#  Wave-Particle Duality of Ultrasound: Acoustic Softening Explained by
Particle Treatment of Ultrasonic Wave

Libin Yang  School of Industrial Automation, Beijing Institute of Technology,
Zhuhai, 519088, China Email<EMAIL_ADDRESS>Lixiang Yang  Department of
Mechanical and Aerospace Engineering, The Ohio State University, Columbus OH,
43210 Email<EMAIL_ADDRESS>
###### Abstract

When ultrasonic wave is irradiated on materials, a small static stress is
required to get materials yielding and flowing. This is called acoustic
softening effect, also known as Blaha effect for a long time. In the past,
this effect was explained by several continuum scale or meso-scale solid
mechanics theories such as stress superposition or energy superposition
theory, or crystal/dislocation plasticity. Due to a lot of microscopic
complexities happening inside the materials during ultrasonic vibration, fully
understanding of acoustic softening effect is not easy. In this paper,
traditional solid mechanics theory is expanded by introducing several concepts
in semi-conductor physics. Four new aspects were introduced to understand
acoustic softening effect. Firstly, contrary to most existed work in acoustic
softening research area which treats ultrasound as waves, it was treated as
particles. Secondly, crystal/dislocation plastic theory was simplified to a
single equation. Thirdly, concepts of photoelectric effect or photo-voltaic
effect were introduced. Analogy of electron movement due to light wave and
defect movement due to ultrasonic wave was illustrated. Particularly, as light
wave is treated as photon, ultrasonic wave is treated as phonon in this paper.
Fourthly, defects such as point defects or line defects are assumed to have
certain bonding energy. Their bonding energies are assumed to be quantized or
discontinuous. The band gap theory used in photo-voltaic theory is embraced to
understand defects movements in solid mechanics due to ultrasonic phonon.

Keywords: Wave-Particle Duality; Ultrasonic Wave; Acoustic Softening,
Photoelectric Effect; Photo-voltaic Effect

##  1 Introduction

Ultrasonic vibration has found a lot applications in industries, such as
ultrasonic cut, ultrasonic cleaning, and ultrasonic welding [1, 2, 3].
Ultrasonic wave is also widely used in biomedical areas such as ultrasonic
imaging, ultrasonic therapy, and ultrasound in dentistry[4]. All of these
applications are related to how to use and control the energy ultrasound
carries. In spite of the widely known Acousto-Plastic Effect (APE) application
in industries and biomedical applications, the basic understanding of the APE
mechanism is far from complete. In ultrasonic cut and welding, it is known
that ultrasound will make materials soft. Experimental findings shows
ultrasound vibration can dramatically make the crystalline materials soft
without significant heating, which is referred as acoustic softening [5]. This
acoustic softening effect is temporary. It only appears during the application
of the high-frequency vibration and will vanish as soon as the vibration
stops. It is shown that acoustic softening can more efficiently reduce
yielding stress compared to heating softening. Acoustic energy is assumed to
be absorbed only by lattice imperfection area (dislocations or grain
boundaries) while heat is distributed homogeneously among all the atoms of the
crystals. Dislocation can be more easily moved by periodic stress than by
constant stresses [6]. Therefore, APE is a more effective way to metal
yielding, because a relatively small stress is needed and the effect is fast.
Strain hardening process assisted by ultrasonic vibration is more effective as
compared to that in conventional strain hardening processes. Experimental
observations show the stress reduction by acoustic softening in different
crystalline materials is proportional to the vibration amplitude [7, 8]. Since
the amplitude of the imposed vibrations is proportional to the input energy,
it is widely accepted that the amplitude of the vibrations is an important
factor for the APE processes [9, 10]. The frequency effect under ultrasonic
vibration is in contradiction. But most researchers thought that frequency
effect is negligible [11, 12]. Moreover, most of research works acknowledged
that the influence of temperature on the APE is negligible.

In more than a half century, various theories have been proposed to describe
the basic mechanisms of the APE process. From historic experimental
development and observation, APE can depend on parameters like the frequency
and amplitude of imposed vibration, and the temperature of environment.
Several hypotheses have been proposed to explain experimental results. These
theories can be classified into three groups: (i) the dislocation / potential-
well hypothesis, (ii) the stress-superposition hypothesis, and (iii) the
energy superposition hypothesis. But a comprehensive mathematical model is
unavailable. In the process of mathematical derivation, many empirical
equations and assumptions were introduced but cannot easily be validated. A
complete review of acoustic softening was given by Liu and Graff [4]. In
chapter 13 of their book, more than 170 articles are collected spanning from
early 1950s to 2023. Following the history line, some representative work in
acoustic softening research area will be given in the following section.

Nevill [11] reported that the decrease of yielding stress in tension in a low-
carbon steel specimen is proportional to the vibrational amplitude and
independent of the frequency of vibration in the frequency range of 15 kilo
cycle to 80 kilo cycle. Study by Blaha and Langenecker [13] showed that sound
waves would change the energy level of the absorbing dislocations. When a
static stress is applied to the metal sample, the dislocation velocity will
increase and plastic deformation would occur. Consequently, the apparent
static stress which is needed for plastic deformation will be reduced.

To further understand ultrasound induced dislocation movements, Langenecker
[14] conducted a series of experiments on zinc, aluminum, beryllium, tungsten,
low-carbon steel and stainless steel. He concluded that sound energy was
absorbed only by lattice imperfections. But thermal energy was absorbed
homogeneously by the whole material. He found that APE is independent of
vibration frequency or the temperature of the material. But APE highly depends
on the amplitude of ultrasound. After continuously testing of the effect of
ultrasound energy level, Langenecker proposed that dislocation theory and
stress wave propagation were not enough to describe the interactions between
ultrasonic waves and the dislocations in metal crystalline grains. He,
instead, proposed an energies superposition hypothesis. The energy required
for plastic deformation is the summation of thermal energy of a dislocation,
the energy produced through internal friction of oscillation dislocations, and
the energy resulting from acoustic strain. Langenecker’s work has a large
impact on the following decades of research work on APE. He had some
suggestions in the future research direction based on quantum mechanics and
phonon-phonon and phonon-electron interaction. But his work stopped there. The
idea of phonon concept and quantum mechanics may be necessary to understand
acoustic softening. At his time, semi-conductor industry and photo-voltaic
industry are not well developed. In fact, energy band gap theory used for
building photo-voltaic solar cells is an energy method based on quantum
mechanics. We will talk about it in the later section.

Series of APE tests were also reported by Kirchner et al [12]. An hourglass
shaped aluminum alloy was subjected to a cyclic loading of 20 KHz. A quasi-
static compression is applied at strain rate of 0.0001/second. And a vibratory
loading at 0.5, 1.0, 10., 50., 20 KHz are simultaneously applied to the metal
samples. Their experimental results showed no frequency influence on the APE.
No microscopic structural changes such as rearrangement of dislocation
structures are observed. The measured mean stress dropped to a lower value
after imposing vibratory loading. The stress will return back to its original
static stress-strain curve after cessation of vibrations. And they realized
that it is very difficulty to measure stress inside the specimen because of
inhomogeneous stress distribution.

