Розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями
The paper investigates the numerical simulation of wave processes in multilayer thin films, which is relevant for understanding their physical properties and optimization for various applications. An integrated mathematical model has been developed that combines Maxwell’s equations, mechanical vibra...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2026
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System research and information technologies| _version_ | 1869472190650384384 |
|---|---|
| author | Bilak, Yurii |
| author_facet | Bilak, Yurii |
| author_institution_txt_mv | [
{
"author": "Yurii Bilak",
"institution": "Uzhhorod National University, Uzhhorod"
}
] |
| author_sort | Bilak, Yurii |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2026-06-30T06:14:59Z |
| description | The paper investigates the numerical simulation of wave processes in multilayer thin films, which is relevant for understanding their physical properties and optimization for various applications. An integrated mathematical model has been developed that combines Maxwell’s equations, mechanical vibrations and thermal conductivity, taking into account the interaction of physical fields in structures with defects. Adaptive algorithms have been proposed for automatic mesh refinement depending on local gradients of physical parameters, which allows to increase the accuracy of modeling in critical zones. A hybrid approach to calculations using CPU and GPU has been implemented, which ensures efficient use of resources for large-scale problems. Software with a modular architecture has been developed that allows integrating numerical methods, optimization and visualization of results in real time. Experimental validation has confirmed the high accuracy and reliability of the model. The results obtained contribute to a deeper understanding of physical processes in thin films and are the basis for the creation of highly efficient multilayer structures in industrial and scientific applications. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2026.2.04 |
| first_indexed | 2026-07-01T01:00:13Z |
| format | Article |
| fulltext |
Yu.Yu. Bilak, 2026
Системні дослідження та інформаційні технології, 2026, № 2 51
TIÄC
ПРОГРЕСИВНІ ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ,
ВИСОКОПРОДУКТИВНІ КОМП’ЮТЕРНІ
СИСТЕМИ
UDC 004.94:004.932:621.373
DOI: 10.20535/SRIT.2308-8893.2026.2.04
DEVELOPMENT OF A MATHEMATICAL MODEL OF WAVE
PROCESSES IN MULTILAYERED STRUCTURES WITH AN
ADAPTIVE ALGORITHM AND HYBRID CALCULATIONS
YU.YU. BILAK
Abstract. The paper investigates the numerical simulation of wave processes in
multilayer thin films, which is relevant for understanding their physical properties and
optimization for various applications. An integrated mathematical model has been
developed that combines Maxwell’s equations, mechanical vibrations and thermal
conductivity, taking into account the interaction of physical fields in structures with
defects. Adaptive algorithms have been proposed for automatic mesh refinement
depending on local gradients of physical parameters, which allows to increase the
accuracy of modeling in critical zones. A hybrid approach to calculations using CPU
and GPU has been implemented, which ensures efficient use of resources for
large-scale problems. Software with a modular architecture has been developed that
allows integrating numerical methods, optimization and visualization of results in real
time. Experimental validation has confirmed the high accuracy and reliability of the
model. The results obtained contribute to a deeper understanding of physical
processes in thin films and are the basis for the creation of highly efficient multilayer
structures in industrial and scientific applications.
Keywords: thin films, numerical simulation, optimization, multiphysics models,
parallel computing, hybrid algorithms, wave processes.
INTRODUCTION
Modern technologies for modeling multilayer structures are an important tool for
researching and optimizing the physicochemical properties of materials. Numerical
modeling of such systems allows for a detailed analysis of complex wave processes,
encompassing electromagnetic, acoustic, and thermal phenomena. Accounting for
the interaction of these processes in multilayer films is particularly relevant for
industry, where the priority is creating materials with specified properties, such as
high-efficiency optical filters, acoustic resonators, and thermoelectric devices.
Most existing approaches focus on highly specialized aspects of modeling [1] or
optimization. The integration of multiphysical models, dynamic mesh optimization,
the use of hybrid computing environments, and the introduction of neural networks
into the numerical modeling process open new opportunities in the analysis of
multilayer structures that have not yet been achieved in a comprehensive manner.
This provides scientific novelty of the work, since such methods have not yet been
implemented in conjunction, which allows solving more complex and large-scale
problems with high accuracy and productivity.
Yu.Yu. Bilak
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 52
The relevance of the research lies in the necessity to enhance the accuracy of
numerical modeling of multiphysical processes. Integration of electromagnetic,
acoustic and thermal phenomena within a single model allows for a better
understanding of the influence of physical parameters on the structure and
functionality of films. Such models contribute to reducing experimental costs and
accelerating the development of innovative technologies. The primary objective is
to create a model capable of accounting for the interaction of wave processes in
real-time. This includes the description of the electromagnetic field according to
Maxwell’s equations, acoustic waves through mechanical vibrations, and thermal
processes based on the heat conduction equation. Prospects for implementing the
research results include creating new technologies for the synthesis of thin films
with specified characteristics, predicting their behavior under various conditions,
as well as the integration of the developed models into modern information systems
for research in materials science and engineering.
LITERATURE ANALYSIS AND PROBLEM STATEMENT
Numerical modeling encompasses a wide range of multiscale methods, such as
a combination of transfer matrix, finite element, and molecular dynamics methods,
which allow for a more precise description of the physical and mechanical
properties of multilayer materials [1–4]. Numerous studies have improved the
efficiency of modeling the electrical conductivity of polymer composites, in
particular using carbon nanotubes and carbon fibers, which allows for the
optimization of their electrical characteristics [5, 6]. In addition, improving the
thermal insulation and mechanical properties of multilayer materials, in particular
vapor-cooled insulating materials and polymer composites, continues to be
a relevant research area [7–9]. The study of multilayer electronic structures, such
as graphene, also allows for a better understanding of their electronic behavior,
including the dependence of the interlayer distance and the effect of pressure on the
electronic properties [10]. The proposed methods and models significantly improve
the accuracy of predicting material properties, but there are still open questions
regarding the adaptation of these models to complex nanostructures, the accuracy
of predicting experimental parameters, and the integration of multiphysical
approaches for more comprehensive modeling.
