The Optimal Membrane Triangle (OPT) element is an isoparametric element and is formulated in the natural triangular coordinate system, $\xi$ and $\eta$. This formulation is analytically integrated, thus it does not require Gauss integration.

## References

- Felippa, C. A. “A Study of Optimal Membrane Triangles with Drilling Freedoms.” Computational Methods in Applied Mechanical Engineering. 192, 2125-2168. 2003.
- Paknahad, M. & Noorzaei, J. “Formulation, Computation, and Application of Optimal Membrane Triangle Element with Drilling Degrees of Freedom.” International Journal of Engineering and Technology. Volume 4, No. 1, 2007. Pages 95-105.

## Degrees of Freedom

The OPT membrane formulation contains 3 degrees-of-freedom (DoFs) at each node:

- x-translation
- y-translation
- z-rotation

There are three nodes defined for this formulation. These nodes are numbered from 1 to 3 in a counter-clockwise direction.

The local stiffness matrix is formulated with the following degree-of-freedom order:

$\begin{bmatrix}u_1 & v_1 & \theta_1 & u_2 & v_2 & \theta_2 & u_2 & v_2 & \theta_3 \end{bmatrix}$

Where:

$u=$Local x-translation

$v=$Local y-translation

$\theta=$Rotation about the local z-axis

The subscript indicates the node number.

## Stiffness Matrix Formulation

The full stiffness matrix is the sum of a basic and a higher order stiffness matrix:

$K = K_b + K_h$