Simulation in MaterialsSummary

Friday, 12/6/2002

MATLAB programmingVisualization:

Stress matrix visualizationStress field visualizationColor expression

Simulation methods:Atomistic simulation

Brownian movementMolecular dynamics (MD)Monte Carlo method (MC)

Continuum SimulationMaterial Point Method (MPM)Finite Element Method (FEM)

Visualization

Stress field visualizationhole under stretchingcrack tip

Stress matrix visualizationhedgehog for 2D stress matrixbean-bag for 3D stress matrix

Color expressiondisplacement distribution in FEM

Stress Distribution Visualization

Crack tip stress distribution

Stress distribution around a hole

Hedgehog Method

€

σ xx σxy

σ yx σyy

⎡ ⎣ ⎢

⎤ ⎦ ⎥ =

1 22 1

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Bean-Bag Method

€

σ11 σ12 σ13

σ21 σ 22 σ 23

σ31 σ 32 σ 33

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ =

1 2 32 2 −13 −1 1

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Visualization of FEM Results

Displacementfield

Pixel:The smallest image-forming unit of a video display.

Atomistic Simulation

Brownian movementMolecular dynamics (MD)Monte Carlo method (MC)

Extension of Random WalkThis model is a two-dimensional extension of a random walk. Displayed is the territory covered by 500 random walkers. As the number of walkers increases the resulting interface becomes more smooth.

Extension of particles from one room to two rooms

Monte Carlo Method1. Current configuration: C(n)

2. Generate a trial configuration by selecting an atom at random and move it.

3. Calculate the change in energy for the trial configuration, U.

Essence of MD

€

ax(i ) =Fx

(i ) +fx(i )

m(i)

ay(i ) =Fy

(i ) +fy(i )

m(i)

€

fx(i ) = fx

(i, j )

j≠i∑

fy(i ) = fy

(i, j )

j≠i∑

Internal forces External forces

€

Fx(i)

Fy(i)

Continuum Simulation

Material Point Method (MPM)Finite Element Method (FEM)

MPM

€

a x(n) = 1M (n) F x(n) + f x(n)( )

a y(n) = 1M (n) F y(n) + f y(n)( )

⎧ ⎨ ⎪

⎩ ⎪

€

v x(n) = 1

M (n) m( p)vx( p)N(n,p)

p∑

v y(n) = 1M (n) m( p)vy

( p)N(n,p)

p∑

⎧

⎨ ⎪

⎩ ⎪

FEM

€

Tx(n) +Fx

(n) = Kxx(n,n')ux

(n') +Kxy(n,n')uy

(n')( )n'∑

Ty(n) +Fy

(n) = Kyx(n,n')ux

(n') +Kyy(n,n')uy

(n')( )n'∑