materials process design and control laboratory stochastic and deterministic techniques for...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


STOCHASTIC AND DETERMINISTIC STOCHASTIC AND DETERMINISTIC TECHNIQUES FOR COMPUTATIONAL TECHNIQUES FOR COMPUTATIONAL

DESIGN OF DEFORMATION PROCESSESDESIGN OF DEFORMATION PROCESSES

Swagato Acharjee

B-Exam

Date: April 13, 2006 Sibley School of Mechanical and Aerospace

EngineeringCornell University




SPECIAL COMMITTEE: Prof. Nicholas Zabaras Prof. Subrata Mukherjee Prof. Leigh Phoenix

FUNDING SOURCES: Air Force Office of Scientific Research (AFOSR), National Science

Foundation (NSF), Army Research Office (ARO) Cornell Theory Center (CTC) Sibley school of Mechanical & Aerospace Engineering

Materials Process Design and Control Laboratory (MPDC)

ACKNOWLEDGEMENTS




OUTLINE

Deterministic design of deformation processes

•Overview of direct and sensitivity deformation problems

•Applications

Stochastic modeling of inelastic deformations

•Probability and stochastic processes

•Generalized Polynomial Chaos Expansions (GPCE)

•Non Intrusive Stochastic Galerkin Approximation

Stochastic optimization

•Robust design of deformation processes

•Applications

Suggestion for future work




Part I - Deterministic design of deformation processes




METAL FORMING PROCESSES

Extrusion

Forging

Rolling

Boeing 747

18,600 forgings




Press forcePress force

Processing temperatureProcessing temperaturePress speedPress speed

Product qualityProduct qualityGeometry restrictionsGeometry restrictions

CostCost

CONSTRAINTSCONSTRAINTSOBJECTIVESOBJECTIVESMaterial usageMaterial usage

Plastic workPlastic work

Uniform deformationUniform deformation

MicrostructureMicrostructure

Desired shapeDesired shape

Residual stressesResidual stresses Thermal parametersThermal parameters

Identification of stagesIdentification of stagesNumber of stagesNumber of stagesPreform shapePreform shapeDie shape Die shape Mechanical parametersMechanical parameters

VARIABLESVARIABLES

BROAD DESIGN OBJECTIVESGiven raw material, obtain final product with desired microstructure and shape with minimal material utilization and costs

COMPUTATIONAL PROCESS DESIGN

Design the forming and thermal process sequenceSelection of stages (broad classification)Selection of dies and preforms in each stageSelection of mechanical and thermal process parameters in each stageSelection of the initial material state (microstructure)

COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES




1. Discretize infinite dimensional design space into a finite dimensional space

2. Differentiate the continuum governing equations with respect to the design variables to obtain the sensitivity problem

3. Discretize the direct and sensitivity equations using finite elements

4. Solve and compute the gradients

5. Combine with a gradient optimization framework to minimize the objective function defined

DEFORMATION PROCESS DESIGN - BROAD OUTLINE




BBn BB

FF e

FF p

FF

FF

Initial configurationInitial configuration Temperature: n

void fraction: fn

Deformed configurationDeformed configuration Temperature: void fraction: f

Intermediate thermalIntermediate thermalconfigurationconfiguration Temperature:

void fraction: fo

Stress free (relaxed) Stress free (relaxed) configurationconfiguration Temperature: void fraction: f

(1) Multiplicative decomposition framework(1) Multiplicative decomposition framework

(3) Radial return-based implicit integration algorithms(3) Radial return-based implicit integration algorithms(2) State variable rate-dependent models(2) State variable rate-dependent models

(4) Damage and thermal effects(4) Damage and thermal effects

Governing equation – Deformation problemGoverning equation – Deformation problem

Governing equation – Coupled thermal problemGoverning equation – Coupled thermal problem

Thermal expansion:Thermal expansion:

FF = I.FF

–1.Hyperelastic-viscoplastic constitutive laws

CONSTITUTIVE FRAMEWORK




ImpenetrabilityImpenetrability ConstraintsConstraints

Coulomb Friction LawCoulomb Friction Law

Inadmissible region

n

τ1

Referenceconfiguration

Currentconfiguration

Admissible region

Contact/friction model

τ2

Continuum implementation of die-workpiece contact. Augmented Lagrangian regularization to enforce impenetrability and frictional stick conditionsContact surface smoothing using Gregory Patches

