Solution of Two Dimensional Quasi-Harmonic Equations with CA Approach

Many computational techniques were applied to solution of heat conduction problem. Those techniques were the finite difference (FD), finite element (FE) and recently meshless methods. FE is commonly used in solution of equation of heat conduction problem based on the summation of stiffness matrix of elements and the solution of the final system of equations. Because of summation process of finite element, convergence rate was decreased. Hence in the present paper Cellular Automata (CA) approach is presented for the solution of heat conduction problem. Each cell considered as a fixed point in a regular grid lead to the solution of a system of equations is substituted by discrete systems of equations with small dimensions. Results show that CA can be used for solution of heat conduction problem.




References:
[1] H. Arzani and M. H. Afshar, "Solving Poisson-s equations by the
discrete least square meshless method." WIT Transactions on Modeling
and Simulation, 2006, 42: p.p.23-31.
[2] A. R. Firoozjaee and M. H. Afshar "Collocation discrete least square
meshless method for the solution of free surface problem." International
Journal of Civil Engineering, 2007, 5(2): p.p.134-143.
[3] S. Ulam, "A collection of mathematical problems--, Inter science
Publishers, New York, 1960.
[4] S. Ulam, "Adventures of a mathematician--, Charles Scribner's Sons,
New York, 1983.
[5] S. Ulam, "Random processes and transformations--, In Proceeding of
International Congress of Mathematics 2, pp 85-87, April 1952.
[6] J. Von Neumann, "Theory of self-reproducing automata--, Univ. of
Illinois Press, Champaign, IL, 1966.
[7] S. Wolfram, "A New Kind of Science--, Wolfram Media, Inc.,
Champaign, IL, USA, 2002.
[8] S. Wolfram, "Cellular automata as models of complexity--, 1984,
Nature, 31 (4), pp 419-424.
[9] S. Wolfram, "Statistical mechanics of cellular automata--, Rev. Mod.
Phys. 55, 601, 1983.
[10] S. Wolfram, "Theory and application of cellular automata--, World
Scientific, Singapore, 1984.
[11] S. Wolfram, "Cellular automata and complexity: collected papers--,
Addison-Wisely Publishing Company, 1994.
[12] B. Chopard and M. Droz, "Cellular Automata Modeling of Physical
Systems--, Collection Alea-Sacley, Cambridge University Press,
Cambridge, UK, 1998.
[13] T. Toffoli, "Cellular Automata as an Alternative to (rather than
approximation of) Differential Equations in Modeling Physics--, Physica
D, 10, 1984, pp 117-127.
[14] J. Weimar, "Simulation with Cellular Automata--, Logos-Verlag, Berlin,
1998.
[15] Z. G├╝rdel and B. Tatting, "Cellular automata for design of truss
structures with linear and nonlinear response--, 41th
AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and
Materials Conference & Exhibit, Atlanta, Georgia, April 2000.
[16] S. Missoum, Z. G├╝rdel and W. Gu, "Optimization of nonlinear trusses
using a displacement based approach--, Struct. Multidisc. Optim. 23, pp
214-221, 2000.
[17] S. Missoum, M. Abdalla and Z. G├╝rdel, "Nonlinear topology design of
trusses using cellular automata--, 44th AIAA/ASME/ASCE/AHS
Structures, Structural Dynamics and Materials Conference, Norfolk,
Virginia 2003.
[18] H. Cortés, A. Tovar, J.D. Mu├▒oz, N.M. Patel and J.E. Renaud,
"Topology optimization of truss structures using cellular automata with
accelerated simultaneous analysis and design--, 6th World Congresses of
structural and multidisciplinary optimization. Reo de Janeiro, Brazil
2005.
[19] A. Tovar, N.M. Patel, A.K. Kaushik and J.E. Renaud, "Optimality
conditions of the hybrid cellular automata for structural optimization--,
AIAA Journal, 2007, 45 (3), pp 673-683.
[20] S. Setoodeh, Z. G├╝rdal and L. T. Watson, "Design of variable-stiffness
composite layers using cellular automata", Computer Methods in
Applied Mechanics and Engineering 195 (9-12), 2006, pp 836-851.
[21] S. Setoodeh, M. M. Abdalla, and Z. G├╝rdal, "Combined topology and
fiber path design of composite layers using cellular automata", Structural
and Multidisciplinary Optimization, 30 (6), 2005, pp 413-421.
[22] S. Setoodeh, D. B. Adams, Z. G├╝rdal and L. T. Watson, "Pipeline
implementation of cellular automata for structural design on messagepassing
multiprocessors", Mathematical and Computer Modeling, 43,
2006, pp 966-975.
[23] E. F. Moore, "Machine models of self-reproducing--, Proc. Symp. Appl.
Math. 14, 17, 1962.