Evolving 3D microstructures using a Genetic Algorithm ... - CiteSeerX

Materials Science Forum Vols. 467-470 (2004) pp 1019-1024 online at http://www.scientific.net © (2004) Trans Tech Publications, Switzerland Online available since 2004/Oct/15

Evolving 3D microstructures using a Genetic Algorithm David Basanta1 , Mark A. Miodownik1 , Elizabeth A. Holm2 and Peter J. Bentley3 1 Mechanical

Engineering Department, King’s College London, Strand, London, UK. National Laboratories, Albuquerque, New Mexico, USA. 3 Computer Science Department, University College London, Gower St., London, UK. 2 Sandia

Keywords: Evolutionary Computing, Genetic Algorithms, Cellular Automata, Stereology, mi-

crostructures

Abstract. We describe a general approach to obtaining 3D microstructures as input to computer simulations of materials properties. We introduce a program called MicroConstructor, that takes 2D micrographs and generates 3D discrete computer microstructures which are statistically equivalent in terms of the microstructural variables of interest. The basis of the code is a genetic algorithm that evolves the 3D microstructure so that its stereological parameters match the 2D data. Since this approach is not limited by scale it can be used to generate 3D initial multiscale microstructures. This algorithm will enable microstructural modellers to use as their starting point, experimentally based microstructures without having to acquire 3D information experimentally, a very time consuming and expensive process. Introduction At the last international meeting on recrystallisation and grain growth there was general consensus of the need to incorporate experimental microstructures as starting configurations into computer models of various kinds. In 2D this is relatively straight forward since microstructures can be imaged at various length scales by a number of well known techniques such as scanning electron microscopy (SEM), transmission electron microscopy (TEM) and orientation image microscopy (OIM). But measuring real 3D information is experimentally very challenging. Serial section methods, for example, are problematic despite recent advances in automation [1]. X-ray synchrotron techniques [2], are promising but involve access to expensive infrastructure and are not suitable for routine analysis. In this paper we introduce MicroConstructor, a tool that can evolve 3D characterisations of microstructures of materials using stereological information obtained from 2D images. Currently, MicroConstructor is limited to two phase single crystal microstructures but the method presented here is general enough that it could be applied not only to other types of microstructures but also in other areas of science and technology where 3D digital characterisations grown from 2D images could be needed. Figure 1 is a schematic overview of the MicroConstructor approach, it shows an example of the type of a 2D microstructure with two visible features (two phase) experimental inputs being discretised and measured to establish a stereological characterisation. It also shows an example of how we evolve binary genotypes, by mapping it onto a 3D microstructure and comparing its stereological parameters with the 2D characterisation. The evolution is carried out using a genetic algorithm and the mapping is carried out using a cellular automaton.

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Fig. 1: A schematic overview of MicroConstructor. (top) 2D input micrographs are discretised and measured stereologically. (bottom) A binary genotype is mapped onto a 3D microstructure using a CA. A GA evolves this system to achieve 3D microstructures that match the 2D experimental microstructures stereologically. Genetic Algorithms Genetic Algorithms (GAs) use survival of the fittest and inheritance with variation in order to evolve a population of potential solutions to a problem, starting from individuals that have been randomly created [3]. One of the features of GAs that differentiate them from other evolutionary algorithms is that the solution and its encoding are different. This means that GAs work in two different spaces: the solution space and the search space [4]. Figure 1 shows an example of the mapping between the search space and the solution space. The representation of the individual in the search space is called genotype and in MicroConstructor it is a binary string. The representation in the solution space is known as phenotype and is a 3D bitmap of a microstructure. Evolution acts on the genotype but fitness competition occurs in solution space. The algorithm of a typical GA is as follows [4]: 1. Randomly create initial population of genotypes. 2. For each individual in the population • Decode the genotype into a phenotype of final representation • Evaluate the fitness of the phenotype 3. While size of next generation < threshold • Select two parents, choosing fitter individuals with increased probability. • Use a selection and mutation to generate two offspring from the parents • Place offspring into new population 4. If acceptable solution not yet found, repeat from step 2.