Oscillatory unidirectional stresses experiments on crystals of KCl, NaCl, and
NaBr by Ohgaku and Takeuchi [15] confirmed that the effect of temperature on
the APE was negligible. The higher amplitudes of imposed ultrasonic vibrations
led to reduction in “static-loading stress”. They claimed that the theorem of
stress superposition cannot explain acoustic softening effect.

Mao et al.[16] studied acoustic softening in aluminum and its alloys. They
found that the reduction in flow stress is linearly proportional to the
applied ultrasound intensity for all three types of aluminum alloys.

Contrast to acoustic softening, residual acoustic hardening may happen if
ultrasound of sufficiently high-stress amplitude is applied on the crystalline
materials in a long period and turned off [17]. This residual hardening will
show up as higher stress in stress-strain curve after ultrasound is stopped.
It will finally get saturated even with very high amplitude ultrasound [18,
19]. It is assumed that dislocations reach the maximum number of dislocations
in the crystals that can be created by ultrasound.

Numerical simulation also becomes a powerful tool for investigate ultrasonic
welding processing [20, 21, 22]. Finite element analysis has been widely used
[23, 24]. In ultrasonic bonding processing, a large elasto-plastic deformation
happens. This observation makes people to think the current solid mechanics
theory which is mostly built on small deformation at Lagrange frame cannot be
used. The fundamental theory from Eulerian frame may be more suitable.
Therefore, conservation of mass, momentum, and energy needed be solved with
finite elastic-plastic constitutive model or elastic-plastic kinetic evolution
equations [25, 1]. Fundamental simulation of finite elastoplastic deformation
was attempted by many researchers [26]. The numerical simulation can clearly
show nonlinear wave stress profile, density evolution, and velocity
distribution during the ultrasonic welding processing. But the limitation is
obvious. Numerical errors by discretization of governing equation and boundary
condition could smash results. Artificial damping is added into numerical
models to stabilize the computation. The influence of huge amount of defects
existed in the materials can hardly be modeled by numerical methods. The
elastic-plastic energy flow may not be correctly modeled by the current
elastic-plastic constitutive models or elastic-plastic kinetic evolution
equations. Mathematical modeling of acoustic softening is even more
challenging. Based on existed crystal plasticity theory, many mathematical
models were built to understand acoustic softening effect. For example, by
adding a softening factor into plastic slip flow rule, Siddiq and Sayed [27,
28] built a model to relate yielding stress softening to ultrasonic intensity.
However, most mathematical structures based on dislocation theory are fairly
complex [29]. Usually many differential equations and more than 20 parameters
are needed to capture acoustic softening [30]. Nearly all mathematical models
are considering to add an additional softening parameter to lower down the
yielding stress. In this paper, a new approach is presented to understand
acoustic softening with the yielding stress ignored. Plasticity of materials
will be determined by Young’s modulus, defect density and defect velocity. The
stress deviates from the linear relationship by defect density changes. Defect
density is assumed to strongly depend on the number of phonons which is
proportional to the amplitude of ultrasonic wave. A concise crystal plastic
equation is used to explain acoustic softening effect in section III. In
section II, the wave-particle duality of light and sound is discussed.
Particularly, particle treatment of ultrasound will be illustrated.

##  2 Wave-Particle Duality of Light and Sound

In the early 20th century, there was a big challenge in physics called
ultraviolet catastrophe. That is, black body will emit infinity energy if
wavelength of light decreases to the ultraviolet range. The challenge was
solved by Max Planck. He assumed that the energy of light is not continuous
and should be discretized. His idea was later adopted by Albert Einstein to
understand photoelectric effect. At that time everyone thought that light is
electromagnetic wave. Its motion should obey Maxwell’s equation. Albert
Einstein, however, explained photoelectric effect by treating light as
particles instead of wave. He stated that light should be viewed as wave in
some cases and particles in the some other cases. When light is irradiating on
a metal surface, electrons will absorb energy from light particles or photons.
If photon energy is larger than the work function of the metal, electrons will
be ejected from the metal. The number of electrons ejected is proportional to
the number of photons. The photon energy is proportional to its frequency
only, e.g., $E=h\nu$, where $E$ is the energy of photons, $h$ is Planck’s
constant, $\nu$ is frequency. Therefore, the maximum kinetic energy of the
photoelectrons is independent of the light intensity. The kinetic energy of
emitted electrons is determined by light frequency. On the other hand, the
number of electrons ejected is proportional to the intensity of the incident
light. Amplitude of electric current is proportional to available free
electrons ejected by photons. So if light is treated as wave, its energy is
related to wave amplitude. If it is treated as photon, energy of photon is
proportional to its frequency. Therefore, photon flux can be defined as the
amount of photons per time per area. For fixed wave length, photon flux $\Phi$
is related to light wave power density $P$ by

$\Phi\>=P\frac{\lambda}{hc}\,$ (2.1)

where $c$ is the speed of light and $\lambda$ is the wave length of light.

The particle treatment of light also plays crucial role in photovoltaic
industry. Similar to photoelectric effect, photovoltaic effect, firstly
discovered by Edmond Becquerel in 1839, now has been utilized to produce solar
cells. When high enough frequency light is shining on solar cells, photon
energy with different frequency will be absorbed by different band gap semi-
conductor materials. Photon energy will be absorbed by covalent electrons.
After covalent electrons gain enough energy from photon, for example, larger
than band gap energy, they then have enough energy to move away from the
constraint of nuclei. They become free and move around inside materials. In an
energy concept, electrons will jump from covalent band to conduction band. A
schematic sketch of band gap concept for electrons is shown in Fig.(1)(a).

Fig. 1:  Schematic sketch of energy band gap theory. (a) Electrons in covalent
band jump to conduction band after absorbing photon energy. (b) Defect atoms
in immobile band jump to mobile band after absorbing phonon energy.

When electrons jump to conduction band, holes will be created in covalent
band. Incident light causes electron and hole pairs to be generated in the
semiconductor. Electrons are free to move in conduction band and holes are
free to move in covalent band. Both the conduction-band electrons and covalent
band holes play important parts in the electrical behavior of photovoltaic
cells. Electrons and holes freed from their positions in the crystal by light
photon are called as light-generated excessive electron-hole pairs. If there
is no internal built electric potential, these light-generated excessive
electrons and holes will recombine immediately when light is turned off, which
means the energy of electrons and holes cannot be utilized. In solar cells,
however, in order to use the electron and hole energy, a build-in electric
potential, e.g., PN junction (P type crystalline silicon and N type
crystalline silicon junction) is needed. PN junction will separate electrons
and holes. Electrons and holes will move in the opposite direction. When
electrons and holes move across an external circuit, their energy can be
collected. That is how solar cell creates power.

