OneCAD-Simulation/tests/oracle/pinched-cylinder

Pinched cylinder with rigid diaphragms

Our model of the published problem: a cylinder of radius 300, length 600 and thickness 3, E = 3 × 10⁶, ν = 0.3, capped by rigid diaphragms (u_y = u_z = rot_x = 0 at the ends) and pinched by two opposite unit forces at mid-length, one on each of two diametrically opposite generators. Reference: 1.8248 × 10⁻⁵ radial displacement under the load (Belytschko, Stolarski, Liu, Carpenter and Ong, Stress projection for membrane and shear locking in shell finite elements, CMAME 51 (1985) 221–258, the shell obstacle course) — the value the case already cited.

The eighth model: x from the loaded section (x = 0, symmetry in x) to the diaphragm (x = 300), φ from the loaded generator (φ = 0, symmetry in y) to the side (φ = 90°, symmetry in z), with R sin φ = y and R cos φ = z on the mid-surface R = 300. A quarter of the unit force, −0.25 in z, acts on the single loaded node (0, 0, 300) — the model the solver's own pinched-cylinder benchmark builds in code.

The mesh

mesh.block's 2D form takes its four corners at z = 0, so it cannot mesh a patch on a cylindrical surface (it can be curved in its own plane, but it is planar): the recipe writes the 1089 nodes and 1024 quad4 elements one by one with mesh.node.add and mesh.element.add, the route docs/verification.md allows ("nodes and elements written one by one"). Node 1 + i + 33 j lies at (300 i/32, 300 sin(90° j/32), 300 cos(90° j/32)); element 1 + i + 32 j is [node(i, j), node(i + 1, j), node(i + 1, j + 1), node(i, j + 1)], counter-clockwise seen from outside, so its normal points away from the axis — the same mesh the solver's own pinched-cylinder benchmark builds in code (crates/solver/tests/benchmarks.rs).

The recipe was generated with the same script as scordelis-lo-roof-shell (python3 cylinder_recipe.py pinched-cylinder 32):

import json, math

def pinched_cylinder(n):
    label = lambda i, j: 1 + i + (n + 1) * j
    commands = []
    for j in range(n + 1):
        for i in range(n + 1):
            x, phi = 300.0 * i / n, (math.pi / 2) * j / n
            commands.append({"command": "mesh.node.add",
                             "params": {"label": label(i, j),
                                        "xyz": [x, 300.0 * math.sin(phi), 300.0 * math.cos(phi)]}})
    for j in range(n):
        for i in range(n):
            nodes = [label(i, j), label(i + 1, j), label(i + 1, j + 1), label(i, j + 1)]
            commands.append({"command": "mesh.element.add",
                             "params": {"label": 1 + i + n * j, "kind": "quad4", "nodes": nodes}})
    return commands + [
        {"command": "model.set.create", "params": {"name": "shell", "elements": list(range(1, n * n + 1))}},
        {"command": "model.set.create", "params": {"name": "diaphragm", "nodes": [label(n, j) for j in range(n + 1)]}},
        {"command": "model.set.create", "params": {"name": "middle", "nodes": [label(0, j) for j in range(n + 1)]}},
        {"command": "model.set.create", "params": {"name": "top", "nodes": [label(i, 0) for i in range(n + 1)]}},
        {"command": "model.set.create", "params": {"name": "side", "nodes": [label(i, n) for i in range(n + 1)]}},
        {"command": "model.set.create", "params": {"name": "pinch", "nodes": [label(0, 0)]}},
        {"command": "model.material.create", "params": {"name": "Mat", "elastic": {"type": "isotropic", "young": 3e6, "poisson": 0.3}}},
        {"command": "model.section.create", "params": {"name": "shell", "set": "shell", "material": "Mat", "kind": {"type": "shell", "thickness": 3.0}}},
        {"command": "model.boundary_condition.create", "params": {"name": "diaphragm", "type": "displacement", "target": {"set": "diaphragm"}, "values": [None, 0, 0, 0, None, None]}},
        {"command": "model.boundary_condition.create", "params": {"name": "middle", "type": "symmetry", "target": {"set": "middle"}, "plane": "x"}},
        {"command": "model.boundary_condition.create", "params": {"name": "top", "type": "symmetry", "target": {"set": "top"}, "plane": "y"}},
        {"command": "model.boundary_condition.create", "params": {"name": "side", "type": "symmetry", "target": {"set": "side"}, "plane": "z"}},
        {"command": "model.load.create", "params": {"name": "pinch", "type": "force", "target": {"set": "pinch"}, "vector": [0, 0, -0.25]}},
        {"command": "model.step.create", "params": {"name": "Load", "procedure": {"type": "static"}, "boundary_conditions": ["diaphragm", "middle", "top", "side"], "loads": ["pinch"]}},
        {"command": "solve.static", "params": {}, "id": "solve"},
    ]

print("[\n " + ",\n ".join(json.dumps(c) for c in pinched_cylinder(32)) + "\n]")

A fixed-mesh benchmark, not a convergence study

The quantity is the deflection under a point load, and there a Reissner–Mindlin (MITC4) shell carries an unbounded transverse shear force: the deflection under the load keeps growing with refinement instead of converging. The published 1.8248 × 10⁻⁵ is a thin-shell value, and this case compares against it on the conventional 32 × 32 eighth mesh — the mesh the solver's own pinched-cylinder benchmark uses — not as a limit. The table shows what refinement actually does (each row is the listing above with the 32 in its last line replaced by the row's divisions and run through cargo xtask oracle as a scratch case):

cells	u	vs reference
8 × 8	−1.3559 × 10⁻⁵	−25.7%
16 × 16	−1.6942 × 10⁻⁵	−7.2%
24 × 24	−1.7735 × 10⁻⁵	−2.8%
32 × 32 (this case)	−1.803845342860719 × 10⁻⁵	−1.15%
48 × 48	−1.8275 × 10⁻⁵	+0.15%
64 × 64	−1.8369 × 10⁻⁵	+0.66%

The sequence rises through the reference between 32 × 32 and 48 × 48 and keeps rising at the two finer meshes, which is what a shear-deformable shell under a point load must do.

The solver's in-code benchmark of the same problem shows the same ascent on its 8 × 8, 16 × 16 and 32 × 32 eighth meshes (it gates 32 × 32 within 2%), so the command route and the in-code route agree on the fixed-mesh comparison.

Tolerance

2% covers the agreement with the thin-shell reference at the 32 × 32 eighth mesh (1.15% low), the conventional mesh of the shell obstacle course and the one the solver's own pinched-cylinder benchmark gates at 2%, so the command route and the in-code route are compared on equal terms. The reference itself is quoted by Belytschko et al. to four figures (1.8248 × 10⁻⁵). Because the point load makes the deflection mesh-dependent rather than convergent (above), this is a fixed-mesh tolerance and not a convergence statement: the two finer meshes sit 0.15% and 0.66% above the reference and stay inside it. The mesh is not marked slow: the case solves in about 0.15 s.

fixture.json is the accepted value at 32 × 32 (−1.803845342860719 × 10⁻⁵) with the runner's default band: the mesh is written by the recipe, not generated from geometry, so it is bit-identical on every platform and the result is frozen.

Running it

cargo xtask oracle --case pinched-cylinder
cargo xtask oracle --bless pinched-cylinder   # only after the case passes