the old well 3/15/2003 ams 2003 spring southeastern sectional meeting 1 an adaptive method for...

3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting1

The Old Well

An adaptive method for coupled continuum-molecular simulation of crack propagation in solids

Sorin [email protected]

http://www.amath.unc.edu/Faculty/mitranThe University of North Carolina at

Chapel Hill

Applied Mathematics Programhttp://www.amath.unc.edu

Incipient failure Rupture

Plots of density of broken bonds


The Old Well

Overview

Computational description of failure Dynamic computation of constitutive relations Fully adaptive algorithm

Salient Features Macroscopic continuum simulated using wave propagation algorithm Local elastic speeds determined by microscopic averaging Thermal motions identified and eliminated using principal component analysis Microscopic dynamics constrained to follow continuum wave modes

Background

Elastic materials fail due to breakdown of microscopic bonds Failure theories are empiric: ► need calibration ► work in calibration region First principles derivation of failure: ► predictive ► major analytical challenge


The Old Well

Related Research

Quasicontinuum analysis – Tadmor, Ortiz & Phillips (Phil. Mag. A73:1529, 1996) ► static ► finite element description of continuum strain ► microscopic deformation given by finite element form functions ► no thermal effects Atomistic-continuum method – E & Huang (JCP 182:234, 2002) ► separate microscopic, continuum domains ► match microscopic phonon modes to macroscopic waves on atomistic-continuum boundaries ► finite difference method for elasticity equations ► no thermal effects Large-scale molecular dynamics – Zhou et al. (Phys. Rev. Lett. 78:479, 1997) ► accurate ► expensive ► useful to determine fracture mechanisms rather than as a practical tool


The Old Well

One-dimensional model of ductile failure in a rod

Model features

Progressive damage Bonds break due to combined thermal, mechanical effect

},...,1,0{

,)(

Nn

nnk

Point masses connected by multiple springs

m mcba x

)(xF

Force-deformation law


The Old Well

Equations of motion

Displacement from equilibrium:

02

0)2(

211

11

l

uuuklu

uuukum

iiii

iiii

/,0 22 lkcucu xxtt

0)()( 12/112/1 iiiiiii uunuunum

0)(2 xxtt uxcu

Continuum limit

With damage

iu

Mass per lattice spacing:

lm /

No damage:

Continuum limit

/)()(2 xnxc

Presence of damage requires microscopic information, i.e. the number of broken springs

iu 1iu1iu

2/12/1 ii nk 2/12/1 ii nk

Zero temperature limit


The Old Well

Two-dimensional model of thin shell failure

cba

0)(

)(

)(

)(

,2/11,2/12/1,2/1

1,2/1,2/12/1,2/1

2/1,2/1,12/1,2/1

2/1,12/1,2/1,2/12/1,

jijiji

jijiji

jijiji

jijijiji

uun

uun

uun

uunum

i

2D lattice of oscillators

j

No damage

0)/(

0)/(

2

2

ijlij

ijlij

vkv

uku

0

022

22

vcv

ucu

tt

ttContinuum limit

With damage

0),(

0),(2

2

vyxcv

uyxcu

tt

ttContinuum limit


The Old Well

Failure scenarios

Bonds break under dynamic loading due to combined thermal (microscopic) and continuum motion

cba

i

Dynamic loading

j

cba

i

Melting

j


The Old Well

Eliminating thermal behavior

Microscopic dynamics

Principal component analysis

0)2( 11 iiii uuukum contains all system information Very little of the information is relevant macroscopically Coarse graining approaches:

-1

-0.5

0

0.5

1

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Principal modes – 32 point masses

Cutoff after three decadesN

Nji

P

n

jnji

ni

ni

nii

rrC

P

uuuuC

Pntuutu

......

