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RESEARCH PAPERS

Topology Optimization Using a Hybrid Cellular Automaton Method With Local Control Rules

[+] Author and Article Information
Andrés Tovar1

Department of Mechanical and Mechatronic Engineering, National University of Colombia, Cr. 30 45-03, Bogotá, Colombiaatovarp@unal.edu.co

Neal M. Patel

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556npatel@nd.edu

Glen L. Niebur

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556gniebur@nd.edu

Mihir Sen

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556msen@nd.edu

John E. Renaud

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556jrenaud@nd.edu

1

Corresponding author.

J. Mech. Des 128(6), 1205-1216 (Jan 07, 2006) (12 pages) doi:10.1115/1.2336251 History: Received March 20, 2005; Revised January 07, 2006

The hybrid cellular automaton (HCA) algorithm is a methodology developed to simulate the process of structural adaptation in bones. This methodology incorporates a distributed control loop within a structure in which ideally localized sensor cells activate local processes of the formation and resorption of material. With a proper control strategy, this process drives the overall structure to an optimal configuration. The controllers developed in this investigation include two-position, proportional, integral and derivative strategies. The HCA algorithm combines elements of the cellular automaton (CA) paradigm with finite element analysis (FEA). This methodology has proved to be computationally efficient to solve topology optimization problems. The resulting optimal structures are free of numerical instabilities such as the checkerboarding effect. This investigation presents the main features of the HCA algorithm and the influence of different parameters applied during the iterative optimization process.

Copyright © 2006 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Neighborhood layouts for a cellular automaton. (a) Empty, N̂=0; (b) von Neumann, N̂=4; (c) Moore, N̂=8; (d) radial, N̂=12; (e) extended, N̂=24.

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Figure 2

Boundary conditions for a cellular automaton. (a) Fixed; (b) adiabatic; (c) reflecting; (d) periodic.

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Figure 3

Hybrid cellular automaton algorithm. The algorithm starts with the definition of the design domain, material properties, load conditions, and the initial design, x(0). A finite element analysis (FEA) is performed to determine the mechanical stimuli and, therefore, the error signal operating on the automata, ei(t). The mass is updated according to the set of rules, x(t+1)=R(x(t),e(t)). The convergence is determined according to the change in the design variables and/or field variables. If there is no convergence, the process continues with a new FEA.

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Figure 4

Design domain for a cantilevered structure composed of 30×30 CAs

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Figure 5

Two-position control with cT=0.10. Final topology at t=17, U=6.909, F=0.525.

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Figure 6

Two-position control with cT=0.25. Final topology at t=13, U=6.552, F=0.558.

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Figure 7

P control with cP=0.25∕Ui*. Final topology at t=25, U=6.330, F=0.580.

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Figure 8

P control with cP=0.5∕Ui*. Final topology at t=19, U=6.417, F=0.571.

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Figure 9

P control with cP=1.0∕Ui*. Final topology at t=15, U=6.420, F=0.569.

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Figure 10

PI control with cP=0.25∕Ui* and cI=0.5∕Ui*. Final topology at t=15, U=6.487, F=0.563.

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Figure 11

PD control with cP=1.0∕Ui* and cD=−0.5∕Ui*. Final topology at t=19, U=6.511, F=0.558.

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Figure 12

PID control with cP=0.25∕Ui*, cI=0.50∕Ui*, and cD=−0.05∕Ui*. Final topology at t=13, U=6.400, F=0.571.

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Figure 13

PID control with cP=1.0∕Ui*, cI=0.5∕Ui*, and cD=−0.5∕Ui*. Final topology at t=10, U=6.0173, F=0.608.

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Figure 14

Initial design xi=ximin. Two-position control with cT=0.1. Final topology at t=25, U=6.495, F=0.566.

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Figure 15

Initial design xi=ximin. P control with cP=0.5∕Ui*. Final topology at t=20, U=6.417, F=0.571.

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Figure 16

Initial design xi=random. Two-position control with cT=0.1. Final topology at t=25, U=6.463, F=0.563.

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Figure 17

Initial design xi=random. P control with cP=0.5∕Ui*. Final topology at t=20, U=6.383, F=0.571.

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Figure 18

Topologies for different neighborhoods. Final topologies at (a) N̂=0, t=24, U=6.894, F=0.521; (b) N̂=4, t=19, U=6.417, F=0.571; (c) N̂=8, t=20, U=6.383, F=0.574; (d) N̂=12, t=19, U=6.259, F=0.591; (e) N̂=24, t=23, U=6.222, F=0.589.

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Figure 19

Topologies for different mesh sizes. The final topologies are defined by (a) 10×10, t=25, U=5.980, F=0.598; (c) 20×20, t=21, U=6.167, F=0.591; (b) 30×30, t=19, U=6.417, F=0.571; (c) 60×60, t=21, U=6.773, F=0.545; (d) 90×90, t=20, U=6.960, F=0.536.

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Figure 20

Topologies for different penalization powers. The final topologies are defined by (a) p=1.0, t=9, U=6.096, F=0.557; (b) p=2.0, t=18, U=6.333, F=0.573; (c) p=3.0, t=19, U=6.417, F=0.571; (d) p=4.0, t=22, U=6.444, F=0.570; (e) p=5.0, t=31, U=6.394, F=0.571.

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Figure 21

Intermediate relative densities for different penalization powers. The number of “gray” elements is (a) p=1.0, 150 or 16.7%; (b) p=2.0, 46 or 5.1%; (c) p=3.0, 38 or 4.2%; (d) p=4.0, 30 or 3.3%; (e) p=5.0, 18 or 2.0%.

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Figure 22

Topologies for different local SED targets Ui*. The final topologies are defined by (a) Ui*=0.0024, t=19, U=4.806, F=0.795; (b) Ui*=0.0048, t=19, U=4.417, F=0.571; (c) Ui*=0.0096, t=17, U=8.629, F=0.412; (d) Ui*=0.0144, t=14, U=10.575, F=0.343; (e) Ui*=0.0192, t=12, U=12.669, F=0.177.

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Figure 23

Variation of final strain energy U and the final mass fraction F with respect to the local SED target Ui*

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Figure 24

Linear relationship between the ratio of strain energy and relative mass U∕F and the local SED target Ui*

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Figure 25

Tradeoff curve between strain energy U and relative mass F for final topologies obtained with the HCA algorithm and the OC heuristic approach developed by Bendsøe (48) and implemented by Sigmund (49)

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Figure 26

Cellular automaton neighborhoods in a 3D model. (a) 3D von Neumann, N̂=6; (b) 3D radial, N̂=18; (c) 3D Moore, N̂=26.

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Figure 27

Three dimensional topologies for different local SED targets Ui*. The final topologies are defined by (a) Ui*=0.75U0i, t=12, U=1.272, F=0.497; (b) Ui*=1.00U0i, t=12, U=1.428, F=0.431; (c) Ui*=1.50U0i, t=13, U=1.666, F=0.359.

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Figure 28

Iterative process of a cantilever topology design with the HCA algorithm. The design domain is composed of 160×40 CAs. Moore neighborhood (N̂=8) and PID control are used. Convergence is achieved in 15 iterations.

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Figure 29

Three dimensional topologies of a cantilevered beam obtained with the HCA algorithm. (a) The design domain is composed of 30×30×60 CAs; (b) optimum topology for a single load condition obtained in nine iterations; (c) optimum topology for a two-load condition obtained in ten iterations. 3D radial neighborhood (N̂=18) and PID control are used.

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