Mass scaling

Mass scaling enables an analysis to be performed economically without artificially increasing the loading rate. Mass scaling is the only option for reducing the solution time in simulations involving a rate-dependent material or rate-dependent damping, such as dashpots. In such simulations increasing the loading rate is not an option because material strain rates increase by the same factor as the loading rate. When the properties of the model change with the strain rate, artificially increasing the loading rate artificially changes the process.

The following equations show how the stable time increment is related to the material density. As discussed in Definition of the stability limit, the stability limit for the model is the minimum stable time increment of all elements. It can be expressed as

Δ⁢t=Lecd,

where Le is the characteristic element length and cd is the dilatational wave speed of the material. The dilatational wave speed for a linear elastic material with Poisson's ratio equal to zero is given by

cd=Eρ,

where E is the Young's modulus and ρ is the material density.

According to the above equations, artificially increasing the material density, ρ, by a factor of f2 decreases the wave speed by a factor of f and increases the stable time increment by a factor of f. Remember that when the global stability limit is increased, fewer increments are required to perform the same analysis, which is the goal of mass scaling. Scaling the mass, however, has exactly the same influence on inertial effects as artificially increasing the loading rate. Therefore, excessive mass scaling, just like excessive loading rates, can lead to erroneous solutions. The suggested approach to determining an acceptable mass scaling factor, then, is similar to the approach to determining an acceptable loading rate scaling factor. The only difference to the approach is that the speedup associated with mass scaling is the square root of the mass scaling factor, whereas the speedup associated with loading rate scaling is proportional to the loading rate scaling factor. For example, a mass scaling factor of 100 corresponds exactly to a loading rate scaling factor of 10.

There are several ways to implement mass scaling in your model using either fixed or variable mass scaling. The mass scaling definition can be changed from step to step, allowing great flexibility. Refer to Mass Scaling for details.