=Paper=
{{Paper
|id=Vol-3646/Short_1.pdf
|storemode=property
|title=The Method of Effective Numerical Solution of the System of Equations of Thermal Conductivity
|pdfUrl=https://ceur-ws.org/Vol-3646/Short_1.pdf
|volume=Vol-3646
|authors=Oleg Kurchenko,Kseniia Dukhnovska,Oksana Kovtun,Anastasiia Nikolaienko,Iryna Yurchuk
|dblpUrl=https://dblp.org/rec/conf/iti2/KurchenkoDKNY23
}}
==The Method of Effective Numerical Solution of the System of Equations of Thermal Conductivity==
<pdf width="1500px">https://ceur-ws.org/Vol-3646/Short_1.pdf</pdf>
<pre>
                         The Method of Effective Numerical Solution of the System of
                         Equations of Thermal Conductivity
                         Oleg Kurchenko, Kseniia Dukhnovska, Oksana Kovtun, Anastasiia Nikolaienko and
                         Iryna Yurchuk
                         Taras Shevchenko National University of Kyiv, Bohdan Hawrylyshyn str. 24, Kyiv, UA-04116, Ukraine

                                         Abstract
                                         The article considers the method of effective numerical solution of the system of equations of
                                         thermal conductivity, which is used for numerical modeling of the forecast of
                                         thermophysiological state of man, and its parallel software implementation on a
                                         multiprocessor system. The method is suitable for building a three-dimensional model for
                                         predicting the thermophysiological state of man. The results of numerical experiments and
                                         parallel calculations show the good suitability of the proposed algorithms for parallel
                                         implementation and their significant potential for creating on their basis applied computer
                                         systems for modeling the prediction of human thermophysiological state.
                                         Keywords 1
                                         System of equations of thermal conductivity, parallel algorithm of the Gaussian method, the
                                         method of finite differences

                         1. Introduction
                             Boundary value problems, which reflect technological or natural processes, have a high
                         computational complexity. To solve them, various analytical or numerical approaches are used,
                         neglecting the accuracy of the solution. Such boundary value problems can also include problems
                         with heat conduction equations.
                             Solving this problem requires significant CPU time to perform calculations. This is especially true
                         for modeling tasks, where the requirements for quality and detail of three-dimensional models that
                         describe the processes of heat transfer, which, in turn, leads to increased computational costs.
                             The use of parallel computing is extremely important for obtaining an efficient and accurate
                         forecast, as it provides the necessary acceleration of algorithm calculations when performing them on
                         powerful multiprocessor systems. However, the development of effective parallel algorithms for
                         multiprocessor systems is a difficult scientific and technical problem and requires solving
                         interdisciplinary research using knowledge of mathematical, architectural and software models of
                         parallel computing.

                         2. Literature review and problem statement
                            Equations of thermal conductivity are quite widely used. In work [1], the flow of viscous
                         dissipation, chemical reactions, and convection heat on the walls of the channel is investigated with
                         the help of heat conduction equations. The Math software package was used to solve the nonlinear
                         system of differential equations. The results of this work can be used to solve the cooling problem in
                         various industries.
                            The work [2] presents the results of nanofluid research in the heat engineering sector. The system
                         of nonlinear differential equations that describes this problem is first converted into a system of

                         Information Technology and Implementation (IT&I-2023), November 20-21, 2023, Kyiv, Ukraine
                         EMAIL: oleg.kurchenko@knu.ua (A.1); kseniia.dukhnovska@knu.ua (A.2); kovok@ukr.net (A.3); n_nastja@ukr.net (A.4);
                         i.a.yurchuk@gmail.com (A.5)
                         ORCID: 0000-0002-3507-2392 (O. Kurchenko); 0000-0002-4539-159X (K. Dukhnovska); 0000-0003-0871-5097 (O. Kovtun); 0000-0002-
                         2402-2947 (A. Nikolaenko); 0000-0001-8206-3395 (I. Yurchuk);
                                    ©️ 2023 Copyright for this paper by its authors.
                                    Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
                                    CEUR Workshop Proceedings (CEUR-WS.org)


