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.

Phase Field Transport

Phase field methods were original proposed to model phase transitions such as crystallization. They have since be used in various applications such as model fracture and charging and discharging of batteries. A phase field equation has the generic form

      [math]\displaystyle{ {\partial \xi\over \partial t} = L\left(\nabla^2 \xi - f(\xi)\right) + g(\xi) }[/math]

where [math]\displaystyle{ \xi }[/math] is the phase field. This diffusion-style equation models formation of the phase field and its growth or decay. The last term, [math]\displaystyle{ g(\xi) }[/math], is a physical terms modeling growth or decay of a phase field. Two examples or energy promoting crack propagation (growth only) or chemical reactions modeling battery processes (charge or discharge). The first term models natural shape of the phase field. If the source termis zero, the steady-state phase field geometry is given by solution to:

      [math]\displaystyle{ \nabla^2 \xi - f(\xi) = 0 }[/math]

For example, phase field fracture replaces a sharp cracks with a diffusive cracking having exponential decay in [math]\displaystyle{ \xi }[/math] from a damage state. A different [math]\displaystyle{ f(\xi) }[/math] in battery phase field modeling replaces a sharp electrode interface with diffusive sigmoidal interface.

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 (V) or Joules/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, the potential should technically scale with units for energy units. It is usually simpler to rescale the potential equation to always be in volts regardless of current energy units. This approach works provided current is in C/(time units)), conductivity in C/(V-(time units)-(length units)), and the effective viscosity is C/(V-(length units)³). The source term should be C/((time units)-(length units)³). The flux has units C/((time units)-(length units)²). For example, Legacy units (which are mm-g-sec) would have conductivity in A/(V-mm), effective viscosity in C/(V-mm³), source term in A/mm³, and flux in A/mm².

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 but should set using Diffusion command with extra parameters.
• poroelasticity (or 2) - poroelasticity diffusion but should set using Poroelasticity command with extra parameters.
• fracture (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.

Note that all transport calculations are often improved by using FMPM(k) methods. For example, fracture phase field diffusion requires it because without FMPM(k) the phase field has poor quality and mechanical response is inaccurate.

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.