IPN Progress Report 42-237•

May 15, 2024

Design of Antennas from Primitive Shapes
Using Genetic Algorithms
Julie Rolla,∗ Bryan Reynolds,† Dylan Wells,† Jacob Weiler,† Amy Connolly,† and Ryan
Debolt†

ABSTRACT. — This report presents developments to a genetic algorithm that evolves

antenna designs from primitive shapes. The gain patterns of individuals are evaluatedusing XFdtd electromagnetic simulation software, and fitness is evaluated bycomparison to a target gain pattern. This work is an update to a prior reportdemonstrating evolution of 3D structures to target geometries. The algorithm was
updated to evolve antennas by adding feeds, early shorting prevention, and simulation
components. Different fitness functions were developed, and differences in convergenceto similar gain patterns was explored. Computational efficiency was also improved byconverting to an asynchronous steady-state algorithm and incorporating, among otherimprovements, an elite method, and a diversity forcing feature. The algorithm wasable to evolve antennas to match the desired gain pattern using each fitness function.
The algorithm was run for a single-frequency (300 MHz) design and for a broadband
case (200 MHz to 800 MHz). Later versions of the algorithm will utilize fitness
functions connected to science simulation software to generate designs optimized toscience outcomes. Future improvements to the algorithm to facilitate more complex
designs and to reduce computation time are also discussed.

I. Introduction
This report introduces an evolutionary algorithm that evolves antennas from primitive
shapes to specified gain patterns without predefined antenna topologies (i.e., dipole orbiconical). It expands upon the previous August 2023 Interplanetary Network
Progress Report “Design of 3D Antenna Geometries Using Genetic Algorithms,”∗ Tracking System and Applications Section.
† Department of Physics, Center for Cosmology and AstroParticle Physics, The Ohio State University.

The research described in this publication was carried out by the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and Space Administration.
© 2024 All rights reserved.

1where a method for evolving 3D structures using genetic algorithms (GAs) was
developed [1]. The initial report demonstrated an algorithm capable of creating 3D
structures that matched predefined target shapes, marking an initial step toward thegoal of designing optimized antennas for specific scientific outcomes. The method
consisted of assembling geometric primitives and evaluating fitness using costfunctions tied to the design’s similarity to a target shape. This work builds on theprevious GA by implementing cost functions that are derived from performance
metrics based on antenna radiation patterns produced in XFdtd, an electromagnetic
finite difference in the time domain simulation software by Remcom [2]. Individuals
are evaluated on their similarity to a predefined target gain pattern or sensitivity in
specified directions. Additionally, several enhancements to the previous GA were madeto improve the speed of converging to a high-quality solution and consequently reduce
computation time. Future versions of the algorithm will evolve toward complex science
goals by integrating experiment simulation software and broadening its capabilities togenerate antennas with more complex designs.
This work was originally motivated by astrophysics experiments dependent on highly
sensitive detectors, although the potential impact goes beyond astrophysics byreducing instrument development overhead while adhering to constraint
considerations. The design of such instruments poses a substantial challenge,
especially when restricted by scheduling and cost limitations. These optimization
tasks typically occur in high-dimensional parameter spaces with multiple objectives
and are computationally prohibitive to solve using traditional techniques; however,using innovative Artificial Intelligence (AI) optimization algorithms, such as GAs, vast
parameter spaces can be realistically explored. This report delves into utilizing GAs todesign antennas optimized for specific antenna responses, with future goals ofmultiobjective optimization tied to sensitivity to science outcomes. GAs arecomputational heuristics that mimic evolutionary principles to efficiently identify
solutions to defined problems. Although this endeavor has the potential to expand toa wide range of detector technologies, the immediate focus is on enhancing sensitivity
for radio observations, including antennas for the Global Navigation Satellite System(GNSS) [3, 4, 5], passive sounding [6, 7, 8], and low-frequency astronomy radio signals,
including radio emissions from extrasolar planets [9, 10, 11], cosmic ray electrons andcosmic magnetic fields [12, 13, 14, 15, 16], and the highly redshifted neutral hydrogen
hyperfine line [17, 18, 19, 20]. By addressing a critical need for innovative early-stage
design optimization tools, this work aims to increase the scientific capabilities of future
astrophysics experiments (ranging from instrumentation on flagship missions to
low-cost SmallSats), which face complex performance, geometry, and budget
requirements.
Evolutionary algorithms have been used in a variety of applications, including antenna
design, and remain a focal point of active research [21, 22, 23, 24]. Antenna array
optimization using GAs has also been an area of recent development [25, 26, 27]. The
Genetically Evolving Neutrino Telescopes (GENETIS) collaboration has pioneered the2integration of GAs with antenna and science simulation software, aiming to designbiconical and horn antennas with improved sensitivity to neutrinosignals [28, 29, 30, 31, 32]. McCarthy et al. utilized a GA to design a horn antenna for
detecting of Cosmic Microwave Background radiation [33]. Furthermore, theLong-Baseline Neutrino Oscillation (LBNO) experiment and the Deep Underground
Neutrino Experiment (DUNE) optimized the design of neutrino beamlines through a
GA using science simulations to determine fitness [34, 35]. GAs have contributed to
optimizing various aspects of detector design, including layout, sensors, shielding,trigger optimization, and antenna arrays [36, 37, 38, 39, 40].
GAs are a powerful heuristic search method capable of finding high-quality solutions
to complex problems [41, 42, 43, 21]. They are effective in problems with manyhigh-cardinality parameters and have been successfully demonstrated in numerous
applications across many disciplines. Rooted in principles of biological evolution, GAs
begin with an initial set (population) of solutions (individuals) that undergo iterative
evolution toward improved results. Individuals are evaluated by a cost or fitness
function, which informs the selection of parents (using techniques called selection
methods) for the generation’s new designs, called children. Children are formed fromparents with various methods of mutating or combining individuals throughtechniques called operators. As in biology, evolutionary pressure induced by selection
methods tied to cost functions results in individuals with enhanced designs. Given
their stochastic nature, GA results will fluctuate when evolving toward the same
solution multiple times. To increase the probability of a successful algorithm,
consideration must be taken in the design of selection methods, genetic operators,
hyperparameters, and, most importantly, the fitness function.
Expanding to build antennas, instead of simple 3D structures shown in our previous
report, required three main developments to the GA. First, the construction of 3Dmodels transitioned to utilize XFdtd in order to simulate the individual gain patterns
required by the fitness calculations, moving away from Blender [44], an open-source
3D modeling software. Second, adjustments were made to the structures to include
the antenna feed and ground components that are necessary to generate an antenna
response. Finally, mechanisms were established to prevent the designs from shorting
across the feed and ground points, as such designs are typically ineffective. These
modifications and the associated challenges are discussed in more detail in
Section II.B.
This report presents initial findings resulting from the use of a GA to constructantennas and evolve individuals optimized for desired antenna responses. Section II
describes a detailed overview of the GA, including constructing an individual, XFdtd
simulations, fitness evaluation, and the selection methods and operators. Section III,
describes the process of calculating various fitness scores that were explored. The
results of the evolution using each fitness function are given in Section IV. Finally,
Section V discusses the significance of the results, the next steps in evolving antenna
designs toward specified science outcomes, and identifies potential challenges.

3II. Genetic Algorithm
A flowchart of the GA used in this analysis is shown in Figure 1, and the steps are
discussed in detail in this section. The loop begins by generating an initial populationof solutions. Following the steps described in Section II.A below, the algorithm
generates individuals and then constructs those designs in XFdtd. XFdtd simulates
the gain response pattern of each individual, which allows fitness scores to be
calculated by comparing each response pattern to a target gain pattern. The fitness
scores are then used to select parents and create offspring. The GA has a numerous
parameters outlined in Appendix I.
Initial Designs

Designs Createdin XFdtd

Fitness Function
Evaluation

Offspring
Creation

ParentSelection

Improved
Designs

Figure 1. Diagram illustrating the basic loop of the GA. Green boxes indicate initialization andtermination. The blue boxes show the steps required to evaluate the fitness function, and the redboxes give the steps for creating the new individuals.

A. Building an IndividualSingle individuals are built from basic geometric primitive shapes connected to form amore complex structure as shown by the example on the left side of Figure 2. Future
iterations of the algorithm will allow for disconnected components that affect theantenna beam pattern, such as reflectors and directors used in Yagi-Uda antennas.
These primitives may be thought of as building blocks taking the form of one of four
primitive shapes: cuboids, cylinders, cones, or spheres. An individual is fully describedby the dimensions, location, orientation, and connections of each component shape. In
the code, an individual is represented by a dictionary tree structure that contains thegenes of all the component shapes. The right side of Figure 2 illustrates the tree
structure for the example individual. Currently, each side of a shape is only allowed tohave one connected shape. All shape types behave consistently, with a maximum of
one attachment allowed in each region corresponding to its front, back, left, right, top,
and bottom. Additionally, when cylinders and cones are added to another shape, they
always connect with their flat top or bottom face. In this analysis, individuals wereallowed to be constructed with up to seven primitive shapes. A more thoroughdiscussion of the basic methodology for individual construction is given in the priorprogress report [1].
For the constructed geometries to perform as antenna designs, several additions were
implemented, including the placement of voltage feed and ground points and theprevention of unwanted shorting across the feed point. The algorithm begins
constructing an individual by first creating a small cube at the origin as a placeholder

4Base Shape● Shape Type: Cuboid● Dimensions = {L= 2, W = 2, H = 2}
● Rotation = {θ= 0, Φ= 0}● Connections = {Side 2, Side 5}

Shape 2
● Shape Type: Sphere● Dimensions = {R = 1}
● Location= {0, 2, 0}● Rotation = {θ= 0°, Φ= 0°}● Connected from = {Base Side 2}● Connections = {None}

Shape 3● Shape Type: Cylinder● Dimensions = {R = 1, H = 2}
● Location= {1, -1, 2}● Rotation = {θ= 0°, Φ= 0°}● Connected from = {Base Side 5}● Connections = {Side 5}

Shape 4
● Shape Type: Cone
● Dimensions = {R1 = 0, R2 = 1, H = 2}
● Location= {-0.579, 0.406, 1.707}● Rotation = {θ= -35°, Φ= -35°}● Connected from = {Shape 3 Side 5}● Connections = {None}

Figure 2. Left: Example individual consisting of each of the four primitive shapes. A base cuboid at theorigin with an attached cylinder and sphere. The cylinder then has an attached cone. Right: Tree
structure of example individual, with the genes of each shape listed.

for the antenna feed and ground assemblies and then builds off of it with newprimitives. In its current state, the algorithm is designed to focus on the constructionof single-feed dipole-shaped antennas. As such, it is required that exactly two
primitives be placed on separate sides of the starting cube in all evolved designs, with
one primitive to be attached to the antenna feed line and the other to the ground.
To prevent potential antenna performance issues caused by shorting across the feed
point, a collision detection algorithm utilizing the open-source Flexible Collision
Library software package was implemented [45]. In this algorithm, a bounding volume
is created that encloses each primitive that makes up an individual. Each time a newprimitive is added or the genes of an existing primitive are changed, the algorithm
disallows any operations that cause overlap between primitives that are electricallyconnected to opposite sides of the feed point. It is acknowledged that universally
disallowing shorting limits the potential antenna designs that the algorithm maycreate; however, this is suitable for the single-feed dipole-like cases that are the focusof this report. Future work will incorporate more complex logic to allow for designs
where the feed and ground are electrically connected, such as loop dipoles or somelog-periodic dipole arrays.

