computational models for stochastic multiscale systems materials process design and control...

Computational models for stochastic Computational models for stochastic multiscale systemsmultiscale systems

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Nicholas Zabaras

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://mpdc.mae.cornell.edu/




Outline of presentationOutline of presentation

1. Stochastic multiscale modeling of diffusion in random heterogeneous media

2. Some aspects of stochastic multiscale modeling of polycrystal materials

x GPCE & support methods for macroscopic models

x Modeling mesoscopic uncertainty using maximum entropy methods

x Information passing – variability in properties induced by microstructural uncertainties

x Robust materials design




MULTISCALE MATERIALS MODELING

grain/crystal

Inter-grain slip

Grain boundary accommodation

Twins

precipitatesatoms

Meso

Micro Nano

Performance

Me

ch

anic

s of slip

MD

Homogenization

Continuum : Process




CLASSIFICATION OF UNCERTAINTY

Uncertainty in engineering systems

Modeling

IntrinsicExtrinsic

Parametric Experimental setup Sensor errors Surroundings

Constitutive relations Assumptions on underlying physics Surroundings

Boundary conditions Process parameters




UNCERTAINTY MODELLING TECHNIQUES Reliability models

Concerned with extreme variations [not in the robust analysis zone] Analysis beyond second order is extremely complicated

Sensitivity derivatives, Perturbation methods Linear uncertainty propagation Cannot address large deviations from mean

Neumann expansions Are accurate for relatively small fluctuations Derivation is complicated for higher order uncertainties

Monte Carlo Most robust of all uncertainty quantification techniques Extremely computation intensive

Can we combine essential features of one or more of the above [Karhunen-Loeve and Generalized polynomial chaos expansions]




STOCHASTIC PROCESSES AS FUNCTIONS

A probability space is a triple comprising of collection of basis outcomes , permutation of these outcomes and a probability measure

FP

A real-valued random variable is a function that maps the probability space to a real line [regions in go to intervals in the real line]

F

X : Random variableX

: ( , , )X F P

A space-time stochastic process is can be represented as

: ( , , )W x t + other regularity conditions




SERIES REPRESENTATION [CONTD] Karhunen-Loeve

1

( , , ) ( , ) ( , ) ( )i ii

W x t W x t W x t

Stochastic process

Mean function

ON random variablesDeterministic functions

The deterministic functions are based on the eigen-values and eigenvectors of the covariance function of the stochastic process.

The orthonormal random variables depend on the kind of probability distribution attributed to the stochastic process.

Any function of the stochastic process (typically the solution of PDE system with as input) is of the form( , , )W x t

1( , , ) fn( , , , , )NW x t x t




SERIES REPRESENTATION [CONTD] Generalized polynomial chaos expansion is used to represent quantities like

0

( , , ) ( , ) (ξ( ))i ii

W x t W x t

Stochastic process

Askey polynomials in inputDeterministic functions

Stochastic input

1( , , ) fn( , , , , )NW x t x t

The Askey polynomials depend on the kind of joint PDF of the orthonormal random variables

Typically: Gaussian – Hermite, Uniform – Legendre, Beta -- Jacobi polynomials

1, , N




MODEL MULTISCALE HEAT EQUATION

( )u

K u ft

Permeability of Upper Ness formation

DDomain

Boundary

gu u

in

on

D

Multiple scale variations in K

K is inherently random

Composites

Diffusion processes

0( ,0) ( )u x u x

Diffusion processes in crystal microstructure

in D




STOCHASTIC GOVERNING PDES

( ( , ) ) ( , ), ( , ) ( , )u

K x u f x x Dt

( , ) ( , ), ( , ) ( , )gu x u x x

Denotes a random quantity

ASSUMPTIONS

The Dirichlet boundary conditions do not have a multiscale nature (i.e. they can be resolved using the coarse grid)

The correlation function of K decays slowly. Thus only a few random variables are required for its approximation

