2. Design of experiments for CAE

In CAE sensitivity analysis and response surface modeling, the design of experiments (DoE) is the bridge between a parametric CAD/CAE model and a statistically defensible surrogate. The DoE determines which input combinations are simulated, and therefore controls how much information each solver run contributes. A poorly chosen DoE wastes expensive compute on redundant points or leaves large regions of the design space unobserved; a well-chosen one extracts maximum sensitivity and curvature information from a minimal number of runs. This section covers the three DoE families that dominate CAE practice — screening designs, space-filling designs, and the sample generation strategies used to feed sensitivity and response surface studies — and explains how to select and size them for real simulation campaigns.

Why CAE DoE differs from physical experimentation

Physical DoE must contend with random error, blocking, and randomization. CAE DoE faces a different reality: the solver is deterministic, so replication adds no information; the cost per run is high (minutes to days), so run count is the binding constraint; and the input space is often high-dimensional (10–50 parameters) with mixed physical ranges. The practical consequence is that CAE DoE prioritizes coverage per run and identifiability of effects over classical replication and blocking. Two questions drive the choice of design: which factors matter? (screening) and how does the response vary across the space? (response surface and sensitivity).

Screening designs: finding the influential few

Screening is used when the parameter list is long and most factors are expected to be inert. The goal is to rank factors by influence at low run cost, then carry only the active subset into the more expensive response surface stage. The workhorse is the Plackett–Burman (PB) design, a two-level fractional design that estimates main effects with N runs for up to N−1 factors. A 12-run PB design screens 11 factors; a 20-run design screens 19. Resolution III designs (e.g., 2⁷⁻⁴) serve the same purpose when factor counts are small.

The critical caveat is aliasing: PB designs confound main effects with two-factor interactions, so a significant main effect may actually be an interaction. Treat screening results as a ranking, not a final model. A practical workflow for a 15-parameter crashworthiness or thermal study looks like this:

1. Define each factor with a low and high level tied to physically realizable bounds (e.g., sheet thickness 1.2–2.0 mm, rib height 15–35 mm).
2. Generate a 20-run PB design (19 columns available, 15 used, 4 reserved as dummy columns).
3. Run the 20 simulations, extract the scalar response (e.g., peak acceleration, max temperature).
4. Fit a main-effects model and inspect the dummy columns: if a dummy column shows an effect comparable to a real factor, the noise floor is high and the ranking is unreliable.
5. Retain factors whose standardized effect exceeds roughly 2–3× the dummy-column effect magnitude; drop the rest.

Space-filling designs: covering the response surface

Once the active factor set is known (typically 4–10 parameters), the objective shifts from ranking to spatial coverage. Response surface methods — polynomial regression, Kriging/Gaussian process, radial basis functions — need samples distributed across the entire design space, including interior points, so that curvature and interaction structure can be resolved. Space-filling designs are the standard answer.

Latin Hypercube Sampling (LHS) is the most widely used. For n samples and k factors, each factor's range is divided into n equal-probability strata and each stratum is sampled exactly once, with the pairing across factors randomized. This guarantees one-dimensional projection uniformity — every factor is sampled across its full range — while allowing arbitrary sample counts independent of factor count. Plain LHS, however, can produce poor multidimensional coverage: two samples may sit close together in the joint space even though each is well-placed marginally.

Maximin and Audze–Eglājs (U-centered) LHS correct this by optimizing the design after generation. Maximin LHS maximizes the minimum pairwise Euclidean distance between samples; Audze–Eglājs minimizes the sum of inverse squared distances. Both are solved by swapping column entries in the LHS matrix and accepting swaps that improve the criterion. In practice, a few thousand swap iterations on a 50-sample, 6-factor design converge quickly and measurably improve coverage. Halton, Sobol, and Hammersley low-discrepancy sequences are an alternative for very large sample counts or when progressive sampling (adding points without discarding earlier ones) is required.

The choice between LHS variants matters most for Kriging surrogates, which are sensitive to sample clustering. For a 6-factor, 60-run study, a maximin LHS typically yields a minimum inter-sample distance 20–40% larger than plain LHS, which translates into a visibly tighter Kriging confidence band in the interior.

Sample generation for sensitivity and response surface studies

The DoE must match the sensitivity method. Three common pairings:

For variance-based global sensitivity analysis, the Saltelli scheme builds two independent matrices A and B of size N×k, then forms k additional matrices A_B⁽ⁱ⁾ in which column i of A is replaced by column i of B. The total run count is N(k+2). For k = 8 and N = 1000, that is 10,000 solver runs — a number that must be checked against available compute before committing. If it is infeasible, reduce N to 300–500 (Sobol indices remain usable, with wider confidence intervals) or switch to a cheaper screening-then-surrogate workflow.

For polynomial response surfaces, the classical Central Composite Design (CCD) combines a two-level factorial or fractional-factorial core with axial (star) points at distance α and center points. For a rotatable CCD with k factors, α = (2^k)^1/4; for k = 3, α ≈ 1.682. Center-point replication (3–5 runs) estimates pure error and tests for lack of fit — the one place where replication is still meaningful in CAE, because it detects solver non-determinism or mesh-induced scatter.

Reading the design before running the solver

Before launching a simulation campaign, inspect the DoE matrix itself. The figure below summarizes the decision path from parameter count and study objective to design family, and the diagnostics to check on the generated matrix.

Three diagnostics catch most DoE mistakes before any solver time is spent. First, plot each factor's sample distribution: an LHS should show one sample per stratum, so a histogram with empty bins indicates a generation bug. Second, compute the minimum pairwise distance in normalized (0–1) factor space; if it is very small relative to the mean distance, the design is clustered and a maximin optimization pass is warranted. Third, compute the Pearson correlation matrix of the factor columns; for an orthogonal or LHS design, off-diagonal entries should be near zero. A correlation of 0.6 between two factors means the design cannot separate their effects, and the sensitivity results will be misleading regardless of solver accuracy.

Practical sizing and a worked example

Consider a thermal CAE study with 12 candidate parameters (material conductivity, convection coefficients, thicknesses, heat-generation rates) and a target of building a Kriging surrogate for peak temperature. The workflow: (1) run a 16-run PB design with 3 dummy columns to screen the 12 parameters; suppose 5 factors show effects above the dummy noise floor. (2) Generate a maximin LHS with n = 10 × 5 = 50 samples in the 5-factor space, normalized to [0,1] and mapped back to physical bounds. (3) Run 50 simulations, fit the Kriging model, and check the leave-one-out cross-validation R² and the standardized residuals. If R² < 0.9 or residuals show structure, add 10–15 infill points in the region of largest prediction variance (sequential design) rather than regenerating from scratch. This staged approach — screen, then fill, then infill — typically reaches a usable surrogate in 60–80 runs instead of the 500+ a full space-filling design over all 12 factors would require.

The key discipline is to treat the DoE as a deliberate information budget: screening buys factor ranking cheaply, space-filling buys coverage for the surrogate, and sequential infill buys accuracy exactly where the model is weakest. Matching the design family to the study objective, and verifying the matrix before committing compute, is what separates a defensible sensitivity or response surface study from an expensive scatter of simulation results.