4. Feature engineering for crashworthiness

Crashworthiness simulation produces a rich but unwieldy data landscape. A single full-frontal LS-DYNA model may contain millions of shell elements, hundreds of material definitions, and dozens of time-history outputs. Before any surrogate model, sensitivity study, or uncertainty-quantification workflow can operate, that raw landscape must be converted into a compact, physically meaningful feature vector. This section treats feature engineering as a deliberate design activity: choosing which geometric, material, and loading variables to expose, how to encode them, and how to reduce their dimensionality without discarding crash-relevant signal.

Why Feature Engineering Determines the Ceiling of the Surrogate

A surrogate model can only learn relationships that are present in its inputs. If a geometric feature is omitted, no amount of Gaussian-process tuning will recover its effect. Conversely, redundant or noisy features inflate the input space, degrade distance metrics in kernel methods, and make sensitivity indices unstable. The practical goal is therefore a feature set that is complete with respect to the design intent, non-redundant, and scaled consistently so that no variable dominates purely because of its unit magnitude.

Geometric Parameterization

Geometric features fall into three tiers. The first tier is direct CAD parameters: rail thickness, bumper beam depth, rocker section height, B-pillar gauge. These are the variables a designer actually controls, and they map one-to-one onto morphing operations. The second tier is derived scalar descriptors that capture shape intent more compactly than raw coordinates — for example, the crush-zone length-to-width ratio, the longitudinal distance from bumper to firewall, or the section second moment of area of a rocker. The third tier is field-based representations, where a component's mid-surface is projected onto a low-dimensional basis.

For the third tier, proper orthogonal decomposition (POD) is the standard tool. Given a snapshot matrix X whose columns are nodal displacement or thickness fields from n design variants, the singular value decomposition X = UΣVᵀ yields modes U. Retaining the first k modes gives a reduced coordinate vector a = U_kᵀ x. In crash applications, 5–15 modes typically capture over 95% of geometric variance for a single stamped panel, and those modal amplitudes become the design variables passed to the surrogate.

Material Card Variables

Material uncertainty is often the dominant contributor to scatter in crash responses, yet it is frequently under-parameterized. For a typical *MAT_PIECEWISE_LINEAR_PLASTICITY (MAT_024) card used for mild steel, the influential variables are:

• Density (RO) — affects inertia and kinetic energy; low scatter but non-zero.
• Young's modulus (E) — controls elastic stiffness and wave speed.
• Poisson's ratio (PR) — usually fixed near 0.3; rarely worth sampling.
• Yield stress (SIGY) — the single most influential plasticity parameter for intrusion and peak force.
• Hardening curve (LCSS / ES–EPS pairs) — the full stress–strain table, not just one point.
• Strain-rate parameters (C, P in Cowper–Symonds) — critical for high-speed events.
• Failure strain (FAIL) — governs element deletion and fracture initiation.

For advanced grades, *MAT_037 (transversely anisotropic) adds R00, R45, R90 Lankford coefficients, and *MAT_224 (tabular Johnson–Cook) requires a full effective-plastic-strain vs. stress vs. strain-rate vs. temperature table. The engineering decision is which of these to treat as uncertain. A defensible default is to sample SIGY, the hardening curve scale factor, and FAIL, while holding E and RO at nominal values unless the material source is known to be variable.

Hardening curves are themselves functional data. Two practical encodings exist: (1) sample a multiplicative scale factor s applied to the entire curve, giving σ(ε) = s · σ₀(ε); or (2) apply POD to a library of measured curves and sample the first two or three modal coefficients. The scale-factor approach is simpler and preserves curve shape; the POD approach captures genuine shape variation from different coil suppliers.

Loading Conditions

Loading features describe the event, not the structure. For a frontal impact they include impact velocity, impact angle (yaw and pitch), barrier type (rigid wall, deformable barrier, pole), and offset percentage. For side impact, add pole height and impactor mass. These variables are usually well-characterized from regulatory test procedures, so their uncertainty is smaller than material scatter — but they are essential as controlled inputs because the surrogate must generalize across the loading envelope, not just one test.

A common mistake is to encode loading conditions as categorical labels (e.g., "FMVSS 208" vs. "Euro NCAP"). This destroys the continuous structure that a surrogate could exploit. Prefer continuous encodings: velocity in m/s, angle in degrees, offset as a fraction of vehicle width.

Dimensionality Reduction

After assembling geometric, material, and loading features, the raw vector may contain 30–80 dimensions. Dimensionality reduction serves two purposes: removing correlation and improving surrogate conditioning. Three methods are appropriate here.

Principal Component Analysis (PCA) is the workhorse. Standardize each feature to zero mean and unit variance first, then compute the covariance matrix and its eigenvectors. Retain components explaining 95% of variance. PCA is linear and interpretable: each component is a weighted combination of original features.

Active subspaces are more targeted for uncertainty quantification. The method computes the gradient of the quantity of interest with respect to each input, forms the matrix C = ∫ ∇f ∇fᵀ ρ(dx), and identifies the directions in input space along which the output varies most. For crash models, the active subspace often has dimension 2–4 even when the input has 40 dimensions, because intrusion and peak force are dominated by a few combinations of rail gauge and yield stress.

Feature importance screening via variance-based Sobol indices or Morris elementary effects provides a complementary filter. Morris screening is cheap (it requires only r(k+1) model evaluations for k inputs and r trajectories) and is well suited to the early elimination of inert variables before a full Sobol analysis.

A Worked Example: Frontal Rail Feature Set

Consider a frontal crash model with a single front rail. The design intent is to study how rail geometry and material grade affect maximum intrusion and peak force. A defensible feature set is:

Eight features, all continuous, all physically interpretable. After standardizing and running PCA on a 200-sample Latin hypercube, suppose the first three components explain 96% of variance. PC1 loads heavily on t and SIGY (the "strength" direction), PC2 on h and v₀ (the "energy" direction), and PC3 on s and FAIL (the "ductility" direction). These three components become the surrogate inputs.

Practical Verification

After reduction, verify the feature set by three checks. First, reconstruction error: project the original data onto the reduced space and back, and confirm that the relative error is below a threshold you set (typically 5% for geometric fields, 2% for scalar features). Second, surrogate cross-validation: train the surrogate on the reduced features and compare leave-one-out error against the full-feature baseline. If reduction degrades accuracy significantly, a physically important direction was discarded. Third, sensitivity stability: recompute Sobol indices on the reduced features and confirm that the ranking of influential variables is unchanged. If the ranking flips, the reduction has introduced a bias.

The output of this stage is a compact, standardized, physically interpretable feature vector — the input contract for every downstream surrogate, sensitivity analysis, and uncertainty-propagation study in the crashworthiness workflow.