Constrained least squares finite element formulation of two-dimensional elastodynamic problems

Abstract

In the present work, a least?squares finite element formulation for two?dimensional elastodynamic problems is developed. This development is based on an earlier formulation reported by Pratap. The earlier formulation was developed by considering only one wave-the P?wave-which finds limited applications. In the present work, both the P?wave and the SV?wave are considered. Unlike the conventional residual methods, this method does not require the approximate function to satisfy the boundary conditions a priori. The approximating function is assumed solely on the basis of the order of the differential equation. The sum of the squares of the errors resulting from the approximate solution is then minimized over the entire domain as well as the boundary in a single functional. However, certain continuity requirements are lost when both the potentials and the normals are taken as the independent degrees of freedom at the nodes. This problem has been addressed by treating the continuity requirements as constraint equations and satisfying them over the entire domain. The residual resulting from these constraint equations is also included in the same least?squares functional. This yields results in good agreement with the analytical solutions, within acceptable numerical errors. A finite element code has been developed. Since the technique incorporates both wave potentials, it can serve as a powerful tool for the non?destructive evaluation of cracked solids.