A novel fast overrelaxation updating method for continuous-discontinuous cellular automaton

• Published in: Applied Mathematical Modelling Vol. 66; pp. 156 - 174
• Main Authors: Yan, Fei, Pan, Peng-Zhi, Feng, Xia-Ting, Li, Shao-Jun, Jiang, Quan
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.02.2019; Elsevier BV
• Subjects: Applied mathematics; Cellular automata; Computing time; Convergence; Iterative methods; Mathematical models; Stiffness; The accelerating factor; The continuous-discontinuous cellular automaton method; The fast successive relaxation updating method; Updating rule; The continuous-discontinuous cellular automaton method; Updating rule; The accelerating factor; The fast successive relaxation updating method; Convergence
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2018.08.025

•A fast successive relaxation updating method for continuous-discontinuous cellular automaton(CDCA) is proposed.•A fast CDCA is developed, and increments of displacement and nodal force are enlarged by the accelerating factor.•A new discontinuity tracking method which combines cell space cutting and cell neighbor searching is proposed.•The optimal value of the accelerating factor is studied, and an adaptive iteration scheme is proposed. Because of its local property, cellular automaton method has been widely applied in different subjects, but the main problem is that the cellular updating is time-consuming. In order to improve its calculation efficiency, a fast successive relaxation updating method is proposed in this paper. Firstly, an accelerating factor ω is defined, and a fast successive relaxation updating theory and its corresponding convergence conditions are developed. In each updating step, the displacement increment is enlarged ω times as a new increment to replace the old one, similarly, the nodal forces for its neighbors caused by this displacement increment are also enlarged by the same accelerating factor, and do those updating operations until the convergence is achieved. By this method, the convergence rate is greatly improved, by a suitable accelerating factor, 90 to 98% of iteration steps are decreased compared to that of the traditional method. Besides, the influence factors for the accelerating factor are studied, and numerical studies show that the suitable accelerating factor is 1.85 < ω < 1.99, which is greatly influenced by cell stiffness, and the optimal accelerating factor is 1.96 < ω < 1.99. Finally, numerical examples are given to illustrate that the present method is effective and high convergence rate compared to the traditional method.