4. Full factorial and response surface designs

Full factorial designs are the natural starting point for restraint DOE because they are transparent: every combination of every factor level appears exactly once, so main effects and all interactions are estimable without aliasing. For a restraint problem with k factors at two levels each, the matrix contains 2^k runs. With three factors — say shoulder-belt anchor height, pretensioner force limit, and airbag vent diameter — that is 8 runs; with five factors it is 32. The cost grows exponentially, and this is precisely where the design decision becomes interesting: full factorials are excellent for screening and for detecting interactions, but they are a poor use of budget when the goal is to locate an optimum on a curved response surface.

Restraint performance responses are rarely linear. Occupant injury criteria such as HIC₃₆, chest deflection, and neck Nij typically respond to belt force limiting and anchor geometry with a pronounced curvature: too little force limiting produces high chest loading, too much produces excessive excursion and head contact. A two-level factorial can only fit a plane through such a response, and it will systematically mislead you about where the minimum lies. This is the motivation for response surface methodology (RSM), which augments the design with center points and axial or edge points so that quadratic terms become estimable.

Why center points come first

Before committing to a full RSM design, add replicated center points to the factorial. Center points serve two purposes. First, they provide a pure-error estimate: the variance among replicates at the same factor settings is not explained by the model, so it gives an honest denominator for lack-of-fit testing. Second, the difference between the average center-point response and the average factorial response is a direct check for curvature. If the factorial runs average HIC₃₆ = 412 and the center points average 468, the gap of 56 is a strong signal that a quadratic model is warranted. In LS-DYNA terms, this is a cheap diagnostic: four to six extra runs of the same keyword deck with only the parameter values changed.

Central composite design (CCD)

The central composite design is the workhorse of RSM. It combines three blocks: a two-level factorial (or a resolution IV/V fraction of it), a set of center points, and a set of axial (star) points placed at distance α from the center along each factor axis. For k factors the run count is 2^k + 2k + n_c for a full-factorial CCD, or 2^k−p + 2k + n_c for a fractional CCD. With k = 3 and n_c = 6, a full CCD requires 8 + 6 + 6 = 20 runs — a very manageable matrix for a restraint sled or CAE study.

The choice of α determines the design's properties. A rotatable CCD sets α = (2^k)^1/4, which for k = 3 gives α ≈ 1.682. Rotatability means the prediction variance depends only on the distance from the design center, not on direction — valuable when you do not yet know which direction the optimum lies. A face-centered CCD (α = 1) keeps all runs inside the original factor box, which matters when the axial points would otherwise push a parameter outside a physically valid range, such as a vent diameter that cannot go negative or a pretensioner force limit beyond the hardware's capability.

Practical CCD construction for a restraint study proceeds as follows. Coded factor levels are −1, 0, +1, with axial points at ±α. Suppose the three factors are:

• A — shoulder anchor height: 1,180 mm to 1,320 mm (center 1,250 mm)
• B — pretensioner force limit: 2.0 kN to 4.0 kN (center 3.0 kN)
• C — airbag vent diameter: 25 mm to 45 mm (center 35 mm)

The coded-to-physical mapping is x_phys = center + (coded × half-range). For factor A the half-range is 70 mm, so the axial point at +1.682 corresponds to 1,250 + 1.682 × 70 ≈ 1,368 mm — outside the stated 1,320 mm limit. This is exactly the situation where a face-centered CCD (α = 1) is the correct engineering choice, keeping every run within the validated parameter envelope. The matrix below shows the structure of a face-centered CCD for these three factors.

Box-Behnken design (BBD)

The Box-Behnken design takes a different geometric approach: it places runs at the midpoints of the edges of the factor cube, never at the corners, plus center replicates. For k = 3 the BBD has 12 edge points plus n_c centers, typically 15–17 runs total. The defining property is that no run combines all factors at their extreme levels simultaneously. For restraint work this is a genuine advantage: a run with maximum anchor height, maximum force limit, and minimum vent diameter may be physically unrealistic or may produce a non-physical occupant response that wastes a simulation. BBD avoids those corner combinations entirely.

The trade-off is that BBD does not extend to the corners of the region, so it cannot estimate the response at the extreme corner combinations, and its prediction variance at the corners is higher than CCD's. It is also not rotatable in the same sense. For a three-factor restraint optimization where the feasible region is a box and corner combinations are suspect, BBD is often the more efficient choice; for a study where you genuinely need corner behavior, CCD is preferable.

From matrix to optimum

Once the runs are complete, fit the quadratic model by least squares and inspect the analysis of variance. A significant model F-test with a non-significant lack-of-fit test is the green light to proceed. Then examine the response surface: contour plots of HIC₃₆ versus force limit and vent diameter at fixed anchor height reveal the shape of the optimum, and the stationary point (found by setting the gradient of the fitted model to zero) gives the candidate optimum. If the stationary point lies outside the experimental region, the optimum is on a boundary and you should either expand the region or accept the boundary solution.

In practice, the restraint engineer's workflow is: screen with a two-level factorial or fractional factorial to identify the active factors, add center points to test for curvature, then augment to a CCD or BBD for the optimization stage. The matrix generation step is where this sequence is encoded — the run table, the coded levels, the block assignment, and the replication pattern all determine whether the subsequent regression can answer the question you actually asked.