TET10 — 10-node quadratic tetrahedron

The quadratic-tet kernel. Four corner nodes + six mid-edge nodes — 10 total, 30 DOFs per element. Indispensable for unstructured-mesh CAD geometries where a hex mesh isn’t buildable.

• Spec: ELEMENTS.TET10

Shape functions

Lagrange in volume coordinates \((L_0, L_1, L_2, L_3)\) with \(L_0 + L_1 + L_2 + L_3 = 1\):

• Corner nodes (\(i = 0, 1, 2, 3\)):

  \[N^{c}_{i} = L_i\, (2 L_i - 1).\]

• Mid-edge nodes (one per edge \((i, j)\)):

  \[N^{m}_{ij} = 4\, L_i\, L_j.\]

The basis interpolates a quadratic polynomial in \(L_i\) exactly across the tet, satisfies Kronecker- delta at all 10 nodes, and is partition-of-unity to machine precision (verified on every reference-element node by the gallery example).

See TET10 shape-function slices on the unit tetrahedron for contour plots on the \(\xi_3 = 0\) face slice and a verified Kronecker-delta basis-matrix check.

Integration

Keast 4-point rule (Keast 1986), used for both stiffness and mass:

\[\boldsymbol{\xi}_q = (a, a, a, 1 - 3a), \qquad a = \tfrac{5 - \sqrt{5}}{20}, \qquad w_q = \tfrac{1}{24},\]

with \(q = 1, \ldots, 4\) and the four points related by the symmetric permutations of the volume coordinates. Exact for polynomials of degree ≤ 2 — sufficient for the quadratic-tet \(\mathbf{B}^{\!\top}\, \mathbf{C}\, \mathbf{B}\) integrand under a constant-Jacobian map.

See Numerical quadrature for Stroud-rule context.

Verification cross-references

• TET10 — prescribed uniform strain on a 10-node tet — quadratic-tet uniaxial-strain test.

• TET10 reference geometry — corner + mid-edge nodes, Keast points — node layout + 4-point Keast Gauss locations.

Implementation: femorph_solver.elements.tet10.

References

• Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002) Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, Table 6.5-1 (TET10 volume- coordinate shape functions).

• Hughes, T. J. R. (2000) The Finite Element Method — Linear Static and Dynamic Finite Element Analysis, Dover, §3.8 (quadratic tetrahedron).

• Zienkiewicz, O. C. and Taylor, R. L. (2013) The Finite Element Method: Its Basis and Fundamentals, 7th ed., §8.8.2 (TET10 basis on the unit tetrahedron).

• Keast, P. (1986) “Moderate-degree tetrahedral quadrature formulas,” Computer Methods in Applied Mechanics and Engineering 55 (3), 339–348.

• Stroud, A. H. (1971) Approximate Calculation of Multiple Integrals, Prentice-Hall (foundational tetrahedral cubatures).