B. Creating Antennas in XF

Once the tree describing an individual is created as a dictionary, it is converted to amodel in XFdtd through the program’s scripting API. The algorithm allows for thematerial properties of the antenna designs to be evolved; however, all studies
presented in this report defined the primitives to be hollow and made from perfectelectrical conductor material. Before XFdtd simulation and fitness evaluation, the5starting cube is removed and replaced with the antenna feed. The algorithm allows for
single-frequency or broadband-frequency designs. For this work, the single-frequency
runs simulated the response at 300 MHz, while the broadband runs used 37 different
frequencies of equal steps from 200 MHz to 800 MHz. The antenna response wassimulated at 5◦ increments of the azimuth-zenith coordinates. The increment size canbe adjusted as needed for specific applications. The gain of an antenna is a measure of
how efficiently it converts received radio waves from a given direction into power.
XFdtd calculates the far-zone realized gain (hereafter referred to as simply “gain”) ofan antenna at a specific (θ, ϕ) coordinate using Eq. 1 [46]:GR =e ϕ)|22πr2 |E(θ,
.
ηPM

(1)

e ϕ)
Here, GR is the realized gain [47] of the antenna in a specific direction in dBi. E(θ,is the complex electric field incident on the antenna from the (θ, ϕ) direction, η is thewave impedance in the medium (377 Ω in free space), r is the distance between the
power source and the sensors in the simulation (1 m), and PM is the power acceptedby the antenna. The gain is equivalent to the directivity multiplied by the antenna’sradiation efficiency.

C. Fitness Function Calculation

Once an individual is simulated in XFdtd, fitness functions are used to compare theresulting gain pattern to the desired goal. Two fitness functions were developed andtested during this analysis: Euclidean distance and directional gain, which aredescribed in detail in Section III.

D. Parent Selection

This analysis used tournament selection as the sole selection method. In tournament
selection, a small subset of the population is chosen at random, and the individual withthe highest fitness score in the subset is selected as a parent [48, 49, 50, 51]. We used atournament size of 7% of the population or 8 individuals for a population of 120.

E. Genetic Operators

This section describes the three main genetic operators that were used to build thenext generation: mutation, crossover, and injection.
Mutation alters one component of a single parent individual to produce an offspring.
Due to the complexity of combining shapes into individuals, two different types ofmutation were implemented: standard mutation, where a single gene is altered, andregenerative mutation, where an entire component shape is altered. In standard
mutation, a single dimension, rotation, or location gene of a single shape is altered by6generating a value from a Gaussian distribution centered on the original value, where
the standard deviation is a predefined percentage of the total range of values. While
standard mutation makes minor changes to individuals, regenerative mutation allows
larger changes to occur. There are four types of regenerative mutation: grow, prune,
replacement, and side switch. For grow mutation, an empty side of one shape on theindividual is selected, and a new shape is generated that connects to that side. In
prune mutation, a shape, and all subsequent connected shapes in the tree, are removed
from the individual. Replacement mutation is the process of exchanging one shapewith another, where the new shape is generated from scratch. During the evolutions
tested in this work, we noticed that some plateaus were the result of a replacement
mutation that rarely performed well, as the new shape would often have geometry very
different from the previous shape. For example, a dipole that approximates a conewith similar dimensions will perform better than a cone with a very different geometry.
To ameliorate this, a proportional replacement mutation was developed and added tothe algorithm during the broadband runs, which causes the new shape to havedimensions that are similar to the shape it is replacing. Finally, side switch mutation
selects a shape and moves it, and all subsequent connected shapes in the tree, to adifferent unoccupied surface of the shape to which it was originally connected.
Components of two parents are combined to produce two offspring in gene or branchcrossover. In gene crossover, two parents exchange a single gene of one shape. The
shapes are restricted to be the same type, unless the gene swapped is rotation or
location. In branch crossover, all shapes in one branch of the tree structure areswapped between the two parents. The algorithm selects a tree depth in each of theparents and then exchanges the branches off that point. The branch to swap from aparent is allowed to be empty, which would effectively trim the branch of the otherparents. As with mutation, the algorithm ensures that the shape is still valid and thatthe constraints are still satisfied.
Injection is the creation of a brand new individual in the next generation. Individuals
made by injection are created using the same method as individuals in the first
generation.

F. Steady-state Evolution

In an improvement to the prior algorithm, this version incorporates an asynchronous
steady-state methodology, where new children are immediately introduced into the
population, instead of having delineated generations [52, 53, 54, 21]. After the initial
population is evaluated, a single new child is created where the type of genetic operator
used is chosen based on a predefined probability. Once the child is created, a random
individual in the population is replaced by the new child. This increases evolutionarypressure by allowing the genetic information from a child to be immediately available
for the population, which reduces computation time by increasing the rate ofconvergence. For comparison with traditional GAs, we define that a new generation

7

occurs when new children equal to the population have been created.
Two additional features were added to complement the change to steady-stateevolution. First, we implemented an elitism constraint, where the individual with thehighest fitness is always preserved, meaning that it cannot be removed from the
population. This feature replaced the reproduction operator, used in the previously
reported algorithm, that directly copied a parent to the next generation. Second, aforced diversity feature was implemented during individual construction, which
requires that there are no duplicate individuals in the population. Prior to
implementing these changes, initial tests of the steady-state algorithm resulted in large
numbers of duplicate individuals and large plateaus that the GA was not able to breakout of due to a lack of diversity in the population.

III. Fitness Function Details
This section details the Euclidean distance and directional gain fitness functions thatwere developed for the purposes of this algorithm.

A. Euclidean Distance Fitness Function

The Euclidean distance is used as a metric to determine the similarity between
N -dimensional vectors by summing in quadrature the corresponding elements of each
vector. A Euclidean distance of zero indicates perfect matching between vectors, andlarger values indicate a greater degree of dissimilarity. The general form for Euclidean
distance between vectors n and m, where i is the Nth component is given by
Equation 2:vuNuXd(⃗n, m)⃗ = t (n − m )2
i

i

(2)

i=1

To compare an individual’s simulated radiation pattern to the target, complex values
that encode the gain and phase in each sampled direction are calculated in thevertical, v, and horizontal, h, polarizations separately, as shown in Equations 3 and 4.

Zv = Gv [cos(Φv ) + i sin(Φv )](3)

Zh = Gh [cos(Φh ) + i sin(Φh )](4)

Gv and Gh represent the gain in each polarization direction, and Φv and Φh represent
the phase in each polarization direction.
This complex value is evaluated at each point, i, sampled in θ and ϕ for both the
evolved and target antenna gain patterns, and, accounting for the vertical and8horizontal polarizations independently and weighting their contributions, theEuclidean distance-based fitness metric comparing the resulting vectors takes the formshown in Equation 5.
v
!
u N h
i
u X
2
2
A |Zvi ,evolved − Zvi ,target | + B |Zhi ,evolved − Zhi ,target |
d=t

(5)

i=1

Here, the weights of the contributions from each polarization are represented by A andB, which are constrained to obey the relationship A + B = 1. For all studies presented
in this report, each polarization was given equal weighting, leading to A = B = 0.5
This metric is the basis for the Euclidean distance fitness function given by Equation 6.

FE =

1
1+d

(6)

Fitness scores resulting from the Euclidean distance-based metric are bound between
FE = 0 and FE = 1. Scores approaching 1 correspond to evolved antenna designs withradiation patterns exhibiting a high degree of similarity with the target radiation
pattern. A fitness score of 1 indicates that the evolved antenna’s radiation pattern andphase perfectly match that of the target.

1. Single-frequency and Broadband Applications

The Euclidean distance fitness function maintains the form presented in Equations 5
and 6 when used to evaluate the similarity between the radiation pattern of an evolved
antenna with that of the target for both single-frequency and broadband cases. The
broadband case requires that the gain and phase values at all sampled points in thesimulated radiation patterns corresponding to each frequency step in the user-definedband be considered in the fitness score calculation. The single-frequency case simply
considers one radiation pattern at a single specified frequency to derive the fitness
score.

B. Directional Gain Fitness Function

A fitness function was devised to maximize and/or minimize gain in user-specified
angular ranges, directing antenna designs to evolve toward an optimal directional
sensitivity profile. This required separate contributions to the fitness score accounting
for gain values in the angular ranges to be maximized, where larger values must
positively impact the fitness score, and gain values in angular ranges to be minimized,
where larger values must negatively impact the fitness score. The highest scoring
individuals will have the highest average gain in maximization regions and lowest
average gain in minimization regions. An equation was then created to meet thesedesired outcomes, which gives the directional gain fitness function, FD , in Equation 7.

