“BLOWN” MODE OF A SQUARE FINITE ELEMENT: A COGNITIVE AND GRAPHICAL ANALYSIS

Abstract

In the problems of restoring functions of two arguments, the main tool is the ancient method of sections. It is traditionally believed that horizontal sections of surfaces (level lines) are more important. In our analysis the level lines have receded into the background. More informative vertical sections formed a strange chain of prominent scientists, such as Bernoulli, Lagrange, Legendre, Leonardo da Vinci, Gauss, Arnold. The golden proportion has step by step returned the nodes of the exotic cubature of trigonometric origin to the traditionally familiar points discovered by Bernoulli and Legendre. It turns out that regardless of the stereometry of the mode Gauss-Léjeandre cubatures are the best for calculating its volume. In this paper we analyze the geometric features and little-known properties of the “blown” mode models of polynomial and trigonometric origin. In English-language sources the “blown” mode refers to a basic unimodal surface associated with the central interpolation node of a square or triangular finite element. This surface is found in problems of restoring functions of two arguments and resembles a soap film. The French engineers prefer the name “pile of sand”. Comparison of the stereometric characteristics of the models illustrates an interesting example of "soft" and “hard” mathematical modeling (in V. Arnold’s terminology). The method of sections of a polynomial surface provides a simple way to determine the nodes of the Gauss-Lejeune (Bernoulli) quadrature. In the case of the trigonometric mode the usual arrangement of the quadrature nodes is disturbed (the “hard” model), and the volume of the “blown” mode unjustifiably increases. The exact value of the volume is restored using the “golden” proportion. In fact the work continues and significantly complements the theme of the “blown” mode of the third-order triangular finite element. Now the soap film covers a square-shaped finite element. Two canonical square-carriers are considered.