Modern approaches to numerical modeling include simulation multi-level
helical structures for coal machines [11], assessment of the safety of rocket
missions using multilayer models [12], and analysis of microfluidic systems for
biomedical applications [13]. The proposed methods allow to significantly reduce
modeling errors, increase prediction accuracy, and optimize calculations, which is
critical for achieving high results in these industries. Furthermore, research in the
digitalization of business models [14] and modeling of thin-film solar cells based
on ZnO/CdS/CuInGaSe2 [15] demonstrate the importance of multilayer models for
assessing the efficiency of technologies in various fields, from business to energy.
Despite the successes achieved, issues related to the adaptation of models for
complex multifunctional structures and the integration of multiphysical approaches
remain open, particularly in the context of the accuracy of predicting experimental
parameters and verifying models in practice.
Along with this, considerable interest is aroused by studies of the spectral
characteristics of plasma discharges [16, 17], which revealed the mechanisms of
Development of a mathematical model of wave processes in multilayered structures with an…
Системні дослідження та інформаційні технології, 2026, № 2 53
formation of excited plasma components that affect the formation of nanostructured
films. It has been shown that the parameters of the electric field and plasma
composition determine the features of the energy distribution and affect the
morphology of the deposited films. However, questions remain open regarding the
precise control of the composition of film-forming particles and the influence of
local plasma inhomogeneities on the uniformity of the coating. Works dedicated to
the influence of the electrode material and discharge conditions on the structure of
films [18, 19, 21] demonstrate the dependence of phase composition and
electrophysical properties of films on the chemical composition of the electrodes,
temperature and gaseous environment. Material transfer process during laser
ablation and under high-voltage discharge conditions have been investigated.
Nonetheless, the mechanisms of interaction of plasma particles with the substrate
and the role of impurities in the stability of film formation require further study.
Some studies [20, 21] focus on the electrophysical properties of deposited films and
their potential applications in sensor and energy devices. It has been established
that the gas-discharge deposition method allows obtaining materials with high
conductivity and controlled structural characteristics. However, the long-term
stability of films and their adaptation to real operating conditions remain relevant
issues.
Another promising research direction is the development of efficient
lithium-ion batteries (LIBs), which are cutting-edge energy devices due to their
environmental friendliness, low self-discharge and long life cycle [22–24].
Accurate determination of the state of charge (SOC) in real conditions is a difficult
task due to the nonlinear characteristics of LIBs, therefore, the study of SOC
estimation methods is highly relevant and has been studied in a significant number
of scientific works. Currently, various methods for SOC estimation are being
actively developed, which are based on models [25]. Models are widely used due
to the optimal balance between computational complexity and accuracy [26]. They
include three stages: model construction, parameter identification, and state
estimation. Electrochemical models, while considering the internal structure
of the battery, have excessively high computational complexity for practical
application [27]. At the same time, equivalent circuit models (ECMs) reflect the
internal parameters of the battery through external components, which makes them
an effective and widely used solution. Current research on SOC estimation of
lithium-ion batteries focuses on improving the accuracy, adaptability, and
computational efficiency of the methods, which is reflected in the works [28–30],
which consider approaches based on adaptive Kalman filters. It should be noted
that when modeling lithium-ion batteries, mathematical models can be used to
predict their behavior, considering various processes, such as thermal or electrical.
Adaptive algorithms and hybrid computing can be used to optimize these models,
which is typical for both wave process modeling and battery efficiency research.
The literature review demonstrated a wide range of directions for the use of
multilayer structures and showed significant progress in the development of models
for the study of thin films, the synthesis of films based on metallic and polymeric
materials, as well as the study of their electrical and mechanical properties. The
effectiveness of many approaches, such as finite element methods, multiscale
modeling and numerical methods, allowing accurate prediction of the properties of
films and their interaction with various substrates and the environment, was noted.
Yu.Yu. Bilak
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 54
However, several unsolved issues remain. While numerous models have been
developed to describe the behavior of thin films, research on their behavior at
different scales (molecular, micro- and macro-levels) still requires improvement in
the accuracy of models for multilayer structures. The precise modeling of the
mechanical properties of thin films under deformation, especially in cases where
films are used in complex multilayer structures, remains an open question. There is
a need of deeper exploration of the environmental impact (temperature, humidity,
chemical reactions) on film characteristics, particularly for their synthesis in
various gas environments. Although there are successful developments in the
creation of nanostructured films, methods for accurate simulation of the process of
their synthesis and behavior at the molecular level remain limited. Integration of
new models is required for more accurate prediction of processes in real conditions.
More attention should be paid to optimizing the synthesis technologies of films
from different materials to improve their efficiency in specific applications
(e.g., for electronic or medical devices, solar cells, lithium-ion batteries). Thus, for
further development of models, it is necessary to focus on improving the accuracy
of multilayer models, developing complex models, automating calculations,
studying the mechanical characteristics of thin films, as well as their synthesis and
the impact of external factors.
RESEARCH AIMS AND OBJECTIVES
The main aim of the work was to develop an integrated numerical modeling
algorithm that accounts for the interaction of wave processes (electromagnetic,
acoustic, thermal) in multilayer structures to accurately predict the physical
properties of thin films, identify the influence of key parameters on wave processes,
as well as validate the obtained results using experimental data.
The research objectives include:
1. Development of a mathematical model to describe wave processes in
multilayer films.
