Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified
This article deals with a mathematical model of the explosion process based on a liquid model. It takes into account the mutual influence of the deformable anisotropic porous medium parameters and the explosive process characteristics. The corresponding boundary value problem is solved using the num...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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| Zitieren: | Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified / A. Bomba, K. Malash // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 18. — С. 5-17. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1621622020-01-04T01:25:42Z Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified Bomba, A. Malash, K. This article deals with a mathematical model of the explosion process based on a liquid model. It takes into account the mutual influence of the deformable anisotropic porous medium parameters and the explosive process characteristics. The corresponding boundary value problem is solved using the numerical quasiconformal mappings method which ensures the possibility of its solution taking into account the presence of the reverse effect, the existence of which essentially complicates the process of solving the problem by other, less «dynamical» methods. Algorithm used in the modelling of similar processes in hydrodynamics and electrodynamics, in particular for the study of filtration processes and electromotography is adapted for solving appropriate boundary value problems. The method of identifying the external boundary of the domain of the explosive process influence is developed by introducing certain changes to the «classical » algorithm for solving such a type of boundary problems for the twice-bounded domain since the last one requires a priori assignment of the inner and outer domain contours. У статті сформовано математичну модель процесу вибуху, яка базується на рідинній. Вона враховує взаємовплив параметрів деформівного анізотропного пористого середовища та характеристик вибухового процесу. Відповідна крайова задача розв’язується з використанням числового методу квазіконформних відображень, що забезпечує можливість її розв’язання з врахуванням наявності зворотного впливу, існування якого суттєво ускладнює процес розв’язування задачі іншими, менш «динамічними», методами. Адаптовано алгоритм розв’язування крайових задач, що використовуються при моделюванні аналогічних процесів у гідродинаміці та електродинаміці, зокрема, для дослідження фільтраційних процесів та електротомографії. Розроблено методику ідентифікації зовнішньої межі області впливу вибухового процесу шляхом внесення певних змін до «класичного» алгоритму для розв’язування такого типу крайових задач для двозв’язної області, оскільки останній вимагає апріорного задання внутрішнього та зовнішнього контурів області. 2018 Article Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified / A. Bomba, K. Malash // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 18. — С. 5-17. — Бібліогр.: 10 назв. — англ. 2308-5916 DOI: 10.32626/2308-5916.2018-18.5-17 http://dspace.nbuv.gov.ua/handle/123456789/162162 517.9 en Математичне та комп'ютерне моделювання. Серія: Технічні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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| description |
This article deals with a mathematical model of the explosion process based on a liquid model. It takes into account the mutual influence of the deformable anisotropic porous medium parameters and the explosive process characteristics. The corresponding boundary value problem is solved using the numerical quasiconformal mappings method which ensures the possibility of its solution taking into account the presence of the reverse effect, the existence of which essentially complicates the process of solving the problem by other, less «dynamical» methods. Algorithm used in the modelling of similar processes in hydrodynamics and electrodynamics, in particular for the study of filtration processes and electromotography is adapted for solving appropriate boundary value problems. The method of identifying the external boundary of the domain of the explosive process influence is developed by introducing certain changes to the «classical » algorithm for solving such a type of boundary problems for the twice-bounded domain since the last one requires a priori assignment of the inner and outer domain contours. |
| format |
Article |
| author |
Bomba, A. Malash, K. |
| spellingShingle |
Bomba, A. Malash, K. Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified Математичне та комп'ютерне моделювання. Серія: Технічні науки |
