Somenath Mukherjee

Conservation of Best-Fit Paradigm at an Element Level

The projection theorem describing finite element analysis shows that the strains/stresses computed by the displacement finite element procedure are a best approximation of the true strains/stresses at a global level. For simple linear elastostatics problems this best-fit paradigm holds at an element level, too. But it has been observed that in a certain class of problems where artificial stiffening takes place due to discretization, the finite element solution deviates from the best-fit solution. In this paper we have generalized the projection theorem at the element level and shown that even for such problems the finite element model continues to perform its computation in the best-fit manner to a modified system, so that now the FEM strain in every element is a best-fit not to the original strain but to a modified strain. The analytical expressions presented here have been substantiated by actual finite element results from the relevant computer codes.

Examining the best-fit paradigm for FEM at element level

Interestingly, an esoteric branch of pure mathematics called “functional analysis”, more general and profound than variational calculus and originally developed by mathematicians, can be employed to explain clearly how the finite element machinery works. In a very abstract way, finite element results can be portrayed as “shadows” or orthogonal projections on predetermined function subspaces of the analytical results, known or unknown. Herein lies the philosophy of the finite element method. In the present work, an effort has been made to validate this important aspect of the finite element method at the element level through a particular differential equation representing a specific case of equilibrium