OPTIMAL SAMPLING POINTS IN TRIANGULAR AND TETRAHEDRAL ELEMENTS FOR STRESS RECOVERY

Abstract

During the hay days of the development of the finite element analysis, in the 1960's and 1970's. when SO many new different kinds of elements were developed, it was observed that stresses at certain points within the elements were more accurate than other points. One of the first attempts to locate these points was taken up by Barlow [I] and thus the Barlow points were born. Prathap [21 came up with the idea of best fit points which seems to look deeper into the heart of the finite element method. But both methods are ingenious in their own respects. For l-D bar. 2-D quadrilaterals and 3-D hexaliedra these methods were effective in finding optimal or super-convergent points and these points are generally taken to be as the Gauss-Legendre sampling points. But for 2-D triangles and 3-D tetrahedra nobody has been able to conic to a consensus on the location of these points. Barlow's method cannot be applied effectively in this case. The best fit method which is considered slightly superior does predict for constant strain triangles, the optimal sampling points. It also gives some points which are good for linear strain triangles. These methods and their results are studied in this work and an attempt is made to locate the optimal sampling points for stress recovery in triangular and tetrahedral elements independently. As the work on this topic progressed, its subtle nature came to light, and it was realized that location of these points to a high precision was difficult in any 2-D and 3-D elements. It became clear that Barlow and best fit points were more of guidelines. These observations have been discussed. Furthermore the suitability of the roots of orthogonal polynomials as possible candidates is also seen