Computer simulation of diffusion transport processes in multilayer nanofilms
The difficulties associated with studying diffusion in multilayer films require the progress of contemporar y modeling methods and software platforms to precisely represent phenomena, taking into account transitions between adjacent layers. In addition to the indispensable role of advanced modeling...
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| author | Petryk, M.R. Doroshenko, A.Yu. Mykhalyk, D.M. Yatsenko, О.A. |
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| description | The difficulties associated with studying diffusion in multilayer films require the progress of contemporar y modeling methods and software platforms to precisely represent phenomena, taking into account transitions between adjacent layers. In addition to the indispensable role of advanced modeling techniques and software in solving the problems of studying diffusion in multilayer films, it is extremely important to admit the key contribution of sophisticated computational approaches. In this paper, the authors attempt to merge intricate mathematical models with optimal software development methodologies to address the challenge of simulating diffusion transport processes in multilayer nanofilms computationally. Based on the experimental findings and employing the suggested model, identification was conducted utilizing the theory of state control for multicomponent systems. With the help of methods of optimal control of the state of multicomponent transport systems, the analytical solution of the model and the data of experimental observations, the distributions of diffusion coefficients for the considered components of nanofilms (samples of aluminum, molybdenum, silicon) were reproduced. Numerical simulation results were compared with experimental observations. The profiles obtained from the modeling closely match the corresponding experimental profiles, especially as the duration of multilayer formation converges to the final stages of completing the protective nanofilm multilayer formation. The maximum observed deviation does not exceed 2–3%, confirming the reliability of the mathematical model and demonstrating the practical value of the results provided. A software framework is developed for the automation of the specified calculations with the possibility of extension to other subject areas with similar tasks of identifying the key factors of the process and further numerical modeling of the time-space characteristics using the obtained results.Prombles in programming 2024; 2-3: 62-68 |
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Комп’ютерне моделювання
62
UDC 619.6 http://doi.org/10.15407/pp2024.02-03.062
M.R. Petryk, A.Yu. Doroshenko, D.M. Mykhalyk, О.A. Yatsenko
COMPUTER SIMULATION OF DIFFUSION TRANSPORT
PROCESSES IN MULTILAYER NANOFILMS
The difficulties associated with studying diffusion in multilayer films require the progress of contemporar y
modeling methods and software platforms to precisely represent phenomena, taking into account trans itions
between adjacent layers. In addition to the indispensable role of advanced modeling techniques and software
in solving the problems of studying diffusion in multilayer films, it is extremely important to admit the key
contribution of sophisticated computational approaches. In this paper, the authors attempt to merge intricate
mathematical models with optimal software development methodologies to address the challenge of simulat-
ing diffusion transport processes in multilayer nanofilms computationally. Based on the experimental find-
ings and employing the suggested model, identification was conducted utilizing the theory of state control
for multicomponent systems. With the help of methods of optimal control of the state of multicomponent
transport systems, the analytical solution of the model and the data of experimental observations, the distri-
butions of diffusion coefficients for the considered components of nanofilms (samples of aluminum, molyb-
denum, silicon) were reproduced. Numerical simulation results were compared with experimental observa-
tions. The profiles obtained from the modeling closely match the corresponding experimental profiles, espe-
cially as the duration of multilayer formation converges to the final stages of completing the protective nan-
ofilm multilayer formation. The maximum observed deviation does not exceed 2–3%, confirming the relia-
bility of the mathematical model and demonstrating the practical value of the results provided. A software
framework is developed for the automation of the specified calculations with the possibility of extension to
other subject areas with similar tasks of identifying the key factors of the process and fur ther numerical
modeling of the time-space characteristics using the obtained results.
Key words: coefficients identification, diffusion, mathematical model, multilayer nanofilm, numerical simu-
lation, oxide film.
