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High order polynomial elements with isoparametric mapping
Author(s) -
Hoppe Viggo
Publication year - 1980
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.1620151204
Subject(s) - hermite interpolation , mathematics , hermite polynomials , interpolation (computer graphics) , finite element method , degree (music) , degree of a polynomial , polynomial , hermitian matrix , element (criminal law) , algebra over a field , pure mathematics , mathematical analysis , computer science , structural engineering , frame (networking) , engineering , telecommunications , physics , acoustics , law , political science
Finite elements with polynomial interpolation functions of a degree higher than 2 are used comparatively little on large FEM systems, except in shell elements. However, the author has had several years' experience in the use of the so‐called ‘isoparametric, reduced Hermitian element’, 4 Which has behaved excellently in an industrial as well as an educational environment. The reason for the interest in the Hermitian concept is that the overcompatibility of the element reduces the number of unknowns, the solution time and the discontinuities in stresses between elements. Explicit formulae for the family of interpolation polynomials of order q and degree p = 2 q + 1 are given and hierarchical Hermite elements are introduced. The families of Hermite and serendipity elements are isomorphic and the latter may thus be extended to arbitrary high order. For some problems the equidistant node configuration in Lagrange elements of degree 3 and higher is not optimal with respect to smoothness, and a new type of element, the ‘Lobatto element’, is introduced. The methods consistently produce results of an accuracy which is above the requirements of usual engineering applications, but in graphics smoothness of curves is important for a convincing representation. The methods are of particular interest in industries working with structures composed of almost linear materials with well‐known properties.