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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Eigenfunction of the Laplace operator in +1-dimentional simplex

Vestnik MGSU 11/2014
  • 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; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • 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), .

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.
  14. Davis H.F. The Laplace Operator. AIP Am. J. Phys. 1964, 32, 318. DOI: http://dx.doi.org/10.1119/1.1970275. Date of access: 25.03.2012.
  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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BASIC FUNCTIONS FOR THE METHOD OF TWO-SIDED EVALUATIONS IN THE PROBLEMS OF STABILITY OF ELASTICNON-UNIFORMLY COMPRESSED RODS

Vestnik MGSU 6/2013
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Мoscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 63-70

The author considers the method of two-sided evaluations in the problems of stability of a one-span elastic non-uniformly compressed rod under various conditions of fixation of its ends.The required minimum critical value of the loading parameter for the rod is the minimum value of the functional equal to the ratio of the norms of Hilbert space elements squared. Using the inequalities following from the problem of the best approximation of a Hilbert space element through the basic functions, it is possible to construct two sequences of functionals, the minimum values of which are the lower evaluations and the upper ones. The basic functions here are the orthonormal forms of the stability loss for a rod with constant cross-section, compressed by longitudinal forces at the ends, which are fixed just so like the ends of the non-uniformly compressed rod.Having used the Riesz theorem about the representation of a bounded linear functional in the Hilbert space, the author obtains the additional functions from the domain of definition of the initial functional, which correspond to the basic functions. Using these additional functions, the calculation of the lower bounds is reduced to the determination of the maximum eigenvalues of the matrices represented in the form of second order modular matrices with the elements expressed in the form of integrals of basic and additional functions. The calculation of the upper bound value is reduced to the determination of the maximum eigenvalue of the matrix, which almost coincides with one of the modular matrices. It is noted that the obtained upper bound evaluations are not worse than the evaluations obtained through the Ritz method with the use of the same basic functions.

DOI: 10.22227/1997-0935.2013.6.63-70

References
  1. Kupavtsev V.V. K dvustoronnim ocenkam kriticheskih nagruzok neodnorodno szhatyh uprugih sterzhnej. [On Bilateral Evaluations of Critical Loading Values in Respect of Non-uniformly Compressed Elastic Rods]. Izvestija vuzov. Stroitel’stvo I arhitektura. [News of Institutions of Higher Education. Construction and Architecture]. 1984, no. 8, pp. 24—29.
  2. Alfutov N.A. Osnovy rascheta na ustojchivost’ uprugih sistem. [Fundamentals of Stability Analysis of Elastic Systems]. Moscow, Mashinostroenie Publ., 1991, 336 p.
  3. Rektoris K. Variatsionnye metody v matematicheskoy fizike I tekhnike. [Variational Metods in Mathematical Physics and Engineering]. Moscow, Mir Publ., 1985, 589 p.
  4. Panteleev S.A. Dvustoronie otsenki v zadache ob ustojchivosti szhatyh uprugih blokov. [Bilateral Assessments in the Stability Problem of Compressed Elastic Blocks]. Izvestyja RAN. MTT. [News of Russian Academy of Sciences. Mechanics of Solids]. 2010, no. 1, pp. 51—63.
  5. Bogdanovich A.U., Kuznetsov I.L. Prodol’noe szhatie tonkostennogo sterzhnja peremennogo sechenija pri razlichnyh variantah zakreplenija torcov [Longitudinal Compression of a Thin-Walled Bar of Variable Cross Section with Different Variants of Ends Fastening (Informftion 1)]. Izvestija vuzov. Stroitel’stvo [News of Institutions of Higher Education. Construction]. 2005, no. 10, pp. 19—25.
  6. Bogdanovich A.U., Kuznetsov I.L. Prodol’noe szhatie tonkostennogo sterzhnja peremennogo sechenija pri razlichnyh variantah zakreplenija torcov [Longitudinal Compression of a Thin-Walled Core of Variable Cross Section with Different Variants of Ends Fastening (Informftion 2)]. Izvestija vuzov. Stroitel’stvo [News of Institutions of Higher Education. Construction]. 2005, no. 11-12, pp. 10—16.
  7. Nicot Francois, Challamel Noel, Lerbet Jean, Prunier Frorent, Darve Felix. Some in-sights into structure instability and the second-order work criterion. International Journal of Solids and Structures. 2012. Vol. 49, no. 1. pp. 132—142.
  8. Aristizabal-Ocha J. Dario. Matrix method for stability and second rigid connections. Engineering Structures. 2012. Vol. 34. pp. 289—302.
  9. TemisYu.M.,Fedorov I.M. Sravnenie metodov analiza ustojchivosti sterzhnej peremennogo sechenija pri nekonservativnom nagruzhenii [Comparing the Methods for Analysing the Stability of Rods of a Variable Cross-section under Non-conservative Loading]. Problems of strength and plasticity [Proceeding sof Nizhni Novgorod University]. 2006, no. 68, pp. 95—106.
  10. Le Grognec Philippe, Nguyen Quang-Hay, Hjiaj Mohammed. Exat buckling solution for two-layer Timoshenko beams with interlayer. International Journal of Solids and Structures. 2012. Vol. 49, ¹ 1. pp. 143—150.
  11. Chepurnenko A.S., Andreev V.I., Yazyev B.M. Energeticheskiy metod pri raschete na ustoychivost’ szhatykh sterzhney s uchetom polzuchesti. [Energy Method of Analysis of Stability of Compressed Rods with Regard for Creeping]. Vestnik MGSU. [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 1, pp.101—108.