2. Implementation of adaptive numerical methods for mesh refinement.
3. Implementation of hybrid computational approaches using CPU and GPU.
4. Analysis of physical phenomena in multilayer structures.
5. Development of software for modeling automation.
6. Experimental validation of modeling results.
These tasks will ensure both scientific novelty and practical value of the
research results, in particular for optimizing the processes of creating multilayer
films with specified characteristics. The practical value of the work lies in the
implementation in the form of software for designing multilayer structures, in
particular for creating optical filters, acoustic resonators, thermoelectric materials,
as well as for diagnosing defects in complex systems.
The object of the research is the processes of numerical modeling of the
interaction of electromagnetic, thermal and acoustic fields in multilayer structures
using an adaptive grid and parallel calculations.
Research hypothesis: the integration of multiphysical processes within
a single numerical approach using an adaptive grid and parallel algorithms will
allow to increase the accuracy of predicting the physical properties of multilayer
structures, optimize computational costs and improve the correspondence of the
model to experimental data.
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MATERIALS AND RESEARCH METHODS
For the numerical solution of the aforementioned problems, discretization methods
are used, particularly finite difference, element and volume methods, which are
selected based on the problem’s geometry and accuracy requirements. Typically,
multilayer structures are integrated into models using methods such as the Transfer
Matrix Method (TMM), which allows for the description of wave processes in
systems with multiple layers, and a multiphysical approach which accounts for the
interaction of electromagnetic, acoustic and thermal processes. One of the key
aspects of the work is the optimization of algorithms, which includes adaptive
meshes for accuracy in critical areas, parallel computing to reduce calculation time
and the use of schemes with high tolerance to errors. Parallel computing is based
on the distribution of tasks between several processor cores (CPU) or graphics
processing units (GPU), which makes it possible to simultaneously process
numerous mesh elements and reduces calculation time. Another important
optimization strategy is the use of linear algebra methods to solve large systems of
equations, such as iterative methods (conjugate gradient descent) and distributed
matrix multiplication methods, which work well in parallel environments. Data
preprocessing algorithms are also used to reduce the dimensionality of the problem
and reduce the complexity of the models, which allows for faster calculations
without losing accuracy. Ultimately, computational optimization ensures the
practical suitability of the developed algorithms for industrial applications and their
integration into automated thin film analysis systems.
Multiphysical approach to numerical modeling of wave processes. In this
work, a multiphysical approach was used to integrate a multilayer structure into the
model, which considers the interaction of electromagnetic, acoustic, and thermal
processes. The main focus was on creating a new algorithm that simultaneously
accounts for electromagnetic, acoustic, and thermal wave processes in multilayer
structures. This approach considers the interaction of various physical fields, which
significantly increases the accuracy of modeling and opens new opportunities for
designing materials with specified properties.
Physical basis. Each of the wave processes is described by its own set of
equations. Electromagnetic waves are modeled by Maxwell’s equations, which
account for the propagation of an electromagnetic field through a medium with
certain dielectric and magnetic properties:
D
, 0B
, BE
T
, DH J
T
,
where E
– electric field strength vector; H
– magnetic field strength vector;
D E
– electric induction (ε – dielectric constant); B H
– magnetic
induction (μ – magnetic permeability). At the boundary of the layers, the conditions
of continuity of the field components are fulfilled:
,1 ,2 ,1 ,2 ,1 ,2 ,1 ,2, , , E E D D H H B B
.
Acoustic waves are described by equations of mechanical vibrations that
determine changes in pressure and particle velocity in a medium depending on its
density and elasticity.
Yu.Yu. Bilak
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 56
2
2
u
t
, ( ) 2u I
,
where u – vector of movement of particles of the medium; σ – stress tensor;
1 ( )
2
Tu u
– strain tensor; λ and μ – Lamé coefficients characterizing the
elasticity of the material.
Thermal processes are based on the heat conduction equation, which models
the temperature distribution in layers, considering thermal conductivity, heat
capacity, and heat transfer between layers:
( )Tc k T Q
t
,
where T – temperature; c – heat capacity; k – thermal conductivity coefficient;
Q – a heat source (e.g. generated by an electromagnetic field).
Integration of physical models. The developed algorithm combines these
equations into a single numerical model by joint solution. Regarding the choice of
common parameters, the models are linked through material properties, i.e.,
a change in temperature affects the dielectric constant (for electromagnetic waves)
and the speed of sound (for acoustic waves). The connection between the equations
is implemented as follows. The electromagnetic field generates heat:
2
Q E
,
where σ – electrical conductivity of the material.
Temperature change affects dielectric constant:
0( )T T ,
where α – temperature dependence coefficient.
Thermal expansion affects acoustic properties:
λ(T)=λ0(1+βT), μ(T)=μ0(1+γT),
where β, γ – coefficients of temperature dependence of mechanical parameters.
To account for temporal and spatial dependencies, a unified discretization of
time and space is used, ensuring accuracy in considering the interaction between
processes. A single grid is implemented for all physical fields with adaptive
refinement in areas of large parameter gradients.
As for the boundary conditions, they are adaptively adjusted for each layer of
the structure depending on the physical properties and type of wave process. That
is, the joint boundary conditions account for the transfer of heat, acoustic waves,
and changes in the electromagnetic field across the layer boundaries.
Algorithm development. As mentioned above, the algorithm implementation
uses a multiphysical approach where all equations are solved in integrated form.
The Finite Element Method (FEM) provides high accuracy on complex geometries.
Here, each equation is discretized using FEM in space and an explicit/implicit
scheme in time. The equations are written in matrix form:
XM KX F
t
,
Development of a mathematical model of wave processes in multilayered structures with an…
Системні дослідження та інформаційні технології, 2026, № 2 57
where M
– mass matrix; K
– stiffness matrix; F
– vector of external influences;
X
– vector of unknowns (fields, temperature, displacement).
