Design Of Experiments

Design of Experiments (vehicle-design/mdo/design-of-experiments)

Use when the task is design of experiments (DOE) for an aircraft or spacecraft conceptual design study: planning a small structured set of simulation runs or wind tunnel experiments that maps the design space, screens the factors, and estimates their effects before optimization. This leaf builds coded design matrices in pure Python stdlib and analyzes the responses. It pairs with vehicle-design/mdo/multidisciplinary-optimization, which consumes the screened factor set and the baseline point that DOE identifies.

Domain quick reference

• Coded levels: a two-level factor codes low as -1 and high as +1, so contrasts are direct; a factor with m >= 3 settings enumerates the levels 0 .. m-1; latin hypercube samples are continuous in [0, 1]^k.
• Full factorial: every combination of the per-factor level counts, 2^k runs for k two-level factors. Supported up to k = 7 (128 runs).
• Half-fraction 2^(k-1), k = 4..7: built from one defining relation word, default I = ABC...K (the first k letters), so the principal fraction satisfies product over the word letters = +1 on every run. The default word length is k, resolution k. At k = 4 the defining relation I = ABCD aliases A with BCD and AB with CD, so all main effects stay clear of the two-factor interactions at half the runs of the full design.
• Latin hypercube: n samples over k factors, each factor column a random permutation of the n strata drawn from a fixed seed, each sample at its stratum midpoint (perm + 0.5) / n. Every factor covers its strata exactly once and all rows are distinct.
• Central composite: 2^k factorial runs plus 2k axial runs at +-alpha on one axis at a time plus center runs at the origin; run count is 2^k + 2k + center. Rotatable alpha = (2^k)^(1/4) is the default ('axial'); alpha = 'faced' puts the axial runs on the cube faces.
• Main effect: mean response at the high level minus the mean at the low level, twice the linear regression coefficient of that factor. Interaction effect: the same contrast on the product x_i * x_j. On y = 2A + 1.5B + A*B with A, B coded -1/+1, the main effect of A is 4.0, of B is 3.0, and the A-by-B interaction effect is 2.0.
• FAR-25 and CS-25 set the certification context (the design point and the loads margins that the screened factors feed); the DOE methods are standard statistical design methodology, summary only.

Workflow

1. Choose the factors and their ranges from the conceptual study (wing planform, loading, thrust, configuration settings) and code each factor to its low and high level.
2. Screen with full_factorial or fractional_factorial_2k when the run budget is tight: a 2^(k-1) half-fraction for k = 4..7 halves the runs and keeps the main effects clear of the two-factor interactions; verify the principal fraction with check_principal_fraction.
3. When the factors are continuous and the budget still exceeds the factorial size, spread the runs through the design space with latin_hypercube at a fixed seed for a reproducible sample.
4. Around a chosen baseline design point, refine with central_composite to fit curvature: factorial plus axial plus center runs, rotatable alpha by default.
5. Run the simulation or experiment at every row of the design and collect the responses in the same order.
6. Rank the factors with analyze_main_effects and check the pairwise coupling with analyze_interactions, then carry the top-ranked factors and the baseline point into the MDO loop.
7. Assemble any design through build_design_matrix, which returns the coded matrix together with the run count for the budget check.

Worked example

Screening a wing planform study with two factors, aspect ratio A and sweep B, each at two levels coded -1/+1. The full factorial has four runs: (-1, -1), (-1, +1), (+1, -1), (+1, +1). Let the aerodynamic response follow y = 2A + 1.5B + A*B, giving y = -2.5, -1.5, -0.5, 4.5. analyze_main_effects returns the A effect 4.0 (high mean 2.0 minus low mean -2.0), the B effect 3.0, and the ranking ["A", "B"]; analyze_interactions returns the A-by-B effect 2.0. The coded levels and these exact numbers are asserted in the contract test.

For a space-filling check, latin_hypercube(10, 2, seed=5) returns 10 distinct rows in [0, 1]^2 whose first row is (0.25, 0.45), and each column covers the 10 strata exactly once. For refinement around the baseline, central_composite(2, center=1) returns 2^2 + 4 + 1 = 9 runs: the four factorial corners, the axial points at +-sqrt(2) = 1.4142 on each axis, and one center run at (0, 0).

Pitfalls

• Reading alias structure off an unverified fraction: the half- fraction is built from one defining relation word (I = ABC...K default), so main effects and interactions alias; verify the principal fraction with check_principal_fraction before interpreting effects (at k = 4, AB aliases CD).
• Forgetting the run budget on a full factorial: 2^k runs explode past the supported window (k = 7 caps at 128 runs); a tight budget calls for the 2^(k-1) half-fraction, not a truncated full design.
• Trusting an unreproducible latin hypercube: the samples come from random.Random with a fixed integer seed, so the matrix only reproduces when the seed is fixed; an unseeded or re-seeded run changes the sample plan.
• Analyzing effects on a non-two-level design: the main and interaction effect contrasts assume coded -1/+1 levels, and the analysis raises ValueError on non-two-level designs.
• Reading the effect as a coefficient: the main effect is twice the linear regression coefficient (mean high minus mean low), so on y = 2A + 1.5B + AB the A effect is 4.0, not 2.0.
• Forgetting the center runs in a central composite: the run count is 2^k + 2k + center and rotatable alpha = (2^k)^(1/4) is the default; a study planned without the center runs cannot fit curvature.

Verification

• Deterministic: all random draws come from random.Random with a fixed integer seed, so every matrix is reproducible run to run.
• Rejection: ValueError on empty level lists or level counts below 2, k outside the supported windows, fractions other than 1 or 2, bad generator words, n_samples below 2, k_factors below 1, center runs below 1, non-positive alpha, non-two-level analysis designs, and response counts that do not match the run count.
• The exact worked-example numbers above are asserted in scripts/test_doe_logic.py together with the principal-fraction identity, the latin hypercube stratification, the central composite run count formula, and every ValueError case.

Related leaves

• vehicle-design/mdo/multidisciplinary-optimization: consumes the screened factor set, the ranked effects, and the baseline point that this leaf produces; DOE maps the design space before the optimizer closes the coupling loop.
• vehicle-design/cost-estimation/parametric-cost: the design point screened here feeds the cost CERs that rank the configurations.
• cross-cutting/numerics/monte-carlo-sampling: draws many random samples to estimate output distributions, where this leaf plans few structured runs to estimate factor effects.
• cross-cutting/numerics/uncertainty-propagation: propagates input uncertainties through the analysis once the influential factors are known.

Behavior contract (gate 3)

Run: python3 skills/vehicle-design/mdo/design-of-experiments/scripts/test_doe_logic.py The stdlib unittest contract covers 33 checks: full factorial coding and run counts, half-fraction construction with the principal-fraction identity for k = 4..7, latin hypercube uniqueness and stratification at fixed seeds, central composite layout and run count formula, the worked-example main effects and interaction effects, dispatch through build_design_matrix, and ValueError rejection of invalid inputs. Offline, deterministic, exit 0 in under a second.

Compliance

• Standards referenced, not reproduced: FAR-25 is US government work (public domain) and CS-25 is a free EASA download; the DOE methods (factorial, fractional factorial, latin hypercube, central composite, effect estimation) are common statistical design methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.