4. Uncertainty quantification (UQ) framework

Crash CAE models are deterministic machines fed by uncertain inputs. Material properties scatter from coil to coil, gauge thickness varies within rolling tolerances, spot-weld nugget diameters are only statistically known, and friction coefficients at impact interfaces are rarely measured with precision. A single LS-DYNA run produces one number for peak intrusion or chest acceleration, but the decision the engineer must make — does this design meet the occupant-safety target across the production population? — is inherently probabilistic. The uncertainty quantification (UQ) framework in this chapter connects three operations that are usually taught separately but must be engineered as one loop: forward propagation of input uncertainty to output distributions, sensitivity analysis to rank which inputs actually drive that spread, and inverse calibration to condition the model on physical crash-test evidence.

Forward propagation: from input distributions to output distributions

Forward propagation answers: given a joint distribution over the input vector X = (yield stress, thickness, weld diameter, friction, …), what is the distribution of the response Y = g(X), where g is the crash solver? The practical constraint is cost. A full frontal sled model may take 40–90 minutes per run on 64 cores, so Monte Carlo with 10,000 samples is out of reach. The standard engineering compromise is a surrogate-assisted workflow: run a space-filling design of experiments (Latin hypercube or Sobol sequence) of 80–200 crash simulations, fit a Gaussian process or polynomial-chaos surrogate, validate it with a hold-out set, then draw 10⁵–10⁶ surrogate evaluations for the output statistics.

A reproducible example: five uncertain inputs — DP600 yield stress σ_y ~ N(420, 25) MPa, B-pillar thickness t ~ N(1.50, 0.05) mm, weld nugget diameter d ~ N(5.0, 0.3) mm, friction μ ~ U(0.10, 0.25), and strain-rate parameter C ~ N(40, 4) s⁻¹. A 128-point Latin hypercube is executed, and a Gaussian-process surrogate is fitted to peak B-pillar intrusion. The surrogate's leave-one-out R² should exceed 0.95 before it is trusted; if it does not, the design is too sparse in the region where intrusion saturates, and points must be added adaptively near the response knee. The propagated result might read: intrusion mean 142 mm, standard deviation 9.8 mm, 95th percentile 159 mm. That 95th percentile — not the nominal run — is what the safety margin must be checked against.

Sensitivity analysis: ranking the drivers

Sensitivity analysis tells you where to spend tolerance money and where to spend simulation effort. Two families matter here. Local (derivative-based) sensitivities — ∂Y/∂X_i — are cheap from the surrogate but misleading when the response is nonlinear or inputs are correlated. Global variance-based Sobol indices decompose Var(Y) into first-order contributions S_i and total-effect indices S_Ti that capture interactions. For the example above, a plausible outcome is S_thickness = 0.52, S_σy = 0.21, S_weld = 0.11, with S_Tweld = 0.19 — the weld diameter matters more through its interaction with thickness than on its own. That distinction changes the engineering response: tightening the weld process alone will not recover the variance; the thickness–weld joint effect must be addressed together.

Compute Sobol indices from the surrogate using Saltelli's pick-freeze estimator with N = 2¹⁴ base samples; the surrogate makes this essentially free. Always report confidence intervals from bootstrap resampling — an index of 0.11 ± 0.06 is not a reliable ranking signal.

Inverse calibration: conditioning the model on test data

Forward propagation assumes the input distributions are known. They usually are not — they are the very thing crash tests can inform. Inverse calibration (Bayesian calibration) treats the uncertain parameters θ as random variables and updates their prior p(θ) with test evidence y^obs through the likelihood p(y^obs | θ), yielding the posterior p(θ | y^obs) ∝ p(y^obs | θ)p(θ). Because each likelihood evaluation requires a simulation, the surrogate is again the enabler: Markov-chain Monte Carlo (e.g., 4 chains × 2,000 draws, No-U-Turn sampler) runs entirely on the GP surrogate, with the surrogate's own discrepancy variance folded into the likelihood so that surrogate error does not masquerade as parameter uncertainty.

A realistic calibration target set: three sled tests measuring peak intrusion (138, 145, 151 mm) and two component tests measuring weld pull-out force. The posterior for thickness might tighten from N(1.50, 0.05) to N(1.49, 0.02) mm, while friction stays broad — the data simply do not constrain it. That asymmetry is itself a finding: it tells the test program where additional instrumentation is needed. After calibration, the posteriors replace the priors and the forward propagation is re-run, closing the loop.

Closing the loop and reading the result

The three operations form a cycle, not a pipeline. Forward propagation with prior inputs gives a P95 that is too wide to certify; sensitivity analysis shows thickness dominates; calibration against the three sled tests tightens the thickness posterior; the re-run forward propagation then yields a narrower P95 — perhaps 152 mm — with the surrogate uncertainty explicitly carried through. The engineering judgment is whether that interval clears the target with margin. If the posterior for friction remains unconstrained, the honest conclusion is that the model cannot yet certify friction-sensitive responses, and the test plan must add a component-level friction measurement rather than pretending the prior was knowledge.

Two failure modes recur in production. First, surrogate over-trust: MCMC explores regions far outside the DOE box, where the GP reverts to its prior mean and the posterior becomes an artifact of the surrogate, not the physics. Constrain the parameter bounds to the design space or add adaptive infill runs. Second, identifiability collapse: two parameters (e.g., yield stress and thickness) produce nearly the same intrusion signature, so the posterior is a narrow ridge rather than a point. Detect this by inspecting posterior correlations; the remedy is additional test observables — a local strain measurement or a force–displacement curve — that break the degeneracy. The UQ framework is only as trustworthy as the evidence fed into it, and its most valuable output is often the explicit statement of what the current test program still cannot resolve.