Numerical methodfor solving dynamic problems of the theory of elasticity in the polar coordinate system similar to the finiteelement method

Vestnik MGSU 7/2013
  • Nemchinov Vladimir Valentinovich - Moscow State University of Civil Engineering (MGSU) Candidate of Technical Sciences, Professor, Department of Applied Mechanics and Mathematics, Mytischi Branch; +7 (495) 602-70-29, Moscow State University of Civil Engineering (MGSU), 50 Olimpiyskiy prospekt, Mytischi, Moscow Region, 141006, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Musayev Vyacheslav Kadyr ogly - Moscow State University of Civil Engineering (MGSU) Doctor of Technical Sciences, Professor, Consulting Professor, Mytischi Branch, Moscow State University of Civil Engineering (MGSU), 50 Olimpiyskiy prospekt, Mytischi, Moscow Region, 141006, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 68-76

The authors consider a dynamic problem solving procedure based on the theory of elasticity in the Cartesian coordinate system. This method consists in the development of the pattern of numerical solutions to dynamic elastic problems within any coordinate system and, in particular, in the polar coordinate system. Numerical solutions of dynamic problems within the theory of elasticity are the most accurate ones, if the boundaries of the areas under consideration coincide with the coordinate lines of the selected coordinate system.The first order linear system of differential equations is converted into an implicit difference scheme. The implicit scheme is transformed into the explicit method of numerical solutions. Using the Galerkin method, the authors obtain formulas for the calculation of both the points of the computational domain and the boundary points.Difference ratios similar to those obtained for a discrete rectangular grid and derived in this paper are suitable to design any geometry, which fact significantly increases the value of the methods considered in this paper.As a test case, the problem of diffraction of a longitudinal wave in a circular cavity, where maximum stresses are obtained analytically, was considered by the authors. The proposed method demonstrated sufficient accuracy of calculations and convergence of numerical solutions, depending on the size of discrete steps. The problem of diffraction of longitudinal waves in a circular cavity was taken for example; however, the proposed method is applicable to any problems within any computational domain.The polar coordinate system is the best one for any research into the diffraction of plane longitudinal waves in a circular cavity, since the boundaries of the computational domain coincide with the coordinate lines of the selected system.

DOI: 10.22227/1997-0935.2013.7.68-76

  1. Nemchinov V.B. Dvukhsloynaya raznostnaya skhema chislennogo resheniya ploskikh dinamicheskikh zadach teorii uprugosti [Bilayer Difference Scheme of a Numerical Solution to Two-Dimensional Dynamic Problems of Elasticity]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2012, no. 8, pp. 104—111.
  2. Fletcher K. Chislennye metody na osnove metoda Galerkina [Numerical Methods Based on the Galerkin Method]. Moscow, Mir Publ., 1988, 352 p.
  3. Sekulovich M. Metod konechnykh elementov [Finite Element Method]. Moscow, Stroyizdat Publ., 1993, 664 p.
  4. Musaev V.K. Primenenie metoda konechnykh elementov k resheniyu ploskoy nestatsionarnoy dinamicheskoy zadachi teorii uprugosti [Application of the Finite Element Method to the Plane Non-stationary Dynamic Problem of the Theory of Elasticity]. Mekhanika tverdogo tela [Solid Body Mechanics]. 1980, no. 1, pp. 167—173.
  5. Sabodash P.F., Cherednichenko R.A. Primenenie metoda prostranstvennykh kharakteristik k resheniyu zadach o rasprostranenii voln v uprugoy polupolose [Application of Method of 3D Characteristics to Problems of Propagation of Waves in an Elastic Half-strip]. Izvestiya AN SSSP. Mekhan. tverdogo tela [News of the Academy of Sciences of the USSR. Solid Body Mechanics]. 1972, no. 6, pp. 180—185.
  6. Gernet Kh., Kruze-Paskal’ D. Neustanovivshayasya reaktsiya nakhodyashchegosya v uprugoy srede krugovogo tsilindra proizvol’noy tolshchiny na deystvie ploskoy volny rasshireniya [Unstable Response of an Arbitrary Thickness Circular Cylinder to the Action of a Plane Expansion Wave]. Prikladnaya mekhanika. Trudy amerikanskogo obshchestva inzhenerov-mekhanikov. Ser. E. [Applied Mechanics. Works of the American Society of Mechanical Engineers. Series E.] 1966, vol. 33, no. 3, pp. 48—60.
  7. Bayandin Yu.V., Naimark O.B., Uvarov S.V. Numerical Simulation of Spall Failure in Metals under Shock Compression. AIP Conf. Proc. of the American Physical Society. Topical Group on Shock Compression of Condensed Matter. Nashville, TN, 28 June — 3 July 2009, vol. 1195, pp. 1093—1096.
  8. Burago N.G., Zhuravlev A.B., Nikitin I.S. Models of Multiaxial Fatigue Fracture and Service Life Estimation of Structural Elements. Mechanics of Solids. 2011, vol. 46, no. 6, pp. 828—838.
  9. Li Y., Liu G.R., Zhang G.Y. An Adaptive NS/ES-FEM Approach for Plane Contact Problems Using Triangular Elements. Finite Elem. Anal. Dec. 2011, vol., 47, no. 3, pp. 256—275.


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