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NUMERICAL CALCULATION OF SPATIALLY
VARIANT ANISOTROPIC METAMATERIALS
ASAD ULLAH HIL GULIB
Master’s Program in Computational Science
APPROVED:
Dr. Raymond Rumpf, Chair
Dr. Rosa Fitzgerald
Dr. Rodrigo Romero
Charles Ambler, Ph.D.
Dean of the Graduate School
Copyright ©
by
Asad Ullah Hil Gulib
2016
Dedication
To my parents, brother and sisters who have been supporting me unconditionally, and be there
for me whenever I needed. To the rest of my family and friends, thanks for always supporting me
and being there for me.
NUMERICAL CALCULATION OF SPATIALLY
VARIANT ANISOTROPIC METAMATERIALS
by
ASAD ULLAH HIL GULIB, B.Sc.EEE
THESIS
Presented to the Faculty of the Graduate School of
The University of Texas at El Paso
in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE
Computational Science Program
THE UNIVERSITY OF TEXAS AT EL PASO
November 2016
v
Acknowledgements
I would like to show my deepest gratitude to all my family members for their
encouragement and support in my higher studies. They have always been there for me, when I
needed them.
My biggest appreciation goes to my mentor and committee chair Dr. Raymond Rumpf who
gave me the opportunity to work with him in Computational Electromagnetics and Computational
Geometry. I am very grateful and lucky to have an awesome professor like him whose guidance,
vast knowledge and experience motivated me in moving forward to achieve my goals. His
excellent mentorship and encouragement helped me to complete research projects and grow as a
future researcher. It was a great pleasure to work under his mentorship.
I am also thankful to Dr. Rosa Fitzgerald and Dr. Rodrigo Romero for agreeing to serve on
my thesis committee and provide me with the mentorship and encouragement to complete this
work. I am also very grateful to all the members of Computation Science Program, especially
Dr. Ming-Ying Leung and Cindy Davis for their support and guidance.
I also gratefully acknowledge the co-operation and help of all my friends and EM Lab
members.
vi
Abstract
3D printing, or additive manufacturing, is rapidly evolving into a mainstream
manufacturing technology that is creating new opportunities for electromagnetics and circuits. 3D
printing permits circuits to fully utilize the third dimension allowing more functions in the same
amount of space and allows the devices to have arbitrary form factors. 3D printing is letting us
discover new physics that is not possible in standard 2D circuits and devices. However, evolving
electromagnetics and circuits into three dimensions introduces some serious problems like thermal
management, interference, and mutual coupling between the components which degrades
performance and hurts signal integrity.
Metamaterials are engineered composites that exhibit extreme electromagnetic properties
and allow extraordinary control over electromagnetic fields. The EM Lab is developing spatially-
variant anisotropic metamaterials (SVAMs) as a solution to mitigate mutual coupling between
components. The concept of SVAMs is to electrically stretch the space between components to
reduce mutual coupling. To do this, alternating layers of different dielectric must bisect adjacent
components. However, the overall dielectric fill must also conform around dozens of electrical
components and be smooth, continuous, and defect free. The research described here is the first
prototype of an algorithm which generates a SVAM infill between all of the electrical components
of a circuit in order to reduce the mutual coupling.
vii
Table of Contents
Acknowledgements ..................................................................................................v
Abstract .................................................................................................................. vi
Table of Contents .................................................................................................. vii
List of Tables ......................................................................................................... ix
List of Figures ..........................................................................................................x
Chapter 1: Introduction ............................................................................................1
Chapter 2: Background Theory and Experiments ....................................................2
2.1 Metamaterials .........................................................................................2
2.2 Anisotropic Metamaterials .....................................................................4
2.3 Space Stretching.....................................................................................7
2.4 Spatially Variant Lattices .....................................................................10
2.5 Field Sculpting In Spatially Varied Anisotropic Medium ...................11
2.6 Prior Research In SVAMs ...................................................................13
2.7 STL File Format ...................................................................................15
Chapter 3: Numerical Methods ..............................................................................16
3.1 Finite-Difference Method ....................................................................16
3.2 Numerical Solution of Laplace Equation .............................................19
3.3 Voxelization .........................................................................................22
3.4 Voronoi Tessellation ............................................................................23
Chapter 4: Algorithm to generate 3D SVAM ........................................................25
4.1 Read STL Files ....................................................................................25
4.2 Voronoi Tessellation ............................................................................27
4.2.2 Voronoi Tessellation ............................................................................29
4.3 Separate Voronoi area of Each Object .................................................30
4.4 Voxelization .........................................................................................30
4.5 Solve Laplace .......................................................................................31
4.6 Generate Iso-Surfaces ..........................................................................32
4.7 Generate SVAM and Create STL File .................................................33
viii
Chapter 5: Conclusion............................................................................................36
5.1 Summary ..............................................................................................36
5.2 Future Work .........................................................................................37
References ......................................................................................................................................38
Vita 41
ix
List of Tables
Table 2.1 [15]: Dielectric Properties of Different Anisotropic Materials ....................................... 6 Table 2.2 [2]: Envelope Correlation Coefficient .......................................................................... 14
x
List of Figures
Figure 1 [15]: Classification of Engineered Materials .................................................................... 3 Figure 2 [15]: Classification of Materials by Size and Frequency ................................................. 4 Figure 3 : Positive and Negative Uniaxial Metamaterial ................................................................ 7 Figure 4 [19] : Two antennas placed in close proximity ................................................................ 8 Figure 5 [19] : Negative Uniaxial Material Parameters in order to stretch the space between two
antennas........................................................................................................................................... 9 Figure 6: Example of Spatially Variant Lattice (a) 3D Spatially Variant all Dielectric Photonic