| author_facet |
Bomba, A. Malash, K. |
| author_sort |
Bomba, A. |
| title |
Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified |
| title_short |
Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified |
| title_full |
Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified |
| title_fullStr |
Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified |
| title_full_unstemmed |
Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified |
| title_sort |
modeling of explosive processes in anisotropic media where boundary of the influence region is identified |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/162162 |
| citation_txt |
Modeling of Explosive Processes in Anisotropic Media where Boundary of the Influence Region is Identified / A. Bomba, K. Malash // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 18. — С. 5-17. — Бібліогр.: 10 назв. — англ. |
| series |
Математичне та комп'ютерне моделювання. Серія: Технічні науки |
| work_keys_str_mv |
AT bombaa modelingofexplosiveprocessesinanisotropicmediawhereboundaryoftheinfluenceregionisidentified AT malashk modelingofexplosiveprocessesinanisotropicmediawhereboundaryoftheinfluenceregionisidentified |
| first_indexed |
2025-07-14T14:42:55Z |
| last_indexed |
2025-07-14T14:42:55Z |
| _version_ |
1837633819246592000 |
| fulltext |
Серія: Технічні науки. Випуск 18
5
UDC 517.9
DOI: 10.32626/2308-5916.2018-18.5-17
A. Bomba, Doctor of Engineering, Professor,
K. Malash, Ph. D. Student
Rivne State Humanitarian University, Rivne
MODELING OF EXPLOSIVE PROCESSES
IN ANISOTROPIC MEDIA WHERE BOUNDARY
OF THE INFLUENCE REGION IS IDENTIFIED
Nowadays, explosive processes are widely used for the optimiza-
tion of extraction minerals processes, in the buildings construction and
industry. This practice allows to significantly increase the speed of the
work and, at the same time, reduce it cost. However, side effects of the
explosion usage can be catastrophic, since its destructive power is ca-
pable of completely demolishing even fairly stable buildings and caus-
ing irreparable damage to the environment, therefore there is a need for
a precise mathematical modeling of the explosive process with a de-
tailed calculation of all its consequences.
One of the models used to investigate the explosion process is
a fluid based on the simulation of an environment in which an ex-
plosion occurs as a filtration fluid. In this case, the velocity field
generated by the explosion is usually considered to be potential.
This article deals with a mathematical model of the explosion
process based on a liquid model. It takes into account the mutual in-
fluence of the deformable anisotropic porous medium parameters
and the explosive process characteristics. The corresponding bounda-
ry value problem is solved using the numerical quasiconformal map-
pings method which ensures the possibility of its solution taking into
account the presence of the reverse effect, the existence of which es-
sentially complicates the process of solving the problem by other,
less «dynamical» methods. Algorithm used in the modelling of simi-
lar processes in hydrodynamics and electrodynamics, in particular for
the study of filtration processes and electromotography is adapted for
solving appropriate boundary value problems. The method of identi-
fying the external boundary of the domain of the explosive process
influence is developed by introducing certain changes to the «classi-
cal» algorithm for solving such a type of boundary problems for the
twice-bounded domain since the last one requires a priori assignment
of the inner and outer domain contours.
Key words: anisotropic medium, complex analysis, explosion
processes, hydrodynamic mesh, identification, mathematical mod-
elling, numerical methods, quasiconformal mapping.
Introduction. Explosive processes are often used in mining, particu-
larly, for grinding the hard rock preventing easily access to the minerals at
the present stage of development of production. Also, explosions are often
© A. Bomba, K. Malash, 2018
Математичне та комп’ютерне моделювання
6
used in buildings construction, for example, to clear the territory or create
large depths. It is extremely important to determine the correct technologi-
cal parameters of the explosion process, since the inaccuracies made may
couth to catastrophic consequences, particularly, the destruction of nearby
buildings, or the other significant damage, the elimination of which re-
quires enormous costs or is impossible. For this purpose, mathematical
modeling of the process is carried out.
