2. LS-DYNA explicit time integration fundamentals

Every explicit LS-DYNA run is governed by one central fact: the solution is advanced by a conditionally stable integrator. If the time step exceeds a stability limit, the calculation does not merely lose accuracy — it diverges, typically within a few hundred cycles, producing runaway nodal velocities and negative volumes. Understanding why that limit exists, how LS-DYNA computes it, and how to control it is the foundation for everything that follows in mesh convergence and time step sensitivity work.

The Central Difference Update

LS-DYNA's explicit solver uses the central difference method, a second-order accurate explicit scheme. The semi-discrete equation of motion at time t_n is

M a_n + C v_n + F^int_n = F^ext_n

where M is the lumped mass matrix, C the damping matrix, F^int the internal (element) forces, and F^ext the applied loads. Because M is lumped (diagonal), the acceleration follows directly without any matrix inversion:

a_n = M^-1 (F_ext_n - F_int_n - C v_n)

v_(n+1/2) = v_(n-1/2) + a_n * dt_n
x_(n+1)   = x_n + v_(n+1/2) * dt_(n+1/2)

This is the key architectural advantage of explicit integration: each cycle costs O(N) operations with no global linear solve. The price is conditional stability. The velocity is stored at half-step points and the displacement at integer steps — the classic staggered (leapfrog) arrangement. Note that dt_n may vary from cycle to cycle; LS-DYNA recomputes the stable step as the mesh deforms.

Where the Critical Time Step Comes From

Linear stability analysis of the central difference operator on a single degree of freedom with natural frequency ω yields the condition

Δt ≤ Δt_crit = 2 / ω_max

where ω_max is the highest natural frequency of the discretized system. For a truss or beam element of length L, density ρ, and Young's modulus E, the highest frequency corresponds to the dilatational wave mode, giving the familiar Courant-type expression

Δt_crit = L / c,    c = √(E/ρ)

with c the speed of sound in the material. The physical interpretation is the Courant condition: information (a stress wave) must not traverse more than one element per time step. If Δt is larger, the numerical domain of dependence fails to contain the physical one, and the scheme amplifies rather than propagates the disturbance.

In practice LS-DYNA applies a safety factor and a characteristic length that depends on element type. For solid elements the default is

dt = 0.9 * L_e / c        (solid elements, default TSSFAC = 0.9)
L_e = characteristic length (e.g. volume/area for hexahedra)

Shells use a similar form with the shell's characteristic length and include a factor for the bending stiffness contribution. The factor 0.9 (controlled by *CONTROL_TIMESTEP, parameter TSSFAC) absorbs nonlinearity and the fact that ω_max grows as elements distort. Reducing TSSFAC to 0.6–0.8 is a common remedy when a model diverges despite a seemingly correct mesh.

What Actually Sets the Step in a Real Model

The global stable step is the minimum over all elements. This single fact explains most time step surprises in practice: one badly shaped element, one tiny sliver in a transition region, or one stiff material inclusion can dominate the entire run. The figure below shows how the global step is assembled from element-level estimates and how a single offending element collapses it.

The practical consequence is severe. In the example above, a single 0.4 mm sliver reduces the global step by more than an order of magnitude, multiplying the cycle count — and wall-clock time — by roughly 12×. This is why time step diagnostics are inseparable from mesh quality: the smallest element, not the average element, controls cost.

Reading the Time Step from LS-DYNA Output

LS-DYNA reports the step in the d3hsp file at initialization and periodically during the run. A typical initialization block looks like:

Time step size......................... 7.03E-08
Time step scale factor................. 9.00E-01
Element  12345  has the smallest time step
  type 1 (solid),  L = 4.02E-04,  c = 5.17E+03

Two things are worth checking immediately after a short test run: the reported step size, and whether the element flagged as critical is where you expect it to be. If the flagged element sits in a region you did not intend to resolve finely — a fillet transition, a contact surface, an imported CAD artifact — the model has a mesh problem, not a physics problem. LS-DYNA also prints a histogram of element time steps under *DATABASE_ASCII / GLSTAT or via *CONTROL_TIMESTEP with DTSTAMP; the distribution tells you whether the step is set by a broad population of small elements (a genuine refinement decision) or by a handful of outliers (a cleanup task).

Mass Scaling: What It Does and Does Not Fix

Because Δt_crit ∝ √(ρ), artificially increasing the density of small elements raises their stable step. This is mass scaling, activated with *CONTROL_TIMESTEP and DT2MS (target step) or MS1ST. The mechanics are simple: any element whose computed step is below the target has its density scaled by (Δt_target/Δt_element)². The trade-off is added inertia, which corrupts the solution when the added mass is a significant fraction of the model's total or when inertial effects matter (e.g., low-speed crushing, springback).

Connecting the Step to Mesh Convergence

The Courant condition ties mesh size directly to cost: halving element size halves Δt and doubles the cycle count, so a uniform refinement in 3D multiplies cost by roughly 16 (8× elements × 2× cycles). This is the economic reason mesh convergence studies are done on coarse-to-fine sequences with careful extrapolation rather than brute-force refinement. It is also why the time step sensitivity study in later chapters must distinguish between two effects: the change in spatial resolution (mesh convergence) and the change in temporal resolution (step sensitivity). They are coupled through Δt_crit, and a result that changes when you refine the mesh may be responding to either — or both.

A sound workflow therefore begins with three checks before any production run: confirm the reported Δt against a hand calculation using the smallest element and its wave speed; identify the element that sets the step and decide whether it is legitimate; and record the added-mass percentage if mass scaling is active. These three numbers — step size, critical element, added mass — form the baseline against which every subsequent convergence and sensitivity result is interpreted.