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Two-sided evaluations based on the variational formulations of integral equations for the stability of elastic rods

Vestnik MGSU 10/2014
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Мoscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 41-47

The author considers the method of two-sided evaluations in solving the problems of stability of one-span elastic non-uniformly compressed rod with variable longitudinal bending rigidity in case of different classic conditions of fixation of the rod ends. The minimum critical value of the loading parameter for the rod is represented as a problem of calculating minimum value of the functional corresponding to the Euler equation, which is the same as the integral equation for the rod stability. Using the inequalities following from the problem of the best approximation of a Hilbert space element through the basic functions, the author constructs two sequences of functionals, the minimum values of which are the lower evaluations and the upper ones for the required value of the loading parameter. The basic functions here are the derivative forms of the stability loss for a rod with constant cross-section, compressed by longitudinal forces applied at the rod ends. The calculation of the lower bounds value is reduced to the determination of the maximum eigenvalues of block matrices. The elements of the aforesaid matrices are expressed through the integrals of basic functions depending on the type of the fixation of the rod ends. The calculation of the upper bound value is reduced to the determination of the maximum eigenvalue of the matrix, which almost coincides with one of the modular matrices. It is noted that the obtained upper bound evaluations are not worse than the evaluations obtained by the Ritz method with the use of the same basic functions.

