A numerical comparison between two unconstrained variational formulations

• Published in: Journal of sound and vibration Vol. 72; no. 4; pp. 491 - 505
• Main Authors: Wu, J.J., Simkins, T.E.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 1980
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/0022-460X(80)90360-0

In an effort to relieve the often cumbersome burden of meeting the requirements of the end conditions and to unify the solution formulation for boundary- and initial-value problems, unconstrained variational statements have been introduced in conjunction with some approximate methods. In the case of a boundary value problem, it is shown in this paper that two different variational statements can be established: one is arrived at by the use of the Lagrange multipliers, the other by energy considerations. The numerical convergence of the solutions associated with finite element schemes involving use of one of these two different variational statements is compared with that of the other. In the case of an initial value problem, both formulations can again be established when the adjoint field variable and the adjoint variational statement are introduced. The numerical data presented here indicate that while both methods generate excellent convergent results for the boundary problem, the method of stiff springs yields results which show much better convergence for the initial value problem than those achieved by Lagrange multipliers.