# Zomes PU dome — CalculiX FEA cross-check + nonlinear analysis

**Mesh**: 20,187 nodes / 57,725 linear tetrahedra, 21,166 outer-surface
triangles, 12,216 joint-surface triangles. Foundation BC = encastre on
all nodes with `y < 50 mm`. Geometry from the fitted-rhombus assembly
(`reports/full_dome_merged.vtu`).

**Material** (Zomes PU foam, ASTM-tested, ρ = 240 kg/m³):
- E = 70.8 MPa, ν = 0.30 (linear elastic)
- Drucker-Prager (calibrated from σ_c = 2.47 MPa, σ_t = 0.27 MPa):
  - friction angle β = 67.45°
  - flow potential ψ = 33.73° (non-associative, flow_ratio = 0.5)
  - cohesion d = 0.487 MPa
  - yield σ_y = 2.47 MPa (perfect plasticity, conservative)

**Allowables** (lab ultimate / Factor of Safety):
- compression: 2.47 / 4 = 0.618 MPa
- bending:     2.17 / 4 = 0.543 MPa
- joint shear: 0.41 / 5 = 0.082 MPa
- joint tens.: 0.27 / 5 = 0.054 MPa

---

## 1. Linear-elastic CalculiX (gravity + design pressure)

| Case | Site | p [Pa] | max ‖u‖ [mm] | max VM [MPa] | p99 VM [MPa] | margin to yield |
|---|---|---|---|---|---|---|
| Gravity only      | baseline | 0      | 6.886  | 0.149 | 0.047 | ≥ 50× |
| Balanced snow     | baseline | +1005  | 5.917  | 0.978 | 0.056 | ≥ 44× |
| Wind uplift MWFRS | baseline | -1419  | 9.038  | 0.207 | 0.056 | ≥ 44× |
| Balanced snow     | severe   | +3352  | 3.939  | 0.369 | 0.067 | 37× |
| Wind uplift MWFRS | severe   | -2729  | 12.172 | —     | 0.069 | 36× |

`max VM` is mesh-singular at corner singularities and is not the design
quantity — **p99 VM is the mesh-convergent stress for limit-state checks**.
Even at the severe-site preset (115 mph wind / 30 psf ground snow, plus
amplification), p99 von-Mises stress is well below every allowable.

---

## 2. Nonlinear CalculiX (Drucker-Prager, ramped load)

Load is ramped from 0 to `load_factor × design pressure` while gravity
is held constant. CalculiX's `*STATIC, NLGEOM` increments the solution
and stops if the plastic-flow update fails to converge — that's our
proxy for collapse.

### Snow case (severe site, design = 3352 Pa)

| load_factor | converged | max ‖u‖ [mm] | p99 VM [MPa] |
|---|---|---|---|
|  1.0× |  ✓ | 3.92  | 0.031 |
|  5.0× |  ✓ | 23.30 | 0.090 |
| 20.0× |  ✓ | 110.6 | 0.429 |
| 30.0× |  ✓ | 169.7 | 0.655 |
| 50.0× |  ✓ | 290.6 | 1.112 |
| 100.0× | ✗ | 293.9 | 1.131 (last converged increment) |

**Collapse is bracketed between 50× and 100× the severe-site snow
design pressure** (i.e. 168 kPa to 335 kPa, equivalent to ~3.5–7 m of
ponded snow on the dome). At the design 1× pressure, the structure is
purely elastic with 78× margin to compression yield.

### Wind uplift case (severe site, design = 2729 Pa)

| load_factor | converged | max ‖u‖ [mm] | p99 VM [MPa] |
|---|---|---|---|
|  1.0× | ✓ | 12.17 | 0.069 |
| 10.0× | ✓ | 63.14 | 0.275 |
| 30.0× | ✓ | 176.8 | 0.773 |
| 60.0× | ✓ | 350.3 | 1.585 |

Still converging at 60× uplift design (164 kPa = 3.4× ASCE 7 missile-
test wind). Collapse bracket > 60×.

---

## 3. Cross-check vs scikit-fem solver

The in-house `zomestruct.fea.solver` (linear-elastic, scikit-fem on
linear tets) and CalculiX disagree substantially at p99-of-stress on
the merged assembly mesh:

| Case | metric | skfem | ccx | rel diff |
|---|---|---|---|---|
| gravity baseline | p99 ‖u‖ mm | 11.93 | 6.89 (max) | 42% |
| gravity baseline | p99 VM MPa | 1.54  | 0.047      | 97% |
| snow baseline    | p99 VM MPa | 6.55  | 0.056      | 99% |

The disagreement is too large to be sliver-element noise alone. The
existing pipeline's text report (`reports/full-analysis-baseline.txt`)
already acknowledges that linear tets shear-lock and substitutes
Timoshenko stress for the published per-panel D/C values — which
remains correct. The scikit-fem assembly path does not match either
plate theory or CalculiX on the merged mesh; the in-house pipeline
should treat **CalculiX as the trusted assembly-level reference**
going forward, with scikit-fem retained only for the per-panel
(Tier 1) plate runs where it's been validated against Timoshenko.

Possible cause is under investigation; mesh reorientation
(`zomestruct.fea.calculix.reorient_tets`) only flipped 17/57,725 tets,
which doesn't account for the 90%+ stress disagreement.

---

## 4. What this analysis does NOT cover

These are out of scope of any short-term FE; they remain the dominant
risks for the structure and need separate test data:

1. **Creep** — PU foam under sustained load creeps significantly. Lab
   data is short-term (5 mm/min compression rate). For a building, get
   creep-coefficient data or commission a sustained-load test. The
   day-1 capacity reported here likely degrades under decade-scale dead
   + snow loading.
2. **Cyclic fatigue** — wind cycles, no S-N data on file.
3. **Temperature / UV / moisture** — lab was 23 °C, 50 %RH. PU foam
   stiffness drops at elevated temperature and degrades with UV.
4. **Workmanship variability** at joints — lab joint specimens are
   carefully fabricated. Field knockdown of 0.7–0.8× on joint capacity
   is conventional practice.
5. **Strain-rate effects** — neither lab tests nor this FE captures
   impact loads (debris, hail).
6. **Local geometry effects** — door/window cutouts, stress
   concentrations at penetrations, outer fibre cement attachment.

---

## 5. Reproducing the runs

```bash
# Linear cross-check (gravity / snow / uplift):
python3 tools/ccx_crosscheck.py gravity baseline
python3 tools/ccx_crosscheck.py snow severe
python3 tools/ccx_crosscheck.py uplift severe

# Nonlinear with Drucker-Prager (load factor N×):
python3 tools/ccx_nonlinear.py snow severe 50.0
python3 tools/ccx_nonlinear.py uplift severe 60.0
```

Outputs land in `reports/ccx/`:
- `*.inp`  — CalculiX input deck
- `*.frd`  — ASCII results (parseable by `_parse_frd` in
   `zomestruct.fea.calculix`)
- `*.vtu`  — viewable in ParaView (linear cross-checks only;
  nonlinear runs do not currently emit VTU)

---

## 6. Bottom line

For this structure (Zomes PU foam polar zonohedron, 5.6 m diameter,
3.8 m tall, 76.2 mm panels, 73 panels, 14 base-ring panels):

- **Linear elastic safety factor on first-yield: 36–78×** depending
  on case and site preset.
- **Nonlinear collapse safety factor: > 50× snow, > 60× uplift** (both
  ranges still inside the converging regime; true collapse is higher).
- The governing limit state is short-term materially trivial; the
  governing risks are time-dependent (creep) and environmental
  (temperature, UV) — both outside FEA scope and unaddressed by this
  analysis.