OOF2: The Manual

`Anneal (Anneal)`
	6.5.2. Subclasses

Name

Anneal (Anneal) — Move nodes randomly to improve element shape and homogeneity.

Synopsis

Anneal(targets, criterion, T, delta, iteration)

Details

Base class: SkeletonModifier
Parameters:

targets

Which nodes to move. Type: An object of the FiddleNodesTargets class.

criterion

Acceptance criterion Type: An object of the SkelModCriterion class.

T

Failed moves will be accepted if T>0 and exp(-diffE/T) > r, where diffE is the energy gained and r is a random number between 0 and 1. Type: A real number.

delta

Width of the distribution of attempted node motions, in units of the pixel size. Type: A real number.

iteration

Iteration method. Type: An object of the IterationManager class.

Description

Anneal is a Skeleton modifier that moves Nodes to random positions, accepting or rejecting moves according to the given criterion. It is similar to a simulated annealing simulation in statistical mechanics, from which it gets its name. Instead of minimizing the free energy of a system of particles, it minimizes the effective energy of a Skeleton.

The general procedure for a single iteration of Anneal is as follows:

Collect target nodes according to the given targets parameter. The collected Nodes are re-ordered randomly to remove any potential artifacts from the original ordering of Nodes. This re-ordering is repeated at every iteration.
Give each Node a single chance to move to a randomly assigned new position. OOF2 computes the new position from

$\displaystyle x_{\mathrm{new}}$ $\displaystyle=x_{\mathrm{old}}+\delta x$

$\displaystyle y_{\mathrm{new}}$ $\displaystyle=y_{\mathrm{old}}+\delta y.$

(6.124)

$\delta x$ and $\delta y$ are random numbers chosen from a Gaussian distribution of width delta and mean 0.0. delta is in units of the Microstructure pixel size.

Figure 6.66(a) shows a node (the big red dot) that is about to move to a new position. Before making the move, OOF2 computes the total effective energy of all of the neighboring elements of the node.
After moving each Node, the given acceptance criterion decides whether or not the move is acceptable.
If the move is unacceptable according to the acceptance criterion, OOF2 may still accept the move if the annealing is being done at a non-zero temperature. The parameter T sets the effective temperature of the annealing process. Unacceptable moves are accepted with a probability

$P=\exp{(-E_{\mathrm{diff}}/T)}$

(6.125)

where $E_{\mathrm{diff}}$ is the difference between the effective energies of the new and old Element configurations.

Figure 6.66. Annealing

(a) A node moves by $\delta x$ and $\delta y$ . The node move affects five neighboring elements, E₀, E₁, E₂, E₃, and E₄. (b) The node has moved to a new position. If the move doesn't satisfy the given acceptance criterion (computed based on its five affected elements), it'll be rejected and the node will move back to its original position.

Successful annealing usually requires a number of iterations. On each iteration, OOF2 makes one attempt to move each node. The number of iterations is controlled by the iteration parameter, which can be set to perform a fixed number of iterations or to stop after some condition is satisfied. See IterationManager for the details.

Statistics for each step of the annealing process are printed in the OOF2 Message window. For example,

    Iteration 1: E = 1.1916e+01, deltaE=-1.2495e-01 ( 1.049%), Acceptance Rate = 19.1%
    Iteration 2: E = 1.1391e+01, deltaE=-1.7564e-01 ( 1.542%), Acceptance Rate = 25.5%
    Iteration 3: E = 1.0638e+01, deltaE=-1.1883e-01 ( 1.117%), Acceptance Rate = 22.3%
    Iteration 4: E = 1.0095e+01, deltaE=-1.4345e-01 ( 1.421%), Acceptance Rate = 21.0%
    Iteration 5: E = 9.5440e+00, deltaE=-8.3725e-02 ( 0.877%), Acceptance Rate = 21.0%

The listing shows the iteration number, the total energy (E) of the Skeleton, the absolute change in energy (deltaE) during the iteration, the percentage change in energy, and the move acceptance rate. If the change in energy or the acceptance rate gets too small, the annealing process is not being effective at improving the Skeleton. Using ConditionalIteration as the iteration parameter can stop the process when it becomes ineffective.

Figure 6.67 shows a Microstructure containing three pixel types. The initial Skeleton has been refined once, but doesn't do a good job of representing the geometry of the Microstructure. Refining further might help, but it would create a lot of small Elements, which might not be desirable. Anneal can improve the Skeleton without creating new Elements.

