4. Sources of discrepancy

In a learning-based crash test to CAE correlation workflow, the residual between the physical measurement and the simulation prediction is never a single number with a single cause. It is the superposition of several independent error mechanisms, each with its own physical origin, its own scaling behavior, and its own signature in the time-history data. The purpose of this chapter is to decompose that residual into five dominant contributors — modeling assumptions, material strain-rate effects, mesh discretization, boundary conditions, and test fixture variability — and to show how each one can be identified, quantified, and either corrected or bounded.

The decomposition matters because the five sources behave differently under refinement. Mesh discretization converges: halving the element size reduces the discretization error by a predictable order. Strain-rate effects do not converge — they are a constitutive modeling problem, and a finer mesh with the wrong rate law will still be wrong. Boundary conditions and fixture variability are epistemic: they are not numerical errors at all but mismatches between what the model assumes and what the laboratory actually did. Treating all five as "model error" and tuning a single global correction factor is the most common failure mode in correlation studies, because it produces a model that matches one test and generalizes to none.

Modeling Assumptions

Every CAE crash model embeds assumptions that are invisible in the input deck but visible in the residual. Typical examples include rigid-body treatment of components that actually deform, omission of secondary load paths, idealized spot-weld or adhesive connections represented as rigid beams, and lumped mass distributions that do not match the as-built hardware. A useful diagnostic is to compare the initial slope of the force–displacement curve. If the model is too stiff at small displacement, the cause is usually a geometric or connection assumption rather than a material issue, because strain-rate and plasticity effects are negligible in the elastic regime.

Assumption errors are best exposed by a hierarchy of models rather than a single high-fidelity model. Build a coarse "assumption-check" model with the same mass, the same primary load path, and deliberately simplified connections. If the coarse model already reproduces the peak force within 10–15%, the assumptions are sound and the remaining residual belongs to the other four categories. If it does not, no amount of mesh refinement will fix it.

Material Strain-Rate Effects

Automotive steels, aluminum alloys, and polymers all exhibit positive strain-rate sensitivity: yield stress increases with the logarithm of strain rate. In a frontal impact, the strain rate in the front rail can range from 10⁰ s^-1 in the crush initiator region to over 10³ s^-1 at the hinge lines. A constitutive model calibrated only from quasi-static tensile tests will systematically underpredict the force plateau.

The standard remedy is a rate-dependent plasticity law. For metals, the Cowper–Symonds relation is widely used:

sigma_y(eps_dot) = sigma_y0 * (1 + (eps_dot / D)^(1/p))

where:
  sigma_y0  = quasi-static yield stress        [MPa]
  eps_dot   = effective plastic strain rate    [1/s]
  D, p      = rate sensitivity parameters      [-]

Example (DP590, illustrative values):
  sigma_y0 = 380 MPa,  D = 8000 s^-1,  p = 5.2
  eps_dot = 100 s^-1  ->  sigma_y = 380 * (1 + (100/8000)^(1/5.2))
                                  = 380 * (1 + 0.0125^0.192)
                                  = 380 * 1.446  = 549 MPa

The parameters D and p must come from split-Hopkinson pressure bar (SHPB) tests at 10²–10³ s^-1, not from curve-fitting to the vehicle-level test. Fitting rate parameters to the full-vehicle residual is a form of inverse crime: it will absorb mesh and boundary-condition errors into the material law and destroy predictive capability.

Mesh Discretization

Shell-element crash models typically use 5–10 mm characteristic length in the crush zone. The discretization error manifests in three ways: artificial stiffening from under-integrated elements in bending, hourglass energy that is not physical, and a shift in the localization point of folding. A practical convergence check is to refine only the crush zone by a factor of two and compare the peak force and the absorbed energy. If the peak force changes by more than 5%, the coarse mesh is not converged.

Two quantitative guards should be monitored in every run. First, hourglass energy should remain below 5% of internal energy; above 10%, the result is not trustworthy. Second, the ratio of internal energy to kinetic energy at the end of the event should be stable — a sudden jump indicates element inversion or contact instability rather than a physical effect.

Boundary Conditions and Fixture Variability

These two are grouped because they share a common root: the model's idealized constraint does not match the physical constraint. In a sled test, the model may apply a prescribed velocity pulse at a node set, while the actual sled imposes a measured acceleration through a hydraulic piston with finite stiffness. The difference appears as a phase lag in the force–time history, typically 2–5 ms, which is easily mistaken for a material delay.

Fixture variability is the stochastic component. Clamping torque, weld nugget diameter, adhesive bead width, and specimen alignment all vary within tolerance, and their effect on the measured force can be 5–15% even for nominally identical tests. The correct treatment is not to tune the model to one specimen but to run a small design-of-experiments matrix — for example, three clamping torques × three weld diameters — and compare the model against the envelope of the physical results.

Separating the Contributions in Practice

A reproducible procedure for attributing the residual is to run a controlled sequence of model variants against the same physical test, changing one factor at a time. The following table gives an illustrative parameter set for a frontal rail component; the numbers are representative of a typical DP590 rail and are intended as a starting configuration, not as measured results.

After R1–R4, the residual that remains is the irreducible combination of unmodeled physics and measurement uncertainty. If that residual is still larger than the fixture envelope, the model has a genuine assumption error that must be addressed at the architecture level, not by further parameter tuning. If it falls inside the envelope, the model is correlation-ready and the remaining work is statistical validation across the test matrix rather than single-point matching.

The practical takeaway is that discrepancy analysis is a decomposition problem, not a fitting problem. Each of the five sources has a distinct physical origin, a distinct numerical behavior under refinement, and a distinct signature in the response data. Identifying which source dominates a given residual — before touching any parameter — is what separates a correlation study that generalizes from one that merely reproduces a single curve.