Nowadays there are several models of explosion processes, each of
which is used by various researchers. Particularly, V. Kravets, V. Koro-
bichychuk, V. Boyko use linear-elastic, elastic-plastic and visco-elastic
ones [1]. V. Bulavatskii, V. Skopetskii, and I. Luchko consider the liquid
model based on the representation of the medium in which the explosion
occurs as an ideal liquid, and the field formed by the explosion is consid-
ered to be a potential. In this case the process modelling is based on the
solution of the corresponding boundary value problem using conformal
mappings. However, their functional does not provide sufficient accuracy
of the solution, since the field formed as a result of the explosion is not
ideal (this is due to the presence of squeezed parts of the soil, as well as
areas close to the cavities); in addition, there is a need to take into account
the reciprocal influence of process characteristics and the medium charac-
teristics. The liquid model of explosive processes was improved by
A. Bomba and his scientific school [4–5]. Quasiconformal mappings
methods are used for solving the corresponding boundary value prob-
lem [6–10]. The impact of the explosive process on an isotropic medium is
investigated in [4], the impact on anisotropic one is described in [5] (the
boundaries of crater, pressed and undisturbed sections of the medium are
determined, and, also, the hydrodynamic grid of the formed field is con-
structed). The area of an explosion influence is considered to be given a
priori in these works (the corresponding mathematical domain is modeled
as a twice-bounded domain).
In this article we propose a solution to the problem of determining
the boundaries of the crater, pressed and undisturbed domains of the medi-
um with the simultaneous identification of the area boundary of the explo-
sion impact on the investigated environment, taking into account the inter-
action of the characteristics of the environment and the process with the
automatic construction of the hydrodynamic grid.
Presenting main material. We consider a certain domain zG
z x iy , where the charge of a given form with a constant quasi-
potential on it is placed, in a medium where an explosion was occur-
red (Fig. 1).
Серія: Технічні науки. Випуск 18
7
Fig. 1. Physical domain of the medium
The particles motion process is described (analogously to [5]) by the
motion equation k grad
and the continuity equation 0div
,
where ( , ), ( , )x yx y x y
is the particle velocity, and ,x y is
the quasi-potential of the corresponding field,
11 12
21 22
( , , , ) ( , , , )
( , , , ) ( , , , )
x y x y
x y x y
k x y k x y
k
k x y k x y
is the conductivity coefficient of the medium (which characterizes the abil-
ity of particles to rise). We consider that the explosion influence area is
bounded by two contours — external and internal ones. The inner contour
is the charge boundary. It is known a priori:
* *: ( , ) 0 { :L z f x y x iy * ( )x x t , * ( )y y t , * *}t .
The outer contour of the domain separates the studied area from the gen-
eral one, and we can’t set it a priori. Physically, it delimits the soil areas
with the same characteristics, because the undisturbed zone of soil from
the studied domain has the same characteristics as the outside one, so the
external contour, which is established solely for the purpose of correct
calculation, should be chosen so that it is situated in the undisturbed zone
(emphasize that the contour that separates the pressed and the undisturbed
soil zone is important to know from a practical point of view). It is inap-
propriate to choose the outer contour so that the undisturbed area within
the studied area is too large, since an increase in the size of the studied
area requires an increase in the number of the partition nodes, which leads
Математичне та комп’ютерне моделювання
8
to higher costs of machine time or to the loss of accuracy; on the other
hand, the choice of the external contour so that it would be outside the
unbroken area will result in distortion of the results. We propose to identi-
fy the outer contour of the investigated area by solving a sequence of simi-
lar tasks, each of which differs from the previous one by outer contour.
After solving each of the following tasks (starting with the second one),
we compare the obtained boundaries of the crater, the pressed and undis-
turbed domains of the soil with those obtained for the previous. If they
coincide with sufficient accuracy, then the contour that we found in the
first of the comparable tasks is the wanted one; otherwise, we go to solv-
ing the next problem, in which we consider a little bit "larger" area. the
process should be repeated until the desired boundaries of the crater,
pressed and undisturbed domains of soil in the two adjacent tasks do not
coincide. So, in each of the subtasks the outer contour is set in such way:
* *: ( , ) 0sL z f x y *{ : ( )sx iy x x t , * ( )sy y t , * *}t ,
where
* *L , *
*
sL , *
* . In each task from the
considered sequence, the functions *( ),sx t *( )sy t have the same structure
and differ only in parameters in such a way that they form a sequence of
so-called «concentric» contours.
