Truss topology and catalogue-section sizing

This is an educational linear-elastic, pin-jointed 2-D truss model. It is not a code-compliant structural design, does not replace connection, fatigue, imperfection, second-order, fire, durability, fabrication, or foundation checks, and must not be used for construction.

Within that boundary, the tutorial tackles a genuinely awkward global-search problem: choose a sparse subset of a ground structure, assign a real circular hollow section to every retained member, move selected nodes, reject mechanisms without inventing response data, and compare mass, displacement, and single-member-removal behavior.

The publication run found:

• a feasible 1,789.319 kg scalar design with differential evolution, down 57.30% from the explicit 4,190.377 kg triangulated seed;
• a three-bar equilibrium/virtual-work oracle agreeing with the FEM to 7.28e-12 N in force and exactly at stored precision in displacement;
• a descriptor-pilot rejection—only 99 / 384 mixed-generator candidates were feasible, 95.96% of feasible designs had zero removal survival, and neither emergent descriptor pair approached the frozen 40% minimum per-arm coverage gate; and
• 32 feasible nondominated population representatives from constrained MODE, spanning 1,664.421–4,191.366 kg and 2.987–7.737 mm, while every retained point still lost at least one load path under single-member removal.

That last pair of results matters. The finite optimization found light, serviceable intact trusses; it did not demonstrate structural robustness.

Frozen reference problem

The deterministic 6 × 3 lattice has 18 nodes over a 12 m span. Every node pair no more than 5 m apart creates one candidate, yielding 75 possible members. The left support is pinned, the right support is a vertical roller, and two interior upper nodes carry the service loads.

Two simultaneous load cases are optimized:

The holdout changes kind rather than merely changing a random seed: both load components are multiplied by 1.10, Young’s modulus is reduced by 10%, and the roller settles by 5 mm. It is used by the descriptor pilot, never by scalar or MODE selection.

Twelve nominal CHS sizes form the section catalogue. The designations and dimensions are representative of the EN 10210 family, while area, second moment of area, radius of gyration, and mass are recomputed from ideal circular geometry in Rust. sections.csv is the checked-in numerical contract and PROVENANCE.md states what is sourced, computed, and merely illustrative.

Mixed discrete and continuous decoding

For M = 75 candidate members and N = 10 movable nodes, the normalized decision vector has

1 + M topology ranks + M section keys + 2N offsets = 171 coordinates.

The first coordinate selects an exact cardinality k ∈ [8, 40]. The k smallest topology ranks become active, with member index resolving ties. Every active member’s section key maps through 12 equal-width bins. Ten non-support, non-load nodes may move by at most ±0.36 m horizontally and ±0.30 m vertically.

This decoder keeps continuous optimizers in a simple box while enforcing exact member count and valid catalogue indices by construction. Tests cover both endpoints, equal bin occupancy, non-finite rejection, tie order, exact cardinality, and fixed load-node geometry.

Structural analysis and its failure contract

Each active bar contributes the standard 2-D axial stiffness

(EA/L) [ c²   cs  −c²  −cs
         cs   s²  −cs  −s²
        −c²  −cs   c²   cs
        −cs  −s²   cs   s² ].

After the three support degrees of freedom are removed, the reference problem has a 33 × 33 reduced stiffness matrix. A symmetric eigensolve establishes:

• numerical rank with tol = n · ε · max(λmax, 1);
• positive definiteness; and
• spectral reciprocal condition rcond = λmin / λmax ≥ 1e-10.

Only then does Cholesky solve the load cases. Physical output includes axial stress, Euler-buckling utilization for compressive bars, displacement, compliance, and per-member governing utilization. Prescribed settlement uses the partitioned right-hand side Ff − Kfc uc.

Disconnected, singular, ill-conditioned, and solve failures are typed. Stress, buckling, and displacement remain absent after such a failure. Finite sentinels exist only to transport constraints through optimizers; they are not published as fictitious physics.

Validation before optimization

The independent oracle is a symmetric three-bar triangle with half-span 2 m, height 3 m, and a 100 kN apex load. Joint equilibrium gives

diagonal force = −P sqrt(a² + h²) / (2h)
base tie force = Pa / (2h).

The apex displacement is derived separately by unit-load virtual work, δ = Σ Nₑ nₑ Lₑ / (EAₑ). The numerical evidence is:

The bit-exact displacement row is expected: the equilibrium/virtual-work path uses analytic forces and lengths, and its result rounds to the same f64 as the FEM solve. Independence is visible in the force rows, whose separately derived values differ by 7.28e-12 N; the zero is not evidence that the two displacement paths share an implementation.

An additional invariant applies the same vertical settlement to both supports: the triangle translates rigidly and develops no member force.

The conditioning threshold is a declared modeling policy, not a hidden solver side effect. The triangulated reference has rcond = 4.07e-3; the checked-in sweep shows which stricter thresholds would alter its classification.

