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An isoparametric Hermitian element for the solution of field problems
Author(s) -
Frind Emil O.
Publication year - 1977
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.1620110604
Subject(s) - hermite polynomials , hermitian matrix , mathematics , element (criminal law) , degrees of freedom (physics and chemistry) , mathematical analysis , field (mathematics) , geometry , finite element method , distortion (music) , hermite interpolation , physics , pure mathematics , amplifier , optoelectronics , cmos , quantum mechanics , political science , law , thermodynamics
Abstract A two‐dimensional isoparametric Hermitian element is investigated for use in potential field problems with irregular geometry. Nodal degrees‐of‐freedom are the potential, the gradients in the directions normal and tangential to the element sides, and the twist. In applying the isoparametric principle a simple approximation is used which in general guarantees C 1 continuity only at the element corners. The present version is restricted to 90‐degree interior angles. The element is applied in the Galerkin solution of the steady‐state continuity equation and several examples are presented. It is found that in cases where the potential field is simple but the element is strongly distorted, gradients obtained from the Hermite are not as accurate as those from the Lagrange‐type cubic element having the same number of degrees‐of‐freedom. When the potential field is complex and the distortion is small, the Hermitian gradients are comparable to those of the Lagrange cubic; with increasing distortion they are comparable at the boundaries but less so in the interior. The errors in the potentials themselves are very similar, both elements giving a high degree of accuracy. Since all of its degrees‐of‐freedom are shared by four elements in the interior of a grid, the Hermite element is approximately 20 per cent more efficient than the Lagrange cubic for all but the smallest grids. Also, since the Hermite uses fewer nodes, data handling is greatly reduced. The element appears to be attractive at least in the area of groundwater flow.