Download - Topology Optimization with ASAND-CA 1/20 Topology Optimization of Truss Structures using Cellular Automata with Accelerated Simultaneous Analysis and Design.

Topology Optimization with ASAND-CA

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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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Vertical displacement of an node (structural analysis)

With accelerating

(1)

(2)


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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


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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.