COMPUTATIONAL MATERIALS ENGINEERING Episode 6 pdf

4.2.1 One-Dimensional RecrystallizationThe preceding introductory example is really as basic as a cellular automaton can get.Surprisingly enough, we can already build a recrystallization

4.2.1 One-Dimensional Recrystallization

The preceding introductory example is really as basic as a cellular automaton can get.Surprisingly enough, we can already build a recrystallization model—admittedly it will bevery much simplified and only one-dimensional, but nevertheless a model that demonstrateshow cellular automata can be straightforwardly used to model microstructure evolution Thefact that it is one-dimensional also allows us to show you the timeline without turning toanimated figures, which is rather difficult on regular paper

Let us say state0represents plastically deformed material and state1represents lized material Algorithm 4-1 thus simulates the nucleation (line 5 in the algorithm) and growth

recrystal-of a single recrystallized grain in a deformed material, as shown in Figure 4-2—it is as simple

as that

Assume a one dimensional material with random position, site saturated nucleation, with anucleation probability of0.01per cell volume To simulate site saturated recrystallization weonly need to replace the initialization in Algorithm 4-1 to setting the state at s[i][0]for alli

to state1with a probability of0.01, resulting in a simulation result similar to the one shown

in Figure 4-3 Note that we could keep track of the recrystallized grains simply by assigning

different states to every recrystallized grain; that is, the state is equal to 0for the deformedmaterial, and equal to a grain identification number for the new grains

PROBLEM 4-1: Multiple Grains in One Dimension

Actually, in the preceding one issue remains unsolved before we can keep track of grains rately What is it? And how would you solve it?

sepa-4.2.2 Before Moving to Higher Dimensions

The example of one-dimensional recrystallization in Figure 4-3 is very limited in that all thegrain boundaries move at the same velocity all of the simulation time Knowing that cellular

automata are discretized in space and time, how can we add velocity variations to the algorithm?

Velocity Variations with Time

Assume the deformed microstructure is recrystallized at an elevated temperatureT To do soone must bring the material to that temperature from the ambient temperature, sayT0, and at theend of the process allow the material to cool down to that same temperatureT0, as illustrated inFigure 4-4 Using the classic model for grain boundary motion we know that the velocity of thegrain boundary is given by the following equation:

FIGURE 4-3 One-dimensional recrystallization with site saturated nucleation using cellular

automa-ton 254.

0 100 200 300 400 500 600 700 800 900 1000

Temperature(°C)

1/T (1/K)

Time (Arbitrary Units)

0 50 100 200 300 400

Time (Arbitrary Units)

FIGURE 4-4 Mobility variation during an annealing cycle The diagram on the left shows the classic

Avrami model for the dependency of the grain boundary mobility on temperature The diagrams on the right plot the typical temperature cycle and the mobility as a function of time.

Let us assume the plastic deformation to be constant over the whole specimen and that thedriving pressurepis constant We know that the grain boundary mobility depends on tempera-ture This dependence is usually assumed to follow an Avrami-type behavior:

m=m0

T e−

Note that in literature one commonly neglects the1/Tdependence of the prefactor and includes

it as a “minor temperature dependence” inm0 The mobility is typically plotted asln mversus

1/Tas shown in Figure 4-4 All this means is that we have to be able to vary the grain boundaryvelocity as a function of time in our cellular automata—or not?

Well, let us analyze this simplified case If indeed the driving forcep is constant and only mobility depends on temperature, then it must follow that the outcome of the simulation is not

dependent on the exact temperature–time path Instead of modifying the cellular automaton, wecan also solve our problem by merely adapting the time step to cellular automaton-step scaling

If our one-dimensional cellular automaton is defined with a constant cell length equal tolCA,

we know that at eachCA step a grain boundary moves over a distancelCA:

and with equations (4.2) and (4.3) one can straightforwardly derive that

Rephrased: a single simulation tells us where grain boundaries go, equation (4.5) tells us when

they will get there

It should be no surprise that the example in the previous section is overly simplified and farfrom real microstructure processing For the larger part of cases equation (4.5) will not work,the reason being that the driving pressurepand/or the mobilitymshow local variation, that is,variation from position to position in the microstructure