As a result of the problem solution it is necessary to construct a hy-
drodynamic grid, to determine the boundaries of the crater, the pressed and
undisturbed sections of the soil (in Fig. 1, these borders are marked by
dashed lines * 0,I I I I ).
We model the explosion process in the same way as [5] — taking in-
to account the interaction of the quasipotential ( , )x y gradient
2 2
x yI and the conductivity tensor 11 12
21 22
k k
k
k k
and the for-
mation of the sections (Fig. 1), in which the correction of the latter is
made, depending on the values I according to the following formula:
* 0 0
0
1 ,
2rs rs rsk k I I I I I I , 1, 2r s , (1)
where 0I , *I are the critical gradient values, which characterize the delay
and separation of particles (the position of the line of the section), the pa-
rameter tensor 11 12
21 22
, which characterizes the medium anisot-
ropy change, is selected based on the physical experiment [3].
Серія: Технічні науки. Випуск 18
9
We introduce a flow function ,x y , that is complex conju-
gate to ,x y (as described in [7]), fix a certain point *A B L
on the inner contour of the domain and make a conditional incision of
the investigated domain along one of the flow lines (unknown yet, it
will be refined in the process of problem solving). The top and bottom
of the section are marked via AА and BВ on Fig. 1. We obtain (in the
case of a fixed k) the problem on a quasiconformal mapping
, i ,z x y x y [6] formed by a single-bounded domain
0 /z zG G AA on the corresponding rectangular domain of a complex
quasipotential :G i * , 0 Q with an un-
known parameter Q :
11 12
21 22
, , , , , , ,
, , , , , ,
x y x y
x y x y
x y x y
x y y
x y x y
x y x
, 0, zx y G . (2)
*
*
*
*, , 0,
sL L AA
*
.y xBB
L
Q dx dy
The corresponding inverse boundary value problem on the quasi-
conformal mapping , ,z z x iy of the domain G on
0
zG and the real ,x x and imaginary ,y y parts equation
(we also require it's execution in the section to account for their «split»
in the transition from the domain zG to G ) of the characteristic flow
function with the unknown position of the section and the value Q is
obsessed as:
11 12
21 22
, , , , , , ,
, , , , , , ,
, , ,
y x y xy x xx y x y
J J J J
y x y xy x yx y x y
J J J J
G J x y x y
(3)
*
* * * *
* * *
( , ), ( , ) 0, ,
( , ), ( , ) 0, 0 ,s
f x y
f x y Q
(4)
*
*( ,0) ( , ), ( ,0) ( , ), ,x x Q y y Q (5)
Математичне та комп’ютерне моделювання
10
11 22 21 12 12
11 11 11
1 x x x
21
11 22
10,x y
(6)
11 22 21 12 21 12
22 22 22
0.y y y
We construct the algorithm for numerical solution of the problem
analogously to [5]. The difference analogs of equations (6), boundary con-
ditions (4), as well as additional conditions for boundary and near-
boundary nodes in the corresponding uniform grid domain
( , ) :i jG
*i i , 0,i n ; j j , 0,j m ;
*
*
n
, Q
m
,
is written, respectively, in the form:
1, 1 1, 1 1, 1 , 1 , 1 , 1 1, 1
