of of

D(x, y) illustrated in Figure I2.9a into rectangular elements, as illustrated in Figure 12.12. The global solution domain D(x, y) is covered by a two-dimensional grid of lines. There are I lines perpendicular to the x axis, which are denoted by the subscript i. There are J lines perpendicular to the y axis, which are denoted by the subscript ). There are (I - 1) x (J - I) elements, which are denoted by the superscript (i, i). Element (i,) starts at node i,) and ends at node i + 1, ) + 1. The grid increments are !'ixi = xi+} - Xi and ~y) = YJ+I - y). _

Let the global exact solution f(x, y) be approximated by the global approximate solution f(x, y), which is the sum of a series of local interpolating polynominals fUJ)(x, y) (i = 1, 2, ... , 1-1,) = 1, 2, ... , J - 1) that are valid within each element. Thus,

I-I J-I f(x, y) = L L fUJ)(x, y)

Let's define the local interpolating polynominal f UJ)(x, y) as a linear bivariate polynomina\. Element (i,) is illustrated in Figure 12.13. Let's use a local coordinate system, where node i,) is at (0.0), node i + 1, ) is at (!'ix, 0), etc. Denote the grid points as 1, 2, 3, and 4. The linear interpolating polynominal,jUJ)(x, y), corresponding to element (i,) is given by

(12.139) where;;, 12, etc., are the values of f(x, y) at nodes I, 2, etc., respectively, and N} (x, y), N2(x, y), etc., are linear interpolating polynominals within element (i,). The interpolating polynominals, N I (x, y), N2(x, y), etc., are called shape functions in the finite element literature. The subscripts of the shape functions denote the node at which the corresponding shape function is equal to unity. The shape function is defined to be zero at the other

j+1

Figure 12.13. Rectangular element (i, j).