М.Р. Петрик, А.Ю. Дорошенко, Д.М. Михалик, О.А. Яценко
КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ ПРОЦЕСІВ
ДИФУЗІЙНОГО ТРАНСПОРТУ В БАГАТОШАРОВИХ
НАНОПЛІВКАХ
Проблеми, пов’язані з дослідженням дифузії в багатошарових плівках, вимагають удосконалення су-
часних методів моделювання та програмного забезпечення для точного зображення явищ, враховую-
чи переходи між сусідніми шарами. На додаток до незамінної ролі передових методів моделювання
та програмного забезпечення у розв’язанні проблем вивчення дифузії в багатошарових плівках, над-
звичайно важливо визнати ключовий внесок складних обчислювальних підходів. У цій роботі автори
зробили спробу поєднати складні математичні моделі з передовим досвідом розробки програмного
забезпечення для розв’язання задачі комп’ютерного моделювання процесів дифузійного транспорту в
багатошарових наноплівках. За результатами експериментальних даних та з використанням запропо-
нованої моделі проведено ідентифікацію з використанням теорії контролю стану багатокомпонент-
них систем. За допомогою методів оптимального керування станом багатокомпонентних транспорт-
них систем, аналітичного рішення моделі та даних експериментальних спостережень відтворено роз-
поділи коефіцієнтів дифузії для розглянутих складових компонентів наноплівок (зразків алюмінію,
молібдену, кремнію). Результати чисельного моделювання порівнюються з експериментальними спо-
стереженнями, що відображають вміст зразка. Профілі, отримані в результаті моделювання, тісно
збігаються з відповідними експериментальними профілями, особливо коли тривалість багатошарово-
го утворення наближається до остаточного періоду завершення утворення захисної наноплівки. Мак-
симальне спостережене відхилення не перевищує 2–3 %, що підтверджує надійність математичної
моделі та вказує на практичну корисність отриманих результатів. Розроблено програмний інструмен-
тарій для автоматизації зазначених розрахунків з можливістю поширення на інші предметні області
зі схожими завданнями ідентифікації ключових факторів процесу. Створено базу для подальшого чи-
сельного моделювання часо-просторових характеристик з використанням отриманих результатів.
Ключові слова: ідентифікація коефіцієнтів, дифузія, математична модель, багатошарова наноплівка,
чисельне моделювання, оксидна плівка.
© M.R. Petryk, A.Yu. Doroshenko, D.M. Mykhalyk, О.A. Yatsenko, 2024
ISSN 1727-4907. Проблеми програмування. 2024. №2-3
Комп’ютерне моделювання
63
Introduction
The challenges associated with
investigating diffusion in multilayer films
necessitate the advancement of modern
modeling techniques and software frameworks
to accurately depict phenomena while
considering transitions between neighboring
layers [1, 2]. One of the most effective ap-
proaches to thoroughly address these issues is
the widely recognized integral transformation.
These methods are utilized to derive solutions
for diverse boundary value problems in
mathematical physics pertaining to
homogeneous structures, encompassing
diffusion scenarios across various environments
and enabling their mathematical representation.
In addition to the indispensable role of
advanced modeling techniques and software
frameworks in addressing the challenges of
studying diffusion within multilayer films, it is
crucial to recognize the pivotal contribution of
sophisticated computational approaches. By
utilizing cutting-edge methodologies in
computer programming and software
architecture, the capabilities of mathematical
approaches, such as integral transformations,
can be elevated to unprecedented levels of
precision and efficiency. This synergy between
mathematical modeling and computational
innovation not only enhances our ability to
accurately represent diffusion phenomena
within multilayer structures but also opens new
avenues for exploration and analysis.
Furthermore, the integration of
modern modeling techniques with state-of-
the-art computational tools facilitates a more
nuanced understanding of diffusion processes
across diverse material compositions and
environmental conditions. By leveraging the
power of numerical simulations and data-
driven analyses, researchers can extract
valuable knowledge on complex diffusion
mechanisms that were previously
inaccessible. This multidisciplinary approach
enables scientists and engineers not only to
solve fundamental questions in materials
science and engineering but also to devise
innovative strategies for optimizing the
performance and functionality of multilayer
films in various technological applications.
Problem formulation. In the pro-
posed research, authors attempt to combine
complex mathematical models with the best
practices of software engineering to solve the
problem of computer simulation of diffusion
transport processes in multilayer nanofilms.
