Discrete Mechanical System Research Articles

Iterative procedures for improving penalty function solutions of algebraic systems

AbstractThe standard implementation of the penalty function approach for the treatment of general constraint conditions in discrete systems of equations often leads to computational difficulties as the penalty weights are increased to meet constraint satisfaction tolerances. A family of iterative procedures that converges to the constrained solution for fixed weights is presented. For a discrete mechanical system, these procedures can be physically interpreted as an equilibrium iteration resulting from the appearance of corrective force patterns at the nodes of ‘constraint members’ of constant stiffness. Three forms of the iteration algorithm are studied in detail. Convergence conditions are established and the computational error propagation behaviour of the three forms is analysed. The conclusions are verified by numerical experiments on a model problem. Finally, practical guidelines concerning the implementation of the corrective process in large‐scale finite element codes are offered.

Zur Umkehrung des Satzes von Lagrange über die Stabilität

The theorem of Lagrange-Dirichlet states that the equilibrium of a discrete conservative holonomous mechanical system is stable if the potential energy assumes a minimum in this position. It seems physically obvious that the equilibrium is unstable in case of a maximum of the potential energy. This question was examined by Liapunov and Chetayev, but until now the inversion of the theorem of Lagrange has only been proved under very restrictive conditions.