PIECE-PLANAR APPROXIMATION ALGORITHM ON POLYGONAL DOMAINS

Abstract

The article deals with the piecewise planar approximation, which was first proposed by R. Courant, initiating the development of the finite element method (FEM). Further generalization of Courant’s basic idea about the simplest basis functions became a decisive step in the FEM technique. The construction of simple elements is useful in itself, but it is even more significant that these elements can convincingly illustrate deep connections between polynomial interpolation on a finite element and probability theory. The paper shows an algorithm for constructing higher-order finite functions on square and triangle-shaped elements using the so-called “half-cap” function. In each triangulation triangle, we implement the idea of randomizing the Courant function, which can be obtained as a geometric probability in each triangle of the Courant cell. The drawback of the standard model is the physical inadequacy of the nodal distribution of uniform mass force (negativism of some nodal loads). One of the founders of the finite element method, O. Zenkevych, described such a distribution as unnatural, despite its theoretical validity. O. Zenkevych’s experience with standard bases of square finite elements of higher orders convinced him that a theoretically justified distribution of nodal loads (even if it is unnatural) guarantees greater accuracy than a uniform (or any other) distribution based on engineering intuition. Therefore, it can be concluded that optimal bases are those bases that implement theoretically and physically justified distributions of nodal loads. The paper shows that this drawback arises when choosing a method for determining basis functions using matrix algebra. Non-matrix methods for constructing basis functions are one of the ways to eliminate “negativism” in the nodal load distributions of higher-order models. As a result, an alternative model of a bicubic finite element of the serendipity family (3rd-order element) was obtained. Courant’s profound and fruitful idea of piecewise-planar approximation of finite functions has received a simple probabilistic interpretation. The proposed algorithm for piecewise-planar approximation on two-dimensional polygonal domains provides a clear and convenient method for constructing alternative bases, which can be generalized to spatial discrete elements.