Transfer coefficient algorithm for small mass nodes in material point method

Xia Ma, Balaji Jayaraman, Paul T. Giguere and Duan Z. Zhang

CartaBlanca TeamTheoretical Division

Fluid Dynamics and Solid Mechanics GroupT-3, B216,

Los Alamos National Laboratory

MPM on material interface

• There have been ample applications for MPM in solid mechanics.

• MPM has been regarded as a method to handle large material deformations.

• Recently we have extended it for multi-material interactions. The formulation is similar, or an extension, to multiphase flow theory.

• The numerical method to handle material interface becomes an important issue.

MPM Random Instability • We use one-D simulation to demonstrate the small mass instability

associated with MPM method• Sometimes we observe instability in our MPM simulation and the instability is

elusive. So we set-up a one-D case in a special way that the instability is not random with our control and we are for sure it is due to small mass nodes.

The last particle of the bar is placed very close to node 41 deliberately.MPM without DCA is un-stable for dt=3.2E-9s and become stable after lower the dt to 3E-9s which is an order of magnitude smaller then the CFL requirement.

Oscillation at the end of barElastic bar

Distribution Coefficients Formulation for Small Mass Nodes(1)

• weak form of momentum equation (node based discretization, number of freedom is based on number of grid points, not on MP)

1 1

( , ) ( )N N

ij i ij jj j

dm S m S dS

dt

jj

us G s n x

Small Small( )iS x

Not small Not small iS

1 1

( )pNN

i ij i p i pj p

m m S dv m S

x1

( , ) ( )pN

i p p i pp

S v S

s s x

For node i=42, Mass is smallFoce is not small

The acceleration (dividing force by mass) is computed artificially big for node 42 and cause instability for MPM method. This instability happens usually in regions with large gradient of the volume faction which is typical near material interface.

Node quantities are obtained from MP

Distribution Coefficients Formulation for Small Mass Nodes(2)

• Introduce distribution coefficient to transfer a part of the force away from the small mass nodes to its neighbor nodes where the mass is not small in such a way that it conforms with the order of accuracy of original MPM.

• In the region of large gradient of volume fraction , the error in the MPM is first order in space and this force is transferred within this order of error.

• Should satisfy conservation of mass, momentum, and energy, which put some requirement on

• Distribution coefficient , if l is not a small mass nodes to conserve second order in interior

li liC

( 0)liC

1 1

( , ) ( )N N

i li l ij jl j

C S m S dS

jf s G s n x

( 0)liC

Conservation laws

• Mass conservation• Momentum conservation

requires that be a constant. Combining with condition for the nodes with sufficient mass, we find .

• Energy conservation

to move particle by within an error of order for energy , we need to weight the node velocity by in calculating the stress terms.

1

1 1 1 1 1

,( )p pN N N N N

n n np p p p j j l li

p p j l i

m m m G t S C t

u u s

1

N

lii

C

li liC

1

1N

lii

C

1 1/2

1 1 1 1

1/2

1 1 1

: ( ) ( ) ( )2

( ) ( ), ,2

p pN N L nN Nn n n n n ni ip p p p p p j j p i p

p p j i

L nN N Nn n n L L n nl l

p l p l li i l li il i i

K K v t m S S t

S C C

u uσ u x b x x

u uu x x u u u u

1 1/2n n np p px x u t 2( )t

liC

Time scales affected by distribution coefficients• Let’s consider 1-D elastic bar in a cell with one material point

at between nodes 0 and 1 . There is no material point beyond node 1. The momentum equation for node 1 is

The solution has a time period If is small, the time period is also small, in turn a high

intrinsic frequency. A small time step is thus needed for the original MPM to prevent a numerical instability.

• With DCA, momentum equation is modified as the time period becomes , if we choose independent of . This suggest a general

form of DCA as , where j is neighbor of l .( , )

/li i jj lC m m

21

1 12

d Emdt x

012 / 2 /pm x E x x E

px

221

1 11 12

d Em Cdt x

21 112 / ( )m x C E 11 1 / pC m m

2 / 2 /pm x E x c px

x

node 0 1

l

j

j j

j

j

px

Numerical examples (1)

Projectile-target interaction

Tail positionsTungsten projectile

Cylinder ofArmor steel

This is a bench mark case for which there are available experimental data. Here we computed again with and without DCA to see the effect of the algorithm for small mass nodes for CartaBlanca.

Air (Large density ratio)

Nose positionTime direction

Numerical examples (2)

dt<1E-20 and exit

An order of magnitude small than CFL requirement

DCA: 25 min

No DCA: 6 h 40min

Although DCA did not increase the accuracy, it is more robust and use less time

Overall the difference is small

Numerical examples (3)

Analytic period is 29.27µs DCA with Δt=0.01µs, 29.28µs without DCA unstable for this Δtreduce Δt to 0.001µs, 29.28ms

To ensure the algorithm produces accurate results in cases with large ratio of boundary nodes to internal nodes, we no consider the vibration of a thin elastic shell (50m/s)

Small mass node are often near material boundary or interfaces

Total energy

Kinetic energy

Numerical examples (4)

• We studied the instability associated with MPM method in detail for one-D

At the trough, last particle moves closer to node 41, reducing mass at 42 further, causing instability.MPM without DCA requires a time step an order of magnitude smaller than that based on CFL=0.75 for wave and material speed, which is dt=3E-8.

Conclusions

• Small mass nodes often appear near material interfaces with large deformation of materials.

• Such a small mass nodes often leads to tiny time steps. Often it results in instability and failure of numerical simulation.

• The algorithm presented here satisfies mass and momentum conservation exactly. The energy conservation error is second order on time and in spatial discretization.

• Numerical examples show the algorithm is very effective and can significantly increase the allowed size of time steps and make the simulation robust.