A construction algorithm for full parametric analytical solutions in the basic mixed problem of elastostatics for~the~simply connected body

V. B. Pen'kov ; O. S. Novikova ; L. Levina

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 3, pp. 586-598

Résumé

Using analytical solutions to analyze the state of the bodies at the research and engineering calculations provides computing resources. We propose a methodology for structuring full parametric solutions to the problems of mathematical physics, including the basic mixed problem of elastostatic. The tool is a relatively new energy method of boundary states based on computer algebra. The method is based on the concept of state of the medium, isomorphism of Hilbert spaces of internal and boundary states of the body. The method is self-sufficient in the sense that, in principle, does not require comparison of the solution of test problems with those constructed by other methods. For inclusion in the solution in an explicit form of the medium constants we recommend saving computing resources method of boundary states with perturbations in which the direct method is combined with approach to A. Poincare. To explicitly include in the decision parameters the boundary conditions we suggested the technology of the reference solutions. Its effectiveness is demonstrated on a concrete example the basic mixed problem of elastostatic. The object of research is a limited simply connected body whose boundary is divided into three sections. At each site held individual method of parameterization of the points of the border: polar, cylindrical, spherical coordinate systems. The calculations are made using the computer algebra of the system “Mathematica” and demonstrated the effectiveness of the developed methodology to achieve this goal. The sequence of steps leading to guaranteed achievement of goal is described. The decision of a concrete task is made. Its results are presented in explicit analytical form containing all the parameters of the boundary value problem of elasticity theory and illustrated graphically after calculation by the analytic solution for a concrete set of parameter values.

Détail

Keywords: method of boundary states, method of boundary states with perturbations, full parametric analytical solutions, basic mixed problem, elastostatics, computer algebra.