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Damping Options

Nairnj — Sun, 12 Apr 2026 19:40:11 GMT

PIC Damping

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	== Introduction ==		== Introduction ==

	Many times it is useful to apply damping to results, such as when the goal to achieve a simulation of quasi-static results. This section explains various ~~artificial~~ damping options that are applied to the particle update in the MPM time step.		Many times it is useful to apply damping to results, such as when the goal to achieve a simulation of quasi-static results. This section explains various damping options that are applied to the particle update in the MPM time step.

	=== Grid and Particle Damping ===		=== Grid and Particle Damping ===
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	<math> \vec V_p^{(n+1)} = \vec V_p^{(n)} + \biggl(\vec a_{g\to p}^{(n)} - \left(\alpha_g-{1\over \Delta t}\right) \vec v_{g\to p}^{(n)} - \left(\alpha_p+{1\over \Delta t}\right) \vec V_{p}^{(n)}\biggr)\Delta t </math>		<math> \vec V_p^{(n+1)} = \vec V_p^{(n)} + \biggl(\vec a_{g\to p}^{(n)} - \left(\alpha_g-{1\over \Delta t}\right) \vec v_{g\to p}^{(n)} - \left(\alpha_p+{1\over \Delta t}\right) \vec V_{p}^{(n)}\biggr)\Delta t </math>

	This damping ~~view~~ of PIC leads to two significant improvements for the use of PIC in simulations:		The extra damping created by adopting PIC methods are seen to be


			PIC Damping = <math>\frac{\vec V_{p}-\vec v_{g\to p}^{(n)}}{\Delta t}</math>

			This damping interpretion of PIC leads to two significant improvements for the use of PIC in simulations:

	* It provides insight and justification for adding PIC. The PIC damping term is related to the extrapolation error between particle velocity and extrapolated grid velocity. This error will be large in elements that contain particles with widely varying velocity (as sometimes happens in MPM simulations) and small in areas with rather uniform velocity. Thus PIC damping will selectively damp out regions with large velocity variations and have little affect on regions with smooth velocities.		* It provides insight and justification for adding PIC. The PIC damping term is related to the extrapolation error between particle velocity and extrapolated grid velocity. This error will be large in elements that contain particles with widely varying velocity (as sometimes happens in MPM simulations) and small in areas with rather uniform velocity. Thus PIC damping will selectively damp out regions with large velocity variations and have little affect on regions with smooth velocities.
	* By implementing PIC as a damping mechanism, it became clear how to modify the position update in PIC as well.<ref name="cutting"/> Some early proposals to use PIC (or to mix a fraction PIC with FLIP) used PIC update for velocity but retained the FLIP update for position.<ref name="Diz"/> The above position update addresses that issue and greatly improves particle displacement results in simulations ~~with a significant amount of~~ PIC.<ref name="cutting"/> Also note that ~~the~~ damping terms in the position update are second order, but because PIC damping is inversely proportional to Δ''t'', the PIC damping ~~terms become~~ first order ~~terms~~ in the position update.		* By implementing PIC as a damping mechanism, it became clear how to modify the position update in PIC as well.<ref name="cutting"/> Some early proposals to use PIC (or to mix a fraction PIC with FLIP) used PIC update for velocity but retained the FLIP update for position.<ref name="Diz"/> The above position update addresses that issue and greatly improves particle displacement results in simulations using PIC.<ref name="cutting"/> Also note that standard damping terms in the position update are second order, but because PIC damping is inversely proportional to Δ''t'', the PIC damping term is a first order term in the position update.

	A drawback of PIC damping is ~~typcially that~~ it damps too much. It can work well is quasi-static problems, but often damps too much in an dynamic problems. A paper on using MPM for animation<ref name="Diz"/> suggested that simulations could be improved by modifying the velocity update to include a fraction β of standard FLIP style MPM velocity updates and a fraction (1-β) of PIC style updates:		A drawback of PIC damping is it usually damps too much. It can work well is quasi-static problems, but often damps too much in dynamic problems. A paper on using MPM for animation<ref name="Diz"/> suggested that simulations could be improved by modifying the velocity update to include a fraction β of standard FLIP style MPM velocity updates and a fraction (1-β) of PIC style updates:


	<math> \vec v_{p}^{(n+1)} = \beta \vec v_{p,FLIP}^{(n+1)} + (1-\beta)\vec v_{p,PIC}^{(n+1)} </math>		<math> \vec v_{p}^{(n+1)} = \beta \vec v_{p,FLIP}^{(n+1)} + (1-\beta)\vec v_{p,PIC}^{(n+1)} </math>

	If this ~~method~~ also mixes fraction β of standard FLIP fraction (1-β) of PIC position ~~updates~~ (although original reference did not mix position ~~options~~<ref name="Diz"/>) it provides ~~and options~~ for controlling the amount of damping. [[NairnMPM]] provides a similar feature, but rather than mix FLIP and PIC on each time step, it intersperses FLIP time steps with PIC time steps. If a PIC time step is done every n<sup>th</sup> time step, it is nearly the same as a simulation with β=(n-1)/n. To create simulations with periodic PIC, add a [[PeriodicXPIC Custom Task]] to you simulation.		If this approach also mixes fraction β of standard FLIP position update with fraction (1-β) of PIC position update (although original reference incorrectly did not mix position updates<ref name="Diz"/>), it provides one option for controlling the amount of damping. [[NairnMPM]] provides a similar feature, but rather than mix FLIP and PIC on each time step, it intersperses FLIP time steps with PIC time steps. If a PIC time step is done every n<sup>th</sup> time step, it is nearly the same as a simulation with β=(n-1)/n. To create simulations with periodic PIC, add a [[PeriodicXPIC Custom Task]] to you simulation.

