Hexahedral elements
numgeo provides hexahedral elements with either 8, 20 or 27 nodes:
Figure 1: Hexahedral elements implemented in numgeo.
The elements differ in the number of nodes and the polynomial order of the shape functions used for interpolation.
- 8 nodes: A linear element with nodes only at the eight corners. It's the simplest 3D solid element.
- 20 nodes: A quadratic "serendipity" element. It has nodes at the eight corners and at the midpoint of each of the twelve edges. It offers higher accuracy than the linear H8 element.
- 27 nodes: A quadratic "Lagrange" element. It includes all nodes of the H20 element, plus a node at the center of each of the six faces and one node at the element's geometric center. This is the highest-order hexahedral element in numgeo.
Shape functions
The interpolation within an element is defined by its shape functions, \(N_i\). These functions map the physical element in global \((x, y, z)\) coordinates to a canonical parent element in natural coordinates \((\xi, \eta, \zeta)\), where \(\xi, \eta, \zeta \in [-1, 1]\). The node numbering for the functions below corresponds to the schemes shown in Figure 1.
8-node hexahedron
The standard trilinear shape functions for a hexahedron with 8 nodes (H8) are:
$$
\begin{aligned}
N_1 &= \frac{1}{8} (1 - \xi)(1 - \eta)(1 - \zeta) \\
N_2 &= \frac{1}{8} (1 + \xi)(1 - \eta)(1 - \zeta) \\
N_3 &= \frac{1}{8} (1 + \xi)(1 + \eta)(1 - \zeta) \\
N_4 &= \frac{1}{8} (1 - \xi)(1 + \eta)(1 - \zeta) \\
N_5 &= \frac{1}{8} (1 - \xi)(1 - \eta)(1 + \zeta) \\
N_6 &= \frac{1}{8} (1 + \xi)(1 - \eta)(1 + \zeta) \\
N_7 &= \frac{1}{8} (1 + \xi)(1 + \eta)(1 + \zeta) \\
N_8 &= \frac{1}{8} (1 - \xi)(1 + \eta)(1 + \zeta)
\end{aligned}
$$
20-node hexahedron
The shape functions of 20-node quadratic hexahedron (H20 - Serendipity) are quadratic and are grouped into corner nodes (1-8) and midside nodes (9-20).
-
Corner Nodes (1-8):
\[ \begin{aligned} N_1 &= -\frac{1}{8}(1-\xi)(1-\eta)(1-\zeta)(2+\xi+\eta+\zeta) \\ N_2 &= -\frac{1}{8}(1+\xi)(1-\eta)(1-\zeta)(2-\xi+\eta+\zeta) \\ N_3 &= -\frac{1}{8}(1+\xi)(1+\eta)(1-\zeta)(2-\xi-\eta+\zeta) \\ N_4 &= -\frac{1}{8}(1-\xi)(1+\eta)(1-\zeta)(2+\xi-\eta+\zeta) \\ N_5 &= -\frac{1}{8}(1-\xi)(1-\eta)(1+\zeta)(2+\xi+\eta-\zeta) \\ N_6 &= -\frac{1}{8}(1+\xi)(1-\eta)(1+\zeta)(2-\xi+\eta-\zeta) \\ N_7 &= -\frac{1}{8}(1+\xi)(1+\eta)(1+\zeta)(2-\xi-\eta-\zeta) \\ N_8 &= -\frac{1}{8}(1-\xi)(1+\eta)(1+\zeta)(2+\xi-\eta-\zeta) \end{aligned} \] -
Midside Nodes (9-20):
