4=∑V%-∑州%=ΣN 2g Σ：y 空N+空+字 元-∑N++g =g+然g X 蜜I 数r VVDO 卸像一 凰骨目国… 影 6 6 Good and bad isoparametric mappings of 4-node quadrilateral from the {ξ, η} reference plane onto the {x, y} physical plane. In formulating element characteristic matrices, various derivatives of the interpolation functions with respect to the global coordinates are required. In isoparametric elements, both element geometry and variation of the interpolation functions are expressed in terms of the natural coordinates of the parent element, so some additional mathematical complication arises. Specifically, we must compute ∂ Ni /∂ x and ∂ Ni /∂ y (and, possibly, higher-order derivatives). Since the interpolation functions are expressed in (r, s) coordinates, we can formally write these derivatives as Jacobian matrix The isoparametric formulation is by no means limited to linear parent elements. Many higher-order isoparametric elements have been formulated and used successfully. Figure above depicts the isoparametric elements corresponding to the six-node triangle and the eight-node rectangle. Owing to the mapping being described by quadratic functions of the parent elements, the resulting elements have curved boundaries, which are also described by quadratic functions of the global coordinates. Such elements can be used to closely approximate irregular boundaries. However, note that curved elements do not, in general, exactly match a specified boundary curve