Problems of Structural Dynamics with Negative Time

A dynamic problem with negative flow of time is formulated. The conventional equations of motion with the addition of initial conditions are sufficient not only for the analysis of motion of a deformable system under the regular, forward flow of time, but they allow to restore the state of the system for past moments of time. The practical use of solving problems with negative time can be found primarily in testing numerical methods for integrating the equations of motion, since forward and backward algorithms are not identical. The proposed technique of testing numerical methods for solving dynamic problems can be applied virtually to any computational scheme of integrating the equations of motion. Two examples of numerical solution based on explicit computational scheme with Adams extrapolation are presented. The addressed problems deal with the plane deformation state of plates under large displacements. Plate regions are partitioned into triangular finite elements with uniform spacing for spatial meshing. The obtained curvilinear boundaries in this case are stepped. The results of the presented test cases demonstrated good accuracy of the tested method. Problems requiring a large number of integration steps (up to 1 million) were considered, and the system returned to the initial state with high accuracy. The second of the given numerical solutions had a computational scheme of 160 000 finite elements, and the dynamic solution of the problem has a pronounced wave-like behavior. In the examples, data on the recovery of elastic displacement, velocity and stress values are given. The main conclusion of the study is that the proposed technique of control of numerical methods can be effectively applied, especially for problems with wave-like solution properties.