Investigating the properties of mixed finite elements

Abstract

The paper considers quadrature-cubic interpolation in problems of restoration of functions of two arguments. The properties of the most common finite elements Q12 (Lagrange version) and Q10 (Serendipity version) are investigated as finite elements. For more than fifty years, researchers of the finite element method have known the procedure of converting the Lagrangian model to the serendipity model. But, as is known, not all results when using this procedure satisfy users, especially proponents of physical interpretation. We are talking about the magnitude of the nodal loads of the uniform mass force of the serendipity finite element. Thus, if we consider the finite element Q10, it receives the physical inadequacy of the "spectrum" as an inheritance from the "parent" pair of Lagrangian finite elements Q8 and Q12. In Pascal's scheme, there is also a hidden connection between the finite element Q10 and the finite element Lagrangian Q12. The analysis of the inherited properties suggests that it is fundamentally possible for a Q10 substitute-base to exist in nature with the same local and integral characteristics. It turns out that the search for such a base goes beyond the capabilities of traditional modeling methods. An alternative substitute-basis Q10 was found by nonmatrix condensation of the prototype of element Q10, i.e., by using the Lagrangian model Q12. The universal nature of the non-matrix transformation of Q12 into Q10 opens up the possibility of designing a model series of mixed finite elements with physically adequate spectra of nodal loads.