WEDGE6 formulation contract

This document is the active WP07 formulation contract. It retains the T1-R mathematical vocabulary and verification obligations, and records the Terra-authorized elemental kernel without creating a qualified public claim.

Reference element

The natural domain is r >= 0, s >= 0, r + s <= 1, -1 <= t <= 1. Nodes 1-3 form the lower triangle at t=-1; nodes 4-6 form the upper triangle at t=+1. The shape functions are:

N1 = 0.5 (1-t) (1-r-s)    N2 = 0.5 (1-t) r    N3 = 0.5 (1-t) s
N4 = 0.5 (1+t) (1-r-s)    N5 = 0.5 (1+t) r    N6 = 0.5 (1+t) s

Their derivatives, identities, and the physical mapping are recorded in qualification/0_2_7/wedge6_formulation_contract.json. In particular, x=sum(Ni*x_i), J=dx/d(r,s,t), grad_x(Ni)=J^-T grad_ref(Ni), and the orientation contract is det(J)>0 at all declared validity controls.

The signed volume is the reference integral of det(J). The future 3-D small-strain matrix uses engineering shear order (xx, yy, zz, xy, yz, xz). Stress recovery must state its measure and sample location; extrapolation is never implicit.

Integration contract

The WP07 production rule is a six-point product rule: degree-2, three-point triangle quadrature times two-point Gauss quadrature through the thickness. A Duffy-transformed Gauss 5x5 triangle rule times four-point thickness rule is the independent 100-point verification reference. This is a predeclared candidate used by the technical kernel. Reduced or selective integration is not implemented or qualified.

The rule must be compared on affine and distorted positive-Jacobian prisms for constant strain, stiffness rank, energy and sensitivity. The two integration points reported for the external C3D6 deck are an external fact, not a QF integration prescription.

Ordering and faces

The asymmetric fixture in qualification/0_2_7/wedge6_mapping_fixture.json prevents a permutation from passing by geometric symmetry. The reference face cycles are:

Face	Nodes	Outward direction
TRI bottom	1, 3, 2	negative t
TRI top	4, 5, 6	positive t
QUAD side 1-2	1, 2, 5, 4	negative s
QUAD side 2-3	2, 3, 6, 5	positive r+s
QUAD side 3-1	3, 1, 4, 6	negative r

No automatic node or normal repair is allowed. Each adapter must record any permutation and verify positive Jacobian, coordinates, normal, area, uniform-pressure resultant and resultant moment.

Quality, stiffness and mass

For linear WEDGE6, the validity certificate reduces det(J) to three quadratics in t: at fixed t, the determinant is affine over the reference triangle, so its minimum is at one of the three triangle vertices. Each quadratic is checked at t=-1, t=+1 and any interior stationary point. The minimum is compared with a machine-epsilon-scaled determinant magnitude; this is a roundoff guard, not a qualification cutoff. Volume quadrature points, face centroids and the prism interior centroid remain additional diagnostics. Degenerate, inverted, non-finite or unresolved orientations fail closed. Distortion and conditioning remain diagnostics without a universal cutoff.

WP07 must predeclare tests for symmetry, rank, six rigid-body modes, affine constant strain, tension, compression, shear, bending, distortion, energy and replay. The consistent-mass formula and modal checks are specified for WP10 only; no static evidence promotes a modal route.

External comparison

The primary future observables are displacement, total reaction and strain energy. Stress is comparable only with the same measure, point, frame and sign convention. The external tolerance candidates are recorded as PROPOSED_OWNER_REVIEW in the JSON contract and are not derived from any observed QF result. The current CalculiX/Code_Aster artifacts remain deck validation only; no QF external correlation is issued by WP07.

WP07 registers COMB-WEDGE6-linear_static with qualification state EXPERIMENTAL. Preflight reports EXPERIMENTAL_ROUTE. This gate does not authorize WP08 or later numerical work, and does not add Gmsh, face-load, user-workflow, modal, nonlinear or external-correlation support.