For steep decays of correlation function, only Monte Carlo methods are viable




STOCHASTIC WEAK FORM

u U Find such that, for all v V

,( , ) ( , ) ( , ); ( , ) : ( , )tu v a u v f v a u v K u v

12

12

: ( ; ( )),

: ( ; ( )), 0

gU u u L H D u u

V v v L H D v

Basic stochastic function space

Derived function spaces

2

2 ( ) ( ) : ( ) dL h h P

Stochastic weak form




VARIATIONAL MULTISCALE METHOD

h

Subgrid scale

solution

Coarse scale

solution

Actual solution

Hypothesis

Exact solution = Coarse resolved part + Subgrid part [Hughes, 95, CMAME]

Induced function space

Solution function space = Coarse function space + Subgrid function space

Idea

Model the projection of weak form onto the subgrid function space, calculate an approximate subgrid solution

Use the subgrid solution to solve for coarse solution




VMS – A MATHEMATICAL INTRODUCTION

C Fu u u

C FU U U C FV V V

u U Find such that, for all v V

,( , ) ( , ) ( , ); ( , ) : ( , )tu v a u v f v a u v K u v

Stochastic weak form

VMS hypothesis

Exact solution Coarse solution Subgrid solution

Induced function spaces

Solution function space Trial function space




SCALE PROJECTION OF WEAK FORM

, ,( , ) ( , ) ( , ) ( , ) ( , )C C F C C C F C Ct tu v u v a u v a u v f v

C Cu UFind such that, for allC Cv VF Fu Uand

andF Fv V

Eliminate the subgrid solution in the coarse weak form using a modeled subgrid solution obtained from the subgrid weak form

Projection of weak form onto coarse function space

Projection of weak form onto subgrid function space

Idea

, ,( , ) ( , ) ( , ) ( , ) ( , )C F F F C F F F Ft tu v u v a u v a u v f v




INTERPRETATION OF SUBGRID SOLUTION

Projection of weak form onto subgrid function space

Assumptions0ˆF F Fu u u

Subgrid solution Homogenous part Affine correction Incorporates all coarse scale information that affects the subgrid solution

Is that part of subgrid solution that has no coarse scale dependence

, ,( , ) ( , ) ( , ) ( , ) ( , )C F F F C F F F Ft tu v u v a u v a u v f v

Find such that, for allandF Fv VC Cu U F Fu U




SPLITTING THE SUBGRID SCALE WEAK FORM

Subgrid homogeneous weak form

Subgrid affine correction weak form

The homogenous subgrid solution is also denoted as the C2S map (coarse-to-subgrid)

We design multiscale basis functions to determine the C2S map

The affine correction is modeled explicitly

Find such that, for allandF Fv VC Cu U 0ˆ ,F F Fu u U

, ,ˆ ˆ( , ) ( , ) ( , ) ( , ) 0C F F F C F F Ft tu v u v a u v a u v

0 0,( , ) ( , ) ( , )F F F F Ftu v a u v f v




EXAMINING DYNAMICS

ũC ūC

ûF

( )A t

1 1

t t

Coarse solution field at start of time step

Coarse solution field at end of time step

( )B t

Time scale for exact solution =

Time scale for coarse solution =

Local time coordinate =

,t t t

t

1, [ , ]n nt t t t

( )eD In each element, we use a truncated GPCE representation for the coarse solution

1 0( , , ) ( ) ( ) ( )Cnbf PC C

s ssu x t u t x




A CLOSER LOOK AT THE COARSE SOLUTION

Coarse solution is entirely specified by the coefficients

1 0( , , ) ( ) ( ) ( )Cnbf PC C

s ssu x t u t x

( )Csu t

Any coarse scale information that is passed on to the subgrid solution can only be through this coefficient field