Parallel computations are implemented by distributing matrix operations
between GPU or CPU cores using the MPI library. The adaptive mesh allows you
to concentrate computational resources in critical areas, for example, at layer
boundaries or in areas with a large temperature gradient. Additionally, automatic
mesh reconstruction is implemented depending on local inhomogeneities, such as
defects or sharp changes in physical parameters, i.e., an adaptive algorithm has been
developed taking into account property gradients. The algorithm block diagram is
shown in Fig. 1.
The program code is implemented in the Python programming language
using several libraries. NumPy is used for calculations, providing functionality for
working with multidimensional arrays and basic mathematical operations.
The SciPy library is applied for solving differential equations and mathematical
optimization. FEniCS is used for numerical solution of equations using the finite
element method, which allows to specify the equation in weak form and
automatically construct the corresponding stiffness matrix. A fragment of the
developed code is given below.
def electromagnetic_eq(T, E):
epsilon_eff = epsilon * (1 + beta * T) # Temperature dependency
return epsilon_eff * inner(grad(E), grad(v)) * dx
def acoustic_eq(T, u):
lambda_eff = alpha * T # Temperature dependency of elastic module’s
return rho * u * v * dx + lambda_eff * div(grad(u)) * div(grad(v)) * dx
def thermal_eq(T, E):
Q = epsilon * E**2 # Heat generation from an electric field
return rho * c * T * v * dx + k * inner(grad(T), grad(v)) * dx - Q * v * dx
# Collecting equations into a single system
a1 = electromagnetic_eq(T, E)
a2 = acoustic_eq(T, u)
a3 = thermal_eq(T, E)
F = a1 + a2 + a3
comm = MPI.COMM_WORLD # Parallel computing
# Solution by Newton-Raphson method
T.assign(T0)
E.assign(E0)
u.assign(u0)
t = 0
while t < T_end:
solve(F == 0, T, solver_parameters={‘newton_solver’:
{‘relative_tolerance’: 1e-6}})
t += dt
print(f"Time: {t:.2f}, Average temperature:
{T.vector().get_local().mean():.2f}")
Yu.Yu. Bilak
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 58
Fig. 1. Algorithm of multiphysical numerical modeling with adaptive mesh
START
SETTING INPUT PARAMETERS
DEFINING INITIAL CONDITIONS
FORMING THE SYSTEM
OF EQUATIONS
USING THE FINITE
ELEMENT METHOD (FEM)
COMPUTING THE GRADIENTS
OF PHYSICAL PARAMETERS
Is mesh refinement necessary?
Mesh Adaptation (AMR)
SOLVING THE SYSTEM OF
EQUATIONS USING
THE NEWTON-RAPHSON METHOD
Is parallel computing required?
Workload is distribution
using MPI or CUDA Standard calculations
Do material parameters change?
Updating parameter values Using a precomputed value
Does the error exceed the
acceptable threshold?
Refinement of the mesh
or reduction of the time step
Checking for calculation completion
based on time?
SAVING RESULTS
END
Development of a mathematical model of wave processes in multilayered structures with an…
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Description of the algorithm. The developed program code implements a
numerical approach for integrating electromagnetic, acoustic and thermal processes
in multilayer structures using the finite element method. It begins with the
definition of physical parameters, such as dielectric permittivity, thermal
conductivity and material density, as well as the geometry of the grid for
calculations. Initial conditions for temperature, electric field and acoustic
displacement are determined. The basis is the equations that describe the interaction
between physical fields. The electromagnetic equation accounts for the effect of
temperature on dielectric permittivity, the acoustic equation considers changes in
mechanical properties due to thermal expansion, and the thermal conductivity
equation includes a heat source generated by the electromagnetic field. All
equations are combined into a common system, which is solved by the Newton-
Raphson method. For this, parallel calculations are used through the MPI library,
which allows for effective load distribution across multiple processors. During the
calculation process, the results for temperature, electric field, and acoustic
displacement are updated at each time step. Upon completion of the simulation,
these results are saved in .pvd format files for further visualization and analysis.
The developed code demonstrates the implementation of the interaction of
multiphysical processes with the ability to scale for more complex structures and
scenarios.
Optimization is implemented through the use of parallel computing, adaptive
numerical methods and efficient algorithms. Parallel computing is provided by the
mpi4py library, which allows distributing computations between multiple
processors, thereby reduces the task execution time. To implement adaptive
algorithms that account for local gradients of physical properties, the adaptive mesh
refinement (AMR) technique is used. This involves analyzing the gradients of
parameters (for example, refractive index, density or temperature) and adapting the
mesh in critical zones.
A snippet of the implemented code for grid adaptation is given below:
mesh = UnitSquareMesh(16, 16) # Initializing the initial mesh
V = FunctionSpace(mesh, "P", 1)
def refine_mesh(mesh, criteria, threshold): # Grid adaptation
cell_markers = MeshFunction("bool", mesh, mesh.topology().dim())
criteria_array = criteria.vector().get_local()
threshold_value = threshold * np.max(criteria_array)
cell_markers.set_all(False)
for cell in cells(mesh):
if criteria(cell.midpoint()) > threshold_value:
cell_markers[cell] = True
return refine(mesh, cell_markers)
As a property coefficient, a variable coefficient is set that models
inhomogeneities (in our case, these are areas with defects). Next, the gradient of
physical parameters is calculated to identify areas with strong inhomogeneities. For
mesh adaptation, a gradient threshold criterion is used to identify mesh elements
that require refinement, and then the mesh is automatically rebuilt in critical areas,
and the problems are re-solved. This enhances the modeling accuracy in areas with
strong gradients of physical properties, optimize computational resources by
concentrating the mesh in critical areas, and improve results for problems with
complex geometry or defects.