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                  ceur-ws.org
Workshop      ISSN 1613-0073
Proceedings
                                                                                                                                               206
ordinary differential equations. Then it is solved using the bvp4c algorithm. Current work has many
applications in various fields of engineering, biotechnology, nanotechnology and medicine.
    The paper [3] examines the heat exchange of the boundary layer fluid. Similarity transformations
are again applied to transform the governing partial differential equations for mass, momentum, and
energy into a system of ordinary differential equations. The MATLAB package is used to obtain the
solution. The effect of variable thermal conductivity on an exponentially stretched surface is
investigated in [4]. The bvp4c algorithm is used to solve this boundary value problem.
    The thermal conductivity equation is used to predict the thermo-physiological state of a person.
The purpose of the article [5] was to review existing thermophysiological models for the entire human
body and its individual segments. The paper provides an overview of the most recognized thermal
models, such as Fiala, Berkeley Comfort Model, Tanabe and ThermoSem.
    The study [6] proposed clothes with fluid circulation, which can effectively reduce the thermal
stress of people in a hot environment. This work used a model of human thermoregulation together
with a model of warmth and comfort. Thus, in the article [7], a complete virtual three-dimensional
model of human thermophysiological reactions is investigated. The numerical solution of the model
was developed using the ANSYS Fluent platform. This platform works due to parallel processes [8].

3. The purpose and objectives of the research
   The purpose of the research is to develop method of effective numerical solution of the system of
equations of thermal conductivity. This will make it possible to obtain a numerical value for problems
that include the heat conduction equation. To achieve the goal, the following tasks were solved:
    to solve the heat conduction equation, apply the finite difference method together with the
parallel Gaussian algorithm;
    to investigate the effectiveness of the method.
   As can be seen from the review, the boundary value problems, which include the heat conduction
equation, are first analytically translated into the Cauchy problem, and then calculated. This is due to
the high computational complexity of boundary value problems. Let's consider one of these.

4. Mathematical model for predicting the thermophysiological state of
   human
    A multi-compartmental model of human thermoregulation and heat transfer is used for
mathematical modeling of human physiological condition prediction [5-11].
    According to the definition given in [9], a compartment is a certain amount of a substance that is
released in a biological system and has the property of unity, so in the processes of transport and
chemical transformations it can be considered as a whole. Models in which the studied system is
presented as a set of compartments are called multi-compartmental.
    The class of multicompartmental dynamic models of human heat exchange and thermoregulation
describes heat production in organs and tissues, heat transfer by blood flow, conduction, convection,
radiation, evaporation from the skin and upper respiratory tract, afferent and efferent
thermoregulatory processes.
    The structure of the models is based on the compartment. The compartment has an energy source,
the ability to receive and transfer heat to neighboring compartments and exchange energy with the
environment. The set of compartments and the task of connections between them determines the
nature and degree of approximation of the human body and the physiological processes occurring in
it. For the sample of a single compartment the cylinder for representation of all parts of a body and
sphere - for the head is taken. Each compartment corresponds to a specific organ, body part or body
tissue. The models take into account human anatomical and physiological parameters: weight, height,
body surface area, biophysical characteristics of tissues and organs, metabolism, oxygen
consumption, blood flow, pulmonary ventilation, heat transfer and heat transfer coefficients, cardiac
output, sweating, fluid loss, etc. [11]. Approximation of the human body can be performed on any
number of geometric parts, for this model the human body was approximated by 13 cylinders and
sphere: head, torso, left and right arm (shoulders, forearms, hands), left and right leg (thighs, legs,
feet). Each cylinder that approximates a part of the human body has nested cylinders that describe the



                                                                                                    207
properties of other tissues of the human body. The heat balance equation is written to each
compartment and the result is the following system of differential equations in partial derivatives:
                                                                                                  (1)




