verification: implicit contact patch test and Hertz line contact

Two oracle cases: the contact patch test on non-matching meshes in small and finite sliding,
and Hertz's cylinder on a flat under force control (peak pressure, pressure distribution,
subsurface stress). docs/fem.md describes the formulation, the literature survey its papers;
ROADMAP M7 and parity are updated.

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

docs/fem.md

@@ -408,7 +408,8 @@ analysis stops naming the node, the face and the depth). Planar rigid walls take
 Commands `interaction.contact.pair|general|edit|delete|find`; Abaqus writes `*SURFACE
 INTERACTION` with `*FRICTION` (exponential decay), `*CONTACT PAIR` and general `*CONTACT`, LS-DYNA
 `*CONTACT_AUTOMATIC_SURFACE_TO_SURFACE` and `_SINGLE_SURFACE` (`fs`, `fd`, `dc`, `sfsa`,
-`sfsb`); Nastran and the implicit solver refuse contact by name (implicit contact is M7).
+`sfsb`); Nastran and the linear implicit analyses refuse contact by name; nonlinear static steps
+take contact pairs (next section).
 
 **Geometry** (`fieldcraft_fem::contact`, shared with the implicit solver to come): the closest
 point of a triangle (exact region tests) or of a bilinear quadrilateral (projected Newton with the
@@ -451,6 +452,76 @@ Hertz (peak force +2.3%, duration −0.4% on a graded quarter-sphere mesh), a fr
 by general contact on all its faces, initial penetrations adjusted or refused, the face cap, and
 bit-for-bit results on one and four threads.
 
