Computational models for stochastic multiscale systems

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Computational models for stochastic multiscale
systems
Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//mpdc.mae.cornell.edu/
Materials Process Design and Control Laboratory
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Outline 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 Laboratory
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MULTISCALE MATERIALS MODELING
Meso
grain/crystal
Homogenization
Twins
Inter-grain slip
Grain boundary accommodation
Continuum Process
Mechanics of slip
Micro
Nano
atoms
precipitates
MD
Performance
Materials Process Design and Control Laboratory
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CLASSIFICATION OF UNCERTAINTY
Uncertainty in engineering systems
Intrinsic
Extrinsic

• Experimental setup
• Sensor errors
• Surroundings

Modeling
Parametric

• Constitutive relations
• Assumptions on underlying physics
• Surroundings

• Boundary conditions
• Process parameters

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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

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STOCHASTIC PROCESSES AS FUNCTIONS

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

• 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

• Random variable

• A space-time stochastic process is can be
  represented as

other regularity conditions
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SERIES REPRESENTATION CONTD

• Karhunen-Loeve

ON random variables
Mean function
Stochastic process
Deterministic 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

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SERIES REPRESENTATION CONTD

• Generalized polynomial chaos expansion is used
  to represent quantities like

Stochastic input
Askey polynomials in input
Stochastic process
Deterministic functions

• The Askey polynomials depend on the kind of
  joint PDF of the orthonormal random variables
• Typically Gaussian Hermite, Uniform
  Legendre, Beta -- Jacobi polynomials

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MODEL MULTISCALE HEAT EQUATION
Boundary
in
Domain
on
in

• Multiple scale variations in K
• K is inherently random

• Diffusion processes in crystal microstructure

• Composites
• Diffusion processes

Permeability of Upper Ness formation
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STOCHASTIC GOVERNING PDES
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

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STOCHASTIC WEAK FORM

• Basic stochastic function space

• Derived function spaces

• Stochastic weak form

such that, for all
Find
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VARIATIONAL MULTISCALE METHOD

• 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

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VMS A MATHEMATICAL INTRODUCTION

• Stochastic weak form

such that, for all
Find

• VMS hypothesis

Exact solution
Coarse solution
Subgrid solution

• Induced function spaces

Solution function space
Trial function space
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SCALE PROJECTION OF WEAK FORM
such that, for all
Find
and
and

• Projection of weak form onto coarse function
  space

• Projection of weak form onto subgrid function
  space

• Idea

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

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INTERPRETATION OF SUBGRID SOLUTION

• Projection of weak form onto subgrid function
  space

such that, for all
Find
and

• Assumptions

Homogenous part
Affine correction
Subgrid solution

• Incorporates all coarse scale information that
  affects the subgrid solution

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

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SPLITTING THE SUBGRID SCALE WEAK FORM
such that, for all
Find
and

• 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

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EXAMINING DYNAMICS

• Time scale for exact solution
• Time scale for coarse solution
• Local time coordinate

• In each element, we use a truncated GPCE
  representation for the coarse solution

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A CLOSER LOOK AT THE COARSE SOLUTION

• Coarse solution is entirely specified by the
  coefficients

• 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

• Multiscale basis functions

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DYNAMICS OF

• After sufficient algebraic manipulation, we get

1
1

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

uC
uC
Coarse solution field at end of time step
Coarse solution field at start of time step
ûF
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MORE RESULTS

• The subgrid homogenous solution can be written
  as

• With some involved derivations, we can show

FEM shape function
Diffusion coefficient with multiple scales
Askey polynomial
Multiscale basis function
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COMPLETE SPECIFICATION OF C2S MAP

• 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

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BOUNDARY CONDITIONS FOR C2S MAP

• On each coarse element, we have

Coarse element boundary G

• Where, we have on the boundary

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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

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BOUNDARY CONDITIONS

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

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

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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)

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COMPUTATIONAL ISSUES

• Based on the indices in the C2S map and the
  affine correction, we need to solve (P1)(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)

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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

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EXAMPLE 1
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EXAMPLE 1 RESULTS AT T0.05
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EXAMPLE 1 RESULTS AT T0.05
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EXAMPLE 1 RESULTS AT T0.2
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EXAMPLE 1 RESULTS AT T0.2
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EXAMPLE 1 ERROR PLOTS
Quasistatic subgrid
Dynamic subgrid

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

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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

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EXAMPLE 2

• The thermal conductivities of the individual
  consituents is not known
• We use a mixture model

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DESCRIPTION OF THE MIXTURE MODEL

• Assume the pure (a) and (b) 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

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EXAMPLE 2 RESULTS AT T0.05
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EXAMPLE 2 RESULTS AT T0.05
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EXAMPLE 2 RESULTS AT T0.2
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EXAMPLE 2 RESULTS AT T0.2
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Modeling uncertainty propagation in large
deformations
Materials Process Design and Control Laboratory
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MOTIVATIONUNCERTAINTY IN FINITE DEFORMATION
PROBLEMS
Metal forming
Forging velocity
Die shape
Die/workpiece friction
Initial preform shape
Texture, grain sizes
Material properties/models
Small change in preform shape could lead to
underfill
Composites fiber orientation, fiber spacing,
constitutive model Biomechanics material
properties, constitutive model, fibers in tissues
Materials Process Design and Control Laboratory
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WHY UNCERTAINTY AND MULTISCALING ?

