Research of the accuracy of the modified method of lines in the calculation of axisymmetric bodies

Dmytro Levkivsky, Kostiantyn Kaverin, Yuliia Sovych


This paper shows the application of a modified method of line for determining the stress-strain state of bodies of rotation under the static load. Differential equations of the theory of elasticity recorded in the cylindrical coordinate system are used for this purpose. Given the axial symmetry, the problem is reduced to plane and considered in the coordinate system , circular coordinates of the desired functions do not change. In the first stage of the method, the dimensionality of initial differential equations and boundary conditions is reduced by the Bubnov-Petrov projection method by  coordinate, using local basis functions. As a result, the system of equations is reduced to a system of plain first-order differential equations that depend on the  coordinate. On the second stage of the method, the reduced system of equations and boundary conditions are solved by S.K. Godunov`s numerical method of discrete orthogonalization in combination with the Runge-Kutta method of Merson.

The study revealed all the main features of reducing initial differential equations and boundary conditions. The limits at which the solutions give an approximate result to the analytical value (classical plate theory) are determined. The example of massive bodies shows that its accuracy is not lower than the finite element method. The results obtained in this work are fundamental for further implementation of MML for calculation of thermal elasticity, dynamics problems (in two- and three-dimensional formulation) for objects pertaining to bodies of rotation.

The accuracy study was performed on the example of an axisymmetric annular plate of different thicknesses. The results are compared with values of the theory of thin plates and the finite element method.


thick plates; axisymmetric bodies; theory of elasticity; projection method; modified method of lines; local basis functions; method discrete orthogonalization S.K. Godunov; boundary conditions


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