Damping Options

NairnMPM has two forms of grid damping. Their most common use to to damp dynamic effects an converge to static solution.

Introduction

Grid damping is damping term applied each node on the grid that alters the grid forces (or accelerations). In brief, the force update on node i is calculated from:

      [math]\displaystyle{ f_{i}(t+\Delta t) = f_{i}(t) - \alpha p_i }[/math]

where f_i is total force on node i from all sources, p_i is nodal momentum, and α is the grid damping constant (in units of 1/sec). The nodal velocity update becomes:

      [math]\displaystyle{ v_{i}(t+\Delta t) = v_{i}(t) +(a_i^* - \alpha v_i^*) \Delta t }[/math]

where a_i^* = f_i/m_i is nodal acceleration without damping, v_i^* = p_i/m_i is nodal velocity, and m_i is nodal mass.

The two types of grid damping differ only in entry end evolution of the damping term α. Simple damping just specifies α as a constant value. Alternatively, feedback damping provides a variant of Nose-Hoover feedback, which implements a scheme to evolve the damping coefficient depending on the current total kinetic energy. This approach can evolve the kinetic energy to zero (for a static result) or to any provided non-zero kinetic energy. In feedback damping, α is initialized to zero and then evolves by kinetic energy evaluated on the grid

[math]\displaystyle{ {d\alpha\over dt} = {2K\over m_{tot}} \sum_{node i} {1\over 2} m_i |v_i|^2 - T_k }[/math]

where K is the gain parameter entered in #1, mtot is total mass, and Tk is the target kinetic energy (entered in optional parameter #2). The (2/mtot) is an arbitrary scaling used to make K similar to the different feedback damping method used in earlier versions of this code and an attempt to make K less dependent on problem size. The maximum value of α can optionally be limited to a specified maximum with parameter #3.

Damping Commands

In scripted files, diffusion is activated with the command

Diffusion (YesOrNo),<(refConc)>

In XML input files, diffusion is activated with the <Diffusion> command, which must be within the <MPMHeader>:

<Diffusion reference = '(refConc)'/>

where

• (YesOrNo) must be "Yes" or "No" to activate or not activate diffusion calculations. In XML input files, the presence of a <Diffusion> command activates diffusion. The default is "No".
• (refConc) is used to set a reference concentration potential (between 0 and 1) that corresponds to zero strain. All diffusion calculations are done in terms of a concentration potential from 0 to 1 where 1 is the saturation concentration of a material type. The default (refConc) is 0.