4. Design variables and parameterization for crashworthiness

Bayesian optimization (BO) does not optimize a vehicle directly; it optimizes a vector of design variables that a parameterization scheme maps onto geometry, material assignment, and component layout. The quality of that mapping determines whether the optimizer can even reach a good design. This section examines the four variable families most common in crashworthiness work — thickness, shape, topology, and material grade — and the multi-component interactions that couple them.

Thickness Variables

Sheet-metal thickness is the most widely used design variable in body-in-white (BIW) optimization because it is continuous, cheap to perturb in a shell model, and directly controls energy absorption through plastic membrane and bending resistance. A typical front-rail study parameterizes 8–15 gauge variables with bounds such as t ∈ [0.8, 2.5] mm, starting from nominal values like 1.2 mm or 1.5 mm.

Two practical constraints matter. First, thickness must respect the rolling gauge series actually available from the supplier; snapping to discrete increments (0.05 mm steps) after optimization avoids designs that cannot be manufactured. Second, thickness changes alter mass, so a mass constraint M(x) ≤ M₀ is usually imposed alongside intrusion or acceleration limits. Because thickness has a near-monotonic effect on both stiffness and mass, it is a well-behaved variable for BO — but it also means the optimizer will quickly saturate bounds unless the mass constraint is active.

Shape Variables

Shape variables move material rather than add it. Common parameterizations include:

• Morphing / basis-vector methods: a set of precomputed perturbation fields is blended with weights w₁…wₙ, so the mesh deforms smoothly without remeshing.
• Cross-section parameters: rail width, height, corner radius, and wall angle, often 4–8 variables per section.
• Bead and rib geometry: bead depth, spacing, and orientation on closures and floor panels.

Shape variables are attractive because they can raise crash performance without adding mass, but they are also the most likely to produce mesh distortion, negative Jacobians, or self-intersection. A robust workflow checks element quality after each morph and rejects or repairs infeasible samples before they reach the solver. In BO terms, these failures become part of the feasibility model rather than silent crashes.

Topology Variables

Topology optimization answers a different question: where should material exist at all? In crashworthiness, this typically means deciding the placement of ribs, bulkheads, or internal reinforcements inside a component, or selecting which of several candidate load paths are active. Because topology is discrete, it is usually encoded as:

• a density field on a coarse grid, thresholded to solid/void;
• a set of binary flags selecting ribs from a library;
• a small number of categorical layout options (e.g., 3-, 4-, or 6-cell cross-section).

Discrete and categorical variables break the smoothness assumptions behind standard Gaussian-process kernels. Practical remedies include continuous relaxation with a threshold, or using a mixed kernel that combines a Matérn component for continuous variables with a Hamming/overlap kernel for categorical ones. Topology variables also tend to have strong interactions with thickness: adding a rib changes the optimal gauge of the surrounding panel.

Material Grade Variables

Material grade selection is categorical but ordered — for example, mild steel, HSLA 340, DP 590, DP 780, and press-hardened 1500 MPa. The variable is not a number, but the grades have a meaningful ranking in yield strength and ductility. Two encodings are common:

Material grade interacts strongly with thickness and shape. A thinner DP 780 panel may match the energy absorption of a thicker mild-steel panel at lower mass, but with reduced elongation and a higher risk of edge cracking. This is why grade should rarely be optimized in isolation; it belongs in the same design vector as gauge and geometry.

Multi-Component Interactions

Real crash structures are assemblies. The front rail, shotgun, bumper beam, and crash box share load, and improving one in isolation can shift failure to another. Three interaction patterns dominate:

1. Load-path coupling: stiffening the rail may overload the bumper-to-rail joint, causing weld or bolt failure.
2. Mass trade: adding gauge to one component may force mass reduction elsewhere to meet the total target.
3. Sequential collapse: the desired folding order depends on relative strengths across components, so grades and gauges must be co-designed.

These interactions mean the design vector should include variables from multiple components simultaneously, and the objective should be a system-level metric (e.g., B-pillar intrusion, chest acceleration, or a weighted injury criterion) rather than a single-part response.

Parameterization Workflow

A reproducible setup for a front-rail study might look like this:

variables:
  t_rail_inner   : [0.8, 2.4] mm, step 0.05
  t_rail_outer   : [0.8, 2.4] mm, step 0.05
  t_crashbox     : [1.0, 3.0] mm, step 0.05
  section_height : [60, 110] mm
  corner_radius  : [4, 16] mm
  bead_depth     : [0, 6] mm
  grade_rail     : {DP590, DP780, PHS1500}
  grade_crashbox : {HSLA340, DP590, DP780}

constraints:
  mass_total     <= 1.00 * M_baseline
  intrusion_max  <= 120 mm
  peak_force     <= 180 kN

objective:
  minimize chest_g (or maximize SEA)

After sampling, each design is meshed, solved with an explicit crash solver (e.g., LS-DYNA), and the responses are extracted. Infeasible or failed runs are labeled and passed to the feasibility classifier. The BO loop then proposes the next batch, balancing exploration of untested grade/thickness combinations against refinement near the current best.

Reading the Results

When the optimization converges, inspect three things. First, the variable importance ranking — if thickness dominates and topology is negligible, the topology encoding may be too coarse. Second, the interaction map — a strong thickness–grade interaction usually signals that the load path is being redistributed, which is often the most valuable design insight. Third, the feasibility boundary — designs near the mass or intrusion limit are the ones worth validating experimentally, because they carry the least margin.

The figure summarizes how the four variable families feed into a coupled assembly and ultimately into system-level crash metrics. The red arrows mark the load-transfer paths where interactions are strongest; these are precisely the couplings that a single-component optimization would miss. When you build the design vector for a BO run, include at least one variable from each interacting component so the surrogate can learn these cross-couplings rather than treating each part in isolation.