Numerical simulation of shock-wave processes in elastic media and structures. Part I: Solving method and algorithms

Abstract

Finite-difference algorithms for solving non-stationary wave problems are presented, which allow to obtain the description of fronts and front zones with a minimal influence of spurious effects of numerical approximation. The principal condition of the construction of calculation algorithms is the convergence of domains of dependence of continual and difference equations. It is shown that the fulfillment of this condition provides a maximally exact wave fronts description. Numerical solutions of several one-dimensional and two-dimensional wave problems are presented. In the second part of the paper, we will give examples of numerical simulation of applied problems of structure dynamics and geodynamics -impact driving of piles into the soil, formation of pendulum waves in a block massif, stress state of a homogeneous massif in the zone of interaction with a punch, fracture of a layered solid under the action of a local pulse, and high-speed penetration of a layered target.Exact calculation of wave fronts and high-frequency disturbances is always of utmost importance for problems of numerical simulation of the dynamics of deformable media and structures. On the one hand, this is due to an attempt to expand the base of test problems formed by analytical solutions, which is an essential aspect of numerical simulation. On the other hand, it is important for the analysis of various applied problems having no analytical solutions, in which the parameters of wave fronts and disturbances in their vicinity may be criterial for the strength and shock resistance estimation. Examples of such problems in mining are shock pulses propagation in block media, dynamic strength of component parts of impact machines, stability of underground structures under explosions and rock bursts, etc. Adequate simulation of discontinuous and high-frequency components of the wave field acquires special importance due to multiple reflections of unit waves from inhomogeneities that are inevitably present in actual media and structures.Apart from scarce and relatively simple models of the dynamics of media and structures, wave processes in structured bodies cannot be described using analytical approaches.Undoubtedly, the idea of adequate numerical simulation of a wave field with discontinuities in inhomogeneous media and compound structures is attractive. Numerical solutions not only essentially supplement the known analytical ones, but also allow to obtain qualitative and quantitative evaluations of the process under study and to explain physical consequences. Besides, numerical