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Finite elements beyond Courant's triangulation

Gautam Dasgupta
Organization: Columbia University
Department: Department of Civil Engineering and Engineering Mechanics
Journal / Anthology

Innovation in Mathematics: Proceedings of the Second International Mathematica Symposium
Year: 1997
Page range: 107-114

The key to the success of the finite element method is that the local test functions are based on element geometry alone-- irrespective of the governing field equation. Originally, Courant proposed a scheme of (generalized) triangulation for second order elliptic partial differential equations where the domain in Rn was divided into finite subregions of (n+1)-vertices. Weights for Ritz' linear test functions which identically satisfy homogeneous differential equations were determined by minimizing an energy integral. “Computational experience” led Courant's mathematically well-founded method into two-dimensional quadrilateral subregions. Associated quadratic polynomial interpolants, intended for higher accuracy, do not satisfy the equilibrium equation pointwise or in the strong sense. MacNeal's theorem establishes that such four-node eight degree-of freedom (two-dimensional membrane) elements cannot reproduce known results of elasticity simultaneously for uniform and bending stresses. The author identifies shortcomings germane in commercial finite elements to pointwise violation of equilibrium equation and non-modal solution scheme, and proposes Tessellica which: (a) selects local interpolants that satisfy field equations exactly; (b) “trains” -- in neuro-network sense -- the equation solver to reproduce known analytical elasticity results exactly with Mathematica. Tessellica is then an extension of Courant's idea beyond triangulation. Besides theoretical interest ir promises commercial potentials for high-end applications where the best possible accuracy is desirable, e.g., in design of automobile and aircraft bodies and in bio-engineering analysis and biomedical clinical practices.

*Applied Mathematics > Numerical Methods
*Engineering > Finite Element Methods