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GENERATION OF IRREGULAR HEXAGONAL MESHES

Vestnik MGSU 4/2012
  • Vlasov Aleksandr Nikolaevich - Sergeev Institute of Environmental Geoscience of the Russian Academy of Sciences (IEG RAS), Institute of Applied Mechanics of the Russian Academy of Sciences (IAM RAS) Doctor of Technical Sciences, Principal Researcher, +7 (495) 523-81-92, Sergeev Institute of Environmental Geoscience of the Russian Academy of Sciences (IEG RAS), Institute of Applied Mechanics of the Russian Academy of Sciences (IAM RAS), 32а Leninskiy prospekt, Moscow, 119334, Russian Federation Building 2, 13 Ulanskiy pereulok, 101000, Moscow, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Volkov-Bogorodskiy Dmitriy Borisovich - Institute of Applied Mechanics of the Russian Academy of Sciences (IAM RAS) Candidate of Physics and Mathematics, Senior Researcher, +7 (499) 160-42-82, Institute of Applied Mechanics of the Russian Academy of Sciences (IAM RAS), 32а Leninskiy prospekt, Moscow, 119334, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Znamenskiy Vladimir Valerianovich - Moscow State University of Civil Engineering (MGSU) Doctor of Technical Sciences, Professor, Professor, Department of Soil Mechanics, Beddings and Foundations, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Mnushkin Mikhail Grigor'evich - Sergeev Institute of Environmental Geoscience Russian Academy of Sciences (IEG RAS) Candidate of Technical Sciences, Principal Researcher, Sergeev Institute of Environmental Geoscience Russian Academy of Sciences (IEG RAS), Building 2, 13 Ulanskiy pereulok, 101000, Moscow, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 78 - 87

In the paper, the authors propose original mesh generation solutions based on the finite element method applicable within the computational domain. The mesh generation procedure contemplates homeomorphic mapping of the initial domain onto the canonical domain. The authors consider mappings generated through the application of differential operators, including the Laplace operator (harmonic mappings) or the Lamé operator. In the latter case, additional control parameter ν is required The following domains are regarded as canonical: a parametric cube (or a square), a cylindrical layer, and a spherical layer. They represent simply connected or biconnected domains.
The above mappings are based on the parametric mesh generated alongside the domain boundary or boundaries dividing heterogeneous elements (inclusions). Therefore, generation of the above mappings is reduced to the resolution of the boundary problems by means of the Laplace or Lamé differential operators. Basically, the proposed approach represents the problem of the theory of elasticity with regard to the prescribed displacement. This problem may have two solutions. The first one is the analytical (meshless) least square solution, and the second one represents consequent mesh refining on the basis of the finite-element discretization of elasticity equations. The least square method assumes decomposition of the initial domain into the system of simply connected sub-domains. In every sub-domain, or a block, numerical/analytical approximation of homeomorphic mapping of the initial domain onto the canonical domain is performed with the help of local representations generated by means of systems of special functions.
Decomposition is performed in a constructive way and, as option, it involves meshless representation. Further, this mapping method is used to generate the calculation mesh. In this paper, the authors analyze different cases of mapping onto simply connected and bi-connected canonical domains. They represent forward and backward mapping techniques. Their potential application for generation of nonuniform meshes within the framework of the asymptotic homogenization theory is also performed to assess and project effective characteristics of heterogeneous materials (composites).

DOI: 10.22227/1997-0935.2012.4.78 - 87

References
  1. Haber R., Abel J.F. Numer. Meth. Eng., 1982, vol. 18, pp. 41—46.
  2. Volkov-Bogorodskiy D.B. Razrabotka blochnogo analitiko-chislennogo metoda resheniya zadach mekhaniki i akustiki [Development of the Block-based Analytical and Numerical Method of Resolution of Problems of Mechanics and Acoustics]. Collected papers, Composite Materials Workshop. Moscow, IPRIM RAN [Institute of Applied Mechanics of the Russian Academy of Sciences], 2000, pp. 44—56.
  3. Eells J., Sampson J. Harmonic Mappings of Riemannian Manifolds. Amer. J. Math., 1964, vol. 86, pp. 109—160.
  4. Thompson J.F., Soni B.K., Weatherill N.P., Harmonic Mappings. Handbook of Grid Generation. CRC Press, 1998.
  5. Godunov S.K., Prokopov G.P. Ob ispol'zovanii podvizhnykh setok v gazodinamicheskikh raschetakh [On the Use of Flexible Grids in Gas-Dynamic Calculations]. Vychislitel’naya matematika i matematicheskaya fizika [Computational Maths and Mathematical Physics]. 1972, vol. 12, no. 2, pp. 429—440.
  6. Spekreijse S.P. Elliptic Grid Generation Based on Laplace Equations and Algebraic Transformations. J. Comput. Phys., 1995, vol. 118, pp. 28—61.
  7. Novatskiy V. Teoriya uprugosti [Theory of Elasticity]. Moscow, Mir Publ., 1975, 872 p.
  8. Volkov-Bogorodsky D.B. On Construction of Harmonic Maps of Spatial Domains by the Block Analytical-Numerical Method. Proceedings of the minisymposium “Grid Generation: New Trends and Applications in Real-World Simulations”, International Conference “Optimization of Finite-Element Approximations, Splines and Wavelets”, St.Petersburg, 25—29 June 2001. Moscow, Computing Centre RAS, 2001, pp. 129—143.
  9. UWay Software. Certificate of State Registration of Software Programme no. 2011611833, issued on 28 February 2011. Compliance Certificate ROSS RU.SP15.N00438, issued on 27 October 2011.
  10. Kompozitsionnye materialy [Composite Materials]. Moscow, Mashinostroenie Publ., 1990, 511 p.
  11. Novikov V.U. Polimernye materialy dlya stroitel'stva [Polymeric Materials for Construction Purposes]. Moscow, Vysshaya Shkola Publ., 1995, 448 p.
  12. Obraztsov I.F., Yanovskiy Yu.G., Vlasov A.N., Zgaevskiy V.E. Koeffitsienty Puassona mezhfaznykh sloev polimernykh kompozitov [Poisson Ratios for Interphase Layers of Polymeric Composites]. Academy of Science Reports, 2001, vol. 378, no. 3, pp. 336—338.
  13. Bakhvalov N.S., Panasenko G.P. Osrednenie protsessov v periodicheskikh sredakh [Averaging of Processes in Periodic Media]. Moscow, Nauka Publ., 1984, 352 p.
  14. Vlasov A.N. Usrednenie mekhanicheskikh svoystv strukturno neodnorodnykh sred. [Averaging of Mechanical Properties of Structurally Heterogeneous Media]. Mekhanika kompozitsionnykh materialov i konstruktsiy [Mechanics of Composite Materials and Structures]. 2004, vol. 10, no. 3, pp. 424—441.

