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Title: Statistical Microstructure Generation and 3D Microstructure Geometry Extraction


1
Statistical Microstructure Generation and 3D
Microstructure Geometry Extraction
Thesis Overview
  • By Stephen D. Sintay
  • Advisor Anthony Rollett

2
Outline
3
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
4
Motivation 2
5
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.
6
Terminology and scope
7
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

8
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.

9
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

10
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.

11
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.

12
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.

13
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

14
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.
15
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

16
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

17
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.

18
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.

19
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.
20
Snap-to-grid method
21
Snap-to-grid results
edge length 0.1
edge length 2.0
22
Dual grid
23
Dual grid
24
Dual grid
25
Dual grid
26
Dual grid center of mass method
27
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
28
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.

29
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
30
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.

31
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32
AA7075-T651 microstructure observations
33
AA7075-T651 grain size data
RD
ND
TD
34
AA7075-T651 Reconstruction results
Number of grains 4893 Scale 3 mm/voxel Voxel
size 390x300x600 (1170x900x1800 mm)
35
Grain size comparisons
36
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.
37
Molecular dynamic shock wave simulations
38
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
39
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
40
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41
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.

42
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?

43
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.

44
Questions
45
Supplemental Slides
46
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47
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
48
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?).
49
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).
50
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
51
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
52
Grow grains from ellipsoid seeds
51 grains 500 ND x 1000 RD x 500 TD mm box.
53
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
54
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

55
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

56
Model vertex identification and connectivity
57
Higher order dual grid methods
58
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.
59
Rolled product nomenclature
60
Grain aspect ratios
61
Constituent particles
62
AA7075-T651 Texture
63
Specimen overview
1) Load RD
  • Loading conditions
  • Kt 3 notch stress
  • Spectrum loading applied (variable amplitude)
  • Laboratory air

64
Data collection procedure
e-
ION-
EBSD
65
Marker bands at peak stress loading
66
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
67
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
68
EBSD data IPF Maps
Bore hole surface
EBSD scan area
69
Crack plane and Crystal lattice
70
Schmid Factor
71
Crack plane and crystal lattice
72
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