Error Analysis In Computational Elastodynamics

The Finite Element Method (FEM) is the mathematical tool of the engineers and scientists to determine approximate solutions, in a discretised sense, of the concerned differential equations, which are not always amenable to closed form solutions. In this presentation, the mathematical aspects of this powerful computational tool as applied to the field of elastodynamics have been highlighted, using the first principles of virtual work and energy conservation.

Interesting geometrical patterns arising from the errors in the computational process in finite element elastodynamic problems have been discussed, and suitably illustrated through simple bar and beam elements. The approximate Rayleigh Quotient is interpreted in a geometrically abstract, but elegant fashion. It has been shown how incorporation of variationally incorrect procedures (like mass lumping or reduced integration) in the element formulations leads to the violation of the general rules of virtual work in elastodynamic analysis. The Timoshenko beam element has been chosen for demonstrating the variational incorrectness introduced through the ‘variational crime’ of using reduced integration to avoid locking.