Crystal [7] (b) Spatially Variant Photonic Crystal [25] ................................................................ 11 Figure 7 [24]: (a) A Microstrip Transmission Line (b) Electric Field Potential of the
Transmission Line ......................................................................................................................... 12
Figure 8 [24]: (a) Field Potential with Anisotropy tilted by 30o (b) Field Potential with Spatially
Variant Anisotropic Medium. ....................................................................................................... 12
Figure 9 [28]: (a) Experimental setup of Microstrip embedded with SVAM (b) Change in
Reflection as the ball is placed in Microstrip with and without SVAM ....................................... 13
Figure 10 [8]: (a) Design of the Cell Phone (b) Antenna Response without SVAM (c) Antenna
Response with SVAM................................................................................................................... 14
Figure 11 [15] : ASCII format of a STL file ................................................................................. 15 Figure 12 [36]: Classification of Finite Difference Method for First Order Derivative ............... 18 Figure 13: Grid Filled With Boundary Values ............................................................................. 20
Figure 14: The Force matrix 𝑓(𝑥, 𝑦)............................................................................................. 21 Figure 15: Numerical Solution of Laplace Equation .................................................................... 22 Figure 16: (a) Stanford Bunny (b) Voxelized picture of Stanford Bunny .................................... 23 Figure 17: Voronoi Tessellation (a) in 2D [11] (b) in 3D [46] ..................................................... 24
Figure 18: Outline of the Algorithm ............................................................................................. 25
Figure 19: Sample Objects ............................................................................................................ 26 Figure 20: Objects after reading STL files ................................................................................... 27 Figure 21: Voronoi Cells Reaching to Infinity ............................................................................. 28
Figure 22: Objects in Bounding Box ............................................................................................ 29 Figure 23: Steps of Voronoi tessellation ....................................................................................... 29
Figure 24: Objects with Their Voronoi-Area................................................................................ 30 Figure 26: Steps of solving Laplace in Big Grid .......................................................................... 32 Figure 28: Steps to Generate Iso-Surfaces around objects in big grid .......................................... 33 Figure 29: Cross section view of generated STL files .................................................................. 34 Figure 30: Cross section of the object after Boolean operation .................................................... 34
Figure 31: Cross section of SVAM and alternating layers together ............................................. 35
Figure 30 : Merging of layers because of closeness of objects..................................................... 36
1
Chapter 1: Introduction
Traditional circuits remain stuck in two dimensions so they are fundamentally limited in
size, weight, form factor and electrical performance. 3D printing is on the edge of revolutionizing
manufacturing [1] and changing the way to design electronic and electromagnetic devices [2].
Because of the high precision of 3D printing, it’s possible to place materials arbitrarily in three
dimensions that enables design of 3D circuits. But, leaving traditional planar topologies also means
leaving the traditional ground plane. Eliminating the ground plane introduces many problems that
will all need to be solved in order to fully realize 3D circuits. One problem is the increased mutual
coupling between circuit elements that will degrade performance and cause noise and interference.
Metamaterials are engineered electromagnetic structures that have opened a new horizon
of possibilities in the field of electromagnetics. Metamaterials have enabled human beings to think
about creating exotic things that we read in science fiction books or see in movies. Metamaterials
show some different exclusive properties like negative refractive index [3], cloaking [4], self-
collimation [5], reversed Doppler Effect [6], etc.
The EM Lab invented a new class of metamaterials called spatially-variant anisotropic
metamaterials (SVAM) [7] [8]. One application of SVAMs is to reduce the mutual coupling
between components that are in close proximity. Anisotropic materials show different dielectric
properties in different directions because their permittivity and/or permeability are tensors instead
of scalar quantities. It was discovered that the near-fields around components embedded in an
anisotropic medium tend to develop furthest in the directions having the highest constitutive
parameters. When the orientation of the anisotropy is varied as a function of position, the near-
field can be sculpted like they were clay [2]. This dissertation is focused on prototyping the
numerical algorithm to calculate the geometry of SVAM in 3D circuits using finite difference
method.
2
Chapter 2: Background Theory and Experiments
2.1 METAMATERIALS
It’s hard to find a universally accepted definition of metamaterials. The simplest definition
that could be used to describe metamaterials is the set of materials that are simply something other
than ordinary materials [9]. Other sources defined metamaterials as: artificial structural elements
designed to achieve unusual electromagnetic properties [10], engineered material that have a
property not found in nature [11], artificial materials that possess unconventional material
parameters [12], artificial structures that display properties not available in natural materials [13],
an engineered electromagnetic structure that may have some exotic electromagnetic properties
over certain frequency bands that are normally not found in nature [14]and exhibit electromagnetic
properties not found in conventional materials [3]. After studying these properties, metamaterials
can be defined like this: ‘Metamaterials are composite materials that is purposely engineered to
provide material properties that are not otherwise attainable with ordinary materials’ [15].
There is no concrete classification of metamaterials [9] . Electromagnetic metamaterials
fall under the region of engineered materials. The lattice spacing of electromagnetic metamaterials
are much smaller than the wavelength of desired operating frequency [2] [15] [9] [3]. Figure 1
shows the classification of engineered materials.
3
Figure 1 [15]: Classification of Engineered Materials
Another classification of metamaterials can be made based on their size and operating
frequency. The photonic band diagram can be used to classify these metamaterials. The ordinary
materials found in nature or synthesized in the lab falls into the extremely subwavelength region.
They are called “pure” materials. The next scale up size is called “mixtures”. Mixtures are the
composition of different ordinary materials, where the ordinary materials still hold their material
properties. Mixtures are still very subwavelength. If the material size is scaled up further, the next
category of engineered material is called “metamaterial”. Metamaterials are composite materials
designed in the lab to serve special purposes. These materials are still subwavelength enough that
waves do not scatter. Metamaterials that have atomic distance≪ 𝜆/4 are called non-resonant
metamaterials and materials that have wavelength ~𝜆/4 are called resonant metamaterials. The
last size scale is photonic crystals where the spacing is large enough to diffract and scatter waves
[15]. In Figure 2, all types of material that takes place in the band diagram according to their
properties are shown.
