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
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/
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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]
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 1 [RESULTS AT T=0.05]
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 1 [RESULTS AT T=0.05]
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 1 [RESULTS AT T=0.2]
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 1 [RESULTS AT T=0.2]
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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?
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 2
The thermal conductivities of the individual consituents is not known
We use a mixture model
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 2 [RESULTS AT T=0.05]
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 2 [RESULTS AT T=0.05]
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 2 [RESULTS AT T=0.2]
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXAMPLE 2 [RESULTS AT T=0.2]
Modeling uncertainty propagation in Modeling uncertainty propagation in large deformations large deformations
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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)
Materials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
UNCERTAINTY ANALYSIS USING SSFEM
Linearized PVW
On integration (space) and further simplification
( ) ( ) ( )i j i j
P
f d Galerkin projection Inner product
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Equivalent Stress (MPa): 20 30 40 50 60 70 80
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)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Equivalent Stress (MPa): 20 30 40 50 60 70 80
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
ROBUST DESIGN AND OPTIMIZATION WITH UNCERTAINTY
Extending functional optimization methods from the deterministic world
Non-intrusive optimization methods (based on the support method)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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 ?
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
+
+
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Non-dimensional time
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+
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Non-dimensional time
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