2. Governing equations for automotive solids and structures

Every physics-informed neural network (PINN) for structural analysis is only as good as the differential operator it embeds in its loss function. Before a single collocation point is sampled or a network is trained, the engineer must decide which governing equations actually describe the physical behavior of the component under study. In automotive body-in-white (BIW) engineering, that decision spans four distinct regimes: linear elasticity for small-strain stiffness problems, elastodynamics for crash and vibration transients, modal analysis for eigenvalue-based characterization, and geometric nonlinearity for thin sheet-metal assemblies that undergo large rotations. This chapter develops each formulation, states its assumptions and validity limits, and shows how each one translates into a residual that a PINN can minimize.

Linear Elasticity: The Baseline Operator

For a homogeneous, isotropic solid under small deformations, the displacement field u(x) satisfies the Navier–Cauchy equilibrium equation in the absence of inertial effects:

∇·σ + f = 0,   σ = λ tr(ε) I + 2μ ε,   ε = ½(∇u + ∇uᵀ)

Here σ is the Cauchy stress tensor, ε the infinitesimal strain tensor, f the body force per unit volume, and λ, μ the Lamé parameters. For automotive steel with Young's modulus E = 210 GPa and Poisson's ratio ν = 0.30, these evaluate to λ ≈ 121.2 GPa and μ ≈ 80.8 GPa. The traction boundary condition σ·n = t̄ and the Dirichlet condition u = ū complete the boundary value problem.

In a PINN, the residual is formed by substituting the network output u_θ(x) into the equilibrium equation and evaluating the divergence via automatic differentiation. The composite loss typically reads:

L = w_pde · ||∇·σ(u_θ) + f||²  +  w_bc · ||σ(u_θ)·n − t̄||²  +  w_data · ||u_θ − u_obs||²

Linear elasticity is appropriate for stiffness and compliance studies of body panels under static loads — door sag, hood bending, or bracket deflection — where strains remain below roughly 0.2% and rotations are negligible. It is the correct starting point for most BIW linear static load cases, and it is the regime in which PINNs have demonstrated the most reliable convergence.

Elastodynamics: Adding Inertia and Time

When inertial effects cannot be neglected — crash pulses, road-induced vibration, or impact loading — the equilibrium statement becomes the elastodynamic equation of motion:

ρ ü = ∇·σ + f

where ρ is the mass density (≈ 7850 kg/m³ for steel) and ü the second material time derivative of displacement. The problem now requires initial conditions u(x,0) = u₀ and u̇(x,0) = v₀ in addition to the boundary conditions. For a PINN, time becomes an additional input coordinate, and the residual includes the acceleration term obtained by differentiating the network twice with respect to t. This is where PINNs face their stiffest challenge: high-frequency content in crash pulses demands dense temporal sampling, and the loss landscape becomes far more difficult than in the static case.

A practical automotive example is a 1 m × 0.8 m × 0.8 mm steel roof panel subjected to a half-sine pressure pulse of 0.5 MPa amplitude and 5 ms duration. The fundamental period of such a panel is on the order of 10–20 ms, so the pulse excites the first few modes. A PINN trained on this problem must resolve both the spatial bending wavelengths and the temporal oscillation, which typically requires a Fourier-feature input encoding or a time-marching curriculum to converge.

Modal Analysis: The Generalized Eigenvalue Problem

Modal analysis extracts the natural frequencies ω_i and mode shapes φ_i from the undamped free-vibration form of the elastodynamic equation. Assuming harmonic motion u(x,t) = φ(x)e^{iωt} and discretizing with finite elements yields the generalized eigenvalue problem:

K φ = ω² M φ

where K is the global stiffness matrix and M the consistent mass matrix. For a typical BIW body, the first elastic mode (often a global bending or torsion mode) falls between 20 and 45 Hz, while local panel modes can reach 80–200 Hz. These targets drive NVH (noise, vibration, harshness) design: a roof panel whose first mode sits at 28 Hz may resonate with engine idle excitation and require a bead or a constrained-layer damper.

