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 $A=A_1+A_2$ is the combined rest area of the two triangles: $$ A = \frac{1}{2}\left(|(\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_0) \times (\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_0)|_2 + |(\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_3) \times (\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_3)|_2\right) $$
Rest Dihedral Angle: $\bar{\theta}$ is the dihedral angle in the rest configuration

Bending Energy

The per-edge bending energy is:

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

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

Because $\bar h=A/(3L_0)$, the reference metric is $L_0/\bar h=3L_0^2/A$. This already contains the complete Discrete Shells geometric weight: the backend does not multiply by $A$ again. Consequently, the metric is invariant under uniform scaling of the rest hinge.

For the formula overload, the mesh stores a one-sided thickness radius $r$. The full material thickness is $h=2r$, and

\[ \kappa=D=\frac{E h^3}{12(1-\nu^2)} =\frac{E(2r)^3}{12(1-\nu^2)}. \]

Attributes

On edges:

bending_stiffness: $\kappa$ in the energy above; it is not a per-area value