locking phenomena in the material point method
TRANSCRIPT
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
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
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, [email protected]
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM 22
August 9-10, 2010Peter Mackenzie-Helnwein, University of Washington, [email protected]
6th MPM Workshop @ U of New Mexico: Locking Phenomena in MPM 23