	An alternative to mixing FLIP with PIC is to use two new advanced methods that enhance PIC-style updates to minimize or eliminate their dissipation. The two methods are called [[~~XPIC~~ Features\|~~XPIC~~(k) and ~~FMPM~~(k)]] for XPIC ~~or FMPM~~ of order k. ~~XPIC~~(1) and ~~FMPM~~(1) are two similar versions of PIC. Orders greater than one provide new methods that can stabilize simulations without causing too much dissipation. These two methods are ~~also~~ added to simulations by using the [[PeriodicXPIC Custom Task]].		An alternative to mixing FLIP with PIC is to use two new advanced methods that enhance PIC-style updates to minimize or eliminate their dissipation. The two methods are called [[FMPM Features\|FMPM(k) and XPIC(k)]] for FMPM or XPIC of order k. FMPM(1) and XPIC(1) are two similar versions of PIC but sample simulations show to FMPM(1) is better than XPIC(1) and both are much better then a PIC method that omits PIC damping in the position update. Orders greater than one provide new methods that can stabilize simulations without causing too much dissipation. These two methods are added to simulations by using the [[PeriodicXPIC Custom Task]].

	== Damping Commands ==		== Damping Commands ==

PeriodicXPIC Custom Task

Nairnj — Sun, 12 Apr 2026 19:13:17 GMT

Introduction

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	A [[MPM Input Files#Custom Tasks\|custom task]] to use [[~~XPIC~~ Features\|~~XPIC~~(k) ~~and FMPM~~(k) methods]] on all or selected time steps		A [[MPM Input Files#Custom Tasks\|custom task]] to use [[FMPM Features\|FMPM(k) (or XPIC(k)) methods]] on all or selected time steps
	__TOC__		__TOC__
	== Introduction ==		== Introduction ==

	The ~~XPIC~~(k)<ref name="~~XPIC~~"/> ~~and FMPM(k)~~<ref name="~~FMPM~~"/><ref name="~~FMPMrev~~"/> ~~methods are~~ [[~~XPIC~~ Features\|advanced ~~methods~~]] ~~that~~ filter out unwanted noise (in the null space) without damping out useful information. ~~Their~~ drawback is that ~~they add~~ calculation time as order <tt>k</tt> increases. Many simulations (especially those with stability issues) will run better by using ~~XPIC~~(k) or ~~FMPM~~(k). The ~~XPIC(k) and FMPM(k)~~ methods can be done every time step or only on periodic time steps. This custom tasks adds ~~XPIC~~(k) or ~~FMPM~~(k) to any simulation and lets you select the frequency of those calculations.		The FMPM(k)<ref name="FMPM"/><ref name="FMPMrev"/> (or XPIC(k)<ref name="XPIC"/>) method is an [[FMPM Features\|advanced method]] makes use of an approximate inverse of the full mass matrix to filter out unwanted noise (in the null space) without damping out useful information. A drawback is that it adds calculation time as order <tt>k</tt> increases. Many simulations (especially those with stability issues) will run better by using FMPM(k) (or XPIC(k)). The methods can be done every time step or only on periodic time steps. This custom tasks adds FMPM(k) (or XPIC(k)) to any simulation and lets you select the frequency of those calculations.

	== Using FMPM(k) or XPIC(k) For Mechanics ==		== Using FMPM(k) or XPIC(k) For Mechanics ==

FMPM Features

Nairnj — Sun, 12 Apr 2026 18:53:15 GMT

XPIC(k) and FMPM(k) Commands

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	FMPM~~(k)~~<ref name="FMPM"/> ~~and~~ XPIC(k)<ref name="XPIC"/> are ~~two improved forms of MPM. The page explains how to use XPIC(k) and~~ FMPM(k) ~~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 <ref name="FMPM"/><ref name="FMPMrev"/>. 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)<ref name="XPIC"/>. They are very similar, but FMPM(k) is now the preferred option.

	== Introduction ==		== Introduction ==
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	The [[Damping Options#PIC Damping\|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 <tt>k</tt>, where <tt>k=1</tt> is PIC, <tt>k>1</tt> is XPIC, and large <tt>k</tt> approaches a method with all null-space noise removed.		The [[Damping Options#PIC Damping\|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 <tt>k</tt>, where <tt>k=1</tt> is PIC, <tt>k>1</tt> is XPIC, and large <tt>k</tt> approaches a method with all null-space noise removed.

