E.V. Konopatskiy, O.A. Shevchuk, A.A. Krysko
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Abstract: The aim of the study is to develop methods of multidimensional interpolation and approximation as tools for mathematical and computer modeling of the stress-strain state of thin-walled shells of engineering structures. To achieve this goal, we improved the differential equation for modeling the stress-strain state of an elastic cylindrical shell under axisymmetric loading for the numerical analysis of the stress-strain state of a cylindrical tank with geometric form imperfections and obtained its numerical solution using geometric interpolators. Also realized a new approach to taking into account the initial conditions of the differential equation, which is a parallel transfer of the numerical solution to the desired point, the coordinates of which correspond to the initial conditions. The proposed approach for the numerical solution of the differential equations is based on the geometrical theory of multidimensional interpolation realized in the point calculus. To compare the results of modeling the stress-strain state of a storage tank for petroleum products with regard for geometric form imperfections, a reference solution was obtained by approximating the values of displacements from the hydrostatic load with regard for geometric and structural nonlinearity in modeling the stress-strain state in the software package of finite element analysis SCAD. Calculations were performed in accordance with the strength theory of octahedral tangential stresses (Huber-Hancke-Miser energy theory). The proposed approach of using geometrical interpolants as finite superelements that include not only geometrical information but also information on physical parameters can be used for numerical solutions and other differential equations of mathematical modeling of multifactor processes and phenomena. The advantage of this approach is that it eliminates the need to coordinate geometric information in the process of interaction between CAD and FEA systems by analogy with the isogeometric method.
Index terms: geometric interpolant, differential equation, numerical solution, stress-strain state, exploited tank.


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