Christian Svensson
a8040063a6
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.
NAFEMS LE7 axisymmetric cylinder/sphere under pressure (axisymmetric quad8)
Our model of the published problem: a thin-walled vessel, a cylinder closed by a hemispherical cap, of mid-surface radius 1 m and wall thickness 0.025 m, E = 210 GPa, ν = 0.3 (m, MN, MPa), under an internal pressure of 1 MPa. In FieldCraft's axisymmetric models x is the radius and y the axis. The cylinder runs from F (y = 0) to the junction C (y = 1.5), the cap from C to the pole A (y = 2.5). The axial displacement is zero at F (the vessel's plane of symmetry: the whole bottom edge) and the radial one at A (the pole edge, on the axis). The target is the axial stress σ_yy on the outer surface at D (R = 1.0125, y = 1.4034), 0.0966 m below the junction, inside the bending boundary layer the cap's different stiffness causes.
The mesh
Four mesh.block quad8 patches through the wall thickness: three cylinder blocks with E
(y = 1.1136) and D as block corners, so D is a node, and the cap, a quarter ring whose two curved
edges are arcs about (0, 1.5) and whose fourth edge lies on the axis. 2 elements through the
wall, about 50 per metre along it (56, 14 and 5 in the cylinder blocks, 79 round the cap); the
blocks share their edges, so mesh.block joins them. The pressure acts on the inner edges of all
four blocks.
The recipe was generated with this script (python3 le7.py 2 50 > recipe.json):
import json, math, sys
def le7(nt, per_m):
"""NAFEMS LE7 cylinder with a hemispherical cap, axisymmetric (m, MN, MPa): x is the radius,
y the axis. Mid-surface radius 1, wall 0.025 thick; the cylinder runs from F (y = 0, the plane
of symmetry) to C (y = 1.5), the cap from C to the pole A (y = 2.5). nt elements through the
wall, about per_m elements per metre along it."""
ri, ro = 1 - 0.0125, 1 + 0.0125
cuts = [0.0, 1.1136, 1.4034, 1.5] # F, E, D, C: E and D are nodes
along = lambda length: max(1, round(length * per_m))
commands = []
for k, (y0, y1) in enumerate(zip(cuts, cuts[1:])):
commands.append({"command": "mesh.block", "params": {"body": f"cylinder_{k + 1}", "kind": "quad8",
"corners": [[ri, y0, 0], [ro, y0, 0], [ro, y1, 0], [ri, y1, 0]], "divisions": [nt, along(y1 - y0)],
"edges": [{"corners": [1, 4], "name": "inside"}, {"corners": [1, 2], "name": "bottom"}]}})
commands.append({"command": "mesh.block", "params": {"body": "cap", "kind": "quad8",
"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)],
"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"}]}})
bodies = ["cylinder_1", "cylinder_2", "cylinder_3", "cap"]
return commands + [
{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": b, "face": "area"} for b in bodies]}},
{"command": "model.set.create", "params": {"name": "inside", "geometry": [{"body": b, "edge": "inside"} for b in bodies]}},
{"command": "model.set.create", "params": {"name": "F", "geometry": [{"body": "cylinder_1", "edge": "bottom"}]}},
{"command": "model.set.create", "params": {"name": "A", "geometry": [{"body": "cap", "edge": "pole"}]}},
{"command": "model.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 210000.0, "poisson": 0.3}}},
{"command": "model.section.create", "params": {"name": "vessel", "set": "all", "material": "Steel", "kind": {"type": "axisymmetric"}}},
{"command": "model.boundary_condition.create", "params": {"name": "F", "type": "symmetry", "target": {"set": "F"}, "plane": "y"}},
{"command": "model.boundary_condition.create", "params": {"name": "A", "type": "symmetry", "target": {"set": "A"}, "plane": "x"}},
{"command": "model.load.create", "params": {"name": "pressure", "type": "pressure", "set": "inside", "magnitude": 1.0}},
{"command": "model.step.create", "params": {"name": "Pressure", "procedure": {"type": "static"}, "boundary_conditions": ["F", "A"], "loads": ["pressure"]}},
{"command": "solve.static", "params": {}, "id": "solve"},
]
nt, per_m = (int(a) for a in sys.argv[1:3]) if len(sys.argv) > 2 else (2, 50)
print("[\n " + ",\n ".join(json.dumps(c) for c in le7(nt, per_m)) + "\n]")
Refinement
Every row is the listing above with the row's divisions, solved and probed at D as the case does; the case mesh is 2 × 50:
| through the wall × per metre | σ_yy at D | against 25.86 | against 25.27 |
|---|---|---|---|
| 2 × 25 | 25.331 | −2.05% | +0.24% |
| 2 × 50 (this case) | 25.283 | −2.23% | +0.05% |
| 4 × 100 | 25.271 | −2.28% | +0.00% |
| 8 × 200 | 25.269 | −2.29% | −0.00% |
Far from the junction the model is exact: at F the axial stress on the outer surface is 19.503 (p r_i²/(r_o² − r_i²)) and the hoop stress 39.006 (Lamé).
Two references, one known failure
The published target, 25.86, comes from axisymmetric thin-shell theory. The ESRD guide solves the
same problem with axisymmetric solid elements and reports 25.27 on two meshes, 2.3% below the
target, and our solid model converges to the same 25.27: the difference is the model, not the
mesh. So the case keeps the NAFEMS target with the
tolerance a shell model would earn, 1%, marked known_failure until FieldCraft has an
axisymmetric shell element, and checks the solid model against the solid reference, also at 1%.
fixture.json holds the passing quantity's 2 × 50 value: the mesh is built by mesh.block from
the recipe's corners, so it is the same on every platform.
Running it
cargo xtask oracle --case nafems-le7
cargo xtask oracle --bless nafems-le7 # only after the case passes