\[ \begin{aligned} N_9 &= \frac{1}{4}(1-\xi^2)(1-\eta)(1-\zeta) \\ N_{10} &= \frac{1}{4}(1+\xi)(1-\eta^2)(1-\zeta) \\ N_{11} &= \frac{1}{4}(1-\xi^2)(1+\eta)(1-\zeta) \\ N_{12} &= \frac{1}{4}(1-\xi)(1-\eta^2)(1-\zeta) \\ N_{13} &= \frac{1}{4}(1-\xi^2)(1-\eta)(1+\zeta) \\ N_{14} &= \frac{1}{4}(1+\xi)(1-\eta^2)(1+\zeta) \\ N_{15} &= \frac{1}{4}(1-\xi^2)(1+\eta)(1+\zeta) \\ N_{16} &= \frac{1}{4}(1-\xi)(1-\eta^2)(1+\zeta) \\ N_{17} &= \frac{1}{4}(1-\xi)(1-\eta)(1-\zeta^2) \\ N_{18} &= \frac{1}{4}(1+\xi)(1-\eta)(1-\zeta^2) \\ N_{19} &= \frac{1}{4}(1+\xi)(1+\eta)(1-\zeta^2) \\ N_{20} &= \frac{1}{4}(1-\xi)(1+\eta)(1-\zeta^2) \end{aligned} \]
27-node hexahedron
The shape functions of the 27-node bi-quadratic hexahedron (H27 - Lagrange) are formed by the tensor product of 1D quadratic Lagrange polynomials.
-
Corner Nodes (1-8):
\[ \begin{aligned} N_1 &= \frac{1}{8} \xi\eta\zeta(\xi-1)(\eta-1)(\zeta-1) \\ N_2 &= \frac{1}{8} \xi\eta\zeta(\xi+1)(\eta-1)(\zeta-1) \\ N_3 &= \frac{1}{8} \xi\eta\zeta(\xi+1)(\eta+1)(\zeta-1) \\ N_4 &= \frac{1}{8} \xi\eta\zeta(\xi-1)(\eta+1)(\zeta-1) \\ N_5 &= \frac{1}{8} \xi\eta\zeta(\xi-1)(\eta-1)(\zeta+1) \\ N_6 &= \frac{1}{8} \xi\eta\zeta(\xi+1)(\eta-1)(\zeta+1) \\ N_7 &= \frac{1}{8} \xi\eta\zeta(\xi+1)(\eta+1)(\zeta+1) \\ N_8 &= \frac{1}{8} \xi\eta\zeta(\xi-1)(\eta+1)(\zeta+1) \end{aligned} \] -
Midside Nodes (9-20):
\[ \begin{aligned} N_9 &= \frac{1}{4} (1-\xi^2)\eta\zeta(\eta-1)(\zeta-1) \\ N_{10} &= \frac{1}{4} \xi(1-\eta^2)\zeta(\xi+1)(\zeta-1) \\ N_{11} &= \frac{1}{4} (1-\xi^2)\eta\zeta(\eta+1)(\zeta-1) \\ N_{12} &= \frac{1}{4} \xi(1-\eta^2)\zeta(\xi-1)(\zeta-1) \\ N_{13} &= \frac{1}{4} (1-\xi^2)\eta\zeta(\eta-1)(\zeta+1) \\ N_{14} &= \frac{1}{4} \xi(1-\eta^2)\zeta(\xi+1)(\zeta+1) \\ N_{15} &= \frac{1}{4} (1-\xi^2)\eta\zeta(\eta+1)(\zeta+1) \\ N_{16} &= \frac{1}{4} \xi(1-\eta^2)\zeta(\xi-1)(\zeta+1) \\ N_{17} &= \frac{1}{4} \xi\eta(1-\zeta^2)(\xi-1)(\eta-1) \\ N_{18} &= \frac{1}{4} \xi\eta(1-\zeta^2)(\xi+1)(\eta-1) \\ N_{19} &= \frac{1}{4} \xi\eta(1-\zeta^2)(\xi+1)(\eta+1) \\ N_{20} &= \frac{1}{4} \xi\eta(1-\zeta^2)(\xi-1)(\eta+1) \end{aligned} \] -
Face Center Nodes (21-26):
\[ \begin{aligned} N_{21} &= \frac{1}{2}\xi(1-\eta^2)(1-\zeta^2)(\xi-1) \\ N_{22} &= \frac{1}{2}\xi(1-\eta^2)(1-\zeta^2)(\xi+1) \\ N_{23} &= \frac{1}{2}(1-\xi^2)\eta(1-\zeta^2)(\eta-1) \\ N_{24} &= \frac{1}{2}(1-\xi^2)\eta(1-\zeta^2)(\eta+1) \\ N_{25} &= \frac{1}{2}(1-\xi^2)(1-\eta^2)\zeta(\zeta-1) \\ N_{26} &= \frac{1}{2}(1-\xi^2)(1-\eta^2)\zeta(\zeta+1) \end{aligned} \] -
Element Center Node (27):
\[ N_{27} = (1-\xi^2)(1-\eta^2)(1-\zeta^2) \]
Numerical integration
Integrals over the element volume are computed numerically using Gaussian quadrature. The following schemes are implemented in numgeo.