4
2.2 ANISOTROPIC METAMATERIALS
Often, the bound charges in a material are more easily displaced in certain directions than
other directions. This means that materials can become polarized in a slightly different direction
than the applied electromagnetic field. This is what causes materials to be anisotropic and exhibit
different electromagnetic properties to fields in different directions. Anisotropic materials provide
many different ways to control and manipulate waves [15]. For this reason, anisotropic
metamaterials are one prominent class of metamaterials [16] , the constitutive values of which are
not scalars but tensors, with their principle components taking different values [12]. Dispersion
relations show elliptic or hyperbolic shapes because of this property [17].
The constitutive relation for materials is expressed as,
�⃗⃗� = 𝜀0�⃗� (2.1.1)
Figure 2 [15]: Classification of Materials by Size and Frequency
5
Where,
�⃗� = 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑓𝑖𝑒𝑙𝑑 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦
�⃗⃗� = 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑓𝑙𝑢𝑥 𝑑𝑒𝑛𝑠𝑖𝑡𝑦
𝑎𝑛𝑑 , 𝜀0 = 𝑝𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑎𝑖𝑟
If the material polarization (𝑃) is incorporated with equation 2.1.1, then the relation
becomes,
�⃗⃗� = 𝜀0�⃗� + �⃗� (2.1.2)
The constitutive relation can be expressed in terms of complex relative permittivity (𝜀�̃�)
and susceptibility(𝜒),
�⃗⃗� = 𝜀0𝜀�̃��⃗� = 𝜀0�⃗� + 𝜀0𝜒�⃗� (2.1.3)
Comparing the equations 2.1.2 and 2.1.3,
�⃗⃗� = 𝜀0𝜀�̃��⃗� = 𝜀0(1 + 𝜒)�⃗�
𝜀�̃� = (1 + 𝜒) (2.1.4)
And,
𝜀0�⃗� + �⃗� = 𝜀0�⃗� + 𝜀0𝜒�⃗�
�⃗� = 𝜀0𝜒�⃗� (2.1.5)
The polarization equation for anisotropic materials can be expressed in this form [15],
[
𝑃𝑥(𝜔)
𝑃𝑦(𝜔)
𝑃𝑧(𝜔)
] = 𝜀0 [
𝜒𝑥𝑥(𝜔) 𝜒𝑥𝑦(𝜔) 𝜒𝑥𝑧(𝜔)
𝜒𝑦𝑥(𝜔) 𝜒𝑦𝑦(𝜔) 𝜒𝑦𝑧(𝜔)
𝜒𝑧𝑥(𝜔) 𝜒𝑧𝑦(𝜔) 𝜒𝑧𝑧(𝜔)
] [
𝐸𝑥(𝜔)
𝐸𝑦(𝜔)
𝐸𝑧(𝜔)
] (2.1.6)
Where, 𝜔 is the singular frequency of the source.
The relation between electric susceptibility(𝜒𝑒) and complex relative permittivity(𝜀�̃�) can
expressed as
[𝜀�̃�(𝜔)] = 1 + [𝜒𝑒(𝜔)] (2.1.7)
6
So the constitutive relation between the electric field (�⃗� ) and electric flux density
(�⃗⃗� ) becomes,
[
𝐷𝑥(𝜔)
𝐷𝑦(𝜔)
𝐷𝑧(𝜔)
] = 𝜀0 [
𝜀�̃�𝑥(𝜔) 𝜀�̃�𝑦(𝜔) 𝜀�̃�𝑧(𝜔)
𝜀�̃�𝑥(𝜔) 𝜀�̃�𝑦(𝜔) 𝜀�̃�𝑧(𝜔)
𝜀�̃�𝑥(𝜔) 𝜀�̃�𝑦(𝜔) 𝜀�̃�𝑧(𝜔)
] [
𝐸𝑥(𝜔)
𝐸𝑦(𝜔)
𝐸𝑧(𝜔)
] (2.1.8)
Similarly due to magnetic susceptibility(𝜒𝑚), the constitutive relation between magnetic
field (�⃗� ) and magnetic flux density (�⃗⃗� ) can be written as,
[
𝐵𝑥(𝜔)
𝐵𝑦(𝜔)
𝐵𝑧(𝜔)
] = 𝜀0 [
𝜇𝑥𝑥(𝜔) 𝜇𝑥𝑦(𝜔) 𝜇𝑥𝑧(𝜔)
𝜇𝑦𝑥(𝜔) 𝜇𝑦𝑦(𝜔) 𝜇𝑦𝑧(𝜔)
𝜇𝑧𝑥(𝜔) 𝜇𝑧𝑦(𝜔) 𝜇𝑧𝑧(𝜔)
] [
𝐻𝑥(𝜔)
𝐻𝑦(𝜔)
𝐻𝑧(𝜔)
] (2.1.9)
Where, 𝜇 is complex relative permeability.
So, in anisotropic materials, the material properties such as permittivity(𝜀) and
permeability(𝜇) becomes tensor which means that the fields are no longer independent of each
other [15]. The table 2.1 shows properties of different kinds of anisotropic metamaterials.
Table 2.1 [15]: Dielectric Properties of Different Anisotropic Materials
Anisotropy Dielectric Properties
Isotropic [𝜀̃ 0 00 𝜀̃ 00 0 𝜀̃
]
Uniaxial
[
𝜀�̃� 0 00 𝜀�̃� 00 0 𝜀�̃�
]
𝜀�̃� > 𝜀�̃� positive birefringence
𝜀�̃� < 𝜀�̃� negative birefringence
Biaxial
[
𝜀�̃� 0 00 𝜀�̃� 00 0 𝜀�̃�
]
𝜀�̃� < 𝜀�̃� < 𝜀�̃�
7
2.3 SPACE STRETCHING
From Table 2.1 , the dielectric property of uniaxial materials show that it has only an
ordinary effective permittivity (𝜀𝑜) and an extraordinary effective permittivity (𝜀𝑒). The difference
between extraordinary and ordinary effective permittivity (∆𝜀) is defined as birefringence.