9



FD = A tanh Ḡ(θmax , ϕmax ) + B

1
1 + Ḡ(θmin , ϕmin )
(7)

Where Ḡ(θ, ϕ) represents the average gain over all sampled points in the user-specified
angular ranges for the sensitivity to be maximized or minimized, which is denoted byθmax and ϕmax , and θmin and ϕmin , respectively. The first term accounts for the
maximizing range and is weighted by A, while the second term describes theminimizing range and is weighted by B. The weights are constrained to A + B = 1.
For all studies using this fitness function in this report, the two terms were evenly
weighted, with A = B = 0.5. This means that maximizing and minimizing gain in therespective angular regions of interest is given equal importance in the determination ofthis metric of fitness.
This fitness function was designed to be bounded between 0 and 1. However, it isworth noting that to achieve a fitness score of 1, an infinite gain would be required in
the region where gain is to be maximized, which is not possible for physical antennas.
High-performing individuals in the studies presented in Section IV.B of this reportachieved fitness scores nearing 0.9 under this metric.

1. Single-frequency and Broadband Applications

The directional gain fitness function maintains the form presented in Equation 7
applied to both single-frequency and broadband cases. The difference between thesecases lies in the calculation of the average gain at points in the angular range(s) where
maximization or minimization is desired, Ḡ(θmax , ϕmax ) and Ḡ(θmin , ϕmin ),respectively. The average gain calculations for the single-frequency case consider gain
values in the specified angular ranges from the simulated antenna radiation pattern ata single specified frequency. The average gain calculations for the broadband caseaccount for gain values in the specified angular ranges from the simulated antennaradiation patterns at each frequency step in the user-specified band. At present,
contributions from the gain at each sampled point in the specified ranges ofmaximization and/or minimization, from all frequency steps in the band, are given
equal weight in the calculation of Ḡ(θmax , ϕmax ) and Ḡ(θmin , ϕmin ). However, future
implementation to allow for increased importance to be placed on performance at
specific in-band frequencies is planned.

IV. Results and Discussion
The GA was tested using both the Euclidean distance and the directional gain fitness
function. For each fitness function, designs were evolved once for a single frequency of
300 MHz and once for a broadband case of 200 MHz to 800 MHz. For the broadband
case, the simulated antenna radiation patterns are evaluated at 37 evenly spaced
discrete frequency steps in the specified band. This section describes the details of the10

evolutions for each of the four tests. For each evolution discussed in this section, thepopulation size was 120 individuals.

A. Euclidean Distance Target Radiation Pattern Evolution
1. Single Frequency

Figure 3 shows the results for the Euclidean distance evolution at a single frequency of
300 MHz. The single-frequency target radiation pattern is that of a simple dipole. The
highest fitness score of each generation is plotted as the solid green line and quartiles
are shown as the other lines. Over 441 generations, the designs evolved to a fitness
score of approximately 0.75. The 3D models corresponding to the most fit individuals(red) as of generations 0, 25, and 441 are shown in comparison to the model (blue)that created the target radiation pattern in Figure 4.

Fitness of Euclidean Single-Frequency Run

1.0

Fitness Score

0.8

0.6
Best Fit
75th Percentile
Median25th Percentile

0.4

0.2

0.0

0

44

88

132 176 220 264 308 352 396 441

Generation
Figure 3. The fitness score of each generation evolving toward a single-frequency radiation patternusing the Euclidean distance fitness function. The maximum fitness score and the three quartiles
are shown.

Figure 5 shows the gain patterns of the most and least fit individuals, represented by
the green and blue lines, respectively, at the start and end of the evolution, along withthe target gain pattern for reference, represented by the pink diamonds. Here, thegain patterns are represented as two-dimensional slices in the zenith and azimuth
directions. Images of the full three-dimensional gain patterns corresponding to theseslices may be seen in Figure 19 in Appendix II. Figure 6 depicts the phase, in thezenith and azimuth directions, of the most and least fit evolved antennas at the
beginning and end of the evolution. The phase of the most and least fit individuals isrepresented by the green and blue lines, respectively, while the phase of the target isshown in pink. Additional gain and phase results of the most and least fit individuals

11(a) Generation 0(b) Generation 25(c) Generation 441

Figure 4. Three example models of the best individual of generations 0, 25, and 441 are shown,
illustrating the evolution of the antenna using the Euclidean distance fitness function. The antennathat produced the target gain pattern is given in blue, and the individual is shown in red.

at an intermediate generation are displayed in Figures 27 and 28 in Appendix III.
Generation 0

Generation 441

Antenna Gain at 300 MHz

Antenna Gain at 300 MHz

ZenithAzimuthZenith

0°

0°

0°

45°

315°

90°

45°

270°

­50

315°

­3

90°

270°

270°

315°

90°

270°

0.5
-50
-30
-10

2.0
0

3
225°

180°

1.5

-20
2

135°

1.0

-40
1

0
225°

45°

-60

-10

180°

­70

90°

0
-20

135°

315°

-1

-30

0°

45°

-2

-40

Azimuth

135°

225°

135°

180°

225°

180°

Figure 5. The gain patterns of the most and least fit individual as of generations 0 (left) and 441(right) of the single-frequency evolution towards a target radiation pattern using the Euclidean distancefitness function. Also overlaid on each is the gain pattern of the target.

Despite the fitness score peaks at approximately 0.7, Figure 5 shows that the most fit
individual has a gain pattern that is closely aligned with the target. The difference
between the evolved antenna’s performance and that of the target lies almost entirelyin the antenna phase near 0◦ as measured in the zenith direction, as evidenced in
Figure 6. This illustrates that a single data point can have a large effect on the fitness
score of an individual.

2. Broadband Target Radiation Pattern Evolution

The broadband target radiation patterns consisted of antenna gain and phaseevaluated at 37 discrete, evenly-spaced frequency steps between 200 MHz and

12Generation 0

Generation 441
180

108

108

Phase (Degrees)

Phase (Degrees)

Antenna Phase at 300 MHz
180

36

−36
−108
−180

Antenna Phase at 300 MHz

36

−36
−108

0

32

65

98

130

163

196

229

261

294

327

360

−180

0

32

Zenith Direction (Degrees)
Antenna Phase at 300 MHz

130

163

196

229

261

294

327

360

327

360

Antenna Phase at 300 MHz

180

108

108

Phase (Degrees)

Phase (Degrees)

98

Zenith Direction (Degrees)

180

36

−36
−108
−180

65

36

−36
−108

0

32

65

98

130

163

196

229

261

294

Azimuth Direction (Degrees)

327

360

−180

0

32

65

98

130

163

196

229

261

294

Azimuth Direction (Degrees)

Figure 6. The antenna phase in the zenith and azimuth directions of the most and least fit individual
as of generations 0 (left) and 441 (right) of the single-frequency evolution towards a target radiation
pattern using the Euclidean distance-based fitness function. Also overlaid on each is the phase of
the target.

800 MHz. The radiation patterns chosen as targets for the broadband evolution
towards a target test case corresponded to the designs of broadband antenna createdthrough the GENETIS collaboration’s studies on optimization of the biconicalantenna designs [28]. Figure 7 shows the results for the Euclidean distance evolution
for the broadband case. Here, the highest fitness score of each generation is plotted as
the solid green line and quartiles are shown as the other lines. After 216 generations,the most fit individual created in the evolution achieved a fitness score of
approximately 0.39. The models corresponding to the most fit individuals as ofgenerations 0, 5, and 216 are shown overlaid with the model corresponding to thetarget radiation pattern in Figure 8.
Figure 9 shows the most and least fit individuals, represented by the green and bluelines, respectively, at the start and end of the evolution at three example frequencies of
200 MHz, 516.73 MHz, and 800.01 MHz, representing the upper, lower, and middle
frequencies in the band. The target gain pattern at each frequency is also shown forreference and is represented by the pink diamonds. The gain patterns are depicted as
two dimensional slices in the zenith and azimuth directions. Additional images of thefull three-dimensional gain patterns from which these slices were taken may be seen in

13

Fitness of Euclidean Broadband Run

1.0

Fitness Score

0.8

0.6
Best Fit
75th Percentile
Median25th Percentile

0.4

0.2

0.0

0

21

43

64

86

108 129 151 172 194 216

Generation
Figure 7. The fitness score of each generation evolving toward a target broadband radiation patternusing the Euclidean distance fitness function. The maximum fitness score and the three quartiles
are shown.

(a) Generation 0(b) Generation 5(c) Generation 216

Figure 8. Three example models of the best individuals from generations 0, 5, and 216 are shown,
illustrating the evolution of broadband antenna using the Euclidean distance fitness function. The
antenna that produced the target gain pattern is given in blue, and the individual is shown in red.

Figures 20-22 in Appendix II.
Figure 10 depicts the phase, in the zenith and azimuth directions, of the most andleast fit evolved antennas at the beginning and end of the evolution. Results are again
shown for the example frequencies of 200 MHz, 516.73 MHz, and 800.01 MHz. The
phase of the most and least fit individuals is represented by the green and blue lines,
respectively, while the phase of the target is shown in pink. Additional gain and phaseresults of the most and least fit individuals at an intermediate generation are displayedin Figures 29 and 30 in Appendix III.

14

Generation 0

Generation 216

Antenna Gain at 200 MHz

Antenna Gain at 200 MHz

ZenithAzimuthZenith

0°

0°

0°

45°

315°

90°

45°

270°

­60

315°

­5

90°

270°

315°

90°

-1

-20

135°

225°

ZenithAzimuthZenith0°0°0°Antenna Gain at 516.73 MHz

315°

­18

90°

-20
-10

Azimuth
0°

315°

90°

45°

270°

­40

315°

90°

-12.5
-10.0
-20

-7.5
-5.0

-10

-6
-4

135°

-2.5

0

-2
225°

135°

0.0
225°

135°

225°

180°

180°

ZenithAzimuthZenithAzimuth

0°

0°

0°

0°

Antenna Gain at 800.01 MHz

45°

315°

­40

315°

­35

90°

180°

Antenna Gain at 800.01 MHz

45°

270°

45°

270°

315°

90°

45°

270°

315°

­12

90°

­30

-30

-30

-15

-10

-8

-20

-20

-6
-10

-4

-10
0

-2

0

-5

0

0
225°

180°

135°

225°

180°

270°
-10

-25
-20

135°

270°

­15.0

-30

0

180°

225°

180°

45°

270°

-16
-14
-12
-10
-8

-30

225°

135°

Antenna Gain at 516.73 MHz

45°

270°

­40

1

0

225°

180°

315°

0

-10
3

135°

180°

135°

-2

2

0

90°

-3

1

-10

45°

270°
-4

-30
0

-20

180°

­5

90°

-40
-1

225°

315°

-50
-2

-30

135°

45°

270°

­60

-3

-40

90°

0°

45°

-4

-50

Azimuth

135°

225°

180°

135°

225°

180°

Figure 9. The gain patterns of the most and least fit individuals in generation 0 (left column) and 216(right column) of the evolution towards a target broadband radiation pattern using the Euclidean
distance-based fitness function at three example in-band frequencies: 200 MHz (first row), 516.73 MHz(second row), and 800.01 MHz (third row). The gain pattern of the target is overlaid at the given
frequency. Note the target and most fit lines of the Azimuth chart at 200 MHz for generation 216 areoverlapping at the center, since the large gain of the least fit indiviudal is increasing the scale of theplot.

