(f(x, h(y) + fie-x) + hex) x)]dxdy

Wk(x, y) (k = I, 2, ...) are chosen to be the shape functions N J (x, y), N2(x, y), etc. specified by Eg. (12.142). Let's evaluate Eg. (12.156) for W(x, y) = N1(x, y). From Eg. (12.142a),

Differentiating Eg. (12.157) with respect to x and y gives

(12.157)

and ( IX)(WI) = --+-y t1y t1y (12.158)

Substituting Egs. (12.157) and (12.158) into Eg. (12.156) yields

1(f(x, y)) = - ~2 J10(-1 + ji)[t; (-I + ji) + fi(1 - ji) + h(y) + J4( -ji)] dxdy - t1~2 J10(-1 + x)[fi(-I + x) + fie -x) + hex) +.14(1 - x)]dxdy - J1o(1-x-ji+xji)FdxdY =O (12.159)

Recall that x=x/t1x, and ji=y/t1y. Thus, dx=t1xdx and dy=t1ydji. Thus, Eg. (12.159) can be written as

1(f(x, y)) = J~ [J~ (- ..)t1xdX] t1ydji (12.160) where the integrand of the inner integral, denoted as (...), is obtained from Eg. (12.159). Evaluating the inner integral in Eg. (12.160) gives

(- ..) = -t1x{ ~2 [fi(1 - 2ji+ i) + fi(-I + 2ji - i) +h(-ji +i) +J4(Y - i)] dx - t1xJ~ t1~2 [fi (I - 2x + xl) +fi(x - xl) + f3( -x + x2) + J4(-I + 2X - xl)] dx