This study extends a recently developed cellular automata (CA) modeling approach (Leamy, 2008, “Application of Cellular Automata Modeling to Seismic Elastodynamics,” Int. J. Solids Struct., 45(17), pp. 4835–4849) to arbitrary two-dimensional geometries via the development of a rule set governing triangular automata (cells). As in the previous rectangular CA method, each cell represents a state machine, which updates in a stepped manner using a local “bottom-up” rule set and state input from neighboring cells. Notably, the approach avoids the need to develop and solve partial differential equations and the complexity therein. The elastodynamic responses of several general geometries and loading cases (interior, Neumann, and Dirichlet) are computed with the method and then compared with results generated using the earlier rectangular CA and finite element approaches. Favorable results are reported in all cases with numerical experiments indicating that the extended CA method avoids, importantly, spurious oscillations at the front of sharp wave fronts.
Issue Section:
Research Papers
1.
Schröder
, C. T.
, 2001, “On the Interaction of Elastic Waves With Buried Land Mines: An Investigation Using the Finite-Difference Time-Domain Method
,” Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA.2.
Leamy
, M. J.
, 2008, “Application of Cellular Automata Modeling to Seismic Elastodynamics
,” Int. J. Solids Struct.
0020-7683, 45
(17
), pp. 4835
–4849
.3.
Pryor
, R. W.
, 2009, Multiphysics Modeling Using COMSOL: A First Principles Approach
, Jones and Bartlett
, Boston
.4.
von Neumann
, J.
, 1966, Theory of Self-Reproducing Automata
, University of Illinois Press
, Urbana, IL
.5.
Gardner
, M.
, 1970, “Fantastic Combinations of John Conway’s New Solitaire Game Life
,” Sci. Am.
0036-8733, 223
(4
), pp. 120
–123
.6.
Raabe
, D.
, 2002, “Cellular Automata in Materials Science With Particular Reference to Recrystallization Simulation
,” Annu. Rev. Mater. Res.
1531-7331, 32
, pp. 53
–76
.7.
dos Santos
, R. M. Z.
, and Coutinho
, S.
, 2001, “Dynamics of HIV Infection: A Cellular Automata Approach
,” Phys. Rev. Lett.
0031-9007, 87
(16
), p. 168102
.8.
Green
, D. G.
, Tridgell
, A.
, and Gill
, A. M.
, 1990, “Interactive Simulation of Bushfires in Heterogeneous Fuels
,” Math. Comput. Modell.
0895-7177, 13
(12
), pp. 57
–66
.9.
Schreckenberg
, M.
, Schadschneider
, A.
, Nagel
, K.
, and Ito
, N.
, 1995, “Discrete Stochastic Models for Traffic Flow
,” Phys. Rev. E
1063-651X, 51
(4
), pp. 2939
–2949
.10.
Honma
, T.
, and Tosaka
, N.
, 2003, “Autonomous Decentralized Finite Element Method and Its Applications
,” Int. J. Numer. Methods Eng.
0029-5981, 57
(6
), pp. 853
–874
.11.
Ryoo
, J.
, Hajela
, P.
, Suhr
, J.
, and Koratkar
, N.
, 2007, “Estimation of Young’s Modulus of Single-Walled Carbon Nanotube Using Cellular Automata
,” Adv. Eng. Software
0965-9978, 38
(8–9
), pp. 531
–537
.12.
Simons
, N. R. S.
, Bridges
, G. E.
, Podaima
, B. W.
, and Sebak
, A. R.
, 1994, “Cellular Automata as an Environment for Simulating Electromagnetic Phenomena
,” IEEE Microw. Guid. Wave Lett.
1051-8207, 4
(7
), pp. 247
–249
.13.
Lan
, Y. J.
, Xiao
, N. M.
, Li
, D. Z.
, and Li
, Y. Y.
, 2005, “Mesoscale Simulation of Deformed Austenite Decomposition Into Ferrite by Coupling a Cellular Automaton Method With a Crystal Plasticity Finite Element Model
,” Acta Mater.
1359-6454, 53
(4
), pp. 991
–1003
.14.
Raghavan
, S.
, and Sahay
, S. S.
, 2007, “Modeling the Grain Growth Kinetics by Cellular Automaton
,” Mater. Sci. Eng., A
0921-5093, 445-446
, pp. 203
–209
.15.
Yang
, B. J.
, Chuzhoy
, L.
, and Johnson
, M. L.
, 2007, “Modeling of Reaustenitization of Hypoeutectoid Steels With Cellular Automaton Method
,” Comput. Mater. Sci.
0927-0256, 41
(2
), pp. 186
–194
.16.
Bernsdorf
, J.
, Durst
, F.
, and Schafer
, M.
, 1999, “Comparison of Cellular Automata and Finite Volume Techniques for Simulation of Incompressible Flows in Complex Geometries
,” Int. J. Numer. Methods Fluids
0271-2091, 29
(3
), pp. 251
–264
.17.
Krafczyk
, M.
, Tölke
, J.
, Rank
, E.
, and Schulz
, M.
, 2001, “Two-Dimensional Simulation of Fluid–Structure Interaction Using Lattice-Boltzmann Methods
,” Comput. Struct.
0045-7949, 79
(22–25
), pp. 2031
–2037
.18.
Das
, S.
, Abbod
, M. F.
