SERENDIPITY FINITE ELEMENTS: GEOMETRIC APPROACHES TO SOLVING THE APPROXIMATION PROBLEM IN THE FINITE ELEMENT METHOD

Abstract

The year 1968 is considered the birth date of serendipity approximations, when I. Ergatoudis, B. Irons, and O. Zienkiewicz demonstrated how a curvilinear finite element can be straightened by means of a coordinate transformation. Today, such a transformation represents a central tool in the finite element method. As is well known, the best boundaries of elemental domains are piecewise polynomial functions, for the same reasons that make them optimal for approximating displacements: they are convenient for computer implementation. It was found that the choice of coordinates can be described by the same class of polynomials from which the trial functions are taken. Such transformations are called isoparametric. It was precisely these transformations that stimulated the emergence of serendipity models in the theory of approximation of functions of two and three variables. Unfortunately, the creators of the serendipity family of finite elements limited themselves to standard models only, which are not free from well-known deficiencies. In terms of their interpolation and computational properties, serendipity elements undoubtedly surpass Lagrange elements. However, the construction of non-standard models free of specific shortcomings proved to be too complex to be solved by traditional methods of matrix algebra. By applying geometric modeling in combination with matrix analysis, the authors develop the theory of multiparametric serendipity approximations using the bicubic serendipity element as an example. The introduction of new ideas and new methods into the theory of serendipity approximations is capable of changing – sometimes radically – certain established notions. The method of interpretations in mathematical modeling involves a special procedure of translating the original problem into another «language» and solving this «different» problem instead of the original one, which makes it possible to explain the «paradoxes» of the standard model.