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
1 parent 5770288f47
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+# Contact patch test (implicit mortar contact, non-matching meshes)
+
+Two unit cubes stacked along z, meshed independently: 3 × 3 × 2 hex8 below (graded in x and y),
+2 × 2 × 2 above, so the nodes of the interface at z = 1 do not match. A frictionless contact pair
+joins the lower block's top (main) and the upper block's bottom (secondary). The faces x = 0 and
+y = 0 of both blocks are symmetry planes, the base z = 0 is held in z, and the top z = 2 is pushed
+down by 0.02 (E = 1000, ν = 0.3) in two nonlinear static steps: `Small` without nonlinear
+geometry (small sliding, mortar integrals in the reference configuration) and `Finite` with it
+(finite sliding, searched and integrated on the deformed faces). The exact state is uniaxial
+stress, uniform in both blocks, with a uniform contact pressure equal to it.
@@ -0,0 +1,432 @@
+{
+ "title": "Contact patch test: two blocks with non-matching meshes in uniaxial compression (implicit mortar 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; passed by mortar methods: Puso & Laursen, CMAME 193:601–629, 2004. Uniaxial stress between frictionless symmetry planes is uniform in both blocks",
+ "quantities": [
+  {
+   "id": "small_lower_corner",
+   "quantity": "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)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      10
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "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)"
+  },
+  {
+   "id": "small_lower_centre",
+   "quantity": "Small: 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": "Small",
+     "elements": [
+      14
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "small_lower_base",
+   "quantity": "Small: σ_zz of the lower block's bottom corner element (element 1) over that of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      1
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "small_upper_top",
+   "quantity": "Small: σ_zz of the upper block's top far corner element (element 26) over that of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      26
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "small_lateral",
+   "quantity": "Small: σ_xx of element 14 over its σ_zz (uniaxial stress: 0)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      14
+     ]
+    },
+    "pointer": "/elements/0/stress/0/0",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       14
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 0.0,
+   "tolerance": {
+    "absolute": 1e-06
+   },
+   "reason": "frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them"
+  },
+  {
+   "id": "small_pressure",
+   "quantity": "Small: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "at": [
+      [
+       0.5,
+       0.5,
+       1.0
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/contact/0/pressure",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2",
+     "scale": -1.0
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface"
+  },
+  {
+   "id": "small_force",
+   "quantity": "Small: the integrated contact pressure over the reaction at the pushed top",
+   "measure": {
+    "from": "small",
+    "pointer": "/contacts/0/normal_force",
+    "over": {
+     "command": "results.reactions",
+     "params": {
+      "step": "Small",
+      "sets": [
+       "top"
+      ]
+     },
+     "pointer": "/sets/0/force/2",
+     "scale": -1.0
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "the pressure carries the whole load across the interface"
+  },
+  {
+   "id": "finite_lower_corner",
+   "quantity": "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)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      10
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "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"
+  },
+  {
+   "id": "finite_lower_centre",
+   "quantity": "Finite: 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": "Finite",
+     "elements": [
+      14
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "finite_lower_base",
+   "quantity": "Finite: σ_zz of the lower block's bottom corner element (element 1) over that of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      1
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "finite_upper_top",
+   "quantity": "Finite: σ_zz of the upper block's top far corner element (element 26) over that of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      26
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "finite_lateral",
+   "quantity": "Finite: σ_xx of element 14 over its σ_zz (uniaxial stress: 0)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      14
+     ]
+    },
+    "pointer": "/elements/0/stress/0/0",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       14
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2"
+    }
+   },
+   "value": 0.0,
+   "tolerance": {
+    "absolute": 0.0002
+   },
+   "reason": "frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them"
+  },
+  {
+   "id": "finite_pressure",
+   "quantity": "Finite: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "at": [
+      [
+       0.5,
+       0.5,
+       1.0
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/contact/0/pressure",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       19
+      ]
+     },
+     "pointer": "/elements/0/stress/0/2",
+     "scale": -1.0
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface"
+  },
+  {
+   "id": "finite_force",
+   "quantity": "Finite: the integrated contact pressure over the reaction at the pushed top",
