Design of adaptive unstructured grids using differential methods

  • Olzhas Nurkonysuly Turar Al-Farabi Kazakh National University
  • D. Zh. Akhmed-Zaki International University of Business


Adaptive generation of computational grids can improve the efficiency of mathematical modeling by increasing the accuracy of numerical approximations. The paper describes a method for constructing unstructured grids with adaptation based on differential methods. The application of these methods ensures a smooth distribution of the geometric characteristics of the grid, i.e. the appearance of adjacent cells that differ greatly in size and shape becomes unlikely. To achieve proper adaptation in unstructured grids we use the novel approach based on methodology of adaptive structured grid construction. This approach uses the method of grid construction based on solving inverted Beltrami equation to create mapping of some sample grid domain to the physical area. This mapping is used to construct point set on which the unstructured grid is constructed using Delaunay triangulation method. Thus, the result is unstructured grid with proper adaptation. Adding fault and fractures or other structure elements may be supported by implementing constrained Delaunay triangulation.


[1] Thompson, Joe F., B. K. Soni, and N. P. Weatherill. Handbook of Grid Generation. Boca Raton, FL: CRC Press, 1999.
[2] Shewchuk, J. R. "Unstructured mesh generation."In U. Naumann and O. Schenk, editors, Combinatorial Scientific Computing. , CRC Press, chapter 10 (2012):257–297.
[3] Lo, S. H. "A New Mesh Generation Scheme for Arbitrary Planar Domains." International Journal for Numerical Methods in Engineering. 21, no. 8 (1985): 1403-426. doi:10.1002/nme.1620210805.
[4] Peraire, J., J. Peir?, and K. Morgan. "Adaptive Remeshing for Three-dimensional Compressible Flow
Computations."Journal of Computational Physics 101, no. 1 (1992): 229. doi:10.1016/0021-9991(92)90068-a.
[5] Shewchuk, J.R. "Tetrahedral mesh generation by Delaunay refinement." Symp. on Comp. Geom. (1998):86–95.
[6] Klein, Rolf, Franz Aurenhammer, and Der-Tsai Lee. Voronoi Diagrams and Delaunay Triangulations. New Jersey:World Scientific, 2013.
ISSN 1563–0285 KazNU Bulletin. Mathematics, Mechanics, Computer Science Series 1(93)2017
Design of adaptive unstructured grids using differential methods. . . 29
[7] Yerry, Mark, and Mark Shephard. "A Modified Quadtree Approach To Finite Element Mesh Generation." IEEE Computer Graphics and Applications. 3, no. 1 (1983): 39-46. doi:10.1109/mcg.1983.262997.
[8] Mitchell, Scott A., and Stephen A. Vavasis. "Quality Mesh Generation in Three Dimensions." Proceedings of the Eighth Annual Symposium on Computational Geometry - SCG. 92 (1992). doi:10.1145/142675.142720.
[9] Frederick, C. O., Y. C. Wong, and F. W. Edge. "Two-dimensional Automatic Mesh Generation for
Structural Analysis."International Journal for Numerical Methods in Engineering 2, no. 1 (1970): 133-44.
[10] Der-Tsai Lee and Arthur K. Lin. "Generalized Delaunay Triangulations for Planar Graphs.„ Discrete & Computational Geometry, 1 (1986):201–217.
[11] Bronson, Jonathan R., Shankar P. Sastry, Joshua A. Levine, and Ross T. Whitaker. "Adaptive and Unstructured Mesh Cleaving." Procedia Engineering. 82 (2014). doi:10.1016/j.proeng.2014.10.389.
[12] Riemslagh, K., and E. Dick. "A multigrid method with unstructured adaptive grids for steady Euler equations." Journal of Computational and Applied Mathematics 67 (1996): 73-93.
[13] Vilsmeier, R., and D. H?nel. "Adaptive Methods on Unstructured Grids for Euler and Navier-Stokes Equations."Computers & Fluids. 22, no. 4-5 (1993): 485-99. doi:10.1016/0045-7930(93)90021-z.
[14] Bronson, Jonathan R., Joshua A. Levine, and Ross T. Whitaker. "Particle Systems for Adaptive, Isotropic Meshing of CAD Models." Proceedings of the 19th International Meshing Roundtable. (2010). doi:10.1007/978-3-642-15414-0_17.
[15] Mavriplis, D. J. "Unstructured Grid Techniques." Annual Review of Fluid Mechanics. 29, no. 1 (1997): 473-514.
[16] Vekua, I. N. Generalized Analytic Functions. Oxford: Pergamon Press, 1962.
[17] Lisejkin, V. D. Grid Generation Methods. Cham: Springer, 2017.
[18] Castillo, Jose E. Mathematical Aspects of Numerical Grid Generation. Philadelphia: SIAM, 1991.
[19] Glasser, A. H., V. D. Liseikin, and I. A. Kitaeva. "Specification of Monitor Metrics for Generating Vector Field-aligned Numerical Grids." Russian Journal of Numerical Analysis and Mathematical Modelling. 20, no. 5 (2005):439-461. doi:10.1515/156939805775186686..
[20] Godunov, S. K., A. V. Zabrodin, and M. Ja. Ivanov. Chislennoe reshenie mnogomernyh zadach gazovoi dinamiki (Numerical solving of multidimensional problems in gas dynamics). Moscow: Nauka, 1976.
[21] Khakimzyanov, G. S., and N. Yu. Shokina. "Equidistribution Method for the Construction of Adaptive Grids." Russian Journal of Numerical Analysis and Mathematical Modelling 14, no. 4 (1999):339-358. doi:10.1515/rnam.1999.14.4.339.
[22] Thompson, Joe F., Z. U. A. Warsi, and C. Wayne. Mastin. Numerical Grid Generation: Foundations and Applications. New York: North-Holland, 1985.
How to Cite
TURAR, Olzhas Nurkonysuly; AKHMED-ZAKI, D. Zh.. Design of adaptive unstructured grids using differential methods. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 98, n. 2, p. 88-97, aug. 2018. ISSN 1563-0277. Available at: <>. Date accessed: 20 feb. 2019. doi:
Keywords computational grid construction algorithm, unstructured mesh, adaptive mesh, differential elliptic equations, reversed Beltrami equation