Modeling three-dimensional elastic wave propagation in circular cylindrical structures using a finite-difference approach.

Wave propagation along circular cylindrical structures is important for nondestructive-testing applications and shocks in tubes. To simulate elastic wave propagation phenomena in such structures the governing equations in cylindrical coordinates are solved numerically. To reduce the required amount of computer memory and the computational time, the stress components are eliminated in the equilibrium equations. In the resulting coupled partial differential equations, in which only the three displacement components are involved, the derivatives with respect to spatial coordinates and time are approximated using second order central differences. This leads to the present new approach, which is both accurate and efficient. In order to obtain a stable scheme the displacements must be allocated on a staggered grid. The von Neumann stability analysis is performed and the result is compared with an existing empirical criterion. Mechanical energies are observed in order to validate the finite-difference code. Since no material damping or energy dissipation is taken into account in the equations of motion, the total energy must remain constant over time. Only negligible variations are observed during long-term simulations. Dispersion relations are used to check the physical behavior of the waves calculated with the proposed finite-difference method: Theoretically calculated curves are compared with values obtained by a spectrum estimation method, applied to the results of a simulation.