Error Analysis for Material Point Method and a case study from Gas Dynamics

Le-Thuy Tran and Martin Berzins

Thanks to DOE for funding from 1997-2008

Material Point Method (MPM) Outline History See Brackbill IJNMF 2005 47 693-705

•1963 Harlow PIC methods then CiC ViC methods•1980s -90s Flip methods Brackbill et al.•1990s Sulsky Brackbill MPM-PIC•2000+ Sulsky et al. + GIMP Bardenhagen et al

Since then proved effective on difficult problems involving large deformations fracture e.g CSAFE [Guilkey et al.] + [Sulsky et al ] + [Brydon] etc etc

A KEY ISSUE NOW IS THE THEORETICAL UNDERPINNINGOF THE METHOD.

1

2

3

4

5

The Material Point Method (MPM)

6

Handles deformation, contact, high strain

Particles with properties(velocity, mass etc) defined on a mesh

Particle properties mapped onto mesh points

Forces, accelerations, velocitiescalculated on mesh points

Mesh point motion calculatedbut only the particles movedby mapping velocities back toparticles

Errors at each stage of this process

Explicit MPM Algorithm Errors• Particle to grid projection Projection Error

• Interpolation from grid back to particles Interpolation Error

• Time integration of particles Time Error

• Finite element + mass matrix lumping FE +Mass Error• Analysis of mesh/particle interplay from [Grigorev,Vshivkov and

Fedoruk PIC 2002] - more particles may not be better on a given mesh. Both cell size and particles per cell matter.

1D : 4-8 particles per cell is optimal

….

MPM Basis Overviewgrid basis fns particle basis fns

Or delta fn ..S (x)i

npip ip

i p p p ipnpp =1 p

ip ipip 1

p G 1a V , , ( )Vm S

i

ip

dSG x dxdx

np

i p pp =1

v ,v ip ipp np

ip ipip 1

m S,

m S

ip

p

1S ( ) ( )V i

i pS x x dx

pointVelocity

point mass

ip

h ρm

Ni p

ip

ip

h ρm

Ni p

ip

Forward Euler time integration

1 1i i

1

1 n+1i

1

v v a , v v a ,

v

nvn n n ni i p p ip

i

nvn np p ip

i

t t S

x x t S

Local errors are second order in time

Global error is first order in time

Second derivatives of x and v are involved in error estimates These terms may be discontinuous

n+1Vn.b. semi-implicit

p i-1 i p i-1 i1

1p i i+1 p 1 i 1 i+1 1

1

1i-1p i

1

If [ , ] then (a) (1 ) ,

If [ , ] then (b) (1 ) ,

Hence differentiating gives

(a) (1 )

p ii i i

i i

p ii i i

i i

i ii i p

i

x xx x x x a a

x xx x

x x x x a ax x

a ax x x v

x

1p 1 i 1 i+1

1

1 1

1 1

(b) (1 )

Hence jump in is [ ] ][i

i

i ii i p

i i

i i i ix p

i i i i

xa ax x x vx x

a a a ax x x v

x x x x

Acceleration continuous but its derivative is not Because of linear basis functions and moving points

Limits globalaccuracyTo first order

p i-1 i p i-1 i1

1p i i+1 p 1 i 1 i+1 1

1

1i-1p i

1

If [ , ] then (a) (1 ) ,

If [ , ] then (b) (1 ) ,

Hence differentiating gives

(a) (1 )

p ii i i

i i

p ii i i

i i

i ii i p

i

x xx x x x a a

x xx x

x x x x a ax x

a ax x x vx

1p 1 i 1 i+1

1

1 1

1 1

(b) (1 )

Hence jump in is [ ] ][i

i

i ii i p

i i

i i i ix p

i i i i

xa ax x x vx x

a a a ax x x vx x x x

Limits globalaccuracyTo first order

Particle Acceleration Derivative Potentially Discontinuous as Particle Crosses Boundary

Particle Methods applied to Euler Equations

1 1

1

1 1 1

)

)

(1

( 1

n

p

n

p

pn np p

pn np p

n n np p p

pt

t

ArtVis

vx

vx

p

0 )(1 p tvx

Energy

Density

Pressure

Positive density requires

Note differing particle velocity gradients in different cells may lead to new extrema when particles cross over cells

1 dpadx

Acceleration

Arbitrary point Quadrature Error [Hickernell]

1/22N

2 21

1 1 (2 1)D (P, N )12(N ) N 2N

ip

ip i ii i i

ip p p

i hz x

12

22,

D (P, N )hip

h

dfdx

1 N

1

1 1( ) ( )h N

ii p

i

x

iiipx

f x dx f z

22D (P, N )h ( )i

pdfdx

Where 1/2

22

112(N )i

p

Ch

1

1

( ) ( )i

i p

i

x N

iiipx

hf x dx f zN

1 14 3

C

Mass Projection Error

12 1 2 2D (P, N )h ( ) D (P, N )h ( )i i

p p

1

1

N

1

( ) ( ) S ( )N

ii p

i

xim i p i ipi

ipx

hE S x x dx x

First order in space for small mesh sizes, and may achieve a second order of convergence for large mesh.

1/2

22 2

1D (P, N )12(N )

ip i

p

Ch

( ) ±1. .h

idS xn bdx

1 14 3

C

2 1 22 1 2 2

( ) ( )D (P, N )h ( ) D (P, N )h ( )i ii ip p

d S d Sdx dx

Momentum projection error1

1

N

1

( ) ( ) ( ) v S ( )N

ii p

i

xiP i p p i ipi

ipx

hE S x x v x dx x

2 1 22 1 2 2

( ) ( )D (P, N )h ( ) D (P, N )h ( )i ii i

p pd S v d S v

dx dx

12 1 2 2D (P, N )h ( )( ) D (P, N )h ( )( )i i

p pv v

1/2

22 2

1D (P, N )12(N )

ip i

p

Ch

Momentum projection error is also first order in space for small mesh sizes, and second order of convergence for large mesh.