Without loss of generality, we assume the following

1 0ˆ ( , , ) ( ) ( , , )Cnbf PF C F

s ssu x t u t x t

Multiscale basis functions




DYNAMICS OF

( ) , ,

( ) , 0

C F Fs s s t

C F Fs s s

u v

K u v

After sufficient algebraic manipulation, we get

( , , )Fs x t

( ) ( ) ( )

( ) ( ) 1

(0) ( ) 1

( ) (0) 0

C C Cs s su t u A t u B t

A t B t

A B t

A t B

ũC ūC

ûF

( )A t

1 1

t t

Coarse solution field at start of time step

Coarse solution field at end of time step

( )B t

Without loss of generality, inside a coarse time step, we assume




MORE RESULTS The subgrid homogenous solution can be written as

1 0ˆ ( , , ) ( ) ( ) ( , , )Cnbf PF C C F

s s ssu x t u A t u B t x t

With some involved derivations, we can show

( ), , ( ), 0F F F Fs s t s sv K v

Askey polynomial

FEM shape

function

Multiscale basis

function

Diffusion coefficient

with multiple scales




COMPLETE SPECIFICATION OF C2S MAP

( ), , ( ), 0F F F Fs s t s sv K v

The above evolution equation requires the specification of an initial condition (in each coarse element) and boundary conditions (on each coarse element boundary).

Before that, we introduce a new variable

This reduces the C2S map governing equation as follows

Fs s s

, , , 0F Fs t sv K v




BOUNDARY CONDITIONS FOR C2S MAP On each coarse element, we have

1x

2x

3x 4x

s s

Coarse element boundary

Where, we have on the boundary

, ,

,

, ,

, , 0

( , ', ) ( )

( ,0, ) ( , , )

F Fs t s

s t s

s t s t n

v K v

x t

x x t




THE AFFINE CORRECTION TERM On each coarse element, we have

The affine correction term originates from 2 sources

Effect of source, sink terms at subgrid level [A]

Effect of the subgrid component of the true initial conditions [B]

The [B] effect is global and is not resolved in our implementation

Since [B] effect is decaying in time, we choose a time cut-off after which the subgrid solutions are accurate and can be used. This is also called the burn-in time

0 0,( , ) ( , ) ( , )F F F F Ftu v a u v f v




BOUNDARY CONDITIONS The affine correction term has no coarse-scale dependence, we can assume it goes to zero on coarse-element boundaries

1x

2x

3x 4x

0 0Fu If we need to avoid complications due to burn-in time and the effect of above assumption, we can use a quasistatic subgrid assumption

0, , 0F

s t tu




MODIFIED COARSE SCALE FORMULATION We can substitute the subgrid results in the coarse scale variational formulation to obtain the following

We notice that the affine correction term appears as an anti-diffusive correction

Often, the last term involves computations at fine scale time steps and hence is ignored

Time-derivatives of subgrid quantities are approximated using difference formulas (EBF)

0 0

, , , , ,

( , ) , , ,

C C C C C Cs t s s s t s s

C F C F Ct

u v u v u K v

f v K u v u v




COMPUTATIONAL ISSUES Based on the indices in the C2S map and the affine correction, we need to solve (P+1)(nbf) problems in each coarse element

At a closer look we can find that

This implies, we just need to solve (nbf) problems in each coarse element (one for each index s)

0 ( )s s




NUMERICAL EXAMPLES

Stochastic investigations

Example 1: Decay of a sine hill in a medium with random diffusion coefficient

The diffusion coefficient has scale separation and periodicity

Example 2: Planar diffusion in microstructures

The diffusion coefficient is computed from a microstructure image

The properties of microstructure phases are not known precisely [source of uncertainty]

Future issues




EXAMPLE 1

0 1

1

0 00

0 0

1

1 11

1 1

0 1

( )

(2 sin(2 / )) (2 sin(2 / ))1

(2 cos(2 / )) (2 sin(2 / ))

(2 sin(2 / )) (2 sin(2 / ))

(2 cos(2 / )) (2 sin(2 / ))