1, 1, 1, , , , 1,
1, 1 1, 1 1, 1 , 1 , 1 , 1 1, 1
2
1, 1 1, 1 1, 1
1 2
i j i j i j i j i j i j i j
i j i j i j i j i j i j i j
i j i j i j i j i j i j i j
i j i j i j
a x a a x a x
a x a a x a x
a x a a x a x
b x b
1, 1, 1, 1, 1
, 1 , 1 , 1 , , , , 11 2
i j i j i j i j
i j i j i j i j i j i j i j
b x b x
b x b b x b x
1, 1 1, 1 1, 1 1, 1, 1, 1, 1
1, 1, 1 1, 1 1, 1, 1 1, 1
, 1 1, 1 1, 1 , 1 1, 1 1, 1 / 4 0,
i j i j i j i j i j i j i j
i j i j i j i j i j i j
i j i j i j i j i j i j
b x b b x b x
c x x c x x
d x x d x x
(7)
1, 1 1, 1 1, 1 , 1 , 1 , 1 1, 1
1, 1, 1, , , , 1,
1, 1 1, 1 1, 1 , 1 , 1 , 1 1, 1
2
1, 1 1, 1 1, 1
1 2
i j i j i j i j i j i j i j
i j i j i j i j i j i j i j
i j i j i j i j i j i j i j
i j i j i j
a y a a y a y
a y a a y a y
a y a a y a y
b y b
1, 1, 1, 1, 1i j i j i j i jb y b y
Серія: Технічні науки. Випуск 18
11
, 1 , 1 , 1 , , , , 11 2 i j i j i j i j i j i j i jb y b b y b y
1, 1 1, 1 1, 1 1, 1, 1, 1, 1i j i j i j i j i j i j i jb y b b y b y
1, 1, 1 1, 1 1, 1, 1 1, 1
, 1 1, 1 1, 1 , 1 1, 1 1, 1 / 4 0,
1, , 1, .
i j i j i j i j i j i j
i j i j i j i j i j i j
c y y c y y
d y y d y y
i m j n
Where
, 11
,
1 ,i j
i j
a ac
11 22 21 12
, , , ,
, 11
,
,i j i j i j i j
i j
i j
b bd
12
,
, 11
,
,i j
i j
i j
c ac
21
,
, 11
,
,i j
i j
i j
d bd
11 22 21 12
, , , ,
, ,22 22
, ,
1 , ,i j i j i j i j
i j i j
i j i j
a ac b bd
21
,
, 22
,
,i j
i j
i j
c ac
12
,
, 22
,
;i j
i j
i j
d bd
, 1,
, ,
2
i j i j
i j
k k
ac k
, , 1
, .
2
i j i j
i j
k k
bd k
, 1 , 1 , 1 , 1
,
, ,
2 ( ) 2 ( )
, .i j i j i j i jrs
i j rs
i j i j
y y x x
J J
, 1, 1, , 1 , 1 , 1 , 1 1, 1, ,
1, , 1, .
i j i j i j i j i j i j i j i j i jJ x x y y x x y y
i m j n
* 0, 0, 1, 0, * 0, 0, 1, 0,
2 2
* 0, 0, * 0, 0,
2 2 2
1, 0, 1, 0, *0,
* *
, 1 , 1 , , 1 , 1 , 1 , , 1
* 2 *
, 1 , 1
, , ( )
, ,
1 cos ,
, , ( )
,
x j j j j y j j j j
x j j y j j
j j j j j
s x i n i n i n i n s y i n i n i n i n
s x i n i n s
f x y y y f x y x x
f x y f x y
x x y y
f x y y y f x y x x
f x y f
2
, 1 , 1
2 2 2 *
, , 1 , , 1 , 1
,
1 cos ,
y i n i n
i n i n i n i n i n
x y
x x y y
(8)
Математичне та комп’ютерне моделювання
12
where
2 2 2
11 12 21 22cos /x x l y y xf f f f f
11 2 2 22 2
11 12 21 22) .y x y x yf f f f f
The approximation magnitude formula is obtained based on the
«quasi-conformal similarity» condition of elementary rectangles [6] of
two domains:
,
, , 1
, 1,, 0
1 ,
1 1
m n
i j i j
i j i ji jm n a a
(9)
where
2 2
, 1, , 1, ,i j i j i j i j i jx x y y ,
2
, 11 , 1 , 12 , 1 ,
1
2 2
21 , 1 , 22 , 1 , .
i j i j i j i j i j
i j i j i j i j
a y y x x
y y x x
The numerical implementation of the algorithm is carried out as fol-
lows. Firstly, we set the domain G partition parameters: n and m, critical
potential values *I , 0I , parameters 1 , 2 , 3 , which characterize the
accuracy of the difference problem solution, parameter 4 , which charac-
terizes the accuracy of the investigated domain boundary identification.
