Solving 3d problems of potential theory in piecewise homogeneous media by using indirect boundary and near-boundary element methods

: pp. 117-127
Lviv Polytechnic National University; Carpathian Branch of Subbotin Institute of Geophysics of National Academy of Sciences
Lviv Polytechnic National University

Effective numerical-analytical approaches for solving the direct problem of electrical prospecting for the media with inclusions of arbitrary shape and constant electrical characteristics are suggested. They are based on the combination of a fundamental solution of Laplase’s equation and principal ideas of the method of boundary integral equations and that of collocation.

Using the indirect boundary and near-boundary element methods, numerical-analytical approaches for solving the problems of potential theory in spatial piecewise homogeneous objects under conditions of an ideal contact between their components are developed. Discrete-continuous models for finding the intensities of unknown sources introduced into the boundary and near-boundary elements, and approximated by constants are reduced to the systems of linear algebraic equations resulted from the satisfaction, in a collocation sense, of the boundary conditions and those of an ideal interface contact.

The software implementation of the approaches proposed in a half-space with inclusions of various shapes and electrical conductivity for the electrical profiling method in a 3D problem of dc electrical prospecting is done. The numerical analysis performed for some mathematical models illustrates high accuracy and potential abilities of the methods suggested. The developed algorithms make it possible to calculate the potential and intensity of an electric field in inhomogeneous media which are characterized by nonplanar boundaries and arbitrary, by depth and lateral distribution, stationary current sources.

An influence of conductivity and depths of inclusions, their shapes, a distance between two spherical inclusions on the apparent resistivity calculated by the difference of potentials measured on a half-space surface is investigated. It is shown that information on a potential field obtained on the surface of the object can be used to identify local foreign inclusions.

The proposed approaches could be the basis for solving inverse problems of geophysics and technical diagnostics in developing methods for the identification of foreign inclusions, voids and defects, and determining their conductivity, dimentions, and location.

  1. L. Zhuravchak and Y. Grytsko, Near-boundary element technique in applied problems of mathematical physics. Lviv, Ukraine: Publishing house of Carpathian Branch of Subbotin Institute of Geophysics, NAS of Ukraine, 1996. (Ukrainian)
  2. P. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science. London, UK: McGraw-Hill Book Comp., 1984.
  3. C.Brebbia, J. Telles, and L. Wrobel, Boundary Element Methods, Translated from English. Moscow, Russia: Mir, 1987. (Russian)
  4. Y.Zhang, W. Qu, and J.Chen, “A new regularized BEM for 3D potential problems”, Sci Sin Phys Mech Astron, no. 43, p. 297, 2013.
  5. W.Qu, W.Chen, and Z.Fu, “Solutions of 2D and 3D non-homogeneous potential problems by using a boundary element-collocation method”, Eng Anal Bound Elem, no. 60, pp.2–9, 2015.
  6. L. Zhuravchak, B. Grytsko, and O. Kruk, “Numerical and analytical approach to the calculation of thermal fields including thermo sensibility material behavior for complex boundary conditions”, Reports of National Academy of Sciences of Ukraine, no. 12. pp. 51-57, 2014. (Ukrainian)
  7. L.Zhuravchak and  O.Kruk, “Mathematical modeling of distribution of a thermal field in parallelepiped with considering complex heat transfer on its boundary and inner sources”, Kompyuterni nauky ta informatsiyni tekhnologiyi, no. 771, pp. 291-302, Lviv, Ukraine: Publishing house of Lviv Polytechnic National University, 2013. (Ukrainian)
  8. L. Zhuravchak  and N. Shumilina, “Recognition of local volume inhomogeneities for transient thermal field”, Reports of National Academy of Sciences of Ukraine, no. 10, pp. 42-47, 2005. (Ukrainian)
  9. Ya. S.Sapuzhak and L. Zhuravchak, “The technique of numerical solution of 2-D direct current modeling problem in inhomogeneous media”, Acta Geo­physica Polonica, vol.XLVII, no 2. pp. 149–163, 1999.
  10. L. Zhuravchak and O. Kruk, “Consideration of the nonlinear behavior of environmental material and a three-dimensional internal heat sources in mathematical modeling of heat conduction”, Mathematical modeling and computing, vol. 2, no. 1, pp. 107–113, 2015.
  11. L.Zhuravchak, O.Kruk, and B.Grytsko, “Mathematical modeling of the distribution of potential field in piecewise homogeneous objects using boundary elements method”, in Proc. XIIIth International Conference “Modern Problems of Radio Engineering, Telecommunications, and Computer Science”, pp.117–120, Lviv-Slavsko, Ukraine, February 23 – 26, 2016.