Multiparametric basis functions on the finite element Q12

The paper gives a constructive proof of the existence of multi-parameter bases of a finite element of the serendipitous family. On the example of a planar element with 12 nodes (bicubic interpolation), an infinite set of interpolation polynomials containing from 12 to 16 parameters is constructed for the first time. The basis functions of serendipitous finite elements given in the literature are called standard basis functions. These functions cope well with the problem of isoparametric mapping of a square into an arbitrary quadrilateral, but the interpolation qualities of the standard basis functions are not always perfect. All standard bases of serendipitous finite elements, except the bilinear one, have drawbacks. These include the presence of negative loads in the node-by-node distribution of uniform mass force, multiple zeros at nodes. It is not possible to get rid of these disadvantages within the framework of traditional matrix analysis.