APPROXIMATIONS ON A FINITE ELEMENT IN THE FORM OF A REGULAR N-GON

Abstract

The serendipity family of elements constitutes a useful class of finite elements. They are widely employed in practical computations, despite being poorly formalized. The main difficulty with serendipity finite elements (SFE) lies in the fact that existing approaches to constructing shape functions (the inverse matrix method, the systematic shape function generation procedure) do not allow researchers to go beyond the framework of standard SFE models. In standard models, the number of parameters of the interpolation polynomial corresponds to the number of nodes on the element boundary. It is well known that standard higher-order finite element models possess certain drawbacks (e.g., negative values of nodal loads, multiple zeros at boundary nodes). Therefore, there is a need to employ alternative methods for constructing shape functions and to improve the existing ones. This study analyzes the possibility of constructing basis functions on a discrete element in the form of an n-gon. A finite element in the form of a regular pentagon is considered, which can be used as a computational template within a circle. So far, the problem of constructing and investigating a unitary basis on a discrete pentagonal element (pentagon) has not been addressed. For constructing bases of discrete elements in the form of n-gons, a geometric method was applied. This method is a modification of the “product of planes” approach, which employs the technique of multiplying equations of planes and second-order surfaces. The new methods substantially simplify the procedure of basis construction (eliminating the need to solve a system of linear algebraic equations of the corresponding order on the element) and make it possible to obtain alternative SFE models. The presence of “extra-nodal” parameters in the models obtained by these new methods allows one to overcome the drawbacks inherent to standard models (such as negative nodal load values). Three systems of basis functions on the pentagon have been constructed using the matrix method, geometric modeling, and a modified “product of planes” method. Further optimization of the properties of basis functions can be achieved by means of weighted averaging of functions. The proposed techniques can be extended to arbitrary n-gons and generalized to the case of spatial discrete elements with bases in the form of regular n-gons.