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Fractal approach applied to calculate hour‐glass eigenvalue of elastic square 4‐node quad
Author(s) -
Molenkamp F.,
Lewis E. B.,
Sharma C. B.,
Sellmeijer J. B.
Publication year - 2000
Publication title -
international journal for numerical and analytical methods in geomechanics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.419
H-Index - 91
eISSN - 1096-9853
pISSN - 0363-9061
DOI - 10.1002/1096-9853(20001210)24:14<1049::aid-nag110>3.0.co;2-q
Subject(s) - eigenvalues and eigenvectors , isotropy , square (algebra) , mathematics , fractal , finite element method , mathematical analysis , node (physics) , geometry , boundary (topology) , structural engineering , physics , engineering , quantum mechanics
It is shown that the property of the scale invariance of the eigenvalues and eigenmodes of a finite element can be used as a basis to calculate good approximations to the analytical magnitudes of eigenvalues. This requires the subdivision of the element into a mesh of small elements with the same shape as the large element, the enforcement of the modal boundary displacements of the large element to the mesh of small elements and finally the application of the conditions of both the nodal equilibrium and the equality of the nodal work at both scales. Due to the self‐similarity of the elements at all scales the authors propose to call this method the fractal approach. The method is applied to calculate the hour‐glass eigenvalue of a plane square 4‐node quad for isotropic linear elastic material. The resulting hour‐glass eigenvalue is shown to be a good approximation of the analytical magnitude as derived in a companion paper. Copyright © 2000 John Wiley & Sons, Ltd.