+## Contact in nonlinear static steps (mortar, augmented Lagrangian)
+
+**Discretisation** (`fieldcraft_solver::contact`): mortar contact, one pass, the pair's
+`secondary` surface the non-mortar side. Each secondary node A carries a pressure p_A and the
+weighted gap g̃_A = Σ ∫ Φ_A (x_m − x_s)·n dA over the overlaps of its faces with the main faces
+facing them, Φ_A the secondary face's own shape functions; ĝ_A = g̃_A / A_A with A_A = ∫ Φ_A over
+the node's faces. The pressure field is interpolated like the displacements, so a uniform
+pressure crosses a non-matching interface as the consistent nodal forces of both meshes: the
+contact patch test. Linear faces (tri3, quad4: tet4, wedge6, hex8); general contact, quadratic
+faces, 2D models and arc-length steps are refused by name.
+
+**Segmentation** (`fieldcraft_fem::mortar`): both faces are projected along the secondary
+face's unit normal onto its plane, the main polygon is clipped by the secondary one
+(Sutherland–Hodgman; a projected quadrilateral that is not convex goes as its two triangles), and
+the overlap's fan of triangles is integrated with Radon's seven-point degree-5 rule. At each point
+both faces' coordinates come from inverting their projected maps (linear, or bilinear by the root
+of its quadratic), so the main point is on the main surface along the normal, warped or not. A
+vertex within 10⁻⁸ of a clip edge's length of its line is on it and never cut: the intersection
+of two nearly coincident edges is ill-conditioned and its derivative swamped by round-off. The
+kernel is written over a generic scalar: `f64` for values, forward-mode `Dual` for gradients and
+`Hyper` (gradient and packed Hessian in the 24 corner coordinates) for the consistent tangent,
+the exact linearisation of the segmentation, normal and projection (`fieldcraft_fem::dual`).
+
+**Enforcement**: augmented Lagrangian. With the multipliers λ_A fixed, p_A = ⟨λ_A − ε_A ĝ_A⟩,
+the derivative of Π = Σ A_A/(2ε_A) ⟨λ_A − ε_A ĝ_A⟩²: internal forces −p_A ∂g̃_A/∂x and the tangent
+Σ_closed (ε_A/A_A) ∂g̃_A ⊗ ∂g̃_A − p_A ∂²g̃_A (the second term the contact's geometric stiffness).
+Once Newton converges an increment, λ_A ← p_A (Uzawa) and the increment is solved again from its
+solution, until every closed node's gap is within 10⁻⁵ of the thickness of the element under it
+and no open node keeps a multiplier (at most 25 augmentations, then a warning). The penalty per
+length is ε_A = s · 10 · 3Ḡ/h (Ḡ the mean shear modulus, h = V/A of the element under the face,
+s the contact's `penalty_scale`): about ten times the local element stiffness, and 3Ḡ rather than
+the constrained modulus keeps it so for nearly incompressible rubber.
+
+**Active set and Newton**: the max() decides at every evaluation which nodes are closed
+(semismooth Newton); an increment whose active set alternates between two states three times is
+cut back (chattering). The tangent's pattern takes the blocks of the closed nodes' face pairs on
+top of the mesh's. With contact, the force scale of the convergence test is that of the
+converged states, the loads and the current state rather than the largest force of every trial
+(a trial can press a body deep into another before the active set settles). The augmentations'
+re-solutions are measured against the increment's start and its first correction; the next
+increment's size follows the increment's own iterations.
+
+**Small and finite sliding**: without nonlinear geometry the face pairs (those within a main
+element's thickness) and their mortar integrals D_ai = ∫ Φ_a N_i, M_aj = ∫ Φ_a N_j are fixed in the
+reference configuration, so g̃_a = n · (Σ M_aj x_j − Σ D_ai x_i) is linear: exact and quadratic-free.
+With nonlinear geometry the pairs are searched (bucket grid of the main faces' boxes grown by
+their depth limit) and integrated on the current configuration at every evaluation; a pair deeper
+than half the main element's thickness has passed through and is left out.
+
+**Initial overclosure**: nodes whose gap starts negative are reported; `adjust` takes the overlap
+out of their weighted gap (an offset: strain-free, nodes are not moved), `refuse` stops naming
+the node and the depth.
+
+**Output**: per secondary node the pressure (its normal contact force over the area it carries,
+∫ Φ_A over the overlaps, which is smoother than the multipliers at the contact's edge), the
+multiplier, the gap ĝ and the status (open, closed); per contact the force on the secondary
+surface, the normal force Σ p_A ∫Φ_A, the deepest penetration and the augmentations.
+`solve.nonlinear` summarises each contact, `solve.probe` gives a secondary node's `contact`
+entries, and the results display draws `contact_pressure`, `contact_gap` and `contact_status`.
+
+**Verification** (`crates/solver/tests/contact.rs`, `tests/oracle/nl-contact-patch`,
+`tests/oracle/nl-hertz-cylinder`): the contact patch test on non-matching hex8 and tet4 meshes
+(uniform σ_zz, pressure equal to it, no lateral stress: to 10⁻⁹ with small sliding, 10⁻⁴ with
+finite sliding, where the gaps the augmentation leaves enter through their geometric terms);
+Hertz's line contact, plane strain (peak pressure within 0.3%, the pressure at two nodes within
+3% and 6%, the vertical stress under the contact within 2%, the half-width bracketed); Hertz's
+sphere on a flat, a quarter model (peak pressure within 4%, contact radius bracketed; heavy);
+Newton's quadratic rate near the solution with finite sliding; initial overclosure adjusted and
+refused; bit-for-bit results on one and four threads; the refusals.
+
 ## 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.
 
-**201 of 207 quantities pass** (6 known failure(s)) in 62 active cases: 53 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
+**222 of 228 quantities pass** (6 known failure(s)) in 64 active cases: 55 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
 
