forked from 1b-agent/fieldcraft-upstream
Christian Svensson
65066b8451
The Z-section cantilever under an end torque as five hex20 blocks (two flanges, two corner blocks, the web; two elements through every wall): axial stress at A, on the mid-surface of the lower flange at its free edge, -109.45 against -108 (2%). The solid model converges to about -109.7, 1.6% beyond the thin-walled reference, as ESRD's converged p-version solid (-109.5) does. The README also records the MITC4 mid-surface shell model of the same problem, which converges to about -111.3 (3.1%), and why the case is the solid. Part of #97.
101 lines
5.9 KiB
Markdown
101 lines
5.9 KiB
Markdown
# NAFEMS LE5 Z-section cantilever (hex20)
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Our model of the published problem: a cantilever 10 m long (x) whose cross-section is a Z of
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0.1 m thick walls, E = 210 GPa, ν = 0.3 (m, MN, MPa). On the mid-surface the web is 2 m wide, in
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the plane z = 0 from y = −1 to y = 1, and the two flanges are 1 m high: the upper one at y = 1
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rises to z = 1, the lower one at y = −1 drops to z = −1. Every displacement is zero at x = 0. A
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torque of 1.2 MN·m acts at x = 10 as two shear forces of 0.6 MN, one along each flange's end
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(+z on the upper flange, −z on the lower), 2 m apart. The target is the axial stress σ_xx at A,
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x = 2.5 on the mid-surface of the lower flange at its free edge (2.5, −1, −1), where the restrained
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warping of the section puts its largest axial stress.
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## The mesh
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The walls are solid hex20 blocks (`mesh.block`), the route LE1 and LE10 take: the issue allows
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quad4 shells or hex20, and the shell model reaches a different limit (below). Five blocks,
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2 elements through every wall: the two flanges (0.95 m outside the web, k = 4 elements), the two
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corner blocks where a flange meets the web (2 × 2), and the web between them (8 elements); 32
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elements along the length. The blocks share their faces, so `mesh.block` joins them. The shear
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force on each flange is a total over the flange block's end face (z from ±0.05 to ±1), split by
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area. A is a node: the mid-plane of the lower flange (y = −1) is a node plane of its 2 elements
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through the thickness.
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The recipe was generated with this script (`python3 le5.py 4 > recipe.json`):
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```python
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import json, sys
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def le5_solid(k):
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"""NAFEMS LE5 Z-section as hex20 solid blocks (m, MN, MPa): mid-surface web 2 wide in z = 0,
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flanges at y = ±1 reaching z = ±1, all walls 0.1 thick, length 10 along x; k sets the density
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(8k cells along the length, k per 0.95 m across the walls, 2 through the thickness)."""
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h = 0.05
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blocks = { # name: (y0, y1, z0, z1, ny, nz)
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"lower_flange": (-1 - h, -1 + h, -1.0, -h, 2, k),
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"lower_corner": (-1 - h, -1 + h, -h, h, 2, 2),
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"web": (-1 + h, 1 - h, -h, h, 2 * k, 2),
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"upper_corner": (1 - h, 1 + h, -h, h, 2, 2),
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"upper_flange": (1 - h, 1 + h, h, 1.0, 2, k),
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}
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commands = []
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for name, (y0, y1, z0, z1, ny, nz) in blocks.items():
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base = [[0.0, y0, z0], [0.0, y1, z0], [0.0, y1, z1], [0.0, y0, z1]]
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corners = base + [[10.0, y, z] for _, y, z in base]
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commands.append({"command": "mesh.block", "params": {"body": name, "kind": "hex20", "corners": corners, "divisions": [ny, nz, 8 * k], "faces": {"1": "root", "2": "tip"}}})
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return commands + [
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{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": b} for b in blocks]}},
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{"command": "model.set.create", "params": {"name": "root", "geometry": [{"body": b, "face": "root"} for b in blocks]}},
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{"command": "model.set.create", "params": {"name": "lower_tip", "geometry": [{"body": "lower_flange", "face": "tip"}]}},
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{"command": "model.set.create", "params": {"name": "upper_tip", "geometry": [{"body": "upper_flange", "face": "tip"}]}},
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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": "z_section", "set": "all", "material": "Steel", "kind": {"type": "solid"}}},
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{"command": "model.boundary_condition.create", "params": {"name": "root", "type": "fixed", "target": {"set": "root"}}},
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# the 1.2 MN·m torque: 0.6 MN along each flange's end face, 2 m apart
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{"command": "model.load.create", "params": {"name": "lower_shear", "type": "force", "target": {"set": "lower_tip"}, "vector": [0, 0, -0.6]}},
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{"command": "model.load.create", "params": {"name": "upper_shear", "type": "force", "target": {"set": "upper_tip"}, "vector": [0, 0, 0.6]}},
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{"command": "model.step.create", "params": {"name": "Torque", "procedure": {"type": "static"}, "boundary_conditions": ["root"], "loads": ["lower_shear", "upper_shear"]}},
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{"command": "solve.static", "params": {}, "id": "solve"},
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]
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k = int(sys.argv[1]) if len(sys.argv) > 1 else 4
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print("[\n " + ",\n ".join(json.dumps(c) for c in le5_solid(k)) + "\n]")
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```
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## Refinement
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Every row is the listing above with the row's k, solved and probed at A as the case does; the
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case mesh is k = 4:
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| k | elements | σ_xx at A | error |
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|---|---|---|---|
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| 1 | 128 | −108.93 | +0.86% |
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| 2 | 384 | −109.27 | +1.17% |
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| 4 (this case) | 1280 | −109.46 | +1.35% |
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| 8 | 4608 | −109.60 | +1.48% |
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The result converges to about −109.7, 1.6% beyond the target. The published value comes from a
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thin-walled idealisation of the section; a solid model of the same section differs from it by
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about that much (ESRD's guide reports −109.5 for its converged p-version solid model on two
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meshes, p. 16). The next refinement of the case mesh moves the value by 0.13%, under a tenth of
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the tolerance.
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A model of quad4 MITC4 shells on the mid-surface (walls meeting at the junction lines, the edge
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shears as consistent nodal forces) converges further away, to about −111.3 (320 elements along the
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length and 16 per metre across the walls: −111.65 and −111.06 at A from the two elements that meet
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there), 3.1% beyond the target (the Abaqus Benchmarks Guide, Table 4.2.5–1, reports −110.38 for its
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S4 shell on the refined mesh); we have not separated how much of the difference is the shell
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idealisation of the junctions. That is why this case is the solid model.
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## Tolerance
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2%: the target is published to three figures, and the solid model's converged value differs from
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the thin-walled reference by about 1.5% (above). `fixture.json` holds the k = 4 value: the mesh is
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built by `mesh.block` from 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-le5
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cargo xtask oracle --bless nafems-le5 # only after the case passes
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```
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