verification: the tie patch test

An oracle case of uniaxial tension across a tie of non-matching meshes; the tie formulation
in docs/fem.md and the literature survey; ROADMAP M7 and parity updated.

Co-Authored-By: Claude Opus 5.5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo

docs/fem.md

@@ -544,6 +544,30 @@ with small sliding and 3% with finite sliding, where the block turns a little, a
 slipping; `tests/oracle/nl-friction-sliding-blocks`); initial overclosure adjusted and
 refused; bit-for-bit results on one and four threads; the refusals.
 
+## Ties (non-matching meshes)
+
+**Model** (`fieldcraft_model::tie`): a tie names a `main` and a `secondary` surface (sets of
+element faces) and an optional position tolerance; commands `interaction.tie|tie.edit|tie.delete`.
+Abaqus writes `*TIE, ADJUST=NO` (secondary surface first, `POSITION TOLERANCE` when given),
+LS-DYNA `*CONTACT_TIED_SURFACE_TO_SURFACE` between segment sets; Nastran refuses ties by name.
+
+**Constraints** (`fieldcraft_solver::tie`): dual mortar. Each secondary face's dual shape
+functions ψ_a = Σ_b A_ab N_b, A = D M⁻¹ (M_ab = ∫ N_a N_b, D = diag ∫ N_a), are biorthogonal to its
+own, so the secondary mortar matrix is diagonal and each secondary node becomes the weighted
+average of the main nodes under it, u_A = Σ_J (M_AJ/Σ_K M_AK) u_J with M_AJ = ∫ ψ_A N_J over the
+overlaps of the reference configuration (`fieldcraft_fem::mortar`). The rows go to the constraint
+elimination like a coupling's, the secondary node the intended dependent, in linear static,
+frequency, buckling and nonlinear steps (small-displacement constraints). Dividing by the sum
+keeps rigid motions exact at a surface's edge where a face is only partly covered. Main faces
+count when the secondary face overlaps them along its normal with a mean gap within the position
+tolerance (default half the main element's thickness); uncovered secondary nodes stay free with a
+warning. Heat transfer and explicit dynamics refuse ties by name.
+
+**Verification** (`crates/solver/tests/contact.rs`, `tests/oracle/tie-patch`): a linear field
+across non-matching hex8/tet4 meshes reproduced to round-off (to 10⁻⁶ on distorted quadrilateral
+faces, where the overlap integration is accurate to its degree) in linear and nonlinear static
+steps; uniaxial tension across a tie, σ_zz = Eε in every element to 10⁻⁹.
+
 ## Hyperelastic materials in explicit dynamics
 
 Hyperelastic materials (neo-Hookean, Mooney–Rivlin, Yeoh) use the shared finite-strain material

docs/oracle.md

@@ -2,7 +2,7 @@
 
 Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the policy is [verification.md](verification.md). Every case is our own model, built with FieldCraft commands from the problem's published description, and compared with the reference value its `expected.json` cites, within the stated tolerance and for the stated reason. Accepted results are frozen in each case's `fixture.json`: a later change that moves one beyond its band (10⁻⁹ of its size) fails CI even while the reference still passes. No reference value was produced with a commercial product.
 
-**225 of 231 quantities pass** (6 known failure(s)) in 65 active cases: 56 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
+**232 of 238 quantities pass** (6 known failure(s)) in 66 active cases: 57 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
 
