Additional Transport Calculations

In addition to coupling with diffusion or poroelasticy, NairnMPM can couple to several other transport equations. Most of these options are in development and therefore only available in OSParticulas. This information will be expanded when ported to NairnMPM.

General Transport Analysis

A generalized transport equation to be solved on the MPM grid for flow of some conserved content, [math]\displaystyle{ \tau }[/math], per unit volume can be cast as:

      [math]\displaystyle{ c_\theta {\partial \theta\over \partial t} = -\nabla \cdot \vec q(\vec x) + \dot q(\vec x) }[/math]

where [math]\displaystyle{ \theta }[/math] is transport "value," [math]\displaystyle{ c_\theta }[/math] is transport "capacity" (defining [math]\displaystyle{ \tau }[/math] per unit transport value per unit volume), [math]\displaystyle{ \vec q(\vec x) = -\kappa\nabla \theta }[/math] is flow of [math]\displaystyle{ \tau }[/math] per unit area with units Length-(units of [math]\displaystyle{ c_\theta }[/math])-(units of [math]\displaystyle{ \theta }[/math])/sec, and [math]\displaystyle{ \dot q(\vec x) }[/math] is a source term with units (units of [math]\displaystyle{ c_\theta }[/math])-(units of [math]\displaystyle{ \theta }[/math])/sec. In the flow term, [math]\displaystyle{ \kappa }[/math] is a ``conductivity (or ``diffusion) tensor with units of Length²-(units of [math]\displaystyle{ c_\theta }[/math])/sec.

Electrical Conduction using Transport Analysis

This option, which is currently only available in a custom branch of OSParticulas solve the Poisson's equation (which a potential source term) dynamically as a transport problem by casting it as:

      [math]\displaystyle{ \nu {\partial \Phi\over \partial t} = \nabla \cdot \mathbf{\sigma} \nabla \Phi + \dot q(\vec x) }[/math]

where [math]\displaystyle{ \Phi }[/math] is electric potential, [math]\displaystyle{ \mathbf{\sigma} }[/math] is electrical conductivity tensor, and [math]\displaystyle{ \nu }[/math] is an "effective" viscosity. Poisson's equation would equate the right hand side to zero. This transport analysis thus evolves to solution to Poisson's equation either in the limit as [math]\displaystyle{ \nu\to0 }[/math] or at steady state where [math]\displaystyle{ \partial\Phi/\partial t }[/math] reaches zero. As long as potential evolves faster than other coupled phenomena, this approach can give coupled solution to Poisson's equation.

The conserved transport content is Coulombs (C). In SI units, potential is volts (C) or J/C. The current is in Amps (A) or C/sec. The electrical conductivity is A/(V m), and the effective viscosity is C/(V m³). The source term should be A/m³. The flux [math]\displaystyle{ \vec q(\vec x) = -\mathbf{\sigma}\nabla \Phi }[/math] has units C/(m² sec) or A/m². When using consistent units or Legacy units (mm-g-sec), potential is (energy units)/C, current is A, conductivity is V/((potential units)-(length units)), and the effective viscosity is C/((potential units)-(length units))). he source term should be A/m³. The flux [math]\displaystyle{ \vec q(\vec x) = -\mathbf{\sigma}\nabla \Phi }[/math] has units C/(m² sec) or A/m²

Activating Additional Transport Equations

In scripted files, additional transport analysis is activated with the command

Diffusion (style)

In XML input files, additional transport analysis is activated with <Diffusion> commands, which must be within the <MPMHeader> element:

<Diffusion style='1'/>

where (style) is the type of alternate transport to include in the MPM analysis with the options being:

• solvent (or 1) - solvent diffusion
• fracture (or 3) - fracture phase field diffusion
• battery (or 4) - battery phase field calculations (special code only)
• conduction (or 5) - battery conduction equation solved by diffusion (special code only)

Note that the solvent style (which for backward compatibility can be "Yes" or "No") is for solvent diffusion. For details on this option refer to Diffusion Calculations and two additional parameters in the command.

All other diffusion options can be used with or without solvent diffusion. Their use depends on material support for their calculations and you must define any required material properties.

Transport Boundary Conditions

When various transport options are activated, the possible boundary conditions are:

• You can set transport value on the grid.
• You can set a transport flux on particle surfaces.