2. Model verification fundamentals

Verification answers a narrow but essential question: are the equations solved correctly? It is distinct from validation, which asks whether the right equations were chosen for the physical reality. In an LS-DYNA capstone, verification must be established before any correlation with test data is attempted, because a validation discrepancy is uninterpretable if the underlying solver has not been shown to reproduce known mathematical behavior. This section covers three complementary verification instruments: analytical benchmarks, order-of-accuracy (convergence) studies, and unit-level checks of material and contact formulations.

Analytical Benchmarks: Comparing Against Closed-Form Solutions

An analytical benchmark drives the code with a problem whose exact solution is known in closed form, then measures the numerical error. For explicit solid mechanics the canonical example is the one-dimensional elastic bar under a step load. For a bar of length L, density ρ, Young's modulus E, fixed at x = 0 and subjected to a constant traction σ₀ at x = L starting at t = 0, the axial stress behind the wavefront is exactly σ₀ and the wavefront travels at the bar wave speed c = √(E/ρ). The arrival time at the fixed end is t* = L/c, after which the reflected wave doubles the stress to 2σ₀.

A reproducible LS-DYNA setup uses SI units throughout: L = 1.0 m, cross-section A = 1.0 × 10⁻⁴ m², ρ = 7850 kg/m³, E = 210 GPa, giving c ≈ 5171 m/s and t* ≈ 1.934 × 10⁻⁴ s. The mesh is a single row of hexahedral elements along the axis, the material is *MAT_ELASTIC, the load is applied with *LOAD_SEGMENT as a pressure of σ₀ = 1.0 MPa, and the run uses *CONTROL_TERMINATION at ENDTIM = 4.0 × 10⁻⁴ s. The critical time step must be checked against the Courant limit Δt ≤ L_e/c; with ten elements, L_e = 0.1 m and Δt_crit ≈ 1.93 × 10⁻⁵ s, so a *CONTROL_TIMESTEP scale factor of 0.9 is appropriate.

Verification criteria are quantitative, not visual. Extract axial stress at a gauge element using *DATABASE_HISTORY_SOLID and compare against the exact step function. Acceptable agreement for a converged mesh is typically within 1–2% of σ₀ in the plateau region, with the wavefront smeared over roughly one element length. A persistent offset in the plateau, a wave speed error exceeding 1%, or stress ringing that does not diminish with refinement indicates a formulation or input error rather than physical behavior.

Order-of-Accuracy Studies

Convergence studies establish that the discretization error decreases at the rate the theory predicts. For a second-order accurate spatial scheme, the L2 error norm should scale as O(h²), so halving the element size should reduce the error by a factor of four. In practice, run the same benchmark at three or four mesh densities (for example h = 0.2, 0.1, 0.05, 0.025 m), compute the error norm against the analytical solution at a fixed physical time, and fit the slope of log(error) versus log(h).

# Example: observed error norms for the elastic bar benchmark (illustrative values)
h (m)     L2 error (Pa)    ratio
0.200     4.8e4            —
0.100     1.2e4            4.0
0.050     3.1e3            3.9
0.025     7.9e2            3.9
# Fitted slope ≈ 1.98 → consistent with second-order spatial accuracy

The fitted slope is the verification evidence. A slope near 2.0 confirms the element formulation and time integration behave as designed; a slope near 1.0 signals a first-order defect (often excessive numerical damping or an incorrect hourglass control setting), and a slope that flattens at fine meshes suggests the error is dominated by the time step rather than the mesh. Note that explicit time integration is only first-order accurate in time, so if the time step is refined together with the mesh, the observed slope may approach 1.0; hold Δt fixed or refine it separately to isolate spatial order.

Unit-Level Checks for Material and Contact Formulations

Unit-level checks isolate a single constitutive or contact law in a minimal model, so that any deviation is attributable to that formulation. For materials, a single-element uniaxial test is the standard vehicle. Prescribe a displacement ramp on one face, fix the opposite face, and compare the resulting stress–strain curve against the analytical response of the chosen model. For *MAT_PLASTIC_KINEMATIC with yield stress σ_y = 300 MPa, hardening modulus E_t = 2 GPa, and E = 210 GPa, the expected tangent modulus in the plastic regime is E_t, and the stress at 2% plastic strain should be σ_y + E_t · 0.02 = 340 MPa. A single hexahedral element with *BOUNDARY_PRESCRIBED_MOTION_RIGID or nodal prescribed motion, run in quasi-static conditions with mass scaling, gives a clean check.

For contact, the classic unit test is two blocks in normal compression. Prescribe a known penetration and verify that the contact force matches the penalty stiffness relationship F = k · δ, where k is set by *CONTACT parameters such as SFS and SOFT. A friction check then applies a tangential displacement and confirms that the friction force saturates at μ · F_normal for *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE with a specified FS. Typical failure symptoms include contact force that grows without bound (penalty too stiff relative to the time step), force that never reaches the friction limit (incorrect FS or FD interpretation), and spurious oscillation at the interface (inadequate SOFT setting or missing SBOPT).

The three tiers are not interchangeable. A benchmark confirms absolute correctness against a known answer; a convergence study confirms the rate at which the discretization approaches that answer; a unit-level check confirms that an individual formulation behaves as its documentation claims. Together they form the evidence chain that must be closed before any occupant-safety correlation work begins, because only then can a discrepancy in a full vehicle model be attributed to modeling assumptions rather than to solver defects.