Numerical stability analysis for the explicit high-order finite difference analysis of rotationally symmetric shells

Abstract

A numerical stability analysis has been formulated to accompany the already developed explicit high-order finite difference analysis of rotationally symmetric shells subjected to time-dependent impulsive loadings. This already developed analysis utilizes a constant nodal point spacing for the spatial finite difference mesh, with the governing field differential equations formulated in terms of the transverse, meridional, and circumferential displacements as the fundamental variables. The remaining quantities which enter into the natural boundary conditions at each edge of the shell are incorporated into the complete system of equations by defining those quantities at each boundary in terms of the displacements. Surface loadings and inertia forces in each of the three displacement directions of the shell have been considered in the governing equations. Ordinary finite difference representations are used for the time derivatives. All loadings and dependent variables in the circumferential direction of the shell are expressed in Fourier series expansions. The complete system of equations is solved implicitly for the first time increment, while explicit relations are used to determine the three primary displacements within the boundary edges of the shell for the second and succeeding time increments. Separate implicit solutions at each boundary are then used to determine the remaining unspecified primary variables on and outside the boundaries. Subsequently, the remaining primary variables within the boundary edges of the shell and all secondary variables are determined explicitly. Numerical stability (or instability) of numerical solutions for given choices of spatial and time increments is determined by evaluation of the eigenvalues of the explicit coefficient matrix and comparing the maximum eigenvalue with the requirements of a stability criterion developed before by the author. Solutions for typical shells and loadings together with results of stability analyses have been included, and comparisons of the stability requirements and solutions with the requirements and solutions based upon ordinary spatial finite difference representations are included.