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STRESS STATE OF TRANSVERSELY ISOTROPIC ROCKS NEAR PRESSURIZED HYDRAULIC TUNNEL OF HORSESHOE CROSS-SECTION

Vestnik MGSU 10/2017 Volume 12
  • Bautdinov Damir Tahirovich - Russian State Agrarian University - Moscow Timiryazev Agricultural Academy (RSAU - MTAA or RSAU - MAA named after K.A. Timiryazev) Candidate of Technical Sciences, Associate Professor, Department of Technical Mechanics and Construction, Russian State Agrarian University - Moscow Timiryazev Agricultural Academy (RSAU - MTAA or RSAU - MAA named after K.A. Timiryazev), 49 Timiryazevskaya str., Moscow, 127550, Russian Federation.
  • Atabiev Umar Ishakovich - Russian State Agrarian University - Moscow Timiryazev Agricultural Academy (RSAU - MTAA or RSAU - MAA named after K.A. Timiryazev) Postgraduate, Department of Hydro-technical Construction, Russian State Agrarian University - Moscow Timiryazev Agricultural Academy (RSAU - MTAA or RSAU - MAA named after K.A. Timiryazev), 49 Timiryazevskaya str., Moscow, 127550, Russian Federation.

Pages 1172-1179

The parametric analysis of the stress state of a transversally isotropic rock mass near a pressurized hydraulic tunnel of a box-shaped form is carried out. Pressurized hydro-technical tunnels of box-shaped cross-section are widely used in the field of hydraulic engineering construction and are one of the complex, labor-intensive and expensive types of structures that make up the main structures of waterworks, melioration systems and water supply systems. As a culvert and water supply facilities they are built underground if the open excavation is impossible or not economical, or when the tunnel runs through a densely populated or densely built-up area, or when landslides, screes, rockfalls are possible. Violation of integrity of the rock mass, in particular, caused by tunneling, modifies the stress-strain state (SSS) of the rock mass, which leads to appearance of tensile stresses in some places, and in some cases, to significant compressive stresses. If these stresses exceed the design strengths of rock to tension and compression, respectively, then the collapse of the working roof and buckling of the side walls and the bottom of the tunnel may occur. Subject: analysis of the stress state of transversally isotropic rocks near the pressurized hydraulic tunnel of horseshoe cross-section caused by the internal head of water. Research objectives: determination of real values of circumferential stresses along the development contour. Materials and methods: solution of the problem of plane deformation of the theory of elasticity for a transversely isotropic medium containing tunnel excavation cannot be obtained by analytical methods, and therefore the stress-strain analysis was carried out by the finite element method using the ANSYS software package, MCE. Results: determination of stresses along the development contour, construction of diagrams and graphs showing the effects of the anisotropy conditions and Poisson’s ratio. The tangential stresses along the contour of hydraulic tunnel development for various values of deformation modulus and Poisson’s ratio are determined, which makes it possible to estimate the strength of the rock mass for different tunnel depths. The analysis of a long hydro-technical tunnel, laid in a strong, transversally isotropic rock, is reduced to the problem of plane deformation of the theory of elasticity for a transversely isotropic medium containing tunnel excavation. The size and type of the finite element suitable for analysis were determined in advance based on the solution of the test problem. Conclusions: it is necessary to determine the physical and mechanical properties of rocky soils more accurately, paying special attention to elastic characteristics; calculations should be performed taking into account the anisotropy of elastic properties.

DOI: 10.22227/1997-0935.2017.10.1172-1179

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