Similarly, there is an analogy between light wave and acoustic wave. They
share the same wave characteristic properties such as wave reflection, wave
refraction and total internal reflection. When studying nano transport theory,
Gang Chen [31] treated electrons, photons, and phonons in a unified way. Just
like light wave, ultrasonic wave should have both wave and particle
properties. Ultrasonic wave is a stress wave with frequency higher than 20000
Hz. It has all wave properties such as wave speed, wave length, amplitude and
frequency. But its particle side is usually ignored by many researchers. By
analogy to photon in light wave, the particle treatment of acoustic wave is
called phonon [32]. Phonon is the smallest energy carrier of ultrasonic wave.
Just like photons can interact with electrons, phonons will interact with
defects such as dislocations when ultrasonic wave is irradiated on materials.
The interaction between ultrasonic wave and defects is very complex. It will
be related to inelastic scattering. However, if ultrasonic wave is treated as
phonons, all complexity can be solved. The power density or amplitude of
ultrasonic wave is related to the number of phonons. The phonon energy is
given by $h\nu$, where $h$ is Planck’s constant and $\nu$ is the frequency of
ultrasonic wave. The number of mobile defects such as mobile dislocations
generated by phonons is proportional to the number of phonons. Since the
mesoscale defects inside materials are generally due to missing or redundancy
of atoms, these defects such as mobile dislocations here are treated as atom-
hole pairs similar to electron-hole pairs generated by light wave. A similar
energy band gap treatment of phonon induced atom-hole pair is sketched in
Fig.(1)b. After obtaining phonon energy, the weakly bonded atoms at defect
area such as dislocation area will jump from immobile band to mobile band and
leave holes behind in the immobile band. So atoms in mobile band and holes in
immobile band can move around. Their movements will cause plastic deformation.
In acoustic softening experiments, people found that the extent of stress
softening of materials is proportional to the amplitude of the ultrasonic wave
applied but independent of wave frequency. If ultrasound is treated as
particles, high amplitude ultrasonic wave means more phonons. It will create
more mobile atom-hole pairs. The activation energy to to free atoms from their
pinned equilibrium positions is the band gap energy which is the minimum
energy to create mobile atom-hole pairs. Different materials have different
band gap energy. Phonon energy need be larger than band gap energy to active
dislocations and create mobile atom-hole pairs. Since phonon energy only
depend on its frequency, high frequency phonon is more like to create atom-
hole pairs. The number of mobile atom-hole pairs is independent of frequency
of ultrasonic wave. The kinetic energy of atom-hole pairs is decided by band
gap energy. The excessive energy (the difference of phonon energy and band gap
energy) absorbed by pinned dislocations will go through thermalization process
and change to heat in a very short time. Therefore, defect density and defect
velocity are independent of ultrasonic frequency. Defect density in materials
will depend on the number of phonons or amplitude of ultrasonic wave. With
more mobile atom-hole pairs generated by ultrasound, materials become easy to
deform. If ultrasound is turned off, mobile atoms and holes will recombine
since there is no built-in potential to separate atoms and holes like PN
junction in semi-conductor materials. Once atoms and holes are recombined,
less mobile atom-hole pairs will remain. Stress will return back to the
original position. A sketched stress-strain curve with ultrasound is given by
Fig.(2).

Fig. 2:  Schematic sketch of stress-strain curve with ultrasound turned on and
turned off.

Both light and sound have wave and particle characterization. In some cases
they will act like wave. In some other cases they will act like particles. But
how do we know it should be treat as wave or particles? Albert Einstein didn’t
give us an answer. One idea is that when wave packet size is much less than
domain size, it be treated as particles. Gang Chen [31] used coherent phase to
explain wave-particle duality. He proposed that if an incident wave loses its
phase during its propagation, it should be treated as particles. We agree with
his idea. A propagating ultrasonic wave may easily lose its phase when it
meets with defects. Its energy will be absorbed by pinned atoms and create
mobile atom and hole pairs. These mobile atoms will move around and collide
with lattices and lose their energy, which leads to thermalization. As a
result, wave energy is lost at the defect areas. This can be sketched in
Fig.(3). So defects will act as an impedance or a barrier to acoustic wave
propagation. If defect density is large enough, wave speed can be taken as
zero for acoustic wave [33]. This is because wave energy is all absorbed by
defects and the phase of acoustic wave is entirely lost. Therefore, acoustic
wave need be treated as phonon when there exists plenty of defects. This
happens to electrons too. It is called quantum tunneling where kinetic energy
of electrons is less than the potential barrier they are trying to pass
through.

Fig. 3:  Schematic sketch of ultrasonic wave interaction with defects (purple
color atoms). Ultrasonic wave loses its energy to defects and also loses its
phase during its propagation.

##  3 A Mathematical Model of Acoustic Softening

As derived for both amorphous and crystalline materials, the one-dimensional
defect based elastic-plastic model can be written as [33]

$\frac{d\sigma}{d\epsilon}\>=\>E\,-\,q\sigma^{n}\epsilon^{m}\,$ (3.1)

where $\sigma$ is stress, $\epsilon$ is strain, $E$ is Young’s modulus, $q$ is
a constant. m an n depend on defect density and defect velocity evolution
respectively. In this stress-strain relationship, no yielding stress is
needed. Defect density will gradually increase with increased external stress.
The second term of Eq.(3.1) will increase with stress and strain. The slope of
stress and strain curve will gradually decrease and a maximum stress can be
reached if slope becomes zero.

In the derivation of Eq.(3.1), two formula are adopted: defect density is
related to strain by $\rho=a\epsilon^{m}$ and defect velocity is related to
stress by $v=b\sigma^{n}$. With these two equations, Eq.(3.1) can be rewritten
as

$\frac{d\sigma}{d\epsilon}\>=\>E\,-\,C\rho v\,,$ (3.2)

where $C$ is a constant, $\rho$ and $v$ are defect density and defect
velocity. Young’s modulus, $E$, depends on time, temperature, hydrostatic
pressure and immobile defect density [34]. As long as velocity of defects
inside materials is zero, the second term of Eq.(3.2) is zero. Eq.(3.2) will
return back to Hooke’s law. When ultrasonic wave is added, phonon will create
excessive atom-hole pair. These atoms and holes will move around. Defect
density will increase by additional term, e.g., $\delta\rho$. Therefore,
stress and strain relationship without ultrasonic vibration, e.g., Eq. (3.2),
need be changed to

$\frac{d\sigma}{d\epsilon}\>=\>E\,-\,C(\rho+\delta\rho)v.$ (3.3)

Eq.(3.3) is stress and strain relationship with ultrasonic vibration. An
excessive defect density $\delta\rho$ is added to the equation. This excessive
defect density strongly relates to the amount of phonon, which is proportional
to intensity of ultrasonic wave. Similar to photon flux, phonon flux is
defined as the amount of phonons per time per area. For fixed wave length,
phonon flux $\Phi$ is related to power density $P$ by

$\Phi\>=P\frac{\lambda}{hC_{s}}\,$ (3.4)

where $h$ is Plank’s constant, $C_{s}$ is speed of ultrasound. Unit of power
density is given as watts per area.

