3. Design of experiments for occupant safety simulations

Surrogate modeling and uncertainty quantification both begin with a finite budget of simulation runs. In occupant safety CAE, each run is expensive: a full frontal sled model with a Hybrid III 50th percentile dummy, belt system, retractor, pretensioner, and airbag may take 30–90 minutes on 32–64 cores, and a robustness study may require hundreds of runs. The design of experiments (DoE) determines which input combinations you actually simulate, and therefore how much information each CPU-hour buys. A poor DoE cannot be rescued by a sophisticated surrogate; a good DoE makes even a modest Kriging or polynomial-chaos model reliable.

This chapter covers three families of designs that matter in occupant safety: space-filling designs, Latin hypercube sampling (LHS), and constraint-aware sampling. The unifying idea is that the design must cover the feasible input region uniformly, not merely the nominal box.

Input Space Definition Comes First

Before choosing a design type, define the input vector and its bounds. A typical frontal-occupant study might use:

Each variable should be normalized to [0, 1] internally so that distance metrics in the design algorithm are not dominated by the variable with the largest physical range. Record the normalization explicitly; it must be inverted when results are reported.

Space-Filling Designs

A space-filling design places points so that no region of the input space is left unobserved and no two points are redundant. The most common metric is the maximin criterion: maximize the minimum pairwise Euclidean distance between design points. A complementary metric is the centered L2 discrepancy, which measures deviation from uniform distribution. In practice, engineers generate many candidate designs and select the one with the best maximin distance or lowest discrepancy.

Why does this matter for occupant safety? Consider a restraint parameter study where the dummy chest deflection response is highly nonlinear near the airbag vent diameter threshold where the bag just contacts the chest. If the DoE clusters points away from that threshold, the surrogate will smooth over the transition and under-predict peak deflection. Space-filling designs reduce this risk by spreading samples across the entire range, including transition regions.

A simple maximin greedy construction works as follows: pick the first point at random, then repeatedly add the candidate point whose distance to the nearest existing point is largest. This is the maximin greedy algorithm. It is fast but order-dependent; running it 50–100 times with different random seeds and keeping the best design is standard practice.

Latin Hypercube Sampling

LHS is the workhorse of occupant-safety DoE. For n samples and d variables, each variable's range is divided into n equal-probability strata, and exactly one sample is placed in each stratum per variable. The strata are then randomly permuted independently across variables. This guarantees one-dimensional uniformity — every variable is sampled across its full range — while the random permutation provides multi-dimensional spread.

A plain LHS does not guarantee good multi-dimensional coverage; two variables can accidentally correlate. The standard fix is to optimize the design after generation using a maximin or correlation criterion. A practical recipe:

1. Generate 1,000 candidate LHS matrices of size n × d.
2. For each, compute the minimum pairwise distance in normalized space.
3. Keep the candidate with the largest minimum distance (maximin LHS).
4. Optionally refine with a column-wise exchange: swap two entries in one column if it increases the minimum distance, repeat until no improvement.

For a 6-variable, 120-run occupant study, a maximin-optimized LHS typically achieves a minimum normalized distance of about 0.15–0.25, compared with roughly 0.05–0.10 for a naive random design of the same size. That difference directly improves surrogate accuracy near the response surface edges.

Constraint-Aware Sampling

Occupant safety inputs are rarely free to vary independently. Physical and regulatory constraints couple them:

• Geometric feasibility: seat track position and seat back angle jointly determine the dummy head position; extreme combinations may place the head outside the airbag deployment zone or cause the model to initialize in contact with trim.
• Regulatory envelopes: belt anchor height and seat back angle must keep the shoulder belt within the FMVSS 208 or Euro NCAP belt-fit corridor.
• Simulation stability: certain pretensioner stroke and vent diameter combinations cause the explicit solver to abort due to negative volume or excessive hourglassing.

If you sample the full box and discard infeasible runs, you waste simulation budget and, worse, you bias the design toward the feasible region's interior, leaving the constraint boundary under-sampled — exactly where occupant responses often change fastest. Constraint-aware sampling addresses this by generating candidates and accepting only those that satisfy the constraints, while still enforcing space-filling or LHS structure.

A practical constraint-aware LHS procedure:

1. Define a feasibility predicate g(x) ≤ 0 for each constraint, evaluated cheaply (geometry checks, belt-fit corridor formulas, or a fast rigid-body pre-check).
2. Generate a large pool of candidate LHS points (e.g., 20× the target n).
3. Filter the pool by the feasibility predicate.
4. From the feasible pool, greedily select n points using maximin distance, starting from the feasible point closest to the centroid.
5. Verify that the final design still has one sample per stratum per variable; if not, regenerate with a larger pool.

For constraints that are expensive to evaluate (e.g., a full simulation that must run to check stability), use a cheap surrogate of the constraint itself — a response surface fitted to a small preliminary run set — as the feasibility predicate. This is sometimes called surrogate-constrained sampling and is standard in robust design workflows.

Choosing Among the Three

The decision is driven by problem structure, not preference:

The SVG below summarizes the constraint-aware LHS workflow and shows how the feasible region, not the bounding box, is the true sampling domain.

Practical Verification of a DoE

Before committing simulation budget, verify the design numerically. Three checks catch most problems:

1. Stratification check: for each variable, confirm exactly one sample per stratum (for LHS). A histogram of normalized coordinates should be flat.
2. Correlation check: compute the Pearson correlation matrix of the normalized design. Off-diagonal entries should be near zero (|r| < 0.1). A high correlation between, say, seat track and belt anchor height means the design cannot separate their effects.
3. Coverage check: compute the minimum pairwise distance and compare it against the theoretical expectation for a uniform design of the same size. If the achieved minimum distance is far below expectation, regenerate.

A common failure symptom is a surrogate that fits the training data well but predicts poorly at the nominal operating point. This often traces to a design that under-samples the region around nominal — a consequence of naive LHS without maximin optimization, or of discarding infeasible runs without re-filling the gap. The fix is to regenerate with constraint-aware maximin selection and to explicitly include the nominal configuration as a design point (or verify that a design point lies within a small radius of it).

With a verified design in hand — typically 120–200 runs for a 6–8 variable occupant model — the next step is fitting the surrogate and quantifying how its prediction uncertainty propagates to injury criteria such as HIC, chest deflection, and femur load. The quality of that downstream analysis is bounded by the design described here.