Advanced computational techniques for materialsbydesign

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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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AFOSR 06 DURIP AWARD MPDC CLUSTER
Hardware

• File server
• 4 Intel Xeon CPUs 3.2 GHz
• 4G Memory
• 1T SCSI RAID Hard drive
• 2 network cards (1G)
• 64 computing nodes
• 2 Intel Xeon CPU 3.8 GHz
• 2G Memory
• 160G SATA Hard drive
• 2 network cards (1G)
• 1G Network connection

Software

• Redhat Linux AS 3
• Open PBS batch job system
• PETSc, ParMetis, MPDCFemLib
• QuantumEspresso,

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OUTLINE

• Multiscale modeling and design
• Control of microstructure-sensitive material
  properties
• Stochastic deformation modeling
• GPCE, stochastic Galerkin (non-intrusive, Smolyak
  collocation, etc.)
• Robust design (non-intrusive stochastic
  optimization of SPDE systems)
• Interrogation of polycrystal microstructures
• Homogenization techniques
• Computing microstructure PDFs using limited
  information
• Maximum entropy methods
• Conclusions

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MULTISCALE MODELING Process-property-structure
triangle
Design properties
Control process parameters
Improve performance
Materials Process Design and Control Laboratory
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Process optimization using phenomenological
material models
Preform optimization of a Ti-64 cross shaft
Flash
Initial Iteration
Underfill
Intermediate Iteration
Maximize volumetric yield
Final
Final Iteration
No Flash
Materials Process Design and Control Laboratory
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PHYSICAL APPROACH TO PLASTICITY

• Crystallographic orientation
• Rotation relating sample
• and crystal axis
• Properties governed by orientation

• Discrete aggregate of crystals
• (Anand et al.)
• Comparing quantifying textures

• Continuum representation
• Orientation distribution function
• (ODF)
• Handling crystal symmetries
• Evolution equation for ODF

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ORIENTATION DISTRIBUTION FUNCTION (ODF)
Why continuum approach for ODF?
EVOLUTION EQUATION FOR THE ODF (Eulerian)
Conservation principle Texture can be described,
quantified compared Based on the Taylor
hypothesis Eulerian Lagrangian forms
v re-orientation velocity how fast are the
crystals reorienting r current orientation of
the crystal. A is the ODF, a scalar field

• Constitutive sub-problem
• Taylor hypothesis
• deformation in each
• crystal of the
• polycrystal
• is the macroscopic
• deformation.
• Compute the
• reorientation velocity
• from the spin

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MULTI-LENGTH SCALE SENSITIVITY ANALYSIS
The velocity gradient depends on a macro design
parameter
A micro-field depends on a macro design
parameter (and) the velocity gradient as
Sensitivity of the velocity gradient driven by
perturbation to the macro design parameter
Sensitivity of this micro-field driven by the
velocity gradient
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MATERIAL POINT SIMULATOR DESIGN FOR SPECIFIC
MATERIAL RESPONSE
Design for the strain rate such that a desired
material response is achieved
Material 99.98 pure f.c.c Al
Materials Process Design and Control Laboratory
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Multilength scale process design control of
microstructure-sensitive properties
Objective Design the initial preform such that
the die cavity is fully filled and the yield
strength is uniform over the external surface
(shown in Figure below). Material FCC Cu
Multi-objective optimization

• Increase Volumetric yield
• Decrease property variation

Materials Process Design and Control Laboratory
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Multi-scale design OFHC Copper closed die
forging Iteration 7
Underfill
Optimal fill
Yield strength (MPa)
Optimal yield strength
Materials Process Design and Control Laboratory
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Extrusion design problem
Objective Design the extrusion die for a fixed
reduction such that the deviation in the Youngs
Modulus at the exit cross section is
minimized Material FCC Cu
Young's Modulus Distribution Material point
sub-problem -
In sample coordinates
In crystal coordinates
Minimize Youngs Modulus variation across
cross-section
ODF
Die design for improved properties
Polycrystal average in sample coodinates
Youngs modulus (along sample x-axis)
Materials Process Design and Control Laboratory
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Multiscale Extrusion Control of Youngs Modulus
Iteration 5
Optimal solution
Materials Process Design and Control Laboratory
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STOCHASTIC MATERIALS MODELING MOTIVATION
Information flow across scales
Material heterogeneity

• Material information inherently statistical in
  nature.
• Atomic scale Kinetic theory, Maxwells
  distribution etc.
• Microstructural features correlation functions,
  descriptors etc.

