explicit: deformable contact and friction, verified by bars, sliding blocks, the patch test and Hertz

Segment-to-segment penalty contact between element faces: each secondary
face and each main face facing it are projected onto the secondary face's
plane, split into triangles, clipped (Sutherland–Hodgman) and integrated
with a three-point rule per sub-triangle (Puso–Laursen segmentation, the
penalty at the integration points as Fischer–Wriggers), so a uniform
pressure gives the consistent nodal forces on both meshes and the contact
patch test passes on non-matching meshes. Every contact runs both ways at
half the penalty: pairs with each surface as the secondary, general and
self contact face against face. A bucket grid of face boxes every
increment, faces integrated in parallel and their forces added in face
order: the same numbers on any thread count.

The penalty per area ε = s γ min(m/A)/Δt² comes from the lumped masses,
each node's directional area and the stable increment, bounded by
Gershgorin's row sums so contact never shortens the increment. Coulomb
friction with exponential decay by a regularised return map at the
secondary nodes; rigid walls take Coulomb impulses. Contact work and
friction dissipation join the energy balance; each contact reports its
peak force, first and last contact, deepest penetration, friction and the
nodes adjusted at the start (initial penetrations adjusted strain-free or
refused with the node, face and depth); a cap on contact faces.

Verified (crates/explicit/tests/contact.rs, tests/oracle/explicit-*):
colliding bars (2L/c +1.2%, ρcAv₀/2, KE kept to 0.005%, contact energy
back to 8e-5 of it), blocks sliding to a stop on a rigid wall and on a
plate under constant and decaying friction (0.005–0.2%), the patch test
(σ_zz uniform to 0.06%), two spheres colliding against Hertz (peak +2.3%,
duration −0.4%; the oracle case waits for a graded sphere mesher), a free
body untouched by general contact, determinism across threads. Oracle
measures gain `over` (ratios). Existing fixtures do not move.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

tests/oracle/explicit-sliding-block-contact/README.md

@@ -0,0 +1,8 @@
+# Block sliding to a stop on a deformable plate (explicit contact, decaying friction)
+
+The block of `explicit-sliding-block-wall` (1 × 1 × 0.5, 4 × 4 × 2 hex8, E = 10⁵, ν = 0.3,
+ρ = 1) rests on a plate 4.5 × 2 × 0.5 of 9 × 4 × 1 hex8 held at its base, the two meshes not
+matching; a contact pair joins the plate's top and the block's bottom with Coulomb friction whose
+coefficient falls with the slip rate, μ = 0.2 + 0.3 e^(−1.5 v). Under gravity g = 10 the block
+is launched along x at v₀ = 3. A rigid block decelerates at g μ(v), stops after 1.2001 and travels
+∫₀^v₀ v dv / (g μ(v)) = 2.01524. Solved to t = 1.56.

tests/oracle/explicit-sliding-block-contact/expected.json

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+{
+ "title": "Block sliding to a stop on a deformable plate, friction decaying with the slip rate (explicit contact)",
+ "kind": "theory",
+ "source": "A rigid block launched at v₀ with friction μ(v) = μk + (μs − μk) e^(−d v) decelerates at g μ(v); it travels ∫₀^v₀ v dv / (g μ(v)) (Simpson's rule, 200 000 intervals) and stops after [v₀ + ln((μk + (μs − μk) e^(−d v₀))/μs)/d]/(g μk) = 1.2001; Meriam & Kraige, Engineering Mechanics: Dynamics, §3/5; the exponential decay law as in Oden & Martins, CMAME 52:527–634, 1985",
+ "quantities": [
+  {"id": "distance", "quantity": "distance travelled (top centre node) against ∫ v dv / (g μ(v)) = 2.01524",
+   "measure": {"command": "solve.probe", "params": {"step": "Slide", "at": [[0.5, 0.5, 0.5]]}, "pointer": "/nodes/0/displacement/0"},
+   "value": 2.015240387808855, "tolerance": {"relative": 0.01},
+   "reason": "the block bounces on the penalty springs and the plate under it shears elastically, so the normal force and the slip rate it feels vary by a little around their rigid-body values"},
+  {"id": "velocity", "quantity": "velocity of the top centre node after the stop",
+   "measure": {"command": "solve.probe", "params": {"step": "Slide", "at": [[0.5, 0.5, 0.5]]}, "pointer": "/nodes/0/velocity/0"},
+   "value": 0.0, "tolerance": {"absolute": 0.03},
+   "reason": "the block sticks; what is left is its elastic vibration, under 1% of v₀"}
+ ]
+}

