3. Element characteristic length and wave speed

In an explicit dynamics solver such as LS-DYNA, the stable time step is not a global constant chosen by the user. It is a local property of every element in the model, recomputed as the solution advances. The controlling relationship is the Courant–Friedrichs–Lewy (CFL) condition, which for the central-difference explicit scheme can be written as:

Δt_stable = 2 / ω_max  ≈  L_e / c

where L_e is a characteristic length of the element and c is the wave speed of the material. The factor of 2 comes from the highest natural frequency the element can represent; in practice LS-DYNA applies an additional scale factor (default 0.9 for solid elements, controlled by *CONTROL_TIMESTEP via TSSFAC) to provide a safety margin. The essential insight is that the stable step is a ratio: it shrinks linearly with element size and inversely with wave speed. Everything else in this chapter follows from that ratio.

Characteristic length: what the solver actually measures

For a solid element, LS-DYNA does not use the edge length directly. It computes a characteristic length from the element volume and its largest face area, which for a hexahedral element reduces to the ratio of volume to the maximum face area. For a well-shaped cube of side a, this gives L_e = a. For a distorted element, the computed length can be substantially smaller than the nominal edge length, and that smaller value governs the step. This is why a single badly collapsed element can slow an entire model: the global step is the minimum over all elements.

Shell elements use a different measure. The characteristic length is based on the element area and the longest diagonal, so a thin, elongated shell element yields a small L_e even if its in-plane dimensions look reasonable. Beam and truss elements use their length, but the relevant wave speed is the axial one, and the rotational inertia of the cross-section can further reduce the stable step through the highest torsional frequency. The practical consequence is that element quality — aspect ratio, warping, Jacobian — matters as much as nominal size, because it enters the step through L_e.

Wave speed: what the material contributes

For a linear-elastic isotropic solid, the dilatational (pressure) wave speed is:

c_d = sqrt( (K + 4G/3) / ρ )

and the shear wave speed is c_s = sqrt(G/ρ), with K the bulk modulus, G the shear modulus, and ρ the density. The dilatational speed is the larger of the two and therefore controls the step for solid elements. For steel with E = 210 GPa, ν = 0.3, and ρ = 7850 kg/m³, the dilatational speed is approximately 5900 m/s. A 5 mm element in that steel therefore gives a stable step near 0.005 / 5900 ≈ 0.85 µs, before the 0.9 scale factor is applied.

Two practical points follow. First, wave speed depends on density, so any modeling choice that changes effective density changes the step. Second, for rate-dependent or hyperelastic materials the tangent stiffness at the current state governs the local wave speed, and LS-DYNA recomputes it as the material deforms. A material that stiffens under compression will progressively reduce the step during the run.

How the two factors combine across a real model

The figure below shows how the local stable step is assembled from element geometry and material wave speed, and how the global step is then selected as the minimum across the mesh. It also illustrates the two levers available to the analyst: increasing L_e (coarsening or improving element quality) and reducing c (mass scaling, discussed in later chapters).

The table embedded in the figure makes the scaling explicit: halving the element size halves the step and doubles the number of cycles needed to advance the same physical time, so the cost grows roughly as the inverse square of element size once the increased element count is also counted. This is the economic reason mesh convergence studies must be planned rather than brute-forced.

Reading the solver's own numbers

LS-DYNA reports the initial time step and the controlling element in the d3hsp file. A typical line looks like:

initial time step = 7.63E-07
element  184223  has the smallest time step

Locate that element in the model and inspect its geometry and material. If its characteristic length is far below the surrounding mesh, the cause is usually distortion introduced during meshing or during an initial contact adjustment, not the intended element size. Correcting the element is almost always cheaper than accepting the reduced step. If the element is well shaped, the step is telling you the true cost of your chosen resolution, and the decision becomes a convergence-versus-cost trade-off rather than a modeling error.

What to carry forward

The stable step is a local ratio L_e / c, scaled by a safety factor, and the global step is its minimum over the mesh. Element quality enters through L_e; material stiffness and density enter through c; and both are recomputed as the solution evolves. Before any convergence study, verify that the controlling element is intentional and well shaped, and confirm the hand estimate against the solver's reported value. That single check separates a mesh-resolution decision from a meshing defect, and it determines whether the next step should be refining the mesh, repairing an element, or reconsidering the material model.