Equal-budget scalar comparison

The scalar objective minimizes mass plus squared penalties for positive constraint residuals. A deterministic 40-member triangulated design with the largest section is the explicit baseline. CMA-ES, differential evolution, and BiteOpt start near that construction; a fourth BiteOpt arm starts uniformly to expose the cost of omitting structural seeding.

Each arm requested 2,048 objective calls over eight retries. Population completion explains the small actual-call overshoot:

The infeasible uniform result is retained as evidence rather than silently discarded. Its low physical mass does not make it a structure. In arms.csv, metrics_available=0 means the typed connectivity or stiffness failure deliberately stopped analysis; NaN in rcond, stress, buckling, and displacement is therefore “not computed,” not missing successful-analysis data.

The lowest-mass feasible scalar design has 36 members, maximum stress utilization 0.9041, buckling utilization 0.7132, maximum displacement 14.428 mm, and rcond = 1.13e-3.

Removal robustness and the descriptor gate

For every active member, the expensive robustness pass removes that member and re-solves every load case. If all removals survive,

degradation = worst removal compliance / intact compliance − 1
survival    = intact compliance / worst removal compliance.

Any failed removal maps to a capped optimizer degradation of 100 and survival zero, while the number of failed removals remains explicit.

The pre-registered pilot considered:

• D1: depth/span × removal survival;
• D2: member-utilization spread × removal survival; and
• D3: active count × mass as a decision-led negative control.

It used the same derived 12 × 10 geometry as a potential 120-cell archive, three deterministic seed arms, and the kind-changing settlement holdout.

The first protocol used only local perturbations of the 40-member baseline and a [0, 1] survival bound. Review correctly showed that this could not separate a weak descriptor from a narrow generator. Protocol revision 2 therefore freezes a mixed generator—75% structured-local candidates and 25% broad candidates with uniform topology ranks and node offsets, maximum cardinality, and conservative catalogue sections. It also scales both survival axes to [0, 0.30] and D2 utilization spread to [0, 0.30]. This is a declared post-v1 diagnostic revision, not an independent confirmation of a new acceptance claim.

The broad component produced 38 / 96 feasible observations, while the local component produced 61 / 288. It widened reachable depth/span to 0.3333–0.3790 and survival to 0–0.2474, so the revised sample really did leave the baseline neighborhood. The rejection nevertheless became more diagnostic: 95 of 99 feasible designs still had exactly zero survival.

The gate requires |ρ| < 0.7, less than 10% clipping on each axis, minimum per-arm coverage above 40%, and both normal and coarse holdout niche retention above 60%. D1 and D2 fail both lower clipping and coverage; D2 also fails holdout retention. The QD run is therefore represented by a schema-valid skipped manifest with actual_evaluations: null and no stale archive. Implementing MAP-Elites does not authorize presenting a weak repertoire as a result; here the failed gate prevents the planned MAP-Elites stage from executing at all.

Constrained multi-objective result

MODE minimizes four quantities:

1. structural mass;
2. maximum intact displacement;
3. single-member-removal compliance degradation; and
4. active-member count.

Connectivity, mechanism, conditioning, stress, and buckling remain explicit constraints feasible at ≤ 0. The 256-call run performed 11,808 factorizations and 22,448 load-case solves because each candidate expands into an intact solve and up to 40 removal analyses.

The retained population contains a 1,785.221 kg point, only 0.23% from the best feasible scalar mass, satisfying the registered mass-consistency check. However, all 32 retained points have capped degradation 100 because at least one removal creates a failed load path. The visible front is therefore mainly a mass–displacement trade-off, not evidence of a redundant truss.

Carbon is an indicative output, not an EPD

The tutorial multiplies steel mass by a fixed 1.70 kg CO2e/kg illustrative factor. It does not model recycled content, production route, fabrication, transport, coating, maintenance, end of life, or project geography. Carbon is therefore perfectly correlated with mass here and is not claimed as an independent optimization result. Replace the factor with verified, project-specific EPD data before making comparative claims.

Reproduce

From this directory:

cargo test --locked
cargo clippy --all-targets --locked -- -D warnings
cargo deny check licenses

cargo run --release --locked -- \
  --mode all --preset publication --workers 0 \
  --output results/publication

../../.venv/bin/python plot_results.py --write
../../.venv/bin/python plot_results.py --check

--workers 0 uses available candidate parallelism. --preset smoke is the short local/CI path. --evaluations N overrides the selected mode’s candidate/sample budget. The native program owns all mechanics, optimization, metrics, and full-precision CSV/JSON. Python only checks those artifacts and renders SVG.

The result layout follows ../RESULT_SCHEMA.md:

results/publication/
  protocol.json
  sections.csv
  validation/{run.json,oracle.csv,condition_sensitivity.csv}
  so/{run.json,arms.csv,convergence.csv,best_members.csv}
  pilot/{run.json,pilot.csv,pilot.md}
  qd/run.json
  mo/{run.json,pareto.csv,convergence.csv}

The implementation was staged from the reviewed truss-sizing-implementation-plan.md. That plan remains a development record outside the published book; this README and the versioned artifacts are the publication contract.