2. Design of experiments for CAE

In CAE-driven sensitivity analysis and metamodeling, the design of experiments (DoE) is the single most consequential decision made before any solver is launched. The DoE determines which points in the design space will be sampled, and therefore it fixes the information content available to every downstream activity: main-effect screening, variance-based sensitivity indices, and the training accuracy of a response surface or surrogate. A poorly chosen DoE cannot be rescued by a sophisticated metamodel; a well-chosen one often makes a simple surrogate sufficient. This chapter treats DoE as an engineering decision problem: which factors to vary, how to distribute samples across the design space, and how to generate and validate the sample set.

From Physical Test DoE to CAE DoE

Classical DoE grew out of physical experimentation, where replication, randomization, and blocking were essential to separate treatment effects from noise. CAE changes the economics: a single run is deterministic (the same input yields the same output), so replication is unnecessary for noise control, but each run may cost minutes to hours. The dominant constraints become run budget and dimensionality. A crashworthiness model with 40 thickness and material variables cannot afford a full factorial design (2⁴⁰ points); the DoE must extract maximum information per run. Randomization and blocking still matter when multiple solver versions, meshes, or hardware configurations are involved, because those introduce systematic bias between batches.

Step 1: Factor Screening

Screening answers a prior question: of the n candidate variables, which few actually influence the responses of interest? Retaining irrelevant factors inflates the sample requirement exponentially and degrades metamodel conditioning. The standard tool is the Plackett–Burman (PB) design, a two-level fractional design that estimates main effects with N runs for up to N−1 factors. For example, a 12-run PB design screens up to 11 factors, each at a low (−1) and high (+1) level. PB designs are resolution III: main effects are aliased with two-factor interactions, so they identify influential factors but cannot resolve interactions.

A practical screening workflow for a CAE model proceeds as follows. First, define factor ranges from engineering judgment or tolerance stacks — for a bracket study, plate thickness might range 2.0–4.0 mm and fillet radius 3–8 mm. Second, run the PB design and fit a main-effects model, typically by least squares on coded variables. Third, rank factors by standardized effect and apply a Pareto plot or a half-normal plot; factors whose effects fall below the noise floor (estimated from the smallest effects or from inactive-factor pooling) are dropped. Fourth, confirm the reduced factor set with a small follow-up design before committing to the expensive space-filling stage.

Step 2: Space-Filling Designs for Metamodeling

Once the active factors are known, the goal shifts from estimating main effects to covering the design space uniformly so that a surrogate can interpolate accurately everywhere. Latin Hypercube Sampling (LHS) is the workhorse: each of the n factors is divided into N equal-probability strata, and each stratum is sampled exactly once per factor, with the pairing across factors randomized. LHS guarantees one-dimensional uniformity but not two-dimensional uniformity — two factors can still be correlated by chance. Maximin LHS or optimal Latin Hypercube designs improve this by maximizing the minimum pairwise distance between points, which directly improves the conditioning of the surrogate's interpolation problem.

For CAE, the choice among space-filling families depends on the response's expected behavior:

A common rule of thumb is N ≈ 10k for a quadratic response surface and 15–20k for a Kriging or radial-basis surrogate, but this must be checked against the actual fit quality rather than assumed. For a 6-factor NVH study, 60–120 LHS points is a reasonable starting budget; if the surrogate's leave-one-out error exceeds the engineering tolerance, add points adaptively rather than restarting.

Step 3: Sample Generation and Validation

Generating the sample set is a scripting task, and reproducibility matters. The following Python snippet builds a maximin-optimized LHS for a 6-factor bracket study, scales it to physical bounds, and reports the minimum pairwise distance as a coverage check:

import numpy as np
from scipy.stats import qmc

# Factor bounds: thickness (mm), fillet radius (mm), material E (GPa), ...
bounds = np.array([[2.0, 4.0], [3.0, 8.0], [190, 210],
                   [0.28, 0.34], [1.5, 3.5], [0.8, 1.6]])
k, N = bounds.shape[0], 80

sampler = qmc.LatinHypercube(d=k, seed=42,
                             optimization="random-cd")
u = sampler.random(N)                 # stratified in [0,1]^k
X = qmc.scale(u, bounds[:, 0], bounds[:, 1])

# Coverage check: minimum pairwise Euclidean distance (normalized)
Xn = (X - bounds[:, 0]) / (bounds[:, 1] - bounds[:, 0])
d = np.linalg.norm(Xn[:, None, :] - Xn[None, :, :], axis=2)
np.fill_diagonal(d, np.inf)
print("min pairwise distance:", d.min())
print("max correlation:", np.abs(np.corrcoef(Xn.T) - np.eye(k)).max())

Two diagnostics decide whether the sample set is acceptable. The minimum pairwise distance (in normalized coordinates) should be as large as possible; values below roughly 0.05 for N = 80, k = 6 indicate clustering and warrant re-optimization. The maximum absolute off-diagonal correlation should be small (typically < 0.1); a high value means two factors are confounded in the sample, which will bias sensitivity indices. If either check fails, increase the optimization effort (optimization="random-cd" with more iterations) or switch to a Sobol sequence with scrambling.

How the DoE Feeds Sensitivity and Metamodeling

The figure below shows the complete data flow from factor definition through screening to the final sample set, and how each stage constrains the next. Screening reduces dimensionality; the space-filling design then allocates the run budget across the surviving factors; the validated sample set is the single input to both the sensitivity analysis (e.g., Sobol indices computed on the surrogate) and the metamodel training. Note that the sensitivity stage and the metamodel stage share the same sample set — this is deliberate, because reusing runs across both analyses is the main cost advantage of a well-planned CAE DoE.

Practical Considerations and Common Failure Modes

Three failure modes recur in CAE DoE practice. First, screening with too few levels: a two-level PB design cannot detect curvature, so a factor with a strong quadratic effect may appear inactive. If curvature is suspected, add center points or move directly to a three-level design for the suspect factors. Second, space-filling without normalization: as noted above, unnormalized distances let large-magnitude factors dominate, producing samples that are well spread in E but clustered in Poisson ratio. Third, ignoring solver determinism assumptions: if the CAE model includes adaptive meshing or iterative convergence tolerances, nominally identical inputs can yield slightly different outputs; in that case, a small replication set (3–5 points) should be added to estimate numerical noise before interpreting sensitivity indices.

A defensible DoE for a sensitivity and metamodeling study therefore has a clear audit trail: the candidate factor list with ranges and units, the screening design and its Pareto ranking, the space-filling design with its seed and optimization criterion, the two coverage diagnostics, and the final frozen sample table. That table — one row per run, one column per factor in physical units — is the deliverable that both the sensitivity analyst and the metamodel developer consume, and it should be version-controlled alongside the CAE model files so that any result can be traced back to the exact sample set that produced it.