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Determine Jacobian of the eight-node rectangular element of Fig. P9.13 in terms of the parameter a.

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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.