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Cellular Automata Development is the mapping between the DNA (genotype) of a biological organism and the complex pattern of cells that represents their structure (phenotype). It is a process of construction and growth in which pattern and structure emerges from the interactions between proteins and genes and cells, with the environment. From an engineering perspective, we might say that development is the natural language of self-assembly. Development is also the key to understanding how complex systems can be evolved [5]. Cellular Automata (CA) are model developmental systems, because complex dynamics and patterns emerge from simple rules and local interactions [6]. CA computational systems are temporally and spatially discrete. Each cell can exist in one of a number states, and transition between states is dependent on rules. Development of the structure occurs by applying the rules to every cell at every time step. Figure 2 shows an example of the CA used in this work. It is a biologically inspired CA, that we have called EmbryoCA, in which the rules applied to each cell combine a condition, that depends on its neighbour states and its own previous states, with an action, such as; move, clone or die. Structure emerges from the collective behaviour of all cells following the same rules.

Fig. 2: An example of the development of a 3D microstructure using an EmbryoCA. The behaviour of every cell in the CA is determined by a set of rules of behaviour each of which combines a condition with an action. In this case the result of the collective behaviour of the cells results in the growth of a large particle and several particle nuclei. Method MicroConstructor is a system consisting of a GA that evolves populations of CA that develop 3D two phase microstructures. The aim of the GA is to find a 3D microstructure with stereological properties that match those of the user-provided 2D input. The most important features of a GA are the characterisation of the individuals in the population and the way these individuals are measured with a fitness function. Characterisation of individuals The individuals in the population of the GA are rule sets of an EmbryoCA, a 3D CA built using the principles of the EfA model [7]. An EmbryoCA is specified with a list of n rules that have the following format:

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4 if (variable = number) then do consequence where variable can be either the internal variable that keeps track of the number of divisions that the automaton has gone through, or the number of neighbours in the Moore neighbourhood [8]. There are two types of consequences in a rule: actions (move, divide and die, see fig. 2) and anti-actions (inhibiting the automaton from either moving, dividing or dying). At a given time, an automaton may have more than one applicable rule. A conflict resolution mechanism, in which actions and anti-actions cancel out, determines the action of such cells. The Embryo has two states representing an α and β phase. The Initial Configuration (IC) of an EmbryoCA is always one single automaton, or zygote of β phase. Fitness function The fitness function of MicroConstructor is a multi objective fitness function in which several stereological tests are performed on both the user provided 2D input, and on the 3D characterisations grown by the individuals of the GA population. The results of these tests provide a measure of the stereological difference between the 2D and 3D microstructural characterisations. The tests use five standard stereological measures: volume (3D) and area (2D) fraction, surface to volume and area fraction, number of particles, particle size distribution and two point correlation[9]. The two point correlation function used is described in the following equation: Ns 1 X f (d) = 2 nd Ns i=0

(1)

where d is the correlation distance, Ns is the total number of cells that belong to a given phase in the matrix and nd is the number of cells of the phase being characterised that are separated at distance d from cell i. Using a multi objective fitness function raises a number of issues about how the different criteria should be compared and weighted. In this work we used the sum of weighted global ratios method to evaluate the multi objective fitness[10], in which all five fitness criteria were weighted equally.

(a) 2D single particle input

(b) 3D single particle evolved structure

Fig. 3: Input and output structures, and stereological parameters for experiment 1

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(a) 2D single particle input

