XPIC Features
XPIC(k)[1] and FMPM(k)[2] are two improved forms of MPM. The page explains how to use XPIC(k) and FMPM(k) features.
Introduction
The PIC method can be described as applying a projection operator that modifies (and filters) particle velocities before updating them with the grid acceleration. The problem with PIC is that its projection operator filters most problems too heavily resulting in significant dissipation of energy. XPIC(k) is a new method that solves the energy dissipation problem, enhances overall stability of MPM, and reduces noise. XPIC(k) defines a series of new projection operators that can significantly reduce the over damping of PIC simulations. XPIC(k) is defined by an order k, where k=1 is PIC, k>1 is XPIC, and large k approaches amethod with all null-space noise removed.
After deriving XPIC(k)[1] methods, another improvement was to show that XPIC style calculations are equivalent to implementing an MPM method that approximates the inverse of the full mass matrix. Use this revised interpretation, the XPIC(k) scheme was modifed to derive another method denoted FMPM(k)[2]. FMPM(1) defines and improved for of PIC (compared to XPIC(1)) and higher orders also appear to further reduce dissipation seen in XPIC(k). This method is still under development, but is expected to evolve into the optional method for MPM simulations.
Performance
The drawback of XPIC(m) is that each higher order of XPIC(m) requires an extra extrapolation. The extra calculations scale with m*N where N is the number of particles in the problem. XPIC is therefore less efficient than PIC or FLIP (or a PIC damping linear combination of FLIP and PIC). In many problems, m=2 already provides much improvement over PIC and reduces undesirable energy dissipation with minimal extra calculations. Larger m is often better with m=5 appearing to provide much benefit without too much extra cost. Very high values of m (e.g., m>40) are typically unstable (due to too many additional extrapolations).
One option to use XPIC(m) efficiently, is to do XPIC(m) projections periodically rather then on every time step.
XPIC(k) and FMPM(k) Commands
XPIC(k)and FMPM(k) simulations are created by scheduling a PeriodicXPIC Custom Task. In brief, this task selects the XPIC order m and selects a fraction XPIC. Normally, the fraction XPIC should be 1 and the update will use the XPIC(m) method. Using fraction XPIC less than 1 will use a linear combination of FLIP and XPIC(m). Selecting fraction XPIC 0 results in pure FLIP calculations. See help on PeriodicXPIC Custom Task for more details. The remainder of this section describe deprecated methods for selecting XPIC, but these methods should no longer be sued because than cannot select periodic XPIC calculations; that option requires use of the PeriodicXPIC Custom Task.
Deprecated XPIC Commands
Before the PeriodicXPIC Custom Task was added, XPIC(m) was selected using damping commands
Damping (alphagVsT),<(fractionPIC)>,<(XPICOrder)> PDamping (alphapVsT),<(fractionPIC)>,<(XPICOrder)>
where the relevant paramters for XPIC(m) are:
- (fractionPIC) is the fraction PIC to use in the simulation. It can vary from 1 for pure XPIC(m) to 0 for pure FLIP. The default is 0.
- (XPICOrder) to set the XPIC(m) order or m. It must be an integer and values less than 1 are set to 1. Note that this parameter is ignored unless PIC is activated with (fractionPIC) > 0. The default is 1, which is standard PIC.
Although (fractionPIC) and (XPICOrder) can be set in either the Damping or the PDamping command, only one setting for each is allowed; a simulation will use whichever setting comes last.
In XML input files, (fractionPIC) and (XPICOrder) can be set with commands in the <MPMHeader> element:
<Damping PIC='(fractionPIC)' function='(alphagVsT)'>(alphagNum)</Damping> <XPIC order='(XPICOrder)'/>
Note that (XPICOrder) is ignored unless (fractionPIC) is greater than zero.