 ## Theory: closed-form solutions, patch tests and mesh invariants
 
@@ -80,6 +80,21 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [lame-cylinder-plane-strain](../tests/oracle/lame-cylinder-plane-strain/) | radial displacement at the bore (10, 0) | 0.0090794 | 0.0090793 | -0.000267% | 0.1% | pass | matches |
 | [lame-cylinder-plane-strain](../tests/oracle/lame-cylinder-plane-strain/) | hoop stress σ_θ = σ_yy at the bore (10, 0) | 166.67 | 166.36 | -0.183% | 1% | pass | matches |
 | [lame-cylinder-plane-strain](../tests/oracle/lame-cylinder-plane-strain/) | axial stress σ_zz = 2νA at the bore (plane strain) | 20 | 20.083 | 0.415% | 1% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's corner element under the interface (element 10) over that of the upper block's corner element over it (element 19) | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's centre element under the interface (element 14) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the lower block's bottom corner element (element 1) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of the upper block's top far corner element (element 26) over that of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_xx of element 14 over its σ_zz (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-6 | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's corner element under the interface (element 10) over that of the upper block's corner element over it (element 19) | 1 | 1 | 0.000151% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's centre element under the interface (element 14) over that of element 19 | 1 | 0.99999 | -0.00129% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the lower block's bottom corner element (element 1) over that of element 19 | 1 | 0.99999 | -0.000942% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_zz of the upper block's top far corner element (element 26) over that of element 19 | 1 | 0.99999 | -0.000872% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: σ_xx of element 14 over its σ_zz (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 2.0000e-4 | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19 | 1 | 0.99998 | -0.00172% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 0.02% | pass | matches |
+| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of element 19 (uniaxial stress E ε = −10 for the 1% squeeze) | -10 | -9.9996 | 0.00426% | 0.01% | pass | matches |
 | [nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/) | radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 | 5 | 5.004 | 0.0793% | 0.5% | pass | matches |
 | [nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/) | radial displacement of the outer wall (20, 0, 0) | 2.9129 | 2.9173 | 0.152% | 0.5% | pass | matches |
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip deflection δ_v/L at P L²/(E I) = 1 | 0.30172 | 0.30226 | 0.179% | 1% | pass | matches |
@@ -90,6 +105,12 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 5 | 0.38763 | 0.38955 | 0.495% | 2% | pass | matches |
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip deflection δ_v/L at P L²/(E I) = 10 | 0.81061 | 0.81421 | 0.444% | 1% | pass | matches |
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 10 | 0.555 | 0.558 | 0.541% | 2% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the largest contact pressure p₀ (at x = 0) | 2884.6 | 2892 | 0.254% | 2% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the force across the contact (half model, 1 mm thick) | 1132.8 | 1133 | 0.0221% | 0.2% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.29594 (node 7) | 2325.1 | 2257.8 | -2.89% | 5% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.36701 (node 8) | 1959 | 2078.4 | 6.09% | 10% | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.63325 (node 11, beyond the half-width b = 0.5) | 0 | ≈ 0 | ≈ 0 | ± 1 | pass | matches |
+| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the vertical stress σ_yy under the centre of contact at depth 0.47127 (node 713) | -2099.2 | -2059.8 | 1.87% | 4% | pass | matches |
 | [nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/) | axial Cauchy stress, stretch λ = 2, ν = 0.4999 | 3.4988 | 3.4988 | 0.000000135% | 1e-6 rel. | pass | matches |
 | [nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/) | lateral displacement at (1, 1, 1): l − 1 | -0.29281 | -0.29281 | < 1e-7% | 1e-6 rel. | pass | matches |
 | [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | axial Cauchy stress at a logarithmic strain of 0.2 | 642.74 | 642.74 | < 1e-7% | 1e-6 rel. | pass | matches |
@@ -259,8 +280,10 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 - **[nafems-le1](../tests/oracle/nafems-le1/)** (benchmark): NAFEMS LE1 elliptic membrane (hex20 plane stress layer). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE1 (target σ_yy = 92.7 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE1: Plane stress elements—elliptic membrane'. Tolerances — syy_D: the target is published to three figures; plane stress modelled by a thin solid layer and the stress extrapolated to the edge add a few tenths of a percent
 - **[nafems-le1-plane-stress](../tests/oracle/nafems-le1-plane-stress/)** (benchmark): NAFEMS LE1 elliptic membrane with 2D plane stress elements (quad8). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE1 (target σ_yy = 92.7 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE1: Plane stress elements—elliptic membrane'. Tolerances — syy_D: the target is published to three figures; the stress extrapolated to the edge on a converged mesh adds a few tenths of a percent
 - **[nafems-le10](../tests/oracle/nafems-le10/)** (benchmark): NAFEMS LE10 thick plate under pressure (hex20). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE10 (target σ_yy = −5.38 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE10: Thick plate under pressure'. Tolerances — syy_D: the target is published to three figures; at D the stress is extrapolated to a corner where the hole edge meets the loaded face, and published converged solutions scatter by 1–2% (Abaqus Benchmarks Guide fine meshes: −5.51 to −5.89)