Yu.Yu. Bilak
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 60
The weak form of the equations used to model electromagnetic, acoustic, and
thermal processes allows for their efficient integration into a single system of
equations, reducing the number of required calculations. The use of the Newton-
Raphson method to solve nonlinear equations provides fast convergence provided
that the initial approximation is correct. This method, combined with optimized
libraries, further increases the speed of calculations. Optimization is also achieved
by minimizing computational operations: the temperature dependence of material
parameters is calculated only in those areas where it is necessary. All these aspects
make the approach effective for modeling complex physical processes in multilayer
structures.
Overall, the developed algorithm has a number of unique features. It provides
a relationship between physical fields, allowing for the consideration of how
temperature changes affect the wave field, how acoustic waves change the local
dielectric constant, and how electromagnetic waves generate local heating. The
algorithm is scalable, which allows it to be adapted to simulate structures of various
sizes – from nanometer films to macroscopic layers. An important advantage is also
the ability to model defects in multilayer structures, such as inhomogeneities,
cracks and other disturbances that affect the behavior of physical fields.
Optimization of calculations. In this article, the concept of an efficient
algorithm is described through the implementation of a multiphysical approach that
integrates the solution of electromagnetic, acoustic and thermal equations in a
single system. FEM is used to discretize the equations in space, and explicit/implicit
schemes are used for discretization in time. The efficiency of the algorithm is
achieved through a comprehensive approach that includes adaptive discretization,
parallel calculations and automatic mesh optimization. Therefore, the optimization
of numerical modeling of multilayer structures aims to minimize computational
costs while maintaining or increasing the accuracy of calculations. Formally, this
can be presented as a minimization problem:
tm iin under the condi o( , , ) ( , ,s )n permissibleT C N P С N P ,
where T(C,N,P) – computing time, which depends on the complexity of the model
(C), grid size (N) and the number of processor elements (P); аnd ε(C,N,P) –
numerical calculation error, which must remain within acceptable limits (εpermissible).
Optimization is achieved through parallel computing, adaptive meshing, and
efficient equation solving algorithms.
Regarding the justification of the selection of algorithms for the problem being
solved, the Newton–Raphson method is used for solving nonlinear equations
describing the interaction of physical fields. Its advantage is rapid convergence with
an appropriate choice of the initial approximation. AMR allows for enhancing local
accuracy by increasing the number of nodes only in critical zones with large
gradients of physical parameters. This reduces the overall load on the processor.
And parallel computing (MPI, CUDA) enables distributing computations between
processor or graphics processor cores, which reduces the calculation time. These
methods interact in such a way as to minimize unnecessary calculations and ensure
rapid convergence of the algorithm.
To select the optimal configuration, 4 series of numerical experiments (on
synthetic data) were conducted, in which different optimization methods were
analyzed:
Development of a mathematical model of wave processes in multilayered structures with an…
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1. Without optimization (basic configuration).
2. Parallel calculations on CPU (MPI + CPU).
3. Calculations on GPU (CUDA).
4. Adaptive mesh (AMR) + GPU calculations.
The results of the comparison of calculation time and relative error are given
in Table 1 and Fig. 2.
T a b l e 1 . Comparison of optimization methods
Optimization
method
Calculation
time, sec
Speedup
(relative to baseline) Relative error, %
Without
optimization 100.0 1.00 (basic level) 5.2
MPI + CPU 35.7 2.8 3.1
GPU + CUDA 21.3 4.7 2.4
AMR + GPU 15.6 6.4 1.8
The results of numerical experiments showed that the use of optimization
methods significantly improves the performance and accuracy of the simulation.
The basic approach without optimization gave the longest calculation time
(100 sec.) and the highest relative error (5.2 %), which is associated with a uniform
mesh and the lack of parallel processing. Using MPI on the CPU reduced the time
to 35.7 sec., but the accuracy improved slightly (error 3.1 %). Switching to GPU
(CUDA) reduced the calculation to 21.3 sec. with an error of 2.4 %, which is
explained by the efficient processing of vector operations and more accurate
calculation of gradients. The best results were achieved when combining GPU and
adaptive mesh, which provided the minimum time (15.6 sec.) and the lowest
error (1.8 %).
Fig. 2. Computation time and relative error for different optimization methods
In general, parallel MPI + CPU calculations provide a 2.8–3.5-fold speedup,
while GPU + CUDA for large problems provides a 4.7–6.2-fold performance
Yu.Yu. Bilak
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 62
increase. Using AMR reduces the number of nodes by 30–50 %, which shortens
the computation time by up to 40 % without losing accuracy. In critical zones with
high gradients, AMR reduces the error by 1.5 times, and the average deviation
between numerical and experimental data does not exceed 1.04 % for temperature
and 1.99 % for electric field strength.
Comparison of data indicates that AMR + GPU achieves an optimal balance
between speed and accuracy, reducing the calculation time by 6.4 times compared
to the baseline method, and the average deviation between numerical and
experimental data does not exceed 1.8 %. Further improvement is possible by
optimizing the mesh adaptation parameters and automating the selection of
refinement criteria.
Formalization of the application of neural networks. Neural networks can
be formally applied to this model, as they are well suited for approximating
complex nonlinear dependencies, optimizing parameters, and accelerating
numerical modeling. They can be integrated into three main aspects:
1. Acceleration of differential equation solving. Neural networks can replace
or complement traditional numerical methods for solving equations of heat
conduction, electromagnetic and acoustic fields. For example, Physics-Informed
Neural Networks (PINNs) can learn to solve equations without explicit
discretization.
2. Adaptive mesh refinement. Neural network models can predict areas with
high gradients of physical parameters, which will allow more effective mesh
adaptation. Instead of standard AMR criteria (gradient or entropy methods), deep
convolutional networks (CNNs) can be used, which analyze the distribution of
parameters and determine where to increase the density of nodes [31].