∆𝜀 = 𝜀𝑒 − 𝜀𝑜 (2.1.10)
If the extraordinary effective permittivity is greater than ordinary effective
permittivity(∆𝜀 > 0), it’s called positive birefringence. A uniaxial lattice with a lower effective
permittivity in two directions and a higher effective permittivity in third direction must exhibit
positive birefringence. This requires the electric fields along two axis to be perpendicular to the
interfaces and field along third axis is parallel to interfaces. Any array of cylinders satisfies these
requirements. If the extraordinary effective permittivity is less than ordinary effective permittivity
(∆𝜀 < 0) is called negative birefringence. A uniaxial lattice with a high permittivity in two
directions and low effective permittivity in the third must exhibit negative birefringence. This
requires the electric fields along two axes to be parallel to interfaces while the field along the third
to be perpendicular to interfaces. Any array of planar slabs satisfies these requirements [15] [18].
Figure 3 : Positive and Negative Uniaxial Metamaterial
8
Space stretching means to magnify the distance between components in order to move
them further apart. Electrical circuits contain components in close proximity which degrades the
performance. It’s possible to electrically stretch the distance between two components by varying
the dielectric properties of the medium between components without varying the physical distance
between them. The process of electrically stretching the space is called transformation optics.
Figure 4 [19] : Two antennas placed in close proximity
Figure 4 is showing two antennas placed in close proximity. In order to stretch the space
in z direction with a factor of ‘a’, the coordinate transform parameters become,
𝑥′ = 𝑥
𝑦′ = 𝑦
𝑧′ = 𝑧/𝑎
Transformation optics is used to calculate permittivity and permeability functions that will
stretch the coordinate in the prescribed manner. So the permeability and permittivity tensors found
from transformation optics are,
[𝜇′] = [𝑎𝜇 0 00 𝑎𝜇 00 0 𝜇/𝑎
] (2.1.11)
9
[𝜀′] = [𝑎𝜀 0 00 𝑎𝜀 00 0 𝜀/𝑎
] (2.1.12)
The tensors have two same elements in diagonal and a different third element. From the
dielectric properties it can be seen that the material is a uniaxial material. If 𝑎 > 0 , then the third
element of diagonal becomes smaller than the other two values and the material becomes negative
uniaxial material [19] [15].
Figure 5 [19] : Negative Uniaxial Material Parameters in order to stretch the space between two
antennas
If 𝜀1 and 𝜀2 are permittivities of alternating layers of negative uniaxial material, then the
expression of effective ordinary permittivity (𝜀𝑜) and effective extraordinary permittivity
(𝜀𝑒) derive from effective medium theory is [15],
1
𝜀𝑒=
𝑓
𝜀1+
1 − 𝑓
𝜀2 (2.1.13)
𝜀𝑜 = 𝑓𝜀1 + (1 − 𝑓)𝜀2 (2.1.14)
Where, 𝑓 is the volumetric fill factor of 𝜀1.
10
Using transformation optics, the effective refractive index of the material (𝜀) and the
stretching factor (𝑎) can be derived in terms of 𝑓, 𝜀1, and 𝜀2
𝜀 = √𝑓𝜀1 + (1 − 𝑓)𝜀2
𝑓𝜀1−1 + (1 − 𝑓)𝜀2
−1 (2.1.15)
𝑎 = √1 +𝑓(1 − 𝑓)(𝜀1 − 𝜀2)2
𝜀1𝜀2 (2.1.16)
The maximum value of the stretching factor (𝑎) is found if the fill factor (𝑓) is 0.5. This
means optimized stretching between components can be obtained without knowing the value of
𝜀1 and 𝜀2 [15].
2.4 SPATIALLY VARIANT LATTICES
Electromagnetic fields cannot be controlled inside a homogeneous medium. In order to
control the fields, there must be some kind of inhomogeneous features such as material interfaces,
curved surfaces, gradients, or more [20]. Macroscopically, photonic crystals [21] [22] and
metamaterials [23] with uniform lattices provide control over electromagnetic fields. To get more
control over electric fields, the lattices have to be made macroscopically inhomogeneous. In order
to do that, the lattices must be bent, twisted or spatially varied. But this must be done without any
deformation of unit cells that would weaken or destroy electromagnetic properties. So, a spatially
variant lattice can functionally vary any attributes of a periodic structure without destroying its
properties [15].
Spatially variant lattice is designed as it spatially varies the orientation of periodic structure
of the device introducing minimum discontinuities throughout the volume. Avoiding
discontinuities reduces scattering and field concentrations and improve the performance of the
device. Directional phenome like band gaps, anisotropy and dispersion are exploited by spatially
varying the orientation of the lattice [7]. Figure 6(a) shows a spatially variant all dielectric photonic
crystal device. The device displays self-collimation effect of flowing an unguided beam around a
90o bend without diffracting or scattering [20] [24]. The lattice shown in Figure 6(b) direct the
11
flow of light around a 90o bend. The device permits one polarization to go through sharp bends
while other polarization propagates straight through the lattice [25].
Figure 6: Example of Spatially Variant Lattice (a) 3D Spatially Variant all Dielectric Photonic
Crystal [7] (b) Spatially Variant Photonic Crystal [25]
2.5 FIELD SCULPTING IN SPATIALLY VARIED ANISOTROPIC MEDIUM
To assess the field sculpting concept, it was first used to sculpt the field around a microstrip
transmission line [26]. Microwave transmission line is a distributed-parameter network, where
voltages and currents can vary in magnitude and phase over its length [27]. Finite difference
method is used to calculate the electromagnetic field surrounding the line. A transmission line is
often schematically represented as a two-wire line since transmission lines have at least two
conductors. One of the conductor of microstrip is ground plane and another is signal line. For this
example, the microstrip was completely embedded in a dielectric medium.
First, the transmission line was embedded within an isotropic dielectric substrate. Since the
substrate is isotropic, the permittivity tensor reduces to a single scalar number because the
permittivity is the same no matter the direction of the applied electric field. Figure 7 shows the
electric near-field around the microstrip line embedded in the isotropic medium.