+- **[nl-contact-patch](../tests/oracle/nl-contact-patch/)** (theory): Contact patch test: two blocks with non-matching meshes in uniaxial compression (implicit mortar contact). Source: The contact patch test: a contact discretisation must transmit a uniform pressure across a non-matching interface exactly; Taylor & Papadopoulos, On a patch test for contact problems in two dimensions, Nonlinear Computational Mechanics, Springer 1991, 690–702; Crisfield, Re-visiting the contact patch test, IJNME 48:435–449, 2000; passed by mortar methods: Puso & Laursen, CMAME 193:601–629, 2004. Uniaxial stress between frictionless symmetry planes is uniform in both blocks. Tolerances — small_lower_corner: small sliding is linear and its mortar integrals exact on these rectangular faces: a uniform stress is the discrete solution up to the convergence tolerance (10⁻¹⁰); small_lower_centre: as for the corner element; small_lower_base: as for the corner element; small_upper_top: as for the corner element; small_lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them; small_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; small_force: the pressure carries the whole load across the interface; finite_lower_corner: finite sliding is exact up to the geometric terms of the gaps the augmentation leaves (10⁻⁵ of the element thickness) and the default convergence tolerances; finite_lower_centre: as for the corner element; finite_lower_base: as for the corner element; finite_upper_top: as for the corner element; finite_lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them; finite_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; finite_force: the pressure carries the whole load across the interface; small_stress: the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness, a few 10⁻⁵ of the squeeze)
 - **[nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/)** (theory): Inflation of a thick neo-Hookean cylinder by a follower pressure (plane strain). Source: Finite inflation of an incompressible neo-Hookean thick-walled tube in plane strain: r² = R² + a² − A², σ_θ − σ_r = μ (λ² − λ⁻²) with λ = r/R, integrated through the wall: p = (μ/2)[(ln λ_a² − λ_a⁻²) − (ln λ_b² − λ_b⁻²)] (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3.2). Tolerances — ua: the closed form is incompressible and our material nearly so (κ/μ = 1000, a volume change of order p/κ ≈ 4·10⁻⁴); hybrid hex20, 6 × 16, follow the wall's strain gradient well inside; ub: as at the bore
 - **[nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/)** (theory): Large-deflection cantilever under a dead tip load against the elastica (solids). Source: The elastica of a cantilever under a vertical end load: K. E. Bisshopp, D. C. Drucker, Large deflection of cantilever beams, Quarterly of Applied Mathematics 3(3):272–275, 1945 (elliptic-integral solution); values computed by integrating θ″ = −(P/EI) cos θ by shooting (our own RK4, 20000 steps), agreeing with the table of K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals, IJNME 17(1):145–153, 1981, to all six digits. Tolerances — v1: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u1: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v2: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u2: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v5: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u5: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v10: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u10: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's
+- **[nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/)** (theory): Hertz line contact: a cylinder pressed on a flat (implicit mortar contact, plane strain). Source: H. Hertz, Über die Berührung fester elastischer Körper, J. reine angew. Math. 92:156–171, 1882; K. L. Johnson, Contact Mechanics, Cambridge University Press 1985, §4.3 (two-dimensional contact of cylindrical bodies, eqs. 4.45–4.49): b = √(4PR/(πE*)), p₀ = 2P/(πb), with 1/E* = 2(1 − ν²)/E for two bodies of one material and R the cylinder's radius on a flat. Tolerances — peak_pressure: p₀ = 2P/(πb) for the load per length P = 2265.57 N/mm, b = 0.5 mm; the contact width spans seven elements of the graded mesh; normal_force: the pressure carries the whole load; the soft springs that hold the indenter before contact take a millionth; pressure_x0296: p₀ √(1 − x²/b²) at the node; the nodal pressure (the node's normal force over the area it carries) averages the distribution over the node's elements; pressure_x0367: as at node 7, nearer the edge, where the elements round off the square-root fall of the pressure to zero at b; pressure_x0633: outside the contact the surfaces are apart: no pressure (an element beyond b, since the edge element carries part of the last closed node's pressure); syy_depth: σ_z = −p₀ / √(1 + z²/b²) on the axis of a half-plane (Johnson 1985, eq. 4.49); the foundation is ten half-widths wide and deep, not a half-plane, and the stress is the nodal average of the elements around the node
 - **[nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/)** (theory): Nearly incompressible hybrid block under prescribed stretch. Source: Uniaxial stress of the decoupled compressible neo-Hookean energy W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)² (P. J. Flory, Trans. Faraday Soc. 57:829–838, 1961): σ = μ J^(−5/3) (λ² − I₁/3) + κ (J − 1) with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.4999: μ = 1.0000667, κ = 5000). Tolerances — sigma: a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding; lateral: a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding
 - **[nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/)** (theory): Uniaxial tension of a J2 bilinear-hardening cube to large strain (implicit, finite strain). Source: J2 plasticity with linear isotropic hardening in uniaxial stress (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) in the logarithmic strain, which Simo's exponential return map (CMAME 99:61–112, 1992) integrates exactly on a proportional path: Kirchhoff stress τ = σ_y + E_t (ε − σ_y/E), Cauchy stress σ = τ/J with ln J = τ/(3K) (Hencky's volume law). Tolerances — sigma: a homogeneous deformation the element represents exactly, reached on a proportional path: the tolerance only admits the Newton tolerance and rounding; plastic_strain: ε_p = (τ − σ_y)/H, as for the stress
 - **[nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/)** (theory): Mooney–Rivlin block in uniaxial, equibiaxial and pure-shear tension. Source: Treloar's homogeneous modes of rubber testing (L. R. G. Treloar, The Physics of Rubber Elasticity, 3rd ed., Oxford 1975, ch. 10) for the decoupled compressible energy W = C₁₀ (Ī₁ − 3) + C₀₁ (Ī₂ − 3) + κ/2 (J − 1)²: principal Cauchy stresses σᵢ = (λᵢ/J) ∂W/∂λᵢ with the free stretch found so the free faces carry no stress (bisection; E = 3, ν = 0.49, C₀₁ = 0.2, C₁₀ = μ/2 − C₀₁). Tolerances — uniaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; uniaxial_lateral: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; equibiaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_xx: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_yy: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding

