We focus on the direct and inverse problems in this work. The solution 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 homogeneous half-space with a cylindrical inclusion by the method of integral representations.
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 extended cylindrical inclusion in a homogeneous medium. The solution of the problem is found in the parameters description of surface boundaries of the cylinder.
Suppose, that in a homogeneous isotropic half-space 0 with a specific electric conductivity 0 in the plane xOy, at a distance z z0 , parallel to the axis Ox the inclusion 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. The solution of the inverse problem in searching the [a, b] parameters of the inclusion 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]: F (S) F1(S) F2 (S) u( A, P, S) ue ( A, P) L2 where S S(a,b) is the function of parametric description of a deformable surface 2 S S e 2 ,
W21 u0 ( A, P, S)
(P A) , P 0 ; of the cylinder, ue ( 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 u1 (P, S ) 0 , P 0 ; u0 (P, S) z
z0
u0 (P, S ) S u1 (P, S ) S ;
(
S u0 (P, S ) 0 , P , 1 u1(P, S ) ;
n S
where - Laplace’s operator, - Dirac’s function, condition (
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) : G(P,A) In the case of a homogeneous half-space Green's function has the following form: 1 4 π (xP xA )2 (yP yA )2 (zP z A )2
1 4 π (xP xA )2 (yP yA )2 (zP z A* )2 u(P) ( 0 1 ) G(P, Q) u(Q) dS 0 n
S
I G(P, A)
(
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].
S u(P) 2 ( 0 1 ) G(P, Q)
u(Q) dS ( 0 1 ) n 2 I G(P, A) ( 0 1 ) .
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 numerical implementation of solutions (11) and (12) are transformed to a discrete form. u(P) K (P, Q) u(P, Q) dS F (P) , 2 0 1 , F(P) 2 I G(P, A) , K (P, Q) G(P, Q) .
0 1 0 1 n (12) (13) Take the formula (13) in discrete form where the points P and Q are located in a discrete area S .
S S , P Pi , Q Qj , i, j 0, N,
j0 j uPi N K Pi , Q j d j uQ j F P , i 0, N
j0 j i
N ui K i, j u j d j Fi , where Ki, j Ki, 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 cylindrical inclusion [6] corresponding the formulas of parametric descriptions of the cylinder: x [x0 ,xn ], [0, 2 ), [a, b] [x0 ,xn ] : X ( , x) x0 x, R sin( ), x [a, b], Y ( , x) y0 R sin( ) L2 (x), x [a, b] Z ( , x) z0 R cos( ). , where i 0, N , L2 (x) - Lagrange Interpolation polynomial function for calculating the deformation of the extended cylindrical inclusion in the segment, which is calculated 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) To solve the inverse problem we use the algorithm of searching for minimum functional 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 parameters. 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 amax , a h a a , если F (a , b~) F (a~, b~), где a max amin , 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 ~ 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 , then the search stops and a (k 1) , b (k 1) is considered as a solution.
Software tool was used for searching the diverse problem solution and carrying out a computational experiment.