12
Figure 7 [24]: (a) A Microstrip Transmission Line (b) Electric Field Potential of the Transmission
Line
Next, the medium around the line was made anisotropic and the orientation of the
anisotropy was rotated counter clockwise by 45 degrees. The resulting near-field is shown in
Figure 8(a). It can be observed that the near-field prefers the direction with highest permittivity.
Last, the orientation of the anisotropy was varied with height above the ground plane. It
can be seen from Figure 8(b) that the field still followed the anisotropy through the spatial variance
[26] [2].
Figure 8 [24]: (a) Field Potential with Anisotropy tilted by 30o (b) Field Potential with Spatially
Variant Anisotropic Medium.
13
2.6 PRIOR RESEARCH IN SVAMS
2.6.1 Microstrip Transmission Line Experiment
Many solutions have been proposed to reduce coupling and cross talk, but the solutions use
metal which can produce even more problems in the framework of a 3D system because the
decoupling structures themselves occupy space and limit how closely components can be spaced
[2].
The scattering parameters of an ordinary microstrip were measured. Again the parameters
were measured with a metal ball paled in close proximity of the microstrip. The near field of the
transmission line gets perturbed affecting the signal in the line. Then the microstrip was embedded
in SVAM with dielectric tensor designed so that the near field tilted away from the ball. The near
field was less perturbed, affecting the transmission line much less. [2] [15] [28].
Figure 9 [28]: (a) Experimental setup of Microstrip embedded with SVAM (b) Change in
Reflection as the ball is placed in Microstrip with and without SVAM
2.6.2 Decoupling Adjacent Antennas
Various methods have been proposed in literatures in order to mitigate coupling of antennas
placed in close proximity. But they introduce more complicated feeding networks to the device
which increase manufacturing cost and time to optimize antenna system [2].
14
Figure 10 [8]: (a) Design of the Cell Phone (b) Antenna Response without SVAM (c) Antenna
Response with SVAM
Two antennas were designed into a mockup cell phone and tested in the lab to observe the
electromagnetic interaction between the antennas. This was repeated with and without the SVAM
in place. Ansys HFSS was used to design the device and a dramatic decrease was observed in the
envelope correlation coefficient (ECC) [2].
The data found from the experiment are summarized in table 2.2
Table 2.2 [2]: Envelope Correlation Coefficient
Condition ECC
Two Antennas without SVAM 0.65
Two Antennas with SVAM 0.018
15
2.7 STL FILE FORMAT
There is a controversy about what “STL” stands for. It’s commonly thought that “STL” is
the abbreviation of the word “Stereolithography”, but sometimes it is referred as “Standard
Triangle Language” or “Standard Tessellation Language”. The founder of “3D Systems” and the
inventor of stereolithography, Chuck Hull, reports that the file extension and acronym stands for
stereolithography [29] [30].
The STL file contains the data for describing the layout of an object in three dimensions.
The surface of the object is tessellated or broken into small triangles which are called facets. Each
triangle facet is described by the outward normal of the facet and three corner co-ordinate points
of the triangle. STL file provides complete listing of the triangles x, y and z co-ordinates and
perpendiculars [30]. The facet normal must follow the right hand rule and point outward from
object. The adjacent triangles must have two common vertices. STL files can be in ASCII or binary
format. Binary files are most common because they are more compact [15]. Figure 11 shows
example of an ASCII file format of STL.
Figure 11 [15] : ASCII format of a STL file
16
Chapter 3: Numerical Methods
3.1 FINITE-DIFFERENCE METHOD
In mathematics, finite-difference methods (FDM) are numerical methods for solving
differential equations by approximating them with difference equations, in which finite differences
approximate the derivatives [31]. Today, FDMs are one of the popular approach to numerical
solutions of partial differential equations [32] because finding approximate solutions are easier
than the exact solution. So numerical techniques become the main and popular way that differential
equations are solved. The algebraic estimate to finite difference equations is to replace a derivative
in deferential equation with an approximation. So differential equations become purely algebraic
equations and the goal is to solve resulting algebraic equations to get approximate solutions [33].
The first order derivative of a function can be defined by taking limit of a finite difference
quotient
𝑓′(𝑥) = 𝑙𝑖𝑚∆𝑥→0
𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥)
∆𝑥 (3.1.1)
For relatively small∆𝑥, we can replace this limit inequality with simply an
approximation. So for small ∆𝑥, the finite difference approximation of 𝑓 at x becomes,
𝑓′(𝑥) ≈𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥)
∆𝑥 (3.1.2)
In conclusion, finite difference method is a difference approximation for the derivative of
a quantity at one grid point in terms of the values at neighboring points [34].