One-Dimensional Recrystallization of a Heterogeneously Deformed

Microstructure

In the example of a one-dimensional microstructure that has been deformed inhomogeneously,the driving pressurepfor recrystallization will vary from position to position It is clear fromequation (4.2) that this means that grain boundaries will move at speeds varying throughoutthe microstructure In the fully deterministic cellular automata model we presented so far such

is impossible: the grain boundary velocity is constant as fixed by the cellular automaton grid:

v = lCA/∆t

The most straightforward way to introduce the capability to model locally varying grain

boundary velocities is to implement a cellular automaton with probabilistic state change rules,

in short a “probabilistic cellular automaton.” As an example, we can change the state changerule given in equation (4.1) to a probabilistic version:

current state = 0 ⇒ new state

sim-2 Usepmaxand equation (4.5) to compute∆tfor the currentCAstep

3 Using the local value forpfor each cell, compute the local value ofPusing

P =plocal

4 Finally, use equation (4.6) in Algorithm 4-1 to simulate the recrystallization process.You should now be able to program your first probabilistic cellular automaton The resultsshould look something like Figure 4-5

Continuous Nucleation versus Site Saturation

The preceding examples all stick to site saturation for the nucleation of new grains in thedeformed microstructure Now let us consider the case in which nucleation is a continuousprocess in which the emergence of new nuclei per units of time and volume is dictated by atemperature dependent nucleation rate ˙n(T ) As this function is different from the temperaturedependence of the grain boundary mobilitym[equation (4.3)], equation (4.5) will not work forall cases Before we can explain why, we must know how to introduce the nucleation processinto the cellular automaton

In the context of a discrete cellular automata model continuous nucleation means that perCAstep a certain number of nucleinCAis generated Assuming theCAmodel consists ofmcells,and at a certain simulation step the remaining number of cells in the deformed state ism d, then

the real, immediate nucleation rate is

˙n(T ) = n ∆t lCA/m d

CA = ˙∆t l nCA

FIGURE 4-5 One-dimensional recrystallization with site saturated nucleation using a probabilistic

cellular automaton, plocal/pmax= 0.4 and randomly located nuclei (2 in 100 cells are nuclei).

Important to note is that nCA must be chosen between1and a number substantially smallerthanm Otherwise ourCAgrid remains empty or is filled with nuclei within fewCAsteps and

we are no longer able to simulate the growth process Also note that a nonprobabilistic cellularautomaton can only approximate a continuously varying nucleation rate with a discrete andfinite subset of nucleation rates This is usually unacceptable, and it is common to implement aprobabilistic nucleation model, in which ˙n PCA is defined as the probability perCAstep and percell that a nucleus is generated This leads to the following relation:

˙n(T ) = ∆t l ˙n PCA

In our one-dimensionalCA-model equation (4.5) defines the relation between∆tandlCA.Whether equation (4.8) or (4.9) is used as a nucleation model, in both cases we only have alimited range from which we can choose ˙nCA or ˙n P

CA to comply Let us have a look at thisproblem in detail with a case study

PROBLEM 4-2: One-Dimensional Recrystallization under Conditions of Continuous Nucleation

Program a one-dimensional cellular automaton, capable of simulating the continuous ation and growth of recrystallizing grains Use a statistical nucleation model, and allow the model to have a table specify the temperature versus time.

nucle-The input variables to our problem are

T (t) Temperature [K] as a function of time This function is, for example, made available inthe form of a tabulated function, and specifies how long our simulation needs to run

˙n(T ) The nucleation rate [mm−1s−1] as a function of temperature, also given as a tabulatedfunction

m(T ) The grain boundary mobilitym, specified by the parametersm0in[Kmm3/Ns]andQ

in [J], and a function of the temperatureT[K] as in equation (4.3)

p Assume homogeneous deformation, specified by a constant driving pressurepin[N/mm2]

To simulate the process our algorithm must perform the top level functions specified inAlgorithm 4-2

ALGORITHM 4-2: Top Level Functionality of a One-Dimensional Recrystallization Simulation

1: Initialize the simulation by setting the number of cellsncand the number ofCAstepsnstoperform

2: Read input variables

3: Determine∆tcorresponding to aCAstep

4: Determine a feasible value forlCAusing ˙nand total process timetps

5: fori = 1 → nsdo

6: Perform nucleation step

7: Perform growth step

8: end for

9: Ouput results

Now let us have a line by line look at the details of how to perform these functions:

Line 1: Reasonable numbers should be picked For example, aiming at an average of 20 cells

per recrystallized grain and expecting a total of 20 grains, we need at least 400 cells and

20CAsteps, so let us use 500 cells and 50CAsteps Evidently, a better way to do thingswould be to test for a simulation to finish and to use a while-loop at line 5 instead of thecurrent for-loop

Line 3: As our driving pressure is constant we only need to determine the max mobilitymmaxobserved in the given temperature range (at the maximum temperature) to find the fol-lowing relation:

