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-contact-patch/expected.json

@@ -0,0 +1,176 @@
+{
+ "title": "Contact patch test: two blocks with non-matching meshes in uniaxial compression (explicit contact)",
+ "kind": "theory",
+ "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. Uniaxial stress between frictionless symmetry planes is uniform in both blocks",
+ "quantities": [
+  {
+   "id": "lower_corner",
+   "quantity": "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)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Squeeze",
+     "elements": [
+      10
+     ]
+    },
+    "pointer": "/elements/0/stress/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Squeeze",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.005
+   },
+   "reason": "a uniform stress is an equilibrium of the discrete system (the integrated contact forces are the consistent nodal forces of a uniform pressure on both meshes); what is left is the ringing of the loading, a hundred wave transits slow"
+  },
+  {
+   "id": "lower_centre",
+   "quantity": "σ_zz of the lower block's centre element under the interface (element 14) over that of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Squeeze",
+     "elements": [
+      14
+     ]
+    },
+    "pointer": "/elements/0/stress/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Squeeze",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.005
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "lower_base",
+   "quantity": "σ_zz of the lower block's bottom corner element (element 1) over that of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Squeeze",
+     "elements": [
+      1
+     ]
+    },
+    "pointer": "/elements/0/stress/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Squeeze",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.005
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "upper_top",
+   "quantity": "σ_zz of the upper block's top far corner element (element 26) over that of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Squeeze",
+     "elements": [
+      26
+     ]
+    },
+    "pointer": "/elements/0/stress/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Squeeze",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.005
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "force",
+   "quantity": "the force across the interface over −σ_zz A (A = 1)",
+   "measure": {
+    "from": "solve",
+    "pointer": "/contacts/0/peak_force",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Squeeze",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/2",
+     "scale": -1.0
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.01
+   },
+   "reason": "the integrated contact pressure carries the whole load; the largest force of the run is the final one in a slow squeeze, up to the ringing"
+  },
+  {
+   "id": "lateral",
+   "quantity": "σ_xx of element 14 over its σ_zz (uniaxial stress: 0)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Squeeze",
+     "elements": [
+      14
+     ]
+    },
+    "pointer": "/elements/0/stress/0",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Squeeze",
+      "elements": [
+       14
+      ]
+     },
+     "pointer": "/elements/0/stress/2"
+    }
+   },
+   "value": 0.0,
+   "tolerance": {
+    "absolute": 0.005
+   },
+   "reason": "frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them"
+  }
+ ]
+}