4. Fundamentals of sensitivity analysis

Sensitivity analysis (SA) answers a deceptively simple question: if an input changes, how much does the output change? In occupant safety, the "inputs" are restraint parameters — belt force limiter level, pretensioner firing time, airbag vent size, seat pan angle, anchor position — and the "outputs" are injury metrics such as HIC15, chest deflection, chest acceleration 3 ms clip, neck Nij, or femur force. Before ranking parameters, you must choose a sensitivity method whose assumptions match the physics and the question you are asking. Choosing the wrong method is the most common reason a ranking looks plausible but is operationally useless.

Local (Derivative-Based) Sensitivity

A local method evaluates the partial derivative of the response y with respect to one input xi at a single nominal design point x0:

Si = ∂y/∂xi |x=x0

Because crash models are usually non-differentiable in closed form, the derivative is approximated by a forward or central finite difference. For a belt force limiter level FL with nominal 4.0 kN and a perturbation Δ = 0.2 kN:

S_FL ≈ [ y(4.2 kN) − y(3.8 kN) ] / (2 × 0.2 kN)   # central difference
# Example (illustrative, not measured): HIC15 = 612 at 3.8 kN, 548 at 4.2 kN
# S_FL ≈ (548 − 612) / 0.4 = −160 HIC per kN

The sign carries physical meaning: a negative value means increasing the force limiter level reduces HIC in this region, because more belt force is absorbed by the limiter rather than transferred to the chest. Local sensitivities are cheap — two runs per parameter — and are ideal for gradient-based optimization, tolerance studies, and quick "which knob matters near my baseline" screening.

Their limitation is equally important: a local derivative is only valid in a small neighborhood of x0. Occupant responses are strongly nonlinear and often non-monotonic. A force limiter that reduces chest deflection at 4 kN may increase it at 6 kN once the occupant loads the airbag differently. Local SA cannot see interactions (e.g., the effect of pretensioner timing depends on force limiter level), cannot detect a parameter whose influence is large only in a corner of the design space, and gives no information about how much of the output variance each input explains.

One-at-a-Time (OAT) Screening

OAT is the practical cousin of local SA: vary one factor across a range while holding all others at baseline, and plot or tabulate the response. It is the default in most restraint CAE studies because it is interpretable and requires only n × k runs for n parameters and k levels. OAT is excellent for building intuition and for detecting non-monotonic behavior within a single parameter's range. It is, however, structurally blind to interaction: if the optimal pretensioner time shifts when the force limiter changes, OAT at a fixed baseline will never reveal it. OAT results should therefore be treated as a screening step, not a final ranking.

Global, Variance-Based Sensitivity

Global methods sample the whole input space and apportion the output variance among inputs. The Sobol decomposition writes:

Var(y) = ΣVi + ΣVij + … + V1…k

from which two indices follow. The first-order index Si = Vi/Var(y) is the fraction of output variance explained by xi alone. The total-effect index STi = 1 − Var~i(E[y|x~i])/Var(y) includes all interactions involving xi. The gap STi − Si is a direct measure of how much of a parameter's influence is mediated by interaction with others — precisely the information OAT and local derivatives discard.

Estimation typically uses Saltelli sampling with a Latin Hypercube or Sobol sequence. For k = 8 parameters, a base sample N = 512, and the Saltelli scheme, the run count is N(k + 2) ≈ 5,120 model evaluations — feasible only with a surrogate. The standard workflow is therefore: sample the design space, run the full FE model (or a validated surrogate such as a Gaussian process or polynomial chaos expansion), then compute Si and STi on the surrogate. Convergence is checked by increasing N until the indices stabilize within a tolerance (e.g., ±0.02); non-converged indices usually indicate too few samples or a surrogate that has not resolved the response surface.

Local versus global sensitivity analysis workflow for occupant restraint parameters Comparison of local finite-difference sensitivity, one-at-a-time screening, and variance-based Sobol global sensitivity, showing inputs, sampling strategy, outputs, and applicability to occupant safety. Local / Finite Difference One-at-a-Time (OAT) Global / Variance-Based Sampling Baseline x0 ± Δ per parameter 2 runs per parameter Others held fixed Sweep one factor over range n × k runs Others at baseline Latin Hypercube / Sobol sequence N(k+2) runs, e.g. 512×10 ≈ 5,120 Full space, surrogate-assisted Output ∂y/∂xi at x0 e.g. −160 HIC per kN Sign + magnitude Response curve y(xi) Monotonicity, knee points No interaction term Si (first-order) and STi (total) STi − Si = interaction share Variance apportionment Best used for Gradient optimization, tolerance Near-baseline screening Cheap, fast Intuition building, screening Detecting non-monotonicity Not a final ranking Final parameter ranking Interaction-aware robustness Design-space exploration Occupant safety rule of thumb: OAT to screen → surrogate → Sobol indices to rank → local derivatives to tune near the chosen optimum.

Applicability to Occupant Safety

The choice of method should follow the decision the ranking will support. Three practical patterns dominate:

  • Early concept screening. With 10–15 candidate restraint parameters and a limited run budget, OAT across physically plausible ranges is the right first pass. It exposes non-monotonic responses (e.g., airbag vent size versus chest deflection) that a single local derivative would miss, and it produces plots engineers can defend in design reviews.
  • Final ranking and robustness. When the deliverable is a defensible ordering of parameters — for example, to prioritize which restraint parameter to lock before hardware freeze — variance-based indices are required. They quantify interaction (a pretensioner whose benefit depends on force limiter level will show STi ≫ Si) and they are valid across the whole design space, not just at the baseline.
  • Optimization and tolerance allocation. Once the ranking is fixed, local derivatives at the chosen optimum give the gradient needed for tuning and for mapping manufacturing tolerance (belt anchor position, pretensioner timing scatter) to injury-metric scatter.
A common failure mode: reporting OAT results as a global ranking. If two parameters interact strongly, OAT at a single baseline can rank the wrong one first. Always check the STi − Si gap before locking a ranking, and state the baseline and ranges explicitly, because a ranking is only meaningful relative to the sampled space.
Practical convergence check for Sobol indices: run the estimator at N and 2N; if any index moves by more than 0.02, increase the sample or improve the surrogate before trusting the ranking. Report Si, STi, and the interaction gap together — a single index hides the most decision-relevant information.

In short: local methods tell you the slope at a point, OAT tells you the shape along one axis, and variance-based global methods tell you how the output's uncertainty is actually distributed across the parameter space. For occupant injury ranking, the defensible sequence is OAT screening, surrogate construction, Sobol ranking, and local derivatives for final tuning — each method used where its assumptions hold.

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