A New Weighted Average Method and It's Applications in Finite Element Method

Abstract

In many problems about stationary or time-variant physical fields of 2D plane or 3D space, application of Lagrange's interpolation is very difficult. Because the purpose of interpolation is to predict function value of unknown points according to that of known and limited points, a new method of polynomial interpolation named as weighted average method is suggested and linear equations about weights based on the physical meaning of interpolation is deduced succinctl by this paper. The equations of this method possess the special and uniform format i.e. row vectors of coefficient matrix and that of the right side of the equations have the same format. Therefore, weighted coefficients are easy to be gotten with the aid of Cramer's rule and then interpolation polynomial is obtained easily. Methods suggested by this paper avoid cockamamie steps in process of constructing interpolation basis functions repeatedly and also avoid complex iteratively solving process of linear equations about polynomial coefficients efficiently. Compared with Lagrange or other traditional interpolating methods which rely on seeking interpolating basis functions or solving equations directly, the weighted average method suggested by this paper is both simpler in deduction and more significant in physics. Furthermore, the weighted average method is also applied into shape functions’ derivations of triangular element and quadrilateral isoparametric element so that the correctness of this method is gained verification. At the same time, from view of deduction process, the method of this paper not only possesses simple deducting steps because of no solving linear equations which takes 6 undetermined coefficients as unknown quantities, but also takes on distinct meaning in physics and geometry and so it is easier to make people understand the cause of shape function in finite element method.