3. Formulating occupant protection optimization

Before any Pareto front can be computed, the restraint-system engineer must translate a physical safety requirement into a formal optimization problem. This chapter builds that formulation for a frontal-impact occupant protection case, using a driver-side restraint system as the working example. The goal is not merely to list variables and responses, but to show how each choice — variable bounds, constraint limits, objective definitions, and the coupling to the simulation model — determines whether the resulting Pareto set is physically meaningful or merely numerically valid.

1. Design variables: what the optimizer is allowed to change

Design variables must be quantities that (a) the engineer can actually control in the restraint hardware or its calibration, and (b) the CAE model exposes as parametric inputs. For a driver frontal restraint system, a representative set is:

• Seat-belt force limiter level — the constant or two-stage force threshold in the retractor/load-limiter, typically 2.0–5.0 kN.
• Pretensioner firing time — pyrotechnic or motorized pretensioner activation, typically 8–25 ms after t=0.
• Airbag vent diameter — effective vent area controlling bag pressure decay, e.g. 25–60 mm equivalent diameter.
• Airbag inflator mass flow scaling factor — a multiplier on the inflator gas-mass-flow curve, 0.85–1.15.
• Airbag tether length — controlling bag stand-off and shape, e.g. 120–200 mm.
• Seat-pan cushion stiffness scale — a multiplier on the seat foam/spring characteristic, 0.8–1.2.

Each variable is bounded by manufacturing feasibility and by the validated range of the underlying CAE sub-models. A load-limiter below 2.0 kN may be unachievable with the chosen retractor hardware; an inflator scaling above 1.15 may push the model outside its validated inflator characterization. Bounds are therefore not arbitrary — they encode engineering credibility. Variables are also normalized to [0, 1] internally so that the optimizer's step sizes are comparable across quantities with different physical units.

2. Objectives: the injury criteria that define "better"

Occupant protection is inherently multi-objective because different body regions respond to conflicting restraint strategies. The standard frontal criteria are:

These criteria are not independent. A stiffer load limiter reduces chest deflection but raises head excursion and HIC; a larger vent reduces bag pressure and HIC but may allow the head to contact the steering wheel. This coupling is precisely why a Pareto formulation is appropriate: the optimizer should expose the trade-off surface rather than collapse it into a single weighted score.

For the working example, three objectives are selected: minimize HIC₁₅, minimize D_max, and minimize F_femur. The remaining criteria (a_3ms, N_ij) are treated as constraints, since they are hard regulatory limits rather than quantities to be traded off.

3. Constraints: the feasible region

Constraints fall into three classes. Regulatory constraints encode the pass/fail limits above; they are hard and must not be violated by any candidate in the final Pareto set. Packaging and geometric constraints ensure the design is physically buildable — for example, the airbag must not exceed a maximum deployed volume, and the pretensioner firing time must precede the occupant's forward excursion by a minimum margin. Model-validity constraints keep the simulation within its validated envelope, such as a maximum belt force that the retractor model can represent without extrapolation.

In the example, the constraint set is:

g1: HIC15        <= 700
g2: D_max        <= 63 mm
g3: a_3ms        <= 60 g
g4: F_femur      <= 10 kN
g5: N_ij         <= 1.0
g6: V_bag        <= 65 L          (packaging)
g7: t_pretension <= 25 ms         (timing feasibility)
g8: F_limiter    <= 5.0 kN        (retractor model validity)

Note that g1–g5 overlap with the objectives for HIC and D_max. This is intentional: the objective is to minimize HIC, but the constraint ensures no candidate exceeds the regulatory ceiling. The optimizer therefore searches for low-HIC designs while never proposing an infeasible one.

4. Problem setup and the simulation-in-the-loop workflow

The optimization problem is stated as:

minimize   f(x) = [HIC15(x), D_max(x), F_femur(x)]
subject to g_i(x) <= 0,  i = 1..8
           x_L <= x <= x_U
where x = [F_limiter, t_pretension, d_vent, m_inflator, L_tether, k_seat]

Each evaluation of f(x) requires a full frontal-impact CAE run — typically a 150–300 ms explicit simulation with a Hybrid III 50th-percentile dummy, taking 20–90 minutes on a compute node. Because the objective function is expensive and non-smooth (contact, bag deployment, belt slip), the practical workflow uses a design-of-experiments surrogate: a Latin hypercube or optimal-space-filling sample of 60–120 runs, a Kriging or radial-basis-function surrogate per response, and then a multi-objective evolutionary algorithm (e.g. NSGA-II) on the surrogate. The final Pareto candidates are re-verified with full CAE runs.

5. Reading the formulation as a whole

The SVG below summarizes how the six design variables feed the CAE occupant model, which produces the injury responses, which are then split into objectives and constraints before entering the Pareto optimizer. The key structural point is that the same simulation output serves both roles: HIC and D_max appear as objectives and as constraints, while a_3ms, N_ij, and the packaging limits act only as constraints.

Two formulation choices deserve emphasis. First, the objective set is deliberately small — three criteria — because Pareto fronts beyond four or five objectives become difficult to visualize and to act on; the remaining criteria are better handled as constraints. Second, the constraint limits are set at the regulatory thresholds, not at the current baseline performance. Setting a constraint at the baseline value would artificially shrink the feasible region and hide designs that improve one criterion at the expense of another while still passing regulation.

6. A worked setup check

As a sanity check on the formulation, consider a candidate with F_limiter = 3.2 kN, t_pretension = 14 ms, d_vent = 40 mm, m_inflator = 1.0, L_tether = 160 mm, k_seat = 1.0. A surrogate evaluation might return HIC15 = 620, D_max = 58 mm, F_femur = 7.8 kN, a_3ms = 54 g, N_ij = 0.82, V_bag = 61 L. All constraints are satisfied, so the candidate is feasible and contributes to the Pareto set. If instead D_max = 66 mm, the candidate is infeasible and must be discarded regardless of how low its HIC is — a point that is easy to lose when reading a scatter plot of objectives alone. The formulation, in other words, is what makes the Pareto front trustworthy: it defines exactly which designs are allowed to compete and on what terms.