Statistical Microstructure Generation and 3D Microstructure Geometry Extraction

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Statistical Microstructure Generation and 3D
Microstructure Geometry Extraction
Thesis Overview

• By Stephen D. Sintay
• Advisor Anthony Rollett

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Outline
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Motivation 1
1. Observe microstructure properties
2. Generate implicit representations of the
microstructure that match the statistics of the
observed properties
3. Extract explicit models of the microstructure
for analysis of properties and performance
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Motivation 2
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Hypothesis
2D images
3D images
Hypothesis It is possible to define an explicit
geometric representation of a grain boundary
network from a 3D image that simultaneously
satisfies (a) maximum deviation between the
explicit boundaries and implicit boundaries of
the image (b) minimum local curvature.
Secondarily, upon using this definition, it is
possible to quantify maximum uncertainty in (a)
the inclination angle of planar interface
segments and therefore interface dihedral angles,
and (b) grain volume, surface area, and mean
width.
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Terminology and scope
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Literature review 1
Saylor, D., J. Fridy, B. El-Dasher, K.-Y. Jung,
and A. Rollett (2004). Statistically
representative three-dimensional microstructures
based on orthogonal observation sections.
Metallurgical and Materials Transactions A 35(7),
19691979.
mbuilder

• Statistical microstructure generation based on
  fully annealed AA1050.
• Grains are represented as ellipsoids, but the
  input and output distribution are not defined.
• Implicit 3D image

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Literature review 2
Brahme, A., M. H. Alvi, D. Saylor, J. Fridy, and
A. D. Rollett (2006). 3D reconstruction of
microstructure in a commercial purity aluminum.
Scripta Materialia 55(1), 7580.
Anisotropic stretching

• Statistical microstructure generation based on
  rolled AA1050 by stretching initially isotropic
  grain structure.
• Explicit geometry defined by grouping Voronoi
  cells.

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Literature review 3
Groeber, M., M. Uchic, D. Dimiduk, Y. Bhandari,
and S. Ghosh (2007). A framework for automated 3D
microstructure analysis and representation.
Journal of Computer-Aided Materials Design 14(0),
6374.
SIRI-3D

• 3D microstructure of Inconel100 is observed by
  FIB milling and EBSD scanning.
• The observed microstructure is reconstructed.
• Statistics of grains and interfaces recorded
  including, size, shape, number of neighbors
• Implicit microstructures generated wherein the
  ellipsoids are generated to statistically match
  the input grains size and shape.
• Explicit solid model is defined using Voroni
  tessellation
• Surface and volume meshes are generated

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Literature review 4
Moore, H. M. (2007). Three-Dimensional
Computational Modeling of Polycrystalline
Materials. Ph. D. thesis, Carnegie Mellon
University.
Voxel subdivision for marching tetrehedra. Up to
12 tetrehedra may be created for each voxel
Original and decimated microstructure model of
reconstructed MgO.

• 3D microstructure of a MgO are reconstructed
• Explicit solid model is defined by first using
  marching tetrahedra.
• The resulting surface mesh is decimated, while
  maintaining triangle quality and grain volume.
• Volume meshes are generated and finite element
  analysis is conducted.

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Literature review 5
Wu, Z. and J. M. J. Sullivan (2003). Multiple
material marching cubes algorithm. International
Journal for Numerical Methods in Engineering
58(2), 189207.
Original
Example of voxel subdivision for generalized
marching cubes. At least 2 triangles generated
for each voxel on the surface of the grain.
Smoothed

• Iso-surface extraction method for multiple
  region data.
• Resolves topology through adding additional
  nodes (and triangles) to the cube face centers
  and the cube body center.
• Typically followed by surface smoothing,
  triangle decimation, and/or surface re-meshing.

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Literature review 6
Wang, C. C. L. (2007). Direct extraction of
surface meshes from implicitly represented
heterogeneous volumes. Computer-Aided Design
39(1), 3550.

• Geometry extraction from non-manifold
  multi-region implicit data using polygon soup
• Following geometry extraction the explicit
  geometries are re-meshed.

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Literature review 7a
Dillard, S. E., J. F. Bingert, D. Thoma, and B.
Hamann (2007). Construction of simpli?ed boundary
surfaces from serial-sectioned metal micrographs.
Visualization and Computer Graphics, IEEE
Transactions on 13(6), 15281535.

