEKO results

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1 GHz

Hybrid method and FEKO results have good comparison.

4 GHz

6 GHz



Multi-scale problem: Receive Case

25

Loop is centered at origin, and has a diameter of 5.625 mm Tilt angle of loop is 45o

Interface has 121 x121 nodesNodes on interface have a spacing of λhigh/20 = 0.1875 cmλhigh @ 8 GHz (highest frequency of interest)λhigh = 3.75 cm

Incident field:θ = 90o

Φ = 0o

Modulated Gaussian pulse

Loop cross section radius = 1100

of the loop circumference

In the frequency domain, the modulated Gaussian pulse has a range of 40 MHz to

8 GHz with 40 MHz frequency step


Hybrid FDTD vs FEKO results

26

Hybrid method and FEKO results have good comparison.

1 GHz 4 GHz

8 GHz



Handling arbitrary shaped surfaces

27

Modified RWG basis function

𝐼𝐼 = 𝐼𝐼𝑚𝑚𝑗𝑗𝑗𝑗𝑗𝑗𝛽𝛽𝜌𝜌±

Example: Sphere discretized into triangular patches

Bent current path is handled using a procedure similar to the one used for bent wires to obtain time compatible scattered fields.

Scattered field expressions from basis function current on a current path:

𝐸𝐸𝑢𝑢1 = −𝑗𝑗30𝐼𝐼𝑚𝑚− �𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅1𝑅𝑅1


𝑟𝑟cos 𝛽𝛽𝛽𝛽− − 𝑗𝑗𝑢𝑢1 sin 𝛽𝛽𝛽𝛽− 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗(

1𝑟𝑟2

+1


𝐸𝐸𝑣𝑣1 =𝑗𝑗30𝐼𝐼𝑚𝑚−

𝑣𝑣1𝑢𝑢1 − 𝛽𝛽−)

𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅1𝑅𝑅1

− 𝑢𝑢1cos 𝛽𝛽𝛽𝛽− 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗

𝑟𝑟−

)𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗(𝛽𝛽𝛽𝛽−

𝛽𝛽𝑟𝑟3𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗(𝑟𝑟𝛽𝛽𝑢𝑢12 + 𝑗𝑗𝑣𝑣12

𝐸𝐸𝑢𝑢2 = −𝑗𝑗30𝐼𝐼𝑚𝑚+ �𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅2𝑅𝑅2


𝑟𝑟cos 𝛽𝛽𝛽𝛽+ + 𝑗𝑗𝑢𝑢2 sin 𝛽𝛽𝛽𝛽+ 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗(

1𝑟𝑟2

+1


𝐸𝐸𝑣𝑣2=𝑗𝑗30𝐼𝐼𝑚𝑚+

𝑣𝑣2�𝑢𝑢2 +𝛽𝛽+ 𝑒𝑒−𝑗𝑗𝑗𝑗𝑅𝑅2

𝑅𝑅2− 𝑢𝑢2cos 𝛽𝛽𝛽𝛽+ 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗

𝑟𝑟+

)𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗(𝛽𝛽𝛽𝛽+

𝛽𝛽𝑟𝑟3𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗(𝑟𝑟𝛽𝛽𝑢𝑢22 + 𝑗𝑗𝑣𝑣22

NUMERICAL RESULTS Hybrid FDTD method for problems involving wire and surface scatterers

28

Problem definitionValidation of the MoM code

29

Voltage gap is located at wire center

x y

z

Wire radius = 150

wire length

Modulated Gaussian pulse spectrum

The modulated Gaussian pulse spectrum has a range of 40 MHz to 8 GHz with 40 MHz

frequency step

Scattered fields are computed in the time-domain along the observation lines, and transformed to the frequency domain.


30

Ez field comparison along observation line 1(x=0, z=0.75 cm).Ex field comparison along observation line 1(x=0, z=0.75 cm).

Modified RWG results agree well with FEKO results.

Results along the observation line are compared at 4 GHz which is the center frequency of the modulated Gaussian pulse.


31

Ez field comparison along observation line 2 (x=3.75cm, z=0.75cm).Ex field comparison along observation line 2 (x=3.75cm, z=0.75cm).




32

Ez field comparison along observation line 3 (x=7.5cm, z=0.75cm).Ex field comparison along observation line 3 (x=7.5cm, z=0.75cm).



Multi-scale problemHelical antenna with ground plate

33

Excitation: delta gap source with modulated Gaussian pulse spectrum (40 MHz – 8 GHz).

The helix has 4 turns, a base diameter of 3.62 cm, and a height of 1.2 cm.

The cross section radius of helical wire is 1100

of the helix height.

Ground plate diameter = 10.2 cm.

