4. Mesh convergence and discretization error assessment

Every LS-DYNA result used as AI training data carries a hidden assumption: that the finite element discretization is fine enough for the quantity being extracted. When that assumption fails, the model learns a numerical artifact rather than physics. This chapter establishes a disciplined procedure for quantifying discretization error, choosing between h- and p-refinement, and screening mesh quality before any simulation output enters a training corpus.

Why Discretization Error Must Be Measured, Not Assumed

In explicit dynamics, the spatial discretization controls how stress waves propagate, how contact interfaces resolve, and how localization phenomena (shear bands, necking, crushing) are captured. A coarse mesh does not merely add noise; it systematically shifts peak forces, delays failure initiation, and smooths energy absorption curves. If a dataset mixes coarse-mesh and fine-mesh runs without labeling, a surrogate model will average across two different physical responses and produce predictions that match neither.

The practical rule is that convergence must be demonstrated for the specific quantity of interest (QoI), not for the model as a whole. A mesh that converges global internal energy may still fail to converge peak contact force or crack-tip stress. Define the QoI before running the study — for crashworthiness data this is typically peak deceleration, intrusion depth, and absorbed energy at a fixed time; for metal forming it is thinning and springback.

h-Refinement: The Primary Convergence Tool

h-refinement reduces element characteristic length h while holding element formulation and order fixed. The standard practice is a three-level geometric sequence, e.g. h, h/2, h/4, with the finest level at least twice as dense as the medium. For shell-dominated automotive models, a typical sequence on a stamped rail might move from 8 mm to 4 mm to 2 mm characteristic length.

Convergence is assessed with a relative error metric between successive levels:

e_rel = |Q_fine - Q_coarse| / |Q_fine|

Example (peak contact force, kN):
  Level 1 (h = 8 mm):  F1 = 142.6
  Level 2 (h = 4 mm):  F2 = 158.3
  Level 3 (h = 2 mm):  F3 = 163.1

  e_12 = |158.3 - 142.6| / 158.3 = 9.9%   -> not converged
  e_23 = |163.1 - 158.3| / 163.1 = 2.9%   -> approaching convergence

A common acceptance threshold for training data is e_rel ≤ 3% between the two finest levels, combined with a monotonic trend. If the error oscillates rather than decays, the cause is usually a localized feature — a sharp fillet, a contact edge, or an hourglass mode — that needs local refinement rather than global densification. Global h-refinement is expensive in explicit dynamics because the stable time step scales with the smallest element; halving h everywhere roughly doubles the cost per cycle and doubles the cycle count, so a 2× refinement can cost 4–8× in wall time.

p-Refinement and Element Formulation Choice

p-refinement increases the polynomial order of the interpolation instead of reducing element size. In LS-DYNA this is realized through element formulation selection rather than a single global order parameter: for solids, *SECTION_SOLID ELFORM 1 (constant stress), 2 (fully integrated), and −1/−2 (reduced integration with hourglass control) represent different effective resolutions; for shells, ELFORM 2 (Belytschko-Tsay) versus ELFORM 16 (fully integrated) versus ELFORM −16 change both accuracy and cost. Higher-order solid elements (ELFORM 1 with 8-node versus 20-node bricks via *ELEMENT_SOLID options) provide genuine p-refinement for bending-dominated problems.

p-refinement is attractive when the solution is smooth and the geometry is well represented, because it improves bending accuracy without shrinking the time step. It is a poor choice near discontinuities — contact interfaces, cracks, and shock fronts — where higher-order shape functions produce oscillations (Gibbs-type behavior) and can trigger negative volumes. For training data dominated by contact and failure, h-refinement remains the workhorse; p-refinement is best used to verify that the chosen element formulation is not the dominant error source.

Mesh Quality Metrics That Matter for Explicit Runs

Convergence studies are meaningless if the mesh contains distorted elements that degrade locally. Before launching a refinement sequence, screen the mesh against the metrics below. Thresholds are typical starting points for shell and solid meshes in explicit crash/impact work and should be tightened for QoIs sensitive to localization.

Hourglass energy ratio deserves special attention because it is both a quality indicator and a convergence diagnostic. If the ratio grows as the mesh is refined, the formulation or hourglass control is inadequate, and further refinement will not converge the QoI. Report it alongside every convergence data point.

Structured Workflow for a Convergence Study

The following sequence produces defensible training data with a documented error bound. It is written for a single load case; repeat per case family.

1. Fix the QoI and extraction method. Example: peak contact force from *DATABASE_ASCII RCFORC, filtered with a 1 kHz low-pass to remove numerical noise consistently across levels.
2. Build the coarse baseline with quality metrics inside the table above. Record minimum element size and resulting initial time step.
3. Generate refinement levels by halving characteristic length in the QoI region; keep the far field unchanged to control cost.
4. Run all levels with identical boundary conditions, material cards, and termination time. Any change other than mesh invalidates the comparison.
5. Compute e_rel and hourglass ratio per level. Plot QoI versus 1/h or element count; a flattening curve indicates convergence.
6. Apply the acceptance rule (e_rel ≤ 3%, monotonic trend, hourglass < 5%). If unmet, refine locally or revisit formulation.
7. Tag the dataset with mesh level, element size, QoI error estimate, and hourglass ratio so downstream training can weight or filter samples.

Reading the Convergence Behavior

The figure below summarizes how the QoI and the hourglass ratio behave across refinement levels, and how the decision to accept or refine is made. The key visual signal is the flattening of the QoI curve coinciding with a stable or decreasing hourglass ratio; divergence of the two indicates a formulation problem rather than a discretization problem.

In the example above, L1→L2 shows a 9.9% change, which is too large to accept. L2→L3 drops to 2.9%, and the hourglass ratio falls from 4.1% to 2.3%, confirming that the formulation is behaving. L3 is therefore accepted as the training-data mesh, with a documented error bound of approximately 3% on peak force. L4 would add cost without meaningful gain and is retained only as a spot-check.

Recording the Error Bound in the Dataset

Convergence evidence is only useful if it travels with the data. Each sample or case family should carry a compact record: QoI definition and filter, mesh level used, characteristic element size, e_rel against the next finer level, hourglass energy ratio, and the refinement decision. This metadata lets a training pipeline exclude under-converged samples, stratify by error level, or attach uncertainty labels. Without it, a surrogate model has no way to distinguish a physically meaningful peak force from a mesh-dependent one, and the numerical verification chain that this course builds collapses at the data layer.