Stress and strain relationship with ultrasonic vibration can also be written
as the following form

$\frac{d\sigma}{d\epsilon}\>=\>E\,-\,q\sigma^{n}(\epsilon^{m}+\epsilon_{U}),$
(3.5)

where excessive strain, $\epsilon_{U}$, is due to excessive defect density
$\delta\rho$ created by ultrasonic wave. With higher intensity of ultrasonic
wave, $\delta\rho$ is larger. Strain is also larger. The slope of stress and
strain curve will become smaller. Stress will become lower. After ultrasound
is turned off, atoms and holes will recombine immediately. Therefore,
excessive defect density, e.g., $\delta\rho$, will become zero. Stress-strain
curve will return back to the original curve without ultrasound imposed.
However, this is just an idealized case. Defect density and locations cannot
be exactly the same as those before ultrasound is exerted on the materials.
Depending on ultrasonic vibration time, the final immobile defect density left
inside materials can be different compared to initial defect density. When
vibration time is short, there is no enough time for atom-hole pairs to
combine. There are more defects left in the materials. The atoms are less
closely packed. Recall that Young’s modulus is determined by number of bonding
of atoms in a unit volume. So Young’s modulus will decrease. This leads to
residual acoustic softening effect. When vibration time is long enough, mobile
atom-hole pairs have enough time to combine. Atoms in the materials are more
closely packed. Bonding density of unit volume will increase. Young’s modulus
will got enhanced, which leads to residual acoustic hardening effect. Young’s
modulus variation by ultrasonic vibration was observed by several researchers
[35]. This can be shown in our mathematical model in the next section. A
schematic sketch of atom redistribution due to ultrasonic wave is shown in
Fig.(4).

Fig. 4:  Atoms are more closely packed after long time ultrasonic vibration.
Young’s modulus will increase due to long time ultrasonic vibration.

To validate the current mathematical model, model results are compared to two
sets of experimental results. Firstly, the model is used to compare with
stress and elongation curve of aluminum single crystals. The experiments were
performed by Langenecker [14]. Ultrasonic wave with 20 k cycle/second was used
in the experiments. He firstly got a stress-strain curve without ultrasonic
vibration. Then he added ultrasonic wave to the tests. By increasing power
density of ultrasonic wave from 15 $watts/cm^{2}$ to 50 $watts/cm^{2}$, he
found the stress that leads to the same amount strain will decrease.
Ultrasonic vibration makes materials easy to deform. His experimental data are
replotted in Fig.(5) to compare with our model prediction. In the model
equation, Eq. (3.5), Young’s modulus of high-purity aluminum single crystals
is set to be 70 GPa. $q$ is choose to be 69.22 GPa. $m=0.00285$ and $n=0$.
Excessive strain, $\epsilon_{U}$, is 0 under no ultrasonic vibration.
Excessive strain, $\epsilon_{U}$ is selected to be 0.0011, 0.0046, 0.0079 for
ultrasonic power density level 15 $watts/cm^{2}$, 35 $watts/cm^{2}$ and 50
$watts/cm^{2}$ respectively. In small strain, e.g., $\epsilon<0.2$, model
prediction is a little deviation from the experimental data. As plastic strain
increases, model prediction becomes better. The overall stress and strain
shapes under different power density ultrasonic vibration are well predicted
by our mathematical model. Since Eq.(3.5) is a nonlinear equation, the
relationship of stress and strain is calculated by using nonlinear equation
solver $ode23s$ in Matlab. In the calculation, stress has real part and
imaginary part. Real part is used and imaginary part is discarded.

Fig. 5:  Experimental data by Langenecker [14] are compared to model
prediction. Experimental data are represented by different shape and color
markers. Model results are shown by using different color solid lines. The
equations and parameters are also shown to predict stress and strain curves
under no ultrasonic vibration (blue line), 15 $watts/cm^{2}$ (green line), 35
$watts/cm^{2}$ (magenta line), 50 $watts/cm^{2}$ (yellow line).

To further validate our model equation, model results are compared to another
set of experimental data given by Yao [30]. Vibration frequency 9.6 $Hz$ was
selected for the commercially pure aluminum (Al 1100, McMaster-Carr)
ultrasonic vibration tests. In the experimental tests, they observed that flow
stress is significantly reduced once ultrasonic vibration is turned on. They
also saw residual hardening of materials after ultrasonic vibration is turned
off. In their tests, residual hardening can be seen when vibration time is 8
seconds and no residual hardening is observed when vibration time is 2
seconds. Their tests also showed that effect of strain rate on residual
hardening is smaller than the effect of ultrasonic vibration time. As we
discussed before, after long time ultrasonic vibration, there is more time for
atoms and holes to recombine. Young’s modulus will increase by a small amount.
In our equation, Eq.(3.5), if let Young’s modulus increase from initial value,
$70000MPa$ to a new value, $70030MPa$, and keep all other parameters
unchanged, flow stress will increase due to this small change of Young’s
modulus. Stress and strain curve with residual hardening obtained by the
mathematical model is plotted against experimental data as shown in Fig. (6).
When strain is less than $0.1$, model results deviate from the experiments.
When strain increases, especially, in large plastic zone, model prediction
gets better. For small strain region, model prediction can be improved if many
low defect energy terms are added to the right hand of Eq.(3.5).

Fig. 6:  Model predictions are compared to experimental data conducted by Yao
et al.[30]. Experimental data are represented by different shape and color
markers. Model results are shown by using different color solid lines. Model
predicted results are shown as blue line (no ultrasonic vibration), green line
(with ultrasonic vibration) and yellow line (residual acoustic hardening).

##  4 Conclusion

In the past, ultrasonic wave propagation theory and dislocation theory have
been continually used to understand acoustic softening. After many years’
development, more and more complex physical concepts and mathematical
structures are added. This made the science behind of acoustic softening even
elusive. In this paper, however, some attempts to simplify its physical
concepts and mathematical structures are presented. The ultrasonic wave is
treated as particles, e.g., phonons. We walk away from traditional dislocation
theory by introducing energy band gap theory used in semi-conductor physics.

Energy of phonon is assumed to be proportional to its frequency. Number of
defects created by ultrasonic wave is proportional to phonon numbers or power
density of ultrasonic wave. Residual softening or hardening after ultrasonic
vibration is turned off is related to the change of Young’s modulus.

Energy band gap theory is originally built to explain electrons jump from
covalent band to conduction band after light irradiation. This theory is
extended to explain atoms jump from immobile band to mobile band after
ultrasonic wave irradiation. Based on atom-hole pair generation and
recombination, ultrasonic wave induced excessive defect density is added to a
crystal plasticity equation. Yielding stress is not required in this
mathematical model. The new constitutive model is bench marked with two
experimental results and shows good consistence.