Note: Although the part of the algorithm detailed so far is not really an essential part of aCAalgorithm, it is worthwhile as it tells us whether the simulation we perform is correctly

sized for the problem we want to analyze before we run the actual simulation Evidently,

this is overkill for the small problem we analyze in this example, but having the discipline

to do so becomes valuable when more complex problems are attacked

Line 6: A nucleation step consists of finding all the remaining cells still in state “deformed”,

and changing the state of these cells to “recrystallized” with probability ˙n PCA Note that

in doing so one forces a nucleus to have the size of one cell One can circumvent thisproblem by introducing a third state “nucleated” next to “deformed” and “recrystallized”,and modifying the growth step to handle this additional state properly

Line 7: A growth step consists of locating all cells changing state from “deformed” to

“recrys-tallized” using the state change rules specified in equation (4.1)

Now all that is left for you to do is to program this algorithm in your favorite language

PROBLEM 4-3: Modification to Problem 4-2

Edit algorithm and source code of Problem 4-2 to include the situation in which a microstructure was deformed heterogeneously, resulting in a locally varying driving pressure p

PROBLEM 4-4: Another Modification to Problem 4-2

Edit algorithm and source code of Problem 4-2 to handle a third state “nucleated”, so that nuclei do not automatically appear as fully recrystallized cells The transition from state “nucle- ated” to “recrystallized” should be handled in the growth step Watch out for cells that become

“nucleated” and that do not grow in the next growth step Do you allow them to be overgrown

by neighboring grains or not? Is there a di fference between simulation results making one or the other choice?

4.3 +2D CA Modeling of Recrystallization

The step from one-dimensional to two- or three-dimensional cellular automata involves onemajor issue In 2D and 3D representations of microstructure, interfaces between neighboringgrains are lines and surfaces and need to be approximated within theCAgrid Grain boundariesplay an important role in the details of microstructure evolution, and an imperfect representa-tion may lead to serious modeling problems We will address these problems in the followingsections

Apart from these interface-representation problems the extension to+2D is straightforwardand only requires substantial work on the programmer’s part We will try to give the reader

a good idea of the work involved, as the one-dimensional CAused so far is only helpful foreducational purposes Whether you choose 2D or 3D when you start writing your own cellularautomaton, be aware that, depending on the problem to be analyzed, either can be the bestchoice, so it is always a good idea to provide your source code with the capabilities for bothtypes of simulations

4.3.1 CA-Neighborhood Definitions in Two Dimensions

Before we can turn our attention to neighborhood definitions, we must realize that a cellularautomaton operates on a space-filling grid of cells The most classic form is a periodic grid ofsquares or cubes Other types of grids are possible, as illustrated in Figure 4-6

FIGURE 4-6 Types of periodical cellular automaton grids in two and three dimensions.

FIGURE 4-7 The classic neighborhood definitions of von Neumann and Moore.

von Neumann and Moore

Classic grids of cells come with classic definitions of neighborhood, and the most known aredefinitely the von Neumann and Moore neighborhood definitions on a 2D square grid of cells

as illustrated in Figure 4-7 In these types of neighborhoods, of which a multitude of shapes can

be imagined, a cell (shaded in the figure) is part of its own neighborhood This is the usual casewhen using cellular automata in microstructure evolution modeling

The central idea of a Margolus-type neighborhood definition is that at different CAsteps,different subzones of a total neighborhood are sampled (Figure 4-8) This allows for neighbor-hoods with a simplified shape, and thus simpler state-change rules As described by Chopardand Droz [CD98], Margolus neighborhood types to some extent prevent long distance effects

4.3.2 The Interface Discretization Problem

The easiest way to explain why aCAgrid has difficulties with modeling lines and surfaces is byhaving the student programming one without correction for this specific problem Therefore, tryProblem 4-5 The algorithm could look like Algorithm 4-3

PROBLEM 4-5: A Simple 2D Cellular Automaton for Recrystallization

Using Algorithm 4-2 as top level, and forgetting about the relation between real time and CA time

to simplify things, design and program a 2D algorithm that allows you to select different types

of neighborhoods at run time, so that you can study the effect of the neighborhood definition on the simulation.

ALGORITHM 4-3: Top Level Functionality of a 2D Recrystallization Simulation

1: Initialize the simulation by setting the number of cellsnc = ncxncy

2: Define the cell state0for unrecrystallized material, and state1for recrystallized

3: Define the neighborhood, for example, von Neumann

4: Define the state change rule so that a cell that has a recrystallized cell in its neighborhoodbecomes (or remains) recrystallized itself at the nextCAstep

5: Assume site saturation, set a number of nucleation sites

6: while there are still cells with status0do

7: Compute recrystallization step

8: end while

9: Ouput results

Margolus

FIGURE 4-8 The Margolus neighborhood definition The neighborhood definition is different for

each increment and switches between the definitions given by the full and the dashed line The state transformation rule depends on relative location of the neighborhood to the focus cell (shaded in the figure) See ref [CD98] for a more detailed description.

