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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;
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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;
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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;
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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;
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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
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Ovchintsev Mikhail Petrovich -
Moscow State University of Civil Engineering (MGSU)
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation;
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Sitnikova Elena Georgievna -
Moscow State University of Civil Engineering (MGSU)
Candidate of Physical and Mathematical Sciences, Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), .
In order to find eigenfunction of the Laplace operator in regular
n+1-dimensional simplex the barycentric coordinates are used. For obtaining this result we need some formulas of the analytical geometry. A similar result was obtained in the earlier papers of the author in a tetrahedron from
R
3 and in gipertetrahedron from
R
4. Let П be unlimited cylinder in the space
R
n, its cross-section with hyperplane has a special form. Let
L be a second order linear differential operator in divergence form, which is uniformly elliptic and η is its ellipticity constant. Let
u be a solution of the mixed boundary value problem in Π with homogeneous Dirichlet and Neumann data on the boundary of the cylinder. In some cases the eigenfunction of the Laplace operator allows us to continue this solution from the cylinder Π to the whole space
R
n with the same ellipticity constant. The obtained result allows us to get a number of various theorems on the solution growth for mixed boundary value problem for linear differential uniformly elliptical equation of the second order, given in unlimited cylinder with special cross-section. In addition we consider
n-1-dimensional hill tetrahedron and the eigenfunction for an elliptic operator with constant coefficients in it.
DOI: 10.22227/1997-0935.2014.11.68-73
References
- Sitnikova E.G. Sobstvennaya funktsiya operatora Laplasa v gipertetraedre [Eigenfunction of the Laplace Operator in the Tetrahedron]. Integratsiya, partnerstvo i innovatsii v stroitel’noy nauke i obrazovanii : sbornik trudov Mezhdunaridnoy nauchnoy konferentsii [Integration, Partnership and Innovations in Construction Science and Education : Collection of Works of International Scientific Conference]. Moscow, MGSU, 2011, pp. 755—758. (In Russian).
- Sitnikova E.G. Neskol’ko teorem tipa Fragmena-Lindelefa dlya ellipticheskogo uravneniya vtorogo poryadka [Several Theorems of Phragmen-Lindelof Type for the Second Order Differential Equation]. Voprosy matematiki i mekhaniki sploshnykh sred : sbornik nauchnykh trudov [Problems of Continuum Mathematics and Mechanics: Collection of Works]. Moscow, MGSU Publ., 1984, pp. 98—104. (In Russian).
- Sitnikova E.G. Sobstvennaya funktsiya operatora Laplasa v tetraedre [Eigenfunction of the Laplace Operator in the Tetrahedron]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 4, pp. 80—82. (In Russian).
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Ovchintsev Mikhail Petrovich -
Moscow State University of Civil Engineering (MGSU)
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation;
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.
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Sitnikova Elena Georgievna -
Moscow State University of Civil Engineering (MGSU)
Candidate of Physical and Mathematical Sciences, Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), .
In the following article the authors continue investigating elliptical equation. Let P be an unlimited cylinder in the space R3, the cross-section of which is a regular dodecagon. The authors have previously estimated linear self-conjugate uniformly elliptic equation of second order in the cylinder and obtained theorems on the growth of the solution in bounded domain. In order to prove the theorems we have to continue solving the differential equation and its coefficients for the whole space Rn.
Let L be a second order linear differential operator in a divergence form which is uniformly elliptic and h is its ellipticity constant. Let u be a solution of the mixed boundary value problem in P for the equation Lu=0 (u>0) with homogeneous Dirichlet and Neumann data on the boundary of the cylinder.
In this paper the solution for mixed boundary value problem is continued from the cylinder to the whole space R3.
The solution of the mixed problem has connection with the notion of the mathematical tessellation. This tessellation is a sum of nonintersecting regular dodecagons and triangles filling the whole space R2
DOI: 10.22227/1997-0935.2014.10.48-53
References
- Sitnikova E.G. Neskol’ko teorem tipa Fragmena — Lindelefa dlya ellipticheskogo uravneniya vtorogo poryadka [Several Theorems of Phragmen-Lindelof Type for the Second Order Differential Equation]. Voprosy matematiki i mekhaniki sploshnykh sred: sbornik trudov [Problems of Mathematics and Mechanics of Continuous Media: Collection of Works]. Moscow, MGSU Publ., 1984, pp. 98—104. (in Russian)
- Landis E.M. O povedenii resheniy ellipticheskikh uravneniy vysokogo poryadka v neogranichennykh oblastyakh [On Solutions Behavior of High Order Elliptic Equations in Unbounded Domains]. Trudy MMO [Works of Moscow Mathematical Society]. Moscow, MGU Publ., 1974, vol. 31, pp. 35—58. (in Russian)
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- Mikhlin S.G. Kurs matematicheskoy fiziki [Course in Mathematical Physics]. Moscow, Nauka Publ., 1968, 576 p. (in Russian)
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