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Canonical principles in FEM
Completeness
Continuity
Consistency
Correctness
Correspondence

The C-complementarities in the finite element method (FEM) are five canonical principles, completeness, continuity, consistency, correctness and correspondence, that guide the finite element method. These canonical virtues cannot be achieved at the same time, these are complementarities. While the first four are prescriptive rules to ensure robust elements, the last is a descriptive rule defining exactly how the procedure works from an organising principle like the least action principle. In science, as well as in art (life), one finds that it is not possible to achieve all virtues at the same time. Confucius noticed that life is full of contradictions and conflicts and what is most important is to seek harmony and balance. It is the same with finite element computation.

History

See also: Finite Element Method

During the development of simple finite elements in the late 70's, there were only two canonical concepts governing the crucial discretization step which is the essence of the finite element method. These were the completeness and continuity requirements on the special functions used to model the deformation within a structural region in an approximate (i.e. numerical) way. It was around 1977, the two cardinal principles of FEM, were summarized thus: the piecewise approximations used over element domains should be based on complete polynomials, these should be compatible and therefore should satisfy the continuity of these functions or their derivatives, where required, across inter-element boundaries; these functions should be able to represent states of constant strain in the limit; and finally, these functions should be able to recover strain-free rigid body motions.

Hence it was once believed that elements satisfying the continuity and completeness requirements would indeed, in the limit of mesh refinement, converge to the actual solution without any difficulty. The host of locking problems and deterioration of solution accuracy with geometric distortion established another canonical virtue called consistency in the 80's. The consistency paradigm had more than heuristic appeal; the strain field definitions in the displacement model must maintain a certain consistency to recover the true constraints in a meaningful way. It was necessary to re-derive the assumed strain field from the kinematically derived strain field only in a variationaly correct way and hence another guiding principle in FEM, the correctness paradigm was introduced.

An excellent tradition using the energy theorems, the virtual work or least action principles, originated with the seminal analysis of the Finite Element Method by Strang and Fix and the projection theorems for elastostatics and elastodynamics, known respectively as Theorem 1.1 and Lemma 6.3. Thus if the finite element fields and the exact fields are tracked separately, it is possible to interpret this projection theorem as relating the approximate stresses/strains to the error in the strains. In other words, finite element computation manages the stress prediction in an optimal fashion, simultaneously at the local (element) and global levels, governed strictly by canonical variational principles. The strains computed by the finite element procedure are a variationally correct ‘best approximation’ of the true state of strain, a uniquely defined correspondence between the approximate stress and true stress exists in finite element computation.

References

1. Hu, H. C. (1955), On some variational methods in the theory of elasticity and palsticity, Scientia Sinica, 4, 33-54.
2. Strang, G. and Fix G. F. (1966), Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, New Jersey.
3. Dym, C. L. and I. H. Shames,(1973), Solid Mechanics: A Variational Approach, McGraw-Hill.
4. Washizu, K. (1982), Variational Methods in Elasticity and Plasticity, Pergamon Pr, ISBN 0-08-026723-8.
5. Cook, R. D. (1981), Concepts and applications of Finite Element Analysis John Wiley and Sons, New York.
6. Zienkiewicz, O. C. & Taylor R. L. (1989), The Finite Element Method, London: McGraw-Hill.
7. Prathap, G. (1993), The Finite Element Method in Structural Mechanics, Kluwer Academic Press, Dordrecht.
8. MacNeal, R. H. (1994), Finite Elements: Their Design and Performance, Marcel Dekker: NY, 264.
9. Bathe, K. J. (1996), Finite Element Procedures, Prentice Hall, ISBN 0-13-301458-4.
10. Ramesh Babu, C., Subramanian, G. and Prathap, Gangan (1987), Mechanics of field-consistency in finite element analysis - A penalty function approach, Computers and Structures, 25 (2). pp. 161-173. ISSN 0045-7949 Full text
11. Prathap, G. (1996), Finite Element Analysis and the Stress Correspondence Paradigm, Sadhana, 21,525-546.
12. Mukherjee, S. and Prathap, G. (2001), Analysis of shear locking in Timoshenko beam element using the function space approach, Communications in Numerical Methods in Engineering, Vol. 17, pp 385-393.
13. Reddy, J. N. (2002), Energy Principles and Variational Methods in Applied Mechanics, John Wiley, ISBN 0-471-17985-X
14. Prathap, G. and Mukherjee, S. (2004), Management-by-stress Model of Finite Element Computation, Research Report CM 0405, CSIR Centre for Mathematical Modelling and Computer Simulation, Bangalore, November 2004.[1]

Category:Continuum mechanics Category:Finite_element_method Category:Calculus of variations Category:Numerical differential equations Category:Structural analysis