Damping

(This section is under construction.)

type_of_damping

  • Type: String

  • Description: The type of damping Hercules should use. Possible options include rayleigh, mass, none, bkt, bkt2, bkt3, and bkt3f.

simulation_velocity_profile_freq_hz

  • Optional: Yes

  • Type: Float

  • Description: This parameter is only effective when type_of_damping is set to bkt, bkt2, bkt3, or bkt3f. It can be a floating number from 0.0 to simulation_wave_max_freq_hz. This is an optional parameter and the default value is 0.0.

use_parametric_q_factor

  • Optional: Conditional

  • Type: Yes or No

  • Description: This parameter is required if type_of_damping is set to bkt, bkt2, bkt3, or bkt3f.

use_infinite_qk

  • Optional: Conditional

  • Type: Yes or No

  • Description: This parameter is required if type_of_damping is set to bkt, bkt2, bkt3, or bkt3f.

parametric_q_factor_constant

  • Optional: Conditional

  • Type: Float

  • Description: This parameter is required if use_parametric_q_factor is set to yes.

parametric_q_factor_alpha

  • Optional: Conditional

  • Type: Float

  • Description: This parameter is required if use_parametric_q_factor is set to yes.

parametric_q_factor_beta

  • Optional: Conditional

  • Type: Float

  • Description: This parameter is required if use_parametric_q_factor is set to yes.

the_threshold_damping

  • Type: Float

  • Description: The threshold to limit the \(\zeta\) damping ratio. If the computed \(\zeta\) is larger than this threshold, it will be set to this threshold.

do_damping_statistics

  • Optional: Yes

  • Type: Boolean

  • Description: Whether Hercules should print damping statistics to the standard output (usually the command line window running Hercules). This is an optional parameter and the default value is False (do_damping_statistics = 0).

Rayleigh Damping

In Hercules, two frequencies used to calculate the Rayleigh damping coefficients are \(0.2 f\) and \(f\), where \(f\) is the maximum frequency of the simulation (defined by simulation_wave_max_freq_hz). The Rayleigh damping coefficients are calculated as follows:

\[\frac{\alpha}{\zeta} = \frac{\omega_1 \omega_2 [ (2(\omega_1^2+\omega_2^2) + 2 \omega_1 \omega_2)(\ln(\omega_1) - \ln(\omega_2)) + 3(\omega_2^2 - \omega_1^2)]}{\omega_1^3 - \omega_2^3 + 3 \omega_2^2 \omega_1 - 3 \omega_1^2 \omega_2}\]
\[\frac{\beta}{\zeta} = \frac{3 [2 \omega_1 \omega_2 (\ln(\omega_2) - \ln(\omega_1)) + \omega_1^2 - \omega_2^2]}{\omega_1^3 - \omega_2^3 + 3 \omega_2^2 \omega_1 - 3 \omega_1^2 \omega_2}\]

where \(\omega_1 = 2 \pi (0.2 f)\) and \(\omega_2 = 2 \pi f\). The damping ratio (\(\zeta\)) is calculated as follows element by element:

\[\zeta = \frac{25.0}{V_S}\]

where \(V_S\) is the shear wave velocity of the element in meters per second. Note that the damping ratio is limited by the threshold damping value (the_threshold_damping). Once the damping ratio exceeds this threshold, it will be set to this threshold.

Mass Damping

Similar to Rayleigh damping, there are also two frequencies used to calculate the mass damping coefficients: \(0.1 f\) and \(8 f\). The mass damping is only proportional to the mass matrix. As a result, the damping coefficients are calculated as follows:

\[\frac{\alpha}{\zeta} = 1.3 \frac{2 \omega_1 \omega_2 \ln(\frac{\omega_2}{\omega_1})}{\omega_2 - \omega_1}\]
\[\frac{\beta}{\zeta} = 0\]