• Reconstructed serial section data of Tantalum.
• Dual grid based method where the dual grid
  constrains the location of nodes and triangles
• Initial iso-surface from marching tetrahedra.
• Decimation and smoothing is conducted
  simultaneously

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Literature review 7b
Dillard, S. E., J. F. Bingert, D. Thoma, and B.
Hamann (2007). Construction of simpli?ed boundary
surfaces from serial-sectioned metal micrographs.
Visualization and Computer Graphics, IEEE
Transactions on 13(6), 15281535.
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Literature review 8
Cohen, J., A. Varshney, D. Manocha, G. Turk, H.
Weber, P. Agarwal, F. Brooks, and W. Wright (1996
of Conference). Simpli?cation envelopes. In
Proceedings of the 23rd annual conference on
Computer graphics and interactive techniques,
Series Simpli?cation envelopes. ACM.

• Re-meshing of surface elements is accomplished
  through mesh decimation
• Final surface is bounded by offsets of the
  original surfaces.
• A local (removal and re-meshing of one vertex)
  and global (removal and re-meshing of multiple
  vertices is accomplished.
• Authors state that it is difficult to know what
  neighborhood size should be used for global
  re-meshing

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Literature review 9
Kuprat, A., A. Khamayseh, D. George, and L.
Larkey (2001). Volume conserving smoothing for
piecewise linear curves, surfaces, and triple
lines. Journal of Computational Physics 172(1),
99118.

• Work with explicit microstructures for
  physics-base simulations of grain growth.
• They conclude that more than one vertex in the
  smoothing algorithm will dramatically increase
  the ability of the algorithm to smooth the object
• There is no constraint on node location and
  ultimately the method will not preserve sharp
  features

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Literature review 10
Wright, S. I. and R. J. Larsen (2002). Extracting
twins from orientation imaging microscopy scan
data. Journal of Microscopy 205(3), 245252.

• 2D Microstructure geometry extraction process in
  TSL OIM Data Analysis Software
• Connects triple junctions first and then deforms
  the line to conform to the boundary
• 3D implementation by S. Lee is termed
  Constrained Line Straightening (CLS)
• Smooths triple lines, and grain boundary surfaces
  independently.

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Microstructure geometric modeling Problem
Statement

• Voxelized 3D images often need to be converted to
  3D finite element meshes.
• The images contain multiple materials and/or
  grains with wide range of sizes and small angles
  between surfaces.
• Conventional Marching Cubes and marching
  tetrahedra algorithms generated very fine meshes.
• The surfaces/interfaces/boundaries need to be
  smoothed and decimated in order to efficiently
  model the geometry of the polycrystalline
  material.
• Generating high quality meshes is challenging,
  although some software solutions exist (not well
  tested).
• Such tools are also needed for modeling
  microstructure-property relationships.

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Microstructure geometry extraction
Solid modeling is concerned with the
construction and manipulation of unambiguous
computer representations of solid objects. These
representations permit us to distinguish the
interior, the boundary, and the complement of a
solid. . . .
Rossignac, J. R. and A. A. G. Requicha (1991).
Constructive non-regularized geometry.
Computer-Aided Design 23(1), 2132.
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Snap-to-grid method
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Snap-to-grid results
edge length 0.1
edge length 2.0
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Dual grid
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Dual grid
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Dual grid
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Dual grid
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Dual grid center of mass method
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Dual grid center of mass results
Snap-to-grid with edge length 2.0
Dual grid with loose corrections-constraint
Dual grid with tight corrections-constraint
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Lines at an angle

• Lines at an arbitrary angle are defined by
  exactly two nodes.
• However, there is discrete set of inclined
  lines that can be reconstructed.

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Uncertainty of inclination angle
Case 1 Line parallel to grid
The uncertainty of the inclination angle, Uq , is
defined as, the allowable angular deviation of a
reconstructed line such that it does not violated
a dual grid node. It is both a function of the
line length and the angle of the line. Uq , can
be defined using the dot product between the
reconstructed line and the lines defining the
rotation limits.
Case 2 Line with more than two risers
Case 3 Line with one riser and two treads
Case 4 Line with one riser and one tread
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Discussion of Dual grid method
Hypothesis It is possible to define an explicit
geometric representation of a grain boundary
network from a 3D image that simultaneously
satisfies (a) maximum deviation between the
explicit boundaries and implicit boundaries of
the image (b) minimum local curvature.
Secondarily, upon using this definition, it is
possible to quantify maximum uncertainty in (a)
the inclination angle of planar interface
segments and therefore interface dihedral angles,
and (b) grain volume, surface area, and mean
width.