Embedding modal analysis into a PINN is less direct than embedding the PDE residual, because the eigenvalue problem is not a pointwise differential equation. Two strategies are common. The first treats ω as a trainable scalar and minimizes the residual ||K φ_θ − ω² M φ_θ|| together with a mass-normalization constraint φ_θᵀ M φ_θ = 1. The second uses a frequency-domain formulation, where the network learns the steady-state harmonic response over a sweep of excitation frequencies and the resonances appear as peaks in the transfer function. The first approach is more faithful to the eigenvalue problem; the second is more robust when damping or broadband excitation is present.

Geometric Nonlinearity in Sheet-Metal Assemblies

Thin automotive panels routinely undergo rotations that exceed the small-strain assumption even when the material remains elastic. A 0.8 mm steel panel that snaps through a shallow oil-canning instability, or a closure panel that deflects 15 mm under a 200 N palm load, exhibits rotations of several degrees — enough that the linear strain-displacement relation ε = ½(∇u + ∇uᵀ) no longer holds. The correct measure is the Green–Lagrange strain:

E = ½(FᵀF − I),   F = I + ∇u

and equilibrium must be written in the reference configuration using the second Piola–Kirchhoff stress S:

∇·(F S) + f = 0

This introduces a quadratic dependence on displacement gradients, which is precisely the kind of nonlinearity that PINNs are well suited to represent — provided the network has enough capacity and the collocation points are distributed where the deformation gradients are steep. In practice, the residual is evaluated in the deformed configuration, and the loss must include a penalty for the incompressibility constraint if the material is nearly incompressible, or a hyperelastic constitutive law if the strain exceeds yield.

For a sheet-metal assembly, the geometric nonlinearity is often localized: a small region near a bead, a hem flange, or a bolt hole carries most of the rotation, while the rest of the panel remains nearly linear. A uniform collocation grid wastes points in the linear region. Adaptive sampling — refining points where the residual is largest — is therefore essential for convergence, and it is one of the main practical differences between a naive PINN and one that works on real automotive geometry.

From Equations to Residuals: A Unified View

The four regimes are not independent; they form a hierarchy of increasing physical fidelity. Linear elasticity is the small-strain, static limit of elastodynamics. Modal analysis is the eigenvalue problem obtained from elastodynamics after harmonic ansatz and discretization. Geometric nonlinearity generalizes linear elasticity by replacing the linear strain measure with the Green–Lagrange strain and the Cauchy stress with the second Piola–Kirchhoff stress. A PINN implementation can therefore share most of its architecture across regimes, changing only the residual computation and the boundary/initial condition terms.

The figure below maps this hierarchy, showing the governing equation, the key kinematic assumption, and the automotive load case each regime serves.

Practical Guidance for Implementation

When implementing these equations in a PINN framework, three practical points deserve attention. First, non-dimensionalize the inputs and outputs. A displacement of 1 mm and a modulus of 210 GPa differ by fourteen orders of magnitude; without scaling, the loss is dominated by the stiffness term and the network learns nothing useful. A common choice is to scale lengths by the panel's largest dimension, displacements by a characteristic deflection, and stresses by E.

Second, choose collocation points where the physics is hardest. For linear elasticity, a uniform grid or Latin hypercube sampling suffices. For geometric nonlinearity, start uniform and then resample toward regions of high residual after every few thousand epochs. For elastodynamics, ensure the temporal sampling resolves the highest frequency of interest — at least ten points per period of the fastest mode.

Third, validate against a known analytical or finite-element solution before trusting the PINN on a new geometry. A cantilever beam under a tip load has a closed-form deflection δ = FL³/(3EI); a simply supported plate has a known first natural frequency. These benchmarks catch sign errors, missing boundary terms, and scaling mistakes before they propagate into a full BIW model.

The governing equations presented here form the physical backbone of every subsequent chapter. Linear elasticity provides the stiffness operator for static compliance studies. Elastodynamics adds the inertial term needed for crash and vibration. Modal analysis extracts the eigenvalues that define NVH targets. Geometric nonlinearity extends the framework to the large-rotation regime that dominates thin sheet-metal assemblies. A PINN that respects these equations — and the assumptions under which each is valid — can serve as a fast surrogate for the finite-element solvers that currently dominate automotive structural analysis.