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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY GPCE
approach
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 GPCE
approach
Dominant effect of material heterogeneity on
response statistics
Load vs Displacement
SD Load vs Displacement
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Effect of random voids on material response
Support space approach
Material heterogeneity induced by random
distribution of micro-voids modeled using KLE
and an exponential kernel. Gurson type model for
damage evolution
Mean
Uniform 0.02
Using 6x6 uniform support space grid
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Effect of random voids on material behavior
Suppose space approach
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PROCESS UNCERTAINTY
Random ? friction
Random ? Shape
Axisymmetric cylinder upsetting 60 height
reduction (Initial height 1.5 mm) Random initial
radius 10 variation about mean (1 mm)
uniformly distributed Random die workpiece
friction U0.1,0.5 Power law constitutive
model Using 10x10 support space grid
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PROCESS STATISTICS Support space approach
SD Force
Force
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PROCESS STATISTICS
Relative Error
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ROBUST DESIGN OF DEFORMATION PROCESSES
Design Objective
Probability Constraint
Norm Constraint
SPDE Constraint
Augmented Objective
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A CONTINUUM STOCHASTIC SENSITIVITY SCHEME (CSSM)
Compute sensitivities of parameters with respect
to stochastic design variables by defining
perturbations to the PDF of the design variables.
CSSM problem decomposed into a set of CSM problems
Decomposition based on the fact that
perturbations to the PDF are local in nature
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NISG APPROXIMATION FOR OBJECTIVE FUNCTION
Design Objective unconstrained case
Set of NelEn objective functions
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BENCHMARK APPLICATION
Flat die upsetting of a cylinder
Case 1 Deterministic problem Case 2 1 random
variable (uniformly distributed) friction 66
variation about mean (0.3) (10x1 grid) 1D
problem Case 3 2 random variables (uniformly
distributed) friction (66) and desired shape
(10 about mean) (10x10 grid) - 2D problem
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OBJECTIVE FUNCTION
Deterministic problem - optimal solution
Deterministic problem
1D problem
2D problem
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DESIGN PARAMETERS
Initial guess parameters
Deterministic problem
2D problem
Mean
SD
1D problem
Mean
SD
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OBJECTIVE FUNCTION
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FINAL FREE SURFACE SHAPE CHARACTERISTICS
Mean
SD
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MULTISCALE NATURE OF MATERIAL HETEROGENEITIES
Present method Assume correlation between macro
points Decompose using KLE
Fine scale heterogeneities
grain size, texture, dislocations
Coarse scale heterogeneities
macro-cracks, phase distributions
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Motivation for Maximum Entropy Approach
Different statistical samples of the manufactured
specimen
When a specimen is manufactured, the
microstructures at a sample point will not be the
same always. How do we compute the class of
microstructures based on some limited information?
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PDF of microstructures (topology) and its features
Given Microstructures at some points Obtain
PDF of microstructures
ODF (a function of 145 random parameters)
Grain size
Know microstructures at some points
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The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure
Geometrical grain size Texture ODFs
Compute a PDF of microstructures
MAXENT
Compute bounds on macro properties
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Generating input microstructures Phase field
method

• Isotropic mobility (L1)
• Discretization
• problem size 75x75x75
• Order parameters
• Q20
• Timesteps 1000
• First nearest neighbor approx.