The primary objectives are:
− modeling diffusion processes: to in-
tegrate complex mathematical models with
contemporary software development method-
ologies, in domain of the diffusion within
multilayer films;
− parameter identification: to employ
the theory of state control for multicomponent
systems to identify diffusion coefficients. It
includes using methods of optimal control to
analyze experimental data and accurately re-
produce the distributions of diffusion coeffi-
cients for the nanofilm components;
− numerical simulation and validation:
to conduct numerical simulations and compare
the results with experimental observations. The
target is to achieve a high degree of corre-
spondence between the modeled and experi-
mental profiles, particularly as the duration of
multilayer formation approaches completion;
− development of a software frame-
work: to create a software framework that au-
tomates the specified calculations and can be
extended to other subject areas with similar
tasks. This framework should facilitate the
identification of key process factors and ena-
ble further numerical modeling of time-space
characteristics based on the obtained results.
By addressing these objectives, this
research seeks to overcome the limitations of
existing methods and provide a robust tool for
studying and optimizing the diffusion pro-
cesses in multilayer films. The ultimate aim is
to enhance the efficiency of experimental
studies and contribute to the development of
new nanomaterials with improved properties
1. Mathematical model
We propose a physical problem and
corresponding mathematical model for the
diffusion mass transfer process in multilayer
films, based on the following multilayer me-
dium comprised of n double layers composed
Комп’ютерне моделювання
64
of two distinct media with differing properties
(Fig. 1).
Fig.1. Schema of multilayer nanofilm
According to this representation, at
each interface within the formed multicompo-
site, there is a mutual diffusion of components
between two adjacent layers of the medium.
The mechanisms governing such mutual
transfer are determined by the variable gradi-
ents and rates of concentration change at the
interface boundaries between layers. By in-
corporating changes in concentrations and
their gradients over time into the boundary
and interface conditions, it becomes feasible
to model the mechanisms of this additional
mutual transport alongside the fundamental
transport equations.
When formulating a mathematical
model for diffusion transfer within oxide
films, a multilayer structure is taken into ac-
count. Assuming that the diffusion of atoms
of constituent components of oxide films
(aluminum, molybdenum, silicon) primarily
drives system mixing, concentration profiles
for such a multilayer system can be derived
from Fick’s equations, incorporating bounda-
ry conditions at the outer layers and contact
conditions between successive layers. This
approach provides a mathematical model de-
scribing the diffusion transfer process within
a planar multilayered medium
2 2
2
0 2 2( , , )
k
k k
k k k z
C CC t x z C D D
z x z
+ = +
(1)
at domain
1
1 0 1
1
0, (0, ),
: ( , ); 0; ,
n
k k n
k
t x R
z z l l l l
+
− +
=
=
1, 2 1, ,j k n= =
where kC is diffusion distribution; kD is dif-
fusion coefficient; 2
k is mass distribution co-
efficient;
The corresponding initial and bounda-
ry conditions for the model:
0 0 0( , , ) ( , ) ( ),
k kk tC t x z C x z C z= =
0 0
0 0
11 11 1( , , ) ( , );z l l
d C t x z C t x
dz
=
+ =
(2)
1 0;n
z
C
z
+
=
=
1 1
2 2 1
0,
k
k k
j j k
k k
j j k
z l
C
z
C
z
+
=
+ − =
− +
And boundary conditions for varia-
ble x
0 10; ( , ),
k
k
x k x R
C C C t z
x = =
= =
(3)
where k
ij , k
ij ; 0,k n= ; , 1, 2,i j = are coef-
ficients determining boundary conditions and
contact conditions; , , x y z are spatial coordi-
nates.
The exact solution of the problem de-
scribed by equations (1)–(3) is directly writ-
ten out by applying integral Fourier trans-
forms [3].
0 0
1
1
1 11
1
1
1
1 11
0 0
1
1 0 0
0
1
,
1 0
( , , )
( ; , ; ) ( , )
( ; , ; , )
( , ) ( )
( ; ; , )
,k
k
k
k
k
k
k
k
t R
l l
lt rn
k ,k
k l
k
ltn
R k
k l
C t x z
W t x z C d d
H t x z
C d d d
W t x z
−
−
+
=
+
+
=
=
− +
+ −
+
+ −
111 ( , )
k kC d d .
Комп’ютерне моделювання
65
Here are Green’s functions:
0
0 ,
0
( , ; , )
cos 2 ( , )(-1) ;
,kl
m m m
l k
mm
W t x z
W t z
R
=
=
= −
1
2
0
1
1
0
0
( ; , ; , )
2 (-1) ( ; , )(-1)
cos
k ,k
m
R
D t m m m
m k ,k
m
m
W t x z
e D t z
R
x.