## Data Availability

Data sharing not applicable - no new data generated.

## Reference

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# Energy-Aware Service Offloading for Semantic Communications in Wireless
Networks

Hassan Saadat1, Abdullatif Albaseer1, Mohamed Abdallah1, Amr Mohamed2, Aiman
Erbad1
1 Division of Information and Computing Technology, College of Science and
Engineering
Hamad Bin Khalifa University, Doha, Qatar
2 Department of Computer Science and Engineering, Qatar University, Doha,
Qatar Email: 1{hasa52143, aalbaseer, moabdallah<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Today, wireless networks are becoming responsible for serving intelligent
applications, such as extended reality and metaverse, holographic
telepresence, autonomous transportation, and collaborative robots. Although
current fifth-generation (5G) networks can provide high data rates in terms of
Gigabytes/second, they cannot cope with the high demands of the aforementioned
applications, especially in terms of the size of the high-quality live videos
and images that need to be communicated in real-time. Therefore, with the help
of artificial intelligence (AI)-based future sixth-generation (6G) networks,
the semantic communication concept can provide the services demanded by these
applications. Unlike Shannon’s classical information theory, semantic
communication urges the use of the semantics (meaningful contents) of the data
in designing more efficient data communication schemes. Hence, in this paper,
we model semantic communication as an energy minimization framework in
heterogeneous wireless networks with respect to delay and quality-of-service
constraints. Then, we propose a sub-optimal solution to the NP-hard
combinatorial mixed-integer nonlinear programming problem (MINLP) by utilizing
efficient techniques such as discrete optimization variables’ relaxation. In
addition, AI-based autoencoder and classifier are trained and deployed to
perform semantic extraction, reconstruction, and classification services.
Finally, we compare our proposed sub-optimal solution with different state-of-
the-art methods, and the obtained results demonstrate its superiority.

###### Index Terms:

semantic communication, 6G networks, classical information theory, energy
minimization, resource allocation

##  I Introduction

In recent years, the purpose of data communication has transformed from being
only concerned with transmitting text, images, videos, and live streaming into
more complex and intelligent applications. For example: 1) virtual reality
(VR), such as metaverse and VR games, 2) holograms, 3) intelligent
transportation, such as self-driving cars, and 4) collaborative robots, such
as in medical and industrial environments. These applications tend to
communicate with very large sizes of live data and require very fast
transmission in terms of milli or even microseconds. Although current fifth-
generation (5G) networks can provide data rates of up to gigabytes/seconds,
they still cannot cope with the requirements of the aforementioned highly-
demanding intelligent applications. Therefore, a smarter and more efficient
way of communication is urged to be deployed in future networks in order to
serve modern applications. One of the promising solutions is to apply the
concept of semantic communication (SemCom).

The current traditional way of communication is based on Shannon’s classical
information theory (CIT)[1]. In his CIT, Shannon believed that the semantics
(meaningful contents) of the transmitted data should not affect the design of
the communication framework. His belief aligns with the technical level of
communication that Weaver defined in [2]. SemCom, on the other side, tries to
move beyond naive communication at the technical level and achieve higher
levels of intelligent communication at the semantic and effectiveness levels.
SemCom is a communication framework that opts to convert communicating nodes
from being naive symbol-exchanging transmitters and receivers into becoming
more intelligent agents that exchange the semantic parts of the data to help
effectively and efficiently achieve the goal of communication [3].

Deploying SemCom in heterogeneous wireless networks has many challenges that
must be encountered. For example, although SemCom guarantees less transmission
energy consumption compared to traditional communication -as less data is
being transmitted-, the extra computational energy consumed during semantic
extraction and reconstruction must be considered. Also, due to the
heterogeneous nature of wireless networks, not all end devices can deploy the
same semantic extraction techniques and respect the same delay constraints. In
addition, SemCom introduces semantic extraction and reconstruction losses that
have to be minimized. Furthermore, a semantic knowledge mismatch between the
transmitter and the receiver could lead to errors that negatively affect the
goal of communication.

To evaluate the capabilities of current networks and determine the required
technologies for SemCom, Yang et al. [4] present AI-enabled SemCom and its 6G
key enablers, discuss its communication- and semantic-related techniques and
challenges, and provide examples of applications where SemCom can be
beneficial. In [3], Chaccour et al. provide a theoretical foundation for
future SemCom. The authors view future SemCom as an intelligent reasoning-
driven framework where communication sides build a common semantic language
and knowledge base. DeepSC, a more practical work, is proposed by Xie et al.
[5]. DeepSC sets the ground for numerous current deep learning (DL)-enabled
SemCom implementations, where joint source-channel encoding is performed, a
generalized transfer learning-based SemCom framework is proposed, and a new
text semantic similarity metric is introduced.

The existing works in the literature have deployed SemCom in various
applications. For example, Tong et al. [6] propose a federated learning-based
audio transmission SemCom framework, and use convolutional neural network-
based autoencoders for the semantic tasks. To apply the concept of
disentangling the data sample into learnable and memorizable parts as
discussed in their earlier work [3], Chaccour et al. [7] use a contrastive
learning-based approach, which managed to reduce the size of transmitted data
by 57%, while maintaining the same semantic performance. In the work of
Farshbafan et al., [8], curriculum and reinforcement learning (RL) were used
to build a common semantic language in a goal-oriented SemCom-based dynamic
environment. In the paper of Wang et al. [9], a RL-based solution is applied
at the base station to optimize the resource allocation and semantic token
selection ratio, such that maximal text reconstruction similarity at the
users’ sides is ensured with respect to delay constraints.

Even though the aforementioned works tackle interesting SemCom-related
problems, none of them takes into consideration the computation and
transmission energy consumption of the system. Also, they either consider a
system that contains one edge/base station and its connected users, or they
assume distance-based user-edge association (i.e., user-edge association is
not a decision variable in their optimization problem). In Yang et al.’s work
[10], a rate splitting multiple access (RSMA)-based [11] SemCom is deployed in
downlink communication between a base station and its users. Here, the goal is
to perform physical resource and semantic extraction ratio allocation to
minimize the total energy consumption of the system with respect to a semantic
accuracy threshold. Xiao et al. [12] apply DeepSC-based uplink SemCom in
healthcare environments. In their system, medical users require intelligent
health services, and the cloud performs user-edge association and bandwidth
allocation, such that the total energy of the system is minimum and quality-
of-service (QoS) is maximum. Noting that each edge, based on its available
capabilities, can serve different services with different qualities.

As will be shown next, our work differs from the discussed works, such that we
apply energy-aware image-based SemCom in wider networks and treat both the
user-edge association and semantic extraction ratio allocation as part of our
optimization decisions.

The objectives of this paper are highlighted as follows:

  1. 1.

Model SemCom as an energy minimization framework in heterogeneous wireless
networks, with respect to minimum QoS and maximum delay constraints.