• Uncertainties introduced across various length
  scales have a non-trivial interaction
• Current sophistications resolve macro
  uncertainties

Micro
Meso
Macro

• Use micro averaged models for resolving physical
  scales

• Imprecise boundary conditions
• Initial perturbations

• Physical properties, structure follow a
  statistical description

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UNCERTAINTY ANALYSIS USING SSFEM
F(?)
xn1(?)
X
Bn1(?)
B0
xn1(?)x(X,tn1, ?,)
Key features Total Lagrangian formulation
(assumed deterministic initial configuration) Spec
tral decomposition of the current configuration
leading to a stochastic deformation gradient
Materials Process Design and Control Laboratory
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TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM
VARIABLES
Non-polynomial function evaluations
Scalar operations
Use direct integration over support space

1. Square root
2. Exponential
3. Higher powers

1. Addition/Subtraction
2. Multiplication
3. Inverse

Use precomputed expectations of basis functions
and direct manipulation of basis coefficients
Matrix Inverse
Matrix\Vector operations
Compute B(?) A-1(?)

• Addition/Subtraction
• Multiplication
• Inverse
• Trace
• 5. Transpose

(PC expansion)
Galerkin projection
Formulate and solve linear system for Bj
Materials Process Design and Control Laboratory
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UNCERTAINTY ANALYSIS USING SSFEM
Linearized PVW
On integration (space) and further simplification
Inner product
Galerkin projection
Materials Process Design and Control Laboratory
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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.
Eigenvectors
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
Dominant effect of material heterogeneity on
response statistics
Load vs Displacement
SD Load vs Displacement
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EFFECT OF UNCERTAIN FIBER ORIENTATION
Aircraft nozzle flap composite material,
subjected to pressure on the free end
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)
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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. Constit
utive problem fully deterministic deterministic
evaluation at quadrature points trivially
extend to damage problems.
True PDF Interpolant
FE Grid
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SUPPORT SPACE METHOD SOLUTION SCHEME
Linearized PVW
Support space
GPCE
Galerkin projection
Galerkin projection
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EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
Mean
Uniform 0.02
Using 6x6 uniform support space grid
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PROBLEM 2 EFFECT OF RANDOM VOIDS ON MATERIAL
BEHAVIOR
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FURTHER VALIDATION
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
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PROCESS UNCERTAINTY
Random ? friction
Random ? Shape
Axisymmetric cylinder upsetting 60 height
reduction Random initial radius 10 variation
about mean uniformly distributed Random die
workpiece friction U0.1,0.5 Power law
constitutive model Using 10x10 support space grid
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PROCESS STATISTICS
SD Force
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
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An Information-theoretic Tool for Property
Prediction Of Random Microstructures
Sethuraman Sankaran and Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email ss524_at_cornell.edu,
zabaras_at_cornell.edu URL http//mpdc.mae.cornell.e
du/
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Idea Behind Information Theoretic Approach
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.
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
Information Theory
Linkage?
Information Theory
Statistical Mechanics
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Information 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.

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MAXENT as an optimization problem
Find Subject to
feature constraints
features of image I
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MAXENT 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?

Process data
Component
Experimental microstructure images
Grain size using Heyn intercept method
Macro scale
Obtain PDF of grain sizes using MAXENT
Sampling from the PDF
Materials Process Design and Control Laboratory
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PARALLEL 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
Choose a random grain
Sample properties for the grain conditioned on
other grains
Do domain decomposition for grains
Pre process
At the level of individual processors
Materials Process Design and Control Laboratory
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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

PDF of grain sizes
Grain sizes Heyns intercept method
Materials Process Design and Control Laboratory
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RECONSTRUCTION OF 3D MICROSTRUCTURES
Microstructure samples from the PDF
Input PDF (grain size distribution)
Materials Process Design and Control Laboratory
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RECONSTRUCTION OF ODFs
Orientation distribution function The
probability distribution of orientation of
individual grains in a microstructure
Average of ODF computed from samples
Statistical samples of ODF
Input ODF (expectation value)
Materials Process Design and Control Laboratory
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Implementation of homogenization scheme
Largedef formulation for macro scale Update macro
displacements
Macro-deformation gradient
Homogenized (macro) stress, Consistent tangent
Boundary value problem for microstructure Solve
for deformation field, determine average stress
Consistent tangent formulation (macro)
meso stress, consistent tangent
meso deformation gradient
Integration of constitutive equations Continuum
slip theory Consistent tangent formulation (meso)
Materials Process Design and Control Laboratory
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Study properties of real microstructures
Materials Process Design and Control Laboratory
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Study property variability in a material
Materials Process Design and Control Laboratory
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• ROBUST DESIGN AND OPTIMIZATION WITH UNCERTAINTY
• Extending functional optimization methods from
  the deterministic world
• Non-intrusive optimization methods (based on the
  support method)

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EXAMPLE SIHCP

• 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

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DESIGNING THE OBJECTIVE

• Need to match extra temperature readings at the
  boundary GI

Stochastic temperature solution at the inner
boundary GI, given a guess for the unknown heat
flux q0
Temperature sensor readings with specified
variability at the inner boundary GI

• Try to match above in mean-square sense

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SOLVING THE OPTIMIZATION PROBLEM

• 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 ?

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CONTINUUM STOCHASTIC SENSITIVITY

• 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

Perturbation in PDF
Temperature at a point
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GRADIENT DEFINITION VIZ ADJOINT

• 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

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INCORPORATING LOSS FUNCTION

• 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

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

• 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

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TRIANGLE FLUX PROBLEM

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

X d
Insulated
X 0
X 1
Termperature sensor

• Data generation
• Flux applied to left end following the profile
  see Fig
• Sensor readings polluted with noise collected
  at location x d

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DATA SIMULATING REAL EXPERIMENT
Sensor readings
Large noise level
Small noise level

• Sensor accuracy Vs estimation results
• Estimation of lower moments like mean