− +
=
=
=
Cauchy’s function
1
1
0
( ; , ; , )
2 ( , , ) cos cos
k ,k
m
k ,k m m
m
H t x z
t z x.
R
=
=
=
2. Experimental data and
coefficients identification
According to the results of experi-
mental data and using the proposed model,
identification was carried out using the theory
of state control of multicomponent systems
(results obtained in [4]) (Fig. 2).
Fig. 2. Coefficients distribution at first sample
The distributions of diffusion coeffi-
cients are reproduced using methods of opti-
mal control of the state of multicomponent
transport systems, analytical solution of the
model, and data of experimental observations,
for the considered constituent components of
nanofilms (samples of aluminum, molyb-
denum, silicon) (Fig. 3).
Fig. 3. Coefficients distribution at the second
sample
The diffusion coefficients identified in
such a way, correspond to real experimental
data and are used as input parameters of the
obtained mathematical solution of the model
for modeling and analysis of concentration
distributions of the main components of nano-
films (aluminum, molybdenum, silicon, etc.).
3. Numerical simulation
Numerical simulation results are com-
pared with experimental observations depict-
ing the sample content. These concentration
distributions are generated for varying time
intervals during the formation of the multi-
layer. The designated time frame corresponds
to the experimental duration of three weeks.
The process of forming the technological
multilayer through molecular diffusion of the
specified components is segmented into 5 pe-
riods, encompassing the inception of the pro-
tective multilayer from the initial period to the
concluding period (Fig. 4).
Fig. 4. Simulated concentrations for case 5
Continuous lines represent simulated
data for different time durations. The dashed
Комп’ютерне моделювання
66
line is experimentally measured data at the fi-
nal duration.
As depicted in the figures above, the
profiles derived from modeling closely align
with the corresponding experimental profiles,
particularly as the duration of multilayer for-
mation approaches the completion period of
the protective nanofilm multilayer formation.
The maximum observed deviation does not
surpass 2–3 %, affirming the reliability of the
mathematical model and indicating the practi-
cal utility of the provided results (Fig. 5).
Fig. 5. Simulated concentrations
4. Software framework
The above results require complex
resource-intensive calculations [5]. For this
purpose, we developed a framework that au-
tomates the specified calculations with the
possibility of extension to other subject areas
with similar tasks of identifying the key
factors of the process and further numerical
modeling of the time-space characteristics
using the obtained results.
The software framework in focus is
implemented in Java programming language.
JVM-based implementation allows us to rid
of target-specific builds and follow the write-
once-run-everywhere idea. At the same time,
the JIT-compilation feature of JVM optimizes
code execution by converting bytecode into
native machine code at runtime. It allows to
have runtime optimizations that are not
possible with ahead-of-time compilation and
are beneficial for CPU-intensive applications.
The framework design follows an
object-oriented approach, and the code base is
divided into major modules, such as Models,
DataProcessing, Identification, and GUI. The
input either can be read from the file or
provided in UI. Calculated results also can be
visualized or saved to the file. All
components are connected with abstract
interfaces, which gives the ability to easily
replace one model implementation with
another implementation (separation of
interface and implementation).
One example is the implementation of
diffusion coefficient identification, which de-
fines the internal kinetic parameters of the
process. To ascertain the coefficient distribu-
tion, we employ the gradient methods, the
mathematical underpinnings of which lay in
the context of parametric identification chal-
lenges in multicomponent distributed sys-
tems [6]. Tailoring our approach to the nano-
films domain, we find the method of mini-
mum errors particularly apt. So, we picked the
implementation of this method in simulations
among others in the software framework. The
coefficients identification algorithm is based
on a gradient-identification procedure where-
in, to ascertain the next approximation of the
diffusion coefficient within the intraparticle
space, we adhere to the following protocol
(Fig. 6).
Fig. 6. Procedure for internal parameters
identification
Комп’ютерне моделювання
67
The determined distributions of diffu-
sion coefficients within the interparticle space
enable the modeling of concentration fields
and integral mass distributions within the cat-
alytic nanoporous layer with a high level of
precision. As evidenced by the concentration
distributions presented (Fig. 2 and Fig. 3), the
profiles derived from the model closely match
those obtained experimentally (Fig. 4 and
Fig. 5), illustrating a noteworthy degree of
consistency between the two sets of data. This
alignment is further emphasized by the com-
plete coincidence observed between the mod-
el and experimental graphs depicting the inte-
gral mass during benzene adsorption.
