### GENERATION OF IRREGULAR HEXAGONAL MESHES

Vestnik MGSU 4/2012

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
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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.
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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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### Eigenfunction of the Laplace operator in +1-dimentional simplex

Vestnik MGSU 11/2014

Pages 68-73

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
1. 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).
2. 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).
3. 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).
4. Mikhaylov V.P. Differentsial’nye uravneniya v chastnykh proizvodnykh [Differential Equations in Partial Derivatives]. Moscow, Nauka Publ., 1976, 391 p. (In Russian).
5. Mikhlin S.G. Kurs matematicheskoy fiziki [Course in Mathematical Physics]. Moscow, Nauka Publ., 1968, 576 p. (In Russian).
6. Lazutkin V.F. Ob asimptotike sobstvennykh funktsiy operatora Laplasa [On Asymptotics of Eigenfunctions of the Laplace Operator]. Doklady AN SSSR [Reports of the Academy of Sciences of the USSR]. 1971, vol. 200, no. 6, pp. 1277—1279. (In Russian).
7. Lazutkin V.F. Sobstvennye funktsii s zadannoy kaustikoy [Eigenfunctions with Preassigned Caustic Curve]. Zhurnal vychislitel’noy matematiki i matematicheskoy fiziki [Computational Mathematics and Mathematical Physics]. 1970, vol. 10, no. 2, pp. 352—373. (In Russian).
8. Lazutkin V.F. Asimptotika serii sobstvennykh funktsiy operatora Laplasa, otvechayushchey zamknutoy invariantnoy krivoy «billiardnoy zadachi» [Asymptotics of Eigenfunctions Series of the Laplace Operator Matching Closed Invariant Curve of a "Billiard problem"]. Problemy matematicheskoy fiziki [Mathematical Physics Problems]. 1971, no. 5, pp. 72—91. (In Russian).
9. Lazutkin V.F. Postroenie asimptotiki serii sobstvennykh funktsiy operatora Laplasa, otvechayushchey ellipticheskoy periodicheskoy traektorii «billiardnoy zadachi» [Asymptotics Creation of Eigenfunctions Series of the Laplace Operator Matching Elliptical Periodic Path of a "Billiard problem"]. Problemy matematicheskoy fiziki [Mathematical Physics Problems]. 1973, no. 6, pp. 90—100. (In Russian).
10. Apostolova L.N. Initial Value Problem for the Double-Complex Laplace Operator. Eigenvalue Approaches. AIP Conf. Proc. 2011, vol. 1340, no. 1, pp. 15—22. DOI: http://dx.doi.org/10.1063/1.3567120.
11. Pomeranz K.B. Two Theorems Concerning the Laplace Operator. AIP Am. J. Phys. 1963, vol. 31, no. 8, pp. 622—623. DOI: http://dx.doi.org/10.1119/1.1969694.
12. Iorgov N.Z., Klimyk A.U. A Laplace Operator and Harmonics on the Quantum Complex Vector Space. AIP J. Math. Phys. 2003, vol. 44, no. 2, pp. 823—848.
13. Fern?ndez C. Spectral concentration for the Laplace operator in the exterior of a resonator. AIP J. Math. Phys. 1985, vol. 26, no. 3, pp. 383—384. DOI: http://dx.doi.org/10.1063/1.526618.
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15. Gorbar E.V. Heat Kernel Expansion for Operators Containing a Root of the Laplace Operator. AIP J. Math. Phys. 1997, vol. 38, no. 3, pp. 1692. DOI: http://dx.doi.org/10.1063/1.531823. Date of access: 25.03.2012.

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### Continuation of the solution of an elliptic equation and mathematical tesselations

Vestnik MGSU 10/2014

Pages 48-53

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
1. 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)
2. 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)
3. Brodnikov A.P. Sobstvennye funktsii i sobstvennye chisla operatora Laplasa dlya treugol’nikov [Eigenfunctions and Eigenvalues of the Laplace Operator for Triangles]. Available at: http://chillugy.narod.ru/Mathematics/laplas/start/start.html. Date of access: 17.02.2014. (in Russian)
4. Kolmogorov A.N. Parkety iz pravil’nykh mnogougol’nikov [Tesselations of the Regular Polygons]. Kvant [Quantum]. 1970, no. 3. Available at: http://kvant.mccme.ru/1970/03/parkety_iz_pravilnyh_mnogougol.htm. Date of access: 17.02.2014. (in Russian)
5. Mikhaylov O. Odinnadtsat’ pravil’nykh parketov [Eleven Regular Tessellation]. Kvant [Quantum]. 1979, no. 2. Available at: http://kvant.mccme.ru/1979/02/odinnadcat_pravilnyh_parketov.htm. Date of access: 17.02.2014. (in Russian)
6. Sitnikova E.G. Prodolzhenie obobshchennogo resheniya kraevoy zadachi [Continuation of the Generalized Solution for the Boundary Value Problem]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2007, no. 1, pp. 16—18. (in Russian)
7. Mikhaylov V.P. Differentsial’nye uravneniya v chastnykh proizvodnykh [Differential Equations in Partial Derivatives]. Moscow, Nauka Publ., 1976, 391 p. (in Russian)
8. Mikhlin S.G. Kurs matematicheskoy fiziki [Course in Mathematical Physics]. Moscow, Nauka Publ., 1968, 576 p. (in Russian)
9. Petrovskiy N.G. Lektsii ob uravneniyakh s chastnymi proizvodnymi [Lections on the Equations with Partial Derivatives]. 3rd edition, Moscow, Fizmatgiz Publ., 1961, 401 p. (in Russian)
10. Lazutkin V.F. Ob asimptotike sobstvennykh funktsiy operatora Laplasa [On the Asymptotics of Eigenfunctions of Laplace Operator]. Doklady AN SSSR [Proceedings of the USSR Academy of Sciences]. 1971, vol. 200, no. 6, pp. 1277—1279. (in Russian)
11. Jiaquan Liu, Zhi-Qiang Wang, Xian Wu. Multibump Solutions for Quasilinear Elliptic Equations with Critical Growth. AIP. J. Math. Phys. 2013, no. 54, 121501. Available at: http://scitation.aip.org/content/aip/journal/jmp/54/12/10.1063/1.4830027. Date of access: 17.02.2014. DOI: http://dx.doi.org/10.1063/1.4830027.
12. Chavey D. Tilings by Regular Polygons—II: A Catalog of Tilings. Computers & Mathematics with Applications. 1989, vol. 17, no. 1—3, pp. 147—165. DOI: http://dx.doi.org/10.1016/0898-1221(89)90156-9.
13. Grünbaum B., Shephard G.C. Tilings And Pattern. New York, W.H. Freeman and Company, 1987, 700 p.
14. Berger R. The Undecidability of the Domino Problem. Memoirs of the American Mathematical Society. 1966, no. 66, pp. 1—72.
15. Penrose R. Pentaplexity: A Class of Non-Periodic Tilings of the Plane. The Mathematical Intelligencer. 1979, vol. 2, no. 1, pp. 32—37. DOI: http://dx.doi.org/10.1007/BF03024384.

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