where c – specific heat, kcal / (kg · ° C); m – mass, kg; T – temperature, ° C; t – time; M –
metabolism, kcal; Q – heat flux, kcal; V – volume, l; W – cardiac output, l / h. Indices: * – initial
value; b – blood; ph – physical activity; RS – heat loss with respiration; sh – cold tremor of skeletal
muscles; K – conduction; air – air; water – water.
   To solve this system requires additional conditions: initial and boundary.
   Initial conditions. At the beginning of time, temperatures, metabolism, blood flow, etc. are set for
each compartment: if t = 0: Tij = T0ij, for each i,j when 0≤i≤N-1, 0≤j≤L-1.
   Boundary conditions. To solve this system, boundary conditions of the fourth kind are used. For
neighboring compartments, conditions must be met at the boundary between them.
            𝑖𝑗
   If 𝑥 ∈ 𝐺𝑥 :
                               𝜕𝑇𝑖𝑗          𝜕𝑇𝑖,𝑗−1                                            (2)
                            𝜆       =𝜆
                                𝑖𝑗 𝜕𝑥    𝑖,𝑗−1 𝜕𝑥    , ∀𝑡 ≥ 0, 𝑗 ≠ 𝑠𝑘,
                                     𝜕𝑇𝑏      𝜕𝑇𝑖,𝑗                                             (3)
                                  𝜆𝑏 𝜕𝑥 = 𝜆𝑖,𝑗 𝜕𝑥 , ∀𝑡              ≥ 0, 𝑗 ≠ 𝑠𝑘.
            𝑖𝑗
   If 𝑦 ∈ 𝐺𝑦 :
                                   𝜕𝑇𝑖𝑗            𝜕𝑇𝑖,𝑗−1                                      (4)
                               𝜆𝑖𝑗 𝜕𝑦 = 𝜆𝑖,𝑗−1        𝜕𝑦
                                                              , ∀𝑡 ≥ 0, 𝑗 ≠ 𝑠𝑘,
                                        𝜕𝑇         𝜕𝑇𝑖,𝑗                                        (5)
                                 𝜆𝑏 𝜕𝑦𝑏 = 𝜆𝑖,𝑗 𝜕𝑦 , ∀𝑡 ≥ 0, 𝑗 ≠ 𝑠𝑘.
            𝑖𝑗
   If 𝑧 ∈ 𝐺𝑧 :
                                   𝜕𝑇𝑖𝑗            𝜕𝑇𝑖,𝑗−1                                      (6)
                               𝜆𝑖𝑗 𝜕𝑧 = 𝜆𝑖,𝑗−1        𝜕𝑧
                                                              , ∀𝑡 ≥ 0, 𝑗 ≠ 𝑠𝑘,
                                        𝜕𝑇         𝜕𝑇𝑖,𝑗                                        (7)
                                 𝜆𝑏 𝜕𝑧𝑏 = 𝜆𝑖,𝑗 𝜕𝑧 , ∀𝑡 ≥ 0, 𝑗 ≠ 𝑠𝑘.
   For j = sk, 𝑥 ∈ 𝐺𝑥 𝑖,𝑠𝑘 :
                               𝜕𝑇𝑖,𝑠𝑘                                                           (8)
                                𝜕𝑥
                                        = 𝑎𝑖𝑗 𝜆𝑖,𝑠𝑘 (𝑇𝑖,𝑠𝑘 − 𝑇 𝑎𝑖𝑟 ), ∀𝑡 ≥ 0,
                                             𝜕𝑇            𝜕𝑇𝑖,𝑠𝑘                               (9)
                                        𝜆𝑏 𝜕𝑥𝑏 = 𝜆𝑖,𝑠𝑘      𝜕𝑥
                                                                    , ∀𝑡 ≥ 0.
   For j = sk, 𝑧 ∈ 𝐺𝑦 𝑖,𝑠𝑘 :
                               𝜕𝑇𝑖,𝑠𝑘                                                          (10)
                                𝜕𝑦
                                        = 𝑎𝑖𝑗 𝜆𝑖,𝑠𝑘 (𝑇𝑖,𝑠𝑘 − 𝑇 𝑎𝑖𝑟 ), ∀𝑡 ≥ 0,



                                                                                                   208
                                            𝜕𝑇𝑏             𝜕𝑇𝑖,𝑠𝑘                               (11)
                                       𝜆𝑏         = 𝜆𝑖,𝑠𝑘            , ∀𝑡 ≥ 0.
                                            𝜕𝑦               𝜕𝑦
   for j = sk, 𝑥 ∈ 𝐺𝑧𝑖,𝑠𝑘 :
                              𝜕𝑇𝑖,𝑠𝑘                                                             (12)
                                       = 𝑎𝑖𝑗 𝜆𝑖,𝑠𝑘 (𝑇𝑖,𝑠𝑘 − 𝑇 𝑎𝑖𝑟 ), ∀𝑡 ≥ 0,
                               𝜕𝑧
                                 𝜕𝑇          𝜕𝑇                                              (13)
                               𝜆𝑏 𝑏 = 𝜆𝑖,𝑠𝑘 𝑖,𝑠𝑘 , ∀𝑡 ≥ 0,
                                  𝜕𝑧          𝜕𝑧
where Gij surface of ij-compartment, Gx, Gy, Gz - the projection of these surfaces on the coordinate
axis [12].