Conclusion
The difficulties associated with study-
ing diffusion in multilayer films require the
progress of contemporary modeling methods
and software platforms to precisely represent
phenomena, taking into account transitions
between adjacent layers. Alongside the essen-
tial function of cutting-edge modeling tech-
niques and software frameworks in addressing
the difficulties of examining diffusion within
multilayer films, it’s vital to acknowledge the
significant input of sophisticated computa-
tional methods.
In this paper, we endeavor to merge
intricate mathematical models with optimal
software development methodologies to ad-
dress the challenge of simulating diffusion
transport processes in multilayer nanofilms
computationally. Based on the experimental
findings and employing the suggested model,
identification is conducted utilizing the theory
of state control for multicomponent systems.
Using the methods of optimal control of the
state of multicomponent transport systems,
the analytical solution of the model, and the
data of experimental observations, the distri-
butions of diffusion coefficients for the con-
sidered components of nanofilms (samples of
aluminum, molybdenum, silicon) were repro-
duced. Numerical simulation results were
compared with experimental observations.
The profiles obtained from the modeling
closely match the corresponding experimental
profiles, especially as the duration of multi-
layer formation approaches the final stages of
completing the protective nanofilm multilayer
formation. The maximum observed deviation
does not exceed 2–3%, confirming the relia-
bility of the mathematical model and demon-
strating the practical value of the results pro-
vided.
A software framework is developed
for the automation of the specified calcula-
tions with the possibility of extension to other
subject areas with similar tasks of identifying
the key factors of the process and further nu-
merical modeling of the time-space character-
istics using the obtained results.
Література
1. J. Kärger, F. Grinberg, P. Heitjans,
Diffusion fundamentals, Leipzig,
Leipziger Unviersite, 2005.
2. M.R. Petryk, A. Khimich, M.M. Petryk,
J. Fraissard, Experimental and computer
simulation studies of dehydration on mi-
croporous adsorbent of natural gas used as
motor fuel, Fuel, 2019. Vol. 239.
P. 1324–1330.
3. М. Р. Петрик, П. М. Василюк, Д. М.
Михалик, Н. В. Бабій, О. Ю. Петрик,
Моделі процесів дифузійного переносу
і методи оцінювання параметрів в бага-
то композитних наноплівках. Терно-
піль, Вид-во ТНТУ імені Івана Пулюя,
2015.
4. В. С. Дейнека, М. Р. Петрик, Иденти-
фикация параметров неоднородных
задач диффузии в наномультикомпо-
зитах с использованием градиентных
методов, Компьютерная математика,
2012. № 1. С. 41–51.
5. M. Petryk, A. Doroshenko, D. Mykhalyk,
P. Ivanenko, O. Yatsenko, Automated
parallelization of software for identifying
parameters of intraparticle diffusion and
adsorption in heterogeneous nanoporous
media. In: Shkarlet, S., et
al. Mathematical Modeling and
Simulation of Systems. MODS 2022.
Lecture Notes in Networks and Systems.
2023. Vol. 667. P. 33–47.
https://link.springer.com/chapter/10.1007
/978-3-031-30251-0_3
6. И. В. Сергиенко, В. С. Дейнека, Систе-
мный анализ многокомпонентных рас-
Комп’ютерне моделювання
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пределенных систем, Киев, Наукова
думка, 2009.
References
1. J. Kärger, F. Grinberg, P. Heitjans, Diffu-
sion fundamentals, Leipzig, Leipziger
Unviersite, 2005.
2. M.R. Petryk, A. Khimich, M.M. Petryk,
J. Fraissard, Experimental and computer
simulation studies of dehydration on mi-
croporous adsorbent of natural gas used as
motor fuel, in: Fuel 239 (2019) 1324–
1330.
3. M. Petryk, P. Vasylyuk, D.Mykhalyk, N.
Babiy, O. Petryk, Models of diffusive
transport processes and methods of pa-
rameter estimation in multi-composite
nanofilms, Ternopil, TNTU, 2015. doi:
10.32626/2308-5916.2015-12.141-160.
