OPTIMIZATION OF CONTROLLED EXPLOSION PROCESSES PARAMETERS USING COMPLEX ANALYSIS METHODS

Abstract. The optimal charge power and position necessary for forming the maximum possible size of the crater along with preservation of the integrity of the two nearby objects with the numerical quasiconformal mapping methods with the alternate parameterization of the of the medium and process character are established. Unambiguously the boundaries of crater, pressed and disturbed soil zones are identified and the corresponding field dynamic grid is built. A number of experiments was held on the basis of the developed algorithm and their results were analyzed.


Introduction
The upper layers of the earth's crust are increasingly exploited in industry, building construction, and infrastructure at the present stage of society's development. There are many semi-underground and completely underground structures such as shopping malls, parking lots, low floors of high-rise buildings. Also, the number of underground transport routes is growing rapidly. Practically all metropolises successfully operate the subway, a number of long underground tunnels even under water are created. There are the Euro-tunnel under the La-Mansh Channel, the under-Bosphorus tunnel in Istanbul exist, the construction of an underground tunnel between Germany and Denmark is planned, and even one of the Hyperloop routes will use an underground tunnel. The minerals extraction "deepens" under the ground more and more, since easily accessible deposits are already largely exhausted.
The formation of large cavities in the upper layers of the crust plays a very important role in carrying out all of the abovementioned works. One of the ways of their formation is the using of explosive processes. It allows you to get the desired result for a rather short period of time and significantly less cost (in comparison, for example, with high-costing drilling). Also, the problem of eliminating excavated soil is automatically eliminated and the problem of strengthening the walls of the formed cavities is solved, as a result of the explosion around the fissure, a pressed zone is formed.
However, the using of an explosion processes requires a clear pre-determination of the necessary parameters, because minor errors can lead to catastrophic consequences. This is especially needed when near the locations where explosive processes are used, there are certain objects situated, such as residential buildings.
Various mathematical models are used by different scientists to investigate explosive processes, the most common of which are solid, solid-liquid and liquid ones [2,8,11,12,15]. The model used in this article is based on the liquid one and is implemented with using the quasi-conformal mappings methods and a step-by-step parameterization of the medium and process characteristics [1,13].

Problem statement
We solve the following problem. In the environment where the explosion should occur, there are some two points M and N, between which it is necessary to create a the crater of the maximum size so that these points are in the unperturbed zone. The form and size of charge are also known a priori. It is necessary to determine the explosive power of charge and location of its position, as well as the boundaries of the sections of the crater, pressed and unperturbed parts of the environment, resulting from the explosion (Fig. 1).

Fig. 1. Schematical physical domain
The idea of solving the problem is as follows. Some initial domain z G   z x iy  is determined. The charge is placed so that it is evenly spaced from the points M and N at a minimum distance from them. The charge contour is known a priori: The process of particles motion of the medium will be described using the equation of motion k grad is the permeability coefficient of the medium (which characterizes the ability of particles to rise) [4,6]. Also, we take into account the inverse effect of the process characteristics on the characteristics of the environment, so in the process of solving the problem, the coefficient k is specified: I , * I are the critical values of the gradient, which characterize the delay and particle separation (the position of the line of the section), the parameter characterizing the change in permeability of the medium, is selected on the basis of the physical experiment [4]. The outer contour of the initial domain is defined as described in [5].
Let's solve the problem using the quasiconformal mappings methods [3,7]. We introduce the function    Fig. 1). We obtain the problem of a quasiconformal mapping

The inverse problem statement
We turn to the inverse problem on the quasi-conformal mapping , , We arrive at the Laplace type equations:

The numerical algorithm of problem solving
The algorithm for numerical solving the problem is constructed as described, for example, in [3]. The difference analogs of equations (6), boundary conditions (4), as well as additional conditions for boundary and near-boundary nodes in the corresponding uniform grid:    will be written in the following form: Here , 1, , , We obtain the formula for the approximation of the quantity  on the basis of the quasi-conformal similarity in the small of the two domains [3]: The "gluing" conditions on a conditional cut have the such form: The numerical realization of developed algorithm is carried out, as described in [4,5].

Determination of explosive charge power
We obtain the outer boundary contour * of the points will belong to the contour. Then we move to paragraph 2). 4) If one of the points (e. g., N) is placed outside the contour, and the other one is placed inside, we reduce the value * 0  by value 0   і and solve the problem with the specified quasipotential value again (similar to paragraph 3)) until one of the points belongs to the contour , and the other is placed inside. Then we move to paragraph 2).

The numerical experiments results
On the basis of the developed algorithm a number of numerical experiments have been carried out, which confirm the expediency of its use for solving such type of problems while the modelling of the impact of the explosive process on the medium. Here

Conclusions
A mathematical model of the explosion impact on the environment was developed using quasiconformal mappings numerical methods and a step-wise parametrization of the environment and process parameters taking into account their interaction. This model provides optimization of the explosive process parameters, namely allows to identify the position and explosive force of the charge of a given shape and size in order to create the maximum crater between the two given points. At the same time, the points themselves are placed in the unperturbed zone. At the same time, the boundaries of the crater section, the pressed and unperturben sections of the environment in which the explosion takes place are determined, and a dynamic grid of the field formed by it is constructed.
The developed algorithm allows to calculate the necessary parameters for carrying out of blasting works nearby important objects for reception of the maximum possible cavity without danger of their damage.
In the long run is optimization of the explosive process parameters to get the crater of the maximum possible size, provided that the integrity of three or more objects adjacent to each other is preserved; determining the optimal shape or size of the required charge; taking into account the possible anisotropy of the medium; the possibility and expediency of using two or more charges; corresponding spatial problems.