docs/parity.md

@@ -55,7 +55,7 @@ products are run, no documentation text is copied). Status: **done**, **started*
 | 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 | 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 contact in M7) |
+| 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: frictionless mortar contact pairs in nonlinear static steps by an augmented Lagrangian, friction and 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 |
 | Decks | Abaqus input writer (`deck.write`, flat deck; reports what Abaqus cannot express) | M3 | | done |
@@ -63,7 +63,7 @@ products are run, no documentation text is copied). Status: **done**, **started*
 | Decks | LS-DYNA keyword writer (implicit; explicit in M8) | M3 | | done (implicit static and eigenvalue; beam sections planned) |
 | Decks | Readers keeping unknown keywords | M9 | | planned |
 | Solve | Own linear static and modal solver, verified against analytic solutions | M4 | yes | started (solid elements: tet4, tet10, wedge6, hex8, hex20; 2D plane stress, plane strain and axisymmetric tri3, tri6, quad4 (full, incompatible modes), quad8 (full, reduced); quad4/tri3 MITC shells and Timoshenko beams with profiles, mixed with solids; reference points, kinematic and distributing couplings, moments; centrifugal loads) |
-| Solve | Own nonlinear implicit solver: Newton–Raphson with line search and automatic increments, total Lagrangian solids and 2D elements, mixed u–p at finite strain, follower pressure | M15 | yes | started (solids, plane strain, axisymmetric; St Venant–Kirchhoff, neo-Hookean and J2 plasticity (Simo's exponential return map at finite strain, radial return at small strain, consistent tangents, the return shared with explicit dynamics); verified by the elastica, tube inflation, patch and objectivity tests, uniaxial tension to large strain, Hill's thick cylinder, a cantilever's collapse and the Bauschinger effect; Mooney–Rivlin and Yeoh; thermal strain at finite and small strain; a symmetric indefinite LDLᵀ factorization with inertia, Newton through indefinite tangents (hybrid elements at ν = 0.4999 under prescribed displacement); linear buckling of solids (Euler columns, Bryan's plate) and imperfections from its modes; arc-length with critical points from the inertia (shallow-arch snap-through); shells, implicit contact next) |
+| Solve | Own nonlinear implicit solver: Newton–Raphson with line search and automatic increments, total Lagrangian solids and 2D elements, mixed u–p at finite strain, follower pressure | M15 | yes | started (solids, plane strain, axisymmetric; St Venant–Kirchhoff, neo-Hookean and J2 plasticity (Simo's exponential return map at finite strain, radial return at small strain, consistent tangents, the return shared with explicit dynamics); verified by the elastica, tube inflation, patch and objectivity tests, uniaxial tension to large strain, Hill's thick cylinder, a cantilever's collapse and the Bauschinger effect; frictionless mortar contact (patch test, Hertz); Mooney–Rivlin and Yeoh; thermal strain at finite and small strain; a symmetric indefinite LDLᵀ factorization with inertia, Newton through indefinite tangents (hybrid elements at ν = 0.4999 under prescribed displacement); linear buckling of solids (Euler columns, Bryan's plate) and imperfections from its modes; arc-length with critical points from the inertia (shallow-arch snap-through); shells and implicit friction next) |
 | Solve | Own explicit dynamics solver (central differences, one-point hex8 with hourglass control, nodal-pressure tet4, bulk viscosity, energy balance), verified against analytic wave and impact solutions | M10 | yes | started (X1 core and X2: large deformation, objective rates, J2 and Johnson–Cook plasticity, damage, erosion; X3: deformable contact and friction, verified by bar collision, sliding blocks, the contact patch test and Hertz; hyperelastic materials on the shared material point; opt-in mass scaling) |
 | Results | Deformed shape, contours, mode shapes, probes, reaction sums | M4 | | started (results in the session, `solve.probe`, reaction sums; display planned) |
 | Results | XY and path plots, cuts, display groups, free-body forces, animation | M9 | | planned |

