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 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
2026-10-14 14:58:32 -04:00
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+# Hertz line contact: cylinder on a flat (implicit mortar contact)
+
+A steel cylinder of radius R = 10 mm pressed on a steel block (E = 210 GPa, ν = 0.3; mm, N, MPa)
+in plane strain: a quarter model of the cylinder's lower part (a 2.5 mm wide block whose bottom
+is the circular arc, graded towards the contact) on a 5 × 5 mm foundation, symmetric about
+x = 0, one element thick with every node held in z. A pressure on the indenter's top carries
+the load P = 2265.57 N/mm (per length of the whole cylinder) chosen so that Hertz's half-width is
+b = 0.5 mm and the peak pressure p₀ = 2884.6 MPa. Soft springs (0.01 N/mm per node) hold the
+indenter's top vertically until the contact closes. One nonlinear static step without nonlinear
+geometry: small sliding, mortar contact of the round face (secondary) on the flat (main) by an
+augmented Lagrangian.
@@ -0,0 +1,109 @@
+{
+ "title": "Hertz line contact: a cylinder pressed on a flat (implicit mortar contact, plane strain)",
+ "kind": "theory",
+ "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",
+ "quantities": [
+  {
+   "id": "peak_pressure",
+   "quantity": "the largest contact pressure p₀ (at x = 0)",
+   "measure": {
+    "from": "solve",
+    "pointer": "/contacts/0/max_pressure/value"
+   },
+   "value": 2884.6153846153843,
+   "tolerance": {
+    "relative": 0.02
+   },
+   "reason": "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"
+  },
+  {
+   "id": "normal_force",
+   "quantity": "the force across the contact (half model, 1 mm thick)",
+   "measure": {
+    "from": "solve",
+    "pointer": "/contacts/0/normal_force"
+   },
+   "value": 1132.7858125924733,
+   "tolerance": {
+    "relative": 0.002
+   },
+   "reason": "the pressure carries the whole load; the soft springs that hold the indenter before contact take a millionth"
+  },
+  {
+   "id": "pressure_x0296",
+   "quantity": "the contact pressure at the secondary node at x = 0.29594 (node 7)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Press",
+     "nodes": [
+      7
+     ]
+    },
+    "pointer": "/nodes/0/contact/0/pressure"
+   },
+   "value": 2325.066800691437,
+   "tolerance": {
+    "relative": 0.05
+   },
+   "reason": "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"
+  },
+  {
+   "id": "pressure_x0367",
+   "quantity": "the contact pressure at the secondary node at x = 0.36701 (node 8)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Press",
+     "nodes": [
+      8
+     ]
+    },
+    "pointer": "/nodes/0/contact/0/pressure"
+   },
+   "value": 1959.0164149439815,
+   "tolerance": {
+    "relative": 0.1
+   },
+   "reason": "as at node 7, nearer the edge, where the elements round off the square-root fall of the pressure to zero at b"
+  },
+  {
+   "id": "pressure_x0633",
+   "quantity": "the contact pressure at the secondary node at x = 0.63325 (node 11, beyond the half-width b = 0.5)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Press",
+     "nodes": [
+      11
+     ]
+    },
+    "pointer": "/nodes/0/contact/0/pressure"
+   },
+   "value": 0.0,
+   "tolerance": {
+    "absolute": 1.0
+   },
+   "reason": "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)"
+  },
+  {
+   "id": "syy_depth",
+   "quantity": "the vertical stress σ_yy under the centre of contact at depth 0.47127 (node 713)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Press",
+     "nodes": [
+      713
+     ]
+    },
+    "pointer": "/nodes/0/stress/1"
+   },
+   "value": -2099.1580228637245,
+   "tolerance": {
+    "relative": 0.04
+   },
+   "reason": "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"
+  }
+ ]
+}
@@ -0,0 +1,29 @@
+{
+  "note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