The run quickly discovered that individuals constructed with two cylinders
approximated the two cones of the design responsible for the target radiation pattern,
as evidenced in generation 5 shown in Figure 8. After over 125 generations, thealgorithm had yet to change the cylinders to cones, because the replacement mutation
was generating new shapes from scratch, resulting in poor fitness individuals. At this15

Generation 0

Generation 216

Antenna Phase at 200 MHz

Phase (Degrees)

Phase (Degrees)

108

108

36

−36
−108
−180

Antenna Phase at 200 MHz

180

180

36

−36
−108

0

32

65

98

130

163

196

229

261

294

327

−180

360

0

32

65

Zenith Direction (Degrees)

108

108

36

−36

196

229

261

294

327

360

327

360

327

360

36

−36

0

32

65

98

130

163

196

229

261

294

327

360

−180

0

32

Zenith Direction (Degrees)

65

98

130

163

196

229

261

294

Zenith Direction (Degrees)

Antenna Phase at 800.01 MHz

Antenna Phase at 800.01 MHz

180

180

108

108

Phase (Degrees)

Phase (Degrees)

163

−108

−108

36

−36

36

−36
−108

−108
−180

130

Antenna Phase at 516.73 MHz
180

Phase (Degrees)

Phase (Degrees)

Antenna Phase at 516.73 MHz
180

−180

98

Zenith Direction (Degrees)

0

32

65

98

130

163

196

229

261

294

Zenith Direction (Degrees)

327

360

−180

0

32

65

98

130

163

196

229

261

294

Zenith Direction (Degrees)

Figure 10. The antenna phase in the zenith direction of the most and least fit individual in generation
0 (left column) and 216 (right column) of the evolution towards a target broadband radiation patternusing the Euclidean distance-based fitness function at three example in-band frequencies: 200 MHz(first row), 516.73 MHz (second row), and 800.01 MHz (third row). Also overlaid on each is the phaseof the target at the given frequency.

point, the proportional replacement mutation operator was added in the middle of therun, which facilitated the increase in fitness around generation 175 to a design withtwo cones. Maximizing the efficiency of the algorithm to avoid plateauing fitness
scores in local maxima continues to be a point of emphasis for future development.

16Generation 0

Generation 216

Antenna Phase at 200 MHz

180

108

Phase (Degrees)

Phase (Degrees)

108

36

−36

36

−36
−108

−108
−180

Antenna Phase at 200 MHz

180

0

32

65

98

130

163

196

229

261

294

327

−180

360

0

32

Azimuth Direction (Degrees)
Antenna Phase at 516.73 MHz

Phase (Degrees)

Phase (Degrees)

163

196

229

261

294

327

360

327

360

327

360

108

36

−36
−108

36

−36
−108

0

32

65

98

130

163

196

229

261

294

327

360

−180

0

32

Azimuth Direction (Degrees)

65

98

130

163

196

229

261

294

Azimuth Direction (Degrees)

Antenna Phase at 800.01 MHz

Antenna Phase at 800.01 MHz

180

180

108

Phase (Degrees)

108

Phase (Degrees)

130

Antenna Phase at 516.73 MHz

180

108

36

−36
−108
−180

98

Azimuth Direction (Degrees)

180

−180

65

36

−36
−108

0

32

65

98

130

163

196

229

261

294

Azimuth Direction (Degrees)

327

360

−180

0

32

65

98

130

163

196

229

261

294

Azimuth Direction (Degrees)

Figure 11. The antenna phase in the azimuth direction of the most and least fit individual ingeneration 0 (left column) and 216 (right column) of the evolution towards a target broadband
radiation pattern using the Euclidean distance-based fitness function at three example in-band
frequencies: 200 MHz (first row), 516.73 MHz (second row), and 800.01 MHz (third row). Also
overlaid on each is the phase of the target at the given frequency.

17

B. Directional Gain Fitness Function Results

The next evolution used the directional gain fitness function, which attempted tomaximize and/or minimize gain in the specified angular ranges. For this run, theangular ranges for gain maximization and minimization were defined to approximately
match the expected sensitivity of a typical dipole antenna. The algorithm targetedmaximizing gain between zenith angles of 45◦ to 135◦ and minimizing between zenith
angles of 0◦ to 45◦ and 135◦ to 180◦ . The gain in the azimuthal direction did notfactor into the fitness score for the purpose of this test. Note that phase did not factor
into the directional gain fitness function but could be added based on project orexperimental requirements.

1. Single Frequency

Figure 12 shows the results of the directional gain evolution at a single frequency of
300 Mhz. The most fit individual after 142 generations from this evolution achieved afitness score of approximately 0.87. Figure 13 shows the most fit individuals as ofgenerations 0, 5, 95, and 142, demonstrating how the most fit solution changed
throughout the evolution. The individual shown from generation 142 is only slightly
improved over generation 95, and while both designs are viable options, the design
differences highlight the potential of the GA to find unique solutions, even in thissimple case. The evolution will continue to run to explore potential improvements.

Fitness of Directional Single-Frequency Run
1.0

Fitness Score

0.9
0.8Best Fit
75th Percentile
Median25th Percentile

0.7
0.6
0.5
0.4

0

14

28

42

56

71

85

99

113 127

142

Generation
Figure 12. The fitness score of each generation of the single-frequency evolution using the directional
gain fitness function. The maximum fitness score and the three quartiles are shown.

Figure 14 depicts the gain patterns of the most and least fit individuals from the
population at the beginning and end of the single-frequency directional gain evolution.
The most fit individuals are represented by the dashed line and the least fit

18

(a) Generation 0(b) Generation 5(c) Generation 95(d) Generation 142

Figure 13. Four example models of the best individual of generations 0, 5, 95, and 142 are shown,
illustrating the evolution of the single-frequency antenna using the directional gain fitness function.
There is no target antenna since the desired gain pattern was not created from an antenna.

individuals by the dotted line. For reference, the zenith angle ranges specified for gain
maximization and minimization are shaded in green and red, respectively. These gain
patterns are presented as two-dimensional slices in the zenith and azimuth direction.
The corresponding three-dimensional gain patterns for the most fit individual in thesegenerations are available in Figure 23 in Appendix II. An additional set of gain
patterns corresponding to the most and least fit individuals in an intermediate
generation of the evolution is shown in Figure 31 in Appendix III.
Generation 0

Generation 142

Antenna Gain at 300 MHz
AzimuthZenith

0°

0°

0°

45°

315°

­30

90°

Antenna Gain at 300 MHz

Zenith

45°

270°

315°

90°

-4
-5

-2
0
225°

225°

180°

90°

135°

-20
-15
-10
-5
0
225°

180°

270°

­25

5

2
135°

315°

0

0
5

45°

270°

-30
-25
-20
-15
-10
-5

-6
-10

180°

­35

90°

-8
-15

135°

315°

-10
-20

0°

45°

270°

­12

-25

Azimuth

5
225°

135°

180°

Figure 14. The gain patterns of the most and least fit individual as of generations 0 (left), and 142(right), of the single-frequency evolution using the directional gain maximization/minimization fitness
function. The regions shaded in green and red represent the angular ranges designated for
maximization and minimization, respectively.

The single-frequency directional gain evolution achieved a maximum fitness score of
approximately 0.88 over 142 generations. The GA quickly ascertained that adipole-like design performed well in the single-frequency directional gain test, which is
expected given that the angular ranges for gain to be maximized were chosen to mimic
the gain pattern of a typical dipole antenna. However, in the later generations, theGA discovered an alternative solution that achieved a slightly higher fitness function19

(0.8791 versus 0.8801)by adding a third primitive cylinder to the side of the dipole
design, as seen in Figure 13. Figure 15 shows the difference in gain patterns between
these two designs. The zenith plot patterns appear to show the response patterns
closely overlapping, with the additional primitive causing slightly larger gain in theminimization region around 15◦ , which would reduce the fitness score. The azimuthal
plot, which is the horizontal slice of the zenith plot and exists entirely in the
maximization region, shows a slight decrease in gain around 90◦ and larger increase in
the maximization region around 270◦ . The additional primitive caused a slightdecrease in some areas but a larger increase in others, resulting in a slightly improved
fitness score. This illustrates that despite symmetrical target regions, the algorithm is
not restricted to symmetrical designs. This addition to the typical dipole design serves
as an example of the GA’s potential to discover effective designs that are outside ofthose that would traditionally be used, and with additional generations new primitives
could continue to be added to improve the fitness.

Antenna Gain at 300 MHz
ZenithAzimuth

0°

0°

45°

315°

­40

90°

45°

270°

315°

90°

270°

0.5

-30

1.0
1.5

-20

2.0
-10

2.5
3.0

0

3.5
135°

225°

180°

135°

225°

180°

Figure 15. The gain patterns of the most fit individual in generation 95 and generation 142 at
300 MHz using the directional fitness function. The regions shaded in green and red represent theangular ranges designated for maximization and minimization, respectively.

2. Broadband Directional Gain Evolution

Figure 16 shows the results of the broadband directional evolution using the same
frequency range of 200 MHz to 800 MHz as the broadband Euclidean distance test. The
most fit individual after 229 generations from this evolution achieved a fitness score of
approximately 0.82. Figure 17 shows the most fit individuals as of generations 0, 5,
and 229, demonstrating how the most fit solution changed throughout the evolution.
Figure 18 depicts the gain patterns of the most and least fit individuals from the
population at the beginning and end of the evolution. The most fit individuals are
represented by the dashed line and the least fit individuals by the dotted line. For
reference, the zenith angle ranges specified for gain maximization and minimization

20

1.0

Fitness of Directional Broadband Run

Fitness Score

0.9
0.8Best Fit
75th Percentile
Median25th Percentile

0.7
0.6
0.5
0.4

0

22

45

68

91

114 137 160 183 206 229

Generation
Figure 16. The fitness score of each generation of the broadband evolution using the directional gain
fitness function. The maximum fitness score and the three quartiles are shown.