1

))((

,...,1)},({)(

21

,11

► Spatial averaging - homogenization

► Fourier mode elimination - RNG


The Old Well

Wave Propagation Method

Used for continuum computation LeVeque (JCP 114, 1997) Method with physical interpretation in terms of eigenmode propagation

0)()( yxt qgqfq

wM

m

mm

nijxxqx

nij

nij

nnij

rQrArAqqfqf

x

t

QQ

dydxtyxqyx

Q

jiji

ijij

FF

qAqAxt

1

correctionorder -Second

Godunovorder First

1

,,)(

),,(1

,,1

n

jiQ ,1n

jiQ ,

x

t

nji

nji

nij QQQ ,1,

2W 1W

1,njiQ1

,1n

jiQ

Local Riemann problem update strategy


The Old Well

Elasticity Eigenmodes

Transform continuum wave equation to a system of first-order PDE’s

Ttyx

tt

srfqfusuru

uyxcu

,,,

0)),((

0

,

100

011

0

,

0

0

,

001

000

00

,

000

001

00 22

c

ccc

R

scrc

c

B

c

A

BqAqq

AA

yx

yxt


The Old Well

Adaptive Computation

For continuum levels Trial step on coarse grid determines placement of finer grids Boundary conditions for finer grids from space-time interpolation

Time subcycling: more time steps (of smaller increments) are taken on fine grids Finer grid values are obtained by interpolation from coarser grid values Coarser grid values are updated by averaging over embedded fine grids Conservation ensured at coarse-fine interfaces (conservative fixups)

t

ionInterpolat

Inject

Averaging

Restrict

t

2/t 2/t

4/t 4/t 4/t 4/t


The Old Well

Direct Simulation Monte Carlo

Full microscopic computation too expensive We need microscopic data to evaluate elastic speed and local damage Use a Monte Carlo simulation to sample configuration space

},1|),,,{( Mjivuvu nij

nij

nij

nij

Unbiased Monte Carlo simulation requires extensive sampling – too expensive

► hierarchical Monte Carlo

► bias sampling in accordance with principal components from immediately coarser level

-1

-0.5

0

0.5

1

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32


The Old Well

Continuum-Microscopic Interaction

Continuum to microscopic injection of values

frequency low

frequencyhigh ,

ij

ijijijij

u

uuuu

low frequency contribution from principal components of coarser grid level high frequency contribution from Maxwell-Boltzmann distribution of unresolvable coarser grid modes

Microscopic to continuum restriction:

► Subtract contribution from thermal and minor modes

► The energy of these modes defines a “temperature” valid for current grid level

► Transport coefficient from standard statistical mechanics

TyxTt ),(

modes thermal- modes elastic -

modesminor - modes principal -

,

,..., 21

TE

MP

TEPMPR

rrrRrCr N


The Old Well

Instantaneous Constitutive Relations

For simple model considered here only constitutive relation is the dependence of elastic speed upon local damage

dydxyxn

yxyxc ),(

/),(2

Update of local damage:

► on each level

► after each time step

► check if displacement has increased beyond current elastic law restriction

cba x

)(xF

Force-deformation law


The Old Well

Two-dimensional example: Rupturing membrane

Simulation parameters 50 initial molecular bonds n<5 ruptured bonds reform initial Gaussian deformation along x direction with amplitude umax for x<0.2 chosen so initial umax does not cause rupture for x>0.2 chosen so initial umax causes rupture zero-displacement boundary conditions adiabatic boundary conditions initial 32x32 grid 6 refinement levels (3 visualized) refinement ratios:

[ 2 2 2 | 8 8 8 ]

Continuum DSMC + PCA 16.7 million atoms

Animation of density of ruptured atomic bonds


The Old Well

Conclusions

Algorithm for systematic reduction of degrees of freedom and computation of constitutive properties Microscopic-continuum interaction Diffusion treatment of thermal motion Local thermodynamic equilibrium instantiation of microscopic states

Open Issues Choice of cutoffs Geometric bias from initial grid More realistic microscopic model Statistical convergence of DSMC method

the old well 3/15/2003 ams 2003 spring southeastern sectional meeting 1 an adaptive method for...

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