solver: mortar contact in 2D models

Plane strain, plane stress and axisymmetric models take contact pairs and general (self) contact on the linear edges of tri3 and quad4 elements: edges oriented out of their elements, the edge mortar of fem with the thickness or 2 pi r in the weights, the element's area over the edge's length as the thickness under it, and the in-plane rows of the tangent blocks. Verified by the contact patch test in plane strain and axisymmetric (to 1e-9 small sliding, 1e-4 finite), Hertz's line contact on quad4 edges (Johnson 1985, 4.3; equal to the hex8 model's to 1e-5), and a hybrid rubber C-ring closing on itself that moves node for node as the one-layer solid ring. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
2026-10-14 16:30:44 -04:00
1 parent 6baa189231
commit 227c4544ed
@@ -0,0 +1,10 @@
+# Contact patch test in plane strain (implicit mortar contact, non-matching edges)
+
+The 2D form of `nl-contact-patch`: two unit squares stacked along y in plane strain (2 mm thick),
+meshed independently: 3 × 2 quad4 below (graded in x), 2 × 2 above, so the nodes of the interface
+at y = 1 do not match. A frictionless contact pair joins the lower block's top edge (main) and
+the upper block's bottom edge (secondary). The edge x = 0 is a symmetry line, the base y = 0 is
+held in y, and the top y = 2 is pushed down by 0.02 (E = 1000, ν = 0.3) in two nonlinear static
+steps: `Small` without nonlinear geometry (small sliding) and `Finite` with it (finite sliding).
+The exact state is uniaxial stress σ_yy (σ_zz = ν σ_yy out of plane), uniform in both blocks,
+with a uniform contact pressure equal to it.
@@ -0,0 +1,432 @@
+{
+ "title": "Contact patch test in plane strain: two blocks with non-matching edges 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; passed by 2D mortar methods: Yang, Laursen & Meng, IJNME 62:1183–1225, 2005. Uniaxial stress between frictionless supports is uniform in both blocks",
+ "quantities": [
+  {
+   "id": "small_lower_corner",
+   "quantity": "Small: σ_yy of element 4 over that of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      4
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "small sliding is linear and its edge mortar integrals exact (degree-3 Gauss on the overlaps): the uniform stress is the discrete solution up to the convergence tolerance"
+  },
+  {
+   "id": "small_lower_far",
+   "quantity": "Small: σ_yy of element 6 over that of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      6
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "small_lower_base",
+   "quantity": "Small: σ_yy of element 1 over that of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      1
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "small_upper_top",
+   "quantity": "Small: σ_yy of element 10 over that of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      10
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "small_lateral",
+   "quantity": "Small: σ_xx of element 6 over its σ_yy (uniaxial stress: 0)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      6
+     ]
+    },
+    "pointer": "/elements/0/stress/0/0",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       6
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 0.0,
+   "tolerance": {
+    "absolute": 1e-06
+   },
+   "reason": "a frictionless interface (and symmetry plane) leaves 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, 1) over −σ_yy of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "at": [
+      [
+       0.5,
+       1.0,
+       0.0
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/contact/0/pressure",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Small",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1",
+     "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/1",
+     "scale": -1.0
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 1e-06
+   },
+   "reason": "the pressure carries the whole load across the interface (both are totals over the thickness)"
+  },
+  {
+   "id": "finite_lower_corner",
+   "quantity": "Finite: σ_yy of element 4 over that of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      4
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "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_far",
+   "quantity": "Finite: σ_yy of element 6 over that of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      6
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "finite_lower_base",
+   "quantity": "Finite: σ_yy of element 1 over that of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      1
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "finite_upper_top",
+   "quantity": "Finite: σ_yy of element 10 over that of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      10
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "as for the corner element"
+  },
+  {
+   "id": "finite_lateral",
+   "quantity": "Finite: σ_xx of element 6 over its σ_yy (uniaxial stress: 0)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "elements": [
+      6
+     ]
+    },
+    "pointer": "/elements/0/stress/0/0",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       6
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1"
