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Ȼɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɣ
ɫɩɢɫɨɤ
1.
Haber R., Abel J.F.
// Numer. Meth. Eng., V. 18, 1982, p. 41—46.
2.
-
. .
//
-
«
».
. :
, 2000. . 44—56.
3
. Eells J., Sampson J.
Harmonic mappings of Riemannian manifolds // Amer. J. Math. 1964.
Vol. 86. P. 109—160.
5.
. .,
. .
//
.
.
.
. 1972. . 12.
№
2. . 429—440.
6.
Spekreijse S.P.
Elliptic grid generation based on Laplace equations and algebraic
transforma-tions // J. Comput. Phys. 1995. Vol. 118. P. 28—61.
7.
.
. . :
, 1975. 872 .
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” in the International conference “Optimization of finite-element
approximations, splines and wavelets” (St.-Petersburg, 25—29 June 2001). Moscow: Computing
Cen-tre RAS, 2001. P. 129—143.
9.
«UWay».
-№
2011611833, 28
, 2011 .
№
RU.
15. 00438, 27
, 2011 .
10.
. . :
, 1990. 511 .
11.
. .
. . :
.
., 1995. 448 .
12.
/ . .
,
. .
,
. .
, . .
//
. 2001. . 378.
№
3.
. 336—338.
13.
. .,
. .
.
. :
, 1984. 352 .
14.
. .
//
. 2004. . 10.
№
3. C. 424—441.
2012 .
:
ȼɥɚɫɨɜ
Ⱥɥɟɤɫɚɧɞɪ
ɇɢɤɨɥɚɟɜɢɱ
—
,
,
ɂɧɫɬɢɬɭɬ
ɩɪɢɤɥɚɞɧɨɣ
ɦɟɯɚɧɢɤɢ
ɊȺɇ
(
ɂɉɊɂɆ
ɊȺɇ
)
, 119334, .
,
-., . 32 ,
,
ɂɧɫɬɢɬɭɬ
ɝɟɨɷɤɨɥɨɝɢɢ
ɢɦ
.
ȿ
.
Ɇ
.
ɋɟɪɝɟɟɜɚ
ɊȺɇ
(
ɂȽɗ
ɊȺɇ
)
, 101000, .
,
, . 13,
. 2, 8 (495) 523-81-92,
[email protected];
ȼɨɥɤɨɜ
-
Ȼɨɝɨɪɨɞɫɤɢɣ
Ⱦɦɢɬɪɢɣ
Ȼɨɪɢɫɨɜɢɱ
—
-
,
-,
ɂɧɫɬɢɬɭɬ
ɩɪɢɤɥɚɞɧɨɣ
ɦɟɯɚɧɢɤɢ
ɊȺɇ
(
ɂɉɊɂɆ
ɊȺɇ
)
, 119334,
.
,
., . 32 , 8 (499) 160-42-82, [email protected];
Ɂɧɚɦɟɧɫɤɢɣ
ȼɥɚɞɢɦɢɪ
ȼɚɥɟɪɢɚɧɨɜɢɱ
—
,
,
ɎȽȻɈɍ
ȼɉɈ
«
Ɇɨɫɤɨɜɫɤɢɣ
ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ
ɫɬɪɨɢɬɟɥɶɧɵɣ
ɭɧɢɜɟɪɫɢɬɟɬ
», 129337, .
,
, . 26,
(495) 589-23-37, [email protected];
Ɇɧɭɲɤɢɧ
Ɇɢɯɚɢɥ
Ƚɪɢɝɨɪɶɟɜɢɱ
—
,
-,
ɂɧɫɬɢɬɭɬ
ɝɟɨɷɤɨɥɨɝɢɢ
ɢɦ
.
ȿ
.
Ɇ
.
ɋɟɪɝɟɟɜɚ
ɊȺɇ
(
ɂȽɗ
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,
101000, .
,
, . 13,
. 2, [email protected].
:
/ . .
, . .
--
, . .
, . .
//
. 2012.
№
4. . 78—87.
A.N. Vlasov, D.B. Volkov-Bogorodskiy, V.V. Znamenskiy, M.G. Mnushkin
GENERATION OF IRREGULAR HEXAGONAL MESHES
In the paper, the authors propose original mesh generation solutions based on the finite ele-ment method applicable within the computational domain. The mesh generation procedure contem-plates homeomorphic mapping of the initial domain onto the canonical domain. The authors consid-er mappings genconsid-erated through the application of diffconsid-erential opconsid-erators, including the Laplace
op-erator (harmonic mappings) or the Lamé opop-erator. In the latter case, additional control parameter
is required. The following domains are regarded as canonical: a parametric cube (or a square), a
conse-quent 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 repre-sentations generated by means of systems of special functions.
Decomposition is performed in a constructive way and, as option, it involves meshless represen-tation. Further, this mapping method is used to generate the calculation mesh. In this paper, the au-thors 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).
Key words: finite element method (FEM), mesh, block method, Laplace operator, Lamé op-erator, effective deformation characteristics, composite material.
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
matemati-cheskaya 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 Transforma-tions. 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 Ap-proximations, 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 Pur-poses]. Moscow, Vysshaya. Shkola Publ., 1995, 448 p.
12. Obraztsov I.F., Yanovskiy Yu.G., Vlasov A.N., Zgaevskiy V.E. Koeffitsienty Puassona
mezhfaz-nykh sloev polimermezhfaz-nykh 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.
A b o u t t h e a u t h o r s : Vlasov Aleksandr Nikolaevich —Doctor of Technical Sciences, Principal
Re-searcher, Institute of Applied Mechanics of the Russian Academy of Sciences (IAM RAS), 32
Le-ninskiy prospekt, Moscow, 119334, Russian Federation; Principal Researcher; Sergeev Institute of
En-vironmental Geoscience of the Russian Academy of Sciences (IEG RAS), Building 2, 13 Ulanskiy pereulok, 101000, Moscow, Russian Federation; [email protected]; +7 (495) 523-81-92;
Volkov-Bogorodskiy Dmitriy Borisovich — Candidate of Physics and Mathematics, Senior
Re-searcher, Institute of Applied Mechanics of the Russian Academy of Sciences (IAM RAS), 32
Le-ninskiy prospekt, Moscow, 119334, Russian Federation; [email protected]; +7 (499) 160-42-82;
Znamenskiy Vladimir Valerianovich — Doctor of Technical Sciences, Professor, Moscow State University of Civil Engineering (MSUCE), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federa-tion; [email protected]; +7 (495) 589-23-37;
Mnushkin Mikhail Grigor'evich — Candidate of Technical Sciences, Principal Researcher, Ser-geev Institute of Environmental Geoscience Russian Academy of Sciences (IEG RAS), Building 2, 13 Ulanskiy pereulok, 101000, Moscow, Russian Federation; [email protected].
