Mass Scaling: Use, Abuse & Verification

What mass scaling does

Mass scaling is the deliberate addition of mass to selected elements of an explicit model in order to increase the stable timestep and reduce the number of increments, and therefore the runtime, of the analysis. Because the stable timestep depends on the ratio of a characteristic element dimension to the wave speed, and the wave speed depends on the square root of stiffness over density, increasing the density of a small element increases its stable timestep. Applied to a handful of governing elements, mass scaling can dramatically reduce the cost of a model whose timestep is controlled by a few small features. Applied indiscriminately, it changes the inertia of the structure and therefore the physical problem being solved.

MASS SCALING IS NOT FREE COMPUTATIONAL SPEED. IT CHANGES THE PHYSICAL MODEL AND MUST BE SHOWN NOT TO CHANGE THE ENGINEERING CONCLUSION.

Forms of mass scaling

Mass scaling can be applied in several ways. Global mass scaling adds mass to every element to reach a target timestep; it is the bluntest instrument and the most likely to change the global inertia and the dynamic response. Selective mass scaling adds mass only to the elements whose timestep falls below a target, typically the small or stiff elements that govern the cost; it is more surgical but still requires verification that the added mass in non-critical regions does not affect the critical response. Timestep targeting lets the solver add only enough mass to reach a specified timestep, which bounds the change but does not eliminate it. Some solvers offer variable mass scaling that changes the scaling strategy during the analysis as elements deform. Each form has a place, but none is exempt from the requirement to verify that the physics is preserved.

What mass scaling changes

Adding mass changes the inertia of the model. In a dynamic event inertia is a first-order effect: it governs how the structure accelerates under the contact load, how the load spreads, and how the deformation mode develops. Changing the mass distribution can therefore change the impact timing, the peak forces, the deformation sequence, the failure mode and the energy balance. The change may be small — if the added mass is a tiny fraction of the total and is located away from the critical region — or it may be large enough to alter the engineering conclusion. The analyst cannot know which case applies without a verification step that compares the mass-scaled model against a baseline.

The baseline versus mass-scaled comparison

The only defensible way to use mass scaling is to compare the mass-scaled model against a baseline model with no added mass — or with the minimum added mass needed to run — and to confirm that the quantities on which the engineering conclusion depends are unchanged. The comparison should cover the total added mass as a percentage of the physical mass, the local mass distribution in the critical regions, the centre of gravity of the structure, the impact timing, the peak contact force, the internal and kinetic energy histories, the deformation mode, and the failure mode. If any of these changes materially, the mass scaling has altered the problem being solved and the faster runtime has been bought at the cost of a different engineering conclusion.

What an acceptable percentage means

A natural question is "what percentage of added mass is acceptable?" There is no universal answer. The acceptable added mass depends on the event, the region where the mass is added, the quantity on which the conclusion depends, and the tolerance of that conclusion to inertia changes. A small percentage of added mass in a non-critical region of a quasi-static explicit run may be entirely acceptable; the same percentage added to an impactor or to a critical load path in a dynamic event may change the result. The only reliable test is the baseline comparison. Quoting a universal acceptable percentage without reference to the verification is a habit, not engineering. Where guidance is given by a programme or certification basis, that guidance should be followed and its basis understood.

IF THE RESULT DEPENDS MATERIALLY ON THE ADDED MASS, THE ACCELERATION IN RUNTIME HAS ALTERED THE PROBLEM BEING SOLVED.

A disciplined workflow

A disciplined mass-scaling workflow begins with a baseline model that uses no added mass, or the minimum needed to run at all. The governing elements are identified, the physical necessity of each is reviewed, and geometric or meshing remedies are applied before mass scaling is considered. Mass scaling is then applied selectively to reach a target timestep, and the mass-scaled model is compared against the baseline on every quantity that feeds the engineering conclusion. If the comparison is satisfactory, the mass-scaled model is used for production runs and the verification is documented. If the comparison fails, the target timestep is reduced, the mass scaling is made more selective, or the model is re-meshed. The workflow treats mass scaling as a controlled approximation with a documented verification, not as a free speed-up.

Baseline and mass-scaled inertia distributions compared.

BASELINE
  mass density ρ  uniform / physical
  CG at physical location
  Δt_stable governed by smallest element

MASS-SCALED
  mass density ρ + Δρ at selected small elements
  CG possibly shifted
  Δt_stable raised toward target

  Verification: compare total mass, local mass, CG, timing,
  peak force, energy histories, deformation and failure mode.
  Engineering conclusion must be unchanged.