If one performs a simulation with an algorithm like Algorithm 4-3, the result should besimilar to what is shown in Figure 4-9 Clearly, the grains grow with a geometry suspiciouslysimilar to the shape of the neighborhood used to model the process This is confirmed if oneanalyzes the distribution of the orientation of the grain boundary surfaces, as depicted inFigures 4-10 and 4-11.

As one can observe that the density of the cellular automaton grid and the reach of the borhood is varying with direction, one could naively come up with a probabilistic solution asdepicted in Figure 4-12, showing a Moore-type neighborhood corrected for its shape to approx-imate a circle The correction is such that a deformed cell still changes state to recrystallizedwhen it finds a recrystallized neighbor, but only with a probability equal to the relative area ofintersection with a circle as drawn in the top half of Figure 4-12 The bottom half illustrateswhat this means for the “shape” of the neighborhood (as it cannot resolve curvature within onecell) Applying this probabilistically corrected state transformation function, the shape of thegrowing grain is unfortunately not circular, and that independent of the resolution at which thesimulation is performed, as illustrated in Figure 4-13

neigh-FIGURE 4-9 Illustration of the influence of the von Neumann neighborhood definition on a 2D

Time averaged boundary orientation distribution

)

) 12x

vs log (t) 1

FIGURE 4-10 Evolution of the distribution of the grain boundary planes orientation during a

recrys-tallization simulation performed with a von Neumann neighborhood definition The orientations at

0◦ ,45◦ ,90◦ , etc are more frequently observed due to the grid on which the cells are placed.

FIGURE 4-11 Figure 4-10 continued.

Corrected (Probabilistic) Moore

FIGURE 4-12 A corrected Moore-type neighborhood and state transformation function.

FIGURE 4-13 Application of the neighborhood and state transformation function as depicted in

Figure 4-12 and for different cellular automaton grid resolutions.

a(1 − a)(1 − b) (a + (1 − a)b)(1 − a)(1 − b)

Now consider a2 × 2grid of cells in which the top left cell has state 1 and the other threecells have state 0 Figure 4-14 partially lists the probabilities of different scenarios that canoccur in two consecutive increments Observe that the cell at the bottom-right corner of the

2 × 2 grid can obtain state 1 along different computational paths in the cellular automaton

To find the values of aandbwith which state transformation function (4.10) leads to lar growth of the recrystallizing grain, the total probability that a cell obtains state 1 may only

circu-depend on its distance from the cell at which the grain was nucleated, and may not circu-depend on

FIGURE 4-15 Application of the neighborhood depicted in Figure 4-12 but with a state

transfor-mation function as in Figure 4-13 (with a = 0.8 and b = 0.25 ) Different cellular automaton grid resolutions are shown, and from left to right the simulation result is averaged over an increasing number of cellular automaton simulations In the last column a perfect circle is plotted in gray over the results.

the relative direction of its location Figure 4-15 shows how close to a perfect circle one canget with trial and error A closer observation also reveals that the edge of the circular grain inthe averaged images shows a broader transition range in the diagonal direction This is due tothe fact that the grid density is lower in those directions Other types of periodic grids, like thefrequently used hexagonal one, may improve on this but will not completely solve the inter-face discretization problem It should be emphasized that any computational method, includingthe ones addressed in other chapters in this book, are all influenced by the grid on which thesimulation is performed

PROBLEM 4-6: Hexagonal Cellular Automata

Design and program a cellular automaton that operates on a hexagonal grid (the cells are the six triangles subdividing a hexagon) The most straightforward neighborhood definition is a von Neumann-type one including four cells: the cell itself and its three facing neighbors Note that the visualization of a nonrectangular grid is not so “easy,” as pixels are no longer “square.”

This interface discretization problem is a very serious one, and its impact only deepens onceone realizes that the evolution of a microstructure, through grain boundary mobility and energy,

depends on the orientation of exactly these lines or surfaces Whether approximations with

probabilistic corrections suffice to model microstructure evolution depends on the details of thatevolution

PROBLEM 4-7: Simply from a 2D to a 3D Cellular Automation

Did you remember to provide for 3D in your solution of Problem 4-5? Good! Now upgrade and see the influence of neighborhood definition in 3D space.