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tests: add the NAFEMS LE7 cylinder/sphere oracle case
The thin-walled cylinder with a hemispherical cap under internal pressure as four axisymmetric quad8 blocks: axial stress on the outer surface at D 25.283. The NAFEMS target, 25.86, is the axisymmetric thin-shell solution; our solid model converges to 25.27, the value ESRD reports for its converged axisymmetric solid. So the NAFEMS quantity keeps a 1% tolerance and is a known failure until there is an axisymmetric shell element, and a second quantity checks the model against the solid reference (pass, 0.05%). Part of #97.
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# NAFEMS LE7 axisymmetric cylinder/sphere under pressure (axisymmetric quad8)
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Our model of the published problem: a thin-walled vessel, a cylinder closed by a hemispherical
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cap, of mid-surface radius 1 m and wall thickness 0.025 m, E = 210 GPa, ν = 0.3 (m, MN, MPa),
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under an internal pressure of 1 MPa. In FieldCraft's axisymmetric models x is the radius and y the
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axis. The cylinder runs from F (y = 0) to the junction C (y = 1.5), the cap from C to the pole A
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(y = 2.5). The axial displacement is zero at F (the vessel's plane of symmetry: the whole bottom
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edge) and the radial one at A (the pole edge, on the axis). The target is the axial stress σ_yy
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on the outer surface at D (R = 1.0125, y = 1.4034), 0.0966 m below the junction, inside the
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bending boundary layer the cap's different stiffness causes.
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## The mesh
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Four `mesh.block` quad8 patches through the wall thickness: three cylinder blocks with E
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(y = 1.1136) and D as block corners, so D is a node, and the cap, a quarter ring whose two curved
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edges are arcs about (0, 1.5) and whose fourth edge lies on the axis. 2 elements through the
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wall, about 50 per metre along it (56, 14 and 5 in the cylinder blocks, 79 round the cap); the
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blocks share their edges, so `mesh.block` joins them. The pressure acts on the inner edges of all
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four blocks.
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The recipe was generated with this script (`python3 le7.py 2 50 > recipe.json`):
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```python
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import json, math, sys
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def le7(nt, per_m):
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"""NAFEMS LE7 cylinder with a hemispherical cap, axisymmetric (m, MN, MPa): x is the radius,
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y the axis. Mid-surface radius 1, wall 0.025 thick; the cylinder runs from F (y = 0, the plane
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of symmetry) to C (y = 1.5), the cap from C to the pole A (y = 2.5). nt elements through the
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wall, about per_m elements per metre along it."""
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ri, ro = 1 - 0.0125, 1 + 0.0125
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cuts = [0.0, 1.1136, 1.4034, 1.5] # F, E, D, C: E and D are nodes
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along = lambda length: max(1, round(length * per_m))
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commands = []
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for k, (y0, y1) in enumerate(zip(cuts, cuts[1:])):
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commands.append({"command": "mesh.block", "params": {"body": f"cylinder_{k + 1}", "kind": "quad8",
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"corners": [[ri, y0, 0], [ro, y0, 0], [ro, y1, 0], [ri, y1, 0]], "divisions": [nt, along(y1 - y0)],
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"edges": [{"corners": [1, 4], "name": "inside"}, {"corners": [1, 2], "name": "bottom"}]}})
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commands.append({"command": "mesh.block", "params": {"body": "cap", "kind": "quad8",
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"corners": [[ri, 1.5, 0], [ro, 1.5, 0], [0, 1.5 + ro, 0], [0, 1.5 + ri, 0]], "divisions": [nt, along(math.pi / 2)],
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"edges": [{"corners": [2, 3], "arc": {"center": [0, 1.5, 0]}}, {"corners": [4, 1], "arc": {"center": [0, 1.5, 0]}, "name": "inside"},
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{"corners": [3, 4], "name": "pole"}]}})
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bodies = ["cylinder_1", "cylinder_2", "cylinder_3", "cap"]
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return commands + [
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{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": b, "face": "area"} for b in bodies]}},
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{"command": "model.set.create", "params": {"name": "inside", "geometry": [{"body": b, "edge": "inside"} for b in bodies]}},
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{"command": "model.set.create", "params": {"name": "F", "geometry": [{"body": "cylinder_1", "edge": "bottom"}]}},