1.8, 0.08, 0.04

K K K

P x yK

P y P x

P x yK

P y P x

P

(x,0, ) sin( )sin( )u x y 0u 0u

0u

0u




EXAMPLE 1 [RESULTS AT T=0.05]




EXAMPLE 1 [ERROR PLOTS]

Time

L2

erro

rin

GP

CE

coef

ficie

nts

0.05 0.1 0.15 0.2

0.001

0.002

0.003

0.004

0.005E0

E1 x 5E2 x 20E3 x 40E4 x 100

TimeL

2er

ror

0.05 0.1 0.15 0.2

0.003

0.006

0.009

0.012

0.015

E0

E1

E2

E3x30E4x40

Quasistatic subgrid

Dynamic subgrid

Note that the L2 error in upscaling is larger in the case of the dynamic subgrid assumption? Why?




QUASISTATIC SEEMS BETTER There are two important modeling considerations that were neglected for the dynamic subgrid model

Effect of the subgrid component of the initial conditions on the evolution of the reconstructed fine scale solution

Better models for the initial condition specified for the C2S map (currently, at time zero, the C2S map is identically equal to zero implying a completely coarse scale formulation)

In order to avoid the effects of C2S map, we only store the subgrid basis functions beyond a particular time cut-off (referred to herein as the burn-in time)

These modeling issues need to be resolved for increasing the accuracy of the dynamic subgrid model




EXAMPLE 2

The thermal conductivities of the individual consituents is not known

We use a mixture model




DESCRIPTION OF THE MIXTURE MODEL Assume the pure () and () phases have the following thermal conductivities

The following mixture model is used for describing the microstructure thermal conductivity

The following initial conditions and boundary conditions are specified

* *0 *1( ) ( )k k k

( , ) ( ( ) ( )) ( ) ( )k x k k I x k

(x,0, ) 0u

( 1) ( 0) ( 0,1)| 0, | 1, | 0x x y

uu u k

n

Modeling uncertainty propagation in Modeling uncertainty propagation in large deformations large deformations




Materials Process Design and Control Laboratory



MOTIVATION:UNCERTAINTY IN FINITE DEFORMATION PROBLEMS

Metal forming

Composites – fiber orientation, fiber spacing, constitutive model

Biomechanics – material properties, constitutive model, fibers in tissues

Initial preform shape

Material properties/models

Forging velocity

Texture, grain sizes

Die/workpiece friction

Die shape

Small change in preform shape could lead to underfill




WHY UNCERTAINTY AND MULTISCALING ?

MacroMesoMicro

Uncertainties introduced across various length scales have a non-trivial interaction

Current sophistications – resolve macro uncertainties

Use micro averaged models for resolving physical scales

Imprecise boundary conditions Initial perturbations

Physical properties, structure follow a statistical description




UNCERTAINTY ANALYSIS USING SSFEM

Key features

Total Lagrangian formulation – (assumed deterministic initial configuration)

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

Bn+1(θ)

xn+1(θ)=x(X,tn+1, θ,)

B0

Xxn+1(θ)

F(θ)

1 1( ) ( )n ni i B B

10 1

( , )( ) ( , )

n

n

tt

x X,F x X,

X1

1( ) ( , ) ( ) nx QF P X

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

11

( )( , )

n

nPx

x

1 1( ) ( )n ni i x x

11( )

i

i i

nnP

xx

( )

QX

X




TOOLBOX FOR ELEMENTARY OPERATIONS ON TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM VARIABLESRANDOM VARIABLES

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(θ) = A-1(θ)

Galerkin projection

Formulate and solve linear system for Bj

(PC expansion)




UNCERTAINTY ANALYSIS USING SSFEM

Linearized PVW

On integration (space) and further simplification

( ) ( ) ( )i j i j

P

f d Galerkin projection Inner product




UNCERTAINTY DUE TO MATERIAL HETEROGENEITY

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 1( ,0, ,0) exp

r

bp pR

2

01

( ) (1 ( ))i n 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.