	After deriving XPIC(k)<ref name="XPIC"/> 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)<ref name="FMPM"/>. 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<ref name="FMPMrev"/>. Although both XPIC(k) and FMPM(k) are supported, ~~simulations~~ results show that FMPM(k) is better. XPIC(k) should only be used for comparison purposes.		After deriving XPIC(k)<ref name="XPIC"/> 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)<ref name="FMPM"/>. 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<ref name="FMPMrev"/>. 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 ==		== 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) therefore less efficient than PIC or FLIP. In many problems, <tt>k=2</tt> already provides much improvement over PIC and reduces undesirable energy dissipation with minimal extra calculations. Larger <tt>k</tt> is often better with <tt>k=4</tt> appearing to provide much benefit without too much extra cost and ~~high~~ <tt>k</tt> eventually provides diminishing benefits. If used, very high <tt>k</tt> can become unstable unless the [[MPM Methods and Simulation Timing#Theory: MPM Time Step\|Courant–Friedrichs–Lewy factor (C)]] is reduced to below about 0.25 <ref name="FMPMrev"/>.		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, <tt>k=2</tt> already provides much improvement over PIC and reduces undesirable energy dissipation with minimal extra calculations. Larger <tt>k</tt> is often better with <tt>k=4</tt> appearing to provide much benefit without too much extra cost and higher <tt>k</tt> eventually provides diminishing benefits. If used, very high <tt>k</tt> can become unstable unless the [[MPM Methods and Simulation Timing#Theory: MPM Time Step\|Courant–Friedrichs–Lewy factor (C)]] is reduced to below about 0.25 <ref name="FMPMrev"/>.

	== XPIC(k) and FMPM(k) Commands ==		== XPIC(k) and FMPM(k) Commands ==
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	PDamping (alphapVsT),<(fractionPIC)>,<(XPICOrder)>		PDamping (alphapVsT),<(fractionPIC)>,<(XPICOrder)>

	where <tt>(fractionPIC)</tt> used to blend FLIP and XPIC(k) and <tt>(XPICOrder)</tt> was order of XPIC(k) calculations. These commands are no longer available the blending option has be removed. If these commands are found in old input files, ~~those commands~~ should convert to scheduling a [[PeriodicXPIC Custom Task]]. The <tt>(XPICOrder)</tt> is recommended to change to using FMPM(k). The <tt>(fractionPIC)</tt> 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 <tt>(fractionPIC)=0.25</tt> to using FMPM(k) every 4<sup>th</sup> time step). The results are similar the periodic methods is much more efficient.		where <tt>(fractionPIC)</tt> used to blend FLIP and XPIC(k) and <tt>(XPICOrder)</tt> 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 <tt>(XPICOrder)</tt> is recommended to change to using FMPM(k). The <tt>(fractionPIC)</tt> 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 <tt>(fractionPIC)=0.25</tt> to using FMPM(k) every 4<sup>th</sup> time step). The results are similar the periodic methods is much more efficient.

	== References ==		== References ==

XPIC Features

Nairnj — Sun, 12 Apr 2026 18:44:04 GMT

Nairnj moved page XPIC Features to FMPM Features To emphasize FMPM is better than XPIC

New page

FMPM(k)<ref name="FMPM"/> and XPIC(k)<ref name="XPIC"/> are two improved forms of MPM. The page explains how to use XPIC(k) and FMPM(k) features.

== Introduction ==

The [[Damping Options#PIC Damping|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 <tt>k</tt>, where <tt>k=1</tt> is PIC, <tt>k>1</tt> is XPIC, and large <tt>k</tt> approaches a method with all null-space noise removed.

After deriving XPIC(k)<ref name="XPIC"/> 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)<ref name="FMPM"/>. 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<ref name="FMPMrev"/>. Although both XPIC(k) and FMPM(k) are supported, simulations 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) therefore less efficient than PIC or FLIP. In many problems, <tt>k=2</tt> already provides much improvement over PIC and reduces undesirable energy dissipation with minimal extra calculations. Larger <tt>k</tt> is often better with <tt>k=4</tt> appearing to provide much benefit without too much extra cost and high <tt>k</tt> eventually provides diminishing benefits. If used, very high <tt>k</tt> can become unstable unless the [[MPM Methods and Simulation Timing#Theory: MPM Time Step|Courant–Friedrichs–Lewy factor (C)]] is reduced to below about 0.25 <ref name="FMPMrev"/>.

== 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 <tt>k</tt> 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 <tt>(fractionPIC)</tt> used to blend FLIP and XPIC(k) and <tt>(XPICOrder)</tt> was order of XPIC(k) calculations. These commands are no longer available the blending option has be removed. If these commands are found in old input files, those commands should convert to scheduling a [[PeriodicXPIC Custom Task]]. The <tt>(XPICOrder)</tt> is recommended to change to using FMPM(k). The <tt>(fractionPIC)</tt> 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 <tt>(fractionPIC)=0.25</tt> to using FMPM(k) every 4th time step). The results are similar the periodic methods is much more efficient.