(a) Generation 0(b) Generation 5(c) Generation 229

Figure 17. Three example models of the best individual of generations 0, 5, and 229 are shown,
illustrating the evolution of broadband antenna using the directional gain fitness function. There is notarget antenna since the desired gain pattern was not created from an antenna.

are shaded in green and red, respectively. These gain patterns are presented as
two-dimensional slices in the zenith and azimuth direction. The corresponding
three-dimensional gain patterns for the most fit individual in these generations areavailable in Figures 24–26 in Appendix II. An additional set of gain patterns
corresponding to the most and least fit individuals in an intermediate generation of
the evolution is shown in Figure 32 in Appendix III.
The broadband directional gain evolution achieved a maximum fitness score of
approximately 0.82 in 229 generations. The GA quickly arrived at a design of an

21

Generation 0

Generation 229

Antenna Gain at 200 MHz

Antenna Gain at 200 MHz

ZenithAzimuthZenith

0°

0°

0°

45°

315°

90°

45°

270°

­30

315°

90°

0°

45°

270°

­14

315°

­60

90°

-10

0

5
135°

135°

180°

ZenithAzimuthZenith

0°

0°

0°

315°

­40

90°

225°

Antenna Gain at 516.73 MHz

45°

270°

-30

0
135°

180°

Antenna Gain at 516.73 MHz

­35

-2
225°

180°

315°

-4
0

225°

180°

45°

-6
-10

-2

0

-25

Azimuth
0°

45°

270°

315°

90°

315°

­17.5

90°

-30
-20

-10

-10

-5
0

-10
0

0

5
225°

135°

225°

180°

135°

270°

-15.0
-12.5
-10.0
-7.5
-5.0
-2.5
0.0

-40
-20

-15

45°

270°

­50

-30
-20

135°

2.5
225°

135°

225°

180°

180°

ZenithAzimuthZenithAzimuth

0°

180°

0°

0°

0°

Antenna Gain at 800.01 MHz

45°

315°

90°

Antenna Gain at 800.01 MHz

45°

270°

­30

315°

90°

45°

270°

­20

-25

90°

315°

-10

225°

180°

-12
-10
-8
-6

-10

-4
-2

0

5
135°

225°

180°

135°

225°

180°

270°

-14

-20
0

0

­16

90°

-30
-5

-5

315°

-40
-10

-15

45°

270°

­50

-15

-20

135°

270°

-14
-12
-10
-8

-20

-4

-5

225°

­16

90°

-30

-6

135°

315°

-40
-8

-15

90°

270°

-10

-20

45°

-50

-12

-25

Azimuth

135°

225°

180°

Figure 18. The gain patterns of the most and least fit individuals in generation 0 (left column) and 229(right column) of the evolution using the directional gain maximization/minimization fitness function at
three example in-band frequencies: 200 MHz (first row), 516.73 MHz (second row), and 800.01 MHz(third row). Also overlaid on each is the gain pattern of the target at the given frequency. The regions
shaded in green and red represent the angular ranges designated for maximization and minimization,
respectively.

individual consisting of a cone and a cylinder in a dipole-like configuration, as
depicted in Figure 17, which achieved a fitness score above approximately 0.8. The
design was then fine-tuned over many generations, and incremental gains in fitness
score were achieved. Given additional generations, an improved solution may likely befound either by additional fine-tuning of the most fit individual’s current design or by
exploring new or additional primitives.

22

V. Conclusions
This report presents a GA that can evolve antennas toward desired gain patterns
using combinations of smaller component shapes. Tests showed that the algorithm was
successful at producing antenna designs based on both the closeness in antenna
response to a target antenna gain pattern and an input-desired directional sensitivity.
Using a Euclidean distance fitness function, the algorithm was able to evolve antennas
to match the gain patterns of dipole antenna. Using a directional gain fitness function
the algorithm was able to design antennas to maximize gain between 45◦ to 135◦ andminimize between 0◦ to 45◦ and 135◦ to 180◦ .

A. Future WorkThe ultimate goal of this research is to connect antenna design to science outcomes byusing fitness functions integrated with science simulation software. Improving theeffectiveness of the GA in designing complex and unique antennas for specificfunctions will also be an area of active development, including allowing other primitivegeometries and adding disconnected shapes to act as reflectors and directors.
Individuals will also have increased complexity by removing the constraint that allows
only one shape per side, by adding additional primitives, and by adding material
optimizations including dielectric or metamaterial antennas, such as those discussed in
References [55, 56, 57, 58, 59]. Improvements to the GA, such as the addition of the
proportional replacement mutation, are in development to improve the speed ofconvergence and reduce plateaus. The ability to optimize arrays or to produce arraysto achieve improved solutions will be added to the GA. Many experiments have uniqueor complex design constraints, such as geometry or weight requirements, that will beincorporated into the algorithm in the future.
As the complexity of the algorithm increases it will be critical to continue to optimize
the speed of the algorithm and efficiency in converging on final solutions. Appendix V
discusses completed and in development improvements to the algorithm by exploring
the speed to reach a solution using the shape evolution previously developed as a test
case. This testing was used to validate the steady-state, elite, and forced diversity
changes to the algorithm. In development is the use of adaptable dynamic genetic
operators, where reinforced learning is used to change how individuals are createdbased on the current environment of the run.
More advanced improvements to the algorithm will also be investigated, including
particle swarms and other heuristic methods [60, 61, 62, 63], or neuralnetworks [64, 65], to improve efficiency. Particle swarms could help reduce the numberof ineffective individuals being generated and help to converge on a solution more
quickly. A neural network could be used to predict the fitness of individuals similar topreviously tested individuals and reduce the computational time needed to simulate
performance and calculate fitness.

23

The work presented in this report demonstrates a genetic algorithm that can evolve
antennas with desired gain patterns from primitive shapes. Continued research in thisarea will allow for optimization of designs for science outcomes with reduced cost andscheduling.

Acknowledgments
The research was carried out at the Jet Propulsion Laboratory, California Institute ofTechnology, under a contract with the National Aeronautics and Space Administration(80NM0018D0004). The authors are grateful to the entire GENETIS collaboration for
their prior work and continued assistance, especially Prof. Wolfgang Banzhaf. We
would also like to thank Remcom for their support, especially Tarun Chawla, Walter
Janusz, and Benjamin Hardy.

References[1] J. Rolla, B. Reynolds, J. Weiler, A. Connolly, R. Debolt, A. Machtay, B. Sipe, and
D. Wells, “Design of 3D Antenna Geometries Using Genetic Algorithms,” TheInterplanetary Network Progress Report, vol. 42-234, pp. 1–26, August 2023.
[2] R. Luebbers, “XFDTD and beyond-from classroom to corporation,” 2006 IEEE
Antennas and Propagation Society International Symposium, pp. 119–122, 2006.
[3] F. Tamjid, F. Foroughian, C. M. Thomas, A. Ghahreamani, R. Kazemi, and A. E. Fathy,“Toward high-performance wideband gnss antennas-design tradeoffs and development of
wideband feed network structure,” IEEE Transactions on Antennas and Propagation,
vol. 68, no. 8, pp. 5796–5806, 2020.
[4] B. R. Rao, GPS/GNSS Antennas.

Artech House, 2013.

[5] J. J. H. Wang, “Antennas for global navigation satellite system (gnss),” Proceedings ofthe IEEE, vol. 100, no. 7, pp. 2349–2355, 2012.
[6] A. Romero-Wolf, S. Vance, F. Maiwald, E. Heggy, P. Ries, and K. Liewer, “A passive
probe for subsurface oceans and liquid water in jupiter’s icy moons,” Icarus, vol. 248, pp.
463–477, 2015. https://www.sciencedirect.com/science/article/pii/S0019103514006009[7] D. M. Schroeder, A. Romero-Wolf, L. Carrer, C. Grima, B. A. Campbell, W. Kofman,
L. Bruzzone, and D. D. Blankenship, “Assessing the potential for passive radio soundingof europa and ganymede with rime and reason,” Planetary and Space Science, vol. 134,
pp. 52–60, 2016. https://www.sciencedirect.com/science/article/pii/S0032063316301465[8] R. Prechelt, A. Romero-Wolf, P. Gorham, E. Costello, R. Ghent, and P. Lucey, “Radio
detection of subsurface lunar ice via the askaryan effect with the cosmic ray lunar
sounder (corals),” AGU Fall Meeting Abstracts, pp. P41C–09, 2021.
[9] W. M. Farrell, M. D. Desch, and P. Zarka, “On the possibility of coherent cyclotron
emission from extrasolar planets,” JGR Planets, vol. 104(E6), pp. 14 025–14 032, June
1999.
[10] P. Zarka, R. A. Treumann, B. P. Ryabov, and V. B. Ryabov, “Magnetically-driven
planetary radio emissions and application to extrasolar planets,” APSS, vol. 277, pp.
293–300, June 2001.