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




EFFECT OF UNCERTAIN FIBER ORIENTATION

Aircraft nozzle flap – composite material, subjected to pressure on the free end

Equiv Stress: 0.006 0.026 0.047 0.068 0.088SD Equiv Stress: 0.003 0.005 0.008 0.011 0.013

Deterministic

Stochastic-mean

Orthotropic hyperelastic material model with uncertain angle of orthotropy modeled using KL expansion with exponential

covariance and uniform random variables

Two independent random variables with order 4 PCE (Legendre Chaos)




SUPPORT SPACE METHOD - INTRODUCTION

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.

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

Easily extend to updated Lagrangian formulations.

Constitutive problem fully deterministic – deterministic evaluation at quadrature points – trivially extend to damage problems.

True PDF

Interpolant

FE Grid




SUPPORT SPACE METHOD – SOLUTION SCHEME

Linearized PVW

Galerkin projection0 0

. .j mi i j k k m

B B

dP dV P dV

u u

X X

. .B B

dP dFdV P dFdV

( ) ( )

( ) ( )( ). ( ).

( ) ( )n nn nB B

dP dV P dV

u u

x x

( ) ( )

( ) ( )( ). ( ) ( ). ( )

( ) ( )n nn nP B P B

dP dVd P dVd

u u

x x

Galerkin projection

GPCE

Support space

1 ( )

1 ( )

( )( ). ( )

( )

( ) ( ). ( )

( )

s

e n

s

e n

nel

e nP B

nel

e nP B

dP dVd

P dVd

u

x

u

x

2i j i j l l lu K B




EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR

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




PROBLEM 2: 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




FURTHER VALIDATION

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




PROCESS UNCERTAINTY

Axisymmetric cylinder upsetting – 60% height reduction

Random initial radius – 10% variation about mean – 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 STATISTICS

Force SD Force

Parameter Monte Carlo (7000 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




Sethuraman Sankaran and Nicholas Zabaras

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected], [email protected]: http://mpdc.mae.cornell.edu/

An Information-theoretic Tool forProperty Prediction Of Random

Microstructures




Idea Behind Information Theoretic ApproachIdea Behind Information Theoretic Approach

Statistical Mechanics

InformationTheory

Rigorously quantifying and modeling

uncertainty, linking scales using criterion

derived from information theory, and

use information theoretic tools to predict parameters in the face

of incomplete Information etc

Linkage?

Information Theory

Basic Questions:1. Microstructures are realizations of a random field. Is there a principle by which the underlying pdf itself can be obtained.2. If so, how can the known information about microstructure be incorporated in the solution.3. How do we obtain actual statistics of properties of the microstructure characterized at macro scale.




Information Theoretic Scheme: the MAXENT principleInformation Theoretic Scheme: the MAXENT principle

Input: Given statistical correlation or lineal path functions

Obtain: microstructures that satisfy the given properties

Constraints are viewed as expectations of features over a random field. Problem is viewed as finding that distribution whose ensemble properties match those that are given.

Since, problem is ill-posed, we choose the distribution that has the maximum entropy.

Additional statistical information is available using this scheme.




Find

Subject to

Lagrange Multiplier optimization

Lagrange Multiplier optimization

feature constraints

features of image I

MAXENT as an optimization problemMAXENT as an optimization problem

Partition Function




MAXENT FRAMEWORKMAXENT FRAMEWORK

• MAXENT: A way to generate the complete probabilistic characterization of a quantity based on limited measurements

• The algorithm is solely based on the information contained in the data (the reconstruction is statistical) [comparisons with kriging? ]

Sampling from the PDF

Component

Macro scale

Experimental microstructure

images

Process dataGrain size using Heyn intercept

method

Obtain PDF of grain sizes using

MAXENT




PARALLEL GIBBS SAMPLINGPARALLEL GIBBS SAMPLING

• Reconstruction of microstructures from grain size PDFs typically involve sampling from a large dimensional random space