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{"command": "model.set.create", "params": {"name": "A", "geometry": [{"body": "cap", "edge": "pole"}]}},
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{"command": "model.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 210000.0, "poisson": 0.3}}},
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{"command": "model.section.create", "params": {"name": "vessel", "set": "all", "material": "Steel", "kind": {"type": "axisymmetric"}}},
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{"command": "model.boundary_condition.create", "params": {"name": "F", "type": "symmetry", "target": {"set": "F"}, "plane": "y"}},
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{"command": "model.boundary_condition.create", "params": {"name": "A", "type": "symmetry", "target": {"set": "A"}, "plane": "x"}},
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{"command": "model.load.create", "params": {"name": "pressure", "type": "pressure", "set": "inside", "magnitude": 1.0}},
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{"command": "model.step.create", "params": {"name": "Pressure", "procedure": {"type": "static"}, "boundary_conditions": ["F", "A"], "loads": ["pressure"]}},
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{"command": "solve.static", "params": {}, "id": "solve"},
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]
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nt, per_m = (int(a) for a in sys.argv[1:3]) if len(sys.argv) > 2 else (2, 50)
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print("[\n " + ",\n ".join(json.dumps(c) for c in le7(nt, per_m)) + "\n]")
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```
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## Refinement
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Every row is the listing above with the row's divisions, solved and probed at D as the case does;
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the case mesh is 2 × 50:
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| through the wall × per metre | σ_yy at D | against 25.86 | against 25.27 |
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|---|---|---|---|
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| 2 × 25 | 25.331 | −2.05% | +0.24% |
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| 2 × 50 (this case) | 25.283 | −2.23% | +0.05% |
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| 4 × 100 | 25.271 | −2.28% | +0.00% |
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| 8 × 200 | 25.269 | −2.29% | −0.00% |
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Far from the junction the model is exact: at F the axial stress on the outer surface is 19.503
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(p r_i²/(r_o² − r_i²)) and the hoop stress 39.006 (Lamé).
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## Two references, one known failure
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The published target, 25.86, comes from axisymmetric thin-shell theory. The ESRD guide solves the
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same problem with axisymmetric solid elements and reports 25.27 on two meshes, 2.3% below the
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target, and our solid model converges to the same 25.27: the difference is the model, not the
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mesh. So the case keeps the NAFEMS target with the
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tolerance a shell model would earn, 1%, marked `known_failure` until FieldCraft has an
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axisymmetric shell element, and checks the solid model against the solid reference, also at 1%.
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`fixture.json` holds the passing quantity's 2 × 50 value: the mesh is built by `mesh.block` from
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the recipe's corners, so it is the same on every platform.
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## Running it
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```sh
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cargo xtask oracle --case nafems-le7
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cargo xtask oracle --bless nafems-le7 # only after the case passes
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```
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{
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"title": "NAFEMS LE7 axisymmetric cylinder/sphere under pressure (axisymmetric quad8)",
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"kind": "benchmark",
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"source": "NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE7 (target: axial stress 25.86 MPa on the outer surface at D, R = 1.0125, Z = 1.4034), as reproduced in the ESRD StressCheck Benchmarks Guide 'The Standard NAFEMS Benchmarks: Linear Elastic Tests' (2018), p. 23, which also gives its converged axisymmetric solid result, 25.27 MPa (p. 24); the Abaqus 2016 Benchmarks Guide, section 4.2.7 'LE7: Axisymmetric cylinder/sphere under pressure', gives the target rounded to 25.9 MPa",
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"quantities": [
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{
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"id": "syy_D",
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"quantity": "axial stress σ_yy at D (1.0125, 1.4034), outer surface",
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"measure": {"command": "solve.probe", "params": {"step": "Pressure", "at": [[1.0125, 1.4034, 0]]}, "pointer": "/nodes/0/stress/1"},
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"value": 25.86,
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"tolerance": {"relative": 0.01},