5. Numerical method of finite differences for solving a system of differential
   parabolic equations
    To solve the model (1), the finite difference method was used. This is a method of solving
differential equations based on the replacement of differential operators with their approximate values
at individual points. When using the finite difference method, the differential problem is replaced by
the difference problem.
    The equations given in problem (1) are parabolic differential equations in partial derivatives. It is
proved that finite difference methods can be used to solve parabolic differential equations. To solve
the system of equations, the method of finite differences with an implicit scheme is taken. For
parabolic differential equations, the solution of such a scheme is stable under any conditions.
    To solve the differential equation by the finite difference method, first the area on which the
solution is sought is replaced by a discrete set of points (nodes of the grid). The difference operators
corresponding to the differential equation are written in the internal nodes of the grid. The difference
operators corresponding to the boundary conditions are written in the boundary nodes of the grid. As
a result, we obtain a system of algebraic equations, the number of which is proportional to the number
of internal nodes of the grid region. To obtain a numerical solution, we need to solve this system of
equations.

6. Numerical method of finite differences for solving a system of differential
   parabolic equations
    To solve the model (1), the finite difference method was used. This is a method of solving
differential equations based on the replacement of differential operators with their approximate values
at individual points. When using the finite difference method, the differential problem is replaced by
the difference problem. The equations given in problem (1) are parabolic differential equations in
partial derivatives. It is proved that finite difference methods can be used to solve parabolic
differential equations. To solve the system of equations, the method of finite differences with an
implicit scheme is taken. For parabolic differential equations, the solution of such a scheme is stable
under any conditions. To solve the differential equation by the finite difference method, first the area
on which the solution is sought is replaced by a discrete set of points (nodes of the grid). The
difference operators corresponding to the differential equation are written in the internal nodes of the
grid. The difference operators corresponding to the boundary conditions are written in the boundary
nodes of the grid. As a result, we obtain a system of algebraic equations, the number of which is
proportional to the number of internal nodes of the grid region. To obtain a numerical solution, we
need to solve this system of equations.

7. Parallel algorithm of the Gaussian method
    To find the temperatures at each step of integration, you need to solve a system of linear algebraic
equations. The Gaussian method was used for this purpose.
    The Gaussian method is based on the possibility of performing transformations of linear equations,
which do not change the solution of the considered system. The Gaussian method involves the
sequential execution of two steps. At the first step - the direct course of the Gaussian method - the
initial system of linear equations by successive exclusion of unknowns is reduced to the upper
triangular form. In the reverse course of the Gaussian method (the second stage of the algorithm) the



