stochastic simulation in multi-scale modeling

Stochastic simulation in multi-scale modeling

Lori Graham-Brady

Associate ProfessorDepartment of Civil Engineering

560.700 – Applications of Science-Based Coupling of Models

September 29-October 1, 2008

GOAL: QUANTIFICATION OF STRUCTURAL GOAL: QUANTIFICATION OF STRUCTURAL RELIABILITY (e.g., probability of failure)RELIABILITY (e.g., probability of failure)

Load variability important: studied extensivelyLoad variability important: studied extensively

Material/geometric variability: studied lessMaterial/geometric variability: studied less

Need for multiNeed for multi--scale analysis that incorporates uncertaintyscale analysis that incorporates uncertainty

OBJECTIVE AND GENERAL OUTLINEOBJECTIVE AND GENERAL OUTLINE

GOAL: QUANTIFICATION OF STRUCTURAL GOAL: QUANTIFICATION OF STRUCTURAL RELIABILITY (e.g., probability of failure)RELIABILITY (e.g., probability of failure)

Load variability important: studied extensivelyLoad variability important: studied extensively

Material/geometric variability: studied lessMaterial/geometric variability: studied less

Need for multiNeed for multi--scale analysis that incorporates uncertaintyscale analysis that incorporates uncertainty

•• (Cursory) overview of multi(Cursory) overview of multi--scale approachesscale approaches

•• Motivation for probabilistic analysisMotivation for probabilistic analysis

•• Simulation of stochastic microstructureSimulation of stochastic microstructure

•• Simulation of material properties (elastic, inelastic, Simulation of material properties (elastic, inelastic, brittle strength)brittle strength)

OBJECTIVE AND GENERAL OUTLINEOBJECTIVE AND GENERAL OUTLINE

Why multi-scale?• Failure of macro-scale

structures often initiates/propagates at micro- or nano-scale

• Full-scale models with explicit small-scale features are infeasible

• Multi-scale models incorporate small-scale information into large-scale models

Multi-scale approaches (deterministic)

Concurrent multi-scale models (e.g., Robbins, Fish, Ghosh, Liu): coupling of scales in a single model

•Many assumptions regarding boundaries between scales

•Assumptions regarding very low temperature, etc.

•Assumes that information is available at all scales –deterministic

•Use when necessary –challenging!

Multi-scale approaches (deterministic)

Hierarchical or information-passing models: calculate effective behavior and pass to large scale models

•Decouples scales: no feedback between scales

•Makes use of single-scale results (experimental, computational, analytical)

•In place for a long time (e.g., homogenization)

•Well suited for probabilistic models

σf, Em,…

σy, E,…

Challenge: material structure varies at multiple scales

Global HomogenizationTraditional analyses assume globally homogenized material properties:

• Effective for capturing average macro-scale (e.g., stiffness)• May not work in the following cases:

o Mesh dependencies ariseo Structural length scale near microstructure length scaleo Attempts to model local behavior (local yield, microcracks, etc.)

Graphite-Epoxy Fiber Tow, courtesy of Kunze & Herakovich

E, ν, σY, …

Mesh dependencyThe assumption of homogeneous material properties in modeling localized failure leads to mesh dependencies

•Observed in models of shear bands – results explicitly connected to mesh size rather than physics of the problem

Medyanik et al, JMPS 2007

Small length scalesP

P

Stiffness of small-scale “beams” vary significantly due to crystal orientation of grains

Inelastic behavior dependent on local effectsInelastic behavior dependent on local effectsLocal concentrations obscured by homogenizationLocal concentrations obscured by homogenization

Models of local behavior

Why probabilistic?Importance of spatially varying behavior highlightedRandom material properties lead to random behavior -

predicting scatter is critical to reliability estimates

failure

Probability density function of strength

Higher mean & higher variance lead to more likely failure than lower mean & lower variance

Why probabilistic?• Predicting scatter is critical to survivability

estimates• Deterministic models may not capture typical

behavior, even when it’s not random

Cracking/shear bands: assumptions on initiation sites –can be idiosyncratic

Yielding: simultaneous yielding at all locations

Why probabilistic?• Predicting scatter is critical to survivability

estimates• Deterministic models may not capture typical

behavior• Localized loadings make small-scale material

fluctuations important

Simple example – cantilever beamCalculate variance of displacement at x=L/2, when stiffness EI(x) varies randomly over length of beam.

