'BLOWN' MODE AS COGNITIVE MODEL OF BUILDING THE TRIANGLE OF THIRD ORDER

Abstract

Triangles play an extremely important role in the finite element method (FEM). The work is devoted to the investigation of the little-known properties of the 'blown' mode − the internal function of the ten-parameter basis of the polynomial interpolation of a triangular finite element. 'Blown' modes are modes that have non-zero amplitudes inside the element and amplitudes equal to zero on its sides. The internal nodes are undesirable in the finite element method, so they are excluded along with the respective shape functions. The first method of exclusion is given in R. Gallagher's monograph and involves the condensation procedure with respect to the stiffness matrix of the element. The second method is to directly modify the functions of the form in such a way as to eliminate the degrees of freedom associated with the internal nodes. E. Mitchell gives examples of excluding internal nodes on complexes and multiplexes. On the triangular element of the third order the tenth node in the barycenter is eliminated, as a rule, according to the 'recipe' of Ciarlet Raviart. As a result of condensation (reduction) the 'blown' mode remains unaddressed by researchers and is not used in practical calculations. We consider 'blown' mode as an independent mathematical model and by cognitive-graphical analysis we discover little-known features of surface formation and useful analogies. The existence of links of the 'blown' mode with Hermite-Koons polynomials, Gauss quadratures (Bernoulli’s version and Legendre’s version), and Prandtl's problem of prismatic rods torsion is proved. In this work the internal mode of a triangular finite element of the third order, like the rest of the basis functions, was first used to realize the polynomial interpolation of the functions of two arguments under the conditions of the Lagrange hypothesis. Cognitivegraphical analysis of the surface of the 'blown' mode allowed to analyze more deeply all the properties of this model and opened the potential to create new bases and optimize existing ones. We have another confirmation of the well-known fact: mathematics always gives more than expected. There is no doubt that 'blown' mode is a bright example of a cognitive model.