Solution of Deformable Solid Mechanics Dynamical Problems with Use of Mathematical Modeling by Grid-Characteristic Method

Abstract

The most suitable method for hyperbolic equation systems, e.g., equations describing elastic waves, is the grid-characteristic method. This method allows us to simulate physically correctly wave processes in heterogeneous media and take into consideration boundary conditions and conditions on contact surfaces. Most fully the advantages of the method are for one-dimensional equations, especially in combination with a fixed difference grid, as in conventional grid-based methods. However, in the multidimensional case using the algorithms of splitting with respect to spatial variables, the author has managed to preserve its positive qualities. The use of method of Runge–Kutta type for hyperbolic equations makes it possible to effectively carry out the generalization of methods developed for linear equations, in nonlinear case. Several important problems of exploration seismology, seismic resistance, global seismic studies on Earth and Mars, medical applications, nondestructive testing of railway lines, and other areas of practical application were numerically solved. A significant advantage of the constructed method is the preservation of its stability and precision at the strains of the environment. This article presents the results of numerical solution based on the grid-characteristic method to such problems as modeling elastic–plastic deformation in traumatic brain injury, direct problem of seismology, and impact task.