6th World Congresses of Structural and Multidisciplinary Optimization

1. 6th 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 Cortés1,a, Andrés Tovar1,a, José D. Muñoz1,b, Neal M. Patel2, John E. Renaud2 (1) National University of Colombia - Bogotá, Colombia a. Department of Mechanical and Mechatronic Engineering, b. Department of Physics Emails: hocortesr@unal.edu.co, atovarp@unal.edu.co, jdmunozc@unal.edu.co (2) University of Notre Dame - Notre Dame, Indiana, USA Department of Aerospace and Mechanical Engineering Emails: npatel@nd.edu, jrenaud@nd.edu

2. Outline • Introduction • Evolutionary Design • Cellular Automaton Paradigm • Components of Cellular Automata • CA Representation of two-dimensional Truss • Structures • Methodology • Evolutionary Rule for Analysis • Accelerated Convergence Technique • Evolutionary Rule for Optimization • Algorithm • Software Implementation • Ten-bar truss example • Results increasing the mesh cell density for • a Ground Truss Problem • Conclusions

3. Introduction • Mimic natural evolution of biological systems for structural design • Evolutionary design often relies on local optimality/decision making of independent parts (e.g., reaction wood, bone remodeling) Bone remodeling • Cellular Automata (CA): Decomposition of a seemingly complex macro behavior into basic small local problems

4. Evolutionary Design of Structures Evolutionary Design Individual Designs Species Genetic Algorithms HCA, ESO, MMD, CA Local Rules for Design, Global Analysis Local Evolution of Analysis and Design HCA, ESO, MMD SAND-Cellular Automata

5. Cellular Automaton Paradigm • Weiner (1946) – Operation of heart muscle, Ulam (1952) von Neumann (1966) • Automata Networks – discrete (t, s) dynamical systems • CA (AN- regular lattice, update mode synchronous) • Idealizations of complex natural systems • Flock behavior • Diffusion of gaseous systems • Solidification and crystal growth • Hydrodynamic flow and turbulence • General characteristics • Locality • Vast Parallelism • Simplicity CA Concept behavior of complex systems

6. Components of Cellular Automata • Regular Lattice of Cells • Cell Definitions (States, Rules) • Neighborhoods • Boundaries

7. Components of Cellular Automata • Two-dimensional Lattice Configurations Rectangular Triangular Hexagonal • Definition for state of a cell and update rule time step Center cell cell ID Neighborhood cells

8. NN NW NE N NW N NE N E EE WW W W E W E S SW SE SW S SE SS S Neighborhood Definition • Rectangular Neighborhoods von Neumann Moore MvonN • Boundaries • Periodic • Location Specific

9. N NE NW uC W C E vC d vSE SW S SE uSE CA Representation of 2D-Truss Structures Ground Structure Cell Cell

10. Methodology • Evolutionary Rule for Analysis Definitions Truss member properties (relative to cell center): index k, length Lk, orientation angle k, displacement [far end (uk, vk), near end (uk, vk)] Total Potential Energy: Π = U + W Total strain energy Potential of work

11. Evolutionary Rule for Analysis Minimize Π Equilibrium Equations

12. Accelerated Convergence Technique Vertical displacement of an node (structural analysis) Without accelerating

13. (2) (1) Accelerated Convergence Technique Vertical displacement of an node (structural analysis) With accelerating

14. Accelerated Convergence Technique EDA: Extrapolated Data in Accelerating Previous Data: Linear Extrapolation:

15.  all Ak(t) Ak(t+1) A Evolutionary Rule for Optimization FSD Approach – Ratio Technique Design Rule

16. Algorithm (ASAND)

17. Software Implementation • Ten-bar truss example

18. t=10 t=30 t=60 t=304 Software Implementation • Ten-bar truss example

19. Results increasing the mesh cell density A Ground Truss Problem Evolution of truss design

20. Results increasing the mesh cell density Results of evolution of truss design

21. Observations In the accelerating stage changing the number of iterations to be skipped (T), slightly influences the efficiency of the algorithm. Similarly, the same effect is caused by changes of the frequency of re-sizing which is named parameter (N) in the design stage. Increasing the degree of mesh density, the final designs could not be necessarily practical truss structures. This is because no redundancy exists for the most critical truss members. This is because no redundancy exists for the most critical truss members due to the formulation of the fully stressed design rules. Nevertheless, other rule definitions can be configurated so that the structure satisfies any constraints that are desired.

22. Conclusions • Cellular Automata techniques for topology optimization of truss structures has been demonstrated. Specifically a considerable • increase in the e±ciency of technique was checked when it was incorporated to the proposed method. This • new formulation is based on the future displacements prediction using gradient information. This gradient information • is used to perform linear extrapolations periodically. • The technique is also easy to implement and is versatile in design of truss topologies. A topic for future work is the mathematical • analysis of the CA behavior in presence of external stimulus to the system (domain plus restrictions and loads). • This method could be used with other CA techniques for conservative systems besides the use with the SAND technique.