Monte-Carlo Simulation of Failure Phenomena using Particle Discretization

7/30/2019 Monte-Carlo Simulation of Failure Phenomena using Particle Discretization

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BACKGROUNDSBACKGROUNDSTwo Models of Deformable Body

zcontinuum deformation expressed in terms of field variables

z rigid-body spring assembly of rigid-bodies connected by spring

Distinct Element Method (DEM)z simple treatment of failure: breakage of spring

z non-rigorous determination of spring constants

deformable body particle modeling DEM analysis

breakage of spring

spring constant need to

be determined in termsof material parameters


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MATHEMATICAL INTERPLETATION OF

RIGID-BODY SPRING MODEL

MATHEMATICAL INTERPLETATION OF

RIGID-BODY SPRING MODEL

Non-Overlapping Functions for Discretization

smooth but overlapping functions are

used for discretization

characteristic functions of domain are

used for discretization

ordinary FEM DEMu1

u2

u1

u2


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PARTICLE DISCRETIZATION FOR

FUNCTION AND DERIVATIVE

PARTICLE DISCRETIZATION FOR

FUNCTION AND DERIVATIVE

Discretization of Functionf(x)

in terms of{

(x)}

Discretization of Derivativef,i(x)

in terms of{

(x)}

= )(f)(fd xx

= V2df ds))(f)(f(})f({E xx

=

Vds

1 dsffV

= )(g)(g i

d

i xx

= fdsgV

i,ds

1i

V

minimize Ef

different from {}

form of discretization form of discretization

optimal coefficients for given Voronoi blocks and Delaunay triangles

derivative of{} is not

bounded but integratable


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1-D PARTICLE DISCRETIZATION1-D PARTICLE DISCRETIZATION

xn xn+1

y n-1

y n y n+1

= )(f)(fd xx

== )(g)(g)(dxdf

xxx

average value taken over interval for

is

coefficient of characteristic function

average slope of interval for

is coefficient of

characteristic function

mother points

middle point of neighboring mother points


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2-D PARTICLE DISCRETIZATION2-D PARTICLE DISCRETIZATION

Voronoi blocksfor function Delaunay trianglesfor derivative

dual domain decomposition

funct ion and derivative are discretized in terms of sets of non-overlappingcharacteristic funct ions, such that function and derivative are uniform inVoronoi blocks and Delaunay triangles,


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COMPARISON OF PARTICLE DISCRETIZATION

WITH ORDINARY DISCRETIZATION

COMPARISON OF PARTICLE DISCRETIZATION

WITH ORDINARY DISCRETIZATION

x2 x3

x1

f2

f1

f3

derivative of particle discretization coincides with

slope of plane which is formed by linearlyconnecting Voronoi mother points

+ + =f1 f2 f3

f1 f2 f3

+ + =

ordinary discretization with linear function

particle discretization

particle discretization of derivative


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PARTICLE DISCRETIZATION TO

CONTINUUM MECHANICS PROBLEM

PARTICLE DISCRETIZATION TO

CONTINUUM MECHANICS PROBLEMConjugate Functional

FEM-: FEM with Particle Discretizationz stiffness matrix of FEM- coincides with stiffness matrix of FEM with uniform

triangular element

z including rigid-body-rotation, FEM- gives accurate and efficient computation forfield with singularity

=

=

=

ijij

ii

Vklijklij21ijij

uu

dsc),u(J

hypo-Voronoi for displacement

Delaunay for stress:

coefficients are analytically obtained by stationarizing J


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FAILURE ANALYSIS OF FEM-FAILURE ANALYSIS OF FEM-

]k[]k[]k[

]k[]k[]k[

]k[]k[]k[

333231

232221

131211

]k[]k[ ijT

ji =

]k[]k[]k[ indirect12direct

1212 +=x3

x1

3

1

2

x2

for direct interaction

for indirect interaction

stiffness matrix ofFEM- strain energy due to relative deformation of1 and 2 through movement of3

cut two springs of direct and indirect interaction together orseparately, according to certain failure criterion of continuum


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FAILURE MODELING BY BREAKING SPRINGSFAILURE MODELING BY BREAKING SPRINGS

BA

C

R

Q

P

O

broken edge

region with reduced

strain energy

1

3

2

0.000480

0.000490

0.000500

0.000510

0.000520

0 20000 40000 60000 80000 100000

Number of Elements

J-integral

Analytical

FEM-beta

FEM

relative error of J-integral is around a few percents


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EXAMPLE PROBLEMEXAMPLE PROBLEMSimulation of Crack Growth

Check of Convergencez displacement

z strain/strain energy

Pattern of Crack Growthcan small difference in initial

configuration cause largedifference in crack growth?

uniform tension


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CONVERGENCE OF SOLUTIONCONVERGENCE OF SOLUTION

2.0 2.1 2.2 2.3 2.40.000

0.002

0.004

0.006

d

d

NDF

102

103

104

105

0.000

0.002

0.004

0.006

......

=2.09

Best Performance

displacement norm w.r.t. NDFdisplacement norm w.r.t. mesh quality

=

B

2

B

2

d

dv||u||

dv||uu||displacement norm:


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CHECK OF CRACK GROWTHCHECK OF CRACK GROWTH

yy

evolution of normal strain distribution

pattern of crack growth


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EXAMPLE: PLATE WITH 3 HOLESEXAMPLE: PLATE WITH 3 HOLES

slight difference in location of 3rd hole

simulation of crack growth: crack stems from holes

case a case b


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DIFFERENCE IN CRACK GROWTH:

DISTRIBUTION OF NORMAL STRAIN

DIFFERENCE IN CRACK GROWTH:

DISTRIBUTION OF NORMAL STRAIN

case a

case b


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SIMULATION OF BRAZILIAN TESTSIMULATION OF BRAZILIAN TEST

Brazilian test


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SIMULATION OF FOUR POINT BENDING TESTSIMULATION OF FOUR POINT BENDING TEST

0.01

-0.01

0.00

11

four point bending test


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FOUR POINT BENDING WITH

IDEARLY HOMOGENEOUS MATERIALS

FOUR POINT BENDING WITH

IDEARLY HOMOGENEOUS MATERIALS

0.01

-0.01

0.00

11

FEM- puts two source of local heterogeneity, 1) mesh quality for particle discretization and 2)crack path along Voronoi boundary. An ideally homogeneous material which is modeled with bestmesh quality sometimes fail to simulate crack propagation.


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CONCLUDING REMARKSCONCLUDING REMARKSParticle Discretization

z discretization scheme using set of non-overlapping characteristic functions

Continuum Mechanics Problemz

essentially same accuracy as FEM with uniform strainz applicable to non-linear plasticity

Failure Analysisz simple but robust treatment of failure

z Monte-Carlo simulation for studying local heterogeneity effects on failure

- candidates of failure patterns are pre-determined by spatial discretization

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Monte-Carlo Simulation of Failure Phenomena using Particle Discretization