24

[11] T. J. W. Lazio and W. M. Farrell, “Radio detection of extrasolar planets: present andfuture prospects,” Planetary Radio Emissions VI, pp. 603–610, 2006.
[12] E. Feenberg and H. Primakoff, “Interaction of cosmic-ray primaries with sunlight andstarlight,” Physical Review, vol. 73(5), pp. 449–469, March 1948.
[13] J. E. Felten and P. Morrison, “Omnidirectional inverse compton and synchrotron
radiation from cosmic distributions of fast electrons and thermal photons,” AstrophysicalJournal, vol. 146, p. 686, December 1966.
[14] M. J. Rees, “Studies in radio source structure-III. Inverse Compton radiation from radiosources,” mnras, vol. 137, p. 429, January 1967.
[15] G. R. Blumenthal and R. J. Gould, “Bremsstrahlung, synchrotron radiation, andCompton scattering of high-energy electrons traversing dilute gases,” Reviews of Modern
Physics, vol. 42(2), pp. 237–271, January 1970.
[16] L. Feretti and G. Giovannini, “Diffuse cluster radio sources (review),” Proceedings of the175th Symposium of the International Astronomical Union, vol. 175, p. 333, January
1996.
[17] S. R. Furlanetto, S. P. Oh, and F. H. Briggs, “Cosmology at low frequencies: the 21 cm
transition and the high-redshift Universe,” Physics Report, vol. 433(4-6), pp. 181–301,
October 2006.
[18] J. R. Pritchard and A. Loeb, “Evolution of the 21cm signal throughout cosmic history,”
Physical Review D, vol. 78(10), p. 103511, November 2008.
[19] J. R. Pritchard and A. Loeb, “Constraining the unexplored period between the darkages and reionization with observations of the global 21 cm signal,” Physical Review D,
vol. 82(2), p. 023006, July 2010.
[20] A. Liu, J. R. Pritchard, M. Tegmark, and A. Loeb, “Global 21 cm signal experiments: adesigner’s guide,” Physical Review D, vol. 87(4), p. 043002, February 2013.
[21] B. Alhijawi and A. Awajan, “Genetic algorithms: Theory, genetic operators, solutions,and applications,” Evolutionary Intelligence, pp. 1–12, 2023.
[22] S. K. Goudos, D. E. Anagnostou, C. Kalialakis, P. Vasant, S. Nikolaou et al.,
“Evolutionary algorithms applied to antennas and propagation: emerging trends andapplications,” International Journal of Antennas and Propagation, vol. 2016, 2016.
[23] B. Liu, H. Hu, H. Han, H. Lv, and L. Li, “Optimization of the design of gas cherenkov
detectors for icf diagnosis,” Nuclear Instruments and Methods in Physics Research
Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 897,
pp. 54–58, July 2018.
[24] A. Liu, A. Bross, and D. Neuffer, “Optimization of the magnetic horn for the nuSTORM
non-conventional neutrino beam using the genetic algorithm,” Nuclear Inst. and Methods
in Physics Research, A, vol. 794, pp. 200–205, 2015.
[25] G. Oliveri, P. Rocca, L. Poli, M. Carlin, E. T. Bekele, A. De Matteis, and A. Massa,“Evolutionary strategies for advanced array optimization,” 2011 IEEE International
Symposium on Antennas and Propagation (APSURSI), pp. 2441–2444, 2011.
[26] F. Grimaccia, M. Mussetta, A. Niccolai, and R. E. Zich, “Comparison of binary
evolutionary algorithms for optimization of thinned array antennas,” 2018 IEEE
Congress on Evolutionary Computation (CEC), pp. 1–8, 2018.

25

[27] Q. Xu, S. Zeng, F. Zhao, R. Jiao, and C. Li, “On formulating and designing antenna
arrays by evolutionary algorithms,” IEEE Transactions on Antennas and Propagation,
vol. 69, no. 2, pp. 1118–1129, 2021.
[28] J. Rolla, A. Machtay, A. Patton, W. Banzhaf, A. Connolly, R. Debolt, L. Deer,
E. Fahimi, E. Ferstle, P. Kuzma, C. Pfendner, B. Sipe, K. Staats, and S. A. Wissel,“Using evolutionary algorithms to design antennas with greater sensitivity to ultra highenergy neutrinos,” Physical Review D, 2023, forthcoming.
[29] B. Reynolds, J. Rolla, A. Machtay, W. Banzhaf, D. Calderon, C. Chen, A. Connolly,
R. Debolt, E. Fahimi, N. King, E. Melotti, A. Patton, B. Sipe, K. Staats, D. Wells,
S. Wissel, and A. Zinn, “Design of antennas with greater sensitivity to ultra-high energy
neutrinos,” National Radio Science Meeting, 1 2023.
[30] A. Machtay, J. Rolla, and A. Patton, “Using genetic algorithms to optimize antenna
designs for improved sensitivity to ultra-high energy neutrinos,” 38th International
Cosmic Ray Conference, 2023.
[31] J. Rolla, D. Arakaki, M. Clowdus, A. Connolly, R. Debolt, L. Deer, E. Fahimi, E. Ferstl,
S. Gourapura, C. Harris, L. Letwin, A. Machtay, A. Patton, C. Pfendner, C. Sbrocco,
T. Sinha, B. Sipe, K. Staats, J. Trevithick, and S. Wissel, “Evolving antennas forultra-high energy neutrino detection,” 37th International Cosmic Ray Conference, 2021.
[32] J. Rolla, A. Connolly, K. Staats, S. Wissel, D. Arakaki, I. Best, A. Blenk, B. Clark,
M. Clowdus, S. Gourapura, C. Harris, H. Hasan, L. Letwin, D. Liu, C. Pfendner,
J. Potter, C. Sbrocco, T. Sinha, and J. Trevithick, “Evolving antennas for ultra-highenergy neutrino detection,” 36th International Cosmic Ray Conference, 2019.
[33] D. McCarthy, N. Trappe, J. A. Murphy, C. O’Sullivan, M. Gradziel, S. Doherty, P. G.
Huggard, A. Polegro, and M. van der Vorst, “The optimisation, design and verificationof feed horn structures for future cosmic microwave background missions,” Infrared
Physics & Technology, vol. 76, pp. 32–37, 2016.
[34] M. Calviani, S. di Luise, V. Galymov, and P. Velten, “Optimization of neutrino fluxes
for future long baseline neutrino oscillation experiments,” Nuclear and Particle Physics
Proceedings, 37th International Conference on High Energy Physics, vol. 273-275, pp.
2681–2683, 2016.
[35] H. Schellman, “LBNF beamline,” 39th International Conference on High Energy
Physics, vol. 340, 2018.
[36] A. G. Baydin, K. Cranmer, P. d. C. Manzano, C. Delaere, D. Derkach, J. Donini,
T. Dorigo, A. Giammanco, J. Kieseler, L. Layer, G. Louppe, F. Ratnikov, G. C. Strong,
M. Tosi, A. Ustyuzhanin, P. Vischia, and H. Yarar, “Toward machine learning
optimization of experimental design,” Nuclear Physics News, vol. 31(1), pp. 25–28, 2021.
[37] D. Kastanya, “ADORE-GA: genetic algorithm variant of the ADORE algorithm for
ROP detector layout optimization in CANDU reactors,” Annals of Nuclear Energy,
vol. 46, pp. 160–168, 2012.
[38] E. B. Flynn and M. D. Todd, “Optimal placement of piezoelectric actuators and sensors
for detecting damage in plate structures,” Journal of Intelligent Material Systems andStructures, vol. 21(3), pp. 265–274, 2010.
[39] N. Kleedtke, M. Hua, and S. Pozzi, “Genetic algorithm optimization of tin–copper
graded shielding for improved plutonium safeguards measurements.” Nuclear Inst. andMethods in Physics Research, A, vol. 988, 2021.

26

[40] S. Abdullin, “Genetic algorithm for SUSY trigger optimization in CMS detector at
LHC,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators,
Spectrometers, Detectors and Associated Equipment, vol. 502(2), pp. 693–695, 2003,
Proceedings of the VIII International Workshop on Advanced Computing and Analysis
Techniques in Physics Research.
[41] J. Frenzel, “Genetic algorithms,” IEEE Potentials, vol. 12(3), pp. 21–24, 1993.
[42] K. Man, K. Tang, and S. Kwong, “Genetic algorithms: concepts and applications [inengineering design],” IEEE Transactions on Industrial Electronics, vol. 43(5), pp.
519–534, 1996.
[43] D. Beasley, D. R. Bull, and R. R. Martin, “An overview of genetic algorithms: Part 1,
fundamentals,” University computing, vol. 15, no. 2, pp. 56–69, 1993.
[44] B. O. Community, Blender - a 3D modelling and rendering package, Blender Foundation,
Stichting Blender Foundation, Amsterdam, 2018. http://www.blender.org[45] J. Pan, S. Chitta, and D. Manocha, “Fcl: A general purpose library for collision andproximity queries,” 2012 IEEE International Conference on Robotics and Automation,
pp. 3859–3866, 2012, accessed at: https://github.com/flexible-collision-library.
[46] “Remcom, personal communication,” 2022.
[47] IEEE Standard for Definitions of Terms for Antennas, 2014.
[48] J. Zhong, X. Hu, J. Zhang, and M. Gu, “Comparison of performance between different
selection strategies on simple genetic algorithms,” International Conference on
Computational Intelligence for Modelling, Control and Automation and International
Conference on Intelligent Agents, Web Technologies and Internet Commerce(CIMCA-IAWTIC’06), vol. 2, pp. 1115–1121, 2005.
[49] A. Shuckla, H. M. Pandey, and D. Mehrotra, “Comparative review of selection
techniques in genetic algorithm.” 2015 International Conference on Futuristic Trends on
Computational Analysis and Knowledge Management (ABLAZE), pp. 515–519. IEEE,
2015.
[50] B. L. Miller, D. E. Goldberg et al., “Genetic algorithms, tournament selection, and theeffects of noise,” Complex systems, vol. 9, no. 3, pp. 193–212, 1995.
[51] D. E. Goldberg and K. Deb, A Comparative Analysis of Selection Schemes Used in
Genetic Algorithms, ser. Foundations of Genetic Algorithms, G. J. Rawlins, Ed. Elsevier,
1991, vol. 1.
https://www.sciencedirect.com/science/article/pii/B9780080506845500082[52] E. O. Scott and K. A. De Jong, “Understanding simple asynchronous evolutionary
algorithms,” Proceedings of the 2015 ACM Conference on Foundations of Genetic
Algorithms XIII, p. 85–98, 2015. https://doi.org/10.1145/2725494.2725509[53] E. Scott, M. Coletti, C. Schuman, B. Kay, S. Kulkarni, M. Parsa, and K. De Jong,“Avoiding excess computation in asynchronous evolutionary algorithms,” 20th UK
Workshop on Computational Intelligence, vol. 1409, 11 2021.
https://www.osti.gov/biblio/1845800[54] E. Scott and K. De Jong, “Initialization matters for asynchronous steady-stateevolutionary algorithms,” Proceedings of the Companion Conference on Genetic andEvolutionary Computation, p. 1570–1578, 2023.
https://doi.org/10.1145/3583133.3596404

