=Paper=
{{Paper
|id=Vol-1638/Paper79
|storemode=property
|title=Solution of the inverse problem for cylindrical inclusion fragment form definition
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper79.pdf
|volume=Vol-1638
|authors=Gulnara R. Shamsutdinova,Sergey V. Viktorov
}}
==Solution of the inverse problem for cylindrical inclusion fragment form definition ==
https://ceur-ws.org/Vol-1638/Paper79.pdf
Mathematical Modeling
SOLUTION OF THE INVERSE PROBLEM FOR
CYLINDRICAL INCLUSION FRAGMENT FORM
DEFINITION
G.R. Shamsutdinova, S.V. Viktorov
Sterlitamak Branch of Bashkir State University, Sterlitamak, Russia
Abstract. We focus on the direct and inverse problems in this work. The solu-
tion of the inverse problems for deformation of extended cylindrical bodies
comes to finding out A. N. Tikhonov’s functional regularization extremal. It is
based on the solution of the direct problem for point source field in a homoge-
neous half-space with a cylindrical inclusion by the method of integral repre-
sentations.
Keywords: direct and inverse problems of electric exploration, method of inte-
gral representations, configuration method, Green's function, regional task.
Citation: Shamsutdinova GR, Viktorov SV. Solution of the inverse problem for
cylindrical inclusion fragment form definition. CEUR Workshop Proceedings,
2016; 1638: 658-663. DOI: 10.18287/1613-0073-2016-1638-658-663
Introduction
Nowadays the problems of searching and exploration of mineral fields are urgent
tasks of geophysics. From the practical point of view, the study of these problems is
extremely important, because electrical methods of exploration are environmentally
friendly for subsoil and enable people to investigate mineral deposits, estimate their
size and shape most effectively. The theoretical problems, arising from the study of
physical and mathematical basis of electric exploration, are searching for the form of
the inclusion, and belong to the class of inverse problems of geophysics.
In this paper we solve the inverse problem of searching a deformation site of an ex-
tended cylindrical inclusion in a homogeneous medium. The solution of the problem
is found in the parameters description of surface boundaries of the cylinder.
Formulation of the problem
Suppose, that in a homogeneous isotropic half-space 0 with a specific electric con-
ductivity 0 in the plane xOy, at a distance z z 0 , parallel to the axis Ox the inclu-
Information Technology and Nanotechnology (ITNT-2016) 658
Mathematical Modeling Shamsutdinova GR, Viktorov SV. Solution of the …
sion 1 of the extended cylindrical shape with a specific electrical conductivity 1
is located. The cylinder has a strain on the segment x[a, b] (shift along the axis
Oy) (Fig. 1). The problem can be solved if to find a and b parameters, which define
the segment of curvature of the extended body. Such deformation may occur as a
result of the shift of soil layers at landslides, earthquakes and other destructive natural
phenomena.
Fig. 1. The cylindrical inclusion in a homogeneous isotropic half-space
The solution of the inverse problem in searching the [a, b] parameters of the inclu-
sion 1 of the extended cylindrical shape with a specific electrical conductivity 1
can be found in search for the extremals of the A. N. Tikhonov’s functional as the
following p1,3]:
2 2
F ( S ) F1 ( S ) F2 ( S ) u ( A, P, S ) u e ( A, P) S Se , (1)
L2 W21
where S S (a, b) is the function of parametric description of a deformable surface
e
of the cylinder, u ( A, P ) - changes in experimental geophysical data, obtained at the
area of "day" surface, representing the value of the potential field of a point source of
intensity of direct current, excited at the source and destination point, u( A, P) - the
model solution of the direct problem of the point source field is considered as the
following boundary value problem of elliptic type [5,7]:
I
u 0 ( A, P, S ) ( P A) , P 0 ; (2)
0
u1 ( P, S ) 0 , P 0 ; (3)
u 0 ( P, S )
0; (4)
z z 0
u 0 ( P , S ) S u1 ( P , S ) S ; (5)
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Mathematical Modeling Shamsutdinova GR, Viktorov SV. Solution of the …
u0 ( P, S ) u ( P, S ) ;
0 1 1 (6)
n S n S
u 0 ( P, S ) 0 , P , (7)
where - Laplace’s operator, - Dirac’s function, condition (4) defines isolation of
the Earth's surface respectively, (5) and (6) - conditions of continuity of the potential
and current density respectively, (7) - regularity condition of the solution at infinity.
For the decision of tasks (2) – (7) we use the method of integral representations, based
on the integral transformation with the construction of the Green's function for ac-
commodating space [9, 10].
