6 th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil Topology Optimization of Truss Structures using Cellular Automata with Accelerated Simultaneous Analysis and Design Henry Cort´ es 1,a , Andr´ es Tovar 1,a , Jos´ e D. Mu˜ noz 1,b , Neal M. Patel 2 , John E. Renaud 2 (1) a. Department of Mechanical and Mechatronic Engineering, b. Department of Physics National University of Colombia - Carrera 30 # 45-03, Bogot´a, Colombia Emails: hocortesr@unal.edu.co, atovarp@unal.edu.co, jdmunozc@unal.edu.co (2) Department of Aerospace and Mechanical Engineering University of Notre Dame - Notre Dame, Indiana 46556, USA Emails: npatel@nd.edu, jrenaud@nd.edu Abstract Cellular automata (CAs) have been used as an alternative to represent macroscopic behavior of continuum systems governed by non-linear partial differential equations (PDEs). This task is generally accomplished using simple rules that represent the micromechanics of the system. CA models are composed of regular lattice of cells or automata. Each automaton may change its state at discrete, fixed times (iterations) according to a local rule. The premise behind a CA model is that an overall global behavior can be computed based on local rules operating on cells that only know local conditions. These rules depend on the present states of the cell and its neighbors within a certain proximity. CAs have been proven to have an inherent massive parallel computation capability which makes CAs an efficient computational tool for large systems. This work introduces a new strategy that uses CA rules for analysis and design, and is applied to solving topology optimization problems of truss structures. Analysis rules are derived from local equilibrium equations. They govern cell displacements, where each cell is a nodal point of the truss structure. The displacements represent the field variables of the analysis. Design rules are derived from optimality conditions. The cross sectional areas of the truss members connected to each cell are used as design variables. In this work, an accelerated convergence technique, based on prediction using gradient information, is formulated and implemented. The computational efficiency of this approach is demonstrated through two-dimensional truss structures. Keywords: Topology optimization, cellular automata, simultaneous analysis and design (SAND), truss structures de- sign. 1. Introduction The conventional techniques to do analysis and design for the industry are based on highly flexible finite element imple- mented in software of numerical analysis. This techniques are useful for accomplishing that, but the need to accomplish it repeatedly during the design cycle does it necessary to automatize the design process, leading to the intensive use of optimization algorithms to drive this task. Nevertheless, the use of optimization together with stages of analysis for finite elements bring computational requests increasingly to fulfill. On one side the increase on the speed and memory capacity of the modern computers has induced them to make increasingly more complex analysis that usually require a lot of degrees of freedom, turn hard to accomplish this in the design stage due to the large numbers of repetitive analysis and the great computational cost. On another side the structures have been increased in complexity, and therefore the grade of the analytical complexity of the analysis too, turning the algorithms often less and less efficient due to the complexity of the solution techniques. To work out great part of the problems mentioned, the CAs were combined with the finite element method (FEM) to solve topology optimization problems of continuous structures. This approach have been referred to as the hybrid cellular automaton (HCA) method. The HCA method uses FEM for structural analysis and local CA rules for design. This technique have been studied in recent investigations, being used to develop algorithms for reproducing the bone remod- eling process [1] and design of compliant mechanisms [2]. The HCAs have offered substantial improvements in the case of efficiency and robustness, striking out problems such as the obtaining of structures with chessboard patterns, among others. The use of FEM to perform the stress analysis and the CAs for optimizing also have been investigated by other authors [3],[4] which dealing with problems such as the not obvious mathematical relationship between the automaton rule and the optimization problem, and the difficulty of introduce the stress constraint conditions to the optimization problem. In the same foundations the evolutionary structural optimization (ESO) scheme [5] was developed. The HCA, ESO and similar schemes add certain improvements to the optimization stage but keep on needing to accomplish a com- plete analysis by FEM before accomplishing an iteration in the design stage. Furthermore, the FEM has been required 1