OPTIMISATION DE FORME DANS LA MECANIQUE DES ...MECANIQUE DES STRUCTURES •Présentation de SimTech •Concepts généraux dans l ’optimisation de structures •Optimisation topologique

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SimTech 2005 All rights reserved

OPTIMISATION DE FORME DANS LA MECANIQUE DES STRUCTURES

•Présentation de SimTech

•Concepts généraux dans l ’optimisation de structures

•Optimisation topologique

•Optimisation paramétrique

•Dans d ’autres domaines ?

Edmondo Di Pasquale

Mai 2005

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SimTech 2005 All rights reserved SimTech: Company Profile

• CREATION: 1993• MISSION: CAE driven design• Expertise: optimization, structural analysis, crash and safety, software development

SimTech added value is the research ofsolutions via analysis and optimisation

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SIMTECH OFFER

SERVICES

Optimal structural designDieface designNon-linear analysis and optimisation (Crash, Manufacturing, Assembly)Training

CUSTOM SOFTWARE

PRODUCTSENKIDOU: vertical application (3D) component library

GENESIS: structural optimisationVISUALDOC: optimisation environmentDOT : optimisation component library

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WHAT IS ENKIDOU ?• A library of components (API) for the development of graphic environments for VERTICAL APPLICATIONS (outils métier)

• Based on Java and OpenGL technologies

•Speeds up development lead time by 3 to 10

•Dedicated to scientific computing applications

•Born out of ten years of struggle with GUIs

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Historical Structural optimisation

Optimisation

Optimization added to

commercial structural analysis

programs

1990

’s -Ge

netic

algo

rithm

s, seq

uential

unconstrain

ed m

inimiza

tion tech

nique

s revis

ited

Simp

lex m

etho

d for

Linea

r Programm

ingGr

adien

t ba

sed me

thod

s

Schmitcombines

optimization and analysis:

2Variables; 1/2 hour on

IBM 653

Sequen

tial u

ncon

straine

d minim

izatio

n

tech

nique

s, seq

uential

line

ar program

ming,

feasible

direction

s

Enha

nced

fea

sible

direction

s me

thod

s

Sequen

tial q

uadr

atic

programm

ing m

etho

ds

Schmitet al introduce

physics based

approximationsVanderplaats et al developed 2nd

generation approximations19501960 1980 2000

1970 1990

Topology optimization codes

appear

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SimTech 2005 All rights reserved COMMERCIAL SOFTWARECOMMERCIAL SOFTWARE

GENESISVisualDOC, DOT, BIGDOT

www.vrand.comVanderplaats R&D

NX.Nastran- - -www.ugs.comUGS PLM

- - -Model Centerwww.phoenix-int.comPhoenix Integration

- - -COwww.oculustech.comOculus Technologies

- - -OptQuestwww.optteck.comOpttek

- - -Optimuswww.noesissolutions.com

Noesis

MSC.Nastran- - -www.mscsoftware.comMSC Software

- - -iSIGHTwww.engineous.comEngineous Software

Ansys-CADOE- - -www.ansys.comAnsys

OptiStructHyperOptwww.altair.comAltair Engineering

Structural Optimization

General OptimizationWeb AddressCompany

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GENESIS: Structural Analysis And Optimization

• Preliminary Design (Topological Optimization)

• Designer interpretation

• Final Design: Shape and Size Optimization

GENESIS is the only code on the market combining

all these features

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• Before 1974

Structural OptimizationStructural Optimization

CONTROLPROGRAM

FEMANALYSIS

SENSITIVITYANALYSIS

OPTIMIZER

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Modern Structural Modern Structural OptimizationOptimization

CONTROLPROGRAM

SENSITIVITYANALYSIS

OPTIMIZER

APPROXIMATEPROBLEM

GENERATORAPPROXIMATE

ANALYSIS

FEMANALYSIS

CONSTRAINTSCREENING

INNER LOOP

OUTER LOOP

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• Variables that Reference Basis Vectors• Properties that are Functions of Design

Variables

• Example:

Intermediate VariablesIntermediate Variables

B AX YH I= =

H

B

B

HT

3

3BHA BH IEI

= =

( )( )2A BH B T H T= − − −

( )( )33 212

BH B T H TI

− − −=

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• Consider a Simple Rod - Mid 1970s Method

– Stress Nonlinear, Implicit

– Let X = 1/A

More Nearly Linear

Linear, Explicit

• Worked well for Rods and Membranes

Intermediate ResponsesIntermediate Responses

P

A

FA

σ =

FXσ =

0 T Xσ σ σ δ= +∇

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SimTech 2005 All rights reserved SHAPE OPTIMIZATION

• Design variables / shape generation (MORPHING)

• Sensitivity• Problem solution

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Method of the Basis Vectors Perturbations: Domain elements

You can apply the method to a portion of the structure

You can consider the perturbation as a design variable.

