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A general theory of finite elements. I. Topological considerations
Author(s) -
Oden J. Tinsley
Publication year - 1969
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.1620010209
Subject(s) - mathematics , finite element method , euclidean space , topological space , function space , pure mathematics , dimension (graph theory) , topology (electrical circuits) , hermite interpolation , hermite polynomials , mathematical analysis , algebra over a field , physics , combinatorics , thermodynamics
This paper presents a general theory of finite elements. The concept of finite elements is cast in a general topological framework valid for spaces of finite dimension. It is shown that the idea of finite element models can be developed in higher‐dimensional spaces, independent of specific co‐ordinate systems, for any type of continuous abstract function defined on the space. Generalizations of the familiar Lagrange and Hermite interpolation functions are presented as well as a general statement of the notion of generalized variables and conjugate fields. It is also shown that admissible finite elements can be developed for non‐Euclidean spaces of finite dimension. Topological properties of finite element models are examined in Part I of the paper. Part II is devoted to certain applications.