3D CONTACT PROBLEM




Continuum problem Differentiate Discretize

Design sensitivity of equilibrium equation

Calculate such that x = x (xr, t, β, ∆β )oo

Variational form -

FFrr and and xxoo o

λ and x o

Pr and F,o

o o

Constitutive problem

Regularized contact problem

Kinematic problem

SENSITIVITY DEFORMATION PROBLEM




Curved surface parametrization – Cross section can at most be an ellipse

Model semi-major and semi-minor axes as 6 degree bezier curves

6

1

51

3 33

2 55

( ) cos

( )

(1.0 ) (1.0 5.0 )

20.0(1.0 )

6.0(1.0 )

i ii

x a

a

6

61

4 22

2 44

66

( ) sin

( )

15.0(1.0 )

15.0(1.0 )

i ii

y b

b

2 /z H

Design vector

1 2 3 4 5 6 7 8 9 10 11 12{ , , , , , , , , , , , }T βa

b

(x,y) =(acosθ, bsinθ)

H

PREFORM DESIGN TO MINIMIZE BARRELING




Optimal preform shape

Final optimal forged productFinal forged product

Initial preform shape

Objective: Design the initial preform for a fixed reduction so that the barreling in the final product in minimized

Material:Al 1100-O at 673 K

0

0.1

0.20.3

0.4

0.5

0.6

0.70.8

0.9

1

0 2 4 6 8

Iterations

No

rma

lize

d o

bje

ctiv

ePREFORM DESIGN TO MINIMIZE BARRELING IN FINAL PRODUCT




Remeshing

•Advanced THEX algorithm for unstructured hexahedral remeshing using CUBIT (Sandia).

•Interface CUBIT with C++ code using NETCDF arrays and FAN utilities

Speed

•Fast solution using Block Jacobi\ ILU preconditioned GMRES solver (PetSc).

•Fully parallel assembly.

•Fully parallel remeshing and data transfer.

EXTENSION TO COMPLEX SIMULATIONS




Reference problem – large flashDie/Workpiece Setup

Objective: Design the initial preform such that the die cavity is fully filled with minimum flash for a fixed strokeObjective Function:

PREFORM DESIGN FOR A STEERING LINK




Preform design for a steering link

First iteration – underfill Intermediate iteration – underfill

PREFORM DESIGN FOR CLOSED DIE FORGING




0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Iterations

No

rma

liz

ed

Ob

jec

tiv

e F

un

cti

on

Preform design for a steering link

Final iteration flash minimized and complete fill

Objective function

PROCESS DESIGN




PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT

Initial Setup

Material Ti-6 Al 4-V

Power law model





FlashUnderfill

Initial iteration





Intermediate iteration





Final iteration

Reduced Flash Minimum

Underfill





0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14

Iterations

Nor

mal

ized

Obj

ecti

ve F

unct

ion

Objective Function:




Kinematic Kinematic sub-problemsub-problem

Direct problemDirect problem(Non Linear)(Non Linear)

Constitutive sub-problemsub-problem

Contact sub-problemsub-problem

Thermal Thermal sub-problemsub-problem

Remeshing sub-problemsub-problem

Constitutive sensitivitysensitivity

sub-problemsub-problem

Thermal Thermal sensitivity sensitivity


Contact sensitivity sensitivity


Remeshingsensitivity sensitivity


Kinematic Kinematic sensitivity sensitivity


Sensitivity Sensitivity Problem (Linear)Problem (Linear)

Design Design SimulatorSimulator

OptimizationOptimization

DEFORMATION PROCESS DESIGN ENVIRONMENT




Part II - Stochastic modeling of inelastic deformations




SOURCES OF UNCERTAINTIES

•Uncertainties in process conditions

•Input data

•Model formulation – approximations, assumptions.

•Errors in simulation softwares

Why uncertainty modeling ?

Assess product and process reliability.

Estimate confidence level in model predictions.

Identify relative sources of randomness.

Provide robust design solutions.

Engineering component

Heterogeneous random

Microstructural features

Fail SafeComponent

reliability

All physical systems have an inherent associated randomness

MOTIVATION




Two

way fl

ow o

f sta

tistic

al in

form

atio

n

1 1e2 1e4 1e6 1e9

Eng

ine

erin

g

Length Scales ( )

Phy

sics

Che

mis

try

Ma

teri

als

0 A

Info

rmati

on flow

Statistical filter

Electronic

Nanoscale

Microscale

Mesoscale

Continuum

Material information – inherently statistical in nature.

