Development of algorithms for constructing two-dimensional optimal boundary-adaptive grids and their software implementation

Introduction. It is noted that the use of adaptive grids in calculations makes it possible to improve the accuracy and efficiency of computational algorithms without increasing the number of nodes. This approach is especially efficient when calculating nonstationary problems. The objective of this study is the development, construction and software implementation of methods for constructing computational two-dimensional optimal boundary-adaptive grids for complex configuration regions while maintaining the specified features of the shape and boundary of the region. The application of such methods contributes to improving the accuracy, efficiency, and cost-effectiveness of computational algorithms.

Materials and Methods. The problem of automatic construction of an optimal boundary-adaptive grid in a simply connected region of arbitrary geometry, topologically equivalent to a rectangle, is considered. A solution is obtained for the minimum set of input information: the boundary of the region in the physical plane and the number of points on it are given. The creation of an algorithm and a mesh generation program is based on a model of particle dynamics. This provides determining the trajectories of individual particles and studying the dynamics of their pair interaction in the system under consideration. The interior and border nodes of the grid are separated through using the mask tool, and this makes it possible to determine the speed of movement of nodes, taking into account the specifics of the problem being solved.

Results. The developed methods for constructing an optimal boundary-adaptive grid of a complex geometry region provides solving the problem on automatic grid construction in two-dimensional regions of any configuration. To evaluate the results of the algorithm research, a test problem was solved, and the solution stages were visualized. The computational domain of the test problem and the operation of the function for calculating the speed of movement of interior nodes are shown in the form of figures. Visualization confirms the advantage of this meshing method, which separates the border and interior nodes.

Discussion and Conclusions. The theoretical and numerical studies results are important both for the investigation of the grids qualitative properties and for the computational grid methods that provide solving numerical modeling problems efficiently and with high accuracy.

1. Downing-Kunz MA, Work PA, Schoellhamer DH. Tidal Asymmetry in Ocean-Boundary Flux and InEstuary Trapping of Suspended Sediment Following Watershed Storms: San Francisco Estuary, California, USA. Estuaries and Coasts. 2021. URL: https://doi.org/10.1007/s12237-021-00929-y (accessed: 31.08.2021).

2. Ballent A, Pando S, Purser A, et al. Modelled transport of benthic marine microplastic pollution in the Nazaré Canyon. Biogeosciences. 2013;10(12):7957–7970. https://doi.org/10.5194/bg-10-7957-2013

3. Kirk B, Lipnikov K, Carey GF. Nested grid iteration for incompressible viscous flow and transport. International Journal of Computational Fluid Dynamics. 2003;17(4):253–262. https://doi.org/10.1080/1061856031000173635

4. Xiaoying Liu, Shi Qi, YuanHuang, et al. Predictive modeling in sediment transportation across multiple spatial scales in the Jialing River Basin of China. International Journal of Sediment Research. 2015;30(3):250–255. https://doi.org/10.1016/j.ijsrc.2015.03.013

5. Barnard PL, Jaffe BE, Schoellhamer DH (eds.). A multi-discipline approach for understanding sediment transport and geomorphic evolution in an estuarine-coastal system: San Francisco Bay. Marine Geology. 2013;345:1–326. https://doi.org/10.1016/j.margeo.2013.09.010

6. Alekseenko E, Roux B, Sukhinov A, et al. Coastal hydrodynamics in a windy lagoon. Computers & Fluids. 2013;77:24–35. https://doi.org/10.1016/j.compfluid.2013.02.003

7. Sukhinov AI. Pretsizionnye modeli gidrodinamiki i opyt ikh primeneniya v predskazanii i rekonstruktsii chrezvychainykh situatsii v Azovskom more. Izvestiya TRTU. 2006;58(3):228–235. (In Russ.)

8. Sukhinov AI, Sidoryakina VV. Development and correctness analysis of the mathematical model of transport and suspension sedimentation depending on bottom relief variation. Vestnik of Don State Technical University. 2018;18(4):350–361. https://doi.org/10.23947/1992-5980-2018-18-4-350-361 (In Russ.)

9. Sidoryakina VV, Sukhinov AI. Well-posedness analysis and numerical implementation of a linearized twodimensional bottom sediment transport problem. Computational Mathematics and Mathematical Physics. 2017;57(6):978–994. https://doi.org/10.1134/S0965542517060124

10. Sukhinov AI, Sidoryakina VV, Sukhinov AA. Sufficient conditions for convergence of positive solutions to linearized two-dimensional sediment transport problem. Vestnik of Don State Technical University. 2017;17(1):5–17. https://doi.org/10.23947/1992-5980-2017-17-1-5-17 (In Russ.)

11. Sukhinov AI, Vasiliev VS. Precise two-dimensional models for shallow water basins. Mathematical Models and Computer Simulations. 2003;15(10):17–34. (In Russ.)

12. Carey GF, Anderson M, Carnes B, et al. Some aspects of adaptive grid technology related to boundary and interior layers. Journal of Computational and Applied Mathematics. 2004;166(1):55–86. https://doi.org/10.1016/j.cam.2003.09.036

13. Owen SJ. A Survey of Unstructured Mesh Generation Technology. In: Proc.7th Int. Meshing Roundtable. Dearborn, MI; 1998. P. 239–269.

14. Skovpen’ AV. Improved algorithm for unstructured quadrilateral mesh generation. Computational Mathematics and Mathematical Physics. 2005;45(8):1506–1528. (In Russ.)

15. Krivtsov AM, Krivtsov NV. Method of particles and its application to mechanics of solids. Far Eastern Mathematical Journal. 2002;3(2):254–276. (In Russ.)

16. Belkin AA. On a modification of the method of molecular dynamics. Journal of Applied and Industrial Mathematics. 2006;9(4):27–32. (In Russ.)

17. Zheleznyakova AL, Surzhikov ST. Triangular mesh generation by molecular dynamics method. PhysicalChemical Kinetics in Gas Dynamics. 2011. Vol. 11. URL: http://chemphys.edu.ru/issues/2011-11/articles/192/ (accessed: 31.08.2021). (In Russ.)