## DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

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

Vestnik MGSU 7/2013

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

References
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2. Fletcher K. Chislennye metody na osnove metoda Galerkina [Numerical Methods Based on the Galerkin Method]. Moscow, Mir Publ., 1988, 352 p.
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### BILAYER DIFFERENCE SCHEME OF A NUMERICAL SOLUTION TO TWO-DIMENSIONAL DYNAMIC PROBLEMS OF ELASTICITY

Vestnik MGSU 8/2012

Pages 104 - 111

Numerical modeling of dynamic problems of the theory of elasticity remains a relevant task.
A complex network of waves that propagate within solid bodies, including longitudinal, transverse,
conical and surface Rayleigh waves, etc., prevents the separation of wave fronts for modeling purposes.
Therefore, it is required to apply the so-called "pass-through analysis".
The method applied to resolve dynamic problems of the two-dimensional theory of elasticity
employs finite elements to approximate computational domains of complex shapes, whereby the
software calculates the speed and voltage in the medium at each step. Preset boundary conditions
are satisfied precisely.
The resulting method is classified as explicit bilayer difference schemes that form special
relationships at the boundary points.
The method is based on an implicit bilayer time-difference scheme based on a system of
dynamic equations of the theory of elasticity of the first order, which is converted into an explicit
scheme with the help of a Taylor series in time, while basic relations are resolved with the help of
the Galerkin method. The author demonstrates that the speed and voltage are calculated with the
same accuracy as the one provided by the classical finite element method, whereby determination
of stresses has to act as a numerically differentiating displacement.
The author identifies the relations needed to calculate both the internal points of the computational
domain and the boundary points. The author has also analyzed the accuracy and convergence
of the resulting method having completed a numerical simulation of the well-known problem
of diffraction of a longitudinal wave speed in a circular aperture. The problem has an analytical
solution.

DOI: 10.22227/1997-0935.2012.8.104 - 111

References
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2. Klifton R.Dzh. Raznostnyy metod v ploskikh zadachakh dinamicheskoy uprugosti [Difference Method for Plane Problems of Dynamic Elasticity]. Mekhanika [Mechanics]. 1968, no. 1 (107), pp. 103—122.
3. Musaev V.K. Primenenie metoda konechnykh elementov k resheniyu ploskoy nestatsionarnoy dinamicheskoy zadachi teorii uprugosti [Application of the Finite Element Method to Solve a Transient Dynamic Plane Elasticity Problem]. Mekhanika tverdogo tela [Mechanics of Solids]. 1980, no. 1, p. 167.
4. Musaev V.K. Vozdeystvie prodol’noy stupenchatoy volny na podkreplennoe krugloe otverstie v uprugoy srede [Impact of the Longitudinal Steo-shaped Wave on a Supported Circular Hole in an Elastic Medium]. All-Union Conference “Modern Problems of Structural Mechanics and Strength of Aircrafts.” Collected abstracts. Moscow Institute of Aviation, 1983, p. 51.
5. Sabodash P.F, Cherednichenko R.A. Rasprostranenie uprugikh voln v polose, sostavlennoy iz dvukh raznorodnykh materialov [Propagation of Elastic Waves in a Band Composed of Two Dissimilar Materials]. Collected works on “Selected Problems of Applied Mechanics” dedicated to the 60th Anniversary of Academician V.N. Chelomey. Moscow, VINITI, pp. 617—624.
6. Clifnon R.J. A Difference Method for Plane Problems in Dynamic Elasticity. Quart. Appl. Mfth. 1967, vol. 25, no. 1, pp. 97—116.