Symbolic Computations in Finite Element Formulation

Use of Mathematica for Convex Polygonal Elements and Concave Quadrilaterals

Gautam Dasgupta

Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027

Abstract

In traditional finite element literature, generalization of Courant's triangulation scheme was carried out (Ref. 1) almost forty years ago but numerical implementation was not possible with conventional procedural languages like FORTRAN or C. Mathematica allowed the development of a single Module[] to compute the shape functions for all n-sided convex polygons. For concave quadrilaterals a projective geometry-based algorithm becomes necessary. All such formulations pass the patch-test (Ref. 2) because the symbolic computation engine permits the employment of the divergence theorem to carry out element level  integration in the closed-form. Finally, the Mathematica code allows material randomness to be incorporated (Ref. 3).

The new RegionFunction provided an elegant way to account for concavity where the shape functions had singularities. All these conceptually necessary mathematical steps can hardly be conceived without Mathematica.

The presentation will include code development and classroom applications. It is anticipated that Mathematica will be employed more frequently in  research and teaching of numerical methods that are used in the field of mechanics of solids and fluids.

References

1. Wachspress, E. L., A Rational Finite Element Basis, Academic Press, 1975
2. Dasgupta, G., Tessellica: A defect-free finite element paradigm,  Logic, History and Educational Computing, Computer Science, Helsinki University of Technology, Helsinki, Finland, 1996, pp. 230-242
3. Dasgupta, G., Stochastic Shape Functions ..., Acta Mechanica, Springer Verlag, 2008, pp. 379-395.