[Shizheng-Wen]

Summary

Adds examples/solid/plasticity_cylinder/: the elastic-plastic thick-walled cylinder under internal pressure (Hill 1950), in plane strain, with J2 plasticity and linear isotropic hardening (perfect plasticity by default).

What it adds over the existing plasticity_strip rung (L-BFGS on an incremental energy):

• Newton's method with the consistent tangent of the radial return (closed form, de Souza Neto et al. box 7.4), so every load step converges quadratically: one update for elastic steps, three to five for plastic ones.
• Quadrature-point state through the new quadrature_data argument (feat(assemble): pass per-quadrature-point data to the element and node assemblers #70): the stress feeds an internal-force NodeAssembler, the tangent moduli a stiffness ElementAssembler.
• Pressure along the facet normal n of a FacetAssembler (fix(assemble): select boundary facets by incidence and add facet normals #69).
• Displacement control with a linear predictor (Condenser.update_dirichlet), which follows the collapse plateau; --control pressure is the load-controlled variant.
• A NumPy-built quarter ring of six-node triangles (no gmsh); the mid-edge nodes of the arcs lie on the circles.

The default run (825 nodes, 30 load steps) takes about 2 s on a CPU. It reaches 199.7 MPa, within 0.2 % of the limit pressure (2/√3) σ_y ln(b/a) = 200.1 MPa, approached from below.

Files

• examples/solid/plasticity_cylinder/plasticity_cylinder.py, README.md
• tests/assemble/test_plasticity_cylinder_example.py
• docs/source/example_gallery/solid/plasticity_cylinder.rst plus the solid-mechanics index (new rung, solver pattern, card), with the figure and card image in _static/solid_mechanics/
• zh_CN: a new catalog for the page; the index catalog is edited by hand, so the diff only touches the changed entries

Tests

tests/assemble/test_plasticity_cylinder_example.py:

• Return map: stress on the yield surface; elastic points untouched; deviatoric flow; tangent equal to torch.func.jvp of the stress update (H = 0 and H > 0).
• Stiffness: assembled tangent against central differences of the residual.
• Mesh: area of the quarter ring, the two circles, the symmetry edges.
• Elastic range: stiffness p / u_r(a) against Lamé, under pressure and displacement control.
• First yield: no plasticity below the first-yield displacement; just above it, only in the innermost elements.
• Limit pressure (perfect plasticity): within 1 %, from below; at most six Newton updates per step.
• CUDA vs CPU: skipped without a GPU.

Results on Clariden, CPU partition, float64:

• 8 passed (the node exposes GPUs, so the CUDA test ran).
• The example ran in all three modes: default, --control pressure and --hardening 2000.
• Docs (EN and zh) built against main: no new warnings once the title underline was fixed.

🤖 Generated with Claude Code

https://claude.ai/code/session_01Dst8zhPLnxVpZMBhR4Pvfs