• Need parallel sampling procedures [but how for Gibbs samplers]

Improper PDF

Choose a sample microstructure

image

Do domain decomposition for

grains

Choose a random grain

Sample properties for the grain conditioned on other grains

Pre process At the level of individual processors

Grain sizes: Heyn’s intercept method

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Grain Size( m)

prob

abili

ty

<Gsz>=10.97

<Gsz2>=124.90

PDF of grain sizes




MAXENT DISTRIBUTION OF GRAIN SIZES MAXENT DISTRIBUTION OF GRAIN SIZES

• Given: Experimental image of Al alloy, material properties of individual components, mean orientation of the grains

• Find the class of microstructures of which the current image is a member




RECONSTRUCTION OF 3D MICROSTRUCTURESRECONSTRUCTION OF 3D MICROSTRUCTURES

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Grain Size( m)

prob

abili

ty

<Gsz>=10.97

<Gsz2>=124.90

Input PDF (grain size distribution)

Microstructure samples from the PDF




RECONSTRUCTION OF ODFsRECONSTRUCTION OF ODFs

Input ODF (expectation

value)

Statistical samples of ODF

Average of ODF computed from

samples

Orientation distribution function: The probability distribution of orientation of individual grains in a microstructure




Implementation of homogenization scheme

Largedef formulation for macro scale

Update macro displacements

Boundary value problem for microstructure

Solve for deformation field, determine average stress

Consistent tangent formulation (macro)

Integration of constitutive equations

Continuum slip theory

Consistent tangent formulation (meso)

Macro-deformation gradient

Homogenized (macro) stress, Consistent tangent

meso stress, consistent tangent

meso deformation gradient

(a) (b)

Macro

Meso

Micro




Study properties of real microstructures

X

Y

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

Equivalent Strain: 0.04 0.08 0.12 0.16 0.2 0.24 0.28

(a)

(c)

(b)

XY

Z

Equivalent Stress (MPa): 19 27 36 45 53 62 70 79(d)

X Y

Z

Equivalent Stress (MPa): 20 30 40 50 60 70 80

XY

Z


XY

Z0

10

20

30

40

50

60

0.000 0.010 0.020 0.030 0.040 0.050 0.060

Equivalent plastic strain

Equ

ival

ent s

tres

s (M

Pa)

Simple shear

Plane strain compression

(a)

(c)

(b)

(d)




Study property variability in a material

FORM Approximation

Design iterations

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Grain Size( m)

pro

ba

bili

ty

<Gsz>=10.97

<Gsz2>=124.90

Input PDF (grain size distribution)

Microstructure samples (Voronoi models) from the PDF

X Y

Z


MAXENT algorithm

HomogenizationStatistical variability in material property (uniaxial stress-

strain curve)

Gibbs sampler

Grain size lower order statistics (average grain sizes, shape data) from

manufacturer

Property statistics




ROBUST DESIGN AND OPTIMIZATION WITH UNCERTAINTY

Extending functional optimization methods from the deterministic world

Non-intrusive optimization methods (based on the support method)




EXAMPLE – SIHCP

0 0

0

( ) ( , , ) ( , , )

ˆ( ,0, ) ( , ) ( , ) ( , )

ˆ ( , , ) ( , , ) ( , , )

ˆ ( , , ) ( , , ) ( , , )

( , , ; ) ( , , ) ( , , ) ( , , )

h

I

C k x t D Tt

x x x D

k f x t x t Tn

k q x t x t Tn

x t q Y x t x t T

0

h

I

Thermal conductivity and heat capacity are stochastic processes

Need to find the unknown flux with variability limits such that the temperature solution is matched with the sensor readings on the internal boundary




DESIGNING THE OBJECTIVE Need to match extra temperature readings at the boundary I

0( , , ; )T x t q

( , , )Y x t

Stochastic temperature solution at the inner boundary I, given a guess for the unknown heat flux q0

