ADVANCED COMPUTATIONAL TECHNIQUES FOR

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ADVANCED COMPUTATIONAL TECHNIQUES FOR
MATERIALS-BY-DESIGN
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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DEFORMATION PROCESS DESIGN
DETERMINISTIC PROCESS DESIGN BASED ON CONTINUUM
SENSITIVITY METHOD (CSM)
Preform design for a steering link
Reference problem large flash
First iteration underfill
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PROCESS DESIGN
Preform design for a steering link
Final iteration flash minimized and complete
fill
Objective function
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DESIGNING MATERIALS WITH TAILORED PROPERTIES
Macro problem driven by the macro-design variable
ß
Multi-scale Computation
Micro problem driven by the velocity gradient L
Bn1
Fn1
B0
L L (X, t ß)
Polycrystal plasticity
L velocity gradient
Data mining techniques
Reduced Order Modes
Database
Design variables (ß) are macro design variables
Processing sequence/parameters
Design objectives are micro-scale averaged
material/process properties
Materials Process Design and Control Laboratory
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LIBRARY FOR TEXTURES
Uni-axial (z-axis) Compression Texture
110 fiber family
Feature
q fiber path corresponding to crystal direction
h and sample direction y
z-axis lt110gt fiber (BB)
Materials Process Design and Control Laboratory
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SUPERVISED CLASSIFICATION USING SUPPORT VECTOR
MACHINES
Multi-stage classification with each class
affiliated with a unique process
Tension (T)
Stage 1
Stage 2
Stage 3
Identifies a unique processing sequence Fails to
capture the non-uniqueness in the solution
Given ODF/texture
Materials Process Design and Control Laboratory
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UNSUPERVISED CLASSIFICATION
Find the cluster centers C1,C2,,Ck such that
the sum of the 2-norm distance squared between
each feature xi , i 1,..,n and its nearest
cluster center Ch is minimized.
Each class is affiliated with multiple processes
Cost function
Feature Space
DATABASE OF ODFs
Clusters
Identify clusters
Materials Process Design and Control Laboratory
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INFORMATION FLOW ACROSS SCALES
Engineering
Information flow
Materials
Filtering and two way flow of statistical
information
Chemistry
Physics
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Length Scales ( )
A
1
10
2
10
4
10
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10
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MOTIVATIONUNCERTAINTY IN FINITE DEFORMATION
PROBLEMS
Process
Material
Model
Process
Forging rate Die/Billet shape Friction Cooling
rate Stroke length Billet temperature
Stereology/Grain texture Dynamic
recrystallization Phase transformation Phase
separation Internal fracture Other heterogeneities
Yield surface changes Isotropic/Kinematic
hardening Softening laws Rate
sensitivity Internal state variables
Dependance Nature and degree of correlation
Small change in preform shape could lead to
underfill
Materials Process Design and Control Laboratory
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MOTIVATION
All physical systems have an inherent associated
randomness
Engineering component

• SOURCES OF UNCERTAINTIES
• Multiscale material information inherently
  statistical in nature.
• Uncertainties in process conditions
• Input data
• Model formulation approximations, assumptions.

Heterogeneous random Microstructural features
Why uncertainty modeling ? Assess product and
process reliability. Estimate confidence level in
model predictions. Identify relative sources of
randomness. Provide robust design solutions.
Component reliability
Safe
Fail
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OVERVIEW OF FINITE DEFORMATION DETERMINISTIC
PROBLEM
Governing equation
(1) Multiplicative decomposition framework
(2) State variable based rate-dependent
constitutive models
Linearized principle of virtual work equation
(3) Total/Updated Lagrangian formulation
(4) Various strain and stress measures
(5) Hyperelastic-viscoplastic constitutive
models
Materials Process Design and Control Laboratory
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GENERALIZED POLYNOMIAL CHAOS EXPANSION - OVERVIEW
(Wiener)
Reduced order representation of stochastic
processes. Subspace spanned by orthogonal basis
functions from the Askey series.
Number of chaos polynomials used to represent
output uncertainty depends on
- Type of uncertainty in input - Distribution of
input uncertainty- Number of terms in KLE of
input - Degree of uncertainty propagation
desired
Materials Process Design and Control Laboratory
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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

