Research of Alternative Models of Serendipity Finite Elements Using Model Problems

Abstract

The finite element method is currently the main method of structural analysis in a number of areas of science and technology. The widespread use of this method is largely due to the simple physical interpretation of its computational operations, great geometric flexibility, and applicability to a wide class of partial differential equations. In the theory of approximation of functions of many variables, the elements of the serendipity family play a very important role. These elements can be built on the basis of Lagrange elements if the internal interpolation nodes are removed. Such extraction does not affect the behavior of the function at the boundary between the elements and at the same time significantly reduces the volume of calculations and the computer memory required for saving information. These functions work well in problems of isoparametric transformation of a square into an arbitrary quadrilateral, but the interpolation qualities of standard bases are not always flawless. All standard bases of serendipitous finite elements, except the bilinear one, have disadvantages. These include the presence of negative loads in the nodal distribution of uniform mass force, as well as multiple zeros in the nodes. The interpolation properties of shape functions play an extremely important role in FEM; therefore, the search for alternative models of finite element approximation that would allow reducing the dimensionality of the problem and improving computational properties is a very relevant issue today. The paper tests alternative bases of the biquadratic serendipity finite element for predicting the quality of the finite element approximation of fields in elliptic-type boundary value problems when this element is used. As a model problem, the Dirichlet problem for the squared Poisson equation was chosen.