The Material Point Method and its Uses in SCI

Institute Related Research Projects

Jim Guilkey

With thanks to a whole lot of folks…

Outline

• Brief overview of the MPM algorithm

• SCI Related projects using MPM

• Algorithmic challenges

• Ongoing development

The Basics

• MPM is a “quasi-meshless” particle method

• Governing equations from the weak form

• Basic algorithm consist of three main parts:•Project particle data to a grid, e.g.:

•Solve equations of motion on the grid

•Update particles’ state based on changes of the nodal values

mi =Sipmp wi =Sipmpup ui =wi

mi

1. Discretize geometry with particles and define

an overlying computational grid. Particles

carry all state data (mass, vel, temp., etc.)

4. Particle positions/velocities updated incrementally from mesh solution.5. Discard deformed mesh.

Define new mesh and repeat

1

4

The Algorithm

2. Project particle state to nodes. Stress at particles computed based on gradient of the mesh velocity.

3. Divergence of particle stress gives an internal force on the nodes. Acceleration computed at nodes and integrated, giving updated mesh velocity and (in principal) position

5

2

3ai = finti + fexti( ) mi

MPM Features (the good ones)

• Discretization of complex

geometry is trivial

• Resetting the grid prevents

distortion issues

• Parallelization is

straightforward

• Contact is simple to implement

and fast• Particles interact via the grid, so cost is roughly linear in the number of particles

Example CalculationsFoam

Part-Tool Interaction

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MPM in (or near) the SCI Institute

• C-SAFE (Parker, Berzins, Kirby,

et al.)

• Angiogenesis (Weiss)

• Virtual Soldier (Berzins, Kirby,

Johnson)

• Visualization•Ray Tracing (Parker, Bigler)•GPU Based Rendering

(Parker, Gribble, Stephenson)

C-SAFE Target Scenario:

MPM Enhancements

• Incorporate MPM into a multi-material, compressible, finite volume CFD code

• One mesh for all material phases

• No surface tracking• Tightly coupled through a common pressure field and interaction terms in the governing equations

Common Reference Frame• ICE is a cell centered finite volume method• MPM uses particles and nodesTo establish a common frame of reference on one grid,solid phase data is projected to cell centers

Particle:Mass, volume,Temperature,Velocity, etc.

Node Centered:Mass, volume,Temperature,Velocity, etc.

Cell Centered:Density,Internal Energy,Momentum, etc.

Results

2003

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Results 2005

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Span timescales separating calculation into three phase

• Run fire to “steady state”

• Use SS heat flux to compute heat conduction and thermal expansion in container

• Switch to full capability for explosion

Results 2005

Span length scaleswith AMR

• Run fire on a coarse level(s)

• Container at a fine level

• Still a ways to go here

Need to use both Strategies by Fall ‘06

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Angiogenesis and The Extracellular

MatrixDay 0

Day 4

Day 5

AngiogenesisMechanical Conditioning

By comparing the in vitro sprouting locations withthe computed stress field for a particular specimen, a

correlation between these phenomena can be found

Actuating Post

Fixed Anchor

Tensile Test AreaMedia

AngiogenesisImage Acquisition

via Confocal Microscopy

Day 8One slice of a 3D stackRed = smooth muscle cellsGreen = endothelial cells

Image Processing

(Note: These two images are from different datasets)

AngiogenesisGeometric Representation

4.2 �m grid cells4 4 1ｴ ｴ

2.1 �m grid cells2 2 1ｴ ｴ

8.4 �m grid cells8 8 1ｴ ｴ

One particle is generated for each voxel. Material type (vesselor collagen) is chosen based on intensity level.

For particle-based (aka “meshless” methods) creation of a suitablerepresentation is simple

4.2m grid cells4x4x1 particles

2.1m grid cells2x2x1 particles

AngiogenesisGeometric Representation

52 slices512 X 512 voxels each

525 m X 525 m X 52 m

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AngiogenesisMPM Enhancements

• Implicit Time Integration• Larger Timesteps• Quasi-static loading

• Use of Fully Lagrangian Mode

4 5

AngiogenesisWhy Fully Lagrangian?

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AngiogenesisResults

4.2 m grid, 2x2x1Original Algorithm

8.1 m grid, 8x8x1Fully Lagrangian

2.1 m grid, 2x2x1Fully Lagrangian

Von Mises Stress

AngiogenesisResults

Virtual SoldierWe need a medic!

• Simulations of projectile interactions with human tissue

• MPM Enhancements include:• Transversely isotropic material models

• Material failure models based on strain

Virtual SoldierMaterial Modeling

Ground substance

x

y

z Direction of material symmetry

Fiber family

• Soft tissue was represented as transversely isotropic hyperelastic9

• Composite comprised of a ground substance reinforced by a single fiber family

9 Weiss, et al., CMAME, 1996

Virtual SoldierMaterial Modeling

• Modes of failure considered:

Matrix failure

Fiber failure

Total failure: both matrix and fibers have failed

Maximum Shear Criterion

Maximum Tensile Strain Criterion

> 50%

> 40%

σ fiber

σ volumetric σ matrix

Virtual SoldierMaterial Modeling

• Matrix and fibers present different failure mechanisms

• A strain-based failure model was developed

• Decoupled stress:

Matrix failure

Fiber failure

σ =σ volumetric + σ matrix + σ fiber

Virtual SoldierMaterial Model

Validation

1) Shear to failure experiment • Square sheet of tissue and rigid clamps • Explicit versus implicit time integration

2) Tensile to failure experiment• Dog bone specimen• Grid reset vs. no grid reset results

Tissue

Sym

met

ry p

lan

e

Tissue

Clamps

Virtual SoldierMaterial Model

Validation• Shear to failure

= matrix failure= fiber failure= total failure

Maximum shear strain

0.0 0.62

Explicit time integration Implicit time integration

Failed particles

Virtual SoldierMaterial Model

Validation• Tensile to failure

= matrix failure= fiber failure= total failure

Maximum fiber strain

0.0 0.4

Failed particles

Grid resetting case

Deforming grid case

Virtual SoldierMaterial Model Testing

• Myocardial material slab penetration experiment • Assessed appearance of wound tract• Influence of grid and particle resolution, contact formulation

Projectile

Myocardial slabFiber orientation*

*180o rotation from epicardial to endocardial surface

Virtual SoldierMaterial Model Testing

• Frictional contact = 0.08

• Bullet speed 150 m/s

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Virtual Soldier Model “Validation”

(aka Jeff and Mike go a huntin’)

Bullet entry wound

Heart

Virtual Soldier

Path Forward

fat layer

lungsheart

viscera hard tissue

fragment path

Torso Injury:

Particle Visualization

Ray TracingLocal Illumination Global Illumination

Particle Visualization

GPU Based Rendering

Conclusions• MPM offers an alternative to FEM for problems involving:

• Complex geometries

• Contact

• Material Failure

• Large Deformation

• MPM suffers from a number of numerical problems and lacks some of the theoretical underpinnings that support FEM

Validation: Pressurization of an Annulus

Steel Annulus

HighPressure

Gas

LowPressure

Gas

Validation: Gas Compressed by Piston

MPM material Eulerian gas

Validation: Supersonic Moving Wedge

Mach Number = 3.0

Density Contour Plots

Evolution of Heated Container, cont.

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Evolution of Heated Container, cont.

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