Let’s choose, according to the method, a homogeneous half-space as a co-holding
cylinder medium, and construct a mathematical model for it - subtask for point-source
function (Green's function) G ( P, Q) :
GQ (Q, P) ( P Q) ; (8)
G 0
0; (9)
z z 0
G( P) 0 при P . (10)
In the case of a homogeneous half-space Green's function has the following form:
1
G(P,A)
4 π (xP x A ) (yP y A )2 (zP z A )2
2
1
4 π (xP x A )2 (yP y A )2 (zP z * )2
A
According to the method we write the integral representation of the solution of the
direct problem and the integral equation for the unknown boundary values of the po-
tential at the boundary of the cylindrical inclusion. In order to it, let’s use this formu-
la:
u(Q) G( P, Q) G(P, Q) u(Q)d
Q
u (Q)
n G(P, Q)
G ( P, Q)
n
u (Q) d Q .
Using it to areas 0 and 1 , we get the formula (11) of the integral representation of
the solution and the Fredholm integral equation of the 2nd kind (12) for the unknown
boundary values of the potential u (Q ) [8].
( 0 1 ) G ( P, Q) I G ( P, A)
u ( P)
0 n u(Q) d S , (11)
S 0
Information Technology and Nanotechnology (ITNT-2016) 660
Mathematical Modeling Shamsutdinova GR, Viktorov SV. Solution of the …
2 ( 0 1 ) G( P, Q) 2 I G( P, A)
u ( P) u (Q) d S . (12)
( 0 1 ) n ( 0 1 )
S
Thus, the solution is determined by the formula (11), which is a definite integral is
calculated in case of the formula (12) values u (Q ) on the boundary S. For the numer-
ical implementation of solutions (11) and (12) are transformed to a discrete form.
u ( P) K ( P, Q) u ( P, Q) d S F ( P) ,
(13)
S
0 1 G ( P, Q)
2 , F ( P) 2 I G( P, A) , K ( P, Q) .
0 1 0 1 n
Take the formula (13) in discrete form where the points P and Q are located in a dis-
crete area S .
S S , P Pi , Q Q j , i, j 0, N ,
N
j 0
u Pi K Pi , Q j d j u Q j F Pi , i 0, N
j
N
u i K i , j u j d j Fi ,
j 0 j
where K i , j K i , j d , i 0, N , N – the overall number of points.
The method, offered in our work, is a universal method of reducing the geometric
complexity of the test environment. Besides, this method can be used to phase the
complication of model geometry [2].
To solve the direct problem we used the procedure for the construction of the cylin-
drical inclusion [6] corresponding the formulas of parametric descriptions of the cyl-
inder:
x [ x 0 , x n ], [0, 2 ), [a, b] [ x 0 , x n ] :
X ( , x) x 0 x,
R sin( ), x [a, b],
Y ( , x) y 0 ,
R sin( ) L2 ( x), x [a, b]
Z ( , x) z R cos( ).
0
where i 0, N , L 2 ( x ) - Lagrange Interpolation polynomial function for calculating
the deformation of the extended cylindrical inclusion in the segment, which is calcu-
lated according to the formula:
c 0.5 (a b), d 0.1 (b a) :
( x c ) ( x b) ( x a ) ( x b) ( x a ) ( x c)
L2 ( x) y0 ( y0 d ) y0
( a c ) ( a b) (c a ) (c b) (b a) (b c)
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Mathematical Modeling Shamsutdinova GR, Viktorov SV. Solution of the …
Fig. 2. The guide cylinder with deformation
To solve the inverse problem we use the algorithm of searching for minimum func-
tional configurations of a variable type, directed to the search for global minimum
strongly ravine functions [4].
When searching for the local inclusion a cut [ a, b] is considered as varying parame-
ters. In this case the algorithm of minimization method is defined as the following.
There is an initial approximation x ( k ) (a ( k ) , b ( k ) ) , k 0 . Let’s find a and b by the
following way:
a , если F (a , b ) F (a~, b ), где a min a max , a h
~ ~
a ,
a , если F (a , b ) F (a , b ), где a max a min , a h
~ ~ ~
b , если F (a~, b ) F ( a~, b ), где b min bmax , b h
~
b .
b , если F ( a , b ) F ( a , b ), где b max bmin , b h
~ ~ ~
~
If F (a, b) F (a~, b ) , the approach taken for the value obtained by the rule
a (k 1) a ~ a) , b(k 1) b~ l (b~ b) , where l – coefficient of step conver-
~ l (a
gence magnification, otherwise step decreases: h h h 2 . This process continues until
the condition does not satisfy the algorithm closure: if the convergence step h ,
( k 1) ( k 1)
then the search stops and a , b is considered as a solution.
Conclusion
Software tool was used for searching the diverse problem solution and carrying out a
computational experiment.
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