Abstract:

The paper considers mixed finite elements (FE) Q12 (Lagrange version) and Q10 (serendipity version), which implement the most common quadrature-cubic interpolation in the problems of restoring the functions of two arguments. Converting the Lagrange model to the serendipity model is certainly a useful procedure that has been known for more than fifty years. But not all results of such transformation satisfy users, especially supporters of physical interpretations. We are talking about the value of nodal loads of uniform mass force ("spectrum" of loads) of serendipity FE. For example, the Q10 mixed model receives the physical inadequacy of the “spectrum” as an inheritance from the “parent” pair Q8 and Q12. In addition, there is a hidden connection in Pascal's scheme between the serendipity mixed FE (10 nodes) and the Lagrange mixed FE (12 nodes). The standard Q10 base can be found by the inverse matrix method, the non-matrix Taylor method or the direct geometric designing. The analysis of hereditary properties suggests that in nature the existence of Q10 substitute-base with the same local and integral characteristics is truly possible. It turns out that the search for such a base goes beyond the capabilities of traditional modeling methods. An alternative Q10 substitute-basis was found by non-matrix condensation of Q10 element prototype, ie using the Lagrange Q12 model.  The universal nature of the non-matrix transformation of Q12 into Q10 opens up the possibility of constructing a model range of mixed FEs with physically adequate spectra of nodal loads.