tests/oracle/explicit-sliding-block-contact/fixture.json

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+{
+  "note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
+  "values": {
+    "distance": {
+      "band": 2.0152403878088553e-9,
+      "value": 2.011776866282693
+    },
+    "velocity": {
+      "band": 0.03,
+      "value": -0.00003285207146964345
+    }
+  }
+}

tests/oracle/explicit-sliding-block-contact/recipe.json

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+[
+ {"command": "mesh.block", "params": {"body": "Block", "kind": "hex8", "corners": [[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 0, 0.5], [1, 0, 0.5], [1, 1, 0.5], [0, 1, 0.5]], "divisions": [4, 4, 2], "faces": {"1": "bottom"}}},
+ {"command": "mesh.block", "params": {"body": "Plate", "kind": "hex8", "corners": [[-0.5, -0.5, -0.5], [4.0, -0.5, -0.5], [4.0, 1.5, -0.5], [-0.5, 1.5, -0.5], [-0.5, -0.5, 0], [4.0, -0.5, 0], [4.0, 1.5, 0], [-0.5, 1.5, 0]], "divisions": [9, 4, 1], "faces": {"1": "base", "2": "top"}, "merge": 0}},
+ {"command": "model.set.create", "params": {"name": "block", "geometry": [{"body": "Block"}]}},
+ {"command": "model.set.create", "params": {"name": "plate", "geometry": [{"body": "Plate"}]}},
+ {"command": "model.set.create", "params": {"name": "bottom", "geometry": [{"body": "Block", "face": "bottom"}]}},
+ {"command": "model.set.create", "params": {"name": "top", "geometry": [{"body": "Plate", "face": "top"}]}},
+ {"command": "model.set.create", "params": {"name": "base", "geometry": [{"body": "Plate", "face": "base"}]}},
+ {"command": "model.material.create", "params": {"name": "Mat", "elastic": {"type": "isotropic", "young": 100000.0, "poisson": 0.3}, "density": 1.0}},
+ {"command": "model.section.create", "params": {"name": "block", "set": "block", "material": "Mat", "kind": {"type": "solid"}, "formulation": {"integration": "reduced"}}},
+ {"command": "model.section.create", "params": {"name": "plate", "set": "plate", "material": "Mat", "kind": {"type": "solid"}, "formulation": {"integration": "reduced"}}},
+ {"command": "interaction.contact.pair", "params": {"name": "slide", "main": "top", "secondary": "bottom", "friction": {"static": 0.5, "kinetic": 0.2, "decay": 1.5}}},
+ {"command": "model.boundary_condition.create", "params": {"name": "hold", "type": "fixed", "target": {"set": "base"}}},
+ {"command": "model.load.create", "params": {"name": "weight", "type": "gravity", "acceleration": [0, 0, -10.0], "set": "block"}},
+ {"command": "model.step.create", "params": {"name": "Slide", "procedure": {"type": "explicit_dynamic", "end_time": 1.56, "initial_velocities": [{"set": "block", "velocity": [3.0, 0, 0]}]}, "boundary_conditions": ["hold"], "loads": ["weight"]}},
+ {"command": "solve.explicit", "params": {"step": "Slide"}, "id": "solve"}
+]