== References ==

<references>

<ref name="XPIC">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). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/XPICPaper.pdf See PDF])</ref>

<ref name="FMPM">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). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/FMPMPaper.pdf See PDF])</ref>

<ref name="FMPMrev">J. A. Nairn, "Improved Implementation of Approximate Full Mass Matrix Inverse Methods into Material Point Method Simulations," arXiv:2604.07307 (2026). ([https://arxiv.org/abs/2604.07307 See PDF])</ref>

</references>

Damping Options

Nairnj — Sun, 12 Apr 2026 18:42:23 GMT

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	[[NairnMPM]] has several forms of artificial damping done during particle updates. Their most common use is to damp dynamic effects such that the solution converges to a static result. Some materials also support [[Common Material Properties#Artificial Viscosity\|artificial viscosity]] as a damping mechanism done on the material model level. In addition, [[NairnMPM]] supports FLIP, PIC, [[~~XPIC~~ Features\|XPIC]],<ref name="XPIC"/> ~~[[XPIC Features\|FMPM]]~~<ref name="FMPM"/> and [[PeriodicXPIC Custom Task\|periodic mixtures]] of FLIP with ~~PIC, XPIC, and~~ FMPM; XPIC ~~and FMPM enhance stability and control noise~~. This section ~~focuses on~~ grid and particle damping.		[[NairnMPM]] has several forms of artificial damping done during particle updates. Their most common use is to damp dynamic effects such that the solution converges to a static result. Some materials also support [[Common Material Properties#Artificial Viscosity\|artificial viscosity]] as a damping mechanism done on the material model level. In addition, [[NairnMPM]] supports FLIP, PIC,[[FMPM Features\|FMPM(k) or XPIC(k)]]<ref name="XPIC"/> <ref name="FMPM"/> and [[PeriodicXPIC Custom Task\|periodic mixtures]] of FLIP with FMPM(k) or XPIC(k). This section section describes only the options for grid and particle damping.

	__TOC__		__TOC__

MPM Input Files

Nairnj — Sun, 12 Apr 2026 18:37:21 GMT

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	* [[MPM Methods and Simulation Timing]] - these commands determine the MPM method to use, the time step, and the maximum simulations time.		* [[MPM Methods and Simulation Timing]] - these commands determine the MPM method to use, the time step, and the maximum simulations time.
	* [[Damping Options]] - these commands set up grid-based damping options.		* [[Damping Options]] - these commands set up grid-based damping options.
	* [[~~XPIC~~ Features\|~~XPIC~~(k) ~~and FMPM~~(k) Features]] - improved MPM ~~with~~ enhanced stability and noise ~~reduction~~		* [[FMPM Features\|FMPM(k) (or XPIC(k)) Features]] - improved MPM that approximates the full mass matrix inverse that can enhanced stability and reduce noise.
	* [[LeaveLimit Command]] - this command determines what happens when particles leave the grid.		* [[LeaveLimit Command]] - this command determines what happens when particles leave the grid.
	* [[DeleteLimit Command]] - this command determines what happens when particles get nan values.		* [[DeleteLimit Command]] - this command determines what happens when particles get nan values.
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	Custom tasks are special calculations that are performed at the end of each time step in MPM calculations. Custom tasks can be written by users working with [[NairnMPM]] source code. These user-defined custom tasks can then be scheduled when needed in some calculations. The current custom tasks are:		Custom tasks are special calculations that are performed at the end of each time step in MPM calculations. Custom tasks can be written by users working with [[NairnMPM]] source code. These user-defined custom tasks can then be scheduled when needed in some calculations. The current custom tasks are:

	* [[PeriodicXPIC Custom Task]] - implement [[~~Damping Options#XPIC~~ Features\|~~XPIC~~(m) simulations]] with an option to run them periodically instead of every time step		* [[PeriodicXPIC Custom Task]] - implement [[FMPM Features\|FMPM(k) simulations]] with an option to run them periodically instead of every time step
	* [[AdjustTimeStep Custom Task]] - to adjust time step during the calculations according to current stress state of the particles.		* [[AdjustTimeStep Custom Task]] - to adjust time step during the calculations according to current stress state of the particles.
	* [[VTKArchive Custom Task]] - to archive results extrapolated to the grid in "VTK Legacy" files.		* [[VTKArchive Custom Task]] - to archive results extrapolated to the grid in "VTK Legacy" files.

