4. Design of Experiments (DOE) fundamentals for restraint systems

Before a single LS-DYNA run is launched, the restraint engineer must decide what to vary, how much to vary it, what to measure, and which matrix of runs will answer the question at acceptable cost. That decision set is the Design of Experiments (DOE). In occupant-safety work the DOE is not a statistical formality — it is the mechanism that converts a handful of expensive, noisy, highly nonlinear explicit simulations into a defensible map from restraint hardware and calibration parameters to injury criteria. This chapter establishes the vocabulary and the design-selection logic that the rest of the course builds on.

Factors: the levers you actually own

A factor is an input variable whose setting you control and whose effect on the response you intend to quantify. In a frontal restraint model the factor set typically spans three families:

• Hardware/geometry factors — seat belt anchor positions, D-ring height, retractor pretensioner stroke, load-limiter force level, airbag module volume, vent-hole area, tether length, knee-bolster stiffness.
• Calibration/algorithm factors — pretensioner firing time, load-limiter force profile (single-stage vs. dual-stage), airbag inflator mass-flow curve scaling, vent-opening pressure, buckle-pretensioner delay.
• Environmental/occupant factors — occupant size (5th-percentile female to 95th-percentile male), seat-track position, recline angle, impact speed, pulse shape, ambient temperature affecting inflator output.

Two disciplines separate a usable factor list from a wish list. First, screen for significance before you commit: a 12-factor full factorial at two levels is 4,096 runs, which is not a screening study, it is a budget crisis. Second, respect the physics of the factor range. A load-limiter force level swept from 2 kN to 6 kN is meaningful; swept from 2 kN to 20 kN it will produce belt pay-out so large that the occupant submarines and the response surface becomes dominated by a regime change rather than a smooth trend. Factor ranges should bracket the plausible design space, not the numerically possible one.

Levels: resolution and the cost of refinement

A level is a specific setting of a factor. Two-level designs (low/high, coded −1/+1) are the workhorse for screening and for estimating main effects and two-factor interactions. Three-level designs add the ability to detect curvature — essential when a response like chest deflection is expected to be non-monotonic in load-limiter force (too soft → excessive excursion; too stiff → excessive belt force). The cost scales as L^k for a full factorial in k factors at L levels, so the level count is a direct budget decision.

For continuous factors, levels are usually coded to a normalized range so that coefficients from different factors are comparable:

x_coded = (x_actual - x_center) / (x_half_range)
# Example: load-limiter force, actual 3.0–5.0 kN
# x_center = 4.0 kN, x_half_range = 1.0 kN
# 3.0 kN -> -1.0 ; 4.0 kN -> 0.0 ; 5.0 kN -> +1.0

Categorical factors (inflator type A/B/C, vent configuration open/closed) cannot be coded this way and must be handled as discrete levels with no interpolation between them.

Responses: what the DOE is actually optimizing

A response is the measured output extracted from each run. In restraint DOE the response set is dictated by the injury criteria and by the constraints you must not violate. A representative frontal set:

Two practical cautions. First, responses must be extracted consistently — same filter class (SAE J211 CFC 600 for head, CFC 180 for chest), same time window, same node/element set — or the DOE will measure extraction noise rather than design sensitivity. Second, include constraint responses (belt force, retractor pay-out, airbag pressure) alongside injury responses; a design that minimizes HIC by over-softening the belt may violate the excursion or belt-force constraint, and the optimizer needs to see both.

Choosing the design: full factorial, fractional factorial, Latin hypercube

The three families answer different questions, and the selection is driven by factor count, expected curvature, and run budget.

Full factorial (2^k or 3^k) estimates all main effects and all interactions without aliasing. It is the reference design and is appropriate when k is small (≤ 4–5) and interactions are expected to matter — for example, a 3-factor, 3-level study of load-limiter force × pretensioner timing × vent area (27 runs) where the interaction between limiter force and vent area governs chest loading. Beyond about five factors the run count becomes prohibitive for explicit occupant models that may take 30–90 minutes per run.

Fractional factorial (2^k−p) trades interaction estimation for run economy by deliberately aliasing higher-order interactions onto main effects. A 2⁷⁻³ resolution IV design screens seven factors in 16 runs, resolving main effects clear of two-factor interactions (though two-factor interactions alias with each other). This is the correct tool for the early screening phase when you have 8–12 candidate factors and need to identify the 3–4 that actually drive injury response before committing to a refined study.

Latin hypercube sampling (LHS) is a space-filling design for continuous 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. LHS does not estimate classical main effects and interactions directly; it is used to build surrogate models (Kriging, polynomial response surfaces, or ML regressors) that are then optimized. It is the right choice when factors are continuous, the response is expected to be strongly nonlinear, and the goal is a global surrogate rather than an effect ranking. A 60-point LHS over six continuous factors is a common budget for a Kriging-based restraint optimization.

Putting the selection into practice

A defensible restraint DOE usually runs in two stages. Stage 1 — screening: list 8–10 candidate factors, assign two levels each, and run a resolution IV fractional factorial (e.g., 2⁸⁻⁴ = 16 runs). Analyze main effects on HIC₁₅, chest deflection, and belt force; retain the 3–5 factors with the largest standardized effects and any factor whose effect approaches the noise floor. Stage 2 — refinement: take the retained factors, assign three levels where curvature is plausible, and either run a full factorial (if ≤ 4 factors) or an LHS of 40–60 points to fit a Kriging surrogate for multi-objective optimization. The screening stage is what makes the refinement stage affordable; skipping it and going straight to a large LHS over ten factors wastes runs on factors that do not move the injury response.

The output of this chapter's workflow is a run matrix — a table of factor settings per run — plus a response extraction plan. That matrix is the input to the surrogate fitting and multi-objective optimization covered later in the course. Getting the factor list, level ranges, response definitions, and design family right at this stage is what determines whether the subsequent optimization converges on a real design improvement or on an artifact of a poorly posed experiment.