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Input microstructural samples
2D microstructural samples
3D microstructural samples
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The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure Geometrical
grain size Texture ODFs
Compute a PDF of microstructures
MAXENT
Compute bounds on macro properties
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Microstructural feature Grain sizes
Grain size obtained by using a series of
equidistant, parallel lines on a given
microstructure at different angles. In 3D, the
size of a grain is chosen as the number of voxels
(proportional to volume) inside a particular
grain.
2D microstructures
Grain size is computed from the volumes of
individual grains
3D microstructures
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The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure Geometrical
grain size Texture ODFs
Tool for microstructure modeling
Compute a PDF of microstructures
MAXENT
Compute bounds on macro properties
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MAXENT as an optimization problem (E.T.Jaynes
1957)
Find
feature constraints
Subject to
features of image I
Lagrange multiplier approach
Lagrange multiplier optimization
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Gradient Evaluation

• Objective function and its gradients
• Infeasible to compute at all points in one
  conjugate gradient iteration
• Use sampling techniques to sample from the
  distribution evaluated at the previous point
  (Gibbs sampler)

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The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure Geometrical
grain size Texture ODFs
Tool for microstructure modeling
Compute a PDF of microstructures
MAXENT
Compute bounds on macroscopic properties
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Microstructure modeling the Voronoi structure
p1,p2,,pk generator points.
Cell division of k-dimensional space
Voronoi tessellation of 3d space. Each cell is a
microstructural grain.
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Stochastic modeling of microstructures
Sampling using grain size distribution
Sampling using mean grain size
Match the PDF of a microstructure with PDF of
grain sizes computed from MaxEnt
Each microstructure is referred to by its mean
value.
Strongly consistent scheme
Weakly consistent scheme
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Algorithm for generating voronoi centers
Given grain size distribution Construct a
microstructure which matches the given
distribution
Generate sample points on a uniform grid from
Sobel sequence
No
Forcing function
Yes
stop
Rcorr(y,d)gt0.95?
Objective is to minimize norm (F). Update the
voronoi centers based on F
Construct a voronoi diagram based on these
centers. Let the grain size distribution be y.
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ODF reconstruction using MAXENT
Representation in Frank-Rodrigues space
Input ODF
Reconstructed samples using MaxEnt PDF of textures
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Ensemble properties
Expected property of reconstructed samples of
microstructures
Input ODF
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The main idea
Experimental microstructures
Phase field simulations
Extract features of the microstructure Geometrical
grain size Texture ODFs
Tool for microstructure modeling
Compute a PDF of microstructures
MAXENT
Compute bounds on macroscopic properties
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3D random microstructures evaluation of
property statistics
Given some known information on lower-order
microstructure moments (limited samples), compute
the grain size and ODF probability distributions
using the MaxEnt technique as well as the PDF of
the homogenized material properties.
Input constraints macro grain size observable.
First four grain size moments, expected value of
the ODF are given as constraints.
Output Entire variability (PDF) of grain size
and ODFs in the microstructure is obtained.
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Grain size distribution computed using MaxEnt
0.25

Grain volume distribution
using phase field simulations
pmf reconstructed using MaxEnt
0.2
0.15
Probability mass function
Comparison of MaxEnt grain size distribution with
the distribution of a phase field microstructure
0.1
K.L.Divergence0.0672 nats
0.05
0

0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Grain volume (voxels)
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Reconstructing strongly consistent microstructures
Computing microstructures using the Sobel
sequence method
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Reconstructing strongly consistent
microstructures (contd..)
Computing microstructures using the Sobel
sequence method
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(First order) homogenization scheme

• Microstructure is a representation of a material
  point at a smaller scale
• Deformation at a macro-scale point can be
  represented by the motion of the exterior
  boundary of the microstructure (Hill 1972)

Materials Process Design and Control Laboratory
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COMPUTING THE PDF OF HOMOGENIZED PROPERTIES
Computing the variability in the strength of a
polycrystal induced from microstructural
uncertainty
Aluminium polycrystal with rate-independent
strain hardening. Pure tensile test.
Statistical variation of homogenized
stress-strain response
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Many important applications of MaxEnt

• Stochastic data-driven simulations
• Account for the propagation of the information
  (uncertainty) not captured in the data
• Stochastic multiscale modeling and design
• Use as a downscaling tool
• Quantify statistical assumptions across
  length-scales
• how (statistical) information propagates across
  scales?
• stochastic homogenization
• Model processes on distributions of
  microstructures rather than on particular
  realizations
• With one simulation capture both heterogeneities
  and randomness
• Diffusion, advection, deformation, fracture