NairnMPM

Nairnj — Sun, 12 Apr 2026 18:34:21 GMT

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	* [[MPM Methods and Simulation Timing\|Several options for particle update methods]]		* [[MPM Methods and Simulation Timing\|Several options for particle update methods]]
	* [[MPM Methods and Simulation Timing\|Several types of shape function including spline-based shaped functions]]		* [[MPM Methods and Simulation Timing\|Several types of shape function including spline-based shaped functions]]
	* ~~A extended PIC methods~~ known as [[~~XPIC~~ Features\|~~XPIC~~]]		* Can implement method that approximates the inverse of the full mass matrix known as [[FMPM Features\|FMPM(k)]]
	* [[Material Models\|Many material models]] including elastic, plastic, isotropic, anisotropic, viscoelastic, small strain, and large strain.		* [[Material Models\|Many material models]] including elastic, plastic, isotropic, anisotropic, viscoelastic, small strain, and large strain.
	* Plasticity materials can use a variety of [[Hardening Laws\|hardening laws]].		* Plasticity materials can use a variety of [[Hardening Laws\|hardening laws]].

XPIC Features

Nairnj — Sun, 12 Apr 2026 18:31:03 GMT

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	~~XPIC~~(k)<ref name="~~XPIC~~"/> and ~~FMPM~~(k)<ref name="~~FMPM~~"/> are two improved forms of MPM. The page explains how to use XPIC(k) and FMPM(k) features.		FMPM(k)<ref name="FMPM"/> and XPIC(k)<ref name="XPIC"/> are two improved forms of MPM. The page explains how to use XPIC(k) and FMPM(k) features.

	== Introduction ==		== Introduction ==

	The [[Damping Options#PIC Damping\|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 <tt>k</tt>, where <tt>k=1</tt> is PIC, <tt>k>1</tt> is XPIC, and large <tt>k</tt> approaches ~~amethod~~ with all null-space noise removed.		The [[Damping Options#PIC Damping\|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 <tt>k</tt>, where <tt>k=1</tt> is PIC, <tt>k>1</tt> is XPIC, and large <tt>k</tt> approaches a method with all null-space noise removed.

	After deriving XPIC(k)<ref name="XPIC"/> 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)<ref name="FMPM"/>. 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<ref name="FMPMrev"/>.		After deriving XPIC(k)<ref name="XPIC"/> 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)<ref name="FMPM"/>. 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<ref name="FMPMrev"/>. Although both XPIC(k) and FMPM(k) are supported, simulations results show that FMPM(k) is better. XPIC(k) should only be used for comparison purposes.

	== Performance ==		== Performance ==

	~~The~~ drawback of ~~XPIC(k)/~~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. ~~XPIC and~~ FMPM ~~are~~ therefore less efficient than PIC or FLIP. In many problems, <tt>k=2</tt> already provides much improvement over PIC and reduces undesirable energy dissipation with minimal extra calculations. Larger <tt>k</tt> is often better with <tt>k=4</tt> appearing to provide much benefit without too much extra cost~~. Very~~ high ~~values of~~ <tt>k</tt> ~~(''e.g~~.'', <tt>k~~>40~~</tt>~~) are typically~~ unstable (~~due~~ to ~~too many additional extrapolations or to changes in effective eigenvalues for the equations)~~.		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) therefore less efficient than PIC or FLIP. In many problems, <tt>k=2</tt> already provides much improvement over PIC and reduces undesirable energy dissipation with minimal extra calculations. Larger <tt>k</tt> is often better with <tt>k=4</tt> appearing to provide much benefit without too much extra cost and high <tt>k</tt> eventually provides diminishing benefits. If used, very high <tt>k</tt> can become unstable unless the [[MPM Methods and Simulation Timing#Theory: MPM Time Step\|Courant–Friedrichs–Lewy factor (C)]] is reduced to below about 0.25 <ref name="FMPMrev"/>.

	== XPIC(k) and FMPM(k) Commands ==		== 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 ~~or FMPM~~ order <tt>k</tt> and the frequency for using the calculations. See help on [[PeriodicXPIC Custom Task]] for ~~more~~ details~~. The remainder of this section describes prior methods for selecting XPIC(k) (but not FMPM(k)). They are no longer available in the latest code~~.		FMPM(k) (or XPIC(k)) simulations are created by scheduling a [[PeriodicXPIC Custom Task]]. In brief, this task selects the FMPM (or XPIC) order <tt>k</tt> and the frequency for using the calculations. See help on [[PeriodicXPIC Custom Task]] for all the details.

	=== Eliminated XPIC Commands ===		=== Eliminated XPIC Commands ===

	~~Before the [[PeriodicXPIC Custom Task]] was added,~~ XPIC(m) ~~was selected using [[Damping Options#Damping Commands\|damping commands]]. Any old input files you have with~~ this ~~command should change those files to use a [[PeriodicXPIC Custom Task]] instead. The deleted command options are~~:		The following two commands used to implement XPIC(k) in all versions of this code:

	Damping (alphagVsT),<(fractionPIC)>,<(XPICOrder)>		Damping (alphagVsT),<(fractionPIC)>,<(XPICOrder)>
	PDamping (alphapVsT),<(fractionPIC)>,<(XPICOrder)>		PDamping (alphapVsT),<(fractionPIC)>,<(XPICOrder)>

	where ~~the relevant paramters for XPIC(m) are:~~		where <tt>(fractionPIC)</tt> used to blend FLIP and XPIC(k) and <tt>(XPICOrder)</tt> was order of XPIC(k) calculations. These commands are no longer available the blending option has be removed. If these commands are found in old input files, those commands should convert to scheduling a [[PeriodicXPIC Custom Task]]. The <tt>(XPICOrder)</tt> is recommended to change to using FMPM(k). The <tt>(fractionPIC)</tt> 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 <tt>(fractionPIC)=0.25</tt> to using FMPM(k) every 4<sup>th</sup> time step). The results are similar the periodic methods is much more efficient.