Taylor series have been widely used to study the behavior of numerical approximation of
differential equations. Taylor series expansion of a function 𝑓(𝑥) is
𝑓(𝑥) = ∑(𝑥 − 𝑥𝑖)
𝑛
𝑛!(𝜕𝑛𝑓
𝜕𝑥𝑛)𝑖
∞
𝑛=0
(3.1.3)
Expanding the function 𝑓 at 𝑥𝑖+1 about the point 𝑥𝑖
𝑇1: 𝑓𝑖+1 = 𝑓𝑖 + ∆𝑥 (𝜕𝑓
𝜕𝑥)𝑖+
(∆𝑥)2
2(𝜕2𝑓
𝜕𝑥2)
𝑖
+(∆𝑥)3
6(𝜕3𝑓
𝜕𝑥3)
𝑖
+ ⋯ (3.1.4)
The equation can be rearranged as follows
17
𝑇1 ⇒ 𝑓𝑖+1 − 𝑓𝑖
∆𝑥− (
𝜕𝑓
𝜕𝑥)𝑖=
∆𝑥
2(𝜕2𝑓
𝜕𝑥2)
𝑖
+(∆𝑥)2
6(𝜕3𝑓
𝜕𝑥3)
𝑖
+ ⋯ (3.1.5)
This form of finite difference equation is called forward Euler formula or forward
difference method. Forward Euler corresponds to truncating the Taylor series after the second term
and the error introduced during truncation is called truncation error. The truncation error can be
defined as the difference between the partial derivative and the finite difference representation
[35]. So, the truncation error𝒪(∆𝑥),
𝒪(∆𝑥) =∆𝑥
2(𝜕2𝑓
𝜕𝑥2)
𝑖
+(∆𝑥)2
6(𝜕3𝑓
𝜕𝑥3)
𝑖
+ ⋯ (3.1.6)
In general, the exact truncation error is not known. So, as∆𝑥 → 0, the behavior of the error
can be characterized as,
(𝜕𝑓
𝜕𝑥)𝑖=
𝑓𝑖+1 − 𝑓𝑖∆𝑥
+ 𝒪(∆𝑥) (3.1.7)
Again, expanding the Taylor series for function 𝑓 at 𝑥𝑖−1 about the point 𝑥𝑖
𝑇2: 𝑓𝑖−1 = 𝑓𝑖 − ∆𝑥 (𝜕𝑓
𝜕𝑥)𝑖+
(∆𝑥)2
2(𝜕2𝑓
𝜕𝑥2)
𝑖
−(∆𝑥)3
6(𝜕3𝑓
𝜕𝑥3)
𝑖
+ ⋯ (3.1.8)
Rearranging the function,
𝑇2 ⇒ 𝑓𝑖 − 𝑓𝑖−1
∆𝑥− (
𝜕𝑓
𝜕𝑥)𝑖= −
∆𝑥
2(𝜕2𝑓
𝜕𝑥2)
𝑖
+(∆𝑥)2
6(𝜕3𝑓
𝜕𝑥3)
𝑖
+ ⋯ (3.1.9)
This is the backward Euler or backward finite difference formula. And we can generalize
the truncation error as ∆𝑥 → 0
(𝜕𝑓
𝜕𝑥)𝑖=
𝑓𝑖 − 𝑓𝑖−1
∆𝑥+ 𝒪(∆𝑥) (3.1.10)
Higher order approximation can be made by combining the forward and backward finite
difference equations.
𝑇1 − 𝑇2 ⇒𝑓𝑖+1 − 𝑓𝑖−1
2∆𝑥− (
𝜕𝑓
𝜕𝑥)𝑖=
(∆𝑥)2
6(𝜕3𝑓
𝜕𝑥3)
𝑖
+ ⋯ (3.1.11)
This is called central difference method. This method converges quadratically for an
equally spaced grid [35]. As ∆𝑥 → 0
18
(𝜕𝑓
𝜕𝑥)𝑖=
𝑓𝑖+1 − 𝑓𝑖−1
2∆𝑥+ 𝒪(∆𝑥2) (3.1.12)
Figure 12 [36]: Classification of Finite Difference Method for First Order Derivative
Similarly, finite difference approximation for higher order derivatives can be obtained. The
approximation for second order derivatives are:
Second order central difference:
(𝜕2𝑓
𝜕𝑥2)
𝑖
=𝑓𝑖+1 − 2𝑓𝑖 + 𝑓𝑖−1
∆𝑥2+ 𝒪(∆𝑥2) (3.1.13)
Second order forward difference:
(𝜕2𝑓
𝜕𝑥2)
𝑖
=𝑓𝑖+1 − 2𝑓𝑖+1 + 𝑓𝑖
∆𝑥2+ 𝒪(∆𝑥) (3.1.14)
Second order backward difference:
(𝜕2𝑓
𝜕𝑥2)
𝑖
=𝑓𝑖 − 2𝑓𝑖−1 + 𝑓𝑖−2
∆𝑥2+ 𝒪(∆𝑥) (3.1.15)
19
3.2 NUMERICAL SOLUTION OF LAPLACE EQUATION
Mathematically the Laplace equation is expressed as
∆2𝑢 = 0 (3.2.1)
For 2D Cartesian co-ordinates, Laplace equation becomes
𝜕2𝑢
𝜕𝑥2+
𝜕2𝑢
𝜕𝑦2= 0 (3.2.2)
And for Cartesian 3D co-ordinates,
𝜕2𝑢
𝜕𝑥2+
𝜕2𝑢
𝜕𝑦2+
𝜕2𝑢
𝜕𝑧2= 0 (3.2.3)
From the above declaration, it can be seen that Laplace equation is a second order partial
differential equation. The implication of second order derivative is that it quantifies curvature. But
as the second order derivative is set to zero, the function satisfying Laplace’s equation varies
linearly [36]. So Laplace equation can be used to fill the grid linearly between two boundary
values. So knowing the boundary values, Laplace equation can be used to fill the intermediate grid
points with linearly varying numbers. By following steps describe how solution to Laplace
equation can be used as number filler inner [36].
Use finite-difference method to express Laplace’s equation in matrix form.
∆2𝑢 = 0
𝜕2𝑢(𝑥, 𝑦)
𝜕𝑥2+
𝜕2𝑢(𝑥, 𝑦)
𝜕𝑦2= 0
𝐷𝑥2𝑢 + 𝐷𝑦
2𝑢 = 0 (3.2.4)
𝐿𝑢 = 0 (3.2.5)
Where,
𝐷𝑥2 =
𝜕2𝑢(𝑥, 𝑦)
𝜕𝑥2 𝐷𝑦
2 =𝜕2𝑢(𝑥, 𝑦)
𝜕𝑦2 (3.2.6)
𝐿 = 𝐷𝑥2 + 𝐷𝑦
2 (3.2.7)
20
Build a function that contains the known values of the grid (boundary values). We can also
call it 𝑏(𝑥, 𝑦) containing the boundary values. Figure 2.6 shows how the function should look like.
The grid contains values of the points where it forced to be the boundary values. All other grid
points are set to zero.
Figure 13: Grid Filled With Boundary Values
Specify the area where Laplace has to interpret the values. In this algorithm, we are calling
it a force matrix 𝑓(𝑥, 𝑦). In the force matrix, the boundary values are set to ‘1’ at a boundary value
and the other grid point values set to ‘0’.
