Determine Jacobian of the eight-node rectangular element of Fig. P9.13 in terms of the parameter a.
Using the coordinate system (x, y), which coincides with the natural coordinate system (ξ, η), we obtain
which is zero at (i) (ξ, η)=(−1, −1) when a = 0.5 (left quarter point) and (ii) (ξ, η) = (1, −1) when a = −0.5 (right quarter point). The Jacobian is negative when the node is placed inside a quarter point and the nearest corner node.