	* <tt>(fractionPIC)</tt> ~~is the [[Damping Options#PIC Damping\|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. Note that this option is inefficient~~ and ~~was therefore removed. You can replace a fraction~~ XPIC(m) ~~of φ in old calculations with XPIC(m) done every~~ <tt>~~int~~(1/</tt>~~φ<tt>~~)~~</tt> time steps~~ in ~~the~~ [[PeriodicXPIC Custom Task]].
	* <tt>(XPICOrder)</tt> to ~~set the XPIC(m) order or <tt>m</tt>. 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 <tt>~~(~~fractionPIC~~)~~</tt> > 0~~. The ~~default is 1, which is standard PIC.~~

	~~Although~~ <tt>(fractionPIC)</tt> ~~and <tt>(XPICOrder)</tt>~~ can be ~~set in either the <tt>Damping</tt> or the <tt>PDamping</tt> command, only one setting for each is allowed; a simulation will use whichever setting comes last.~~

	~~In <tt>XML</tt> input files, <tt>(fractionPIC)</tt> and <tt>~~(~~XPICOrder~~)~~</tt> can be set~~ with ~~commands in the [[MPM Input Files#MPM Header\|<tt><MPMHeader></tt> element]]:~~

	~~<Damping PIC='~~(~~fractionPIC)~~' ~~function=~~'~~(alphagVsT)~~'~~>(alphagNum)</Damping>~~
	~~<XPIC order=~~'~~(XPICOrder)'/>~~

	~~Note that~~ <tt>(~~XPICOrder~~)</tt> ~~is ignored unless~~ <tt>~~(fractionPIC)~~</tt> is ~~greater than zero~~.

	== References ==		== References ==

MPM Methods and Simulation Timing

Nairnj — Sun, 12 Apr 2026 18:09:18 GMT

Theory: MPM Time Step

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	<math> C = { v\Delta t\over \Delta x} \qquad {\rm or} \qquad\Delta t =C { \Delta x\over v}</math>		<math> C = { v\Delta t\over \Delta x} \qquad {\rm or} \qquad\Delta t =C { \Delta x\over v}</math>

	The entered ''C'' must be less then 1.0. The default in [[NairnMPM]] is ''C'' = 0.5. Lower values may be ~~are~~ needed ~~to resolve instabilities around~~ [[Setting Velocity and Transport Values\|fixed-velocity boundary conditions]] on the grid or created by [[Rigid Material\|rigid particles]] when modeling [[Multimaterial MPM\|contact]], ~~and~~ when modeling brittle materials using [[Material Models#Softening Materials\|damage mechanics]]. An alternative to very low ''C'' is to try [[#Restarting Time Steps\|restarting time steps]] that develop high accelerations on the grid.		The entered ''C'' must be less then 1.0. The default in [[NairnMPM]] is ''C'' = 0.5. Lower values may be needed depending on simulations features such as [[Setting Velocity and Transport Values\|fixed-velocity boundary conditions]] on the grid or created by [[Rigid Material\|rigid particles]] when modeling [[Multimaterial MPM\|contact]], when modeling brittle materials using [[Material Models#Softening Materials\|damage mechanics]], or when using [[PeriodicXPIC Custom Task\|high order FMPM(k)]]. An alternative to very low ''C'' is to try [[#Restarting Time Steps\|restarting time steps]] that develop high accelerations on the grid.

	When transport tasks are activated ([[Thermal Calculations\|conduction]], [[Diffusion Calculations\|diffusion]], or [[Poroelasticity Calculations\|poroelasticity]]), convergence of transport calculations requires the time step to satisfy		When transport tasks are activated ([[Thermal Calculations\|conduction]], [[Diffusion Calculations\|diffusion]], or [[Poroelasticity Calculations\|poroelasticity]]), convergence of transport calculations requires the time step to satisfy

XPIC Features

Nairnj — Fri, 10 Apr 2026 18:05:12 GMT

References

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	The [[Damping Options#PIC Damping\|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 <tt>k</tt>, where <tt>k=1</tt> is PIC, <tt>k>1</tt> is XPIC, and large <tt>k</tt> approaches amethod with all null-space noise removed.		The [[Damping Options#PIC Damping\|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 <tt>k</tt>, where <tt>k=1</tt> is PIC, <tt>k>1</tt> is XPIC, and large <tt>k</tt> approaches amethod with all null-space noise removed.

	After deriving XPIC(k)<ref name="XPIC"/> 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)<ref name="FMPM"/>. 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).		After deriving XPIC(k)<ref name="XPIC"/> 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)<ref name="FMPM"/>. 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<ref name="FMPMrev"/>.