3. Prediction of numerical results. Neural networks can learn from
previous numerical calculations and be used to quickly predict the distribution
of temperature, electric field or other parameters. Recurrent neural networks
(RNN, LSTM) [32, 33] can predict the temporal dynamics of changes
in physical parameters without the need for detailed calculations at each
time step.
Formal requirements for applying neural networks to a model include the
availability of a sufficient training sample, which can be generated through
previous numerical experiments. Furthermore, it is necessary to consider the
formalization of physical constraints so that the network does not violate
fundamental physical laws, which can be achieved using approaches such as PINN
or special loss functions. A comparative analysis of accuracy should also be
performed to determine whether neural network approaches really contribute to
improving speed and accuracy compared to classical methods. Thus, neural
networks can be applied to improve the computational efficiency of the proposed
model, but their implementation requires a separate study to assess the accuracy
and computational cost.
RESEARCH RESULTS
Synthetic data. To demonstrate the model’s performance, we present graphical
results for various physical conditions obtained on synthetic data. The results are
saved in .pvd format and visualized using the matplotlib and Pyvista libraries.
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a b
c
Fig. 3. a – temperature field distribution in a multilayer structure; b – electric field
distribution in the structure; c – acoustic displacement distribution in a multilayer structure
Fig. 3, a shows the distribution of the temperature field in a multilayer
structure. We observe that the maximum temperature is in the central part of the
region, which corresponds to the zone of local heating caused by the
electromagnetic field. The temperature distribution smoothly decreases from the
center to the edges, demonstrating heat exchange with the surrounding
environment. The graph (Fig. 3, b) illustrates the distribution of the electric field.
It has the nature of a wave function with periodic changes in intensity. This shows
the behavior of the electromagnetic field in the region, in particular its interaction
with boundary conditions, which may be a consequence of reflection or interference
of waves. Fig. 3, c shows the distribution of acoustic displacements in the structure.
Acoustic waves generated by the interaction of physical fields propagate in the
region with certain periodic patterns. Maximum displacements are observed in
certain zones, which corresponds to local resonances or intense interaction of
waves. In general, the graphs illustrate the interaction of physical processes within
the multiphysical model, including heat transfer, propagation of electromagnetic
waves, and their effect on acoustic displacements.
Experimental data. The results of the developed information-numerical model
were tested on experimental data [21]. The model successfully reproduced the
physical regularities described in the article: in particular, the localization of high
temperatures at high pressure (101.13 kPa) and a wider heating zone at low pressure
(13.3 kPa). The electric field was simulated (with high accuracy) considering the
influence of voltage and discharge geometry, which confirms the adequacy of the
numerical approach for predicting the conditions for film formation. The simulation
results are shown in Fig. 4.
Yu.Yu. Bilak
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 64
а b
c d
Fig. 4. a, b – temperature distribution in the discharge gap at high pressure p = 101.13 kPa
and low pressure p = 13.3 kPa (respectively); c, d – electric field distribution at high
pressure p = 101.13 kPa and low pressure p = 13.3 kPa (respectively)
The graphs depict the physical processes that occur during the formation of
tungsten oxide films under different discharge gap conditions. Fig. 4, a, which
displays the temperature distribution at high pressure (101.13 kPa), shows intense
heating in the central zone of the discharge. The maximum temperature is reached
near the axis of the discharge gap, where the energy concentration is the highest.
This is due to the high voltage and significant energy contribution to the plasma.
The heating decreases exponentially from the center to the periphery,
demonstrating the characteristic energy dissipation in high-pressure plasma.
The temperature distribution at low pressure, which is displayed in Fig. 4, b
(13.3 kPa), demonstrates a lower temperature peak in the central zone, which is
explained by the decrease in energy contribution and lower particle density in the
plasma. However, the heating zone is wider, since the energy is dissipated more
evenly due to the lower ionization resistance at reduced pressure. The electric field
distribution at high pressure (Fig. 4, c) shows an intense field in the center, which
gradually decreases with distance from the axis of the discharge gap. The high
voltage creates a strong field gradient, which promotes intensive ionization of
molecules and the formation of plasma in a narrow central zone. This ensures the
efficient formation of films with high density and uniformity in the central
deposition region. The electric field distribution at low pressure (Fig. 4, d) indicates
a smoother field decay, which is due to the lower voltage. The field intensity is
lower, but the zone of its action is wider. This corresponds to conditions where
films can be formed over a larger area, but their physicochemical properties may
be less stable due to the lower plasma concentration. The obtained results confirm
that the discharge conditions significantly affect the temperature and electric field
distribution, which, in turn, determine the characteristics of the obtained films.
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High pressure and tension promote the formation of dense and uniform films, while
low pressure can ensure the formation of films over a larger area but with less
structural stability.
Model accuracy assessment and error analysis. The modeling accuracy was
assessed by analyzing the deviations between the calculated and experimental
values of parameters such as temperature, electric field distribution, and structural
properties of thin films.
The average relative deviation was chosen as the criteria for assessing the
model accuracy, calculated by the formula:
model exp
exp
1
1 100 %
N
i i
i i
X X
N X
,
where model
iX – value, obtained using the model; exp
iX – experimental value;
N – number of comparison points. The maximum error between numerical and
experimental data and the coefficient of determination R2, which measures what
proportion of the variation in the dependent variable is explained by the
independent variables in the model, were also calculated. The results of the model
accuracy assessment are given in Table 2.
T a b l e 2 . Assessment of the accuracy of the numerical model
Parameter Numerical
results
Experimental
results
Absolute
error
Relative
error, %
Coefficient of
determination R2
Maximum
temperature (K)
at 101.13 kPa
795.5 798.0 2.5 0.31 0.992
Maximum
temperature (K)
at 13.3 kPa
370.3 374.2 3.9 1.04 0.985
Electric field
strength (V/m)
at 101.13 kPa
29 600 30 200 600 1.99 0.978
Electric field
strength (V/m)
at 13.3 kPa
11 880 12 050 170 1.41 0.981
From the table we see that the coefficient of determination for temperature
varies within 0.985–0.992, which indicates a high agreement between numerical
and experimental values. This indicates that the model predicts the temperature
distribution in the discharge gap well. For the electric field, the coefficient
of determination is somewhat lower, but still maintains a good accuracy of
0.978–0.981. This may be a consequence of unaccounted plasma inhomogeneities
or possible experimental errors in the measurements of the electric field strength.