12 cm


xy

z

Examples with human model

External antennaInternal

antenna

FDTD region MoM region

Hybrid FDTD vs FEKO results at 1 GHz

34

Ex comparison Ey comparison

Ez comparison


Number of operations using hybrid method > 102 Hybrid method and FEKO results have good comparison.


35


Ez comparison


Number of operations using hybrid method > 102 Hybrid method and FEKO results have good comparison.


36


Ez comparison

Hybrid method and FEKO results have good comparison.Efficiency factor =Number of operations using I.F.T. method at interface


NOVEL FINITE DIFFERENCE METHODCapacitance matrix computation for coated multi-conductor problems

37

Proposed Finite Difference method

38

Laplace′s Equation: 𝛻𝛻2∅ = 0

∅ 1 =∅ 2 + ∅ 3 + ∅ 4 + ∅ 5

4

Handling non – cartesian geometries:

2D stencil for finite difference method

Handling dielectric coated structures

39

Step 1: We express the potential at node 1 in terms of the potential at nodes 1*, and theknown potential ‘V’. Boundary condition at node 1* is used.

Step 2: We express the potential at node 1* in terms of potential at nodes 1, 2, and 3 using quadratic interpolation.

Handling two layer dielectric coated structures

40

εr1 ∅ 𝑃𝑃 −∅ 𝑄𝑄

𝑡𝑡1= εr2

∅ 𝑄𝑄 −∅ 𝑅𝑅𝑡𝑡2

εr2 ∅ 𝑄𝑄 −∅ 𝑅𝑅

𝑡𝑡2= ∅ 𝑅𝑅 −∅ 1

∆𝑝𝑝

ɸ 1 =𝛥𝛥𝑥𝑥 2𝛥𝛥𝑥𝑥

𝛥𝛥𝑥𝑥 + 𝛥𝛥𝛥𝛥 2𝛥𝛥𝑥𝑥 + 𝛥𝛥𝛥𝛥ɸ 𝑅𝑅 +

𝛥𝛥𝛥𝛥 2𝛥𝛥𝑥𝑥𝛥𝛥𝑥𝑥 + 𝛥𝛥𝛥𝛥 𝛥𝛥𝑥𝑥

ɸ 2 −𝛥𝛥𝛥𝛥 𝛥𝛥𝑥𝑥2𝛥𝛥𝑥𝑥 𝛥𝛥𝑥𝑥

ɸ 3

∅ 𝑅𝑅 = 𝑎𝑎∅ 1 + 𝑏𝑏

ɸ 1 =𝛥𝛥𝑥𝑥 2𝛥𝛥𝑥𝑥

𝛥𝛥𝑥𝑥 + 𝛥𝛥𝛥𝛥 2𝛥𝛥𝑥𝑥 + 𝛥𝛥𝛥𝛥(𝑎𝑎∅ 1 + 𝑏𝑏) +

𝛥𝛥𝛥𝛥 2𝛥𝛥𝑥𝑥𝛥𝛥𝑥𝑥 + 𝛥𝛥𝛥𝛥 𝛥𝛥𝑥𝑥

ɸ 2 −𝛥𝛥𝛥𝛥 𝛥𝛥𝑥𝑥2𝛥𝛥𝑥𝑥 𝛥𝛥𝑥𝑥

ɸ 3

Three trapezoid shaped conductors with two layer dielectric coating: Problem definition

41

Narrow channel

Handling narrow channel region

An example of a practical problem in packaging applications.

Potential and Electric Field distribution1V, 0V, 0V excitation of conductors

42

Potential distribution Electric field distribution

𝑄𝑄1 = 𝐶𝐶11∅1 + 𝐶𝐶12∅2 + 𝐶𝐶13∅3

Electric field at nodes on contour enclosing the conductor is used to find enclosed charge.

𝑄𝑄2 = 𝐶𝐶21∅1 + 𝐶𝐶22∅2 + 𝐶𝐶23∅3𝑄𝑄3 = 𝐶𝐶31∅1 + 𝐶𝐶32∅2 + 𝐶𝐶33∅3

Potential and Electric Field distribution0V, 1V, 0V excitation of conductors

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Potential distribution Electric field distribution

𝑄𝑄1 = 𝐶𝐶11∅1 + 𝐶𝐶12∅2 + 𝐶𝐶13∅3

Electric field at nodes on contour enclosing the conductor is used to find enclosed charge.

𝑄𝑄2 = 𝐶𝐶21∅1 + 𝐶𝐶22∅2 + 𝐶𝐶23∅3𝑄𝑄3 = 𝐶𝐶31∅1 + 𝐶𝐶32∅2 + 𝐶𝐶33∅3

Capacitance matrix results

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FD results obtained using coarse mesh and interpolation have good accuracy.