1-point integration
1-point integration, degree of precision = 1
| Integration point | \(\xi\) | \(\eta\) | \(\zeta\) | \(w\) |
|---|---|---|---|---|
| 1 | 0.0 | 0.0 | 0.0 | 8.0 |
8-point integration
8-point integration (\(2\times2\times2\) rule), degree of precision = 3
| Integration point | \(\xi\) | \(\eta\) | \(\zeta\) | \(w\) |
|---|---|---|---|---|
| 1 | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | 1.0 |
| 2 | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | 1.0 |
| 3 | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | 1.0 |
| 4 | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | 1.0 |
| 5 | \(-1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | 1.0 |
| 6 | \(1/\sqrt{3}\) | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | 1.0 |
| 7 | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | 1.0 |
| 8 | \(-1/\sqrt{3}\) | \(1/\sqrt{3}\) | \(1/\sqrt{3}\) | 1.0 |
14-point integration
14-point integration, degree of precision = 5
| Integration point | \(\xi\) | \(\eta\) | \(\zeta\) | \(w\) |
|---|---|---|---|---|
| 1 | -0.758786911 | -0.758786911 | -0.758786911 | 0.335180055 |
| 2 | -0.758786911 | -0.758786911 | 0.758786911 | 0.335180055 |
| 3 | -0.758786911 | 0.758786911 | -0.758786911 | 0.335180055 |
| 4 | -0.758786911 | 0.758786911 | 0.758786911 | 0.335180055 |
| 5 | 0.758786911 | -0.758786911 | -0.758786911 | 0.335180055 |
| 6 | 0.758786911 | -0.758786911 | 0.758786911 | 0.335180055 |
| 7 | 0.758786911 | 0.758786911 | -0.758786911 | 0.335180055 |
| 8 | 0.758786911 | 0.758786911 | 0.758786911 | 0.335180055 |
| 9 | -0.795822426 | 0.0 | 0.0 | 0.886426593 |
| 10 | 0.795822426 | 0.0 | 0.0 | 0.886426593 |
| 11 | 0.0 | -0.795822426 | 0.0 | 0.886426593 |
| 12 | 0.0 | 0.795822426 | 0.0 | 0.886426593 |
| 13 | 0.0 | 0.0 | -0.795822426 | 0.886426593 |
| 14 | 0.0 | 0.0 | 0.795822426 | 0.886426593 |
15-point integration
15-point integration (special nodal scheme)
| Integration point | \(\xi\) | \(\eta\) | \(\zeta\) | \(w\) |
|---|---|---|---|---|
| 1 | -1.0 | -1.0 | -1.0 | 0.5 |
| 2 | -1.0 | -1.0 | 1.0 | 0.5 |
| 3 | -1.0 | 1.0 | -1.0 | 0.5 |
| 4 | -1.0 | 1.0 | 1.0 | 0.5 |
| 5 | 1.0 | -1.0 | -1.0 | 0.5 |
| 6 | 1.0 | -1.0 | 1.0 | 0.5 |
| 7 | 1.0 | 1.0 | -1.0 | 0.5 |
| 8 | 1.0 | 1.0 | 1.0 | 0.5 |