	== Performance ==		== Performance ==
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	<ref name="FMPM">J. A. Nairn and C. C. Hammerquist, "Material point method simulations using an approximate full mass matrix inverse," <i>Computer Methods in Applied Mechanics and Engineering</i>, in press (2021). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/FMPMPaper.pdf See PDF])</ref>		<ref name="FMPM">J. A. Nairn and C. C. Hammerquist, "Material point method simulations using an approximate full mass matrix inverse," <i>Computer Methods in Applied Mechanics and Engineering</i>, in press (2021). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/FMPMPaper.pdf See PDF])</ref>

			<ref name="FMPMrev">J. A. Nairn, "Improved Implementation of Approximate Full Mass Matrix Inverse Methods into Material Point Method Simulations," <i>arXiv:2604.07307</i> (2026). ([https://arxiv.org/abs/2604.07307 See PDF])</ref>

	</references>		</references>

Analysis Command

Nairnj — Fri, 10 Apr 2026 18:02:59 GMT

References

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	When used in standard FLIP MPM, a revised extrapolation to the grid follows methods first presented by Wallstedt and Guilkey.<ref name="WGVG"/> The method used to track velocity gradients, however is different. The best approach to FLIP analysis with velocity gradient tracking is under evaluation.		When used in standard FLIP MPM, a revised extrapolation to the grid follows methods first presented by Wallstedt and Guilkey.<ref name="WGVG"/> The method used to track velocity gradients, however is different. The best approach to FLIP analysis with velocity gradient tracking is under evaluation.

	Another paper relevant to velocity gradient tracking is the APIC method.<ref name="APIC"/> APIC is equivalent to using the [[PeriodicXPIC Custom Task]] with <tt>FMPMOrder</tt> 1 (''i.e.'', PIC style) every time step combined with with [[MPM Methods and Simulation Timing#Classic MPM Methods\|<tt>Classic</tt> or <tt>B2Spline</tt>]] grid-based shape functions (the APIC paper is not clear which they used, but it was probably <tt>B2Spline</tt>). Like all PIC-style methods, APIC can dissipate much energy in certain problems. Using higher <tt>FMPMOrder</tt> while tracking velocity gradient extends APIC to reduce dissipation using approximate full mass matrix methods.<ref name="FMPM"/> Switching to other available [[MPM Methods and Simulation Timing\|shape function types]] extends APIC to methods that work better with large deformations. These extensions to APIC are under evaulation.		Another paper relevant to velocity gradient tracking is the APIC method.<ref name="APIC"/> APIC is equivalent to using the [[PeriodicXPIC Custom Task]] with <tt>FMPMOrder</tt> 1 (''i.e.'', PIC style) every time step combined with with [[MPM Methods and Simulation Timing#Classic MPM Methods\|<tt>Classic</tt> or <tt>B2Spline</tt>]] grid-based shape functions (the APIC paper is not clear which they used, but it was probably <tt>B2Spline</tt>). Like all PIC-style methods, APIC can dissipate much energy in certain problems. Using higher <tt>FMPMOrder</tt> while tracking velocity gradient extends APIC to reduce dissipation using approximate full mass matrix methods.<ref name="FMPM"/><ref name="FMPMrev"/> Switching to other available [[MPM Methods and Simulation Timing\|shape function types]] extends APIC to methods that work better with large deformations. These extensions to APIC are under evaulation.

	== Notes ==		== Notes ==
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	<ref name="FMPM">J. A. Nairn and C. C. Hammerquist, "Material point method simulations using an approximate full mass matrix inverse," <i>Computer Methods in Applied Mechanics and Engineering</i>, <b>377</b>, 113667 (2021). (2021). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/FMPMPaper.pdf See PDF])</ref>		<ref name="FMPM">J. A. Nairn and C. C. Hammerquist, "Material point method simulations using an approximate full mass matrix inverse," <i>Computer Methods in Applied Mechanics and Engineering</i>, <b>377</b>, 113667 (2021). (2021). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/FMPMPaper.pdf See PDF])</ref>

			<ref name="FMPMrev">J. A. Nairn, "Improved Implementation of Approximate Full Mass Matrix Inverse Methods into Material Point Method Simulations," <i>arXiv:2604.07307</i> (2026). ([https://arxiv.org/abs/2604.07307 See PDF])</ref>

	<ref name="ASPaper">J.A. Nairn and J.E. Guilkey, "Axisymmetric Form of the Generalized Interpolation Material Point Method,"> <i>Int. J. for Numerical Methods in Engineering</i>, <b>101</b>, 127-147 (2015) ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/AxisymGIMP.pdf See PDF]).</ref>		<ref name="ASPaper">J.A. Nairn and J.E. Guilkey, "Axisymmetric Form of the Generalized Interpolation Material Point Method,"> <i>Int. J. for Numerical Methods in Engineering</i>, <b>101</b>, 127-147 (2015) ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/AxisymGIMP.pdf See PDF]).</ref>