(b) 3D single particle evolved structure

Fig. 4: Input and output structures, and stereological parameters for experiment 2 Experiments In these preliminary experiments, we tested the ability of MicroConstructor with two examples of a two phase microstructures; (i) α phase matrix with a single β phase particle; (ii) α phase matrix with two β phase particles. The inputs and their stereological parameters are shown in figures 3 (a) and 4 (a). The parameters EmbryoCA were as follows: n=500 rules; lattice of 20x20x20; max. time step of 30. The parameters of the GA were: population size of 100; Tournament (3) selection operator [3]; Mutation rate of 0.05; Maximum of number of generations of 500; 2 point crossover; 90% of the individuals in each generation were created by crossing over the fittest individuals of the previous generation whereas the remaining 10% is taken directly from the 10% best candidates of the previous generation. Results and Discussion The solutions for each input are shown in figures 3 (b) and 4 (b). MicroConstructor has managed to reconstruct in each case a 3D microstructure that is stereologically similar to the inputs provided. In the first case, the best output has a single β phase particle, with the same particle size as the input, an identical surface area fraction and a similar 2 point correlation function. It did not match the surface fraction. The overall fitness was 0.7944, in a scale of 0 to 1, where 1 would represent an exact match. In the second case the algorithm faired less well, providing two β phase particles, and a similar 2 point correlation function, but not matching the other three criteria as well as the first case. The overall fitness was 0.763 (this cannot be compared directly with experiment 1 because we use weighted global ratios). These are preliminary results which show the potential of this approach. It should be stressed that the initial EmbryoCA rules are randomly created, thus the algorithm starts with no inherent ability to construct particles or to change their size, or to arrange them at different distances from each other. Only evolution guided by the fitness function produced these CAs that can grow two phase microstructures with different morphologies. The obtaining of a maximum of 80 % fitness in these experiments suggests that changing the maximum number of generations should be the first step to improve the results. Investigation of the influence of

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6 the multi-objective weighting scheme on the algorithm performance also needs to be carried out. Although we have only considered very simple input microstructures in these preliminary experiments, the algorithm is general and we are optimistic that it is only computational power that limits the complexity of the two phase 3D microstructures that can be evolved with this method. Certainly if the algorithm can evolve the capacity to produce one or two particles, there is no obvious reason why it should not be able to construct arrays of particles. Conclusions MicroConstructor has evolved 3D microstructures whose morphologies are stereologically similar to 2D inputs representing a limited variety of two phase microstructures. MicroConstructor uses a general method of pattern construction that does not require knowledge, intervention or tuning from the user. This feature makes MicroConstructor a promising tool for material scientists and engineers, who are non-expert microscopists, but who need to model in 3D microstructures from easily obtained 2D micrographs. Acknowledgements This work was performed in part at Sandia National Laboratories, a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. References [1] J. E. Spowart, H. M. Mullens, and B. T. Puchala. Collecting and analyzing microstructures in three dimensions: A fully automated approach. JOM-J MIN MET MAT, 55:35–37, 2003. [2] S.F. Nielsen, E.M. Lauridsen, Juul Jensen D., and H.F. Poulsen. A 3d x-ray diffraction microscope for deformation studies of polycrystals. Mater. Sci. Eng. A, 319-321:179–181, 2001. [3] David E. Goldberg. Genetic algorithms in search, optimization and machine learning. Addison Wesley, Berlin, Germany, 1989. [4] P. J. Bentley. Evolutionary design by computers. Morgan Kaufmann, San Francisco, 1999. [5] S. Kumar and P. J. Bentley. An introduction to computational development. In On Growth, form and development. Editors Kumar, S. and Bentley, P. J. Academic Press, 2003. [6] P. J. Bentley and S. Kumar. The abcs of evolutionary design: Investigating the evolvability of embryogenies for morphogenesis. Genetic and Evolutionary Computation Conference (GECCO ’99) July 14-17, 1999, Orlando, Florida, USA, 1999. [7] J. Lohn and J. Reggia. Discovery of self-replicating structures using a genetic algorithm. 1995 IEEE International Conference on Evolutionary Computing., 1995. [8] Edward F. Moore. Machine models of self reproduction. American Mathematical Society Proceedings of Symposia in Applied Mathematics 14 17-33, 1962. [9] Ervin E. Underwood. Quantitative Stereology. Addison Wesley, Reading, Massachusetts, 1970. [10] P. J. Bentley and J. P. Wakefield. Finding acceptable solutions in the pareto-optimal range using multiobjective genetic algorithms. Chawdhry, P.K., Roy, R., and Pant, R.K. (eds) Soft Computing in Eng. Design and Manufacturing. Springer Verlag, Part 5, 231-240., 1997.