  2. 2.

Propose a sub-optimal solution to the NP-hard combinatorial mixed-integer
nonlinear programming problem (MINLP) by utilizing efficient techniques such
as discrete optimization variables’ relaxation.

  3. 3.

Train and deploy DL-based autoencoder and classifier to perform semantic
extraction, reconstruction, and classification on the exchanged data.

  4. 4.

Compare the proposed SemCom framework with state-of-the-art user-edge
association methods, and the results show the superiority of our approach.

Figure 1: System Model

##  II System Model & Problem Formulation

In the system shown in Fig. 1, we have a cloud server acting as a control
unit, a set of edge nodes $N$ having a common semantic decoder with input size
$\theta$ and providing a common service type (i.e., object classification),
and a set of end users $U$ having a common semantic encoder with output size
$\theta$ and requiring the service offered by the edge nodes. At each round,
before the communication starts, the cloud performs user-edge association
($\mathbf{X}\in\\{0,1\\}$), semantic extraction ratio allocation
($\mathbf{\Delta}\in(0,1]$), user CPU frequency allocation
($\mathbf{F}\in\Re^{+}$), and edge CPU frequency allocation
($\mathbf{H}\in\Re^{+}$). The size of each decision matrix above is
$|U|\times|N|$.

Once the decision matrices are set, each user $u$ semantically encodes its
image sample $D_{u}$ and randomly selects a fraction from the encoded image
that is equal to the allocated $\delta_{un}\in(0,1]$, such that the total size
of the data to be transmitted by each user $u$ becomes
$\delta_{un}\times\theta$. Higher $\delta_{un}$ means that more semantic
information is sent from user $u$ to edge $n$, which intuitively means edge
$n$ can more accurately reconstruct user $u$’s image and perform the
classification service. If we call the quality (classification accuracy) of
service (QoS) provided by edge $n$ to user $u$ as $Q_{un}(\delta_{un})$, then
the total QoS of the system can be calculated as:

$Q=\sum_{u=1}^{|U|}\sum_{n=1}^{|N|}x_{un}Q_{un}(\delta_{un}),$ (1)

where $x_{un}$ is the user-edge association variable, such that $x_{un}=1$ if
user $u$ is associated with edge $n$, and $x_{un}=0$ otherwise.

The semantic extraction and selection operations by user $u$ take a
computational time of:

$t_{un}^{comp_{u}}=\frac{y_{1un}(|D_{u}|,\theta,\delta_{un})}{f_{un}},$ (2)

and consume a computational energy calculated as:

$E_{un}^{comp_{u}}=\kappa y_{1un}(|D_{u}|,\theta,\delta_{un})f_{un}^{2},$ (3)

where $\kappa$ is the effective switched capacitance coefficient, $y_{1un}$ is
the number of CPU cycles needed by user $u$ to encode its image and select
$\delta_{un}$ part of the encoded image, and $f_{un}$ is the allocated local
CPU frequency of user $u$ when associated with edge node $n$.

Next, the transmission time needed to send the semantic information from user
$u$ to edge $n$ is:

$t_{un}^{trans}=\frac{\delta_{un}\theta}{r_{un}},$ (4)

where $r_{un}=b_{un}\log_{2}(1+\frac{p_{un}g_{un}}{b_{un}N_{0}})$ is the
upload transmission rate, $b_{un}$ is the given bandwidth, $g_{un}$ is the
channel gain between user $u$ and edge $n$, $N_{0}$ is the noise spectral
density, and $p_{un}$ is the given transmission power for user $u$ to send its
data to edge $n$. The transmission energy consumed can be calculated as:

$E_{un}^{trans}=t_{un}^{trans}p_{un}=\frac{\delta_{un}\theta}{b_{un}\log_{2}(1+\frac{p_{un}g_{un}}{b_{un}N_{0}})}p_{un}.$
(5)

After that, the computational time consumed by edge $n$ to reconstruct user
$u$’s data and perform the classification service is calculated as:

$t_{un}^{comp_{e}}=\frac{y_{2un}(|D_{u}|,\theta,\delta_{un})}{h_{un}}.$ (6)

and the computational energy is:

$E_{un}^{comp_{e}}=\kappa y_{2un}(|D_{u}|,\theta,\delta_{un})h_{un}^{2},$ (7)

where $y_{2un}$ is the number of CPU cycles required by edge $n$ to serve user
$u$, and $h_{un}$ is the CPU frequency dedicated on edge $n$ to serve user
$u$.

The total energy consumed by the system can be calculated as:

$E=\sum_{u=1}^{|U|}\sum_{n=1}^{|N|}x_{un}(E_{un}^{comp_{u}}+E_{un}^{trans}+E_{un}^{comp_{e}}).$
(8)

To this end, we can define our goal as performing user-edge association,
semantic extraction ratio allocation, user CPU frequency allocation, and edge
CPU frequency allocation, such that the system’s total energy is minimized,
with respect to global minimum QoS and maximum delay thresholds (set by the
cloud), as formulated in the following optimization problem:

$\displaystyle\min_{\mathbf{X,\Delta,F,H}}\quad E,$ (9)

s.t.

$x_{un}Q_{un}(\delta_{un})\geq Q^{min},\quad\forall u\in{U}\>\&\>\forall
n\in{N},$ (10) $\quad
x_{un}(t_{un}^{comp_{u}}+t_{un}^{trans}+t_{un}^{comp_{e}})\leq
T^{max},\quad\forall u\in{U}\>\&\>\forall n\in{N},$ (11) $x_{un}f_{un}\leq
F_{u}^{max},\quad\forall u\in{U}\>\&\>\forall n\in{N},$ (12)
$\sum_{u=1}^{|U|}x_{un}h_{un}\leq H_{n}^{max},\quad\forall n\in{N},$ (13)
$0<\delta_{un}\leq 1,\quad\forall u\in{U}\>\&\>\forall n\in{N},$ (14)
$0<f_{un},h_{un},\quad\forall u\in{U}\>\&\>\forall n\in{N},$ (15)
$x_{un}\in\\{0,1\\},\quad\forall u\in{U}\>\&\>\forall n\in{N},$ (16)
$\sum_{n=1}^{|N|}x_{un}=1,\quad\forall u\in{U}.$ (17)

Constraint (10) ensures that each user gets served with a minimum QoS of
$Q^{min}$. Constraint (11) ensures that the maximum delay of the system is
below $T^{max}$. Constraints (12) and (13) ensure that the allocated CPU
frequencies of each user $u$ and edge node $n$ do not exceed their maximum
capacity. Constraint (14) guarantees that the size of data to be sent by each
user is within the range $(0\times\theta,1\times\theta]$. Constraint (15)
guarantees that each user gets served (gets allocated local and edge CPU
frequencies). Finally, constraints (16) and (17) allow each user $u$ to be
associated with one edge node only.