27

[55] S. M. Hanham, T. S. Bird, A. D. Hellicar, and R. A. Minasian, “Evolved-profile
dielectric rod antennas,” IEEE Transactions on Antennas and Propagation, vol. 59(4),pp. 1113–1122, 2011.
[56] M. A. Belen and P. Mahouti, “Design of nonuniform substrate dielectric lens antennas
using 3d printing technology,” Microwave and Optical Technology Letters, vol. 62(2), pp.
756–762, 2020. https://onlinelibrary.wiley.com/doi/abs/10.1002/mop.32065[57] M. Tam and R. D. Murch, “Half volume dielectric resonator antenna designs,”
Electronics Letters, vol. 33(23), pp. 1914–1916, 1997.
[58] R. W. Ziolkowski, P. Jin, and C.-C. Lin, “Metamaterial-inspired engineering ofantennas,” Proceedings of the IEEE, vol. 99(10), pp. 1720–1731, 2010.
[59] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically small
antennas,” IEEE Transactions on Antennas and Propagation, vol. 56(3), pp. 691–707,
2008.
[60] Y. Zhang, S. Wang, and G. Ji, “A comprehensive survey on particle swarm optimization
algorithm and its applications,” Mathematical Problems in Engineering, vol. 2015, p.
931256, Oct 2015. https://doi.org/10.1155/2015/931256.
[61] D. Wang, D. Tan, and L. Liu, “Particle swarm optimization algorithm: an overview,”
Soft computing, vol. 22, pp. 387–408, 2018.
[62] G. Papazoglou and P. Biskas, “Review and comparison of genetic algorithm and particle
swarm optimization in the optimal power flow problem,” Energies, vol. 16(3), p. 1152,
2023.
[63] R. Thangaraj, M. Pant, A. Abraham, and P. Bouvry, “Particle swarm optimization:
hybridization perspectives and experimental illustrations,” Applied Mathematics andComputation, vol. 217(12), pp. 5208–5226, 2011.
[64] Y. Jin, “A comprehensive survey of fitness approximation in evolutionary computation,”
Soft computing, vol. 9(1), pp. 3–12, 2005.
[65] F. van den Bergh and A. Engelbrecht, “A cooperative approach to particle swarm
optimization,” IEEE Transactions on Evolutionary Computation, vol. 8(3), pp. 225–239,
2004.
[66] J. Quevedo, M. Abdelatti, F. Imani, and M. Sodhi, “Using reinforcement learning fortuning genetic algorithms,” Proceedings of the Genetic and Evolutionary Computation
Conference, New York, NY, USA, p. 1503–1507, 2021.
https://doi.org/10.1145/3449726.3463203

28

APPENDICES

I. Individual Genes and Genetic Algorithm Parameters
Table 1. Parameters used in Genetic Algorithm

Category

Parameters

Run

Num. Individuals, Num. Generations, Fitness
Function, Target pattern, Steady-state flag,
Forced diversity flagIndividualMax. Tree Depth, Max. Shapes, Shell thickness

XFdtd

Frequency range, Frequency step

CuboidShape allowed, Length, Width, and Height Ranges

SphereShape allowed, Radius Range

CylinderShape allowed, Radius, and Height Range

Cone

Shape allowed, Radius 1, Radius 2, and Height Range

Tournament

Percent of parents, Group Size

Selection

Roulette

Percent of parents

Methods

Rank

Percent of parents

Dim. Mutation

Percent of children, St. Dev. %Rotation Mutation

Percent of children, St. Dev. %Location Mutation

Percent of children, St. Dev. %Grow Mutation

Percent of children

Prune Mutation

Percent of children

Genetic

Regen. Mutation

Percent of children

Operators

Side Switch

Percent of children

Gene Crossover

Percent of children

Branch Crossover

Percent of children

Reproduction

Percent of children

Injection

Percent of children

General

Shape

29

Type

Table 2. Primitive shape genes

Gene

ValuesShape Type

Cuboid, Cylinder, Cone, Sphere

Dimensions

Varies by shape type (See Table 3)

Location

Cartesian coordinates of shape midpoint

Rotationθ, ϕConnected From

Shape built from

Connected To

Side that shapes are attached toTable 3. Dimensions for each shape type

30

Shape Type

Dimension Genes

CuboidLength, Width, Height

CylinderRadius, Height

Cone

Inner Radius, Outer Radius, Height

SphereRadiusII. 3-Dimensional Gain Patterns(a) Generation 0(c) Generation 441(b) Generation 25(d) Target

Figure 19. The antenna gain pattern of the most fit individual as of generations 0 (a), 25 (b), and441 (c) of the single-frequency evolution towards a target radiation pattern (d) using the Euclidean
distance-based fitness function.

31

(a) Generation 0
200 MHz(c) Generation 216
200 MHz(b) Generation 5
200 MHz(d) Target
200 MHz

Figure 20. The antenna gain pattern of the most fit individual as of generations 0 (a), 5 (b), and 216 (c)of the broadband evolution towards a target radiation pattern (d) using the Euclidean distance-based
fitness function. Shown here are the gain patterns corresponding to a frequency of 200 MHz.

32

(a) Generation 0516.73 MHz(c) Generation 216
516.73 MHz(b) Generation 5
516.73 MHz(d) Target
516.73 MHz

Figure 21. The antenna gain pattern of the most fit individual as of generations 0 (a), 5 (b), and 216 (c)of the broadband evolution towards a target radiation pattern (d) using the Euclidean distance-based
fitness function. Shown here are the gain patterns corresponding to a frequency of 516.73 MHz.

33

(a) Generation 0800.1 MHz(c) Generation 216
800.1 MHz(b) Generation 5
800.1 MHz(d) Target
800.1 MHz

Figure 22. The antenna gain pattern of the most fit individual as of generations 0 (a), 5 (b), and 216 (c)of the broadband evolution towards a target radiation pattern (d) using the Euclidean distance-based
fitness function. Shown here are the gain patterns corresponding to a frequency of 800.1 MHz.

34

(a) Generation 0(b) Generation 5(c) Generation 95(d) Generation 142

Figure 23. The antenna gain pattern of the most fit individual as of generations 0 (a), 5 (b), and 95 (c)of the single-frequency evolution using the directional gain fitness function.

35

(a) Generation 0
200 MHz(b) Generation 5
200 MHz(c) Generation 229
200 MHz

Figure 24. The antenna gain pattern of the most fit individual as of generations 0 (a), 5 (b), and229 (c) of the broadband evolution using the directional gain fitness function. Shown here are the gain
patterns corresponding to a frequency of 200 MHz.

36

(a) Generation 0516.73 MHz(b) Generation 5
516.73 MHz(c) Generation 229
516.73 MHz

Figure 25. The antenna gain pattern of the most fit individual as of generations 0 (a), 5 (b), and229 (c) of the broadband evolution using the directional gain fitness function. Shown here are the gain
patterns corresponding to a frequency of 516.73 MHz.

37

(a) Generation 0800.1 MHz(b) Generation 5
800.1 MHz(c) Generation 229
800.1 MHz

Figure 26. The antenna gain pattern of the most fit individual as of generations 0 (a), 5 (b), and229 (c), of the broadband evolution towards a target radiation pattern (d) using the Euclidean
distance-based fitness function. Shown here are the gain patterns corresponding to a frequency of
800.1 MHz.

38

III. Additional Results
Generation 25
Antenna Gain at 300 MHz
ZenithAzimuth

0°

0°

45°

315°

90°

45°

270°

­50

315°

­10

90°

270°
-8

-40

-6
-30

-4
-20

-2
-10

0
2

0
135°

225°

135°

180°

225°

180°

Figure 27. The gain patterns of the most and least fit individual in generation 25 of thesingle-frequency evolution towards a target radiation pattern using the Euclidean distance-based fitness
function. Also overlaid is the gain pattern of the target.

Generation 25

Antenna Phase at 300 MHz

180

Antenna Phase at 300 MHz
180

108

Phase (Degrees)

Phase (Degrees)

108

36

−36

−36
−108

−108
−180

36

0

32

65

98

130

163

196

229

261

294

Zenith Direction (Degrees)

327

360

−180

0

32

65

98

130

163

196

229

261

294

327

360

Azimuth Direction (Degrees)

Figure 28. The antenna phase in the zenith and azimuth direction of the most and least fit ingeneration 25 of the single-frequency evolution towards a target radiation pattern using the Euclidean
distance-based fitness function.

39

Generation 5
Antenna Gain at 200 MHz

Antenna Gain at 516.73 MHz

ZenithAzimuthZenith

0°

0°

0°

45°

315°

90°

45°

270°

­60

315°

90°

270°

315°

90°

270°

315°

90°

-15
-20

-2

-20
-10

-10
-10

-5

-1
0

0

0

0
225°

135°

225°

180°

135°

225°

180°

270°

­20

-30
-3

-30

45°

­40

-4

-40

135°

0°

45°

­5

-50

Azimuth

135°

180°

225°

180°

Antenna Gain at 800.01 MHz
ZenithAzimuth

0°

0°

45°

315°

90°

45°

270°

­30

315°

­30

90°

270°
-25
-20

-20

-15

-10

-10
0

-5
0

135°

225°

180°

135°

225°

180°

Figure 29. The gain patterns of the most and least fit individuals in generation 5 of the evolution
towards a target broadband radiation pattern using the Euclidean distance-based fitness function at
three example in-band frequencies: 200 MHz (a), 516.73 MHz (b), and 800.1 MHz (c). Also overlaid
on each is the gain pattern of the target at the given frequency.

40

Generation 5
ZenithAzimuthAntenna Phase at 200 MHz

108

Phase (Degrees)

Phase (Degrees)

108

36

−36
−108
−180

Antenna Phase at 200 MHz

180

180

36

−36
−108

0

32

65

98

130

163

196

229

261

294

327

−180

360

0

32

Zenith Direction (Degrees)

108

108

36

−36

163

196

229

261

294

327

360

327

360

327

360

36

−36
−108

−108

0

32

65

98

130

163

196

229

261

294

327

−180

360

0

32

Zenith Direction (Degrees)

65

98

130

163

196

229

261

294

Azimuth Direction (Degrees)

Antenna Phase at 800.01 MHz

Antenna Phase at 800.01 MHz

180

180

108

108

Phase (Degrees)

Phase (Degrees)

130

Antenna Phase at 516.73 MHz
180

Phase (Degrees)

Phase (Degrees)

Antenna Phase at 516.73 MHz

36

−36
−108
−180

98

Azimuth Direction (Degrees)

180

−180

65

36

−36
−108

0

32

65

98

130

163

196

229

261

294

Zenith Direction (Degrees)

327

360

−180

0

32

65

98

130

163

196

229

261

294

Azimuth Direction (Degrees)

Figure 30. The antenna phase in the zenith and azimuth direction of the most and least fit individualin generation 5 of the evolution towards a target broadband radiation pattern using the Euclidean
distance-based fitness function at three example in-band frequencies: 200 MHz (a,b), 516.73 MHz (c,d),and 800.1 MHz (e,f). Also overlaid on each is the phase of the target.

