4. Full Factorial DOE for Seat Belt Anchor Height and Pretensioner Force

Why a Full Factorial Here

Seat belt anchor height and pretensioner force are two restraint parameters that rarely act independently. Anchor height sets the belt geometry across the chest and shoulder, which governs how the shoulder belt loads the rib cage and how much the belt can slide laterally during the ride-down. Pretensioner force determines how aggressively the belt is pulled in during the early crash phase, which changes both the timing of chest loading and the residual slack available for slip. Because the two effects interact — a high anchor combined with a strong pretensioner can pull the belt into a different path than either parameter would produce alone — a full factorial design is the correct choice when the factor count is small and the interaction is physically plausible.

With two factors at three levels each, a full factorial requires 3 × 3 = 9 runs. That is cheap enough to run exhaustively in LS-DYNA and gives a clean estimate of both main effects and the two-way interaction without confounding. This case uses that 3² design to quantify how anchor height and pretensioner force jointly affect chest deflection and belt slip in a frontal crash restraint model.

Factor and Response Definition

The design uses the following factor levels. Anchor height is expressed as a vertical offset from the nominal D-ring position in the baseline model; pretensioner force is the peak retraction force applied by the buckle pretensioner during the pretensioning window.

Two responses are extracted from each run: peak chest deflection (mm, from the chest deflection transducer on the Hybrid III 50th percentile dummy) and total belt slip (mm, measured as the relative displacement between the belt and the shoulder reference point over the crash pulse). Both are read from the LS-DYNA nodout/glstat and belt element output at the same time step to keep the comparison consistent.

Building the 3² Run Matrix

The nine runs are the Cartesian product of the three anchor levels and three pretensioner levels. Each run is a separate LS-DYNA keyword deck differing only in the two factor values, so the rest of the model — mesh, contact, material cards, initial velocity, and the 120 ms frontal pulse — stays identical. The run matrix below is the deliverable that drives the batch submission.

The center point (R5) is not a replication in the strict sense, but it anchors the interaction estimate and gives a sanity check that the model is not drifting between runs. If R5 is run twice and the two responses differ by more than a few percent, the model has a non-determinism problem (contact instability, hourglassing, or a time-step issue) that must be fixed before the DOE results mean anything.

Parameterizing the Deck

Rather than editing nine decks by hand, drive the two factors from a single parameterized keyword file. In LS-DYNA, *PARAMETER and *PARAMETER_EXPRESSION let you define the anchor offset and pretensioner force once and reference them in the belt geometry and the pretensioner card. A minimal parameter block looks like this:

*PARAMETER
ANCHOR_OFF     0.0
PT_FORCE       3.5
*PARAMETER_EXPRESSION
ANCHOR_Y       ANCHOR_OFF
PT_LOAD        PT_FORCE*1000.0

The anchor node set is then translated by ANCHOR_Y in the belt routing definition, and the pretensioner *MAT_... / load curve peak is scaled by PT_LOAD. A small shell loop rewrites the two parameter lines per run and submits each deck, which keeps the nine runs traceable to a single source model.

Reading the Interaction

The point of the full factorial is not the nine numbers in isolation but the structure they reveal. Plot chest deflection against anchor height separately for each pretensioner level, and do the same for belt slip. If the three curves are parallel, the factors act additively and the interaction term is negligible. If they cross or diverge, the interaction is real and must be carried into any subsequent response surface or optimization.

The SVG below shows the 3² grid with the expected qualitative behavior: chest deflection tends to fall as pretensioner force rises (more ride-down coupling) but the benefit saturates and can reverse at high anchor positions where the belt path rides up; belt slip falls with pretensioner force but is sensitive to anchor height because a higher anchor shortens the free belt span. The exact numbers must come from your runs — the figure encodes the pattern to look for, not measured values.

Interpreting the Results

Once the nine runs complete, compute the main effects as the difference between the average response at the high and low levels of each factor, and the interaction as the difference of differences. A practical check: if the interaction term for chest deflection is larger than roughly 20% of the larger main effect, the factors cannot be treated independently and the full factorial was the right call. For belt slip, expect the anchor height main effect to dominate at low pretensioner force and the pretensioner main effect to dominate at high anchor positions — that crossover is the interaction signature.

The output of this case is the 9-run matrix, the two response columns, and the main-effect/interaction summary. That summary is the input to the next stage of the restraint DOE — whether that is a response surface fit or a direct optimization over the two factors.