The 15-node pentahedron introduces mid-side nodes to produce a solid element with quadratic shape functions. This element is significantly more accurate than the linear, 6-node pentahedron, at the cost of computation time.

References

1. Abaqus Theory Manual - Triangular, Tetrahedral, and Wedge Elements

Element Geometry

The 15-node pentahedron is formulated as an isoparametric element such that element geometry is inconsequential. The local isoparametric coordinate axes are $\xi,\eta,$ & $\gamma$. It contains six vertex nodes (that define the physical element) and nine mid-side nodes. The mid-side nodes appear in the stiffness matrix and are added to the analysis model. Thus the total number of degrees of freedom is dramatically increased from the 6-node formulation.

Where:

$0≤\xi≤1$

$0≤\eta≤1$

$-1≤\gamma≤1$

Degrees of Freedom

The 15-node pentahedron formulation contains three degrees-of-freedom (DoFs) at each node:

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

There are 15 nodes defined for this formulation. These nodes are numbered from 1 to 15 in a counter-clockwise direction, bottom surface first, then top surface. Vertex nodes are numbered first, then mid-side nodes. This element formulation does not contain rotational degrees of freedom.

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

$\begin{bmatrix}x_1 & y_1 & z_1 & x_2 & y_2 & z_2 & ..... & x_{15} & y_{15} & z_{15} \end{bmatrix}$

Where:

$x_k=$ X-translation at node $k$