•Atomic scale – Kinetic theory, Maxwell’s distribution etc.

•Microstructural features – correlation functions, descriptors etc.

Information flow across scales

Material heterogeneity

MOTIVATION




Initial preform shape

Material properties/models

Forging velocity

Texture, grain sizes

Die/workpiece friction

Die shapeSmall change in preform shape

could lead to underfill

Material ModelForging rate

Die/Billet shape

Friction

Cooling rate

Stroke length

Billet temperature

Stereology/Grain texture

Dynamic recrystallization

Phase transformation

Phase separation

Internal fracture

Other heterogeneities

Yield surface changes

Isotropic/Kinematic hardening

Softening laws

Rate sensitivity

Internal state variables

Dependance Nature and degree

of correlation

Process

MOTIVATION:UNCERTAINTY IN METAL FORMING PROCESSES




Issues with stochastic analysisExtremely complex phenomena – nonlinearities at all stages - large deformation plasticity, microstructure evolution, contact and friction conditions, thermomechanical coupling and damage accumulation – standard RBDO methods do not work well.

Lack of robust and efficient uncertainty analysis tools specific to metal forming.

High levels of uncertainty in the system

Possibility of reusing already developed legacy codes.

Earlier works

1. Kleiber et. al. – IJNME 2004 Response surface method for analysis of sheet forming processes2. Sluzalec et. al. – IJMS 2000 Perturbation type methods 3. Doltsinis et. al. – CMAME 2003,2005 Perturbation type methods – avoided all strong nonlinearities

UNCERTAINTY IN METAL FORMING PROCESSES




The statistical average of a function

Ω

X dyyfygXgE )()()]([

For a stochastic process W (x,t, )

Covariance

Definition – Probability space

The sample space Ω, the collection of all possible events in a sample space F and the probability law P that assigns some probability to all such combinations constitute a probability space (Ω, F, P )

Stochastic process – function of space, time and random dimension.

( , , ) , ,x t x X t T W

)],','(),,([)',',,( txWtxWtxtx C

RANDOM VARIABLES




n

iii txWtxW

0

)(),(~

),,(

Stochastic process

Chaos polynomials

(random variables)Reduced order representation of a stochastic processes.

Subspace spanned by orthogonal basis functions from the Askey series.

Chaos polynomial Support space Random variable

Legendre [] Uniform

Jacobi [] Beta

Hermite [-∞,∞] Normal, LogNormal

Laguerre [0, ∞] Gamma

Number of chaos polynomials used to represent output uncertainty depends on

- Type of uncertainty in input - Distribution of input uncertainty- Number of terms in KLE of input - Degree of uncertainty propagation desired

(Wiener,Karniadakis,Ghanem)

GENERALIZED POLYNOMIAL CHAOS EXPANSION - OVERVIEW




Key features

Total Lagrangian formulation – (assumed deterministic initial configuration)

Spectral decomposition of the current configuration leading to a stochastic deformation gradient

Bn+1()

xn+1()=x(X,tn+1, ,)

B0

Xxn+1()

F()

1 1( ) ( )n ni i B B

10 1

( , )( ) ( , ) n

n

tt

x X,

F x X,X

11( ) ( , ) ( )nx Q

F P X

1 1( )i i i iQ Q F F = P P

11

( )( , ) n

nP

xx

1 1( ) ( )n ni i x x

11( )

i

i i

nnP

xx

( )

Q

XX

FINITE DEFORMATION UNCERTAINTY ANALYSIS USING SSFEM




Scalar operations

Matrix\Vector operations

1. Addition/Subtraction

2. Multiplication

3. Inverse

1. Addition/Subtraction

2. Multiplication

3. Inverse

4. Trace

5. Transpose

Non-polynomial function evaluations

1. Square root

2. Exponential

3. Higher powersUse precomputed expectations

of basis functions and

direct manipulation

of basis coefficients

Use direct integration

over support space

Matrix InverseCompute B() = A-1()

Galerkin projection

Formulate and solve linear system for Bj

(PC expansion)

TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM VARIABLES




Linearized PVW

On integration (space) and further simplification

( ) ( ) ( )i j i j

P

f d Galerkin projection Inner product

UNCERTAINTY ANALYSIS USING SSFEM




State variable based power law model.

State variable – Measure of deformation resistance- mesoscale property

Material heterogeneity in the state variable assumed to be a second order random process with an exponential covariance kernel.