• Primary
• Maximum deviation is satisfied by the
  reconstruction algorithm that places the nodes of
  the line at the center of mass of the dual grid
  and that clearly defines corrections conditions
  and correction procedures.
• Minimum local curvature is satisfied by linking
  the center of mass of the dual grid for all
  interfaces with a constant tread size.

• Secondary
• Inclination angle uncertainty is defined for all
  cases of lines reconstructed using the center of
  mass of the linear elements of the dual grid.
• It remains to address the geometric measures of
  grain volume, surface area, and mean width.

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AA7075-T651 microstructure observations
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AA7075-T651 grain size data
RD
ND
TD
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AA7075-T651 Reconstruction results
Number of grains 4893 Scale 3 mm/voxel Voxel
size 390x300x600 (1170x900x1800 mm)
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Grain size comparisons
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AA7075 geometry extraction results
Pseudo 3D image generated by slicing the implicit
model in 3 orthogonal directions, extracting the
geometry, and then re-stacking the 2D data.
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Molecular dynamic shock wave simulations
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Progressive MD segmentation
Atom data as input
Initial coarse segmentation with VF 1.0
Progressive segmentation VF 1.0 0.0
Final after constrained MC grain growth
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Pseudo 3D MD geometry results
Pseudo 3D geometry constructed by slicing the MD
voxel data along 3 orthogonal directions Voxel
dimensions 102x102x193 Number of slices
42x42x76
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Research plan

• Code development
• Implementation of 3D snap to grid methods
• 3D snap to grid methods will provide a rapid
  alternative solution for 3D geometry extraction
  while trading off accuracy.
• Implementation of 3d dual grid methods
• 3D dual grid methods will provide an error free
  geometry extraction strategy from implicit data
  with clear quantification of the uncertainty of
  geometric features.

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Research plan

• Evaluation of geometry extraction methods
• Evaluate 2D and 3D methods on reconstruction of
  known explicit geometries. The geometries will be
  rotated and sampled at different grid/size
  ratios.
• Test standard geometries for accuracy
• 2D standard tests
• circle, square, single dihedral angle, real
  grain boundary network
• 3D standard tests
• sphere, cube, single dihedral angle with
  surfaces, real 3D grain boundary network
• Measure discrepancies as a function of grid step
  size for
• volume, surface area area, mean width, interface
  dihedral angles.
• Characterize the efficiency of geometric
  representation
• Compare node count with other extraction methods
• Can you mesh the structure efficiently?

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Research Plan

• Generating explicit 3D geometries of
  polycrystalline materials
• AA7075-T651
• Generate models suitable for finite element
  meshing to conduct studies of Microstructurally
  Small Fatigue Crack incubations, nucleation, and
  growth.

• Molecular Dynamic data
• Generate models to study the temporal evolution
  of shock induced solid state bcc -gt hcp phase
  transformation.

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Questions
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Supplemental Slides
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Stereology of mbuilder I

• Dispersion of mono-spheroid, mono-size, low
  volume fraction features. i.e. Particles are only
  spheroid (sphere, oblate, or prolate spheroid)
  with known size (r, a and b, or a/b) . Solutions
  are known 1,2 for f(r) or f(a,c)
• Dispersion of poly-spheroid, mono-size, low
  volume fraction features. i.e. Particles are one
  of many spheroids, with known sizes (r, a and b,
  or a/b). Numerical methods are applicable to
  determine f(r) and f(a,c)
• Dispersion of poly-spheroid, poly-size, low
  volume fraction features. i.e. Particles are one
  of many spheroids, with many unknown sizes (ri,
  ai and bi, or ai/bi). Numerical methods are
  applicable to determine f(ri) and f(ai,ci )
• Dispersion of poly-ellipsoid, poly-size, low
  volume fraction features. i.e. Particles are one
  of many ellipsoids, with many unknown sizes
  (ai,bi, and ci). Numerical methods are applicable
  3,4 to estimate f(a,b,c)
• Dispersion of poly-shape, poly-size, low volume
  fraction features. i.e. Particles are real. No
  numerical methods have been determined.
• Dispersion of poly-shape, poly-size, High volume
  fraction (space filling) features. i.e. Grains
  are real. No numerical methods have been
  determined.