Temperature sensor readings with specified variability at the inner boundary I

Try to match above in mean-square sense

0

2

0 0min ( ) ( , , ; ) ( , , ) d d dI

qT

J q x t q Y x t t P




SOLVING THE OPTIMIZATION PROBLEM

0

0

0 0 0 0 0

0

( ) ( ) ( ) d dtd

( , , ; ) ( , , ) d dtdI

T

q

T

J q q J q J q q P

x t q Y x t D P

0 ( ) ( ) ( )2 0 2 2

2

0 0 0 0( , , ; ) ( , , ; )L L T L

qx t w q q x t w q D O q

00 0( , , ; , ) qx t q q D

Obtain the gradient of the objective function in a distributional sense

We have the definition of continuum stochastic sensitivity embedded in the definition of the gradient

But what is the physical reasoning behind a stochastic sensitivity ?




CONTINUUM STOCHASTIC SENSITIVITY

0 0

( ) ( , , ) ( , , )

( ,0, ) 0 ( , ) ( , )

0 ( , , ) ( , , )

( , , ) ( , , )

h

C k x t D Tt

x x D

k x t Tn

k q x t Tn

Temperature at a point

Perturbation in PDF 0q

Generic definition: Change in output for an infinitesimal change in the design variable

Here: Change in output PDF for an infinitesimal change in PDF of the design variable




GRADIENT DEFINITION VIZ ADJOINT

2 2 22 2 2

*

( ) ( ) ( )( ) ( ) ( ), ( ) ( ),

L D L T LL D L T LL L

2 2 2

*

( ) ( ) ( ), ( ) dxd d

L D L T LT D

L C k t Pt

0

0

0 0 0( ) d dtd ( , , ; ) ( , , ) d dtdI

q

T T

J q q P x t q Y x t D P

The definition of the gradient is implicit in the following equality

We use the adjoint based approach for defining the gradient in an indirect manner

Simplifying the above equation leads to an adjoint problem using which the gradient can be obtained




INCORPORATING LOSS FUNCTION

0

0

0

d d d d d d

( , , ; ) ( , , )

II

I

I

T T

q t P k t Pn

k x t q Y x tn

In particular, we obtain the following residual term that we equate to the loss function [ difference in temperature solution and sensor readings at the internal boundary ]

0

h

I

Loss function is used as a flux boundary condition for the adjoint problem




ADJOINT EQUATIONS

max

0

( ) ( , , ) ( , , )

( , , ) 0 ( , ) ( , )

0 ( , , ) ( , , )

( , , ; ) ( , , ) ( , , ) ( , , )I

I

C k x t D Tt

x t x D

k x t Tn

k x t q Y x t x t Tn

maxt t

0 0 0( , , ; ) ( ) ( , , ) ( , , )x t q J q x t T

The final adjoint equation is obtained as follows

The above unstable backward-diffusion equation is converted into a stable diffusion equation using the transformation

The gradient of the objective function is




TRIANGLE FLUX PROBLEM

Non-dimensional time

flux

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Insulated

X = 0

Triangle flux estimation: Beck J.V and Blackwell. B

X = 1

X = d

Termperature sensor

Data generation

Flux applied to left end following the profile [see Fig]

Sensor readings [polluted with noise] collected at location x = d




DATA SIMULATING REAL EXPERIMENT

0 0.25 0.5 0.75 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.1 0.2-0.05

-0.025

0

0.025

0.05

0 0.1 0.2-0.05

-0.025

0

0.025

0.05

Sensor readingsLarge noise level Small noise level

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


Mea

nflu

xan

dfir

stP

CE

term

0 0.25 0.5 0.75 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 MeanFirst PCE term+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


Mea

nflu

xan

dfir

stP

CE

term

0 0.25 0.5 0.75 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 MeanFirst PCE term+

Sensor accuracy Vs estimation results

Estimation of lower moments like mean

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