Force Projection Error1 1

1 1 1

1 1( ) ( ) ( ) ( )i i i

i i i

x x xproj

i ipi ix x x

F p x G x dx p x dx p x dxh h

1

11

1: :

1 1 1 1( ) ( )i i

p i p ii i

x xiF p pi i

p x I p x Ip px x

E p x dx p p x dx ph N h N

Using Hickernel:

12 1 2 2D (P, N )h ( ) D (P, N )h ( )i i i

F p pdP dPEdx dx

Force Projection Error is first order in space for small mesh sizes, and may achieve up to second order of convergence for large meshes.

Force Projection Error(cont.)

Nodal Acceleration Projection Error1

1

1

( ) ( )

( ) ( )

i

i

i

i

x

ipxi

i xi

ix

p x G x dxFam

x S x dx

i ii i

i i m mF Fa i

i i i i i

F E EE EE am m m m m

Acceleration projection error is one order less in space than the order of convergence for Force projection error and mass projection error, since mass is first order in space.

Nodal Velocity projection (particle to grid) error1

1

1

( ) ( ) ( )

( ) ( )

i

i

i

i

x

ipxi

i xi

ipx

x v x S x dxP

vm

x S x dx

iii mPv i

i i

EEE vm m

Let:1

i exactv i iE v v

1

i iii exact im mPv i v

i i i

E EEE v Em m m

1ivE is first order in space so the order of velocity

projection error is determined by ii

exact mPi

i i

EE vm m

Nodal velocity projection error(cont)

So velocity projection error has constant order as the mesh size get smaller.

The velocity error for different mesh sizes1ivE

Velocity gradient projection error2

1 12

1

( ) ( ) . .2

i i i ip p p

i i

v v x xv vx x x H O Tx x x dx

1 2

* * 12 ( )

2

i ip v v i i

VG p pE E x x vE x x

h x

1* *

i ii v vVG

E EEh

*

i i iv v aE E tE

11

i ii i iv vVG a a

E E tE E Eh h

1i ii v vVG

E EE

h

Velocity gradient projection error(cont)Velocity projection error at nodes for h:

2iv h

E 1iv h

E iv h

E

Velocity projection error at nodes for 2h:

1

2

iv h

E 2

iv h

E

With:i pvE Ch ,We have:

2

2i p iv vh h

E E

1 2

22i p i

v vh hE E

Velocity gradient projection error

2

1 2

22 2 22

i iv vi h hi i i i

vv v v vhp p ph h h h

E EEE E E E

h h h

1 1

2 222

i i i iv v v vp h h h h

E E E E

h h

Therefore:

So velocity gradient projection error is the same order as velocity projection error.

Final Error Combination

Region 1 and Region 3 capture the error at left and right rarefaction

Region 4 captures the error at contact discontinuity Region 6 captures the error at shock-frontRegion 2 is the region of smooth solution of density and velocityRegion 5 contains the error propagation from the shock-front

Final Error Combination(cont.) h L1-Norm L2-Norm0.01 0.00830833 0.0158701

0.005 0.00433578 0.0104553

0.0025 0.00230507 0.0075895

0.00125 0.00126351 0.0057596

0.000625 0.00109963 0.0061870

0.0003125 0.00101256 0.0062115

Burgers’ Problem2

2( , )v v vv x tt x x

( , ) [0,2] [0.0, )x t T

Solving with MPM:

i ip pp

v vj

ip pip j

ip pp

S NS N

1

1 12 2 1

: :

1 1p i p i

i i i ii i i

p x I p x Ip p

v v v va

h N h N

Follow the consequence steps of standard MPM method.

Burgers’ Problem(cont.)

h=0.01 h=0.005 h=0.0025 h=0.00125 h=0.000625

UpdateVelocity UsingExactAcceleration

ε=0.045 2.67e-4 1.24e-4 6.07e-5 3.03e-5 1.52e-5

ε=0.04 2.67e-4 1.25e-4 6.09e-4 3.06e-5 1.53e-5

ε=0.03 2.78e-4 1.27e-4 6.19e-4 3.09e-5 1.55e-5

ε=0.02 2.97e-4 1.24e-4 6.16e-4 3.10e-5 1.55e-5

ε=0.01 5.34e-4 1.58e-4 6.18e-4 3.10e-5 1.55e-5

UpdateVelocity UsingCalculatedAcceleration

ε=0.045 9.82e-4 5.39e-4 2.84e-4 1.42e-4 7.09e-5

ε=0.04 8.54e-4 4.61e-4 2.35e-4 1.17e-4 5.86e-5

ε=0.03 5.97e-4 3.20e-4 1.61e-4 8.05e-4 4.03e-4

ε=0.02 3.73e-4 1.93e-4 9.91e-5 4.95e-4 2.47e-4

ε=0.01 4.12e-4 1.51e-4 6.52e-5 3.20e-4 1.60e-5

Error at T=0.5 for different mesh sizes:

ConclusionsNodal mass, momentum and force projection error converges with order

Nodal acceleration projection error is order

(1 2)ph p

(0 1)ph p

Nodal velocity projection error is order (0 1)ph p

Nodal velocity gradient projection error is the same order with velocity projection error

The overall error at particles as density, energy… will stop converging as the mesh size get smaller.