We set the outer contour * *
1 : ( , ) 0L z f x y *
1{ : ( )x iy x x t ,
*
1 ( )y y t , * * }t so that it is at a short distance from the inner one
for the first problem. We set the initial approximation of the boundary
nodes coordinates (0) (0)
0, 0,,j jx y , (0) (0)
, ,,n j n jx y (with monitoring the fulfill-
ment of conditions (4)) and inner nodes (0) (0)
, ,,i j i jx y (for example, evenly
dividing the segments with the ends at the points (0) (0)
0, 0,,j jx y ,
(0) (0)
, ,,n j n jx y ). Then we find the initial approximation (0) (0) (0)
, ,,i j i jx y
of the quasiconformal invariant by formula (9). Next, we perform the
Серія: Технічні науки. Випуск 18
13
refinement of the internal nodes coordinates ( ) ( )
, ,,i j i jx y with the given
accuracy 1 ( — the total iteration number) using iterative circuits such
as the «cross» obtained by solving (7) with respect to ,i jx and ,i jy . In this
case, the necessary values of the pressure gradient and the permeability
tensor k in the grid nodes G
are calculated from the values ,i jx , ,i jy
from the previous iteration step. We correct the boundary nodes, solving
approximately the system of equations (8), for example, by Newton's
method. If the value of the nodes displacement on the boundary for the
performed -th total iteration
2 2( ) ( 1) ( ) ( 1)
, , , ,,
max i j i j i j i ji j
S x x y y
( ( , )i j are indexes of the boundary nodes coordinates) is greater than 2,
then we return to refine the internal nodes. Otherwise, we find new ap-
proximations ( )LQ and ( )L quantities Q and y the formula (9) and the
condition for the connection between them: Q m . If
( ) ( 1)
3
L LQ Q , then we return to the refinement of the internal nodes,
otherwise we calculate the resulting grid «quasi-conformality» non-
connection 2 2
1 2 , where 1, 2 are the equations approximations
incompatibilities (3):
1, 1
1 1, 1, , , 1 , 1, 1
1, 1
2 1, 1, , , 1 , 1, 1
max | ( ) ( ) |,
max | ( ) ( ) | .
n m
i j i j i j i j i ji j
n m
i j i j i j i j i ji j
x x k y y
y y k x x
Then we determine the position of the section lines of the hole,
pressed and undamaged sections of the soil and set the outer contour
* *
2 : ( , ) 0L z f x y *
2{ : ( )x iy x x t , *
2 ( )y y t , * *}t so
that it is somewhat «bigger» from the previous, and solve the problem for
this contour. We compare the position of the section lines of the pressed
and undisturbed soil plots for the first and second tasks. If the difference
between them does not exceed 4 , then the contour *
1L is sought, other-
wise we go to the next task with the contour *
3L , the results of which are
compared with the results of the second one.
Математичне та комп’ютерне моделювання
14
а)
b)
Fig. 2. Distribution of zones formed by explosion
The process should be repeated until the difference between the sec-
tion lines of the crater, the pressed and undisturbed zones for some s and
1s tasks will not satisfy the given accuracy. The contour *
sL is consid-
ered to be sought.
