3. Sources of uncertainty in occupant injury prediction

An occupant injury prediction model is a chain of transformations: crash pulse → vehicle deceleration → restraint loading → occupant kinematics → tissue-level response → injury metric. Uncertainty enters at every link, and the links are not independent. A 10% variation in seat belt stiffness and a 10% variation in occupant mass can either reinforce or cancel each other depending on the crash mode. Before any quantification method is applied, the analyst must know which sources dominate, how they are represented statistically, and where they interact. This section classifies the four principal sources — material properties, geometry variability, boundary conditions, and model form uncertainty — and shows how each is translated into a quantitative input for a UQ study.

Material properties

Biological and restraint materials are the most variable inputs in the chain. Soft tissues exhibit rate dependence, anisotropy, and large inter-specimen scatter; published constitutive parameters for organs such as liver or brain frequently span a factor of two or more across studies. Restraint hardware is more controlled but still carries manufacturing tolerance: webbing elongation at 10 kN may be specified as 8% ± 1%, and retractor pretensioner gas generatives vary batch to batch.

The practical question is not "what is the material property" but "what distribution should be assigned, and at what level." Three levels are commonly used:

  • Specimen-level scatter — variation between coupons or tissue samples of the same nominal material. Captured by a coefficient of variation (CoV) and a distribution shape, often lognormal for positive-definite stiffness parameters.
  • Model-calibration uncertainty — the residual error when a constitutive model is fitted to test data. Even a perfect specimen has a parameter set that is only an estimate; report the posterior or the confidence interval, not a single value.
  • Population variability — the spread across the real population the model must represent (age, sex, BMI, tissue condition). This is the level that matters for injury risk curves.

For a teaching example, consider a rib cortical bone modulus used in a thorax FE model. Suppose literature and coupon tests suggest a mean of 12 GPa with a CoV of 0.15, modeled as lognormal. In a Monte Carlo run this becomes a sampled input; in a polynomial chaos expansion it becomes a random variable with a specified marginal. The choice of distribution matters: a normal distribution with the same mean and CoV would assign non-trivial probability to negative modulus, which is physically inadmissible. Always check that the support of the chosen distribution matches the physical domain.

When only a range is available (e.g., "8–14 GPa from three studies"), do not default to a uniform distribution without justification. A uniform assumption asserts that every value in the range is equally likely and that nothing outside it is possible — a strong claim. A triangular or truncated normal distribution is often more defensible, and the choice should be documented as an assumption.

Geometry variability

Geometry uncertainty operates at two scales. At the occupant scale, anthropometry drives injury outcome directly: stature, sitting height, chest depth, and limb length determine how the occupant interacts with the restraint and interior. At the component scale, manufacturing tolerance and assembly variation shift hard points — a seat pan angle tolerance of ±2°, a steering column tilt tolerance, a cushion thickness range.

Two representations are common in practice. The first is parametric morphing: a baseline FE mesh is warped by a small set of shape parameters (e.g., chest depth scale, rib cage aspect ratio) and the parameters are treated as random variables. This is efficient because the mesh topology is preserved and the number of variables stays low. The second is statistical shape modeling: a principal component analysis of a scanned population produces modes whose scores are sampled. This captures realistic correlated shape variation but requires a sufficiently large scan database and a registration pipeline.

A frequent error is to sample anthropometric dimensions independently. Stature and chest depth are correlated in the real population; independent sampling produces implausible occupants (very tall with very shallow chest) that inflate the tails of the injury response distribution. Use a correlation matrix or sample from a joint model.

Boundary conditions

Boundary conditions include the crash pulse, occupant posture and position, restraint routing and anchor locations, contact friction, and the interior environment. These are often the largest single contributor to output variance, and they are the hardest to characterize because they are scenario-dependent rather than material-dependent.

The crash pulse is a good illustration. A frontal sled pulse is not a single deterministic waveform; it varies with vehicle, impact speed, barrier type, and structural response. In UQ studies the pulse is frequently parameterized by a small set of features — peak deceleration, pulse duration, rise time, or the coefficients of a simplified pulse shape — and these features are sampled. The choice of parameterization is itself a modeling decision: two pulses with the same peak and duration can produce different occupant responses if their frequency content differs.

Posture is equally consequential and frequently under-represented. Out-of-position occupants, reclined seatbacks, and pre-impact bracing change the belt-to-body load path and can shift peak chest deflection by tens of percent. If the study population includes real-world postures, the posture distribution must be sampled jointly with anthropometry, because posture and body size are correlated.

