LE4: Pressurised hyperbolic shell

Status: failed

Linear elasticity / LE4

Scope

Check meridional and hoop stress in a pressure-loaded hyperbolic shell.

Geometry

Waist radius 1 m; height 1 m; radius at height z is sqrt(1 m² + z²). Uniform thickness 0.01 m. K is at the waist, z=0.

Material

Isotropic elasticity E=210 GPa and Poisson ratio 0.3, following the manual material specification. The linked esa2sfsf.inp instead contains E=2.10 GPa; uniform elastic-modulus scaling changes displacements but not these force-controlled linear-elastic stress observables.

Boundary conditions and loading

Suppress axial translation at K (the waist ring in the three-dimensional model), leaving its radial displacement and bending rotation free. Apply 1 MPa uniform internal pressure and meridional symmetry on the cut planes.

Assumptions

Static small-strain thin-shell response with no thermal or inertial loading.

AM-DT formulation

AM-DT S4 quarter-shell with analytical midsurface normals. Source discrepancy: the manual prose locates the axial support at B, whereas esa2sfsf.inp constrains axial DOF 2 at node 21, the waist K. The approved interpretation uses K, not B. The two published signed stress targets remain unchanged.

Measurement location and recovery

At K and the angular midpoint, recover the midsurface section stress from the two incident cells. Project onto the axial meridional direction and the circumferential tangent separately; do not substitute a stress magnitude.

Acceptance policy

Predeclared AM-DT policy: 2% error and 2% refinement change for each component independently. Some published SAX2 hoop results differ by 3.3%; retain the stated target and fail this policy rather than silently adopting those computed values as references.

Original schematic of the benchmark geometry and loading
Original AM-DT schematic; dimensions and loading follow the cited specification.

Reference observables

meridionalStressAtK_Pa = -5e+07 Pa

hoopStressAtK_Pa = 5e+07 Pa

AM-DT solver results

meridionalStressAtK_Pa

ElementsDOFsAM-DT valueReference
10246534-50140171.7 Pa-50000000
409625350-50087160.1 Pa-50000000

Normalized error: 0.1743%; refinement change: 0.1060%.

hoopStressAtK_Pa

ElementsDOFsAM-DT valueReference
1024653448506226.1 Pa50000000
40962535048478290.6 Pa50000000

Normalized error: 3.0434%; refinement change: 0.0559%.

Run UTC: 2026-10-07T02:51:00.835972+00:00; wall time: 5.57 s.

Reproducibility: revisions and artifact hashes
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    "libdtConstantsServer.so": "8f9f9ba6dfee5f46be83995f442fc51ec94b01c1c3d08fdedad4e39a26a1e057",
    "libdtCrystalPlasticity.so": "6130b4e7c59129b4e8bd37ae445f0c6e73c6fd6e255a52662d71eb8f8a9a9e8f",
    "libdtFiniteElementMethod.so": "f595989347e56b55925e0d6bae52406d4756392392a816dcc951af01f727af50",
    "libdtLITRISOL.so": "202a10e6de1058a033ebeb7a1febbdf73002c24bc75493e38b2b0fb668ff9aaa",
    "libdtMechanicalSolver.so": "6e574945666c4e1df2b5c9b8b37ab089c3414ffb3e8bc94e325380d851798d33",
    "libdtPartScaleMechanicalServer.so": "5a2ea08c7eff48703eec23fc1584f860aebf7ffe722e0951a74e33888f851ae6",
    "libdtPartScaleThermalServer.so": "bca1dcad80e389f5df3e250803771d7630d6b0405997c6b77036ef2bc9f29050",
    "libdtScanPathGeneratorServer.so": "fef36783aebc2f5f4edb3358e727524ecb9cc41f1cdd76b326467a6ba0eeb91f",
    "libpython3.11.so.1.0": "f72bbb1a96cb12d3f39d0a7acd58320b5bf8681d4d26bfc05f43edb4a1cfdc66",
    "libpython3.9.so.1.0": "f10e20c9adbff54099aa873247183816d9250e35fb3100b58766d15493ba7a35"
  }
}

Source: NAFEMS LE4 in the Abaqus benchmark manual.