Integral transformations for isoparametric quadrilateral elements

Question

Suppose I have a reference quadrilateral on $[-1, -1] \times [-1, 1]$ with reference coordinates $\xi, \eta$ and a mapping to an isoparametric quadrilateral in 'physical space' described by coordinates $x, y$.

For a triangular element, the isoparametric mapping is handled the following way for the global nodal basis functions $\phi_i$ in 'physical space': \begin{align} \begin{bmatrix} \frac{\partial \phi_i}{\partial x} \\ \frac{\partial \phi_i}{\partial y} \end{bmatrix} = \begin{bmatrix} \frac{\partial \xi}{\partial x} & \frac{\partial \eta}{\partial x} \\ \frac{\partial \xi}{\partial y} & \frac{\partial \eta}{\partial y} \\ \end{bmatrix} \begin{bmatrix} \frac{\partial \phi_i}{\partial \xi} \\ \frac{\partial \phi_i}{\partial \eta} \end{bmatrix} = J^{-1} \begin{bmatrix} \frac{\partial \phi_i}{\partial \xi} \\ \frac{\partial \phi_i}{\partial \eta} \end{bmatrix} \end{align}

Via the chain rule, so that I can calculate \begin{align} \frac{\partial \phi_i}{\partial x} &= \frac{\partial \phi_i}{\partial \xi}J^{-1}_{11} + \frac{\partial \phi_i}{\partial \eta}J^{-1}_{12} \\ \frac{\partial \phi_i}{\partial y} &= \frac{\partial \phi_i}{\partial \xi}J^{-1}_{21} + \frac{\partial \phi_i}{\partial \eta}J^{-1}_{22} \end{align}

I would then build $J$ by first constructing

\begin{align} J = \begin{bmatrix} \frac{\partial x}{\partial \xi} & \frac{\partial y}{\partial \xi} \\ \frac{\partial x}{\partial \eta} & \frac{\partial y}{\partial \eta} \\ \end{bmatrix} = \begin{bmatrix} \sum x_j^K\frac{\partial \psi_j}{\partial \xi} & \sum y_j^K\frac{\partial \psi_j}{\partial \xi} \\ \sum x_j^K\frac{\partial \psi_j}{\partial \eta} & \sum y_j^K\frac{\partial \psi_j}{\partial \eta} \end{bmatrix} \end{align}

where $x^K$ are the interpolation points on the 'physical space' element $K$, and $\psi_i$ are the nodal shape functions on the master element. I then invert $J$ to find $J^{-1}$.

Does this procedure also work for calculating $J^{-1}$ for isoparametric quads? Or does the chain rule expression become more complicated / nonlinear?

EDIT: The idea is to be able to calculate integrals like $$ \int_K \nabla \phi_j \cdot \nabla \phi_i \, dx = \int_K \left( \frac{\partial \phi_i}{\partial x} \frac{\partial \phi_j}{\partial x} + \frac{\partial \phi_i}{\partial y} \frac{\partial \phi_j}{\partial y}\right)\, dx $$

Answer 1

The rule is exactly the same. The only difference is that the matrix $J$ still depends on the coordinates $\xi,\eta$ because the functions $\psi_j$ are not linear but bilinear. Other than that, everything remains the same.