[in Ukrainian]
4. V. S. Deineka, M. R. Petryk,
Identification of parameters of
inhomogeneous diffusion problems in
nanomulticomposites using gradient
methods, in: Computer mathematics
1 (2012) 41–51. [in Russian]
5. M. Petryk, A. Doroshenko, D. Mykhalyk,
P. Ivanenko, O. Yatsenko, Automated
parallelization of software for identifying
parameters of intraparticle diffusion and
adsorption in heterogeneous nanoporous
media, in: Shkarlet, S., et al. Mathemati-
cal Modeling and Simulation of Systems.
MODS 2022. Lecture Notes in Networks
and Systems, 2023, Vol. 667. P. 33–47.
https://link.springer.com/chapter/10.1007
/978-3-031-30251-0_3
6. V. Sergienko, V. S. Deineka, System
analysis of multicomponent distributed
systems, Kyiv, Naukova Dumka, 2009.
[in Russian]
Одержано: 10.04.2024
Внутрішня рецензія отримана: 20.04.2024
Зовнішня рецензія отримана: 27.04.2024
Про авторів:
1Петрик Михайло Романович,
доктор фізико-математичних наук,
професор.
http://orcid.org/0000-0001-6612-7213.
2Дорошенко Анатолій Юхимович,
доктор фізико-математичних наук,
професор, провідний науковий
співробітник.
http://orcid.org/0000-0002-8435-1451.
1Михалик Дмитро Михайлович,
кандидат технічних наук, доцент.
https://orcid.org/0000-0001-9032-695X.
2Яценко Олена Анатоліївна,
кандидат фізико-математичних наук,
старший науковий співробітник.
http://orcid.org/0000-0002-4700-6704.
Місце роботи авторів:
1Тернопільський національний технічний
університет імені Івана Пулюя,
Тел. (+38) (044) 526-20-08
E-mail: softeng@tntu.edu.ua
Сайт: https://tntu.edu.ua
2Інститут програмних систем
НАН України,
тел. +38-044-526-60-33
E-mail: a-y-doroshenko@ukr.net,
oayat@ukr.net
Сайт: https://iss.nas.gov.ua
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| id | pp_isofts_kiev_ua-article-620 |
| institution | Problems in programming |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T09:42:02Z |
| publishDate | 2024 |
| publisher | PROBLEMS IN PROGRAMMING |
| record_format | ojs |
| resource_txt_mv | ppisoftskievua/6a/3710522951f4c877fcd0622d67c0fd6a.pdf |
| spelling | pp_isofts_kiev_ua-article-6202025-02-14T11:09:46Z Computer simulation of diffusion transport processes in multilayer nanofilms Комп’ютерне моделювання процесів дифузійного транспорту в багатошарових наноплівках Petryk, M.R. Doroshenko, A.Yu. Mykhalyk, D.M. Yatsenko, О.A. coefficients identification; diffusion; mathematical model; multilayer nanofilm; numerical simulation; oxide film UDC 619.6 ідентифікація коефіцієнтів; дифузія; математична модель; багатошарова наноплівка; чисельне моделювання; оксидна плівка УДК 619.6 The difficulties associated with studying diffusion in multilayer films require the progress of contemporar y modeling methods and software platforms to precisely represent phenomena, taking into account transitions between adjacent layers. In addition to the indispensable role of advanced modeling techniques and software in solving the problems of studying diffusion in multilayer films, it is extremely important to admit the key contribution of sophisticated computational approaches. In this paper, the authors attempt to merge intricate mathematical models with optimal software development methodologies to address the challenge of simulating diffusion transport processes in multilayer nanofilms computationally. Based on the experimental findings and employing the suggested model, identification was conducted utilizing the theory of state control for multicomponent systems. With the help of methods of optimal control of the state of multicomponent transport systems, the analytical solution of the model and the data of experimental observations, the distributions of diffusion coefficients for the considered components of nanofilms (samples of aluminum, molybdenum, silicon) were reproduced. Numerical simulation results were compared with experimental observations. The profiles obtained from the modeling closely match the corresponding experimental profiles, especially as the duration of multilayer formation converges to the final stages of completing the protective nanofilm multilayer formation. The maximum observed deviation does not exceed 2–3%, confirming the reliability of the mathematical model and demonstrating the practical value of the results provided. A software