DOI: 10.22227/1997-0935.2014.10.41-47

References
  1. Kupavtsev V.V. Variatsionnye formulirovki integral'nogo uravneniya ustoychivosti uprugikh sterzhney [Variational Formulations of the Integral Equation of Stability of Elastic Bars]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2012, no. 9, pp. 137—143. (in Russian)
  2. Rzhanitsyn A.R. Ustoychivost' ravnovesiya uprugikh system [Stability of Equilibrium of Elastic Systems]. Moscow, GITTL Publ., 1955, 475 p. (in Russian)
  3. Alfutov N.A. Osnovy rascheta na ustoychivost' uprugikh system [Fundamentals of the Stability Analysis of the Elastic Systems]. 2-nd edition. Moscow, Mashinostroenie Publ., 1991, 336 p. (in Russian)
  4. Kupavtsev V.V. Bazisnye funktsii metoda dvustoronnikh otsenok v zadachakh ustoychivosti uprugikh neodnorodno-szhatykh sterzhney [Basic Functions for the Method of Two-sided Evaluations in the Problems of Stability of Elastic Non-uniformly Compressed Rods]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 6, pp. 63—70. (in Russian)
  5. Panteleev S.A. Dvustoronnie otsenki v zadachakh ob ustoychivosti szhatykh uprugikh blokov [Bilateral Assessments in the Stability Problem of Compressed Elastic Blocks]. Izvestiya RAN. Mekhanika tverdogo tela [News of the Russian Academy of Sciences. Solid Body Mechanics]. 2010, no. 1, pp. 51—63. (in Russian)
  6. Santos H.A., Gao D.Y. Canonical Dual Finite Element Method for Solving Post-Buckling Problems of a Large Deformation Elastic Beam. International Journal Non-linear Mechanics. 2012, vol. 47, no. 2, pp. 240—247. DOI: http://dx.doi.org/10.1016/j.ijnonlinmec.2011.05.012.
  7. Manchenko M.M. Ustoychivost' i kinematicheskie uravneniya dvizheniya dinamicheski szhatogo sterzhnya [Dynamically Loaded Bar: Stability and Kinematic Equations of Motion]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 6, pp. 71—76. (in Russian)
  8. Bogdanovich A.U., Kuznetsov I.L. Prodol'noe szhatie tonkostennogo sterzhnya peremennogo secheniya pri razlichnykh variantakh zakrepleniya tortsov. Soobshchenie 1 [Longitudinal Compression of a Thin-Walled Bar of Variable Cross Section with Different Variants of Ends Fastening (Information 1)]. Izvestiya vuzov. Stroitel'stvo [News of Institutions of Higher Education. Construction]. 2005, no. 10, pp. 19—25. (in Russian)
  9. Bogdanovich A.U., Kuznetsov I.L. Prodol'noe szhatie tonkostennogo sterzhnya peremennogo secheniya pri razlichnykh variantakh zakrepleniya tortsov. Soobshchenie 2 [Longitudinal Compression of a Thin-Walled Core of Variable Cross Section with Different Variants of Ends Fastening (Information 2)]. Izvestiya vuzov. Stroitel'stvo [News of Institutions of Higher Education. Construction]. 2005, no. 11, pp. 10—16. (in Russian)
  10. Selamet S., Garlock M.E. Predicting the Maximum Compressive Beam Axial During Fire Considering Local Buckling. Journal of Constructional Steel Research. 2012, vol. 71, pp. 189—201. DOI: http://dx.doi.org/10.1016/j.jcsr.2011.09.014.
  11. Vo Thuc P., Thai Huu-Tai. Vibration and Buckling Of Composite Beams Using Refined Shear Deformation Theory. International Journal of Mechanical Sciences. 2012, vol. 62, no. 1, pp. 67—76. DOI: http://dx.doi.org/10.1016/j.ijmecsci.2012.06.001.
  12. Kanno Yoshihiro, Ohsaki Makoto. Optimization-bazed Stability Analysis of Structures under Unilateral Constraints. International Journal for Numerical Methods in Engineering. 2009, vol. 77, no. 1, pp. 90—125.
  13. Doraiswamy Srikrishna, Narayanan Krishna R., Srinivasa Arun R. Finding Minimum Energy configurations for constrained beam buckling problems using the Viterbi algorithm. International Journal of Solids and Structures. 2012, vol. 49, no. 2, pp. 289—297.
  14. Rektoris К. Variational methods in Mathematics, Science and Engineering. Prague, SNTL-Publ., Techn. Liter., 1980. (in Russian)
  15. Kupavtsev V.V. Variatsionnye formulirovki zadach ustoychivosti uprugikh sterzhney cherez izgibayushchie momenty [Variational Formuliations of Stability Problems of Elastic Rods Using Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 285—289. (in Russian)

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Continuation of the solution of an elliptic equation and mathematical tesselations

Vestnik MGSU 10/2014
  • 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; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • 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), .

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)
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