A computer program was developed and numerical calculations were
performed based on the algorithm. For input data 70 100n m ,
* 0.008I , 0 0.004I . * *: ( , ) 0 { :L z f x y x iy 10 6cos( )x t ,
5 5sin( )y t , 0 2t is charge contour, * 0 , * 1 ,
Серія: Технічні науки. Випуск 18
15
0.1 0.15
0.05 0.06
, the original outer contour * *
0 : ( , ) 0L z f x y
: 48cos( )x iy x t , 39sin( )y t , 0 2t , 0
3 4
1 4
k
. The cor-
responding situational condition (in particular, the hydrodynamic field grid)
resulting from the explosion in such medium is depicted in Fig. 2 a).
We see that 3 depreciated zones were created: A is a crater, B is a
pressed zone, C is an undisturbed zone. The outer contour of the studied
area is also identified: * *: ( , ) 0sL z f x y : 160cos( )x iy x t ,
130sin( )y t , 0 2t . For comparison Fig. 2b) shows the result of
calculations for a predetermined external contour * *: ( , ) 0L z f x y
: 176cos( )x iy x t , 143sin( )y t , 0 2t . We see that the
boundaries of the crater, pressed and undisturbed domains of the soil coincide.
Fig. 3. Distribution of zones for an isotropic medium
Note that the algorithm works for an isotropic medium. Fig. 3 shows
the results for anisotropy.
The input data is the same as for Fig. 2 only 0
3 0
0 3
k
(since the
medium is isotropic). The outer contour is identified *L
*: ( , ) 0z f x y : 144cos( )x iy x t , 117sin( )y t , 0 2t .
We see that the anisotropy of the medium significantly influences the for-
mation of a crater, the pressed and undisturbd domains of the soil, but the
developed algorithm works for both the case of isotropy and for anisotropy.
Математичне та комп’ютерне моделювання
16
Conclusions. The mathematical model of the explosion process that
takes into account the interaction of the process characteristics (quasi-
potential) and a deformable porous anisotropic medium based on the use of
the numerical method of quasi-conformal mappings (as well as the algorithm
for solving the corresponding nonlinear boundary-value problems) and aims to
determine the position of the boundaries of the extruded, and unpolluted soil
zones, is generally summarized in the case of identification and the boundary
of the zone of influence of the explosion process on the medium. The solution
of the corresponding boundary value problem occurs using the special proce-
dure of inverse mapping and the stepwise parametrization of the medium
characteristics and process, as well as the ideas of the block iteration method.
This allows to automatically build a hydrodynamic grid and a speed field.
The results of the numerical experiments developed on the basis of
the algorithm developed showed the feasibility of using it for modelling of
explosive processes in anisotropic deformable porous medium in order to
determine the position of the section lines of the crater, the pressed and
undisturbed zones of the soil with the simultaneous identification of the
boundary of the explosion impact zone. It is shown that the developed
algorithm works for both anisotropic medium and for isotropic.
In the perspective is identification of the explosion process parame-
ters, in particular, finding the location and the shape of the charge, as well
as solving the corresponding spatial problems.
References:
1. Kravets V. G. Physical processes of applied geodynamics of an explosion:
monograph / V. G. Kravets, V. V. Korobyichuk, V. V. Boiko. — Zhytomyr :
ZSTU, 2015. — 408 p.
2. Bulavatskii V. M.. Some inverse problems of the pulsed-hydrodynamic theory
of explosion on the discharge / V. M. Bulavatskii, I. A. Luchko // Investiga-
tions on boundary value problems of hydrodynamics and thermophysics. —
Kiev, 1979. — P. 53–64.
3. Bomba A. Ya. Nonlinear mathematical models of geohydrodynamics pro-
cesses / A. Ya. Bomba, V. M. Bulavatskii, V. V. Skopetskii. — Kiiv : Naukova
dumka, 2007. — 308 p.
4. Bomba A. Ya. Using quasi-conformal mappings to mathematical modeling of
explosion processes / A. Ya. Bomba, A. M. Sinchuk // Volynskii matematych-
nii visnyk. Serie «Applied mathematics». — Ed. 8. — P. 32–41.
