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Incompressibility and axisymmetry: A modified mixed and penalty formulation
Author(s) -
Jinka Ashoka G. K.,
Lewis Roland W.
Publication year - 1994
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620371002
Subject(s) - rotational symmetry , finite element method , penalty method , compressibility , mathematics , bilinear interpolation , displacement (psychology) , displacement field , cylinder , mathematical analysis , geometry , mechanics , mathematical optimization , physics , structural engineering , engineering , psychology , statistics , psychotherapist
This paper describes a modified penalty and two‐field mixed formulation for the analysis of incompressible axisymmetric field problems, A discussion of the existing difficulties with incompressibility and axisymmetric formulations is covered in detail. To facilitate the exact satisfaction of incompressibility in axisymmetric bodies during finite element modelling procedures, a modified shape function is used in the description of radial displacement. Further, the modification to the radial displacement interpolation function overcomes the lacunae of the existing mixed and penalty formulations by eliminating the bias of the isoparametric element behaviour in a divergence free continuum state. A six‐noded triangular element, with a central bubble function, is considered to amplify the issues connected with incompressibility in axisymmetric geometries. The bilinear displacements, along with three reduced/selective integration rules in the case of the modified penalty method and three discontinuous pressure nodes in the element for the case of the modified mixed method, are shown to be equivalent in the case of a numerical study of the incompressible cylinder expansion problem. The displacements, stresses and pressure values obtained from the modified methods will be shown to be in close agreement with exact solutions available from the literature. The numerical treatment proposed in the present study clearly demonstrates the performance and the need for modified methods in the analysis of incompressible axisymmetric engineering problems.