How is the surface Jacobian determinant calculated in FEM?

Question

I am currently trying to evaluate surface forces on a structure. I came across P356 in Bathe's Finite Element Procedures 2014 (example 5.8) in which he related the edge derivative from the global to natural coordinates through this equation:

$dl = det(\mathbf{J}^S) \ dr$

where $r$ is the natural coordinate along the edge of interest. I am not sure how $\mathbf{J}^S$ or its determinant are calculated! I am not sure how to relate that to the edge length for example!

Thanks in advance.

Answer 1

As NNN kindly answered in the comments, it has to do with how the Jacobian is originally defined from the volume/surface/line differential element. More specifically, it's well explained in Section 5.6 P155 in this document. Let me rewrite a simple explanation here:

Let $\vec{r}$ be the position vector of an arbitrary point within a finite element, $\vec{r}$ is defined in terms of the orthogonal basis vectors $\vec{e}_x$, $\vec{e}_y$, and $\vec{e}_z$ of a right-handed Cartesian coordinate system as follows: \begin{equation} \vec{r} := x \vec{e}_x + y \vec{e}_y + z \vec{e}_z \end{equation}

By definition, the differential volume $dV$ can be represented as a scalar triple product in terms of the derivatives of the position vector as follows: \begin{equation} dV := \left( \frac{\partial \vec{r}}{\partial x} dx \times \frac{\partial \vec{r}}{\partial y} dy \right) \cdot \frac{\partial \vec{r}}{\partial z} dz = \left( \vec{e}_x \times \vec{e}_y \right) \cdot \vec{e}_z \ dx \ dy \ dz = dx \ dy \ dz \end{equation}

It's usually easier to transform any integrals from the global coordinates $x,y,z$ to the natural coordinates $\xi, \eta, \zeta$. The partial derivatives can be transformed as follows: \begin{equation} \frac{\partial \vec{r}}{\partial \xi} = \frac{\partial \vec{r}}{\partial x} \frac{\partial x}{\partial \xi} + \frac{\partial \vec{r}}{\partial y} \frac{\partial y}{\partial \xi} + \frac{\partial \vec{r}}{\partial z} \frac{\partial z}{\partial \xi}, \tag{1} \end{equation}

\begin{equation} \frac{\partial \vec{r}}{\partial \eta} = \frac{\partial \vec{r}}{\partial x} \frac{\partial x}{\partial \eta} + \frac{\partial \vec{r}}{\partial y} \frac{\partial y}{\partial \eta} + \frac{\partial \vec{r}}{\partial z} \frac{\partial z}{\partial \eta}, \tag{2} \end{equation}

\begin{equation} \frac{\partial \vec{r}}{\partial \zeta} = \frac{\partial \vec{r}}{\partial x} \frac{\partial x}{\partial \zeta} + \frac{\partial \vec{r}}{\partial y} \frac{\partial y}{\partial \zeta} + \frac{\partial \vec{r}}{\partial z} \frac{\partial z}{\partial \zeta}. \tag{3} \end{equation}

Now, to write the new expression for the differential volume in the natural coordinates: \begin{equation} dV := \left( \frac{\partial \vec{r}}{\partial \xi} d\xi \times \frac{\partial \vec{r}}{\partial \eta} d\eta \right) \cdot \frac{\partial \vec{r}}{\partial \zeta} d\zeta \end{equation}

and using Eqs. 1-3 and representing the cross-dot multiplication as a matrix determinant, we get:

\begin{equation} dV = \begin{vmatrix} \dfrac{\partial x}{\partial \xi} & \dfrac{\partial y}{\partial \xi} & \dfrac{\partial z}{\partial \xi} \\ \dfrac{\partial x}{\partial \eta} & \dfrac{\partial y}{\partial \eta} & \dfrac{\partial z}{\partial \eta} \\ \dfrac{\partial x}{\partial \zeta} & \dfrac{\partial y}{\partial \zeta} & \dfrac{\partial z}{\partial \zeta} \end{vmatrix} d\xi \ d\eta \ d\zeta = |\mathbf{J}| d\xi \ d\eta \ d\zeta \end{equation}

and that's how the Jacobian of the 3D differential volume is derived.

For a differential surface $dS$ (or a differential volume with constant thickness), a similar expression to $dV$ becomes: \begin{equation} dS := \frac{\partial \vec{r}}{\partial \xi} d\xi \times \frac{\partial \vec{r}}{\partial \eta} d\eta \end{equation}

and using the surface version of Eqs. 1-3 (by discarding the third terms in Eqs. 1 and 2 and the entirety of Eq. 3), we get: \begin{equation} dS := \begin{vmatrix} \dfrac{\partial x}{\partial \xi} & \dfrac{\partial y}{\partial \xi} \\ \dfrac{\partial x}{\partial \eta} & \dfrac{\partial y}{\partial \eta} \end{vmatrix} d\xi d\eta = |\mathbf{J}| d\xi d\eta \end{equation}

It can be seen that the surface Jacobian is basically the volume Jacobian by setting the third components perpendicular to the differential surface $dS$ to zero (that's why it's usually written using the same symbol).

Finally, for a differential edge $dL$ (or a differential surface with constant thickness), and assuming $\xi$ is the natural coordinate along that edge and $\eta$ is constant (can be +1 or -1 depending on the location within the finite element): \begin{equation} dL := \left| \frac{\partial \vec{r}}{\partial \xi} \right| d\xi \end{equation}

Going back to Eqs. 1-3, the edge version is obtained by discarding the third term in Eq. 1 and the entirety of Eqs. 2 and 3. Notice that instead of using a determinant, we use the Euclidean norm of the vector from Eq. 1 as follows: \begin{equation} \left| \frac{\partial \vec{r}}{\partial \xi} \right| = \sqrt{ \left( \dfrac{\partial x}{\partial \xi} \right)^2 + \left( \dfrac{\partial y}{\partial \xi} \right)^2 }. \end{equation}