docs/solvers/literature.md

@@ -1299,3 +1299,34 @@ from (cited again in the module docs); docs/fem.md describes the formulation.
   impact of spheres, H. Hertz, J. reine angew. Math. 92:156–171, 1882, and K. L. Johnson,
   *Contact Mechanics*, Cambridge 1985, §11.4; the sliding block, any dynamics text (J. L. Meriam,
   L. G. Kraige, *Engineering Mechanics: Dynamics*, §3/5).
+
+## Part R: Contact in nonlinear static steps (M7), as implemented
+
+The papers `fieldcraft_fem::mortar`, `fieldcraft_fem::dual` and `fieldcraft_solver::contact` are
+written from (cited again in the module docs); docs/fem.md describes the formulation.
+
+- Mortar contact, segment to segment, with nodal weighted gaps and the pressure interpolated by
+  the non-mortar side's shape functions: M. A. Puso, T. A. Laursen, *A mortar segment-to-segment
+  contact method for large deformation solid mechanics*, CMAME 193:601–629, 2004; T. W. McDevitt,
+  T. A. Laursen, *A mortar-finite element formulation for frictional contact problems*, IJNME
+  48:1525–1547, 2000; the segmentation (projection on the non-mortar plane, clipping, integration
+  cells) and the linearisation it needs, A. Popp, M. W. Gee, W. A. Wall, *A finite deformation
+  mortar contact formulation using a primal–dual active set strategy*, IJNME 79:1354–1391, 2009.
+  Chosen over two-pass node-to-surface because mortar passes the contact patch test on
+  non-matching meshes (R. L. Taylor, P. Papadopoulos 1991; M. A. Crisfield 2000, as in Part Q).
+- The consistent tangent by automatic differentiation of the mortar integrals rather than by
+  hand: J. Lengiewicz, J. Korelc, S. Stupkiewicz, *Automation of finite element formulations for
+  large deformation contact problems*, IJNME 85:1252–1279, 2011; forward-mode (truncated Taylor)
+  arithmetic, A. Griewank, A. Walther, *Evaluating Derivatives*, 2nd ed., SIAM 2008.
+- Augmented Lagrangian with Uzawa updates after each converged increment: J. C. Simo, T. A.
+  Laursen, *An augmented Lagrangian treatment of contact problems involving friction*, Comput.
+  Struct. 42(1):97–116, 1992; P. Wriggers, *Computational Contact Mechanics*, 2nd ed., Springer
+  2006, ch. 6 and 10; T. A. Laursen, *Computational Contact and Impact Mechanics*, Springer 2003.
+- 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.
+- Integration of the overlap cells: J. Radon's seven-point degree-5 rule on triangles (A. H.
+  Stroud, *Approximate Calculation of Multiple Integrals*, Prentice-Hall 1971, T2:5-1).
+- Verification: Hertz's contact of cylinders and spheres, H. Hertz, J. reine angew. Math.
+  92:156–171, 1882; K. L. Johnson, *Contact Mechanics*, Cambridge 1985, §4.2–4.3 (eqs. 4.45–4.49
+  for the line contact's pressure and subsurface stresses).