FMPM Features
FMPM refers to variant of MPM that uses an approximation to the full mass matrix inverse rather then rely on lump mass matrix methods [1][2]. It is referred to as FMPM(k) where k is the order of approximation used to find the full mass matrix inverse. FMPM(k) is an incremental improvement an an early methods known as XPIC(k)[3]. They are very similar, but FMPM(k) is now the preferred option.
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. The XPIC(k) methods was developed to reduce energy dissipation problem thereby filtering out noise that should enhances overall stability of MPM. 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 a method with all null-space noise removed.
After deriving XPIC(k)[3] methods, an incremental 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)[1]. FMPM(1) defines and improved form of PIC (compared to XPIC(1)) and higher orders also appear to further reduce dissipation compared to XPIC(k). Recent revisions for FMPM(k) have fixed issues related to boundary conditions and contact mechanics[2]. Although both XPIC(k) and FMPM(k) are both supported, simulation results show that FMPM(k) is better. XPIC(k) should only be used for comparison purposes.
Performance
A drawback of FMPM(k) is that each higher order requires an extra extrapolation. The extra calculations scale with k*N where N is the number of particles in the problem. FMPM(k) is therefore less efficient than PIC or FLIP. In many problems, k=2 already provides much improvement over PIC and reduces undesirable energy dissipation with minimal extra calculations. Larger k is often better with k=4 appearing to provide much benefit without too much extra cost and higher k eventually provides diminishing benefits. If used, very high k can become unstable unless the Courant–Friedrichs–Lewy factor (C) is reduced to below about 0.25 [2].
XPIC(k) and FMPM(k) Commands
FMPM(k) (or XPIC(k)) simulations are created by scheduling a PeriodicXPIC Custom Task. In brief, this task selects the FMPM (or XPIC) order k and the frequency for using the calculations. See help on PeriodicXPIC Custom Task for all the details.
Eliminated XPIC Commands
The following two commands used to implement XPIC(k) in all versions of this code:
Damping (alphagVsT),<(fractionPIC)>,<(XPICOrder)> PDamping (alphapVsT),<(fractionPIC)>,<(XPICOrder)>
where (fractionPIC) used to blend FLIP and XPIC(k) and (XPICOrder) was order of XPIC(k) calculations. These commands are no longer available the blending option has been removed. If these commands are found in old input files, they should convert to scheduling a PeriodicXPIC Custom Task. The (XPICOrder) is recommended to change to using FMPM(k). The (fractionPIC) option can be replaced by periodically using FMPM(k) with FLIP on other time steps rather then blending them on every time steps (i.e., change (fractionPIC)=0.25 to using FMPM(k) every 4th time step). The results are similar the periodic methods is much more efficient.
References
- ↑ 1.0 1.1 J. A. Nairn and C. C. Hammerquist, "Material point method simulations using an approximate full mass matrix inverse," Computer Methods in Applied Mechanics and Engineering, in press (2021). (See PDF)
- ↑ 2.0 2.1 2.2 J. A. Nairn, "Improved Implementation of Approximate Full Mass Matrix Inverse Methods into Material Point Method Simulations," arXiv:2604.07307 (2026). (See PDF)
- ↑ 3.0 3.1 C. C. Hammerquist and J. A. Nairn, "A new method for material point method particle updates that reduces noise and enhances stability," Computer Methods in Applied Mechanics and Engineering, 318, 724– 738 (2017). (See PDF)