##  III Proposed Solution

To make our optimization problem in (9) convex and solvable, the following
limitations must be encountered:

  1. (a)

The relationships between $Q_{un}$ and $\delta_{un}$ in (1), $y_{1un}$ and
($|D_{u}|,\theta,\delta_{un}$) in (3), and $y_{2un}$ and
($|D_{u}|,\theta,\delta_{un}$) in (7) need to be identified.

  2. (b)

Due to the binary user-edge association variable $x_{un}$, the optimization in
(9) is a combinatorial mixed-integer nonlinear programming problem (MINLP)
which is NP-hard [13].

Figure 2: Classification accuracy of reconstructed images by trained
autoencoders with variable output size $\theta$. Figure 3: Classification
accuracy of reconstructed images by our autoencoder with $\theta=0.5(\times
28\times 28)$ and variable semantic extraction ratio $\delta_{un}$.

To encounter the limitation mentioned in (a), first, we introduce the common
autoencoder used in the system. On the users’ side, we have an encoder that
takes a $28\times 28$ image (MNIST image [14]) in the input layer and has a
fully-connected output layer of size $\theta$. At each edge node’s side, there
exists a decoder that takes an encoded image of size $\theta$, and
reconstructs it to its original size of $28\times 28$. Then, each edge node
$n$ classifies the reconstructed image of its connected user(s) $u$ with a
classification accuracy of $Q_{un}$ using a pre-trained MNIST classifier. The
classifier has an input layer of $28\times 28$ nodes, a hidden layer of 400
nodes, and an output layer of 10 nodes (labels). To select the encoder output
size $\theta$, we train ten autoencoders for 5 epochs on the MNIST training
dataset, each with a different value of $\theta$. From Fig. 2, we can see that
the autoencoder’s reconstruction classification accuracy starts converging to
97% at $\theta=0.5(\times 28\times 28)$. Therefore, to reduce the size of the
transmitted images on the network, it is safe to select and deploy the
autoencoder with an output size of $\theta=0.5(\times 28\times 28)$ in our
system.

Then, to find the relationship between the QoS metric $Q_{un}$ and the
semantic extraction ratio $\delta_{un}$, we encode the 10k-MNIST testing set
using our selected encoder, then decode and classify the images based on a
randomly selected $\delta_{un}$ portion of the encoded images. Note that
regardless of the value of $\delta_{un}$, the decoder architecture is fixed
and still expects an input size of $\theta$. Therefore, the receiver fills the
missing $1-\delta_{un}$ portion with zero pixels. As can be seen in Fig. 3, a
linearly perfect positive relationship between $Q_{un}$ and $\delta_{un}$ is
observed, where $Q_{un}=0.873\delta_{un}+0.1006$. This relationship is because
sending more semantic information to the receiver results in more accurate
image reconstruction and classification. Based on this conclusion, (1) can be
re-written as:

$Q=\sum_{u=1}^{|U|}\sum_{n=1}^{|N|}x_{un}(0.873\delta_{un}+0.1006)$ (18)

Figure 4: Average computation time of semantic extraction and selection
$t_{un}^{comp_{u}}$ at the user side, and reconstruction and classification
$t_{un}^{comp_{e}}$ at the edge side.

To identify the relationships $y_{1un}(|D_{u}|,\theta,\delta_{un})$ and
$y_{2un}(|D_{u}|,\theta,\delta_{un})$, we need to bring our attention to the
fact that $|D_{u}|$ in our system is fixed as $28\times 28$ pixels (MNIST
image), and $\theta$ is fixed as $0.5\times|D_{u}|$. Therefore, the number of
CPU cycles $y_{1un}$ needed by user $u$ to perform semantic extraction and
selection, and CPU cycles $y_{2un}$ needed by edge node $n$ to perform
semantic reconstruction and classification, are only affected by the variable
$\delta_{un}$. Now, to identify the relationships $y_{1un}(\delta_{un})$ and
$y_{2un}(\delta_{un})$, we know from (2) and (6) that for fixed CPU
frequencies $f_{un}$ and $h_{un}$, the computation latencies
$t_{un}^{comp_{u}}$ and $t_{un}^{comp_{e}}$ are directly proportional to
$y_{1un}(\delta_{un})$ and $y_{2un}(\delta_{un})$, respectively. Therefore, we
infer our autoencoder on the 10k-MNIST testing set, and measure
$t_{un}^{comp_{u}}$ and $t_{un}^{comp_{e}}$ on different values of
$\delta_{un}$. We can see in Fig. 4 that $t_{un}^{comp_{u}}$ and
$t_{un}^{comp_{e}}$ are constants even with variable $\delta_{un}$. We can
reason these findings to the fact that the $\delta_{un}$ portion selection
happens after $D_{u}$ is encoded. Also, the edge nodes fill the missing
$1-\delta_{un}$ portion with zero pixels and then perform reconstruction on
$\theta$ pixels in all cases. Also, the $\delta_{un}$ portion is randomly
selected with constant complexity. All in all, the same number of semantic
computations are performed. Hence, $y_{1un}$ and $y_{2un}$ are constants and
independent from $\delta_{un}$. The spikes in Fig. 4 represent negligible
periodic operations on the experiment’s host computer.

Finally, to sub-optimally solve the combinatorial MINLP introduced in (b), we
relax the user-edge association matrix X to be any real number $\in[0,1]$.
Then, after the sub-optimal problem is solved, each user $u$ is associated
with edge node $n$ that corresponds to the highest $x_{un}$.

Now that we have countered the limitations defined in (a) and (b), we can
reformulate our original optimization problem in (9) into a sub-optimal
problem as follows:

$\displaystyle\min_{\mathbf{X,\Delta,F,H}}\quad\sum_{u=1}^{|U|}\sum_{n=1}^{|N|}x_{un}(\kappa
y_{1un}f_{un}^{2}+$ (19)
$\displaystyle\frac{\delta_{un}\theta}{b_{un}\log_{2}(1+\frac{p_{un}g_{un}}{b_{un}N_{0}})}p_{un}+\kappa
y_{2un}h_{un}^{2}),$

s.t.

$\frac{Q^{min}-0.1006}{0.873}\leq\delta_{un}\leq 1,\quad\forall
u\in{U}\>\&\>\forall n\in{N},$ (20) $\quad
t_{un}^{comp_{u}}+t_{un}^{trans}+t_{un}^{comp_{e}}\leq T^{max},\quad\forall
u\in{U}\>\&\>\forall n\in{N},$ (21) $0\leq f_{un}\leq F_{u}^{max},\quad\forall
u\in{U}\>\&\>\forall n\in{N},$ (22) $\sum_{u=1}^{|U|}h_{un}\leq
H_{n}^{max},\quad\forall n\in{N},$ (23) $0\leq h_{un},\quad\forall
u\in{U}\>\&\>\forall n\in{N},$ (24) $x_{un}\in[0,1],\quad\forall
u\in{U}\>\&\>\forall n\in{N},$ (25) $\sum_{n=1}^{|N|}x_{un}=1,\quad\forall
u\in{U}.$ (26)

The sub-optimal problem in (19) is convex and is an instance of geometric
programming optimization problems, and can be solved using the GPkit library
on python [15]. As seen in constraint (20), $\delta_{un}$ boundaries are
redefined by utilizing the relationship found in Fig. 3. Also, $x_{un}$ is
taken out from constraints (21), (22), and (23), in order to guarantee that
these constraints are not violated after performing de-relaxation on $x_{un}$.
Furthermore, we can see the relaxation of the user-edge association variable
$x_{un}$ in (25).

