Discrete Shell Bending

Discrete Shell Bending is a constitutive model for simulating the bending behavior of thin shell structures. This model captures the dihedral angle-based bending energy between adjacent triangular faces in a shell mesh.

Reference:

Discrete Shell

#17 Discrete Shell Bending

For a shell bending element defined by four vertices at positions $\mathbf{x}_0$, $\mathbf{x}_1$, $\mathbf{x}_2$, and $\mathbf{x}_3$, where $(\mathbf{x}_1, \mathbf{x}_2)$ forms the shared edge between two adjacent triangular faces, we define:

Dihedral Angle:

The dihedral angle $\theta$ is the angle between the two triangular faces sharing the edge $(\mathbf{x}_1, \mathbf{x}_2)$. The first triangle is formed by vertices $\mathbf{x}_0$, $\mathbf{x}_1$, $\mathbf{x}_2$, and the second triangle is formed by vertices $\mathbf{x}_1$, $\mathbf{x}_2$, $\mathbf{x}_3$.

Rest Configuration Parameters:

From the reference configuration with rest positions $\bar{\mathbf{x}}_0$, $\bar{\mathbf{x}}_1$, $\bar{\mathbf{x}}_2$, and $\bar{\mathbf{x}}_3$:

Rest Length: $L_0 = \|\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_1\|_2$ is the length of the shared edge in the rest configuration
Average Height: $\bar{h}$ is computed as: $$ \bar{h} = \frac{A}{3 L_0} $$ where $A$ is the average area of the two triangles: $$ A = \frac{1}{2}\left(|(\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_0) \times (\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_0)|_2 + |(\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_0) \times (\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_3)|_2\right) $$
Rest Dihedral Angle: $\bar{\theta}$ is the dihedral angle in the rest configuration

Bending Energy Density

The bending energy density of the discrete shell element is given by:

\[ E = \kappa \frac{(\theta - \bar{\theta})^2 L_0}{\bar{h}} \]

where:

$\kappa$ is the bending stiffness parameter
$\theta$ is the current dihedral angle
$\bar{\theta}$ is the rest dihedral angle
$L_0$ is the rest length of the shared edge
$\bar{h}$ is the average height parameter from the rest configuration

The energy penalizes deviations of the dihedral angle from its rest value, scaled by the edge length and inversely by the average height, which accounts for the local geometry of the shell element.