, Zhu
, Q.
, Palmiere
, E. J.
, Howard
, I. C.
, and Linkens
, D. A.
, 2007, “A Combined Neuro Fuzzy-Cellular Automata Based Material Model for Finite Element Simulation of Plane Strain Compression
,” Comput. Mater. Sci.
0927-0256, 40
(3
), pp. 366
–375
.19.
Rothman
, D. H.
, 1987, “Modeling Seismic P-Waves With Cellular Automata
,” Geophys. Res. Lett.
0094-8276, 14
(1
), pp. 17
–20
.20.
Hajela
, P.
, and Kim
, B.
, 2001, “On the Use of Energy Minimization for CA Based Analysis in Elasticity
,” Struct. Multidiscip. Optim.
1615-147X, 23
(1
), pp. 24
–33
.21.
Slotta
, D. J.
, Tatting
, B.
, Watson
, L. T.
, Gürdal
, Z.
, and Missoum
, S.
, 2002, “Convergence Analysis for Cellular Automata Applied to Truss Design
,” Eng. Comput.
0177-0667, 19
(8
), pp. 953
–969
.22.
Eugenio
, A.
, and Rasetti
, M.
, 1996, “A Cellular Automaton for Elasticity Equations
,” Int. J. Mod. Phys. B
0217-9792, 10
(2
), pp. 203
–218
.23.
Abdellaoui
, M.
, El Jai
, A.
, and Shillor
, M.
, 2002, “Cellular Automata Model for a Contact Problem
,” Math. Comput. Modell.
0895-7177, 36
(9–10
), pp. 1099
–1114
.24.
Zhong
, Y.
, Shirinzadeh
, B.
, Alici
, G.
, and Smith
, J.
, 2006, “A Cellular Neural Network Methodology for Deformable Object Simulation
,” IEEE Trans. Inf. Technol. Biomed.
1089-7771, 10
(4
), pp. 749
–762
.25.
Psakhie
, S. G.
, Horie
, Y.
, Ostermeyer
, G. P.
, Korostelev
, S. Y.
, Smolin
, A. Y.
, Shilko
, E. V.
, Dmitriev
, A. I.
, Blatnik
, S.
, Špegel
, M.
, and Zavšk
, S.
, 2001, “Movable Cellular Automata Method for Simulating Materials With Mesostructure
,” Theor. Appl. Fract. Mech.
0167-8442, 37
(1–3
), pp. 311
–334
.26.
Rappaz
, M.
, and Gandin
, C. A.
, 1993, “Probabilistic Modeling of Microstructure Formation in Solidification Processes
,” Acta Metall. Mater.
0956-7151, 41
(2
), pp. 345
–360
.27.
Fabero
, J. C.
, Bautista
, A.
, and Casasus
, L.
, 2001, “An Explicit Finite Differences Scheme Over Hexagonal Tessellation
,” Appl. Math. Lett.
0893-9659, 14
(5
), pp. 593
–598
.28.
Psakhie
, S. G.
, Smolin
, A. Y.
, Stefanov
, Y. P.
, Makarov
, P. V.
, and Chertov
, M. A.
, 2004, “Modeling the Behavior of Complex Media by Jointly Using Discrete and Continuum Approaches
,” Tech. Phys. Lett.
1063-7850, 30
(9
), pp. 712
–714
.29.
Hirsekorn
, M.
, Delsanto
, P. P.
, Leung
, A. C.
, and Matic
, P.
, 2006, “Elastic Wave Propagation in Locally Resonant Sonic Material: Comparison Between Local Interaction Simulation Approach and Modal Analysis
,” J. Appl. Phys.
0021-8979, 99
(12
), p. 124912
.30.
Kwon
, Y. W.
, and Hosoglu
, S.
, 2008, “Application of Lattice Boltzmann Method, Finite Element Method, and Cellular Automata and Their Coupling to Wave Propagation Problems
,” Comput. Struct.
0045-7949, 86
(7–8
), pp. 663
–670
.31.
Zheng
, C.
, Xiao
, N.
, Li
, D.
, and Li
, Y.
, 2008, “Microstructure Prediction of the Austenite Recrystallization During Multi-Pass Steel Strip Hot Rolling: A Cellular Automaton Modeling
,” Comput. Mater. Sci.
0927-0256, 44
(2
), pp. 507
–514
.32.
Delsanto
, P. P.
, Schechter
, R. S.
, Chaskelis
, H. H.
, Mignogna
, R. B.
, and Kline
, R.
, 1994, “Connection Machine Simulation of Ultrasonic Wave Propagation in Materials. II: The Two-Dimensional Case
,” Wave Motion
0165-2125, 20
, pp. 295
–314
.33.
Hopman
, R.
, 2008, “Arbitrary Geometry Cellular Automata for Elastodynamics
,” Master's thesis, Georgia Institute of Technology, Atlanta, GA.34.
Fujimoto
, R. M.
, 1990, “Parallel Discrete Event Simulation
,” Commun. ACM
0001-0782, 33
(10
), pp. 30
–53
.35.
Idesman
, A.
, Samajder
, H.
, Aulisa
, E.
, and Seshaiyer
, P.
, 2009, “Benchmark Problems for Wave Propagation in Elastic Materials
,” Comput. Mech.
0178-7675, 43
(6
), pp. 797
–814
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