+    }
+   },
+   "value": 0.0,
+   "tolerance": {
+    "absolute": 0.0002
+   },
+   "reason": "a frictionless interface (and symmetry plane) leaves 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, 1) over −σ_yy of element 7",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Finite",
+     "at": [
+      [
+       0.5,
+       1.0,
+       0.0
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/contact/0/pressure",
+    "over": {
+     "command": "solve.probe",
+     "params": {
+      "step": "Finite",
+      "elements": [
+       7
+      ]
+     },
+     "pointer": "/elements/0/stress/0/1",
+     "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/1",
+     "scale": -1.0
+    }
+   },
+   "value": 1.0,
+   "tolerance": {
+    "relative": 0.0002
+   },
+   "reason": "the pressure carries the whole load across the interface (both are totals over the thickness)"
+  },
+  {
+   "id": "small_stress",
+   "quantity": "Small: σ_yy of element 7 (plane strain uniaxial stress E ε/(1 − ν²) = −10.989 for the 1% squeeze)",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Small",
+     "elements": [
+      7
+     ]
+    },
+    "pointer": "/elements/0/stress/0/1"
+   },
+   "value": -10.989010989010989,
+   "tolerance": {
+    "relative": 0.0001
+   },
+   "reason": "the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness)"
+  }
+ ]
+}
@@ -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": 1.0000000000022103e-9,
+      "value": 1.0000000000022102
+    },
+    "finite_lateral": {
+      "band": 0.0002,
+      "value": -1.8586502985840345e-6
+    },
+    "finite_lower_base": {
+      "band": 1e-9,
+      "value": 0.9999968799408236
+    },
+    "finite_lower_corner": {
+      "band": 1.0000007080194146e-9,
+      "value": 1.0000007080194144
+    },
+    "finite_lower_far": {
+      "band": 1e-9,
+      "value": 0.9999913032274111
+    },
+    "finite_pressure": {
+      "band": 1e-9,
+      "value": 0.9999924972272739
+    },
+    "finite_upper_top": {
+      "band": 1e-9,
+      "value": 0.9999957165083484
+    },
+    "small_force": {
+      "band": 1e-9,
+      "value": 0.9999999999999859
+    },
+    "small_lateral": {
+      "band": 1e-6,
+      "value": -8.925527731170228e-15
+    },
+    "small_lower_base": {
+      "band": 1e-9,
+      "value": 0.9999999999999801
+    },
+    "small_lower_corner": {
+      "band": 1.0000000000000059e-9,
+      "value": 1.0000000000000058
+    },
+    "small_lower_far": {
+      "band": 1e-9,
+      "value": 0.9999999999999546
+    },
+    "small_pressure": {
+      "band": 1e-9,
+      "value": 0.9999999999999142
+    },
+    "small_stress": {
+      "band": 1.098901098901099e-8,
+      "value": -10.988443575433445
+    },
+    "small_upper_top": {
+      "band": 1e-9,
+      "value": 0.9999999999999762
+    }
+  }
+}
@@ -0,0 +1,22 @@
+[
+ {"command": "mesh.block", "params": {"body": "Lower", "kind": "quad4", "corners": [[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0]], "divisions": [3, 2], "bias": [2.0, 1.0], "edges": [{"corners": [1, 2], "name": "bottom"}, {"corners": [3, 4], "name": "top"}, {"corners": [4, 1], "name": "x0"}]}},
+ {"command": "mesh.block", "params": {"body": "Upper", "kind": "quad4", "corners": [[0, 1, 0], [1, 1, 0], [1, 2, 0], [0, 2, 0]], "divisions": [2, 2], "edges": [{"corners": [1, 2], "name": "bottom"}, {"corners": [3, 4], "name": "top"}, {"corners": [4, 1], "name": "x0"}], "merge": 0}},
+ {"command": "model.set.create", "params": {"name": "lower", "geometry": [{"body": "Lower", "face": "area"}]}},
+ {"command": "model.set.create", "params": {"name": "upper", "geometry": [{"body": "Upper", "face": "area"}]}},
+ {"command": "model.set.create", "params": {"name": "lower_top", "geometry": [{"body": "Lower", "edge": "top"}]}},
+ {"command": "model.set.create", "params": {"name": "upper_bottom", "geometry": [{"body": "Upper", "edge": "bottom"}]}},
+ {"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Lower", "edge": "x0"}, {"body": "Upper", "edge": "x0"}]}},
+ {"command": "model.set.create", "params": {"name": "base", "geometry": [{"body": "Lower", "edge": "bottom"}]}},
+ {"command": "model.set.create", "params": {"name": "top", "geometry": [{"body": "Upper", "edge": "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": "plane_strain", "thickness": 2.0}}},
+ {"command": "model.section.create", "params": {"name": "upper", "set": "upper", "material": "Mat", "kind": {"type": "plane_strain", "thickness": 2.0}}},
+ {"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": "base", "type": "displacement", "target": {"set": "base"}, "values": [null, 0, null, null, null, null]}},
+ {"command": "model.boundary_condition.create", "params": {"name": "push", "type": "displacement", "target": {"set": "top"}, "values": [null, -0.02, null, 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", "base", "push"], "loads": []}},
+ {"command": "model.step.create", "params": {"name": "Finite", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["x0", "base", "push"], "loads": []}},
+ {"command": "solve.nonlinear", "params": {"step": "Small"}, "id": "small"},
+ {"command": "solve.nonlinear", "params": {"step": "Finite"}, "id": "finite"}
+]