simulation in materials summary
DESCRIPTION
Simulation in Materials Summary. Friday, 12/6/2002. MATLAB programming. Visualization: Stress matrix visualization Stress field visualization Color expression Simulation methods: Atomistic simulation Brownian movement Molecular dynamics (MD) Monte Carlo method (MC) - PowerPoint PPT PresentationTRANSCRIPT
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
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σ xx σxy
σ yx σyy
⎡ ⎣ ⎢
⎤ ⎦ ⎥ =
1 22 1
⎡ ⎣ ⎢
⎤ ⎦ ⎥
Bean-Bag Method
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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
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ax(i ) =Fx
(i ) +fx(i )
m(i)
ay(i ) =Fy
(i ) +fy(i )
m(i)
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fx(i ) = fx
(i, j )
j≠i∑
fy(i ) = fy
(i, j )
j≠i∑
Internal forces External forces
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Fx(i)
Fy(i)
Continuum Simulation
Material Point Method (MPM)Finite Element Method (FEM)
MPM
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a x(n) = 1M (n) F x(n) + f x(n)( )
a y(n) = 1M (n) F y(n) + f y(n)( )
⎧ ⎨ ⎪
⎩ ⎪
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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
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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'∑