PIECEWISE-PLANAR MODELING OF BASES OF MIXED SERENDYPITY ELEMENTS

Abstract

The first models of serendipity finite elements had the same number of boundary nodes in the Ox and Oy directions. Q8 (biquadratic interpolation) and Q12 (bicubic interpolation) elements are the most widespread in practical calculations. These elements are quite suitable and convenient for the tasks of restoring functions in an isotropic environment. Mixed models of serendipity elements are required for problems in an orthotropic environment. As an example of a mixed model we analyze the serendipity element Q10 (quadratic-cubic interpolation). In the direction of the Ox axis the function changes according to the law of the cubic parabola, and along the Oy axis - according to the law of the quadratic parabola. The paper considers classical and non-traditional methods of constructing the bases of a mixed finite element Q10, which consists of elements: Q8 and Q12. As expected, the classical approaches (inverse matrix method and non-matrix Taylor method) showed that the mixed model Q10 inherits the imperfections of the 'ingredients' Q8 and Q12. We are talking about the physical inadequacy of the spectra of equivalent nodal loads from a unit mass force. The standard Q10 model has negative loads in the carrier corner nodes. This unnatural phenomenon of 'gravitational repulsion' was called the paradox of Zienkiewicz, who in 1971 first drew attention to the undesirable feature of standard serendipity FEs. According to Zienkiewicz this imperfection cannot be eliminated, it should be accepted. The paper shows that there are alternatives. A simple and visual method of geometric modeling is proposed for constructing mathematically grounded and physically adequate bases of the Q10 element. The algorithm uses only fragments of planes. Portraits of zero-level lines contain only segments of straight lines. Construction begins with such portraits. It remains to perform the procedure of Wachspress - product of planes. Portraits of zero-level lines significantly simplify the cognitive-graphic analysis of base surfaces contour. The authors deliberately constructed two additional incompatible models of the Q10 element, which successfully passed piecewise testing.