21
Figure 14: The Force matrix 𝑓(𝑥, 𝑦)
The Force matrix has to be incorporated with boundary values into Laplace’s equation. To
do the incorporation, we need an identity matrix of the size of matrix L and take the diagonal of
the force matrix. The incorporation is the by the equation,
𝐿′ = 𝐹 + (𝐼 − 𝐹)𝐿 (3.2.8)
Where,
𝐼 = 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑚𝑎𝑡𝑟𝑖𝑥 𝐿
𝐹 = 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝑓(𝑥, 𝑦)
And
𝑏′ = 𝐹 (3.2.9)
This is the last step to solve Laplace’s equation.
𝐿′𝑢 = 𝑏′ (3.2.10)
22
𝑢 = (𝐿′)−1𝑏′ (3.2.11)
Equation 3.2.11fills the intermediate points between boundaries to fill with linearly varying
numbers. Figure 15 shows how the grid point looks like after solving the Laplace equation.
Figure 15: Numerical Solution of Laplace Equation
3.3 VOXELIZATION
Voxelization describes the process of turning a scene representation consisting of discrete
geometric entities (e.g. triangles) into a three-dimensional regular spaced grid. Each cell of the
grid encodes specific information about the scene. The information carried by voxelization
depends on the type of voxelization. Binary voxelization cell stores whether geometry is present
in this cell or not, multi-valued voxelization cell store information like material or normal,
boundary voxelization encodes the object surfaces only, whereas solid voxelization captures the
interior of a model as well [37]. Slicing method [38] [39], custom rasterization [40], binary scene
23
voxelization [41], depth peeling method [42], conservative rasterization [43], Atlas based
voxelization [44] are different methods of representation of a vexelized scene.
Figure 16: (a) Stanford Bunny (b) Voxelized picture of Stanford Bunny
3.4 VORONOI TESSELLATION
Voronoi algorithm is the process of partitioning a plane into sections depending on the
distance of the points in an explicit subset of the plane. The set of points are specified early, and
for each point Voronoi algorithm finds a region consisting of all points closer to that point than to
any other. The regions are called Voronoi cells [45]. The Voronoi diagram includes all the Voronoi
cells.
24
Figure 17: Voronoi Tessellation (a) in 2D [11] (b) in 3D [46]
25
Chapter 4: Algorithm to generate 3D SVAM
The circuit components are given input to the algorithm in STL file format. Then the STL
files of the objects are read and separate the area between the objects. After that, Laplace is solved
inside the Voronoi area and generate alternating layers of dielectric material. The flow diagram of
the algorithm is shown below.
Figure 18: Outline of the Algorithm
4.1 READ STL FILES
Matlab is used to build the algorithm. Several Matlab functions are available to extract
information from STL files. This algorithm used the function “stlRead.m” to read the STL files
[47]. The function returns vertex, facet and facet normal information after reading the STL file.
26
As a sample, we have taken the STL file of a torus, cylinder, cube, cone, and sphere as input.
Blender [48] is used to generate the STL files. Figure 19 shows the sample input files.
Figure 19: Sample Objects
The algorithm works with one object at a time and reads its vertex and facet information.
The output of each object in Matlab is shown in Figure 20.
27
Figure 20: Objects after reading STL files
4.2 VORONOI TESSELLATION
Voronoi algorithm is used to separate the area between objects which will be filled with
alternating layers of dielectrics. While separating the areas, it’s important to ensure that the area
of one object doesn’t intersect with others. Area overlapping can create problems while 3D printing
the model.
To perform Voronoi tessellation, the vertices of all objects should be saved in an array.
Matlab has a built-in function “voronoin.m” that performs the 3D Voronoi tessellation. The
function returns Voronoi vertices and Voronoi cells of the Voronoi diagram for an input array of
3D points [49]. But while performing Voronoi algorithm with input objects, some Voronoi cells
reach to infinity.
28
Figure 21: Voronoi Cells Reaching to Infinity
The solution of this problem is to create a bounding box around objects that will restrict
the Voronoi cells reaching to infinity.
4.2.1 Create Bounding Box
There are two ways to add a bounding box around objects. First option is to take STL file
of a box as input along with STL files of objects. This option needs to make sure that the box
doesn’t intersect with the edges of the objects. There must be safer distance between the facets of
the bounding box and the edges of the objects. Unless, it will create problem generating SVAM
layers and as well as 3D printing the final design.
The second way to create a bounding box is by taking maximum and minimum axis values
from the vertex information of the objects. Then bounding box is created by taking twice of the
maximum and minimum values along each axis. This is the safer way to create the bounding box,
because a safe distance between the edges of objects and facets of bounding box is ensured. In this
algorithm, the second method is followed to create the bounding box. Figure 22 shows the objects
inside the bounding box.
29
Figure 22: Objects in Bounding Box
4.2.2 VORONOI TESSELLATION
The bounding box limits the Voronoi cells going to infinity. So now it’s safer now to
perform Voronoi tessellation. Figure 23 shows different steps of Voronoi tessellation.
Figure 23: Steps of Voronoi tessellation
30
4.3 SEPARATE VORONOI AREA OF EACH OBJECT
The Voronoi diagram contains Voronoi area of all objects, it doesn’t show the separated
area for each object. In order to separate the Voronoi area of each object, we need to find out the
corresponding Voronoi vertices and Voronoi cells for each vertex of the object. shows the objects
with their corresponding Voronoi area after separating from Voronoi diagram.
Figure 24: Objects with Their Voronoi-Area
4.4 VOXELIZATION
In order to solve Laplace using finite difference method, the object needed to be defined
into a regularly spaced grid. Voxelization coverts surface meshes of the components into objects
on a 3D grid. The grid needs to be defined in order to voxelize the object. For Voxelization, small
grids were created for each of the objects and each Voronoi areas. Finally, a big grid was created
to place the objects and their Voronoi area.
