Using of partly-boundary elements as a version of the indirect near-boundary element method for potential field modeling

2021;
: pp. 1–10
https://doi.org/10.23939/mmc2021.01.001
Received: May 13, 2020
Accepted: September 25, 2020
1
Lviv Polytechnic National University
2
Carpathian Branch of Subbotin Institute of Geophysics, National Academy of Sciences

In this paper, the partly-boundary elements as a version of the indirect near-boundary element method has been considered.  Accuracy and effectiveness of their using for 2D problems of potential theory have been investigated.  It is shown that using of partly-boundary elements for objects of canonical shape (circle, square, rectangle, ellipse) and arbitrary polygons allows us to achieve the solution accuracy, which is comparable with the accuracy of the indirect near-boundary element method, and its order of magnitude is higher than in the indirect boundary element method.  In this case, the computation time is reduced by 2–2.5 times than in the near-boundary element method case.  The software of the proposed approach has been implemented in Python.  Practical testing was carried out for the tasks of electrical profiling and vertical electrical sounding in the half-plane with inclusion as a polygon.  The recommendations for application of the partly-boundary elements in geophysical practice have been given.

  1. Cartz L.  Nondestructive Testing: Radiography, Ultrasonics, Liquid Penetrant, Magnetic Particle, Eddy Current.  ASM International (1995).
  2. Holst B., Piskur J., Kostrobiy P. P., Markovych B. M., Suchorski Yu.  Field ionization of helium in a supersonic beam: Kinetic energy of neutral atoms and probability of their field ionization.  Ultramicroscopy. 109 (5), 413–417 (2009).
  3. Kostrobiy P. P., Markovych B. M., Suchorski Y.  Revisiting local electric fields on close–packed metal surfaces: theory versus experiments.  Solid State Phenomena. 128, 219–224 (2007).
  4. Mikhlin S. H.  Multidimensional singular integrals and integral equations.  Moscow, Fizmatgiz (1962), (in Russian).
  5. Muskhelishvili N. Y.  Singular integral equations.  Moscow, Nauka (1968), (in Russian).
  6. Kupradze V. D.  Potential methods in the elasticity theory.  Moscow, Fizmatgiz (1963), (in Russian).
  7. Banerjee P. K., Butterfield R.  Boundary element methods in engineering science.  London, McGraw-Hill (1981).
  8. Brebbia C. A., Telles J. C. F., Wrobel L. C.  Boundary Element Techniques. Theory and Applications in Engineering.  Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1984).
  9. Zhang Y., Qu W., Chen J.  A new regularized BEM for 3D potential problems.  SCIENTIA SINICA Physica, Mechanica \& Astronomica. 43 (3), 297–308 (2013).
  10. Qu W., Chen W., Fu Z.  Solutions of 2D and 3D non-homogeneous potential problems by using a boundary element-collocation method.  Engineering Analysis with Boundary Elements. 6, 2–9 (2015).
  11. Katsikadelis J. T.  The boundary element method for engineers and scientists: theory and applications.  Academic Press, Oxford (2016).
  12. Zhuravchak L. M., Hryts'ko E. H.  Near-boundary element method in applied problems of mathematical physics.  Lviv, The Carpathian Branch of S. I. Subbotin Institute of Geophysics of the NAS of Ukraine (1996), (in Ukrainian).
  13. Zhuravchak L. M., Zabrodska N. V.  Solving of elastic dynamical problem in a porous fluid-saturated piecewise-homogeneous half-space by the indirect method of near-boundary elements.  Radio Electronics, Computer Science, Control. 4 (43), 40–48 (2017), (in Ukrainian).
  14. Zhuravchak L. M.  Mathematical modelling of non-stationary processes in the piecewise-homogeneous domains by near-boundary element method.  In: Shakhovska N., Medykovskyy M. O. (eds)  Advances in Intelligent Systems and Computing IV. CSIT 2019. Vol. 1080, pp. 64–77, (2020).
  15. Zhuravchak L. M.  Comparison of solutions of the problems of the elasticity theory for different near-boundary elements.  Materials Science. 38 (6), 859–867  (2002).
  16. Hryts'ko E. H.  Modeling of synthesis of mathematical methods and the theory of partly-boundary elements.  Modern problems of mechanics and mathematics. Mater. International science conf. Lviv, IPPMM. 291–292 (1998),  (in Ukrainian).
  17. Zhuravchak L. M.  Theoretical aspects of indirect methods of near-boundary and partly-boundary elements.  Boundary value problems for differential equations: Collection of scientific works.  Chernivtsi, Chernivtsi National University. Iss. 8, pp. 256–265 (2002), (in Ukrainian).
  18. Electrical exploration. Handbook of geophysics. In 2 books.  Moscow, Nedra (1989), (in Russian).
Mathematical Modeling and Computing, Vol. 8, No. 1, pp. 1–10 (2021)