+   "measure": {
+    "from": "finite",
+    "pointer": "/contacts/0/normal_force",
+    "over": {
+     "command": "results.reactions",
+     "params": {
+      "step": "Finite",
+      "sets": [
+       "top"
+      ]
+     },
+     "pointer": "/sets/0/force/2",
+     "scale": -1.0
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "the pressure carries the whole load across the interface"
+  },
+  {
+   "id": "small_stress",
+   "quantity": "Small: σ_zz of element 19 (uniaxial stress E ε = −10 for the 1% squeeze)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      19
+     ]
+    },
+    "pointer": "/elements/0/stress/0/2"
+   },
+   "value": -10.0,
+   "tolerance": {
+    "relative": 0.0001
+   },
+   "reason": "the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness, a few 10⁻⁵ of the squeeze)"
+  }
+ ]
+}
@@ -0,0 +1,65 @@
+{
+  "note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
+  "values": {
+    "finite_force": {
+      "band": 1e-9,
+      "value": 0.9999999999974551
+    },
+    "finite_lateral": {
+      "band": 0.0002,
+      "value": 1.223600676667172e-6
+    },
+    "finite_lower_base": {
+      "band": 1e-9,
+      "value": 0.9999905821256437
+    },
+    "finite_lower_centre": {
+      "band": 1e-9,
+      "value": 0.999987114528879
+    },
+    "finite_lower_corner": {
+      "band": 1.000001514518151e-9,
+      "value": 1.000001514518151
+    },
+    "finite_pressure": {
+      "band": 1e-9,
+      "value": 0.9999828003418422
+    },
+    "finite_upper_top": {
+      "band": 1e-9,
+      "value": 0.9999912761681571
+    },
+    "small_force": {
+      "band": 1.0000000000000607e-9,
+      "value": 1.0000000000000606
+    },
+    "small_lateral": {
+      "band": 1e-6,
+      "value": 7.880554574306202e-15
+    },
+    "small_lower_base": {
+      "band": 1e-9,
+      "value": 0.999999999999944
+    },
+    "small_lower_centre": {
+      "band": 1e-9,
+      "value": 0.9999999999999398
+    },
+    "small_lower_corner": {
+      "band": 1e-9,
+      "value": 0.9999999999999855
+    },
+    "small_pressure": {
+      "band": 1e-9,
+      "value": 0.9999999999999953
+    },
+    "small_stress": {
+      "band": 1e-8,
+      "value": -9.99957382191768
+    },
+    "small_upper_top": {
+      "band": 1e-9,
+      "value": 0.9999999999999263
+    }
+  }
+}
@@ -0,0 +1,24 @@
+[
+ {"command": "mesh.block", "params": {"body": "Lower", "kind": "hex8", "corners": [[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1]], "divisions": [3, 3, 2], "faces": {"1": "bottom", "2": "top", "3": "y0", "6": "x0"}, "bias": [2.0, 0.6, 1.0]}},
+ {"command": "mesh.block", "params": {"body": "Upper", "kind": "hex8", "corners": [[0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1], [0, 0, 2], [1, 0, 2], [1, 1, 2], [0, 1, 2]], "divisions": [2, 2, 2], "faces": {"1": "bottom", "2": "top", "3": "y0", "6": "x0"}, "merge": 0}},
+ {"command": "model.set.create", "params": {"name": "lower", "geometry": [{"body": "Lower"}]}},
+ {"command": "model.set.create", "params": {"name": "upper", "geometry": [{"body": "Upper"}]}},
+ {"command": "model.set.create", "params": {"name": "lower_top", "geometry": [{"body": "Lower", "face": "top"}]}},
+ {"command": "model.set.create", "params": {"name": "upper_bottom", "geometry": [{"body": "Upper", "face": "bottom"}]}},
+ {"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Lower", "face": "x0"}, {"body": "Upper", "face": "x0"}]}},
+ {"command": "model.set.create", "params": {"name": "y0", "geometry": [{"body": "Lower", "face": "y0"}, {"body": "Upper", "face": "y0"}]}},
+ {"command": "model.set.create", "params": {"name": "z0", "geometry": [{"body": "Lower", "face": "bottom"}]}},
+ {"command": "model.set.create", "params": {"name": "top", "geometry": [{"body": "Upper", "face": "top"}]}},
+ {"command": "model.material.create", "params": {"name": "Mat", "elastic": {"type": "isotropic", "young": 1000.0, "poisson": 0.3}}},
+ {"command": "model.section.create", "params": {"name": "lower", "set": "lower", "material": "Mat", "kind": {"type": "solid"}}},
+ {"command": "model.section.create", "params": {"name": "upper", "set": "upper", "material": "Mat", "kind": {"type": "solid"}}},
+ {"command": "interaction.contact.pair", "params": {"name": "interface", "main": "lower_top", "secondary": "upper_bottom"}},
+ {"command": "model.boundary_condition.create", "params": {"name": "x0", "type": "displacement", "target": {"set": "x0"}, "values": [0, null, null, null, null, null]}},
+ {"command": "model.boundary_condition.create", "params": {"name": "y0", "type": "displacement", "target": {"set": "y0"}, "values": [null, 0, null, null, null, null]}},
+ {"command": "model.boundary_condition.create", "params": {"name": "z0", "type": "displacement", "target": {"set": "z0"}, "values": [null, null, 0, null, null, null]}},
+ {"command": "model.boundary_condition.create", "params": {"name": "push", "type": "displacement", "target": {"set": "top"}, "values": [null, null, -0.02, null, null, null]}},
+ {"command": "model.step.create", "params": {"name": "Small", "procedure": {"type": "nonlinear_static", "nonlinear_geometry": false, "tolerances": {"force": 1e-10, "displacement": 1e-08, "energy": 1e-16}}, "boundary_conditions": ["x0", "y0", "z0", "push"], "loads": []}},
+ {"command": "model.step.create", "params": {"name": "Finite", "procedure": {"type": "nonlinear_static", "nonlinear_geometry": true}, "boundary_conditions": ["x0", "y0", "z0", "push"], "loads": []}},
+ {"command": "solve.nonlinear", "params": {"step": "Small"}, "id": "small"},
+ {"command": "solve.nonlinear", "params": {"step": "Finite"}, "id": "finite"}
+]