	</references>		</references>

PeriodicXPIC Custom Task

Nairnj — Fri, 10 Apr 2026 18:00:25 GMT

References

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	== Introduction ==		== Introduction ==

	The XPIC(k)<ref name="XPIC"/> and FMPM(k)<ref name="FMPM"/> methods are [[XPIC Features\|advanced methods]] that filter out unwanted noise (in the null space) without damping out useful information. Their drawback is that they add calculation time as order <tt>k</tt> increases. Many simulations (especially those with stability issues) will run better by using XPIC(k) or FMPM(k). The XPIC(k) and FMPM(k) methods can be done every time step or only on periodic time steps. This custom tasks adds XPIC(k) or FMPM(k) to any simulation and lets you select the frequency of those calculations.		The XPIC(k)<ref name="XPIC"/> and FMPM(k)<ref name="FMPM"/><ref name="FMPMrev"/> methods are [[XPIC Features\|advanced methods]] that filter out unwanted noise (in the null space) without damping out useful information. Their drawback is that they add calculation time as order <tt>k</tt> increases. Many simulations (especially those with stability issues) will run better by using XPIC(k) or FMPM(k). The XPIC(k) and FMPM(k) methods can be done every time step or only on periodic time steps. This custom tasks adds XPIC(k) or FMPM(k) to any simulation and lets you select the frequency of those calculations.

	== Using FMPM(k) or XPIC(k) For Mechanics ==		== Using FMPM(k) or XPIC(k) For Mechanics ==
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	<ref name="FMPM">J. A. Nairn and C. C. Hammerquist, "Material point method simulations using an approximate full mass matrix inverse," <i>Computer Methods in Applied Mechanics and Engineering</i>, <b>377</b>, 113667 (2021). (2021). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/FMPMPaper.pdf See PDF])</ref>		<ref name="FMPM">J. A. Nairn and C. C. Hammerquist, "Material point method simulations using an approximate full mass matrix inverse," <i>Computer Methods in Applied Mechanics and Engineering</i>, <b>377</b>, 113667 (2021). (2021). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/FMPMPaper.pdf See PDF])</ref>

			<ref name="FMPMrev">J. A. Nairn, "Improved Implementation of Approximate Full Mass Matrix Inverse Methods into Material Point Method Simulations," <i>arXiv:2604.07307</i> (2026). ([https://arxiv.org/abs/2604.07307 See PDF])</ref>

	<ref name="transport">J. A. Nairn, "Coupling Transport Equations to Mechanics in the Material Point Method Using an Approximate Full Capacity Matrix Inverse," ''Computer Methods in Applied Mechanics and Engineering'', '''420''', 116757 (2024). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/TransportPaper.pdf See PDF])</ref>		<ref name="transport">J. A. Nairn, "Coupling Transport Equations to Mechanics in the Material Point Method Using an Approximate Full Capacity Matrix Inverse," ''Computer Methods in Applied Mechanics and Engineering'', '''420''', 116757 (2024). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/TransportPaper.pdf See PDF])</ref>

	</references>		</references>

MPM Grid Generation

Nairnj — Fri, 10 Apr 2026 17:28:26 GMT

Cell Sizes

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	<ul>		<ul>
	<li><tt>ratio</tt> gives ratio of adjacent cells in that direction outside AOIs. If x direction is not given, it is set to the golden ratio ((1+√5)/2=1.618). If y or z ~~directions~~ are not provided, they are set to the x direction ratio. Any ratio less than or equal to 1 is same as not provided.</li>		<li><tt>ratio</tt> gives ratio of adjacent cells in that direction outside AOIs. If x direction <tt>ratio</tt> is not given, it is set to the golden ratio ((1+√5)/2=1.618). If y or z direction <tt>ratio</tt>s are not provided, they are set to the x direction ratio. Any ratio less than or equal to 1 is same as not provided.</li>
	<li><tt>style</tt> has two options:		<li><tt>style</tt> has two options:
	<ol>		<ol>

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	* [[MPM Methods and Simulation Timing\|Several options for particle update methods]]		* [[MPM Methods and Simulation Timing\|Several options for particle update methods]]
	* [[MPM Methods and Simulation Timing\|Several types of shape function including spline-based shaped functions]]		* [[MPM Methods and Simulation Timing\|Several types of shape function including spline-based shaped functions]]
	* ~~A extended PIC methods~~ known as [[~~XPIC~~ Features\|~~XPIC~~]]		* Can implement method that approximates the inverse of the full mass matrix known as [[FMPM Features\|FMPM(k)]]
	* [[Material Models\|Many material models]] including elastic, plastic, isotropic, anisotropic, viscoelastic, small strain, and large strain.		* [[Material Models\|Many material models]] including elastic, plastic, isotropic, anisotropic, viscoelastic, small strain, and large strain.
	* Plasticity materials can use a variety of [[Hardening Laws\|hardening laws]].		* Plasticity materials can use a variety of [[Hardening Laws\|hardening laws]].