 ## Theory: closed-form solutions, patch tests and mesh invariants
 
@@ -211,6 +211,13 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [rotating-disc-plane-stress](../tests/oracle/rotating-disc-plane-stress/) | hoop stress σ_θ = σ_yy at the bore (20, 0) | 65.312 | 65.188 | -0.19% | 1% | pass | matches |
 | [rotating-disc-plane-stress](../tests/oracle/rotating-disc-plane-stress/) | radial stress σ_r = σ_xx at r = √(ab), its maximum | 20.724 | 20.683 | -0.2% | 1% | pass | matches |
 | [rotating-disc-plane-stress](../tests/oracle/rotating-disc-plane-stress/) | hoop stress σ_θ = σ_yy at the rim (100, 0) | 16.328 | 16.352 | 0.149% | 1% | pass | matches |
+| [tie-patch](../tests/oracle/tie-patch/) | σ_zz of the lower block's corner element under the tie (element 10) | 10 | 10 | < 1e-7% | 1e-9 rel. | pass | matches |
+| [tie-patch](../tests/oracle/tie-patch/) | σ_zz of the lower block's centre element under the tie (element 14) | 10 | 10 | < 1e-7% | 1e-9 rel. | pass | matches |
+| [tie-patch](../tests/oracle/tie-patch/) | σ_zz of the lower block's bottom corner element (element 1) | 10 | 10 | < 1e-7% | 1e-9 rel. | pass | matches |
+| [tie-patch](../tests/oracle/tie-patch/) | σ_zz of the upper block's corner element over the tie (element 19) | 10 | 10 | < 1e-7% | 1e-9 rel. | pass | matches |
+| [tie-patch](../tests/oracle/tie-patch/) | σ_zz of the upper block's top far corner element (element 26) | 10 | 10 | < 1e-7% | 1e-9 rel. | pass | matches |
+| [tie-patch](../tests/oracle/tie-patch/) | σ_xx of element 14 (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
+| [tie-patch](../tests/oracle/tie-patch/) | u_z at the tied secondary node (0.5, 0.5, 1) | 0.01 | 0.01 | < 1e-7% | 1e-9 rel. | pass | matches |
 | [tube-torsion-coupling](../tests/oracle/tube-torsion-coupling/) | rotation of the reference point about z | 0.056071 | 0.056071 | 0.000619% | 0.5% | pass | matches |
 | [tube-torsion-coupling](../tests/oracle/tube-torsion-coupling/) | surface shear stress τ_θz = σ_yz at (10, 0, 20) | 1078.3 | 1081.4 | 0.288% | 1% | pass | matches |
 
@@ -311,6 +318,7 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 - **[rotating-cylinder-axisymmetric](../tests/oracle/rotating-cylinder-axisymmetric/)** (theory): Long rotating solid cylinder, axisymmetric. Source: Timoshenko and Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970, §32 (long rotating cylinder, ε_z = 0). Tolerances — radial_axis: stress extrapolated to the axis, where σ_r = σ_θ: under 1% with sixteen quadratic elements across; hoop_rim: stress extrapolated to the surface: under 1%
 - **[rotating-disc-plane-stress](../tests/oracle/rotating-disc-plane-stress/)** (theory): Rotating disc with a central hole, plane stress. Source: Timoshenko and Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970, §32 (rotating disc of uniform thickness with a hole). Tolerances — hoop_bore: stress extrapolated to the bore surface, where the a²b²/r² term is steepest: 16 radial quadratic elements graded towards the bore leave about 0.2% (8 uniform: −2.4%; 32 graded: −0.05%); radial_max: the nearest node to √(ab) is within half an element of it, where σ_r is stationary (an error of order (h/r)², under 0.5%), and nodal averaging adds a few tenths of a percent; hoop_rim: as at the bore
 - **[scordelis-lo-roof-solid](../tests/oracle/scordelis-lo-roof-solid/)** (benchmark): Scordelis–Lo roof with one layer of hex20 reduced-integration solids. Source: MacNeal and Harder, 'A proposed standard set of problems to test finite element accuracy', Finite Elements in Analysis and Design 1 (1985) 3–20: Fig. 8 (p. 8) geometry, material and load; Table 5(a) (p. 11) reference displacement 0.3024; Belytschko et al., Computer Methods in Applied Mechanics and Engineering 51 (1985) 221–258 (shell obstacle course). Tolerances — uz: MacNeal and Harder grade an element A within 2% of the theoretical value on this prescribed coarse mesh (p. 14); MacNeal and Harder normalise by 0.3024 (the deep-shell value quoted in their [5] is 0.3086), and a solid model of R/t = 100 differs from shell theory by O(t/R)
+- **[tie-patch](../tests/oracle/tie-patch/)** (theory): Tie patch test: two blocks with non-matching meshes tied together in uniaxial tension. Source: The patch test for mesh tying: a tie across non-matching meshes must transmit a constant stress exactly; B. I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal. 38(3):989–1012, 2000; M. A. Puso, A 3D mortar method for solid mechanics, IJNME 59:315–336, 2004. Uniaxial stress between frictionless symmetry planes is uniform in both blocks. Tolerances — szz_10: uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes; szz_14: uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes; szz_1: uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes; szz_19: uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes; szz_26: uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes; sxx_14: the tie must not hold the lateral contraction of either mesh; uz_interface: the tied node follows the uniform strain field
 - **[tube-torsion-coupling](../tests/oracle/tube-torsion-coupling/)** (theory): Torsion of a tube through a kinematically coupled reference point. Source: Saint-Venant torsion of a circular tube, φ = TL/(GJ), τ = Tr/J with J = π(b⁴ − a⁴)/2; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §104. Tolerances — twist: Saint-Venant torsion of a circular tube is exact in elasticity; the rigid end coupling only stiffens a short zone near the tip; quadratic elements with 16 divisions around are within 0.1%; shear: stress extrapolated to the outer surface and averaged where two blocks meet: about 0.3% with four radial quadratic elements (two gave 1.1%)
 
