Six-node prism. The reference triangle is crossed with a linear coordinate through thickness.
01 Model and scope
The prism basis is the product of a linear triangle basis and a linear through-thickness basis. The tensor-product quadrature combines three triangle points with two line points; orientation and Jacobian checks remain necessary at the integration points.
Route assumptions and calculation checks
- The production stiffness rule is a three-point triangle rule times two-point Gauss through thickness.
- The bounded static record is for the declared Gmsh Prism 6 and isotropic small-strain scope.
- The modal record is limited to the documented homogeneous consistent-mass route and first three modes.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- (r,s)
- Reference-triangle coordinates.
- Reference coordinate through prism thickness.
- Three lower and three upper prism shape functions.
- Triangle-by-line quadrature product.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Product topology and six shape functions
WEDGE6 is the product of a linear triangle and a linear line. Nodes 1–3 lie on t=−1 and nodes 4–6 on t=+1. Each node has three translations, so the element vector contains 18 DOFs. The triangle coordinates satisfy r≥0, s≥0, r+s≤1 while −1≤t≤1.
The lower/upper node pairing and prism orientation determine the sign and validity of the map.
2 · Prism Jacobian and geometry certificate
The six-node interpolation is isoparametric. At each point, J maps (r,s,t) into physical coordinates; its inverse transpose maps gradients to physical space. The implementation also runs a certified Jacobian geometry check on the triangular-prism mapping, rejects coincident nodes, and requires positive volume.
This certificate is stronger than checking only the six production points, but it remains a statement about the code's declared geometric contract and tolerance.
3 · Six-point production rule and stiffness
Production integration is a three-point triangle rule crossed with two-point Gauss through thickness. Each of the six points has reference weight 1/6. At each point the code builds B, uses det(J), and accumulates the 18×18 stiffness contribution. A richer 100-point Duffy/Gauss rule is retained for verification comparison, not a second default production route.
The production mass is described as exact for the polynomial mass integrand of an affine six-node prism. That statement does not extend to arbitrary warped geometry or every material response.
4 · Verification and route scope
The WP07 kernel checks shape functions, derivatives, orientation, affine strains, rigid modes, energy and distorted positive-Jacobian examples. Static and modal maturity are separate records: the card's bounded evidence is for named Gmsh Prism 6, isotropic small-strain static cases and a limited homogeneous consistent-mass modal route (first three modes). The source kernel's presence does not promote general contact, nonlinear, or mixed-material use.
04 Algorithm and implementation
Source excerpt · v0.2.11
The production rule or the explicit reference rule is selected by name. Each point returns its weight, physical strain matrix, and Jacobian determinant before stiffness assembly.
Functions shown: stiffness
def stiffness(self, coords: np.ndarray, quadrature: str = "production") -> np.ndarray:
stiffness = np.zeros((self.dof_count, self.dof_count), dtype=float)
for _, weight, b_matrix, determinant in self.integration_data(coords, quadrature):
stiffness += weight * determinant * (
b_matrix.T @ self.material.elasticity_matrix @ b_matrix
)
return symmetrize(stiffness)05 Verification and limitations
QUALIFIED_BOUNDED · static and first-three-mode scopes
Prism orientation and affine-strain checks, then only the recorded static/modal cases.
Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.