📚 Course ContentsChapter 2 Design of Experiments (DOE) fundamentals for crash simulation

2. Design of Experiments (DOE) fundamentals for crash simulation

Crash CAE models routinely contain hundreds of thousands of elements and dozens of uncertain inputs — sheet gauge, material yield stress, spot-weld failure strain, barrier offset, airbag trigger time. Running a full-factorial study over even ten such factors at three levels requires 59,049 simulations, which is economically impossible. Design of Experiments (DOE) is the discipline that lets you extract the maximum information about a response surface from a strictly limited number of runs. In the crash context, DOE serves three distinct purposes that must not be conflated: screening (which factors matter at all), space-filling (building a surrogate over the whole feasible region), and optimization (finding extrema or robust designs). Each purpose implies a different sampling strategy, and choosing the wrong one is the most common source of wasted compute in crash UQ workflows.

Factor Screening: Separating Signal from Noise

Screening designs answer a narrow question: among k candidate factors, which ones produce a response change larger than the numerical and physical noise floor? The classical tool is the two-level fractional factorial, typically a resolution IV design such as a 2^7-3 (16 runs for 7 factors). Resolution IV means main effects are aliased only with three-factor and higher interactions, so main effects remain estimable if you are willing to assume three-factor interactions are negligible — a defensible assumption in crash response, where the dominant nonlinearity is usually pairwise (e.g., gauge × material grade).

For crash models, however, two-level designs have a specific weakness: many crash responses are strongly non-monotonic in the factor. A classic example is the relationship between spot-weld pitch and peak intrusion — too few welds and the joint separates, too many and the heat-affected zone embrittles the sheet. A two-level design samples only the endpoints and can completely miss the interior optimum. The practical remedy is a three-level Plackett–Burman or a definitive screening design (DSD). A DSD with 2k+1 runs estimates all main effects free of two-factor-interaction confounding and can detect curvature, which makes it the preferred screening tool for crash models with 6–20 factors.

A concrete screening workflow for a frontal-impact model with 12 candidate factors:

Define the response metrics before sampling: peak B-pillar acceleration, maximum intrusion, and occupant chest deflection (or the CAE surrogate for it).
Set factor ranges from manufacturing tolerance and material certification data, not from arbitrary ±10% guesses. Gauge, for example, should span the full coil tolerance band.
Run a 25-run DSD (2×12+1) with three center points to estimate pure error.
Fit a main-effects-plus-curvature model and rank factors by standardized effect (effect divided by its standard error).
Retain factors whose standardized effect exceeds roughly 2.0, and drop the rest before the space-filling stage.

Screening is not free. Each run is a full crash simulation, so a 25-run DSD on a 4-hour model costs about 100 core-hours. Budget this explicitly; a screening study that eliminates 8 of 12 factors typically pays for itself by shrinking the subsequent space-filling design from 200 runs to 60.

Space-Filling Designs and Latin Hypercube Sampling

Once the factor set is reduced, the goal shifts from ranking to coverage. You want a design that spreads points uniformly through the k-dimensional box so that a surrogate (Kriging, polynomial chaos, or a neural network) can interpolate without large unsampled voids. The workhorse is Latin Hypercube Sampling (LHS). In LHS, each of the k factors is divided into n equal-probability strata, and each stratum is sampled exactly once per factor, with the n samples randomly permuted independently across factors. This guarantees one-dimensional uniformity — every marginal is perfectly stratified — while leaving the joint distribution free.

The freedom is also the danger. A naive random LHS can produce clusters and diagonal alignments that leave large empty regions in two- and three-dimensional projections. The fix is to optimize the LHS by a space-filling criterion. The two standard criteria are:

Maximin distance — maximize the minimum pairwise Euclidean distance between design points. This directly prevents clustering and is the most intuitive criterion for Kriging.
Centered L₂ discrepancy — minimize the deviation of the empirical point distribution from a uniform measure. This is the criterion of choice when the surrogate is polynomial chaos or when you need good low-order projection properties.

In practice you generate, say, 1,000 candidate LHS matrices and keep the one with the best criterion value. This is cheap (seconds) compared with the simulation cost, so there is no reason to accept an unoptimized LHS.

Optimal Criteria for High-Dimensional Crash Models

When the factor count grows beyond roughly 15–20, or when the feasible region is constrained (e.g., gauge and material grade are coupled by a manufacturing constraint), pure LHS becomes awkward. Here optimal design criteria take over. The three classical alphabetic criteria are:

Criterion	Objective	Best used when
D-optimal	Maximize det(X^TX), minimizing coefficient variance for a specified model	You have a fixed polynomial or regression model and want precise coefficients
A-optimal	Minimize trace((X^TX)^-1), minimizing average prediction variance	Average prediction accuracy matters more than individual coefficients
I-optimal	Minimize integrated prediction variance over the design region	Surrogate prediction over the whole domain is the goal — the usual case in crash UQ

For crash surrogate construction, I-optimal designs are generally preferred over D-optimal because the objective aligns with the actual use case: predicting intrusion or acceleration at unsampled points. D-optimal designs tend to place points at the extremes of the region, which is good for coefficient estimation but leaves the interior sparsely covered. I-optimal designs distribute points more evenly and produce lower average prediction error.

A practical high-dimensional workflow for a 20-factor crash model:

# Pseudocode — I-optimal design generation
factors = 20
runs    = 120          # ~6x factor count, typical for Kriging
model   = quadratic    # main + 2FI + pure quadratic terms
criterion = "I-optimal"
candidates = 5000      # candidate set from maximin LHS
design = select_optimal(candidates, runs, model, criterion)
assert design.min_pairwise_distance > 0.05   # sanity check
assert design.condition_number(X) < 1e6       # avoid ill-conditioning

The condition-number check is not optional. With 20 factors and a quadratic model, the design matrix has 231 columns; a poorly conditioned design will produce unstable surrogate coefficients that look fine in-sample but fail catastrophically at validation points. Always hold out 10–15% of runs as a validation set and report the normalized root-mean-square error (NRMSE) on those points before trusting the surrogate.

Do not mix screening and space-filling runs into a single dataset and fit one surrogate. Screening designs concentrate points at factor extremes and will bias the surrogate toward the corners of the domain. Keep the screening dataset separate, use it only for factor ranking, and build the surrogate exclusively from the space-filling design.

Choosing the Right Design in Practice

The decision is driven by factor count and objective, not by convention. For fewer than 8 factors with a clear optimization target, a face-centered central composite design gives good quadratic resolution. For 8–20 factors with a surrogate goal, a maximin-optimized LHS with 5–10× the factor count in runs is the robust default. For more than 20 factors, screen first with a DSD, then apply I-optimal design on the surviving subset. In every case, record the design matrix, the random seed used for LHS generation, and the optimization criterion value alongside the simulation results — without these, the study is not reproducible, and a crash DOE that cannot be reproduced cannot be defended in a design review.