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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, Colombiaa. Department of Mechanical and Mechatronic Engineering, b. Department of
PhysicsEmails: [email protected], [email protected], [email protected]
(2) University of Notre Dame - Notre Dame, Indiana, USADepartment of Aerospace and Mechanical Engineering
Emails: [email protected], [email protected]
6th World Congresses of Structural and Multidisciplinary OptimizationRio de Janeiro, 30 May - 03 June 2005, Brazil
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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
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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
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Evolutionary Design of Structures
EvolutionaryDesign
GeneticAlgorithms
HCA, ESO, MMD, CA
HCA, ESO,MMD
SAND-CellularAutomata
SpeciesIndividual
Designs
Local Rules for Design, Global Analysis
Local Evolution of Analysis
and Design
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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
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Components of Cellular Automata
Regular Lattice of CellsCell Definitions (States, Rules)NeighborhoodsBoundaries
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Components of Cellular Automata
Two-dimensional Lattice Configurations
Rectangular Triangular Hexagonal
Definition for state of a cell and update ruletime step
cell ID Neighborhood cells
Center cell
TCS ],[ )()()1( t
Nt
Ct
C SSRS
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Rectangular Neighborhoods
von Neumann Moore MvonN
N
S
EW
N
S
EW
SE
NENW
SW
N
S
EW
SE
NENW
SW
EE
SS
WW
NN
Boundaries Periodic Location Specific
Neighborhood Definition
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CA Representation of 2D-Truss Structures
uC
vC
C
N
S
E
NW NE
SWSE
uSE
vSE
W
Cell
Cell
Ground Structurevu
}},{},,{},,{{)(yx
tC FFVariablesSizingMaterialvuS
iyxt
i FFAAAEvuS }},{},,...,,,{},,{{ 821)(
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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
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Evolutionary Rule for Analysis
Minimize Π
Equilibrium
Equations
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Accelerated Convergence Technique
Vertical displacement of an node (structural analysis)
Without accelerating
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Accelerated Convergence Technique
Vertical displacement of an node (structural analysis)
With accelerating
(1)
(2)
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Accelerated Convergence Technique
EDA: Extrapolated Data in Accelerating
Previous Data:
Linear Extrapolation:
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Evolutionary Rule for Optimization
FSD Approach – Ratio Technique
Design Rule
A
all
Ak(t+1
)
Ak(t)
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Algorithm (ASAND)
Start
Update Cells Displacement usingCA Structural Analysis Rule
Penalize displacements to fulfil restricctions
End
x (0)=0Ak(0)=AL ; AOut of Domain=0
x (t+1)=AR(x (t))
x (t+1)=P0 *x (t+1)
Yes
Take gradient information?t=t* or t=t*+2?
Is the first data?t=t*?
Take informationt, x (t)
Take information: t, x (t)
Do the prediction (linear extrapolation)
Penalize displacements to fulfil restricctions
xE(t)=f (t*,x (t*); t, x (t), T)
x (t+1)=P0 *xE(t)
t*=t+2
Make the resizing?t is multiple of FR?
Update Areas usingCA Design Rule
Ak(t+1)=DR(Ak(t))DAkmax.=|Akmax.(t+1)-Akmax.(t)|
Dxmax.< x?
and DAmax< A?
Dxmax.=|xmax.(t+1)-xmax.(t)|
No
Yes
No
YesNo
Second data
Yes
No
Structural Analysis
Accelerating convergence
Optimization
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Software Implementation Ten-bar truss example
t=10 t=30 t=60 t=304
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Results increasing the mesh cell density
A Ground Truss Problem
Evolution of truss design
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Results increasing the mesh cell density
Results of evolution of truss design
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ObservationsIn 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 becauseno redundancy exists for the most critical truss members.
This is because no redundancy exists for the most critical trussmembers due to the formulation of the fully stressed design rules. Nevertheless, other rule definitions can be configuratedso that the structure satisfies any constraints that are desired.
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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.