# Integral transformations for isoparametric quadrilateral elements

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 $$\int_K \nabla \phi_j \cdot \nabla \phi_i \, dx = \int_K \left( \frac{\partial \phi_i}{\partial x} \frac{\partial \phi_j}{\partial x} + \frac{\partial \phi_i}{\partial y} \frac{\partial \phi_j}{\partial y}\right)\, dx$$

## 1 Answer

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.

• Of course the functions on "triangles" are also generically nonlinear for orders greater than one, it's just that people tend to use low order triangular meshes. Commented Apr 13, 2018 at 16:54
• Yes, that is correct. Commented Apr 13, 2018 at 17:40