• Square root
• Exponential
• Higher powers

• Addition/Subtraction
• Multiplication
• 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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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY-MC
RESULTS
MC results from 1000 samples generated using
Latin Hypercube Sampling (LHS). Order 4 PCE used
for SSFEM
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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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EFFECT OF UNCERTAIN FIBER ORIENTATION MC
COMPARISON
Nozzle tip displacement
MC results from 1000 samples generated using
Latin Hypercube Sampling
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MODELING INITIAL CONFIGURATION UNCERTAINTY
xn1(?)x(XR,tn1, ?,)
F(?)
X(?)
xn1(?)
FR(?)
Bn1(?)
B0
XR
F(?)
BR
Introduce a deterministic reference configuration
BR which maps onto a stochastic initial
configuration by a stochastic reference
deformation gradient FR(?). The deformation
problem is then solved in this reference
configuration.
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STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION
UNCERTAINTY
Deterministic simulation- Uniform bar under
tension with effective plastic strain of 0.7 .
Power law constitutive model.
Initial configuration assumed to vary uniformly
between two extremes with strain maxima in
different regions in the stochastic simulation.
Plastic strain 0.7
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STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION
UNCERTAINTY
Stochastic simulation
Results plotted in mean deformed configuration
Plastic strain 0.7
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STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION
UNCERTAINTY
Point at centerline
Point at top
Plastic strain 0.7
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MERITS AND PITFALLS OF GPCE

• Reduced order representation of uncertainty
• Faster than mc by at least an order of magnitude
• Exponential convergence rates for many problems
• Provides complete response statistics
• But.
• Behavior near critical points.
• Requires continuous polynomial type smooth
  response.
• Performance for arbitrary PDFs.
• How do we represent inequalities/eigenvalues
  spectrally ?
• Needs complete code overhaul ?

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NON INTRUSIVE STOCHASTIC GALERKIN (NISG) METHOD
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). Can be
applied to bounded and unbounded spaces (infinite
elements) Can be applied on top of any legacy
code without changing the internal structure
NISG REPRESENTATION OF A 1-D PDF
True PDF Interpolant
FE Grid
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MORE ON SUPPORT-SPACE METHODOLOGY

• We assume that the stochastic input has been
  represented in the KL expansion
• represents the stochastic input vector
• represents the joint PDF of
  inputs
• is
  called the stochastic support space characterized
  by positive input joint PDF
• Any stochastic output can be represented as a
  function defined on this input space
• We consider a finite element piecewise
  representation

Each nodal point on the spatial mesh is linked to
a support-space grid
Statistics of relevance or complete PDF passed on
to the spatial mesh
Spatial domain with finite element grid
Grid on support-space
Importance spaced grid
Support-space of input
Two-level grid approach
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NISG - OVERVIEW
GIVEN STOCHASTIC PROCESS (Finite deformation
problem)
EVALUATION OF MOMENTS
Quadrature integration
Deterministic evaluation at each support space
integration point

• EVALUATION OF PDF
• Least squares projection to nodes
• Subsequent interpolation and MC sampling

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FULL ORDER RELIABILITY ANALYSIS

• Traditional methods FORM/SORM developed in
  the 70s and 80s
• First/Second order approximation of limit state
  function - Complexity increases with higher
  orders
• Reliability index used as an interpretation of
  failure probabilities crude but
  conservative approximation

Full order analysis using NISG
Limit state function
Failure probability
can be evaluated using MC sampling from the
generated PDF
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EXTENSION TO CONTINUUM DAMAGE
Stochastic finite deformation damage evolution
based on Gurson-Tvergaard-Needleman
model. Updated Lagrangian formulation (Anand,
Zabaras et. al.). Material heterogeneity induced
by random distribution of micro-voids modeled
using KLE and an exponential kernel.
Constitutive model
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PROBLEM 2 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
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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
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PROCESS DESIGN IN THE PRESENCE OF UNCERTAINTY
STOCHASTIC OPTIMIZATION USING CONTINUUM
STOCHASTIC SENSITIVITY METHOD ( CSSM )
Problem statement Compute the predefined random
process design parameters which lead to desired
objectives with acceptable (or specified) levels
of uncertainty in the final product.
STOCHASTIC OBJECTIVE FUNCTION
DISCRETE APPROXIMATION
Deterministic evaluation at each support space
integration point corresponds to CSM based
design
Explicit consideration of uncertainty
stochastic design variables with specified
uncertainty Sensitivities computed by
perturbation to the PDF of the stochastic design
variables Objective design in the presence of
uncertainty , not design to minimize uncertainty
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INFORMATION THEORY AND MULTISCALE MODELING?

