Investigation of Alternative Serendipity Models for Solving the Problem of Torsion of Prismatic Rods

The article discusses the testing of new alternative models of the biquadratic finite element of the serendipity family using the problem of torsion of a non-circular cross-section rod and compares the results obtained with the exact solution. The conversion of the Lagrange model to the serendipity model is undoubtedly a useful procedure that has been known for over fifty years. However, not all results of such a transformation satisfy users, especially supporters of physical interpretations. This concerns the value of nodal loads of uniform force (mass) (load ‘spectrum’). For a long time, it was believed that there was a single basis for each serendipity finite element – a standard one, which was obtained algebraically. Using a new approach that employs a combined algebraic-geometric method for constructing basis functions on serendipity finite elements, it has been possible for the first time to obtain alternative bases with a control parameter on a biquadratic finite element. The presence of a parameter in the basis functions of serendipity finite elements allows optimising the computational qualities of the obtained alternative models. The results show that in the problem of torsion of a square section rod using the finite element method, when using new alternative models of the biquadratic finite element, we can obtain higher accuracy compared to the known standard model of this element. Alternative bases of the biquadratic serendipity finite element also have an advantage over the traditional triangulation procedure, since more triangular finite elements must be used to obtain the specified accuracy.