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, 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

Introduction

Last year’s workshop:

Modeling fluid as nearly incompressible viscous

material

Burst of water into a glass

The Locking Problem

Proposed solution for locking

Kinematic constraints on the

background grid relaxed by

smoothened volume change

∈=⇒

/tr ε

∂==⇒

Upp CV

Volumetric

locking

intrinsic

to the MPM

Anti-Locking Strategy

Volumetric

locking

resolved !

Standard MPM

Modified MPM

Sources of Locking

Shape change >> “Volumetric Locking”

0div >u

0div <u

0)(div2

1 2 >Ω= ∫Ω

Low-order kinematics

Excess energy

0div >u

0div <uVolumetric Locking – cont’d

0)div( =−∫CV

dmwuθ

θρkp =⇒

∫∫=⇒CVCV

dmdmudivθ

1 2 →== Ω∫ mkdmpECV

Sources of Locking

Vibrating beam: initial velocity proportional to 1st mode shape

(stress-free)

Period MPM = 0.92 T (~7% error)

Sources of Locking

Bending deformation >> “Shear Locking”

φγ sst =

φε ts −=

Low-order kinematics

bending

h EEν

5.1Excess energy

Sources of Locking

Convection of a stressed body in quasi-static

equilibrium

Known as “cell crossing error”

2211 mmh

fa σσ += )(1

2211 mmh

fb σσ +−=

Sources of Locking

equilibrium

Known as “cell crossing error”

2211 ≠+−= mmh

fc σσThe black sheep

Sources of Locking

equilibrium

2211 ≠+−= mmh

fc σσ 0thatsuchfind =∆⇒ cf

2σσσ

+→ 2

2σσσ

+−→

equilibrium

πξξ cos)( 0bb =

LξLtvx ξ+= 0O

( )1cos)(2

00 −+= πξ

πξπ

ξσ sin)( 0

Lb−=

Analytic solution :

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

Standard MPM

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

MPM + cell averaging

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

∫∫∫∫ →−+⋅−⋅−∂ m

h stationarydmdSdmdm ))((:)( εuεσχtχbε

444 3444 21

αSε ][

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

∑∈

pc m][][:][ SSH

∑∈

pc m][: σSR

ccc RHβ−=⇒

For each cell c

For each particle p in cell c

⇒→ ][ cpp βSσ

pp σCε −=⇒

][ cpp αSε →

elastic

plastic

Adding Smoothing to the Basic Algorithm

particle

The Ultimate Challenge

))(3(8

rarr −+= νρω

( )222

)31()3(8

ra ννρω

σθθ +−+=

0=θσ r

ωθ ru

Spinning Disk

Initial conditions

The Ultimate Challenge: Spinning Disk

CPDI+stress smoothing

MPM+stress smoothing

Standard MPM

?.constrr =σ

After almost one full revolution

TARGET

The Ultimate Challenge: Spinning Disk

CPDI+stress smoothing

MPM+stress smoothing

Standard MPM

?0=θσ r

After almost one full revolution

TARGET

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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