In general, the R2 values confirm the high accuracy of the numerical model and its
ability to reliably predict the physical parameters of the discharge process. Table 2
demonstrates the stability of the numerical model when changing the parameters of
the environment. For low pressure conditions, both the temperature values and the
electric field have slightly larger deviations, which may be due to the influence of
the extended plasma region, which is more difficult to accurately model. However,
Yu.Yu. Bilak
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 66
even in this case, the model results are in good agreement with the experiment.
In general, the numerical model reliably predicts the main physical characteristics
of the process, which makes it an effective tool for analyzing plasma discharges
and their impact on the formation of thin films.
The analysis of errors shows that the main sources of deviations are the
discretization of space and time, the physical assumptions of the model, and the
errors of experimental measurements. The use of an adaptive grid reduces the
numerical errors, but underestimation of local changes is possible with sharp
parameter gradients. Assumptions of plasma homogeneity can simplify the real
picture of the processes, since in real conditions there are inhomogeneities
associated with variations in pressure, temperature, and interaction with the
substrate. In addition, the errors of experimental measurements, due to the limited
accuracy of the sensors and the conditions of the experiment, can affect the
discrepancies between the numerical and real values. Accounting for these factors
during model validation enhances the accuracy of results interpreting and
establishes the reliability range of predictions.
DISCUSSION OF RESULTS
The developed information-numerical model was tested on experimental data and
showed high accuracy in reproducing physical patterns. The localization of high-
temperature zones at high pressure (101.13 kPa) was reproduced, as well as a wider
heating zone at low pressure (13.3 kPa), which is consistent with experimental
observations. The electric field was modeled considering the influence of voltage
and discharge geometry. The electric field distribution showed characteristic
periodic changes in intensity, which may be a consequence of interference effects
and the influence of boundary conditions. This confirms the correctness of the
approach to considering the interaction of electromagnetic waves in multilayer
structures.
Analysis of the results demonstrates the stability of the numerical model when
changing the parameters of the environment. At low pressure, the temperature
distribution and electric field have slightly larger deviations, which may be due to
the influence of the extended plasma region, which is more difficult to accurately
model due to spatial inhomogeneities. However, even in this case, the model results
are in good agreement with the experiment. The use of an adaptive grid allowed to
increase the accuracy of calculations in critical zones, however, at very high
parameter gradients, certain errors may occur due to insufficient discretization.
The proposed model using hybrid CPU-GPU calculations provided a reduction in
calculation time by 40–60 % depending on the complexity of the problem and
allowed to obtain a 3–5-fold increase in performance in the case of large-scale
calculations compared to traditional approaches.
Regarding the uniqueness and advantages of the developed model, it is worth
highlighting the integration of multiphysical processes that account for the
interaction of electromagnetic, thermal and acoustic effects. AMR ensures accuracy
in critical zones, and parallel calculations (MPI, GPU) reduces time costs by
40–60 % and increases productivity by 3–5 times. The model also accounts for the
influence of external conditions, which makes it effective for modeling
nanostructured films.
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The main limitations of the model are the complexity of solving nonlinear
equations when scaling, the dependence of accuracy on the grid discretization, as
well as the need for validation on a wider set of experimental data. The model does
not account for quantum effects, which can affect the accuracy of predictions in
nanoscale structures. In general, the model reliably predicts the main physical
characteristics of the process, which is confirmed by the high correlation between
numerical and experimental data. The relative errors for temperature do not exceed
1.04 %, and for the electric field – 1.99 %, which indicates the adequacy of
numerical modeling.
CONCLUSIONS
The research developed an integrated mathematical model for modeling wave
processes in multilayer thin films, which combines Maxwell’s equations,
mechanical vibrations and thermal conductivity. An adaptive algorithm for mesh
refinement was implemented, which allows it to be automatically rebuilt depending
on local gradients of physical parameters, such as density, temperature or defects,
ensuring high accuracy of modeling in critical zones. A hybrid computational
approach was proposed, utilizing both CPUs and GPUs, which significantly
reduced computational costs for solving large-scale problems. Software with
a modular architecture was developed, which provides integration of numerical
methods, computation optimization, and real-time results visualization.
Experimental validation confirmed the high accuracy and reliability of the model,
allowing to assess the influence of physical parameters on the behavior of
multilayer films. The obtained results contribute to a deeper understanding of wave
processes and create a basis for optimizing multilayer structures in various fields
of science and industry.
In the future, to enhance the model, it is possible to implement neural networks
for predicting computational results and reducing modeling time, integrate
molecular dynamics methods for precise analysis of film formation at the atomic
level, etc. Expanding the work in these areas will increase its uniqueness, provide
new scientific results, and make a significant contribution to the development of
numerical modeling of multilayer thin films.
Conflict of Interest. The authors declare that there is no conflict of interest in
this study, including financial, special, authorship, or any other nature that could
influence the research and the results presented in this article.
Finance. The study was conducted without financial support.
