A variational basis for error analysis in finite element elastodynamic problems

Abstract

In this paper, the variational basis for finite element analysis of elastodynamic problems has been examined using the principle of virtual work. Using the principle of virtual work, two fundamental important theorems on errors in variationally correct formulations in computational elastodynamics have been discussed and illustrated with simple one-dimensional elements. A geometric interpretation of the behavior of these errors in approximate solutions from a variationally correct formulation has been presented using the frequency-error-hyperboloid. It has been shown that derivation of a complete and accurate mathematical description of the nature of errors in free vibration analysis involves a simultaneous consideration of errors in both displacement and strains. This is in sharp contrast to the error analysis in elastostatic problems where the variational basis involves only strains. Furthermore, it has been observed that variationally correct and conforming formulations satisfy the projection theorems that result from the weak forms of elastodynamic problems by virtue of the virtual work principle. These formulations involve consistent mass matrices and yield eigenfrequencies that are always higher than the analytical values, independent of domain discretisation. This is not necessarily true for variationally incorrect lumped mass formulations in which no guarantee of the boundedness of the eigenvalues with respect to the exact ones can be given. With the help of sweep tests, it has been demonstrated that the computed eigenvalues with lumped mass formulations can be higher than, equal to, or lower than the exact values, depending on the finite element mesh.