Maximum percentage difference between FD results and Ansys 2D results = 2.7%.

C33 = C11, C23 = C32, C13 = C31 using symmetry

Efficiency factor =Time resource utilized using 2x fine mesh

Time resource utilized using coarse mesh and interpolation > 4

Observations and Conclusions• Magnitude and phase of the scattered field obtained using the proposed hybrid method for multi-

scale problems demonstrated are in close agreement to that obtained using commercial method of moments code.

• The hybrid method provides accurate results for the transmit as well as receive mode of operation.

• Hybrid results have best accuracy at the center frequency of the band (β = β0) , and a range for which the accuracy is found to deviate by about 10% from the accuracy at center frequency is found.

• Results have good accuracy when scatterer (arbitrary wire or wire and surface combination problem) spans multiple FDTD cells.

• The proposed hybrid method is much more efficient (factor>102) when compared to the past approach where Inverse Fourier Transform (I.F.T.) is used at the interface to obtain time domain fields used as source for the FDTD region.

• The proposed finite difference algorithm provides good accuracy and efficiency (factor >4 when compared to 2x fine mesh) for capacitance matrix calculation for dielectric coated, multi-conductor problems where conductor boundaries do not align with grid lines.

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Future Research• Extend the hybrid method to utilize the near field region for arbitrary shaped wires

and surfaces.

• Extend the finite difference code to 3 dimensional problems when longitudinal dimensions are finite.

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References• [1] R. Garg, Analytical and computational methods in electromagnetics. London: Artech House, 2008.• [2] A. F. Peterson, S. L. Ray, and R. Mittra, Computational methods for electromagnetics. New Jersey: IEEE Press, 1997.• [3] D. K. Cheng, Field and wave electromagnetics. Addison-Wesley, 1989.• [4] R. F. Harrington, Field computation by moment methods. New York: The Macmillan Company, 1968.• [5] R. Mittra, Computer techniques for electromagnetics. New York: Hemisphere Publishing Corporation, 1987.• [6] R. F. Harrington, Time-harmonic electromagnetic fields. New Jersey: IEEE Press, 2001.• [7] Kane S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. On Antennas

and Propagation, vol.14, no.3, pp. 302-307, May 1966.• [8] Jean-Pierre Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves”, Journal of Computational Physics 114, 185-200

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Proceedings of the Fourth European Conference on Antennas and Propagation (EuCAP 2010).• [12] Chiara Pelletti, Kadappan Panayappan, Raj Mittra, Agostino Monorchio, “On the hybridization of RUFD algorithm with the DM approach for solving

multiscale problems”, 2010 URSI International Symposium on Electromagnetic Theory (EMTS).• [13] R.G. Martin, A. Salinas, A.R. Bretones, “Time-domain integral equation methods for transient • analysis”, IEEE Antennas and Propagation Magazine, vol. 34, no. 3, June 1992. • [14] E. C. Jordan and W. L Everitt, "Acoustic Models of Radio Antennas," Proc. IRE, 29, 4, 186 (1941).• [15] C. A. Balanis, Advanced Engineering Electromagnetics, 2nd edition, 2012.• [16] J.N. Bringuier, “Multi-Scale Techniques in Computational Electromagnetics”, PhD dissertation, May 2010.• [17] S. Rao, D. Wilton, A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape”, IEEE Transactions on Antennas and Propagation, vol. 30,

issue 3, May 1982.• [18] L. Gurel, O. Ergul, “Singularity of the magnetic-field Integral equation and its extraction”, IEEE Antennas and Wireless Propagation Letters, vol. 4,

issue 1, August 2005.• [19] J.R. Mautz, R.F. Harrington, “A Combined-Source Solution for Radiation and Scattering from a Perfectly Conducting Body”, IEEE Transactions on

Antennas and Propagation, Vol. Ap-27, No. 4, July 1979.• [20] Z. G. Qian, W.C. Chew, “A quantitative study on the low frequency breakdown of EFIE”, • microwave and optical technology letters, Volume 50, Issue 5, May 2008.• [21] C. Pelletti, G. Bianconi, R. Mittra, A. Monorchio, K. Panayappan, “Numerically Efficient • Method of Moments Formulation Valid over a Wide Frequency Band including Very Low Frequencies”, IET Microwaves, Antennas and Propagation, 2012,

Vol. 6, Issue 1, pp. 46-51. • [22] Pasi Yla – Oijala and Matti Taskinen, “Calculation of CFIE Impedance Matrix Elements with RWG and n x RWG Functions”, IEEE Transactions on

Antennas and Propagation, Vol. 51, No. 8, August 2003.• [23] William H. Hayt, John Buck, Engineering Electromagnetics, Mc Graw Hill, 1981.

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