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"reason": "the target is published to four figures for the thin-shell model of the vessel, and a model of the same theory converges well within 1%",
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"known_failure": "the target is the axisymmetric thin-shell solution and FieldCraft has no axisymmetric shell element; our axisymmetric solid model converges to 25.27 (−2.3%), the value ESRD reports for its converged axisymmetric solid (quantity syy_D_solid)"
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},
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{
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"id": "syy_D_solid",
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"quantity": "axial stress σ_yy at D (1.0125, 1.4034), outer surface, against the converged solid model",
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"measure": {"command": "solve.probe", "params": {"step": "Pressure", "at": [[1.0125, 1.4034, 0]]}, "pointer": "/nodes/0/stress/1"},
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"value": 25.27,
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"tolerance": {"relative": 0.01},
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"reason": "the same model idealisation (axisymmetric solid) as the reference, which ESRD reports converged to under 1% on two meshes; ours moves by 0.05% on the next refinement (see README)"
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}
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]
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}
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{
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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.",
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"values": {
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"syy_D_solid": {
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"band": 2.5282545696283336e-8,
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"value": 25.282545696283336
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}
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}
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}
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[
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{"command": "mesh.block", "params": {"body": "cylinder_1", "kind": "quad8", "corners": [[0.9875, 0.0, 0], [1.0125, 0.0, 0], [1.0125, 1.1136, 0], [0.9875, 1.1136, 0]], "divisions": [2, 56], "edges": [{"corners": [1, 4], "name": "inside"}, {"corners": [1, 2], "name": "bottom"}]}},
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{"command": "mesh.block", "params": {"body": "cylinder_2", "kind": "quad8", "corners": [[0.9875, 1.1136, 0], [1.0125, 1.1136, 0], [1.0125, 1.4034, 0], [0.9875, 1.4034, 0]], "divisions": [2, 14], "edges": [{"corners": [1, 4], "name": "inside"}, {"corners": [1, 2], "name": "bottom"}]}},
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{"command": "mesh.block", "params": {"body": "cylinder_3", "kind": "quad8", "corners": [[0.9875, 1.4034, 0], [1.0125, 1.4034, 0], [1.0125, 1.5, 0], [0.9875, 1.5, 0]], "divisions": [2, 5], "edges": [{"corners": [1, 4], "name": "inside"}, {"corners": [1, 2], "name": "bottom"}]}},
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{"command": "mesh.block", "params": {"body": "cap", "kind": "quad8", "corners": [[0.9875, 1.5, 0], [1.0125, 1.5, 0], [0, 2.5125, 0], [0, 2.4875, 0]], "divisions": [2, 79], "edges": [{"corners": [2, 3], "arc": {"center": [0, 1.5, 0]}}, {"corners": [4, 1], "arc": {"center": [0, 1.5, 0]}, "name": "inside"}, {"corners": [3, 4], "name": "pole"}]}},
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{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "cylinder_1", "face": "area"}, {"body": "cylinder_2", "face": "area"}, {"body": "cylinder_3", "face": "area"}, {"body": "cap", "face": "area"}]}},
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{"command": "model.set.create", "params": {"name": "inside", "geometry": [{"body": "cylinder_1", "edge": "inside"}, {"body": "cylinder_2", "edge": "inside"}, {"body": "cylinder_3", "edge": "inside"}, {"body": "cap", "edge": "inside"}]}},
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{"command": "model.set.create", "params": {"name": "F", "geometry": [{"body": "cylinder_1", "edge": "bottom"}]}},
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{"command": "model.set.create", "params": {"name": "A", "geometry": [{"body": "cap", "edge": "pole"}]}},
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{"command": "model.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 210000.0, "poisson": 0.3}}},
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{"command": "model.section.create", "params": {"name": "vessel", "set": "all", "material": "Steel", "kind": {"type": "axisymmetric"}}},
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{"command": "model.boundary_condition.create", "params": {"name": "F", "type": "symmetry", "target": {"set": "F"}, "plane": "y"}},
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{"command": "model.boundary_condition.create", "params": {"name": "A", "type": "symmetry", "target": {"set": "A"}, "plane": "x"}},
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{"command": "model.load.create", "params": {"name": "pressure", "type": "pressure", "set": "inside", "magnitude": 1.0}},
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{"command": "model.step.create", "params": {"name": "Pressure", "procedure": {"type": "static"}, "boundary_conditions": ["F", "A"], "loads": ["pressure"]}},
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{"command": "solve.static", "params": {}, "id": "solve"}
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]
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