41

Generation 5
Antenna Gain at 300 MHz
ZenithAzimuth

0°

0°

45°

315°

45°

­30

90°

270°

315°

­15.0

90°

-25

270°

-12.5
-20

-10.0
-15

-7.5
-10

-5.0
-5

-2.5
0

0.0
5

135°

2.5

225°

135°

225°

180°

180°

Figure 31. The gain patterns of the most and least fit individual as of generation 5 of thesingle-frequency evolution using the directional gain fitness function. The regions shaded in green andred represent the angular ranges designated for maximization and minimization, respectively.

Generation 5
Antenna Gain at 200 MHz

Antenna Gain at 516.73 MHz

ZenithAzimuthZenith

0°

0°

0°

45°

315°

90°

45°

270°

­35

315°

5
135°

270°

315°

90°

225°

135°

270°

315°

90°

-15

-20
-15

-10

-10

-5

-5
0

0

5
135°

225°

180°

270°

­20

-25

225°

180°

45°

­30

-15.0
-12.5
-10.0
-7.5
-5.0
-2.5
0.0

0

0°

45°

­17.5

90°

-30
-25
-20
-15
-10
-5

Azimuth

135°

180°

225°

180°

Antenna Gain at 800.01 MHz
ZenithAzimuth

0°

0°

45°

315°

90°

45°

270°

­30

315°

90°

-20

-20

-15

-15
-10

-10
-5

-5

0

0

5
135°

225°

180°

270°

­25

-25

135°

225°

180°

Figure 32. The gain patterns of the most and least fit individuals in generation 5 of the evolution using
the directional gain maximization/minimization fitness function at three example in-band frequencies:
200 MHz (a), 516.73 MHz (b), and 800.1 MHz (c). Also overlaid on each is the gain pattern of thetarget at the given frequency. The regions shaded in green and red represent the angular ranges
designated for maximization and minimization, respectively.

42

IV. Computation Time Per Generation

This section summarizes an analysis conducted on the GA’s computation time. Given
the large number of generations and individuals that may be necessary to evaluate inorder to converge on a solution, it is important to quantify the computation of thealgorithm. The study discussed here analyzed the total computation time per
generation using data collected during evolutions towards a target broadband
radiation pattern using the Euclidean distance fitness function as well as both
single-frequency and broadband evolutions using the directional gain fitness function.
A number of factors are expected to influence the typical computation time of ageneration, including the number of individuals per generation, number of frequency
steps evaluated per individual, overall dimensions and number of shapes making up
the individuals, and the number of simulations able to be conducted in parallel. Eachof the evolutions that the data presented in this section is drawn from contained 120
individuals per generation, and the other parameters are not expected to deviatesignificantly between the evolutions towards a target radiation pattern using the
Euclidean distance-based fitness function and the directional gain fitness function,with the exception of the number of simulations set to run in parallel. This factor wasa user-defined input that was limited by the number of XFdtd software licenses
available. For the data presented in this section, the evolutions using the Euclidean
distance fitness function utilized the capability to simulate 9 individuals in parallel,
while the directional gain evolutions were only capable of simulating 5 individuals inparallel. As a result, the data presented here is grouped accordingly.
Another consideration to note is that the data presented in this section includes thetime that jobs spent waiting in the queue of the Ohio Supercomputer Center (OSC),
where the antenna simulations were performed. These queue times are an unavoidable
component of the overall computation time, and may be inconsistent due to theirdependence on resource availability.
The vast majority of computation time necessary to create and evaluate an individualis due to the antenna simulation and queue time. The other stages of the process,including parent selection, application of genetic operators, writing genetic information
to file, and modeling the individual, are completed on the order of a few seconds. Thisis a negligible amount of time relative to the time that it takes to simulate a fullgeneration worth of individuals, which will be seen in the analysis presented here to beon the order of approximately 1 hour or more for a typical generation.
Figure 33 displays a histogram of the computation time per generation from datacollected during the evolution towards a target broadband radiation pattern using the
Euclidean distance fitness function. The majority of generations completed withinapproximately 100 minutes. The generations seen to take significantly longer may be
attributed to abnormally long queue times due to periods of high traffic at thesupercomputing center.

43

Time per 120 Individuals in Euclidean Runs
40

Generations

35
30
25
20
15
10
5
0

0

50

100

150

200

250

300

350

400+

Time (minutes)
Figure 33. Histogram of the per generation computation time observed during the evolution to a target
broadband radiation pattern using the Euclidean distance-based fitness score.

Time per 120 Individuals in Directional Runs

40
35

Generations

30
25
20
15
10
5
0

0

50

100

150

200

250

300

350

400+

Time (minutes)
Figure 34. Histogram of the per generation computation time observed during the evolutions using thedirectional gain fitness score, including both single-frequency and broadband evolutions.

Figure 34 shows a histogram of the computation time per generation from datacollected during evolutions using the directional gain fitness function, including datafrom both single-frequency and broadband evolutions. Most generations still
completed within approximately 100 minutes, but the increased computation time due
to a lesser amount of parallelization is evident by the increased fraction of generations

44

with longer durations. The generations seen to take significantly longer may be still be
attributed to abnormally long queue times due to periods of high traffic at thesupercomputing center.
The results of the analysis presented agree with the expectation that increased
parallelization of the simulation of individuals, known to be the most significant
contributor to the overall computation time of the evolutions, generally leads to ashorter per generation computation time. Outlier generations significantly longer induration were observed but are attributed to unavoidable increases in queue time due
to limited resources at the computing facilities where the simulations were run.

V. Genetic Algorithm Developments and Performance Studies
Various techniques were explored for reducing total computation time required toconverge to a solution by improving the rate of evolution in the GA. These studies
were conducted using the direct dictionary comparison fitness function for ashape-to-shape evolution to a log periodic antenna geometry presented in the previous
progress report [1].
Figure 35 compares the different steps taken to improve the GA. The baseline result isthe evolution presented in the prior report and is shown in the black line. Since thisresult was obtained, a number of improvements were made to algorithm. Specifically,
improvements to the method for recalculating a shape’s midpoint after it or the shapethat it is connected to has been operated upon and the addition of the side switching
regenerative mutation operator significantly increased the probability of individuals
finding improved designs. The version of the generational algorithm with these
improvements implemented is shown in pink. Implementation of the steady-statealgorithm is shown in green and further improves on the rate of convergence. Finally,
the fastest version of the GA, shown in yellow, utilized an additional forced diversity
constraint and elitism. The forced diversity constraint requires that no child produced
be a duplicate of an individual already in the population, and elitism requires that themost fit individual in the population at the current generation be ineligible to be
selected for removal.
Additional studies were conducted that allowed the genetic operator probabilities tobe changed dynamically in each generation depending on the evolution’s current state.
This method used reinforcement learning [66] to tune the genetic operator
probabilities based on the fitness score growth rate and diversity of the population ineach generation. Results of initial tests of the dynamic operator addition to the GA
are shown in Figure 36, alongside the performance of various versions of the GA
without dynamic operators applied.
The reinforcement learning method for dynamic operators requires model training inorder to discover the optimal operator probabilities. This training may either be
conducted prior to the start of an evolution and imported, or it may be conducted45

Log Periodic Fitness
1.0

IPNPR 42-234
GGA Mean
GGA Runs
SSGA Mean
SSGA Runs
Elite FD-SSGA Mean
Elite FD-SSGA Runs

Fitness Score

0.8

0.6

0.4

0.2

0

50

100

150

200

250

Generation
Figure 35. Comparisons of the evolutions using the direct dictionary comparison fitness function for ashape-to-shape evolution to a log periodic antenna geometry [1] using various verisons of the GA. Each
result shown is the mean fitness score per generation over 10 trials. Results tested include the original
generational GA from the previous progress report in yellow (IPNPR), the updated generational GA in
pink (GGA), the steady-state GA in green (SSGA), and the forced diversity steady-state GA in blue
(FDSSGA). The lighter lines are individual test runs, and the darker lines are the mean.

using data from the current evolution itself, allowing the algorithm to attempt to learn
optimal parameters as it progressed. The pretrained result shown in the orange line on
Figure 36 was trained on 200 runs of 250 generations apiece. As seen in Figure 36, it
was found that the pretrained test led to faster evolution; however, the existence of
ample training data may be restrictive in cases where the per generation run time islong. Further studies of the implementation of dynamic parameter tuning in the GA
are planned, including investigation of the impact of allowing various additionalparameters to dynamically change throughout an evolution, as well as studies ofpotential improvements to the reinforcement learning portion of the algorithm.

46

Fitness Score

Mean Log Periodic Fitness
1.0

GGA
SSGA
DO-SSGA (Untrained)

0.8

DO-SSGA (Trained)
FD-SSGA
Elite FD-SSGA

0.6

0.4

0.2
0

50

100

150

200

250

Generation
Figure 36. Comparisons of the mean fitness score per generation over 10 trial evolutions for various
additions to the GA. Each evolution used the direct dictionary comparison fitness function for ashape-to-shape evolution to a log periodic antenna geometry, as initially described in a previous
progress report [1]. Versions of the GA tested include the Generational GA (GGA) in pink,
Steady-State GA (SSGA) in green, Steady-State GA with Dynamic Operators without pre-training(DO-SSGA [Untrained]) in red, Steady-State GA with Dynamic Operators with pre-training (DO-SSGA[Trained]) in organge, Forced Diversity Steady-State GA (FD-SSGA) in dark blue, and Forced Diversity
Steady-State GA with Elitism (Elite FD-SSGA) in light blue.

47

JPL CL#24-2236