Definition of the stability limit

The stability limit is defined in terms of the highest frequency in the system (ωmax). Without damping the stability limit is defined by the expression

Δ⁢tstable=2ωmax,
and with damping it is defined by the expression
Δ⁢tstable=2ωmax⁢(1+ξ2-ξ),
where ξ is the fraction of critical damping in the mode with the highest frequency. (Recall that critical damping defines the limit between oscillatory and non-oscillatory motion in the context of free-damped vibration. Abaqus/Explicit always introduces a small amount of damping in the form of bulk viscosity to control high-frequency oscillations.) Perhaps contrary to engineering intuition, damping always reduces the stability limit.

The actual highest frequency in the system is based on a complex set of interacting factors, and it is not computationally feasible to calculate its exact value. Alternately, we use a simple estimate that is efficient and conservative. Instead of looking at the global model, we estimate the highest frequency of each individual element in the model, which is always associated with the dilatational mode. It can be shown that the highest element frequency determined on an element-by-element basis is always higher than the highest frequency in the assembled finite element model.

Based on the element-by-element estimate, the stability limit can be redefined using the element length, Le, and the wave speed of the material, cd:

Δ⁢tstable=Lecd.

For most element types—a distorted quadrilateral element, for example—the above equation is only an estimate of the actual element-by-element stability limit because it is not clear how the element length should be determined. As an approximation the shortest element distance can be used, but the resulting estimate is not always conservative. The shorter the element length, the smaller the stability limit. The wave speed is a property of the material. For a linear elastic material with a Poisson's ratio of zero

cd=Eρ,

where E is Young's modulus and ρ is the mass density. The stiffer the material, the higher the wave speed, resulting in a smaller stability limit. The higher the density, the lower the wave speed, resulting in a larger stability limit.

Our simplified stability limit definition provides some intuitive understanding. The stability limit is the transit time of a dilatational wave across the distance defined by the characteristic element length. If we know the size of the smallest element dimension and the wave speed of the material, we can estimate the stability limit. For example, if the smallest element dimension is 5 mm and the dilatational wave speed is 5000 m/s, the stable time increment is on the order of 1 × 10−6 s.