Eigen decomposition of the kernel using KLE.

0

n

fs

21 2( ,0, ,0) exp

r

b

p pR

2

01

( ) (1 ( ))i i ii

s s v

p p

V20.3398190.2390330.1382470.0374605

-0.0633257-0.164112-0.264898-0.365684-0.466471-0.567257

V10.4093960.3958130.382230.3686460.3550630.3414790.3278960.3143130.3007290.287146

Eigenvectors Initial and mean deformed config.

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Displacement (mm)

SD

Loa

d (N

)

Homogeneous materialHeterogeneous material

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6

8

10

12

14

Displacement (mm)

Load

(N)

Mean

Load vs Displacement SD Load vs Displacement

Dominant effect of material heterogeneity on response statistics

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY




MC results from 1000 samples generated using Latin Hypercube Sampling (LHS). Order 4 PCE used for SSFEM

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY-MC RESULTS




Bn+1()B0

X() xn+1()F()

xn+1()=x(XR,tn+1, ,)

XR

F*()

BR

FR()

Introduce a deterministic reference configuration BR which maps onto a stochastic initial configuration by a stochastic reference deformation gradient FR(θ). The deformation problem is then solved in this reference configuration.

MODELING INITIAL CONFIGURATION UNCERTAINTY




Deterministic simulation- Uniform bar under tension with effective plastic strain of 0.7 . Power law constitutive model.

Eq. strain1.266561.194461.122351.050240.9781360.9060290.8339220.7618160.6897090.617602

Eq. strain0.7495560.7081580.6667590.6253610.5839620.5425640.5011650.4597670.4183680.37697

Initial configuration assumed to vary uniformly between two extremes with strain maxima in different regions in the stochastic simulation.

INITIAL CONFIGURATION UNCERTAINTY




Stochastic simulation

SD Eq. strain0.2680860.2425070.2169280.1913490.165770.1401910.1146120.08903270.06345370.0378747

Eq. strain0.7566080.7497910.7429750.7361580.7293420.7225250.7157090.7088920.7020760.69526

SD Eq. strain0.2680860.2425070.2169280.1913490.165770.1401910.1146120.08903270.06345370.0378747

SD uy

0.1178790.1060910.09430280.0825150.07072710.05893930.04715140.03536360.02357570.0117879

SD ux

0.05046060.04541450.04036850.03532240.03027640.02523030.02018420.01513820.01009210.00504606

Results plotted in mean deformed configuration





Point at top

Outer boundary plot

0

0.5

1

1.5

2

2.5

3

3.5

4

0.264 0.266 0.268 0.27 0.272 0.274 0.276

x (mm)

y (m

m)

Mean-MC

Mean - SSFEM - o4

Mean -Deterministic

Point at centerline





• Reduced order representation of uncertainty

• Faster than mc by at least an order of magnitude

• Exponential convergence rates for many problems

• Provides complete response statistics and convergence in distribution

But….

• Behavior near critical points.

• Requires continuous polynomial type smooth response.

• Performance for arbitrary PDF’s.

• How do we represent inequalities, eigenvalues spectrally ?

• Can we afford to rewrite complex metal forming codes ?

MERITS AND PITFALLS OF GPCE




Non Intrusive Stochastic Galerkin Method (NISG)




Parameters of interest in stochastic analysis are the moment information (mean, standard deviation, kurtosis etc.) and the PDF.

For a stochastic process

Definition of moments

NISG - Random space discretized using finite elements to

Output PDF computed using local least squares interpolation from function evaluations at integration points.

( , , ) ( , , ) , ,g x t g x t x X t T

( ( , , )) ( )ppM g x t f d

h

1 1

( ( , , )) ( ) ( ( , , )) ( )h

nel nh h p h h p h

p i e ie i iee i

M g x t f d w g x t f

1 1

( ( , ))nel nint

h h p hp i ei ei

e i

M w g x t f

ie

Deterministic evaluations at fixed points

NISG - FORMULATION




Finite element representation of the support space.

Inherits properties of FEM – piece wise representations, allows discontinuous functions, quadrature based integration rules, local support.

Provides complete response statistics.

Decoupled function evaluations at element integration points.

True PDF

Interpolant

FE Grid

Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h , p versions).