• Random spatial variability can be significant to structural response!

How do probabilistic hierarchical models work?

• Deterministic model passes continuum-based parameter(s) to the next larger scale

Deterministic: predict continuum-based conditions for onset of shear bands, fracture, other quantity of interest (eg, failure stress)

How do probabilistic hierarchical models work?

• Deterministic model passes continuum-based parameter(s) to the next larger scale

• Probabilistic model passes distributions of continuum-based parameter(s) – may be spatially varying (need spatial correlation measures from small scale) – HOW TO GET?

Probabilistic: predict distribution on continuum-based conditions for onset of fracture, yielding, etc.

Probabilistic Framework

Large scale numerical models

Small-scale observations

Identify scatter in large scale results

Proper prediction of failure

Microstructure varies randomly from location to location

Distribution/scatter of parameters for large-scale model


Probabilistic FrameworkSmall-scale observations


Experimental results

Largedatasets available


Wouldn’t it be nice to have this for all materials?!


Morphology Characterization…such as shapes, sizes, spatial distributions, networks, volume fractions

Techniques to quantitatively describe the microstructure

Left: alumina/spinel nanostructured ceramic composite; Right: Alumina/titanium diboride composite - variations in alumina phase (shown in grey) sizes/shapes/spatial distribution (Logan 2002).

Experimental results


Distribution/scatter of parameters for large-scale

model

Simulate sample morphologies

Apply numerical models to obtain

sample parameters

Numerical model prediction, e.g. E(1), σf

(1)

Sample 1

Morphology Characterization



model



sample parameters

E(1), σf(1)

Numerical model prediction, e.g. E(2), σf

(2)

Sample 2




model



sample parameters

E(1), σf(1)

E(2), σf(2)

E(N), σf(N)

Sample N




model



sample parameters

E(1), σf(1)

E(2), σf(2)

E(N), σf(N)




Simulate sample morphologiesApply numerical

models to obtain sample parameters



Microstructure Local E(x,y)




Simulate random parameters & apply to large scale




Generate Sample 1 of material properties (depends on (x,y))

Damage D1








Generate Sample 2 of material properties (depends on (x,y))

Damage D2



sample parameters



N damage levels


N damage levels


Morphology CharacterizationSmall-scale observations






N damage levels

Estimate of scatter in performance




Use stochastic simulation to evaluate scatter in structural behavior

Two possible ways to proceed•Explicit simulation of microstructure

•Simulation of randomly varying material properties

Connect Morphology & Material properties

Some microstructures are fairly straightforward to generate –special cases

Spherical/Cylindrical inclusions, randomly distributed with some level of clustering

Voronoi tessellation, assuming uniform grain growth

Microstructural Simulation

Short-range correlation scheme (Graham-Brady & Xu 2005)•Advantage: broad applicability, matches additional probabilisticcharacteristics of the medium •Disadvantage: computationally intensive for large # of samples

OriginalSimulated –3rd order

correlation

Ideally we need an efficient technique to incorporate high order correlation functions – significant challenge!

Simulated –2nd order correlation

Microstructural simulation

Put each simulated microstructure into a finite element model (or other computational mechanics models) of the structureCalculate statistics on the response of interest…Model is simple to envision, but simulation not so simple

OriginalSimulated –3rd order

correlationSimulated –2nd order

correlation

Using microstructural simulation




Calls for statistics of randomly varying material properties: probability density function, correlation functions


Collect E at every point (x,y) – place in a bin

Draw a histogram from the bin

Fit a probability density function to the histogram

Probability density function

Simulation of a single value of E is simple

Use random number generation (many subroutines available for various PDFs)



Simulation of a many values of E is simple, if none of the values are correlated!

Correlation function ρ(τx, τy) measures the correlation between values of E at a distance τx and τy apart

Points close together well-correlated (ρ near 1)

Points far apart poorly correlation (ρ near 0)

Correlation function




Calls for statistics of randomly varying material properties: probability density function, correlation functions

NEED A SAMPLE SET OF RANDOMLY VARYING MATERIAL PROPERTIES!