Model form uncertainty

Model form uncertainty is the error introduced by the structure of the model itself — the governing equations, the constitutive law family, the contact algorithm, the injury criterion — independent of any parameter value. It is the most difficult source to quantify and the most often ignored.

Three practical manifestations deserve attention:

  • Constitutive model form — a linear elastic rib model and a rate-dependent plastic rib model can both be calibrated to the same test, yet predict different responses under a new loading rate. The discrepancy is model form error.
  • Injury criterion form — chest deflection, viscous criterion (VC), and chest compression rate are different metrics with different sensitivities. Selecting one over another is a model form decision that propagates into the risk curve.
  • Numerical discretization — mesh density, time step, and contact stiffness introduce a numerical component that should be separated from physical uncertainty. A mesh convergence study bounds this contribution; if it is not negligible relative to the physical scatter, it must be included.

Model form uncertainty is usually estimated by comparison against validation data: run the model and the experiment under matched conditions, compute the residual, and treat the residual as a bias or an additive error term. Where multiple model forms are plausible, model averaging or a Bayesian model selection framework can propagate the structural ambiguity explicitly.

How the sources combine

The four sources are not additive in their effect on the output. They interact through the physics of the load path, and the relative contribution depends on the output metric and the crash mode. The figure below shows the propagation structure: each source enters at a specific point in the chain, and the injury metric at the end inherits the combined effect.

Material properties modulus, yield, rate dep. Geometry variability anthropometry, tolerance Boundary conditions pulse, posture, friction Model form constitutive law, criterion Restraint loading belt force, airbag pressure Occupant kinematics displacement, velocity Tissue response stress, strain, deflection Injury metric Chest defl., VC, AIS correlated inputs Propagation notes direct physical propagation model form error applied at output stage Dashed grey links: statistical dependence between sources (e.g., stature–chest depth)

Two consequences follow from this structure. First, sensitivity analysis should be run before a full UQ campaign: a variance-based screening (e.g., Sobol indices or a Morris screening) identifies which sources dominate for the metric of interest, and effort can then be concentrated there. Second, the sources must be sampled jointly when correlations exist. Independent sampling of correlated inputs overstates output variance and can produce physically inconsistent occupant models.

A worked screening example

Suppose a frontal sled study targets peak chest deflection for a mid-size male occupant. The analyst defines four inputs: rib modulus (lognormal, mean 12 GPa, CoV 0.15), chest depth scale (normal, mean 1.0, SD 0.05), pulse peak deceleration (uniform, 35–45 g), and belt stiffness (normal, mean nominal, CoV 0.08). A Morris screening with 20 trajectories and 10 levels requires 20 × (4+1) = 100 model runs. The elementary effects are ranked by mean absolute value and by standard deviation; a high standard deviation flags non-linearity or interaction.

In a typical result of this kind (illustrative, not measured), pulse peak and belt stiffness often rank highest for chest deflection, with rib modulus contributing strongly through interaction with belt stiffness rather than as a main effect. That finding changes the study design: the analyst should not reduce the rib modulus to a single mean value, because its interaction term is significant, but can afford a coarser sampling of chest depth if its elementary effect is small. The screening output is a ranked list, not a final variance decomposition — it tells you where to spend the expensive runs.

Do not treat model form uncertainty as a residual to be absorbed into parameter scatter. If a constitutive model systematically under-predicts deflection at high rate, adding that bias to the parameter distribution inflates the apparent physical variability and corrupts the injury risk curve. Estimate model form error separately, against validation data, and propagate it as a distinct term.

Documenting the uncertainty budget

Every UQ study should carry an uncertainty budget: a table listing each source, its representation (distribution, range, or bias term), its basis (test data, literature, expert elicitation, assumption), and its estimated contribution to output variance. The budget is the audit trail that lets a reviewer judge whether the reported confidence interval is credible. Sources that could not be quantified should be listed explicitly as unquantified, with a note on the expected direction of their effect. A prediction interval that silently omits a known source is worse than one that states its scope.

The four sources described here are the inputs to every subsequent method in this course. Sensitivity analysis, surrogate modeling, and Bayesian calibration all operate on the representations established at this stage; if the distributions, correlations, and model form terms are poorly specified, no downstream method can recover the lost information.

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