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

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On the use of polar coordinate system in the projective graphic drawings

Vestnik MGSU 11/2016

Pages 124-131

Projective graphics is a polyhedra simulation method, which is based on the use of trace diagrams of initial polyhedron. Previously developed computer software allows using Cartesian coordinates. In some cases it is advisable to use polar coordinate system for description of projective graphics drawings. Using the example of icosahedron the authors analyzed the advantages of using projective graphics drawings in the polar coordinate system. The transition to the polar coordinate system is a tool that allows using certain patterns of projective graphics drawings in the process of calculation. When using polar coordinate system the search of Polar correspondence for the directs is simplified. In order to analyze the two lines in the polar coordinate system it is enough to compare the corresponding coefficients of the equations of these lines. The authors consider a diagram of the icosahedron in polar coordinates, and a corresponding fragment of calculation program in the Mathematica system. Some examples of forming based on icosahedrons are offered. Optimization of computer programs using polar coordinate system will simplifies the calculations of projective graphics drawings, accelerates the process of constructing three-dimensional models, which expand the possibilities of selecting original solutions. Finally, the authors conclude that it is appropriate to use the polar coordinate system only in the construction of projective graphics diagrams of the planes system having rich symmetry. All Platonic and Archimedean solids, Catalan solid possess this property.

DOI: 10.22227/1997-0935.2016.11.124-131