Cellular Automata Space Research Articles

Statistical modelling of fracture using cellular atomata finite element

Fracture is an inherently statistical phenomenon as it is a function of micro-structural heterogeneities such as distributed defects and inclusions. This is evidenced by scatter in the toughness of seemingly identical specimens. Therefore, deterministic approaches do not give full picture of scatter in fracture behaviour. More suitable probabilistic methods have been devised to describe the scatter associated with fracture. While the probabilistic approaches provide a sound scientific basis for capturing the scatter in the fracture data through assuming a probability for the presence of fracture initiators, their microstructurally agnostic assumptions can limit their predictive capability. This is because there is no information on the microstructure such as grain size and morphology, texture, and other important features considered in them. An alternative class of models which take into account the distribution of toughness is cellular automata finite element models (CAFE). CAFE models are stronger in simulating the scatter in the fracture data through their ability to represent the microstructure although so far, they have been limited to fully brittle or quasi-brittle materials. In addition, the CAFE models they are computationally expensive, and their running time can be prohibitive for their application to large scale engineering components thus reducing their appeal. In this study, a CAFE model was developed to take advantage of the microstructural fidelity of CAFE but presented within the context of a probabilistic fracture approach. The CAFE based model calculates the macroscopic strain from the continuum FE model. The strain is then used to load a model which is defined in the cellular automata space. The CAFE model then simulates the initiation and propagation of fracture in the microstructure to fully capture the heterogeneity of the material at the lower length scale. The critical stress acting normal to the cleavage plane of each grain is calculated in the CAFE model and used to decide the onset of cracking in a probabilistic manner; the stress depends on the orientation of the grain in which microcrack initiates as well as depending on the orientation of the surrounding grains. To evaluate its performance, the model was calibrated using a set of experimental fracture toughness data and the results of its prediction were compared with an intendent and separate set of warm prestress experiments of the same material. Good agreement between the prediction and the experiment of the second set was observed giving confidence in the model.

Cellular Automata model of carbonitrides precipitation process in steels

Cellular Automata (CA) model of carbonitride precipitation in microalloyed steels is presented in the paper. The model accounts for an increase of dislocation density due to plastic deformation and predicts kinetics of precipitation as well as stereological parameters of precipitates. Precipitation of compounds plays crucial role in controlling of properties of alloys. In low alloyed steels the microalloying elements: Ti, Nb, V are added in order to control their microstructure and mechanical properties. The most important tool in development of new processing technologies is numerical modelling. Modelling is a mathematical description of the relation between the main process variables and the resulting material properties. Many thermodynamics models were developed in the second half of the 20th century. For example the model of carbonitrides precipitation in microalloyed steels is considered works of Dutta & Sellars [1], Dutta et al. [2], Dutta et al. [3]. The models allow calculate kinetics of precipitation and stereological parameters of precipitates as a function of processing parameters. CA model proves to be very efficient in modelling various phenomena in materials science. The transition rules transfer the mathematical model and the knowledge regarding precipitation into the cellular automata space.

Inferring the Limit Behavior of Some Elementary Cellular Automata

Cellular automata locally define dynamical systems, discrete in space, time and in the state variables, capable of displaying arbitrarily complex global emergent behavior. One core question in the study of cellular automata refers to their limit behavior, that is, to the global dynamical features in an infinite time evolution. Previous works have shown that for finite time evolutions, the dynamics of one-dimensional cellular automata can be described by regular languages and, therefore, by finite automata. Such studies have shown the existence of growth patterns in the evolution of such finite automata for some elementary cellular automata rules and also inferred the limit behavior of such rules based upon the growth patterns; however, the results on the limit behavior were obtained manually, by direct inspection of the structures that arise during the time evolution. Here we present the formalization of an automatic method to compute such structures. Based on this, the rules of the elementary cellular automata space were classified according to the existence of a growth pattern in their finite automata. Also, we present a method to infer the limit graph of some elementary cellular automata rules, derived from the analysis of the regular expressions that describe their behavior in finite time. Finally, we analyze some attractors of two rules for which we could not compute the whole limit set.

Fracture modeling in dual-phase steel grades based on the random cellular automata finite element approach

The development of a parallel version of the fracture model dedicated for multi-phase materials based on a combination of the finite element model and random cellular automata approach is the overall goal of this study. Dual-phase (DP) steel, commonly used in the automotive industry, is selected as a case study for the present investigation. Firstly, various fracture modes that can occur during deformation in DP steel grade microstructures are presented from an experimental point of view. To consider explicitly microstructure features that play a significant role during initiation and subsequent failure propagation, the digital material representation concept is used. Then, details of the developed random cellular automata model, fully embedded within the finite element framework, are discussed. The cellular automata space definition, internal variables, state variables and transition rules replicating investigated fracture modes are presented in detail and discussed. The concept of data transfer and parallelization based on the Message Passing Interface methodology in such an innovative hybrid numerical model is also clearly presented. The final section of the paper is devoted to examples of obtained results highlighting model predictive capabilities.

NP problems are tractable in the space of cellular automata in the hyperbolic plane

In this paper, we define cellular automata on a grid of the hyperbolic plane that is based on the tessellation obtained from the regular pentagon with right angles. Owing to the properties of that grid, we show that 3-SAT can be solved in polynomial time in that peculiar setting; then we extend that result for any NP problem. On this ground, several directions are indicated.

Mapping the rule table of a 2-D probabilistic cellular automaton to the chemical physics of etching and deposition

Molecules undergo only a finite number of possible chemical reactions. They can therefore be modeled as finite state machines and an array of them, in a cellular automaton, can be used to study molecular dynamics and self organization at the nanotechnology scale. The cellular automaton rule table is, in essence, a map to the molecular dynamics. We would like to use heuristic programming to design new molecules and structures in the space of cellular automata. To that end we have been investigating mapping relations between a cellular automata rule table and the chemical physics of real-world materials. We demonstrate our ideas in the domain of crystal growth and dissolution.

A Polynomial Solution for 3-SAT in the Space of Cellular Automata in the Hyperbolic Plane

A Polynomial Solution for 3-SAT in the Space of Cellular Automata in the Hyperbolic Plane

Emergence of self-replicating structures in a cellular automata space

Past cellular automata models of self-replication have always been initialized with an original copy of the structure that will replicate, and have been based on a transition function that only works for a single, specific structure. This article demonstrates for the first time that it is possible to create cellular automata models in which a self-replicating structure emerges from an initial state having a random density and distribution of individual components. These emergent self-replicating structures employ a fairly general rule set that can support the replication of structures of different sizes and their growth from smaller to larger ones. This rule set also allows “random” interactions of self-replicating structures with each other and with other structures within the cellular automata space. Systematic simulations show that emergence and growth of replicants occurs often and is essentially independent of the cellular space size, initial random pattern of components, and initial density of components, over a broad range of these parameters. The number of replicants and the total number of components they incorporate generally approach quasi-stable values with time.