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Received 09.01.2025
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INFORMATION ON THE ARTICLE
Yurii Yu. Bilak, ORCID: 0000-0001-5989-1643, Uzhgorod National University, Ukraine,
e-mail: yuriy.bilak@uzhnu.edu.ua
РОЗРОБЛЕННЯ МАТЕМАТИЧНОЇ МОДЕЛІ ХВИЛЬОВИХ ПРОЦЕСІВ
У БАГАТОШАРОВИХ СТРУКТУРАХ ІЗ АДАПТИВНИМ АЛГОРИТМОМ
ТА ГІБРИДНИМИ ОБЧИСЛЕННЯМИ / Ю.Ю. Білак
Анотація. Досліджено чисельне моделювання хвильових процесів
у багатошарових тонких плівках, що є актуальним для розуміння їхніх фізичних
властивостей та оптимізації для різних застосувань. Розроблено інтегровану
математичну модель, яка поєднує рівняння Максвелла, механічних коливань
і теплопровідності з урахуванням взаємодії фізичних полів у структурах
із дефектами. Запропоновано адаптивний алгоритм для автоматичного
уточнення сітки залежно від локальних градієнтів фізичних параметрів, що
дозволяє підвищити точність моделювання в критичних зонах. Упроваджено
гібридний підхід до обчислень із використанням CPU і GPU, що забезпечує
ефективне використання ресурсів для задач великого масштабу. Розроблено
програмне забезпечення з модульною архітектурою, яке дозволяє інтегрувати
чисельні методи, оптимізацію та візуалізацію результатів у реальному часі.
Експериментальна валідація підтвердила високу точність і надійність моделі.
Отримано результати, які сприяють глибшому розумінню фізичних процесів
у тонких плівках і є основою для створення високоефективних багатошарових
структур у промислових і наукових застосуваннях.
Ключові слова: тонкі плівки, чисельне моделювання, оптимізація,
мультифізичні моделі, паралельні обчислення, гібридні алгоритми, хвильові
процеси.
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| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-01T01:00:13Z |
| publishDate | 2026 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
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| spelling | journaliasakpiua-article-3652552026-06-30T06:14:59Z Development of a mathematical model of wave processes in multilayered structures with an adaptive algorithm and hybrid calculations Розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями Bilak, Yurii thin films numerical simulation optimization multiphysics models parallel computing hybrid algorithms wave processes тонкі плівки чисельне моделювання оптимізація мультифізичні моделі паралельні обчислення гібридні алгоритми хвильові процеси The paper investigates the numerical simulation of wave processes in multilayer thin films, which is relevant for understanding their physical properties and optimization for various applications. An integrated mathematical model has been developed that combines Maxwell’s equations, mechanical vibrations and thermal conductivity, taking into account the interaction of physical fields in structures with defects. Adaptive algorithms have been proposed for automatic mesh refinement depending on local gradients of physical parameters, which allows to increase the accuracy of modeling in critical zones. A hybrid approach to calculations using CPU and GPU has been implemented, which ensures efficient use of resources for large-scale problems. Software with a modular architecture has been developed that allows integrating numerical methods, optimization and visualization of results in real time. Experimental validation has confirmed the high accuracy and reliability of the model. The results obtained contribute to a deeper understanding of physical processes in thin films and are the basis for the creation of highly efficient multilayer structures in industrial and scientific applications. Досліджено чисельне моделювання хвильових процесів у багатошарових тонких плівках, що є актуальним для розуміння їхніх фізичних властивостей та оптимізації для різних застосувань. Розроблено інтегровану математичну модель, яка поєднує рівняння Максвелла, механічних коливань і теплопровідності з урахуванням взаємодії фізичних полів у структурах із дефектами. Запропоновано адаптивний алгоритм для автоматичного уточнення сітки залежно від локальних градієнтів фізичних параметрів, що дозволяє підвищити точність моделювання в критичних зонах. Упроваджено гібридний підхід до обчислень із використанням CPU і GPU, що забезпечує ефективне використання ресурсів для задач великого масштабу. Розроблено програмне забезпечення з модульною архітектурою, яке дозволяє інтегрувати чисельні методи, оптимізацію та візуалізацію результатів у реальному часі. Експериментальна валідація підтвердила високу точність і надійність моделі. Отримано результати, які сприяють глибшому розумінню фізичних процесів у тонких плівках і є основою для створення високоефективних багатошарових структур у промислових і наукових застосуваннях. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2026-06-30 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/365255 10.20535/SRIT.2308-8893.2026.2.04 System research and information technologies; No. 2 (2026); 51-70 Системные исследования и информационные технологии; № 2 (2026); 51-70 Системні дослідження та інформаційні технології; № 2 (2026); 51-70 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/365255/350706 |
| spellingShingle | тонкі плівки чисельне моделювання оптимізація мультифізичні моделі паралельні обчислення гібридні алгоритми хвильові процеси Bilak, Yurii Розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями |
| title | Розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями |
| title_alt | Development of a mathematical model of wave processes in multilayered structures with an adaptive algorithm and hybrid calculations |
| title_full | Розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями |
| title_fullStr | Розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями |
| title_full_unstemmed | Розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями |
| title_short | Розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями |
| title_sort | розроблення математичної моделі хвильових процесів у багатошарових структурах із адаптивним алгоритмом та гібридними обчисленнями |
| topic | тонкі плівки чисельне моделювання оптимізація мультифізичні моделі паралельні обчислення гібридні алгоритми хвильові процеси |
| topic_facet | thin films numerical simulation optimization multiphysics models parallel computing hybrid algorithms wave processes тонкі плівки чисельне моделювання оптимізація мультифізичні моделі паралельні обчислення гібридні алгоритми хвильові процеси |
| url | https://journal.iasa.kpi.ua/article/view/365255 |
| work_keys_str_mv | AT bilakyurii developmentofamathematicalmodelofwaveprocessesinmultilayeredstructureswithanadaptivealgorithmandhybridcalculations AT bilakyurii rozroblennâmatematičnoímodelíhvilʹovihprocesívubagatošarovihstrukturahízadaptivnimalgoritmomtagíbridnimiobčislennâmi |