2. PAM-CRASH solver fundamentals for crash modeling

Quality assessment in PAM-CRASH is not an abstract checklist exercise; it is grounded in how the explicit solver advances time, how contact is detected and enforced, and how material models convert strain into stress. A model that looks clean in the pre-processor can still produce non-physical results if the time step is misjudged, if contact thickness or penalty stiffness is inconsistent with the mesh, or if a material law is driven outside its calibrated range. This section establishes the solver-level knowledge needed to interpret quality metrics and to decide whether a suspicious result is a modeling error or a genuine physical response.

Explicit time integration and the stable time step

PAM-CRASH uses an explicit central-difference scheme. Nodal accelerations are obtained from the assembled force balance at the beginning of each cycle, then velocities and displacements are updated. Because no global stiffness matrix is factorized, each cycle is cheap, but the scheme is only conditionally stable. The critical step is governed by the Courant–Friedrichs–Lewy condition, which for a shell or solid element can be approximated as:

dt_crit = L_c / c
c = sqrt(E / rho)          (elastic bar wave speed)
L_c = characteristic element length

For shell elements, L_c is derived from the element area and its longest side, so a single distorted or sliver element can dominate the global step. PAM-CRASH reports the actual step and the controlling element in the output listing; the ratio of actual to critical step is the first quality indicator to inspect. A common practical target is a mass increase below 1–2 % when the step is scaled up, but the acceptable threshold depends on the program. The following snippet shows a typical time-step control block:

/TIME
  0.0   0.120   0.0   0.0   0.0
/DT/NODA
  0.9e-6   0.0   0.0   0.0

Here the simulation runs to 120 ms with a nodal time step of 0.9 µs. If the solver internally reduces the step below this value, the model contains an element that violates the intended mesh quality. The recommended diagnostic sequence is: read the minimum step and its element ID from the listing, locate that element, check its aspect ratio and Jacobian, and only then consider mass scaling. Blanket mass scaling hides the root cause and can alter the inertia of thin-walled structures, which directly affects crash energy absorption.

Do not use mass scaling to compensate for a handful of degenerate elements. Adding mass to a localized region changes the kinetic energy balance and can make a model appear stable while producing wrong intrusion and acceleration results.

Contact algorithms and their quality implications

Crash models are dominated by self-contact, especially in folded sheet-metal parts. PAM-CRASH offers penalty-based and kinematic contact treatments. Penalty contact is the default for most automotive applications because it conserves momentum and is robust for large sliding. The penalty force is proportional to penetration depth and a stiffness factor:

F_penalty = k * delta
k = alpha * K * A / t

where K is the bulk modulus, A the segment area, t the shell thickness, and alpha a user scale factor. If alpha is too small, segments interpenetrate visibly; if it is too large, the contact stiffness raises the local stability limit and can trigger spurious high-frequency noise. The contact thickness used for shells must match the physical thickness, otherwise the gap at which contact initiates is wrong. A typical single-surface definition is:

/SURF/SEG
  part_set_1
/INTER/TYPE7
  surf_id   0.0   0.0   0.0   0.0   0.0   0.0
  0.0       0.0   0.0   0.0   0.0   0.0   0.0

Quality checks for contact include: verifying that no node penetrates more than half the shell thickness, confirming that contact energy is positive and does not exceed a small fraction of internal energy, and checking that the number of active contacts is stable rather than oscillating between zero and a large value. Friction coefficients should be justified from test data; a default value of 0.1–0.2 is common for steel-to-steel, but painted or coated surfaces may differ.

Material models relevant to quality assessment

The material law determines whether the predicted force–displacement response is trustworthy. For crashworthiness, the most frequently used models are elastoplastic metals with isotropic hardening, strain-rate sensitivity, and failure. In PAM-CRASH these are typically LAW 2 (Johnson–Cook), LAW 36 (strain-rate dependent plasticity), or LAW 32 (Ladveze–Bouvier) for sheet metal. Each requires a consistent set of parameters: density, Young's modulus, Poisson's ratio, yield curve, and a failure criterion.

A frequent quality defect is extrapolating a tensile test curve beyond the uniform elongation point without adjusting the true stress–strain conversion. The engineering-to-true conversion is:

sigma_true = sigma_eng * (1 + eps_eng)
eps_true   = ln(1 + eps_eng)

If this conversion is skipped, the flow stress is underestimated at large strains, and the model will underpredict energy absorption. Similarly, the failure strain must be specified in the same measure as the input curve. For shell elements, element deletion based on effective plastic strain is common, but the deletion strain should be calibrated against a coupon test that matches the element size, because the apparent failure strain is mesh-dependent.

When reviewing a material card, always cross-check the density and elastic modulus against the actual alloy. A density error of 5 % propagates directly into kinetic energy and into the stable time step, producing a model that is both physically and numerically inconsistent.

Integrated view of solver fundamentals

The three pillars are coupled. The time step depends on element size and wave speed, which depend on the material model. Contact stiffness depends on thickness and bulk modulus, which also feed the stability limit. A quality assessment therefore cannot treat them independently. The diagram below summarizes the dependencies and the primary checks at each stage.

PAM-CRASH Solver Fundamentals: Coupled Quality Checks Material Model • Density ρ, modulus E • True stress–strain curve • Strain-rate sensitivity • Failure / deletion strain • Mesh-size calibration Explicit Time Integration • dt_crit = L_c / c • c = sqrt(E/ρ) • Controlling element ID • Mass scaling < 1–2 % • Energy balance check Contact Algorithm • Penalty stiffness α • Shell contact thickness • Penetration < t/2 • Friction coefficient • Contact energy stability E, ρ → c t, K → k contact stiffness raises local stability limit material wave speed sets dt_crit Quality Verdict • Stable step, no hidden mass scaling • Contact penetration within tolerance • Material response within calibrated range • Energy balance and hourglass energy acceptable combined assessment

The practical takeaway is that solver fundamentals are not background theory; they are the reference against which every quality metric is judged. When a model shows excessive hourglass energy, the first question is whether the time step or contact stiffness forced an artificially high stiffness. When a force–displacement curve is too soft, the first question is whether the material curve was converted to true stress and true strain. When contact penetrates, the first question is whether the shell thickness and penalty scale are consistent with the mesh. Answering these questions requires the solver-level understanding developed in this section, and it is this understanding that separates a defensible crash model from one that merely runs to completion.

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