NISG - DETAILS




SD Eq. strain0.2680860.2425070.2169280.1913490.165770.1401910.1146120.08903270.06345370.0378747

Mean

Initial Final

Using 6x6 uniform support space grid

SD-Void fraction0.01860.01720.01580.01430.01290.01150.0101

Void fraction0.04190.03880.03570.03250.02940.02630.0231

SD-Void fraction0.00980.00960.00940.00920.00910.00890.0087

Uniform 0.02

Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution

2

01

ˆ ˆ( ) (1 ( ))i n ii

f f v

p p

EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR




Load displacement curves

Displacement (mm)

Lo

ad

(N)

0.1 0.2 0.3 0.4

1

2

3

4

5

6

Mean

Mean +/- SD

Displacement (mm)

SD

Lo

ad

(N)

0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR




Comparison of statistical parameters

Parameter Monte Carlo (1000 LHS samples)

Support space 6x6 uniform grid

Support space 7x7 uniform grid

Mean 6.1175 6.1176 6.1175

SD 0.799125 0.798706 0.799071

m3 0.0831688 0.0811457 0.0831609

m4 0.936212 0.924277 0.936017

Final load values

VALIDATION




Axisymmetric cylinder upsetting – 60% height reduction (Initial height 1.5 mm)

Random initial radius – 10% variation about mean (1 mm)– uniformly distributed

Random die workpiece friction U[0.1,0.5]

Power law constitutive model

Using 10x10 support space grid Void fraction: 0.002 0.004 0.007 0.009 0.011 0.013 0.016

Random ? Shape

Random ? friction

PROCESS UNCERTAINTY




Force SD Force

PROCESS STATISTICS




0.00E+00

5.00E-05

1.00E-04

1.50E-04

2.00E-04

2.50E-04

3.00E-04

3.50E-04

0 2 4 6 8 10 12 14

Grid resolution (Number of elements per dimension)

Re

lativ

e e

rro

r

Parameter Monte Carlo (20000 LHS samples)

Support space 10x10

Mean 2.2859e3 2.2863e6

SD 297.912 299.59

m3 -8.156e6 --9.545e6

m4 1.850e10 1.979e10

Final force statistics

Convergence study

PROCESS STATISTICS

Relative Error




FORM Approximation

SORM Approximation

Actual limit state surface Full order reliability method

gDesign point

Safe state Z(g)>0

Unsafe state Z(g)<0

β

Objective: Design the forging press forthe process on the basis of the maximum force required based on a probabilityof failure of 0.0002.- β = 3.54

Minimum required force capacity vs Stroke for a press failure probability of 0.0002 Minimum design force = 2843 N

Limit state function

Probability of failure

RELIABILITY BASED DESIGN




Axisymmetric flashless closed die forging

Void frac: 3.3E-03 6.6E-03 9.9E-03 1.3E-02 1.6E-02

Same process with initial void fraction 0.03

Deterministic Simulation

Decrease in void fraction in the billet during the process leads to unfilled die cavity

Initial preform volume same as volume of die cavity

STOCHASTIC ESTIMATION OF DIE UNDERFILL




2

01

ˆ ˆ( ) (1 ( ))i n ii

f f v

p p

Stochastic Simulation

Assumed void fraction using KLE 21 1( ,0, ,0) exp

r

bp pR

PDF of die underfill

Using 10x10 uniform support space grid

STOCHASTIC ESTIMATION OF DIE UNDERFILL




• Both provide complete response statistics and convergence in distribution.

• GPCE fails for systems with sharp discontinuities. (inequalities).

• Seamless integration of NISG into existing codes. Ideal for complex simulations with strong nonlinearities. (Finite deformations – eigen strains, inequalities, complex constitutive models).

• GPCE needs explicit spectral expansion and repeated Galerkin projections.

• NISG can handle completely empirical probability density functions due to local support with no change in the convergence properties (convergence is based on number of elements used to discretize the support-space and the order of interpolation).

• Curse of dimensionality – both methods are susceptible.

NISG is the way to go

REVIEW OF NISG AND GPCE




Part III - Robust Design of Deformation Processes




• Robustness limits on the desired properties in the product –

acceptable range of uncertainty.

• Design in the presence of uncertainty/ not to reduce uncertainty.

• Design variables are stochastic processes or random variables.

• Consider all ‘important’ process and material data to be random processes – by itself a design decision.

• Design problem is a multi-objective and multi-constraint optimization problem.

KEY ISSUES

PROBLEM STATEMENTCompute the predefined random process design parameters which lead to a desired objectives with acceptable (or specified) levels of uncertainty in the final product and satisfying all constraints.

