Locking Phenomena in the Material Point Method

Locking Phenomena in the Locking Phenomena in the Locking Phenomena in the Locking Phenomena in the Material Point MethodMaterial Point MethodMaterial Point MethodMaterial Point Method

Peter Mackenzie-Helnwein,

Pedro Arduino

Civil and Environmental Engineering

University of Washington, Seattle

August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, [email protected]

6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM

Overview

Introduction

Sources of Locking

Low order shape functions

Convection of a stressed body

Problem Resolution Approach

Volumetric Locking

Shear Locking

Summary and Conclusions

2

Introduction

Last year’s workshop:

Modeling fluid as nearly incompressible viscous

material



Burst of water into a glass

3



The Locking Problem

Proposed solution for locking

Kinematic constraints on the

background grid relaxed by

smoothened volume change

∑

∑

∈

∈=⇒

CVp

pp

CVp

ppp

m

m

ρ

ρ

θ/

/tr ε

θ

θρ

∂

∂==⇒

)(:

Upp CV

Volumetric

locking

intrinsic

to the MPM

4



Anti-Locking Strategy

Volumetric

locking

resolved !

Standard MPM

Modified MPM

5

Sources of Locking

Shape change >> “Volumetric Locking”



0div >u

0div <u

0)(div2

1 2 >Ω= ∫Ω

dkE u

Low-order kinematics

Excess energy

6

0div >u

0div <uVolumetric Locking – cont’d



0)div( =−∫CV

dmwuθ

θρkp =⇒

∫∫=⇒CVCV

dmdmudivθ

02

1div

2

1 2 →== Ω∫ mkdmpECV

θu

7

Sources of Locking

Vibrating beam: initial velocity proportional to 1st mode shape

(stress-free)



Period MPM = 0.92 T (~7% error)

8

Sources of Locking

Bending deformation >> “Shear Locking”



φγ sst =

φε ts −=

Low-order kinematics

s

t

bending

h EEν

ν

+

+=

1

5.1Excess energy

9

Sources of Locking

Convection of a stressed body in quasi-static

equilibrium

Known as “cell crossing error”



)(1

2211 mmh

fa σσ += )(1

2211 mmh

fb σσ +−=

10

Sources of Locking


equilibrium

Known as “cell crossing error”



0)(2

2211 ≠+−= mmh

fc σσThe black sheep

11

Sources of Locking


equilibrium



0)(2

2211 ≠+−= mmh

fc σσ 0thatsuchfind =∆⇒ cf

2

21

21

21

211

2σσσ

mm

m

mm

mm

+−

+

+→ 2

21

211

21

12

2σσσ

mm

mm

mm

m

+

++

+−→

12



equilibrium



πξξ cos)( 0bb =

LξLtvx ξ+= 0O

( )1cos)(2

2

00 −+= πξ

πξ

EA

Lbtvu

πξπ

ξσ sin)( 0

A

Lb−=

Analytic solution :

13


Convection of α L in 1000 steps / L (5 cells/L)



Standard MPM

14


Convection of α L in 1000 steps / L (5 cells/L)



MPM + cell averaging

15


2D/3D >> Filtering of strain/stress fields



∫∫∫∫ →−+⋅−⋅−∂ m

hhh

Vmm

h stationarydmdSdmdm ))((:)( εuεσχtχbε

σ

ψ

β

S

σ

::

00100

0010

0001

:

5

4

3

2

1

=

=

→

=

β

β

β

β

β

σ

σ

σ

444 3444 21

s

t

st

tt

ss

h

αSε ][

2

: →

=

st

tt

ss

h

ε

ε

ε

16


2D/3D >> Filtering of strain/stress fields



∑∈

=cp

pp

t

pc m][][:][ SSH

∑∈

=cp

pp

t

pc m][: σSR

][1

ccc RHβ−=⇒

For each cell c

For each particle p in cell c

⇒→ ][ cpp βSσ

][1

pp σCε −=⇒

][ cpp αSε →

elastic

plastic

17



Adding Smoothing to the Basic Algorithm

particle

cell

node

18

The Ultimate Challenge



ω

))(3(8

222

rarr −+= νρω

σ

( )222

)31()3(8

ra ννρω

σθθ +−+=

0=θσ r

ωθ ru

ur

=

= 0

Spinning Disk

Initial conditions

19

The Ultimate Challenge: Spinning Disk



CPDI+stress smoothing

MPM+stress smoothing

CPDI

Standard MPM

?.constrr =σ

After almost one full revolution

TARGET

20

The Ultimate Challenge: Spinning Disk



CPDI+stress smoothing

MPM+stress smoothing

CPDI

Standard MPM

?0=θσ r

After almost one full revolution

TARGET

21

Summary & Conclusions We identified that, thanks to using identical shape

functions, MPM inherited locking from the FEM.

We interpret the “cell crossing error” as “convection

locking”.

We propose a cell-based smoothing strategy that

filters/removes locking stresses/strains from the

numerical solution.

The proposed technique improves solutions for both

standard MPM and CPDI but does not completely

eliminate the problem (yet) .


6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM 22


6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM 23

Locking Phenomena in the Material Point Method

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