                                                                                                     209
values of the unknowns are determined. The calculations performed on the elements of the matrix A
and the vector b are determined by the following relations:
                                     ′
                                                  𝑎𝑘𝑗                                         (14)
                                   𝑎𝑘𝑗   = 𝑎𝑘𝑗 −      ∗ 𝑎𝑖𝑗 ,
                                                  𝑎𝑖𝑖
                                       ′
                                                 𝑎𝑘𝑗                                          (15)
                                     𝑏𝑘 = 𝑏𝑘 −        ∗ 𝑏𝑖 ,
                                                  𝑎𝑖𝑖
where 𝑖 ≤ 𝑗 ≤ 𝑛 − 1, 𝑖 ≤ 𝑘 ≤ 𝑛 − 1, 0 ≤ 𝑖 ≤ 𝑛 − 1.
   After bringing the matrix of coefficients to the upper triangular form, it becomes possible to
determine the values of the unknown. From the last equation of the transformed system the value of
the variable xn-1 can be calculated, then from the penultimate equation it becomes possible to
determine the variable xn-2, etc. In general, the calculations of the reverse Gaussian method can be
represented by the relations:
                                                 𝑏𝑛−1                                         (16)
                                       𝑥𝑛−1 =            ,
                                               𝑎𝑛−1,𝑛−1
                                             𝑛−1                                              (17)
                               𝑥𝑖 = (𝑏𝑖 − ∑ 𝑎𝑖𝑗 𝑥𝑗 )⁄𝑎𝑖𝑖 ,
                                            𝑗=𝑖+1
where i = n-2,…,0.
   Multicore computers are now commonplace. Multiprocessor machines have almost completely
supplanted single-core counterparts and are used everywhere.
   Many users do not like working with braking applications. Also annoying is when running a task
in an application reduces the performance or sensitivity of another part of the program. Thus, the
requirements for improving the efficiency of software are growing every day, which in turn
necessitates mechanisms that allow rational use of available resources of modern equipment.
   The solution of problem (1) is reduced to the solution of a system of algebraic systems. The
dimension of such a system is equal to the number of integration steps in variable directions. The
computational complexity of the Gaussian algorithm is of the order of O(n 3). In order to speed up the
algorithm, it was decided to use a parallel calculation of the Gaussian algorithm.
   Since the solution by the Gaussian method is reduced to a sequence of similar computational
operations of multiplication and addition over the rows of the matrix, as subtasks can be taken
calculations related to the processing of one or more rows of the matrix A and the corresponding
element of the vector b. Each iteration associated with the solution of the next subtask begins with the
selection of the leading line. The row with the largest absolute value among the elements of the i-th
column corresponding to the excluded variable xi is searched.
   Because the rows of the matrix A are assigned to different subtasks, to find the maximum value in
the subtask column numbered k, the elements must be exchanged for the variable xi, which is
excluded. After collecting all these coefficients, it can be determined which of the subtasks contains
the leading line and which value is the leading element.To continue the calculations, the leading
subtask must send its row to the matrix A, which determines the coefficients of the system of linear
equations and the corresponding element of the vector b to all other subtasks with numbers k, xki.
   When performing the reverse of the Gaussian method, the subtasks perform the necessary
calculations to find the values of the unknowns. As soon as any subtask i, 1≤i≤n, which determines
the value of its variable xi, this value must be used by all subtasks with numbers k, k <i: subtasks
substitute the obtained value of the new unknown and perform value adjustments for elements of the
vector b. Selected basic subtasks are characterized by the same computational complexity. However,
the size of the matrix that describes the system of linear equations is significantly larger than the
number of threads in the program (p << n), and the basic subtasks can be enlarged by combining
several rows of the matrix in one subtask. The use of a sequential data separation scheme for the
parallel solution of systems of linear equations will lead to an uneven computational load between the
flows: as the exclusion (forward) or determination (reverse) unknown in the Gaussian method for
most flows all necessary calculations will be completed and they will stand still. A possible way to
solve the problem of balancing calculations may be to use a tape loop scheme for the distribution of
data between the enlarged subtasks: in this case, the matrix A is divided into sets (strips) of rows of
the form (Figure 2):


                                                                                                    210
Figure 1: Tape scheme

                                                        𝑇
                                     𝐴 = (𝐴1 , 𝐴2 , … , 𝐴𝑝 ) ,                                 (18)
                                                                          𝑛
                 𝐴𝑖 = (𝑎𝑖1 , 𝑎𝑖2 , … , 𝑎𝑖𝑘 ), 𝑖𝑗 = 𝑖 + 𝑗𝑝, 1 ≤ 𝑗 ≤ 𝑘, 𝑘 = 𝑝,                   (19)
where where n is the number of unknowns in the system of linear equations; p is the number of
processes that the operating system allows to create a system of linear equations to solve.
   Comparing the data division scheme and the order of calculations in the Gaussian method, it can
be noted that the use of a cyclic method of forming tapes allows to better balance the computational
load between subtasks. Therefore, after analyzing the sequential version of the Gaussian algorithm,
we can conclude that parallelization is possible for the following computational procedures:
    search for the leading line;
    subtraction of the leading line from all lines to be processed;
    execution of the return stroke.
   This approach significantly speeds up data processing than sequential computation.

