2. Mathematical definitions and normalization

Multi-objective optimization of occupant restraint systems only becomes tractable once every injury metric is expressed in a common, dimensionless currency. Raw physical quantities — acceleration in g, moment in N·m, deflection in mm, force in kN — cannot be summed, weighted, or Pareto-ranked against one another. This section establishes the exact computational definitions of the four metric families used throughout the course, and shows how each is normalized into a risk scale so that a single objective vector can be assembled.

Head Injury Criterion: HIC15 and HIC36

HIC is defined over a sliding time window t₁…t₂ of the resultant translational acceleration at the head center of gravity, expressed in units of g:

HIC = max over (t1, t2) of  [ (t2 − t1) · ( (1/(t2 − t1)) ∫ a_res(t) dt )^2.5 ]
    subject to  t2 − t1 ≤ T_max

HIC15:  T_max = 15 ms
HIC36:  T_max = 36 ms

Two implementation details dominate practical results. First, the integral is the mean resultant acceleration over the window, so the exponent 2.5 amplifies peaks disproportionately: a 10 % increase in mean acceleration raises HIC by roughly 27 %. Second, the window is searched over all admissible start times, not anchored at t = 0. A naive implementation that evaluates only the full 15 ms or 36 ms window from impact will systematically under-report HIC when the true peak occurs later in the pulse.

HIC15 and HIC36 are not interchangeable. HIC36 admits longer windows and therefore captures lower-amplitude, longer-duration loading; it is the classical FMVSS 208 / ECE R94 criterion. HIC15 is more sensitive to short, high-amplitude spikes and is the criterion used in many consumer and motorsport protocols. In an optimization loop, mixing the two silently changes the feasible design space, so the choice must be fixed before any DOE is run. A robust evaluation strategy is to compute the cumulative integral once, then sweep candidate windows with a two-pointer scan in O(n) rather than the naive O(n²) double loop.

Nij: Normalized Neck Injury Criterion

Nij combines four neck load channels — axial compression Fz, axial tension, and the two sagittal bending moments — into a single scalar by normalizing each against its own tolerance limit:

Nij = ( Fz / Fz_crit ) + ( My / My_crit )

with sign-dependent critical values:
  compression:  Fz_crit = Fzc   (Fz < 0 in the SAE J211 compression convention)
  tension:      Fz_crit = Fzt   (Fz > 0)
  flexion:      My_crit = Myf
  extension:    My_crit = Mye

Nij = max( |Nij_combined| ) over the impact duration

The critical values are not universal constants; they depend on the dummy or human body model, the impact direction, and the regulatory protocol. Typical values for a mid-size male Hybrid III in frontal loading are on the order of Fz ≈ 4.0 kN compression, 4.5 kN tension, and My ≈ 190 N·m flexion, 200 N·m extension — these are illustrative magnitudes for teaching, and any real study must take the limits from the applicable regulation or validated model documentation.

Because Nij is a sum of two normalized terms, it is inherently a linear interaction model: a load case at 60 % of the compression limit and 60 % of the flexion limit yields Nij = 1.2, i.e. predicted injury even though neither channel alone is critical. This is precisely why Nij is useful as an optimization constraint — it couples axial and bending load paths that a restraint designer might otherwise tune independently.

Chest Deflection Metrics

Chest injury is quantified by the maximum sternal deflection D (mm), and in more detailed protocols by the deflection rate and the viscous criterion:

D_max = max( |D(t)| )                       [mm]
Viscous criterion:  VC = max( (D(t) · dD/dt) )   [m/s]

Deflection is measured along the sternum-to-spine axis; the sign convention again follows SAE J211, with compression typically negative. The viscous criterion multiplies instantaneous deflection by its rate, so it penalizes fast, shallow compressions that a pure deflection limit would pass. For normalization, deflection is divided by the protocol limit (commonly 63 mm for a 50th-percentile male in frontal testing, again protocol-dependent), and VC by its own limit (commonly 1.0 m/s).

Femur Force Limits

Femur loading is a single-channel axial force criterion. The axial compressive force Fz measured along the femoral shaft is compared against a limit that is itself a function of the axial force, because bone tolerance decreases as the load vector deviates from pure compression:

F_femur_limit = 7.65 kN   for pure axial compression (illustrative)
Normalized femur load:  N_femur = Fz_measured / F_femur_limit

In practice the femur criterion is often reported as the peak axial force per leg, with a separate bending moment check where the protocol requires it. For optimization purposes the normalized ratio is what matters: it places femur loading on the same 0–1+ scale as Nij and the chest metrics.

Assembling the Normalized Objective Vector

With each metric divided by its limit, the occupant state at any design point becomes a vector of normalized injury ratios. The figure below shows how the four raw measurement channels flow through their respective normalizations into a single comparable risk space, and where the sign-dependent branch for Nij sits in that chain.

Once every channel is normalized, the optimization problem can be stated cleanly: minimize a weighted sum or find the Pareto front of the vector r, subject to r_i ≤ 1 for all i. The normalization step is therefore not cosmetic — it defines the geometry of the feasible region and determines which designs are considered equivalent trade-offs.

Worked Example: Evaluating One Design Point

Consider a hypothetical restraint configuration whose simulated response (illustrative values, not measured data) yields: peak resultant head acceleration producing HIC15 = 620; neck loads of Fz = −3.1 kN and My = +95 N·m; chest deflection 52 mm with VC = 0.72 m/s; and peak femoral force 5.4 kN. Using the limits quoted above:

HIC ratio   = 620 / 700          = 0.886
Nij         = 3.1/4.0 + 95/190   = 0.775 + 0.500 = 1.275   → VIOLATION
D ratio     = 52 / 63            = 0.825
VC ratio    = 0.72 / 1.0         = 0.720
Femur ratio = 5.4 / 7.65         = 0.706

The design passes every single-channel check except Nij, which exceeds unity because the axial and bending contributions add. This is the characteristic failure mode of Nij-coupled optimization: a restraint that looks acceptable on each load channel individually can still violate the combined criterion. The corrective action is not to reduce Fz alone but to examine the interaction — for instance, a slightly softer load limiter may reduce My at the cost of a small increase in chest deflection, moving the design back inside the feasible region while keeping the other ratios below 1.

Practical Consistency Rules

• Filter before integrating. Apply the protocol-specified CFC filter (commonly CFC 1000 for head acceleration, CFC 600 for neck loads) before computing HIC or Nij; unfiltered signals inflate peaks and make results incomparable across runs.
• Fix the window search. Compute HIC over all admissible start times, and record which window produced the maximum — it is a useful diagnostic for pulse shaping.
• Bind limits to the protocol. Store critical values in a configuration file keyed by dummy type and impact direction, never hard-coded in the metric function.
• Declare signs once. Document the SAE J211 convention at the top of the evaluation module and assert it in unit tests with synthetic signals of known sign.

With these definitions fixed, the normalized vector r becomes the common language linking restraint design variables to injury outcomes, and the multi-objective search in later chapters operates on a well-posed, physically consistent objective space.