framework is developed for the automation of the specified calculations with the possibility of extension to other subject areas with similar tasks of identifying the key factors of the process and further numerical modeling of the time-space characteristics using the obtained results.Prombles in programming 2024; 2-3: 62-68 Проблеми, пов’язані з дослідженням дифузії в багатошарових плівках, вимагають удосконалення сучасних методів моделювання та програмного забезпечення для точного зображення явищ, враховуючи переходи між сусідніми шарами. На додаток до незамінної ролі передових методів моделювання та програмного забезпечення у розв’язанні проблем вивчення дифузії в багатошарових плівках, надзвичайно важливо визнати ключовий внесок складних обчислювальних підходів. У цій роботі автори зробили спробу поєднати складні математичні моделі з передовим досвідом розробки програмного забезпечення для розв’язання задачі комп’ютерного моделювання процесів дифузійного транспорту в багатошарових наноплівках. За результатами експериментальних даних та з використанням запропонованої моделі проведено ідентифікацію з використанням теорії контролю стану багатокомпонентних систем. За допомогою методів оптимального керування станом багатокомпонентних транспортних систем, аналітичного рішення моделі та даних експериментальних спостережень відтворено розподіли коефіцієнтів дифузії для розглянутих складових компонентів наноплівок (зразків алюмінію, молібдену, кремнію). Результати чисельного моделювання порівнюються з експериментальними спостереженнями, що відображають вміст зразка. Профілі, отримані в результаті моделювання, тісно збігаються з відповідними експериментальними профілями, особливо коли тривалість багатошарового утворення наближається до остаточного періоду завершення утворення захисної наноплівки. Максимальне спостережене відхилення не перевищує 2–3 %, що підтверджує надійність математичної моделі та вказує на практичну корисність отриманих результатів. Розроблено програмний інструментарій для автоматизації зазначених розрахунків з можливістю поширення на інші предметні області зі схожими завданнями ідентифікації ключових факторів процесу. Створено базу для подальшого чисельного моделювання часо-просторових характеристик з використанням отриманих результатів.Prombles in programming 2024; 2-3: 62-68 PROBLEMS IN PROGRAMMING ПРОБЛЕМЫ ПРОГРАММИРОВАНИЯ ПРОБЛЕМИ ПРОГРАМУВАННЯ 2024-12-17 Article Article application/pdf https://pp.isofts.kiev.ua/index.php/ojs1/article/view/620 10.15407/pp2024.02-03.062 PROBLEMS IN PROGRAMMING; No 2-3 (2024); 62-68 ПРОБЛЕМЫ ПРОГРАММИРОВАНИЯ; No 2-3 (2024); 62-68 ПРОБЛЕМИ ПРОГРАМУВАННЯ; No 2-3 (2024); 62-68 1727-4907 10.15407/pp2024.02-03 en https://pp.isofts.kiev.ua/index.php/ojs1/article/view/620/672 Copyright (c) 2024 PROBLEMS IN PROGRAMMING |
| spellingShingle | coefficients identification diffusion mathematical model multilayer nanofilm numerical simulation oxide film UDC 619.6 Petryk, M.R. Doroshenko, A.Yu. Mykhalyk, D.M. Yatsenko, О.A. Computer simulation of diffusion transport processes in multilayer nanofilms |
| title | Computer simulation of diffusion transport processes in multilayer nanofilms |
| title_alt | Комп’ютерне моделювання процесів дифузійного транспорту в багатошарових наноплівках |
| title_full | Computer simulation of diffusion transport processes in multilayer nanofilms |
| title_fullStr | Computer simulation of diffusion transport processes in multilayer nanofilms |
| title_full_unstemmed | Computer simulation of diffusion transport processes in multilayer nanofilms |
| title_short | Computer simulation of diffusion transport processes in multilayer nanofilms |
| title_sort | computer simulation of diffusion transport processes in multilayer nanofilms |
| topic | coefficients identification diffusion mathematical model multilayer nanofilm numerical simulation oxide film UDC 619.6 |
| topic_facet | coefficients identification diffusion mathematical model multilayer nanofilm numerical simulation oxide film UDC 619.6 ідентифікація коефіцієнтів дифузія математична модель багатошарова наноплівка чисельне моделювання оксидна плівка УДК 619.6 |
| url | https://pp.isofts.kiev.ua/index.php/ojs1/article/view/620 |
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