The Shapes of a Three Dimensional Grain Growth Microstructurelazar/TMS.pdf · 2012-07-17 · The...

The Shapes of a Three DimensionalGrain Growth Microstructure

Emanuel A. Lazar1

Jeremy K. Mason2, Robert D. MacPherson1, David J. Srolovitz3

1Institute for Advanced Study (Princeton, NJ)2Lawrence Livermore National Laboratory

3A*Star, Singapore

1st International Conference on 3D Materials ScienceSeven Springs, Pennsylvania

July 2012

The Shapes of a 3D Microstructure Lazar, Mason, MacPherson, and Srolovitz

1. New idea

2. Cool data


The challenge


Two-dimensional characterization

Stainless steel

Yoda et. al., Materials Characterization 61, 913 (2010)


Two-dimensional characterization


Three Dimensions


Three-dimensional characterization

Simulated 3D grain-growth microstructure



ISIJ Internationa], Vol. 38 (1998), No. 6

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Variation of microstructural entropy, S,with time.

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The variation in the distribution of thenumberof faces, Nf' of individual grainswith isotropic grain boundary.

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Fig. 8.

The variation in the distribution of thenumberof faces, Nf' of individual grainswith anisotropic grain boundary,

N-faced grain is proportional to the numberof faces.This result is equivalent to that reported by Andersonet al.4) As shownin Fig. 12, the effect of anisotropy onthe Nf~average size relationship is not significant. So, the

broader distribution curve of the numberof the faces inthe grain with anisotropic grain boundaryenergy is likely

to reflect the grain size distribution.Therelationship betweenthe average numberof faces

563 C 1998 ISIJ

Saito, ISIJ International 38, 6 (1998)



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A problem



Grains with six faces


A solution



Botanical Society of Americahttp://www.jstor.org/stable/2437492 .

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Matzke, American Journal of Botany 33, 1 (1946)



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Another problem



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The solution!



IEEE TRASSlCTIONS ON CIRCUIT THEORY VOL. CT-13, NO. 2 JKJNE 1966

that for unity G. and the coordinate point for Mmin is located between those for G,,., and Fmi,.

REFERENCES

VII. CONCLUSIONS

In this paper it has been shown that the noise measure of linear two-ports can be expressed in terms of the gain parameters, the noise parameters, and the source admittance. Constant M loci have been represented in the source admittance plane as well as constant G, and constant F loci. Transformations of these loci into the Smith- chart have been made, since this is a very convenient way of representing a wide range of immittances. Such a mapping is useful when the gain and noise performance of amplifiers is investigated, especially in the microwave region. By looking at the noise measure chart, for example, it is easy to deduce what source admittance gives the best overall noise performance and how it deteriorates as the admittance differs from its optimum value. These effects were demonstrated using a microwave transistor as an example.

PI 121 [3!

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DOI

“IRE Standards on Electron Tubes: Methods of Testing,” 1962, 62 IRE 7 Sl, pt. 9: “Noise in linear twoports.” H. Rothe and W. Dahlke, “Theory of noisy fourpoles,” Proc. IRE, vol. 44, pp. 811-815, June 1956. H. A. Haus and R. B. Adler, Circuit Theory 0s Linear Noisy Networks. New York: Wiley, 1959. -, “Invariants of linear noisy networks,” 1956 IRE Conv. Rec., pt. 2, pp. 53-67, 1956. J. G. Linvill and J. F. Gibbons, Transistors and Active Circuits, New York: McGraw-Hill, 1961. The authors gave a wrong expression for available power gain in their book, probably due to typographical errors. J. C. Slater, “Microwave Electronics,” Rev. Mod. Phys., vol. 16, pp. 441-512, October 1946. P. H. Smith. “Transmission-line calculator,” Electronics, vol. 12, pp. 2%31, January 1939; and “An improved transl mission-line calculator,” Electronics, vol. 17, p. 130, January 1944. H. A. Haus and R. B. Adler, “An extension of the noise figure definition,” Proc. IRE (Correspondence), vol. 45, pp. 690-691, May 1957. 1% Fukti-, “The characteristics of Esaki diodes at microwave frequencies,” 1961 ISSCC Digest of Tech. Papers, pp. 16-17, 1961; and J. IECE (Japan), vol. 43, pp. 1351-1356, November 1960 (in Japanese). -, [‘The noise performance of microwave transistors,” IEEE Trans. on Electron Devices, vol. ED-13, pp. 329341, March 1966.

A Simple and Isomorphism

Efficient Algorithm for Determining of Planar Triply Connected Graphs

LOUIS WEINBERG, FELLOW, IEEE

Absfract-A problem that arises in many practical applications of linear graphs and in some purely mathematical applications is the determination of the isomorphism of two given graphs; such applications include the automatic retrieval of information, machine translation of languages, pipeline and electric networks, pattern recognition, graph-theoretic enumeration problems, the four-color conjecture, and the problem of “squaring the rectangle.” Up to the present time only heuristic programs have been developed for solving this problem for general graphs. In this paper, a solution for a class of graphs is given in terms of a simple and computationally efficient algorithm, which is ideally suited for computer program- ming; a program in MAD language has been written but has not yet been run. The class of graphs considered are planar and triply connected; this class arises, for example, in the four-color problem and in the problem of squaring the rectangle. The algorithm is applicable to both directed and undirected graphs and to simple graphs and multigraphs. The algorithm is based on TrGmaux’s procedure for generating an Euler path in a graph. Application of the algorithm to a plane triply connected graph yields a set of vector codes which are ordered lexicographically in a code matrix.

