Skip to main content
SpringerLink
Log in

Simulation of ideal material blocks using cellular automata

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We consider deterministic and probabilistic cellular automata to study and describe certain types of patterns in idealized material blocks. We have particular interest in patterns similar to fractures. The internal structure of these material blocks is assumed to be unknown and probabilistic cellular automata are used to obtain distributions for the referred internal structure. We consider the 1D case. Certain deterministic elementary rules are identified as elementary ideal fracture rules and the probabilistic rules are introduced as probabilistic interpolation of these elementary rules. The initial conditions are obtained from the visible borders of the surface (2D block). Therefore, each visible edge is giving additional information and a probabilistic fracture type pattern. Different methods to combine these patterns, into a final one, are discussed. Moreover, we introduce refinement techniques of the CA rules to improve the probabilities distributions. This refinement process may consider prescribed behaviour or empirical data, and, therefore, the CA rules behaviour becomes adjustable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data Availability

The authors declare the manuscript has no data associated.

References

  1. Ramos, C., Riera, M.: Evolutionary dynamics and the generation of cellular automata. Iteration theory (ECIT ’08), 219–236, Grazer Math. Ber., 354, Institut für Mathematik, Karl-Franzens-Universität Graz, Graz, (2009)

  2. Castellaro, S., Mulargia, F.: A simple but effective cellular automaton for earthquakes. Geophys. J. Int. 144, 609–624 (2001)

    Article  Google Scholar 

  3. Ktitarev, D., Godert, G., Hutter, K.: Cellular automaton model for recrystallization, fabric, and texture development in polar ice. J. Geophys. Res. 107(B8), 2165 (2002)

    Article  Google Scholar 

  4. Ishida, T.: Emergence of Turing patterns in a simple cellular automata-like model via exchange of integer values between adjacent cells. Discrete Dyn. Nat. Soc. 2020, 1–12 (2020)

    Article  MathSciNet  Google Scholar 

  5. Ramos, C., Carapau, F., Correia, P.: Cellular Automata describing non-equilibrium fluids with non-mixing substances. In: Carapau, F., Vaidya, A. (eds.) Recent Advances in Mechanics and Fluid-Structure Interaction with Applications. Advances in Mathematical Fluid Mechanics, pp. 229–245. Birkhäuser, Cham (2022)

    Google Scholar 

  6. Oliver, J., Huespe, A.E., Pulido, M.D.G., Chaves, E.: From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng. Fracture Mech. 69(2), 113–136 (2002)

    Article  Google Scholar 

  7. Abdelaziz, Y., Hamouine, A.: A survey of the extended finite element. Comput. Struct. 86(11–12), 1141–1151 (2008)

    Article  Google Scholar 

  8. Wu, J., Zhang, H., Zheng, Y.: A concurrent multiscale method for simulation of crack propagation. Acta Mech. Solida Sin. 28(3), 235–251 (2015)

    Article  Google Scholar 

  9. Litwiniszyn, J.: Stochastic methods in mechanics of granular bodies. In: Stochastic Methods in Mechanics of Granular Bodies, pp. 5–9. Springer, Vienna (1974)

  10. Szczepiński, W.: On the movement of granular materials in bins and hoppers part I: two dimensional problems. Engng. Trans. Pol. Acad. Sci. 51(4), 419–431 (2003)

    Google Scholar 

  11. Hoek, E., Martin, C.D.: Fracture initiation and propagation in intact rock—a review. J. Rock Mech. Geotech. Eng. 6(4), 287–300 (2014)

    Article  Google Scholar 

  12. Feng, X.T., Pan, P.Z., et al.: Simulation of the rock micro fracturing process under uniaxial compression using an elasto–plastic cellular automaton. Int. J. Rock Mech. Min. Sci. 43(7), 1091–1108 (2006)

    Article  Google Scholar 

  13. Pan, P.-Z., Yan, F., Feng, X.-T.: Modeling the cracking process of rocks from continuity to discontinuity using a cellular automaton. Comput. Geosci. 42, 87–99 (2012)

    Article  Google Scholar 

  14. Gomez, R., Castro, R.: Stress modelling using cellular automata for block caving applications. Int. J. Rock Mech. Min. Sci. 154, 105124 (2022)

    Article  Google Scholar 

  15. Tucker, G.E., Hobley, D.E.J., Hutton, E., Gasparini, N.M., Istanbulluoglu, E., Adams, J.M., Nudurupati, S.S., Lab-CTS, C.: continuous-time stochastic cellular automaton modeling using Landlab. Geosci. Model Dev. 9(2016), 823–839 (2015)

    Google Scholar 

  16. Bandeira, L., Martinho, M.J., Ramos, C.C.: Interval maps associated to the cellular automaton rule 184. Chaos Solitons Fractals 41(3), 1501–1509 (2009)

    Article  MathSciNet  Google Scholar 

  17. Bandeira, L., Ramos, C.C.: Transition matrices characterizing a certain totally discontinuous map of the interval. J. Math. Anal. Appl. 444(2), 1274–1303 (2016)

    Article  MathSciNet  Google Scholar 

  18. Ramos, C.: Elements of Symbolic Dynamics: Complex Systems. Lecture Notes—Complex Systems. University of Évora, Évora (2022)

Download references

Funding

The work was partially supported by project BROQ-ALT20-03-0247-FEDER-017659, and MT work was partially supported by (FCT) ICT-UIDB/04683, CR and SF work was partially supported by (FCT) CIMA-UIDB/04674/2020.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações, Universidade de Évora, 7000-671, Évora, Portugal

    C. Correia Ramos & Sara Fernandes

  2. Departamento de Engenharia Mecatrónica, Instituto de Ciências da Terra (Instrumentation and Control Laboratory at ICT), Universidade de Évora, 7000-671, Évora, Portugal

    Nada El Bouziani & Mouhaydine Tlemçani

Authors
  1. C. Correia Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nada El Bouziani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mouhaydine Tlemçani
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sara Fernandes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Correia Ramos.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Correia Ramos, C., El Bouziani, N., Tlemçani, M. et al. Simulation of ideal material blocks using cellular automata. Nonlinear Dyn 111, 22381–22397 (2023). https://doi.org/10.1007/s11071-023-09016-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09016-2

Keywords

Mathematics Subject Classification

Search

Navigation