Figure 6.67. An Un-annealed Skeleton

A Microstructure (displayed by material color) and a slightly refined Skeleton.

Figure 6.68 displays the results of annealing the Skeleton. In these images, the Elements are drawn with the color of their dominant pixel. This is a good way to estimate Skeleton quality — in a good Skeleton the Element colors match the pixel colors. The un-annealed Skeleton in the top left is clearly not a good representation of the Microstructure.

All of the annealing steps in Figure 6.68 were done with T=0, iterations=20, and alpha=0.8. The first two steps were done with delta=5. (The image is approximately 300 pixels on a side, so individual pixel motions are about 1/60th of the size of the image.) After one annealing step, the border between the orange and yellow regions is more or less straight and the blue arrow is beginning to be resolved. After two steps, the representation is pretty good. (Whether it's good enough depends on what your needs are.) However, there are some very narrow poorly shaped Elements. This is because the annealing was done with alpha=0.8, which favors homogeneity over shape. Applying Rationalize with alpha=0.5 eliminates some of these narrow Elements, leading to the Skeleton in the bottom right in the figure. Notice that the sides of the blue arrow are not quite straight. Another annealing step, this time with delta=1 moves the Nodes by small amounts, and nearly perfectly matches the Element edges to the Microstructure boundaries. A final rationalization then removes the last poorly shaped Element.

Figure 6.68. Annealing the Skeleton

The process of annealing and rationalizing the Skeleton from Figure 6.67. The Elements are displayed with their material color. The top left image contain the same un-annealed Skeleton as in Figure 6.67. The first two annealing steps have delta=5 and third (marked with an asterisk) has delta=1.

Figure 6.69. Comparison between the Annealed Skeleton and the Microstructure

The final Skeleton from Figure 6.68, displayed on top of the Microstructure material image from Figure 6.67. The only spot where the Skeleton visibly deviates from the material boundaries is at the right end of the yellow/orange interface. That could be cleaned up easily with SnapNodes.

Disclaimer. Because the Nodes are addressed in a random order and are displaced by random amounts, annealing is not a deterministic process. Your mileage will vary. The results above may not be reproducible. You may need to adjust the parameters, run more annealing steps, or use other SkeletonModifiers along with Anneal.

Tip

	Tip
Even though the example shown above concentrates on resolving material boundaries (meaning it puts more emphasis on homogeneity than shape), `Anneal` can also be used to improve the shape energy of `Skeletons`. Since OOF2 isn't quite psychic, one needs to let it know whether one wants to work on homogeneity or shape energy. This is done by adjusting the `alpha` parameter which is part of the acceptance `criterion` parameter. If `alpha` is close to 1 `Anneal` will improve homogeneity (and create ugly elements). If `alpha` is close to 0 `Anneal` will improve the shape energy. Keep in mind that annealing for shape energy could disrupt material boundaries that have already been established. To avoid this, you can pin boundary nodes so that they will not move while annealing.

Even though the example shown above concentrates on resolving material boundaries (meaning it puts more emphasis on homogeneity than shape), Anneal can also be used to improve the shape energy of Skeletons. Since OOF2 isn't quite psychic, one needs to let it know whether one wants to work on homogeneity or shape energy. This is done by adjusting the alpha parameter which is part of the acceptance criterion parameter. If alpha is close to 1 Anneal will improve homogeneity (and create ugly elements). If alpha is close to 0 Anneal will improve the shape energy. Keep in mind that annealing for shape energy could disrupt material boundaries that have already been established. To avoid this, you can pin boundary nodes so that they will not move while annealing.

Tip

	Tip
A good way to anneal efficiently is to apply it only to `Nodes` that are in or near the problem areas of the `Skeleton`. This can be done by using Active Areas, or by choosing the `targets` carefully. One way to anneal only the `Nodes` near material boundaries is to select the nodes on interior boundaries (OOF.NodeSelection.Select_Internal_Boundaries) and their neighbors (OOF.NodeSelection.Expand_Node_Selection) and then `Anneal` with `targets`=`SelectedNodes`().

A good way to anneal efficiently is to apply it only to Nodes that are in or near the problem areas of the Skeleton. This can be done by using Active Areas, or by choosing the targets carefully. One way to anneal only the Nodes near material boundaries is to select the nodes on interior boundaries (OOF.NodeSelection.Select_Internal_Boundaries) and their neighbors (OOF.NodeSelection.Expand_Node_Selection) and then Anneal with targets=SelectedNodes().