Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Conjugate approximation functions in finite-element analysis

Authors: H. J. Brauchli and J. T. Oden
Journal: Quart. Appl. Math. 29 (1971), 65-90
MSC: Primary 41.30
DOI: https://doi.org/10.1090/qam/288470
MathSciNet review: 288470
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Abstract: In this paper a theory of conjugate approximations is developed which provides a fundamental basis for most methods of continuous piecewise approximation. It is shown that for a given finite set of base functions used in an approximation there corresponds another set of conjugate functions which play a significant role in approximate methods of analysis. In the case of finite-element approximations, it is shown that the domain of the conjugate functions includes the entire assembly of elements, and, consequently, the established method of computing stresses locally in elements based on displacement approximations is not strictly valid. Indeed, the domain of such ``local'' stress fields is the entire connected system of elements. Procedures for computing derivatives and discrete analogues of linear operators which are consistent with the theory of conjugate functions are also discussed. For a given linear operator equation, the significance of the conjugate approximations in connection with the adjoint problem is also discussed.

References [Enhancements On Off] (What's this?)

  • [1] J. T. Oden, A general theory of finite elements. I: Topological considerations, Internat. J. Numerical Methods Engrg. 2, 205-221 (1969)
  • [2] J. S. Archer, Consistent matrix formulation for structural analysis using finite element techniques, AIAA J. 3, 1910-1918 (1965)
  • [3] Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966

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DOI: https://doi.org/10.1090/qam/288470
Article copyright: © Copyright 1971 American Mathematical Society

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