ai bi known, ci known
and
ri known
a , b , c unknown
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Stereology of mbuilder II
1. Grains are assumed to be ellipsoidal.
2. Observe the distribution of ellipses from
orthogonal planes to define fg(a?, b?),
fg(a?, c?), fg(b?, c?).
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Generating ellipsoids for grains
Method 2 1) Randomly sampling Fgen(a?, b?) and
generating a set of ellipsoids, ei.
The purpose of ellipsoid generation is to define
a set of ellipsoids such that fgen(a?, b?)
fdata(a?, b?), Where , gen, indicates the model
distribution.
2) Define fdata(a?, b?) by slicing ei many
times.

• Method 1
• Assume ltagt ap/4, and ltbgt bp/4.
• This allows direct method to define f(a,b,c).
• Sample f(a,b,c) until desired number of
  ellipsoids, ei, are generated.

4) Assume that the optimized set of ei is an
accurate representation of f(a,b,c).
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Population of Ellipsoids for grain structure I
4B,5B Or else use a CA to fill space based on
ellipsoid centers, sizes
1. Geometric Configuration
Step 1. Randomly selected points in a box
Step 4A. Voronoi tessellationset of
perpendicular bisecting planes, delimiting cells
Step 2. Population of Ellipsoids Monte Carlo
(simulated annealing) to minimize overlap, gaps
Step 3. A subset of ellipsoids ? each point
belongs to one ellipsoid only
Step 5A. Final configurationgrains represented
by sets of cells
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Population of ellipsoids for grain structure II
Transformations on ellipsoidsAddSubtractSw
apSubstitute
Overlap cost (energy) a Overlap
encouragement (0.95) w Zero penalty (1.0)
z Ellipsoid function
E
EllipsoidCenter x,y,zSemi-axes a,b,c
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Grow grains from ellipsoid seeds
51 grains 500 ND x 1000 RD x 500 TD mm box.
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Monte Carlo (Potts) Method
Triangular 2D grid
Square 3D grid
1-6 six 1st nearest neighbors 7-18 twelve 2nd
nearest neighbors 19-26 eight 3rd nearest
neighbors
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MC grain growth on CA grains

• Pros
• Relaxes ellipsoid geometry
• Grain boundaries are more natural
• Grain size distribution more natural
• Cons
• - Grains shapes quickly loose anisotropy

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Partial Entity Structure 1

• Three non-manifold topology conditions in
  multiple region data.
• Two regions intersect with a common face
• Three regions intersect with a common line
• Two or more regions intersect with a common point.

• Three partial entities to represent the
  non-manifold conditions
• Partial Face (pf1,pf2)
• Partial edge (pe1, pe2, pe3)
• Partial vertices

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Model vertex identification and connectivity
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Higher order dual grid methods
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Island region connectivity exception
Step 1 Identify island region exists by finding
grain that only has two neighbors Step 2
Eliminate on candidate termination point by
region IDs Step 3 Of the remaining locations
find the appropriate termination point based on
the permutation of regions IDs. Only vertices
with opposite permutations can be connected.
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Rolled product nomenclature
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Grain aspect ratios
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Constituent particles
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AA7075-T651 Texture
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Specimen overview
1) Load RD

• Loading conditions
• Kt 3 notch stress
• Spectrum loading applied (variable amplitude)
• Laboratory air

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Data collection procedure
e-
ION-
EBSD
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Marker bands at peak stress loading
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Data collection procedure
IOM milling Position
ION Imaging
e-
ION-
EBSD
ION Milling parameters Beam Current 7 nA Slice
200 mm x 0.5 mm Time 6 min
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Data collection procedure
EBSD Position
SEM imaging
EBSD collection parameters Scan 150 mm x 50 mm
(0.5 mm step) Points/sec 120 Time 4 min
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EBSD data IPF Maps
Bore hole surface
EBSD scan area
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Crack plane and Crystal lattice
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Schmid Factor
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Crack plane and crystal lattice
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