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.


  1. Felippa, C. A. “A Study of Optimal Membrane Triangles with Drilling Freedoms.” Computational Methods in Applied Mechanical Engineering. 192, 2125-2168. 2003.
  2. 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:

  1. x-translation
  2. y-translation
  3. 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:

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


$u=$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:

$K = K_b + K_h$