• Field of mathematics founded by
• Shannon in 1948
• Revolutionized the outlook towards communication
  of information (rigorous mathematical standpoint)
  
• Information Theory used to link simulations at
  various scales in multiscale simulations
• Try to transfer as much information as possible
  about parameters of interest (displacements,
  stresses, strains etc)

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AN INFORMATION THEORETIC VIEWPOINT
Informational entropy of parameter ensembles
during upscaling reduces due to averaging out of
fine details present at micro scale
How much information is required at each scale
and what is the acceptable loss of information
during upscaling to answer performance related
questions at the macro scale ?
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AN INTERESTING ANALOGY
Wavelet Basis at lower scale
Wavelet Basis at higher scale
Information Upscaling Channel
Wavelet based coding of parameters
Information Theoretic upscaling of
wavelet coefficients
Decoding of wavelet parameters
macro scale
micro scale
Source information
Received information
Information lost here
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Motivation Behind MAXENT approach
Motivation 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 extraction tool for features such as
volume fraction, correlation functions, grain
sizes and orientation distribution function
Generate elastic and plastic property statistics
Extract information (limited) about
microstructure
Generate statistics of microstructure
MAXENT toolbox
E.T. Jaynes 1957
The principle of maximum entropy (MAXENT) states
that amongst the probability distributions that
satisfy our incomplete information about the
system, the probability distribution that
maximizes entropy is the least-biased estimate
that can be made. It agrees with everything that
is known but carefully avoids anything that is
unknown.
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Samples of microstructure generated from MAXENT
distribution
Distribution obtained from two-point probability
function alone. Statistics show a uniform trend.
Distribution obtained from two-point and lineal
path functions alone. Statistics show a sharp
trend.
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MAXENT distribution (contd)
Grain size features
Extract features (average grain size and
variance) using Heyn intercept method. Find the
maximum entropic distribution that matches these
Experimental image AA3002 Al alloy
Maximum entropic distribution of grain sizes
Post-processing by reconstructing microstructures
that match the MAXENT grain size distribution
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STOCHASTIC VMS APPLIED TO POLYCRYSTALS
Large scale information captured
Variational consistent upscaling
3c
3b
3a
Meso-to-macro upscaling
Fine scale detail filtered using entropy based
information filter, updation of wavelet
coefficients
Wavelet homogenized properties, state variable
solutions passed from microstructure to subgrid
Macro-state variable statistics obtained by
upscaling the subgrid statistics using a
consistent VMS approach
Spectral stochastic FEM, RFB, Greens function
2a
2b
2c
Spectral stochastic FEM at macro level
Macro-to-meso downscaling
Statistics of large scale solutions for
macro-state variables represented in wavelet
expansions
Evolution of subgrid statistics driven by macro
residuals
Evolution of microstructure driven by
micromechanical models
1a
1b
1c
Component at start of processing stage
Approximate regions with multiscale physics
identified
Subgrid element mapped to microstructure clouds
Subgrid scale model equations
Three-scale VMS

• Three scale stochastic variational multiscale
  framework
• Upscaling and downscaling of statistical
  information is based on information theoretic
  concepts
• Wavelets are used instead of a finite element
  representation for the support-space output
  representation
• Stochastic homogenization applied to upscale
  from microstructure to subgrid scale

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Microstructure homogenization at large strains
Deformation gradient at micro-scale
Homogenized deformation gradient at macro scale
Assumed averaging Law

Fluctuation due to the presence of microstructure
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Microstructure work averaging laws
Boundary condition
Linear boundary condition satisfies deformation
gradient averaging law
Work averaging relation (Hill-Mandel relation)
Work done by external forces on the microstructure
Internal work performed by homogenized stresses
on virtual displacements of the microstructure

Homogenized stresses
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Homogenization of an idealized polycrystal
Evolution of equivalent stresses in a 3D
Idealized microstructure (FCC Al) 512 grains
under simple shear
Homogenized response
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Interrogation of microstructures
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Microstructure sensitive design

• The mechanical properties of polycrystalline
  materials depend upon meso-structural features.
• Can micro-structural features be tailored in
  large scale thermo-mechanical processes to
  achieve desired properties?

• Design methodology
• Statistical learning
• Gradient based continuum sensitivity
• analysis

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Statistical learning based design
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DESIGNING ALLOYS THROUGH STATISTICAL LEARNING
Meshing and virtual experimentation (OOF)
Diffusion coefficients
Phase field model
Thermodynamic variables (CALPHAD) Mobilities
Interfacial energies
Nucleation Models
Property statistics
Design problems 1) Determine the compositions
that give optimum properties
2) Design process sequences to obtain
desired properties
Materials Process Design and Control Laboratory
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Hyperplanes quantify correlation of local length
scale features with the objective and higher
length scale effects
DFT
Electron scale database
Alloy systems
Alloy Al-Ni Composition ?? (Atomistic) Process
variables ?? (Meso-Macro)
Phase Field
Statistical features at the local length scales
sYmax
Homogenized property (OOF)
Processes Microstructures (Phase field)
Desired strength distribution
Objective
Design decisions