5. Bomba A. Ya. Modeling of the explosion process in an anisotropic medium
with quasiconformal mapping methods/ A. Ya. Bomba, K. M. Malash // Trans-
actions оf Kremenchuk Mykhailo Ostrohradskyi National University. — Kre-
menchuk. — 2017. Ed. 4 (105). — P. 28–33.
6. Bomba A. Ya. Methods of complex analysis: monograph / A. Ya. Bomba, S. S. Kash-
tan, D. O. Prigornytskii, S. V. Yaroshchack. — Rivne : NUWM, 2013. — 415 p.
7. Blair David E. Inversion theory and conformal mapping / David E. Blair. —
American Mathematical Sciety, 2000. — 152 p.
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8. Nearling James. Mathematical tools for physics / James Nearling. — Miami,
2008. — 594 p.
9. Prigornitskii D. O. Modification of the algorithm for numerical solving a class of
nonlinear modeling boundary value problems on quasi-conformal mappings in two-
coupling deformable media / D. O. Prigornitskii // Volynskii matematychnii
visnyk — Serie «Applied mathematics ». — Ed. 9. — 2002. — P. 60–66.
10. Bomba A. Ya. Numerical solution of nonlinear modeling boundary value prob-
lems on quasi-conformal mapping under conditions of interaction of gradients
of potential and environmental characteristics / A. Ya. Bomba, V. V. Sko-
petskii, D. O. Prigornitskii // Visnyk Kiivskoho Universitetu. Serie: «Physics
and mathematics». — 2003. — Ed. 1. — P. 126–135.
МОДЕЛЮВАННЯ ВИБУХОВИХ ПРОЦЕСІВ
В АНІЗОТРОПНОМУ СЕРЕДОВИЩІ
З ІДЕНТИФІКАЦІЄЮ МЕЖІ ЗОНИ ВПЛИВУ
У наші дні для оптимізації процесів видобування корисних копалин, у
будівництві та промисловості досить поширеною є практика застосуван-
ня вибухових процесів. Вона дозволяє значно підвищити швидкість ви-
конання робіт і, водночас, знизити їх вартість. Проте, побічні дії застосу-
вання вибухівки можуть бути катастрофічними, оскільки її руйнівна сила
здатна повністю зносити навіть досить стійкі споруди та завдавати непо-
правної шкоди навколишньому середовищу, тому є необхідність попере-
днього точного математичного моделювання вибухового процесу з дета-
льним прорахуванням усіх його наслідків.
Однією з моделей, котрі застосовуються для дослідження вибухо-
вого процесу, є рідинна, яка базується на моделюванні середовища, у
якому відбувається вибух, як нестискуваної фільтраційної рідини.
При цьому поле швидкостей, породжене вибухом, як правило, вважа-
ється потенціальним.
У статті сформовано математичну модель процесу вибуху, яка базу-
ється на рідинній. Вона враховує взаємовплив параметрів деформівного
анізотропного пористого середовища та характеристик вибухового про-
цесу. Відповідна крайова задача розв’язується з використанням числово-
го методу квазіконформних відображень, що забезпечує можливість її
розв’язання з врахуванням наявності зворотного впливу, існування якого
суттєво ускладнює процес розв’язування задачі іншими, менш «динаміч-
ними», методами. Адаптовано алгоритм розв’язування крайових задач,
що використовуються при моделюванні аналогічних процесів у гідроди-
наміці та електродинаміці, зокрема, для дослідження фільтраційних про-
цесів та електротомографії. Розроблено методику ідентифікації зовніш-
ньої межі області впливу вибухового процесу шляхом внесення певних
змін до «класичного» алгоритму для розв’язування такого типу крайових
задач для двозв’язної області, оскільки останній вимагає апріорного за-
дання внутрішнього та зовнішнього контурів області.
Ключові слова: анізотропне середовище, вибухові процеси,
гідродинамічна сітка, ідентифікація, квазіконформне відображення,
комплексний аналіз, математичне моделювання, числові методи.
Отримано: 20.11.2018
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