Existence of Natural Frequencies of Systems With Artificial Restraints and Their Convergence in Asymptotic Modelling

Negative penalty functions in the element‐free Galerkin method

“…In previous works by Ilanko [1][2][3], it has been argued and shown that convergent results can be obtained in certain constrained variational formulations taking positive as well as negative penalty parameters. Furthermore, positive and negative penalty parameters may be used to bracket the exact solution.…”

The use of negative penalty functions in solving partial differential equations

“…It was shown that by using a combination of a pair of positive and negative penalty parameters it is possible to determine the maximum error due to the asymptotic modelling. Justiÿcations for the use of a special case of negative and positive penalty functions, namely artiÿcial restraints with large positive and negative sti ness, have been published recently [2,3]. These are applicable for calculating natural frequencies and critical loads only.…”

Tyings in linear systems of equations modelled with positive and negative penalty functions

“…However, it has recently been shown that negative penalty parameters can also lead to convergent and accurate results. This was first shown 1164 H. ASKES, S. PIERCY AND S. ILANKO by Ilanko for frequency-domain vibration analysis of structural systems [3,4]. More recently, it was established that negative penalty functions can also be used in linear system of equations, see, for instance, [5,6].…”