##  IV Performance Evaluation

In this section, we compare the performances of three different user-edge
association methods: 1) our proposed sub-optimal solution as in (19), 2)
minimum distance-based user-edge association, and 3) random user-edge
association. In the minimum distance and random associations, problem (19) is
solved to determine F, H, and $\mathbf{\Delta}$, while X is treated as a given
parameter and is no longer an optimization variable. For the sake of
comparison, we run two experiments: 1) measuring the total energy consumption
with variable delay threshold $T_{max}$, and 2) measuring the total energy
consumption with variable QoS threshold $Q_{min}$.

###  IV-A Simulation settings

In our experiments, we consider a system with $|U|=20$ users and $|N|=4$ edge
nodes. The users are randomly distributed in a circle of 500 meter radius, and
the edge nodes are randomly located in a ring with an inner radius of 500
meters and an outer radius of 1000 meters of the same center [12]. Each user
$u$ has a maximum local CPU frequency capacity of
$F_{u}^{max}\in\\{0.5,1.0,1.5,2.5\\}$ GHz. All edge nodes have the same
maximum CPU frequency capacity, that is, $H_{n}^{max}=4$ GHz, $\forall n\in
N$. The effective switched capacitance coefficient is set as $\kappa=10^{-28}$
[10]. As for the network resources, each edge node can allocate a total
transmission power of $P_{n}^{max}=30$ dBm to the users and a total bandwidth
of $B_{n}^{max}=20$ MHz. The transmission power and bandwidth of each edge
node $n$ are divided among all users in fixed values proportional to their
distance $d_{un}$.

###  IV-B Energy Consumption with Variable $T^{max}$

In the experiment shown in Fig. 5, we measure the system’s total energy
consumption at a fixed minimum QoS threshold $Q^{min}=0.8$ while varying the
maximum delay threshold $T^{max}=1,2,\dots,10$ seconds. Since $Q^{min}$ is
fixed at 0.8, we know from the relationship found in Fig. 3 that the lower
bound of $\delta_{un}$ should be
$\frac{0.8-0.1006}{0.873}=0.8011\leq\delta_{un}$. Also, from (19), in order to
minimize the transmission energy, the optimization decision will allocate the
minimum allowed $\delta_{un}$ to all users that guarantee to achieve
constraint (20), which is 0.8011. In other words, for a fixed $Q^{min}=0.8$
and variable $T^{max}$, the transmission energy $E^{trans}_{un}$ and
transmission time $t^{trans}_{un}$ stay constant as the amount of data to be
transmitted is fixed as $0.8011\times\theta$. Therefore, as shown in Fig. 5,
as the time constraint gets stricter (as $T^{max}$ decreases), the
optimization decision needs to allocate more CPU frequencies (higher $f_{un}$
and $h_{un}$) in order to reduce $t^{comp_{u}}_{un}$ and $t^{comp_{e}}_{un}$,
which results in increasing the user and edge computational energy consumption
by a factor of $f_{un}^{2}$ and $h_{un}^{2}$, respectively.

###  IV-C Energy Consumption with Variable $Q^{min}$

In the experiment shown in Fig. 6, we measure the system’s total energy
consumption at a fixed maximum delay threshold of $T^{max}=10$ seconds while
varying the minimum QoS threshold $Q^{min}=0.2,0.3,\dots,0.9,0.9736$ (maximum
possible $Q_{un}$ when $\delta_{un}=1$). Since $Q^{min}$ increases, the lower
bound of $\delta_{un}$ also increases. And we know from Fig. 4 that the user
and edge computational latencies ($t^{comp_{u}}_{un}$ and $t^{comp_{e}}_{un}$)
are independent of the value of $\delta_{un}$ and $Q^{min}$, which also means
that the user and edge computational energy consumption ($E^{comp_{u}}_{un}$
and $E^{comp_{e}}_{un}$) are independent of the value of $Q^{min}$. Therefore,
we conclude that the increase in the total energy as a result of increasing
$Q^{min}$ (Fig. 6) is solely reasoned to an increase in the transmission
energy $E^{trans}_{un}$. Also, the linearity can be reasoned to the fact that
the transmission energy is linearly and directly proportional to the size of
data to be transmitted ($\delta_{un}\theta$), i.e.,
$E^{trans}_{un}\propto\delta_{un}$, which is clearly shown in equation (5).

In both experiments, we can see that our sub-optimal solution outperforms the
other baselines as it consumes less energy. The distance-based association
provides a competitive performance. The reason behind that is, as stated
earlier, the only decision that differentiates between the performances of the
user-edge association methods is X, while all the other decision matrices F,
H, and $\mathbf{\Delta}$, are determined by solving the sub-optimal problem
(19). In other words, the differences in the total energy consumption is much
more significantly affected by the transmission energy than the computational
energy. Therefore, the minimum distance association method performance is
close to the sub-optimal solution because a shorter distance between
communication sides generally consumes less transmission energy and, hence,
less total energy consumption. Finally, the random association consumes the
highest total energy because it does not consider the distances and channel
states between the users and edge nodes.

Figure 5: Total system energy consumption with variable threshold $T^{max}$
Figure 6: Total system energy consumption with variable threshold $Q^{min}$

##  V Conclusion

In this paper, the SemCom concept was briefly introduced, its deployment
challenges were concisely discussed, and several SemCom-related literature
works were reviewed. Then, an energy minimization SemCom framework that
considers delay and QoS constraints was modeled. In addition, a DL-based
autoencoder and classifier trained on the MNIST dataset were deployed for the
semantic task. After that, regression and relaxation techniques were utilized
to propose a sub-optimal solution to the formulated combinatorial MINLP
optimization problem. Next, comparison experiments between the proposed sub-
optimal solution and baseline user-edge association methods at different delay
and QoS constraints were conducted. The results show that the proposed sub-
optimal solution efficiently managed to reduce the energy consumption of the
system while maintaining the defined constraints, compared to the other
baseline methods. Finally, further enhancements can be done in future work,
such as 1) making the solution more abstract and generalizable rather than
being specific to our defined application, and 2) adding the transmission
power and bandwidth to the optimization problem as decision variables.

##  VI Acknowledgment

This work was made possible by GSRA grant # GSRA9-L-1-0519-22025 from the
Qatar National Research Fund (a member of Qatar Foundation). The findings
achieved herein are solely the responsibility of the authors.

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