4. Design of experiments (DOE) for crash simulations

Before a single response surface can be fitted, you must decide which simulation runs to execute. In crash CAE this decision is expensive: a single full-vehicle frontal impact model with roughly 3–5 million elements may take 8–20 hours on a 64-core node, and a design matrix of 200 runs is simply not fundable. Design of Experiments (DOE) is the discipline that lets you extract the maximum information about main effects and interactions from the minimum number of runs. This chapter covers the three DOE families most frequently used in crash optimization workflows — full factorial, fractional factorial, and Plackett-Burman — and explains when each is the correct engineering choice.

Factors, levels, and the cost of exhaustiveness

Every DOE begins with a factor list. In a typical frontal-occupant-protection study you might vary sheet gauge (t_rail), material grade (YS_rail), trigger bead depth (d_bead), and bumper beam thickness (t_beam). Each factor is assigned two or more levels — usually a low (−1) and high (+1) coded level, occasionally a center point (0). The full factorial design runs every possible combination: with k factors at two levels, that is 2^k runs. Four factors give 16 runs; seven factors give 128; ten factors give 1024. The full factorial is the gold standard because it estimates every main effect and every interaction independently, with no confounding. Its weakness is purely economic: cost grows exponentially, and crash models rarely tolerate more than a few dozen runs.

For a four-factor, two-level study, the full factorial is entirely practical and is the recommended starting point. The design matrix below shows the coded levels; the actual values are mapped from the factor ranges you define (for example, t_rail from 1.4 mm to 2.0 mm).

Run  t_rail  YS_rail  d_bead  t_beam
 1     -1       -1       -1      -1
 2     +1       -1       -1      -1
 3     -1       +1       -1      -1
 4     +1       +1       -1      -1
 5     -1       -1       +1      -1
 6     +1       -1       +1      -1
 7     -1       +1       +1      -1
 8     +1       +1       +1      -1
 9     -1       -1       -1      +1
10     +1       -1       -1      +1
11     -1       +1       -1      +1
12     +1       +1       -1      +1
13     -1       -1       +1      +1
14     +1       -1       +1      +1
15     -1       +1       +1      +1
16     +1       +1       +1      +1

With 16 runs you can fit a first-order model plus all six two-factor interactions and still have degrees of freedom for lack-of-fit testing. The practical caution is that crash responses are strongly nonlinear — intrusion and acceleration peaks do not vary linearly with gauge — so a two-level full factorial is best used for screening or for a narrow, well-behaved design space, not for extrapolation.

Fractional factorials: trading interactions for runs

When the factor count rises to six or more, the full factorial becomes unaffordable and the fractional factorial becomes the workhorse. A 2^k−p design executes only a carefully chosen fraction of the full grid. The cost is aliasing: certain effects become confounded with one another, meaning the fitted coefficient represents a sum of effects that cannot be separated without additional runs.

The resolution of a design quantifies this trade-off. A Resolution III design confounds main effects with two-factor interactions — acceptable only for pure screening. Resolution IV confounds two-factor interactions with each other but keeps main effects clean. Resolution V keeps two-factor interactions clean. For crash work, a Resolution IV 2⁶⁻² design (16 runs for six factors) is a common compromise: main effects are estimable, and you accept that some interaction estimates are pooled.

Constructing a fractional factorial by hand is error-prone; the standard approach is to write the full factorial in the first k−p factors and derive the remaining columns from defining relations. For a 2⁶⁻² design with generators E = ABC and F = ABD, columns E and F are computed as the product of the coded signs in A, B, C and A, B, D respectively. The resulting 16-run matrix is then checked for its resolution before any solver time is committed.

Plackett-Burman: maximum screening per run

Plackett-Burman (PB) designs are saturated two-level screening designs that estimate main effects only, using N runs for up to N−1 factors, where N is a multiple of four. A 12-run PB design screens up to 11 factors; a 20-run design screens up to 19. This efficiency is unmatched when the goal is to identify which of a large candidate list — say 15 gauge, material, and topology variables — actually influence the crash responses, so that the expensive subsequent optimization can focus on the vital few.

The price is severe confounding: every main effect is partially aliased with two-factor interactions, and PB designs are Resolution III at best. They are therefore inappropriate for building a predictive response surface. Their correct role is the first pass of a two-stage strategy: screen 15 factors down to 4–6 significant ones with a 20-run PB design, then run a full factorial or a Resolution IV fractional factorial on the survivors to estimate interactions and fit the RSM.

The table below summarizes the engineering trade-offs that drive the selection.

Reading the design before you run it

The single most valuable habit in crash DOE is to inspect the design matrix and its alias structure before committing solver time. The SVG below maps the decision logic and the information each design family delivers, so you can match the design to the question you are actually asking.

Practical execution and verification

Once the design is chosen, the workflow is mechanical but unforgiving. Map coded levels to physical values, generate the run deck, and execute the simulations. For a 16-run fractional factorial on a full-vehicle frontal model, expect roughly 16 × 12 h ≈ 8 days of wall-clock time on a single node, or under a day if 16 nodes are available in parallel — which is why DOE planning and cluster scheduling must be coordinated.

After the runs complete, verify the design before fitting. Check that the response data show no failed or non-converged runs (a common crash symptom is hourglass energy exceeding 5% of internal energy, which invalidates the run and must be corrected by adjusting the hourglass control, not by discarding the point silently). Then inspect the main-effect plots: if a factor's effect is smaller than the run-to-run numerical scatter, it is not a real effect regardless of what the regression p-value suggests. For PB designs, confirm that the significant factors survive a follow-up confirmation run at their predicted optimum before committing to the refined design.

The output of this stage is not a response surface yet — it is a validated design matrix, a clean response table, and a ranked list of significant factors. That ranked list determines whether the next chapter's RSM fit uses a full quadratic model on four factors or a reduced model on the vital few, and it is the reason DOE discipline pays for itself many times over in saved solver hours.