Microstructure Local Material Properties (Exx(x,y))

2 specific approaches considered here:

•Moving-window homogenization (elastoplasticcomposite)

•Local flaw distributions (brittle material)

Link microstructure to material properties

•Pixelized image with distinct phases

•Coupled with a micromechanical homogenization method

•Overlapping windows

•Larger window size leads to more homogenization

Moving-window homogenization

• Rule of Mixtures: Assume stiffness is volume average of stiffness, or flexibility is volume average of flexibility –different results unless volume is very large!

• Mori-Tanaka: Represent each window as a single fiber with volume that satisfies volume fraction

vf=4/9=.444 vf=4/9=.444

Both approaches ignore configuration of phases

Micromechanics

Finite elements are one option:• Need explicit boundary conditions (tractions,

displacements, or mixed) – affects the resulting effective properties

• Quite restrictive – requires absolute continuity of displacements, making FEM computationally expensive!

An alternative, more efficient option is the generalized method of cells (GMC)

Micromechanics - FEM

Strain in Composite

Strain in Subcell

Stress in Subcell

Stress in Composite

Input Constitutive

Relation

Strain Concentration

Volumetric Averaging

Conditions/Assumptions:•Periodic•Continuous Fiber

•Traction Continuity•Displacement Continuity

Paley and Aboudi, 1992

Micromechanics - GMC

Moving-window homogenization

Represent each window by homogenized properties:

•Elastic tensor contains Exx, Eyy, Ezz, νxy, νxz, νyz

•Plastic behavior anisotropic: yield stress and hardening parameters differ in all 6 directions

Local anisotropy can make model very complex!

Sample

Ematrix=91 GPa

Efiber=414 GPa

φfiber=25%

νmatrix=νfiber=0.2

Moving-window Mori Tanaka

Moving-window FEMMoving-window GMC

Effect of micromechanics model on local Exx(x,y) (minimal)

Sample

20%x20% Window2.5%x2.5% Window

How to pick?

Ultimately depends on scale that has most effect on response of interest

Effect of window size on Exx(x,y) – significant!

Microstructure digitized into rectangular pixels – direct application to FEM model leads to artificial stress concentrations.

Single circular fiber in matrix leads to a maximum tensile stress of 1.26 (relative to average tensile stress of 1)

Model based on resolution above leads to stress of 1.62 – error!

Example: stress around fiber

In the elastic range, a window size of approximately the inclusion size gives the best solution for maximum stress (true for any resolution)

Window size – single fiber

Local Material Properties (Exx(x,y)) Response to loading (stress)

Locally, stress is affected by variations in elastic modulus – higher than predictions from model with fully homogenized properties

Provide material properties to FEM model

Stresses, using FE model and material property fields from a 5%x5% window size

σyy σxx

p(x)=1 MPa

1.62

Example – Local stress distribution

Exx(x,y) from given material microstructure (left)

Exx(x,y) generated by stochastic simulation (right)

Simulation of elastic properties

Results from 1000 simulations


Homogenized prediction


Design threshold

Microstructural observations

2 specific approaches considered here:

•Moving-window homogenization (elastoplastic material)

•Local flaw distributions (brittle material)


Flaw density

Clustering of flaws

Flaw size distribution

Probability density function (PDF) of local strain-rate-dependent constitutive parameters

Strain rate dependence in brittle materials

Impact problems: stress waves serve to “localize” loads



Strength of brittle materials driven by flaws (no plasticity)

Flaws occur randomly (location, size, shape all random)



Strength of brittle materials driven by flaws (no plasticity)

Flaws occur randomly (location, size, shape all random)

Focus here on fixed size, penny-shaped flaws of uniform orientation

Effect of random fluctuations in flaw density (uniaxial compression)

Mesh dependency in models of failure

The assumption of homogeneous material properties in modeling localized failure leads to mesh dependencies

•Brannon’s work shows improved mesh independency when strength is allowed to vary spatially

In FEM, each element has a different strength – varies randomly!




How to decide what are appropriate distributions of strength?

Shape of PDF of strength (Weibull)?

Mean/Variance of strength?