 ## Pending cases

docs/parity.md

@@ -54,7 +54,7 @@ products are run, no documentation text is copied). Status: **done**, **started*
 | Model | Reference points, kinematic and distributing coupling | M3 | | done |
 | Model | Steps: linear static, frequency; output presets | M3 | | done |
 | Model | Steps: nonlinear static, explicit dynamics, heat transfer; amplitudes | M7–M8, M15 | | started (nonlinear static with or without nonlinear geometry, increments, tolerances and a load path (cyclic loading), written to Abaqus `*STEP, NLGEOM` and LS-DYNA implicit (the load path as LS-DYNA's curve; refused by Abaqus for now), refused by Nastran; explicit dynamics; heat transfer; amplitudes per load planned) |
-| Interaction | Tie between non-matching meshes | M7 | | planned |
+| Interaction | Tie between non-matching meshes | M7 | | done (`interaction.tie`: dual mortar constraints in the implicit analyses, the tie patch test; Abaqus `*TIE`, LS-DYNA tied contact) |
 | Interaction | Contact (general and pairs), friction, automatic pair detection | M7, M10 | | started (explicit: pairs, general and self contact, segment to segment with a penalty that keeps the time increment, Coulomb friction with exponential decay, friction on rigid walls, `interaction.contact.find`, Abaqus and LS-DYNA contact keywords; implicit: mortar contact pairs in nonlinear static steps by an augmented Lagrangian with Coulomb friction, self contact next) |
 | Interaction | Springs and dashpots (ground or two-point; DOF, vector or line), point masses with rotary inertia | M4 | | done (implicit; explicit translational; all three decks with what each cannot take reported) |
 | Interaction | Bolt pretension, connectors | M7 | | planned |

docs/solvers/literature.md

@@ -1329,6 +1329,11 @@ written from (cited again in the module docs); docs/fem.md describes the formula
   deformations*, CMAME 193:4891–4913, 2004; C. A. Coulomb's law, *Théorie des machines simples*,
   1785. The tangent is symmetrised like the follower pressure's (the non-symmetric coupling to
   the pressure is left out; the residual is exact).
+- Ties by dual mortar constraints: B. I. Wohlmuth, *A mortar finite element method using dual
+  spaces for the Lagrange multiplier*, SIAM J. Numer. Anal. 38(3):989–1012, 2000; A. Popp, B. I.
+  Wohlmuth, M. W. Gee, W. A. Wall, *Dual quadratic mortar finite element methods for 3D finite
+  deformation contact*, SIAM J. Sci. Comput. 34(4):B421–B446, 2012; mesh tying in solid
+  mechanics, M. A. Puso, *A 3D mortar method for solid mechanics*, IJNME 59:315–336, 2004.
 - The active set inside Newton (the max() of the regularised pressure, a semismooth Newton
   method): P. Alart, A. Curnier, *A mixed formulation for frictional contact problems prone to
   Newton like solution methods*, CMAME 92:353–375, 1991; Popp et al. 2009.