8. The study materials and methods
   The characteristics of an average person were used for calculations. The work [17] was the source
of data. It describes the thermophysical characteristics of a person with a body weight of 74.1 kg and
a body surface area of 1.89 m2. In order to obtain these average characteristics, a study was conducted
on 10 men and 10 women (Table 1).
Table 1
Values for Surface Areas of Men and Women [17]
                                                 Surface Area (m2)
    Segment                     10 men                                     10 women
      Head                      0.1326                                       0.1129
     Trunk                      0.6824                                       0.6279
      Arms                      0.2536                                       0.2210
     Hands                      0.0946                                       0.0783
      Legs                      0.5966                                       0.5904
      Feet                      0.1299                                       0.1100

    The work [17] also shows that 25 % of the tissues in the human body are skin and fat, 11 % are
internal organs, 3 % are nervous tissue, 43 % are muscles, and 18 % are the skeleton. At the same
time, the specific heat capacity of the skeleton is 0.5 kcal/g∙°С, the heat capacity for fat is
0.6 kcal/g∙°С, and 0.9 kcal/g∙°С for all other tissues. Thermal conductivity is given in Table 2.
Calculations were performed on the language platform Visual C++ using the library MPI. The choice




                                                                                                   211
of programming language depends on its speed. MPI is the most widely used software development
interface standard in parallel programming.
Table 2
Thermal conductivity between compartments [17]
     Segment                     Сompartment                       Thermal conductivity (kcal/°С∙h)
       Head                         Core                                        1.38
                                   Muscle                                       11.4
                                     Fat                                        13.8
                                    Skin                                        -----
      Trunk                         Core                                        1.37
                                   Muscle                                       4.75
                                     Fat                                        19.8
                                    Skin                                        -----
      Arms                          Core                                        1.20
                                   Muscle                                       8.90
                                     Fat                                       26.20
                                    Skin                                        -----
      Hands                         Core                                        5.50
                                   Muscle                                       9.65
                                     Fat                                        9.90
                                    Skin                                        -----
       Legs                         Core                                        9.00
                                   Muscle                                      12.40
                                     Fat                                       64.00
                                    Skin                                        -----
       Feet                         Core                                       14.00
                                   Muscle                                      17.70
                                     Fat                                       14.10
                                    Skin                                        -----

  This library is used to develop programs for clusters and supercomputers. The primary means of
communication between MPI processes is by passing messages to each other.

9. Results
   To study the acceleration of the parallel Gaussian algorithm, 10 computational experiments on
solving a system of linear equations by the Gaussian method for different dimensions of these systems
were performed and the results presented in Table 3 were obtained.
   Table 3
   Comparison of the results of calculating the operation of a series and parallel Gaussian algorithm
    № test      Dimensionality     Real execution time of serial      Real execution time of parallel
                                            algorithm                    algorithm (4 processes)
       1              50                      0.001                            0.00013751
       2              100                      0.01                             0.0039775
       3              250                     0.188                             0.0461096
       4              500                     2.621                              0.26025
       5             1000                    13.853                              2.03732
       6             1500                    74.842                              6.75569
       7             3000                                                        56.0002
       8             5000           Calculations require better                  243.781
       9             8000               technical support          Calculations require better technical
      10            10000                                                         support




                                                                                                           212
   As can be seen from the first diagram (Figure 2) at a dimension of the matrix of 50, the operating
time of the parallel Gaussian algorithm is greater than the operating time of the sequential algorithm.
This is because the parallelization process is too expensive because the process itself takes time. But
with increasing dimension of the matrix (Figure 3 - Figure 4), the parallel algorithm justifies itself and
works three times faster, as seen in the last diagram.
   According to the table, diagrams are constructed for better visualization (Figure 2 - Figure 4).




Figure 2: The operating time of a series and parallel Gaussian algorithm at a matrix dimension of 50.




Figure 3: The operating time of a series and parallel Gaussian algorithm at a matrix dimension of
100.




Figure 4: The operating time of the series and parallel Gaussian algorithm at a matrix dimension of
500.


                                                                                                      213
10.     Conclusions
   When using the finite difference method to solve the boundary value problem with heat
conduction equations, it is advisable to use the parallel Gaussian algorithm, as it speeds up the
calculation process by three or more times. The method is suitable for building a three-dimensional
model for predicting the thermophysiological state of a person.

11.     References
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[3] M. Ferdows, F. Alzahrani, S. Sun, Dual solution of boundary-layer flow driven by variable plate
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[4] T. Hayat, S. Nadeem, Aspects of developed heat and mass flux models on 3D flow of Eyring-
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