locking phenomena in the material point method
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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, pmackenz@uw.edu
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
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
Burst of water into a glass
3
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
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
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
Anti-Locking Strategy
Volumetric
locking
resolved !
Standard MPM
Modified MPM
5
Sources of Locking
Shape change >> “Volumetric Locking”
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
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
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
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)
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
Period MPM = 0.92 T (~7% error)
8
Sources of Locking
Bending deformation >> “Shear Locking”
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
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”
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
)(1
2211 mmh
fa σσ += )(1
2211 mmh
fb σσ +−=
10
Sources of Locking
Convection of a stressed body in quasi-static
equilibrium
Known as “cell crossing error”
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
0)(2
2211 ≠+−= mmh
fc σσThe black sheep
11
Sources of Locking
Convection of a stressed body in quasi-static
equilibrium
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
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
Problem Resolution Approach
Convection of a stressed body in quasi-static
equilibrium
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
πξξ cos)( 0bb =
LξLtvx ξ+= 0O
( )1cos)(2
2
00 −+= πξ
πξ
EA
Lbtvu
πξπ
ξσ sin)( 0
A
Lb−=
Analytic solution :
13
Problem Resolution Approach
Convection of α L in 1000 steps / L (5 cells/L)
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
Standard MPM
14
Problem Resolution Approach
Convection of α L in 1000 steps / L (5 cells/L)
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
MPM + cell averaging
15
Problem Resolution Approach
2D/3D >> Filtering of strain/stress fields
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
∫∫∫∫ →−+⋅−⋅−∂ 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
Problem Resolution Approach
2D/3D >> Filtering of strain/stress fields
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
∑∈
=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
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
Adding Smoothing to the Basic Algorithm
particle
cell
node
18
The Ultimate Challenge
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
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
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
CPDI+stress smoothing
MPM+stress smoothing
CPDI
Standard MPM
?.constrr =σ
After almost one full revolution
TARGET
20
The Ultimate Challenge: Spinning Disk
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM
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) .
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM 22
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, pmackenz@uw.edu
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM 23
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