Mean/Variance of strength affected by:

Element size

Flaw density, clustering, flaw size distribution

Effect of element size on local flaw density

0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4x 10-3 PDF of flaws/unit volume for different area fractions of bulk

A=1x1A=.25x.25A=0.1x0.1A=.025x.025 Effect of

clustering?

large,small →→ σV

Assuming flaw locations are independent, then local flaw densityis Poisson-distributed

Bulk flaw density=104

flaws/unit volume

Clustering model

Generate N parent points (Poisson process with parameter λpA)

Use Matern Poisson cluster process (1971)

Clustering model

Put remaining flaws around parent flaws, following a Poisson process on area of cluster Ac


Clustering model

Larger Ac reduces clustering.

Larger λp/λalso reduces clustering


Clustering model

Diggle (1978) shows how to estimate λpand Ac for given microstructure using nearest neighbor distribution


COV of Flaw Density vs Sample Size

Bulk η=107/m2, Ac=10-5 m2


NOTE: COV is the standard deviation divided by the mean


Bulk η=107/m2, Ac=10-5 m2

Can characterize random local flaw density – how to predict local strength?


Microstructural observations for given element size, clustering level, average flaw density


Mean & variance of flaw density

Flaw size distribution (deterministic for now)

Probability density function (PDF) of local strain-rate-dependent constitutive parameters – strength under uniaxial compression

Under compression, frictional sliding of flaws produce tension cracks at the tips

Cracks grow in direction of maximum compression, causing axial splitting

σ1

σ1

σ1

σ1

Ashby & Hallam, Acta met. 1986

Brittle failure under compression

Modeling Dynamic Failure with Flaws (Paliwal & Ramesh)

• Stress rate calculation at given strain, strain rate

• E, ν = elastic modulus, Poisson’s ratio (known)Ω=crack density parameter

εννπεννπσ Ω−+−Ω−+−= &&& )45)(1(30

))45)(1(30

1(22

EE

jjj

N

jj llssg && Δ=Ω ∑

=

)(21

η




εννπεννπσ Ω−+−Ω−+−= &&& )45)(1(30

))45)(1(30

1(22

EE

jjj

N


=

)(21

η

Flaw density




εννπεννπσ Ω−+−Ω−+−= &&& )45)(1(30

))45)(1(30

1(22

EE

jjj

N


=

)(21

η

Probability of flaw size in interval around sj (input parameter) – assume fixed flaw size for simplicity at this point




εννπεννπσ Ω−+−Ω−+−= &&& )45)(1(30

))45)(1(30

1(22

EE

jjj

N


=

)(21

η

Crack size (growth modeled using self-consistent model)




Given flaw density, flaw size distribution, and elastic properties:Predict uniaxial σ−ε at a given strain rateTake maximum σ as strength at given strain rate

εννπεννπσ Ω−+−Ω−+−= &&& )45)(1(30

))45)(1(30

1(22

EE

jjj

N


=

)(21

η

Fixed flaw size: strength predictions

3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 1010 Axial strength vs Flaw density

log10(flaw density in 1/m2)

Axi

al s

treng

th in

Pa

Strain rate=104/s

Strain rate=105/s

Strain rate=106/s

C.O.V. of Strength vs Element Size/s

/s

C.O.V. of Strength vs Element Size




In FEM, each element has a different strength – varies randomly!

••Random variations in material properties can lead to variabilityRandom variations in material properties can lead to variability in in structural behavior (e.g., local stress in structural behavior (e.g., local stress in elastoplasticelastoplastic composite)composite)

••Ignoring random variations in local strength can lead to numericIgnoring random variations in local strength can lead to numerical al problems (e.g., mesh dependency in shear bands, dynamic failure problems (e.g., mesh dependency in shear bands, dynamic failure of brittle materials)of brittle materials)

••Tools have been developed for quantifying the random variations Tools have been developed for quantifying the random variations in these properties, using movingin these properties, using moving--window approachwindow approach

••Stochastic simulation an effective tool for quantifying probabilStochastic simulation an effective tool for quantifying probability of ity of failurefailure

••Windowing of Windowing of elastoplasticelastoplastic composites still under studycomposites still under study

••Finite element analyses based on random strength of brittle Finite element analyses based on random strength of brittle materials yet to be developedmaterials yet to be developed

CONCLUSIONSCONCLUSIONS

stochastic simulation in multi-scale modeling

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