A Two-dimensional Meshless Algorithm Based On Cellular Automaton

Cellular automata(C A) is a dynamic syste m of d iscrete time and space, and it is a numerica l method used to deal w it h comp lex syste m. In t his study, a CA- based 2-dime ns iona l numer ical a lgor ithm is developed for the simulat ion of elastostat ics.By this algorit hm, a 2- dimens iona l elast ic doma in is discretized into a grid of nodes distr ibuted randomly. As a result, each node has several ne ighboring nodes located in its ne ighborhood. The ne ighboring nodes are covered by a polygo n which is used to ind icate the influence zone of the specific node. The polygo n’s centre overlaps wit h the specific node, but the s ize and the orientat ion of the polygon may change according to the d istrib ut ion of the neighboring nodes. The mecha nical relat ions hip between an arbitrary node and its ne ighboring nodes in the 2- dimens iona l cont inuum is establis hed by e mplo ying the concept of interpo lation used in finite e le ment method(FEM). Further, such mecha nical relatio nship is defined as the local rule based on which a C A is constructed. Taking the boundary condit ions into account, the C A evolves auto matica lly and the original elast ic proble m can be solved when the disp lacement at all t he random nodes converge. Because the displaceme nts at the vertices of the virtua l polygo nal cover can be deduced fro m the disp lacements of a specific node and its neighboring nodes, the nodal stress at the specific can be evaluated by conve nt iona l FEM scheme. As examp les, a plate wit h a hole subjected to different loadings is analyzed. The numerica l results show that the proposed C A method is simp le and correct. Moreover, the C A method can be incorporate d w ith FEM eas ily to make the numerica l simulat ion more convenient.In the study, the paralle l computability of the C A method is also invest igated and a paralle l approach is proposed too. The numer ical exa mp le proves that the current C A method has great po tent ia l in paralle l processing. Because the C A met hod is based on random nodes, instead of mesh, the errors caused by mesh distortion in numerica l modeling can be avoided. Therefore, the proposed C A method is a meshless algorit hm. O n the other hand, this meshless algor it hm can be transfor med into the FEM form convenie nt ly so that it exhibits a more general concept for numerica l simulat ion.