Manuscript received August 17, 1965; revised February 17, 1966. This research was jointly supported by the Radar Laboratory under Project Michigan, Department of the Army, Contract DA 2%043-AMC-00013(E), and by the Institute of Science and Tech- nology, University of Michigan, Ann Arbor, Mich.

The author is with the Department of Electrical Engineering, City College of City University of New York, N. Y. He was formerly with the Department of Electrical Engineering and The Institute of Science and Technology, University of Michigan.

Two isomorphism theorems are then stated and proved; one theorem asserts that two-plane triply connected graphs are isomorphic if and only if their code matrices are equal. Finally, two significant examples are used as illustrations of how the basic procedure may be shortened by combining it with a check on some simple necessary conditions so that it becomes almost computationally trivial, that is, large graphs can be tested quickly “by hand.”

I. INTRODUCTION

N THIS PAPER we solve a problem that arises in many practical and theoretical applications of linear graphs. Familiar practical applications are electrical

or pipeline networks; less familiar ones are computer programs and computer storage-allocation schemes. Some recent work on pattern recognition has also used linear graphs, where given patterns are matched with pre- viously available classified patterns. For example, the pattern for a chemical molecule may be compared with a dictionary of patterns of molecules; patterns of language structure are represented by trees so that syntactic analysis and automatic translation of languages may be carried out; finally, in document retrieval and the automatic retrieval of information, a linear graph representation of the desired information is compared with the linear graphs

142



1966 WEINBERG : ISOMORPHISM OF TRIPLY CONNECTED GRAPHS 147

Fig. 1. The first pair of graphs to be tested for isomorphism.

X’@ , A’) = 1231341451567818287929697654321. (2) The corresponding Euler paths are given in Figs. 2(a) and 2(b), respectively. As the reader should check for himself, the code for (E’, A’) in the mirror-image graph of G’, like the code in (2), does not equal the code in (1). This proves that the two graphs are not isomorphic. Thus a difficult problem [4] even for a computer has been converted to an extremely simple one; the generation of only three codes has sufficed to check the isomorphism of these two large graphs.

The pair of graphs for the second example are shown in Fig. 3. We desire to use fewer than 4b = 60 codes in our test; we therefore check valences of nodes and lengths of meshes. We find that (p3 = cpi = 10; we also determine by inspection that p, = pi = 2, pa = pi = 2, pS = p: = 2, and ps = p; = 1, where p, and p; are used to designate the number of k-meshes in G and G’, respectively. Thus some information is given for reducing the required number of codes. We could write a code for an arbitrary branch in a S-mesh of G and then determine the 24 codes for the branches of the two 3-meshes in G’ and in its mirror-image graph. A necessary and sufficient condi- tion for isomorphism is that at least one of the 24 codes be equal to the code written for G. However, we are in- terested in writing a much smaller number of codes.

Consequently, we now examine the size of the two meshes that each branch is a part of; this is the dual of what was done in the previous example, or, in other terms, it is the same as the inspection of the valences of the two nodes of each branch in the dual graphs. Now a significant fact emerges: branch h is the only branch in G that is a part of two 5-meshes and branch c’ is the only branch in G’ with this property. Hence, the graphs are isomorphic only if h and c’ are similar branches. In addition, by the incidental information about the automorphism group of the graph in Fig. 3(a) mentioned nreviouslv. we see that h can onlv corresnond to itself.

5 (4

Fig. 2. Euler paths for the graphs ‘in Fig. 1.

5

0,)

Fig. 3. A pair of graphs to be tested in the second example.

(a)

and thus the order of the automorphism group of G cannot exceed four. The further test on necessary conditions subsequently considered decreases the maximum possible order to two, and then the use of the codes shows that it is precisely two.

At this point we need four codes with h as an initial branch, using each of the possible directions in the graph of Fig. 3(a) and its mirror-image graph, and we need any one code with c’ as an initial branch. To reduce the required number of codes even further we inspect the length of each other mesh that each branch on the 5- meshes is a part of. In graph G the five-meshes, where the faces are encircled clockwise, are given by the ordered sets of branches



(a) A grain. (b) Schlegel diagram


Results


Characterizing 3D microstructures

Grain growth microstructure

combinations of faces frequency

(0, 0, 0, 4, 4, 0, 0, 0...) 2.83%(0, 0, 0, 4, 4, 2, 0, 0...) 2.38%(0, 0, 0, 3, 6, 0, 0, 0...) 1.86%(0, 0, 0, 4, 4, 3, 0, 0...) 1.86%(0, 0, 0, 4, 4, 1, 0, 0...) 1.63%(0, 0, 0, 5, 2, 0, 0, 0...) 1.48%(0, 0, 0, 3, 6, 3, 0, 0...) 1.45%(0, 0, 0, 3, 6, 1, 0, 0...) 1.43%(0, 0, 1, 3, 3, 0, 0, 0...) 1.39%(0, 0, 0, 3, 6, 2, 0, 0...) 1.38%



Most popular topological typesin grain growth microstructure

2.83% 1.86% 1.63% 1.53%

1.48% 1.43% 1.39% 1.38%



Most popular topological typesin random Vorornoi microstructure

0.28% 0.17% 0.15% 0.13%

0.13% 0.10% 0.10% 0.10%



0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20 25 30 35 40

freq

uenc

y

Distinct topological types

(0,0,1,3,4,3,1,0...)



Conclusions

3D grain characterization is not obvious

It can be done! It can be done!

Can help us characterize microstructures

Forthcoming:“Complete topology of cells, grains, and bubbles inthree-dimensional microstructures”, Phys. Rev. Lett.


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