Stable Timestep & Explicit Solution Control
Why explicit integration is conditionally stable, how a single small or stiff element can govern the cost of the entire model, and how to control the stable timestep without corrupting the physics.
Conditional stability
Explicit time integration is conditionally stable: the timestep must be smaller than a critical value, or the numerical solution will diverge and the energy will grow without bound. The critical timestep is set by the speed at which information — a stress wave — can travel across a single element in one increment. If the timestep is too large, a disturbance can "jump" an element in a single step, the numerical wave outruns the physical wave, and the solution becomes unstable. Unlike an implicit scheme, where instability usually appears as a convergence failure, explicit instability appears as explosive, unphysical growth of displacement and energy. The stability limit is not a guideline that can be relaxed for speed; it is a hard bound imposed by the mathematics of the method.
The stable timestep expression
In conceptual form the stable timestep is the characteristic element dimension divided by the relevant wave speed. The characteristic dimension is the smallest distance a wave must cross within an element, which for a distorted element is not simply the shortest edge. The wave speed is the fastest relevant wave speed in the element, which for a solid is the dilatational wave speed and for a structural element depends on the bending and membrane wave speeds as well as the material wave speed. The expression is illustrative; every solver computes the stable timestep from its own element formulations and stability theory, and the reported timestep should be reviewed against the smallest and stiffest elements in the model.
Δt_stable ≈ L_char / c where Δt_stable = stable (critical) timestep for the explicit scheme L_char = characteristic element dimension of the smallest critical element c = relevant wave speed in the element material The smallest and stiffest element in the model typically governs the global timestep. The exact expression is solver- and element-specific.
One small element can govern the whole model
Because the stable timestep is a global property — every node advances at the same timestep — a single very small, very stiff or very distorted element can force the entire model to run at a tiny timestep and multiply the number of increments, and therefore the computational cost, many times over. This is one of the most important practical facts about explicit analysis. A meshing feature, a fillet, a bolt hole modelled with tiny elements, a sliver element produced by poor geometry clean-up, or a stiff contact spring can each become the governing element. The cost is borne by the entire model, while the benefit of that small element may be confined to a region that does not even control the engineering result.
IN EXPLICIT ANALYSIS, A SINGLE VERY SMALL ELEMENT CAN GOVERN THE COMPUTATIONAL COST OF THE ENTIRE MODEL.
What controls the stable timestep
The stable timestep is governed by the element size, the material wave speed (through stiffness and density), the contact stiffness, the element formulation and any mass scaling applied. Smaller elements reduce the timestep; stiffer materials (higher modulus) increase the wave speed and reduce the timestep; denser materials reduce the wave speed and increase the timestep; stiff contact can introduce a contact stiffness that behaves like a stiff spring and reduces the effective timestep; mass scaling artificially increases density and increases the timestep at the cost of changing the inertia. Each of these has a physical meaning and a numerical consequence, and each must be controlled deliberately rather than accepted as a by-product of the mesh or the solver defaults.
| Factor | Effect on stable timestep | Computational implication | Mitigation approach |
|---|---|---|---|
| Element size | Smaller elements → smaller L_char → smaller timestep | Dominant cost driver; governs total increments | Mesh the critical regions finely and the rest coarsely; remove sliver elements |
| Material wave speed | Stiffer material (higher E) → higher c → smaller timestep | Stiff materials in the model raise the global cost | Use the correct material; do not over-stiffen; consider submodeling |
| Density | Higher density → lower c → larger timestep | Artificially increasing density is mass scaling | Apply only with verification that inertia changes are negligible |
| Stiffness (contact, springs) | Stiff contact or connector springs reduce the effective timestep | Over-stiff contact penalises the whole model | Tune contact stiffness to the minimum that controls penetration |
| Element formulation | Some formulations have a stricter stability limit than others | Element choice interacts with timestep and with hourglass control | Select formulations appropriate to the deformation mode |
| Mass scaling | Added mass increases density → larger timestep | Reduces increments but changes the physical model | Verify added mass does not change the engineering conclusion |
Diagnosing the governing element
Every explicit solver reports the stable timestep and, on request, the element that governs it. The first diagnostic step in any slow-running explicit model is to identify that element and to ask whether it is physically necessary. If it is a sliver produced by poor geometry, clean up the geometry. If it is an unnecessarily fine feature, coarsen it or remove it. If it is a physically necessary small element in a critical region, accept the cost or use selective mass scaling with verification. If it is a stiff contact or connector, tune the stiffness. The point is that the cost driver should be a deliberate engineering choice, not an accident of meshing.
A VERY SMALL ELEMENT CREATED BY POOR MESH QUALITY OR AN UNNECESSARY GEOMETRIC FEATURE CAN MULTIPLY THE COMPUTATIONAL COST OF THE ENTIRE ANALYSIS WITHOUT IMPROVING THE RESULT.
Controlling cost without corrupting the physics
There are several legitimate ways to control the cost of an explicit analysis without changing the physics. Mesh grading — fine where the physics demands it, coarse elsewhere — is the most powerful. Removing unnecessary geometric detail that does not influence the result avoids creating governing sliver elements. Selective mass scaling applied only to non-critical elements, with verification that the engineering conclusion is unchanged, can reduce the number of increments. Submodelling can resolve a critical local region without forcing the global model to run at that region's timestep. None of these is free; each requires a verification step to confirm that the physics being represented is still the physics of interest. The article on mass scaling treats that verification in detail.
FE mesh with one small element governing the stable timestep. ┌────────────────────────────┐ │ □ □ □ □ □ □ □ │ │ □ □ □ □ □ □ □ │ │ □ □ □ ▪ □ □ □ │ ← tiny sliver / stiff element │ □ □ □ □ □ □ □ │ governs Δt_stable for the whole model └────────────────────────────┘ Δt_stable ≈ L_char(sliver) / c → forces every node to advance at this tiny increment Cost ∝ number of increments ∝ 1 / Δt_stable