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Explicit Rayleigh Damping

Explicit Rayleigh Damping#

This note summarizes the mass-proportional Rayleigh damping formulation used by the explicit solver.

Damping model#

Comfor uses the Rayleigh model restricted to its mass-proportional part:

\[ \mathbf{C}^d = \alpha \mathbf{M} \]

where \(\alpha\) is the nodal damping coefficient.

The explicit solver advances the staggered velocity with a central-difference scheme. For a variable time step, the kick interval is:

\[ \Delta t^n = \frac{1}{2}\left(\Delta t^{n-1/2} + \Delta t^{n+1/2}\right) \]

Damped translational update#

At time \(t^n\), the translational equilibrium is written as:

\[ \mathbf{M}\mathbf{a}^n + \mathbf{C}^d \mathbf{v}^{n-1/2} = \mathbf{f}^n \]

Using \(\mathbf{C}^d = \alpha \mathbf{M}\) gives:

\[ \mathbf{a}^n + \alpha \mathbf{v}^{n-1/2} = \mathbf{M}^{-1}\mathbf{f}^n \]

In Comfor, acceleration-type loads such as gravity are assembled in the physical acceleration term:

\[ \mathbf{a}_{\mathrm{phys}}^n = \mathbf{M}^{-1}\mathbf{f}^n + \mathbf{a}_{\mathrm{load}}^n \]

The damped kick is then written as:

\[ \mathbf{a}_{\mathrm{alg}}^n = \frac{\mathbf{a}_{\mathrm{phys}}^n - \alpha \mathbf{v}^{n-1/2}} {1 + \frac{1}{2}\alpha \Delta t^n} \]

The quantity stored by the solver is therefore the algorithmic acceleration used by the kick, not only the undamped physical term.

Damped rotational update#

The same structure is applied to rotational motion. Let \(\boldsymbol{\omega}^{n-1/2}\) be the staggered angular velocity, \(\mathbf{I}\) the nodal inertia tensor, and \(\mathbf{m}^n\) the nodal moment. The physical angular acceleration includes the gyroscopic term:

\[ \boldsymbol{\alpha}_{\mathrm{phys}}^n = \mathbf{I}^{-1}\left(\mathbf{m}^n - \boldsymbol{\omega}^{n-1/2} \times (\mathbf{I}\boldsymbol{\omega}^{n-1/2})\right) + \boldsymbol{\alpha}_{\mathrm{load}}^n \]

The damped rotational kick becomes:

\[ \boldsymbol{\alpha}_{\mathrm{alg}}^n = \frac{\boldsymbol{\alpha}_{\mathrm{phys}}^n - \alpha \boldsymbol{\omega}^{n-1/2}} {1 + \frac{1}{2}\alpha \Delta t^n} \]

Remarks#

  • The formulation is consistent with the leapfrog central-difference update.
  • Translation and rotation use the same damping denominator.
  • The stored nodal accelerations are algorithmic quantities tied to the explicit update.