7. Mass shift

It is possible to enter a coefficient \(\mathrm{coef}\) behind the COEF_MASS_SHIFT operand in the in the keyword SCHEMA_TEMPS in order to « shift » the mass matrix \(M\) which becomes: \(M\text{'}=M+\mathrm{coef.}K\).

The introduction of this coefficient makes it possible to invert the mass matrix dynamically with an explicit diagram on a physical basis when it has zero terms for certain specific degrees of freedom, for example the pressure for the HM modeling elements. It should be noted that this shift is not necessary in dynamics with an explicit schema on a modal basis (presence of the keyword PROJ_MODAL) because in this case the matrix to be inverted is the generalized mass matrix which is always invertible.

Entering this coefficient also makes it possible to greatly improve convergence in dynamics with an implicit or explicit schema regardless of the type of modeling by imposing a cutoff frequency that is inversely proportional to the value of \(\mathrm{coef}\). This practice, similar to the « selective mass scaling » method proposed by Lars Olovsson in multiple publications, makes it possible to satisfy the convergence criteria with larger time steps at the cost of a distortion of set of natural frequencies of system, light however for low frequencies.

An original natural pulsation \(\omega\) then becomes \(\omega \text{'}\) such that: \(\omega {\text{'}}^{2}\mathrm{=}\frac{{\omega }^{2}}{1+\mathit{coef}\mathrm{.}{\omega }^{2}}\)

From this expression, we can see that \(\omega \text{'}\) tends towards a maximum value \({\mathit{coef}}^{\mathrm{-}0.5}\).

Thus, for example, for a maximum value in practice for \(\mathrm{coef}={10}^{-6}\), the corrected maximum natural pulsation of the \(\omega \text{'}\) system is equal to \(1000r{\mathrm{d.s}}^{-1}\), and a value for an original natural frequency of \(30\mathrm{Hz}\) is corrected by this method to \(\mathrm{29,481}\mathit{Hz}\) by this method.

Remarks:

It should be noted that even if the mass « shift » option is activated, for the calculation of the initial acceleration, it is always the non-shifted mass matrix that is used. The fact that it can be non-invertible does not pose a problem, however, because in this case, the initial acceleration is imposed at 0 at all points.

The stiffness matrix used for this « shift » is in practice the one that is calculated during the prediction phase of the Newton algorithm.