+  "values": {
+    "normal_force": {
+      "band": 1.133036636011328e-6,
+      "value": 1133.0366360113278
+    },
+    "peak_pressure": {
+      "band": 2.891953889449524e-6,
+      "value": 2891.953889449524
+    },
+    "pressure_x0296": {
+      "band": 2.325066800691437e-6,
+      "value": 2257.8357909118563
+    },
+    "pressure_x0367": {
+      "band": 2.0784005834495077e-6,
+      "value": 2078.4005834495074
+    },
+    "pressure_x0633": {
+      "band": 1.0,
+      "value": -0.0
+    },
+    "syy_depth": {
+      "band": 2.0991580228637247e-6,
+      "value": -2059.819639241533
+    }
+  }
+}
@@ -0,0 +1,23 @@
+[
+ {"command": "mesh.block", "params": {"body": "Indenter", "kind": "hex8", "corners": [[0, 0, 0], [2.5, 0.3175416344814579, 0], [2.5, 3.0, 0], [0, 3.0, 0], [0, 0, 1.0], [2.5, 0.3175416344814579, 1.0], [2.5, 3.0, 1.0], [0, 3.0, 1.0]], "divisions": [20, 10, 1], "bias": [8, 8, 1], "edges": [{"corners": [1, 2], "arc": {"center": [0, 10.0, 0]}}, {"corners": [5, 6], "arc": {"center": [0, 10.0, 1.0]}}], "faces": {"3": "round", "5": "top", "6": "sym"}}},
+ {"command": "mesh.block", "params": {"body": "Foundation", "kind": "hex8", "corners": [[0, -5, 0], [5, -5, 0], [5, 0, 0], [0, 0, 0], [0, -5, 1.0], [5, -5, 1.0], [5, 0, 1.0], [0, 0, 1.0]], "divisions": [24, 14, 1], "bias": [10, 0.1, 1], "faces": {"3": "base", "5": "flat", "6": "sym"}, "merge": 0}},
+ {"command": "model.set.create", "params": {"name": "indenter", "geometry": [{"body": "Indenter"}]}},
+ {"command": "model.set.create", "params": {"name": "foundation", "geometry": [{"body": "Foundation"}]}},
+ {"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Indenter"}, {"body": "Foundation"}]}},
+ {"command": "model.set.create", "params": {"name": "round", "geometry": [{"body": "Indenter", "face": "round"}]}},
+ {"command": "model.set.create", "params": {"name": "flat", "geometry": [{"body": "Foundation", "face": "flat"}]}},
+ {"command": "model.set.create", "params": {"name": "top", "geometry": [{"body": "Indenter", "face": "top"}]}},
+ {"command": "model.set.create", "params": {"name": "base", "geometry": [{"body": "Foundation", "face": "base"}]}},
+ {"command": "model.set.create", "params": {"name": "sym", "geometry": [{"body": "Indenter", "face": "sym"}, {"body": "Foundation", "face": "sym"}]}},
+ {"command": "model.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 210000.0, "poisson": 0.3}}},
+ {"command": "model.section.create", "params": {"name": "indenter", "set": "indenter", "material": "Steel", "kind": {"type": "solid"}}},
+ {"command": "model.section.create", "params": {"name": "foundation", "set": "foundation", "material": "Steel", "kind": {"type": "solid"}}},
+ {"command": "interaction.contact.pair", "params": {"name": "hertz", "main": "flat", "secondary": "round"}},
+ {"command": "model.boundary_condition.create", "params": {"name": "sym", "type": "displacement", "target": {"set": "sym"}, "values": [0, null, null, null, null, null]}},
+ {"command": "model.boundary_condition.create", "params": {"name": "plane", "type": "displacement", "target": {"set": "all"}, "values": [null, null, 0, null, null, null]}},
+ {"command": "model.boundary_condition.create", "params": {"name": "base", "type": "displacement", "target": {"set": "base"}, "values": [null, 0, null, null, null, null]}},
+ {"command": "model.spring.create", "params": {"name": "hold", "type": "spring", "first": {"set": "top"}, "direction": {"dof": 2}, "stiffness": 0.01}},
+ {"command": "model.load.create", "params": {"name": "press", "type": "pressure", "set": "top", "magnitude": 453.1143250369893}},
+ {"command": "model.step.create", "params": {"name": "Press", "procedure": {"type": "nonlinear_static", "nonlinear_geometry": false, "max_iterations": 60}, "boundary_conditions": ["sym", "plane", "base"], "loads": ["press"]}},
+ {"command": "solve.nonlinear", "params": {"step": "Press"}, "id": "solve"}
+]