The dimension of each axis is required to define a grid. The small grid was created by
measuring the difference between maximum and minimum axis values of each object. Small grids
for Voronoi areas were created by using the same method. From the maximum and minimum axis
31
values of the Voronoi areas of objects, the maximum and minimum values were taken to define
the dimension of big grid.
Mathworks website has some excellent function for converting surface meshes to points
on a 3D grid. This algorithm used “VOXELISE.m” function which performs voxelization of a
closed triangular polygon mesh [50]. The function takes coordinate values of the grid and STL file
of the object (and Voronoi area) as input and returns voxelized output of the object (and Voronoi
area) in the defined grid.
4.5 SOLVE LAPLACE
Solution to Laplace equation requires second order derivative of each grid point. EM Lab
has developed a function called ‘fdder3d.m’ which solves first and second order derivatives for
Cartesian co-ordinate system. This function uses finite difference approximation to solve the
derivatives. It uses the most efficient method to solve the finite difference approximations.
The objects and Voronoi areas were brought into the big grid and Laplace is solved in the
big grid. The grid points of all objects are set to ‘1’ and Voronoi areas are set to ‘0’. Previously
described numerical method is used to solve Laplace in the big grid. Figure 25 shows some steps
of solving the Laplace equation. The yellow color is representing ‘1’ where the object is present
and the outside voronoi grid is blue color representing grid value ‘0’. The intermediate grid points
are filled with faded yellow and blue representing linearly varying numbers from ‘1’ to ‘0’. Laplace
is solved not for the whole big grid, it’s solved only in the Voronoi area regions.
32
Figure 25: Steps of solving Laplace in Big Grid
4.6 GENERATE ISO-SURFACES
After solving Laplace, the intermediate grid points between object and Voronoi area are
filled with numbers from ‘0’ to ‘1’. It enables to create iso-surfaces around objects using different
iso-values from ‘0’ to ‘1’. The iso-surfaces will be used to create alternating layers of anisotropic
materials. Figure 26 shows different steps of creating iso-surfaces around the objects in the big
grid.
33
Figure 26: Steps to Generate Iso-Surfaces around objects in big grid
4.7 GENERATE SVAM AND CREATE STL FILE
Several Matlab functions are available that converts the vertices and facets of objects into
STL files. In this algorithm, ‘stlwrite.m’ function is used to make STL files [51]. This function
takes the facet and vertices information and the name of the output STL file as input. Facet and
vertex information can be given separately or in FV structure format.
Using the vertex and facet information of iso-surfaces a STL file of the final SVAM object
is created. Figure 27 shows the cross section of the created STL files.
34
Figure 27: Cross section view of generated STL files
Blender is used to visualize the STL files [48]. A Boolean operation is done in order to fill
the alternating layers of the STL file with ‘1’. Figure 28 shows the alternating layers after Boolean
operation.
Figure 28: Cross section of the object after Boolean operation
The combination of two STL files together looks like this,
35
Figure 29: Cross section of SVAM and alternating layers together
36
Chapter 5: Conclusion
5.1 SUMMARY
The work shown in this dissertation develops an algorithm to build 3D circuits surrounded
by SVAM. STL files of the circuit components are taken as input and after reading the files, the
area between objects were divided using Voronoi Algorithm. Laplace is solved in order to fill the
grid points with linearly varying numbers. Then iso-surfaces were created around objects which
will be filled with alternating layers of anisotropic material (SVAM). After that, STL file is created
for the objects with their iso-values.
The main drawback of this 3D circuit is the irreplaceability of the circuit components. As
the components are surrounded by SVAM materials, so it’s not possible to replace a single
component of the system if it gets damage. The whole system needs to be replaced if a single
component is broken.
As the objects come closer, the distance between them decreases and so there is not enough
space to generate SVAM layers. The layers start merging as the objects get close and closer. Figure
30 shows the merging effect of layers due to closeness of the objects.
Figure 30 : Merging of layers because of closeness of objects
37
In Figure 30(a), the distance between objects are enough that SVAM layers are generated
properly. As we decreased the distance between objects, Figure 30(b) shows the SVAM layers get
merged with the layers of neighboring objects. Figure 30(c) shows that there are only one layer
around the object in the middle due to decrease of distance between objects.
5.2 FUTURE WORK
This design only provides the algorithm to generate SVAM around objects in 3D. But to
complete the circuit, electric wiring between the circuit components need to be done. It’s also
necessary to make a connection of circuit components with ground and power supply. It will be a
great challenge to make electric connection of the devices because outside area of the devices is
filled with SVAM.
It’s also not efficient to use finite difference method to make the alternating layers because
it doesn’t give smooth shape of the layers. High resolution simulation was done to get smooth
shapes but the result was not satisfactory. Also the merging effect increases as the objects come
closer. So, Finite Element Method can be solution to avoid merging effect and get smoother shape
of the final design.
38
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41
Vita
Asad Ullah Hil Gulib, the youngest son of Md. Rafiq Ullah and Shahela Begum, was born
on November 01, 1987 in Bangladesh. He completed his Bachelor in Electrical and Electronic
Engineering from Ahsanullah University of Science & Technology, Bangladesh in 2010. His
undergraduate research was ‘Design of a dual mode Instant Power Supply (IPS)’. After graduation,
he worked at Ahsanullah University of Science and Technology as a Lecturer (part-time) and at
North South University as a Lab Instructor. He pursued his Masters in Computational Science
from year 2016 at University of Texas at El Paso. He has been a teaching assistant for several
undergraduate and graduate courses and also been a tutor on Mathematics Resource Center for
Student (MaRCS) at University of Texas at El Paso.
Asad’s research is focused on Computational Electromagnetics and Computational
Geometry. He is working on developing an algorithm to design a 3D circuit enclosed by SVAM
that will minimize the mutual coupling between circuit components.
Permanent address: 716 W Yandell Dr. Apt#1
El Paso, Texas, 79902
This dissertation was typed by the author.
