Piecewise planar method of bicubic interpolation

Abstract

The article on the example of finite elements of bicubic interpolation presents the key ideas and technology of the piecewise planar method (PPM) for the construction of basic functions.The effcient use of triangular plane fragments characterizes PPM as one of the simple and visual methods of constructive theory of serendipity approximations. A model series (7 models) of Q12 elements with special ”portraits” of zero level lines has been built. Appropriate influence functions (Lagrange coeffcients) for angular and intermediate nodes were obtained. The first attempt to use PPM based on triangular simplexes is to construct triangular finite elements of higher order (complexes). As is known, in these cases, higher order Lagrangian polynomials are obtained by directly multiplying first order polynomials. Generalizing the basic idea of Courant about linear bases was a decisive step in the finite elements technique. The authors will show how PPM works based on the example of bicubic interpolation. A series of computer experiments were carried out to test different formulas for determining the spectrum of nodal loads on Q12 from a single uniform mass force. Typically, such a spectrum is determined by the double integration of serendipity polynomials (basis functions). Authors suggest instead of double integration to use stratified averaging of the surface applicators (rule of 9 applicators). Analysis of the stratified sample of the applicators revealed a simple relationship between the average surface applicator and the applicator in the barycenter base. This frees up the double integration and immediately gives the total body volume between the base and the surface.