Damping_global_data

Data Structure: Damping_global_data

Description

"Global Damping data" data structure for the geomechanical field

Usage

Damping_global_data     NUM=ival   where ival  is the data structure number

Description

Overview

The Damping_global_data data structure defines percentage damping that is applied to all nodes in the problem domain based on an estimated value of the lowest vibration frequency computed using the Rayleigh quotient.

Notes

•Only one Damping_global_data data structures may be specified and re-specification will overwrite the existing data structure

Several damping models are available:

Mass Proportional Damping

It is defined via Percentage_damping and works in the same way as defined in Damping_nodal_data but applied to all the nodes in the domain. The prescribed percentage of damping is a mass proportional damping defined in the form:

C =α*M

Or in given the diagonal form of the mass matrix:

The lowest frequency ( Ωmin) of the current deformation mode can be estimated by the Raleigh quotient  defined as:

The upper bound of the oscilation frequencies is defined by the critical time step as:

The frequency used in the evaluation of mass proportional damping is the minimum frequency evaluated by the two methods above. This constraint is applied to prevent inappropriate frequencies that may result if the approximation fort Kij is poor (e.g. due to local instabilities). The proportionality constant ( α) is evaluated every time step so that the fluctuating effective stiffness of the system is accounted for. Higher frequencies are damped more strongly than lower frequencies and so diminish first and consequently α tends to a constant value as the quasi-static solution is approached.

Notes

•Only one Damping_global_data data structure may be specified and re-specification will overwrite the existing data structure

•Percentage damping is specified in the range 0 < damp < 1, where 0.1 corresponds to 10% damping of the lowest eigenvalue

•Generally percentage damping is specified in the range 0.01 - 0.03. If larger values of damping are specified they can result in significant over-damping of higher modes

Bulk Viscosity Model

It is defined via the keyword Bulk_damping_model. The standard dynamic bulk viscosity concept can also be used to provide volume damping in quasi-static simulations solved using an explicit solver. In this case the objective is to minimise oscillations in the effective mean stress and only the linear term of the standard bulk viscosity concept is required; i.e. the dynamic bulk viscosity ( q) for each element is defined as:

Damping is only applied in compression, so assuming elasticity, the contribution of the volumetric stress to the element internal forces is evaluated as:

This increases the resistance to deformation in compression effectively providing damping during volume decrease and no damping during volume increase. Two important properties of the algorithm are:

•1) The model does not damp rigid body motion, so that the gross downward movement resulting from compaction of a column, or upward movement on a thrust, remain little affected.

•2) The damping term is transitory (i.e. it is not accumulated) so if the overall deformation pattern is compressive then this mode of deformation is damped but not prevented.

Notes

•Bulk viscosity damping increases the stiffness of the system and consequently reduces the critical time step necessitating more time steps

•Qlin =0.5 is a suitable value for many quasi-static problems

Artificial Bulk Viscosity Model

Artificial bulk viscosity (Stowe et al., 2015 for review) is a bulk (or volume) based damping designed to smooth high frequency oscillations in the dynamic simulation of shock waves. It is applied by default in all ParaGeo simulations unless Bulk_damping is defined. The formulation is based on von Neumann et al. (1950) who introduced a pressure viscosity term in their work with one dimensional shock propagation:

This dynamic bulk viscosity is positive for compressive strain rates (i.e. δu/δx<0) and negative for rarefactions. Typically, numerical implementations of such shock viscosities only activate the term for compressive strain rates. The one-dimensional viscosity of Von Neumann and Richtmyer is both effective and deceptively simple. In one dimension the strain rate across the shock, and hence velocity jump, is defined as:

In expanding this concept to two or three dimensions, the element length is replaced by the square root of the element area in two dimensions, or the cube root of the volume in three dimensions. The strain rate across the shock is approximated as the trace of the strain rate tensor:

The standard form of the shock viscosity used in many explicit codes today is a combination of the quadratic term proposed by Von Neumann and Richtmyer for strong shocks and a linear term for treating small oscillations that occur after the shock:

Notes

•By default in ParaGeo the dimensionless constant are set to Qlin=1.5 and Qquad=0.06. It is not recommended to change these values.

Examples

demonstrating the usage of Damping_global_data include:

•damp_2d4n_001c

•damp_2d4n_001e