| 9 | -1.0 | 0.0 | 0.0 | 1/6 |
| 10 | 1.0 | 0.0 | 0.0 | 1/6 |
| 11 | 0.0 | -1.0 | 0.0 | 1/6 |
| 12 | 0.0 | 1.0 | 0.0 | 1/6 |
| 13 | 0.0 | 0.0 | -1.0 | 1/6 |
| 14 | 0.0 | 0.0 | 1.0 | 1/6 |
| 15 | 0.0 | 0.0 | 0.0 | 2.0 |
27-point integration
27-point integration (\(3\times3\times3\) rule), degree of precision = 5
Let \(a = \sqrt{3/5} \approx 0.774596669\)
| Integration point | \(\xi\) | \(\eta\) | \(\zeta\) | \(w\) |
|---|---|---|---|---|
| 1 | \(-a\) | \(-a\) | \(-a\) | \(125/729\) |
| 2 | \(0\) | \(-a\) | \(-a\) | \(200/729\) |
| 3 | \(a\) | \(-a\) | \(-a\) | \(125/729\) |
| 4 | \(-a\) | \(0\) | \(-a\) | \(200/729\) |
| 5 | \(0\) | \(0\) | \(-a\) | \(320/729\) |
| 6 | \(a\) | \(0\) | \(-a\) | \(200/729\) |
| 7 | \(-a\) | \(a\) | \(-a\) | \(125/729\) |
| 8 | \(0\) | \(a\) | \(-a\) | \(200/729\) |
| 9 | \(a\) | \(a\) | \(-a\) | \(125/729\) |
| 10 | \(-a\) | \(-a\) | \(0\) | \(200/729\) |
| 11 | \(0\) | \(-a\) | \(0\) | \(320/729\) |
| 12 | \(a\) | \(-a\) | \(0\) | \(200/729\) |
| 13 | \(-a\) | \(0\) | \(0\) | \(320/729\) |
| 14 | \(0\) | \(0\) | \(0\) | \(512/729\) |
| 15 | \(a\) | \(0\) | \(0\) | \(320/729\) |
| 16 | \(-a\) | \(a\) | \(0\) | \(200/729\) |
| 17 | \(0\) | \(a\) | \(0\) | \(320/729\) |
| 18 | \(a\) | \(a\) | \(0\) | \(200/729\) |
| 19 | \(-a\) | \(-a\) | \(a\) | \(125/729\) |
| 20 | \(0\) | \(-a\) | \(a\) | \(200/729\) |
| 21 | \(a\) | \(-a\) | \(a\) | \(125/729\) |
| 22 | \(-a\) | \(0\) | \(a\) | \(200/729\) |
| 23 | \(0\) | \(0\) | \(a\) | \(320/729\) |
| 24 | \(a\) | \(0\) | \(a\) | \(200/729\) |
| 25 | \(-a\) | \(a\) | \(a\) | \(125/729\) |
| 26 | \(0\) | \(a\) | \(a\) | \(200/729\) |
| 27 | \(a\) | \(a\) | \(a\) | \(125/729\) |
General remarks
- 8-node hexahedra elements: 1-point integration is a reduced integration scheme prone to hourglassing and requires the use of stabilisation methods. The standard full integration is the 8-point (\(2\times2\times2\)) rule.
- 20-node hexahedra elements: The 8-point rule acts as a reduced integration scheme, hourglass modes are not likely to propagate through the mesh, stabilisation is not required in most cases.
- 27-node hexahedra elements: The 27-point (\(3\times3\times3\)) rule is the standard scheme for full integration, no reduced-integration rule exists.