ROBUST DESIGN ENVIRONMENT




Design Objective

Probability Constraint

Norm Constraint

SPDE Constraint

Augmented Objective

ROBUST DESIGN PROBLEM FORMULATION




CSSM problem decomposed into a set of CSM problems

Compute sensitivities of parameters with respect to stochastic design variables by defining perturbations to the PDF of the design variables.

Decomposition based on the fact that perturbations to the PDF are local in nature

A CONTINUUM STOCHASTIC SENSITIVITY SCHEME (CSSM)




Design Objective – unconstrained case

Set of NelE*n objective functions

NISG APPROXIMATION FOR OBJECTIVE FUNCTION




BENCHMARK APPLICATION

Case 1 – Deterministic problem

Case 2 – 1 random variable (uniformly distributed) – friction – 66% variation about mean (0.3) (10x1 grid) – 1D problem

Case 3 – 2 random variables (uniformly distributed) – friction(66%) and desired shape (10% about mean) (10x10 grid) - 2D problem

Flat die upsetting of a cylinder




Deterministic problem - optimal solution

Deterministic problem

1D problem

2D problem

OBJECTIVE FUNCTION




DESIGN PARAMETERS

Deterministic problem

2D problem

1D problem

Initial guess parameters

Mean

SD

Mean

SD




OBJECTIVE FUNCTION

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

1 2 3 4 5 6 7 8 9 10

Iterations

Obj

ecti

ve F

unct

ion

1D

Deterministic

2D




FINAL FREE SURFACE SHAPE CHARACTERISTICS

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

6.50E-01 7.50E-01 8.50E-01 9.50E-01 1.05E+00 1.15E+00 1.25E+00

x (mm)

y (m

m)

1D

Deterministic

2D

Mean

SD

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

4.00E-02 5.00E-02 6.00E-02 7.00E-02 8.00E-02 9.00E-02 1.00E-01 1.10E-01

x (mm)

y (m

m)

1D

2D




Suggestions for future work




Fine scale heterogeneities

Coarse scale heterogeneities

•Nature of randomness differs significantly between scales, though not fully uncorrelated.

•Need a multiscale evaluation of the Correlation Kernels

21 2( ,0, ,0) exp

r

b

p pR

Present method

Assume correlation between macro points

Decompose using KLE 01

( ) (1 ( ))n

i i ii

s s v

p p

grain size, texture, dislocations

macro-cracks, phase distributions

MULTISCALE NATURE OF MATERIAL HETEROGENEITIES




As the number of random variables increases, problem size rises exponentially.

1

10000

1E+08

1E+12

1E+16

1E+20

0 5 10 15 20 25

No. of variables

Fu

nct

ion

eva

luat

ion

s

(assume 10 evaluations per random dimension)

CURSE OF DIMENSIONALITY




A PRIORI ADAPTIVITY• Initial sensitivity analysis with respect to random parameters.

• Sensitivities used to a priori refine/coarsen grid discretization along each random dimension.

• Easily implemented using version of earlier CSM analysis

PROPOSED SOLUTIONS




ADAPTIVE DISCRETIZATION BASED ON OUTPUT STOCHASTIC FIELD

• Refine/Coarsen input support space grid based on output defined control parameter (Gradients, standard deviations etc.)

• Applicable using standard h,p adaptive schemes.

Support-space of input Importance spaced grid

PROPOSED SOLUTIONS




DIMENSION ADAPTIVE QUADRATURE (Gerstner et. al. 2003)

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Full grid Scheme Sparse grid Scheme Dimension adaptive Scheme

Very popular in computational finance applications.

Has been used in as high as 256 dimensions.

PROPOSED SOLUTIONS




S. Acharjee and N. Zabaras "A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues space with applications to the control of material properties", Acta Materialia, Vol. 51, pp. 5627-5646, 2003.

S. Acharjee and N. Zabaras "The continuum sensitivity method for the computational design of three-dimensional deformation processes", Computer Methods in Applied Mechanics and Engineering, in press.

S. Acharjee and N. Zabaras, "Uncertainty propagation in finite deformation plasticity -- A spectral stochastic Lagrangian approach", Computer Methods in Applied Mechanics and Engineering, in press.

S. Acharjee and N. Zabaras, "A concurrent model reduction approach on spatial and random domains for stochastic PDEs", International Journal for Numerical Methods in Engineering, in press.

S. Acharjee and N. Zabaras, "A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes ", Computers and Structures, in preparation.

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