Numerical Analysis and Applications

, Volume 11, Issue 4, pp 298–310 | Cite as

On the Calculation of Border and Contact Nodes by a Grid-Characteristic Method on Non-Periodic Tetrahedral Grids

  • A. O. KazakovEmail author


A grid-characteristic method for the numerical simulation of wave processes in continuum mechanics was initially proposed, and has been successfully applied to periodic hexagonal computational grids. Later it was proposed to adapt this method to non-periodic triangle and tetrahedral grids, and wide computational experience has been gained. However, this approach encounters some difficulties in the calculation of border and contact points when applied to various grid configurations in areas with complex geometries. In this paper, limitations of the method which cause such problems are considered, and some improvements to overcome them are proposed.


grid-characteristic method non-periodic computational grids tetrahedral and triangle grids 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Magomedov, K.M. and Kholodov, A.S., The Construction of Difference Schemes for Equations of Hyperbolic Type on the Basis of Characteristic Ratios, Zh. Vych. Mat. Mat. Fiz., 1969, vol. 9, no. 2, pp. 373–386.MathSciNetGoogle Scholar
  2. 2.
    Petrov, I.B. and Kholodov, A.S., Numerical Investigation of CertainDynamical Problems of theMechanics of a Deformable Solid Body by the Grid-CharacteristicMethod, Zh. Vych. Mat. Mat. Fiz., 1984, vol. 24, no. 5, pp. 722–739.Google Scholar
  3. 3.
    Golubev, V.I., Petrov, I.B., Khokhlov, N.I., and Shul’ts, K.I., Numerical Computation of Wave Propagation in Fractured Media by Applying the Grid-Characteristic Method on Hexahedral Meshes, Comput. Math. Math. Phys., 2015, vol. 55, iss. 3, pp. 509–518.Google Scholar
  4. 4.
    Golubev, V.I., Petrov, I.B., and Khokhlov, N.I., Compact Grid-Characteristic Schemes of Higher Orders of Accuracy for a 3D Linear Transport Equation, Math. Mod. Comput. Simul., 2016, vol. 8, iss. 5, pp. 577–584.Google Scholar
  5. 5.
    Favorskaya, A.V., Petrov, I.B., and Khokhlov, N.I., Numerical Modeling of Wave Processes during Shelf Seismic Exploration, Proc. Comput. Sci., 2016, vol. 96, pp. 920–929.CrossRefGoogle Scholar
  6. 6.
    Magomedov, K.M. and Kholodov, A.S., Setochno-kharakteristicheskie chislennye metody (Grid- Characteristic NumericalMethods),Moscow: Nauka, 1988.zbMATHGoogle Scholar
  7. 7.
    Agapov, P.I. and Chelnokov, F.B., Comparative Analysis of Difference Schemes forNumerically Solving Two- Dimensional Problems of Deformable Solid Body Mechanics, in Modelirovanie i obrabotka informatsii, Moscow:Moscow Institute of Physics and Technology, 2003, pp. 19–27.Google Scholar
  8. 8.
    Chelnokov, F.B., Explicit Expression of Grid-Characteristic Schemes for Elasticity Equations in 2D and 3D, Mat. Model., 2006, vol. 18, no. 6, pp. 96–108.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Petrov, I.B., Favorskaya, A.V., Muratov, M.V., Biryukov, V.A., and Sannikov, A.V., Grid-Characteristic Method on Unstructured Tetrahedral Grids, Dokl.Mat., 2014, vol. 90, no. 3, pp. 781–783.CrossRefzbMATHGoogle Scholar
  10. 10.
    Beklemysheva, K.A., Danilov, A.A., Petrov, I.B., Salamatova, V.Yu., Vassilevski, Y.V., and Vasyukov A.V., Virtual Blunt Injury of Human Thorax: Age-Dependent Response of Vascular System, Russ. J. Num. An. Math.Model., 2015, vol. 30, no. 5, pp. 259–268.MathSciNetzbMATHGoogle Scholar
  11. 11.
    CGAL. Computational Geometry Algorithms Library, Scholar
  12. 12.
    Flötotto, J., 2D and Surface Function Interpolation, CGAL User and Reference Manual, 4.8 ed., CGAL, 2016.Google Scholar
  13. 13.
    Devillers, O., Pion, S., and Teillaud, M., Walking in a Triangulation, Proc. 17th Comp. Geom. (SoCG), ACMPress, 2001, pp. 106–114 (INRIA; RR-4120).zbMATHGoogle Scholar
  14. 14.
    Beklemysheva, K.A., Grigoriev, G.K., Kulberg, N.S., Kazakov, A.O., Petrov, I.B., Salamatova, V.Yu., Vassilevski, Yu.V., and Vasyukov, A.V., Transcranial Ultrasound of Cerebral Vessels in Silico: Proof of Concept, Russ. J. Num. An. Math.Model., 2016, vol. 31, no. 5, pp. 317–328.MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Moscow Institute of Physics and Technology (State University)DolgoprudnyiRussia

Personalised recommendations