SHAPE OPTIMIZATIONSHAPE OPTIMIZATION

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Sizing and Shape Optimization

• Objective:– Minimize Mass

• Constraints:– Von Mises

constraints

– Frequency Constraint – Mass reduced 30%

– Frequency: 23 Hz to 45 Hz

• Design Variables:– Stiffener Dimensions

– Panel and Stiffeners thickness

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We could think to change:We could think to change:

The width of the The width of the upper truss of the upper truss of the reinforcementreinforcement

The width of the The width of the lower truss of the lower truss of the reinforcementreinforcement

The height of the The height of the reinforcementreinforcement

Each design variable has an initial, a maximal and a Each design variable has an initial, a maximal and a minimal value.minimal value.

SHAPE OPTIMIZATIONSHAPE OPTIMIZATIONProblem:Problem: we have to change the shape of this part we have to change the shape of this part in order to minimize its mass.in order to minimize its mass.

First step: definition of the design variablesFirst step: definition of the design variables

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SHAPE OPTIMIZATION•• Second step: define a network of Domain Elements Second step: define a network of Domain Elements

and define all nodes associated to it.and define all nodes associated to it.

A domain element to A domain element to control the width of control the width of the lower trussthe lower truss

Three domain elements Three domain elements to control the height of to control the height of the vertical trussthe vertical truss

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SHAPE OPTIMIZATIONA domain element for A domain element for each cylindrical truss, each cylindrical truss, to control the diameter to control the diameter (height of the (height of the reinforcement)reinforcement)

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• The domain elements have to be linked to the design variables. This link consists in defining the weight that each design variable has on the perturbation of the nodes belonging to the domain element

• For each node k, the displacement of the node k is:

dk=Σi βi vi

Where Where βi is the weight of each design variable, and is the weight of each design variable, and vvi i is the current value of the design variableis the current value of the design variable

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•• First design variable:First design variable: width of the lower truss:

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•• Second design variable:Second design variable: height of the system of trusses:

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•• Third design variable:Third design variable: width of the higher truss:

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SHAPE OPTIMIZATIONAN EXAMPLE

•• The mechanical definition of the The mechanical definition of the problemproblem

•• 11\\4 of a tube.4 of a tube.•• Blocking boundary Blocking boundary

conditions assure conditions assure the coherence of the coherence of behavior with the behavior with the rest of the tuberest of the tube

•• No applied forcesNo applied forces•• Normal mode Normal mode

analysis analysis

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•• The formulation of the optimization The formulation of the optimization problemproblem•• VARIABLES:VARIABLES:

width of the lower trusswidth of the lower trussheight of the truss systemheight of the truss systemwidth of the lower trusswidth of the lower truss

•• OBJECTIVE:OBJECTIVE: Minimization of the mass of the Minimization of the mass of the mechanical systemmechanical system

•• CONSTRAINTS:CONSTRAINTS:The first normal mode > 40 HzThe first normal mode > 40 HzThe second normal mode > 40 HzThe second normal mode > 40 HzThe third normal mode > 40 HzThe third normal mode > 40 HzThe fourth normal mode > 40 HzThe fourth normal mode > 40 HzThe fifth normal mode > 40 HzThe fifth normal mode > 40 Hz

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•• The solution of the optimization The solution of the optimization problem:problem:

•• Increase the Increase the height and height and decrease the decrease the width of the width of the truss systemtruss system

•• 16 Iterations 16 Iterations to reach the to reach the optimal optimal configurationconfiguration

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• Discretized Equilibrium Equations for Linear Analysis

– where K, C, M are Functions of the Design Variables

– U and F are also a Function of Time

Fundamental Analysis Fundamental Analysis EquationsEquations

( )1 ig KU CU MU F+ + + =& &&

Transient Response

Frequency Response

Buckling

Natural Vibration

Statics

0k k kK MλΦ − Φ =

KU F=

0k k G kK KλΦ − Φ =

( ) ( )21 e e r i r iig K i C M U iU F iF + + Ω −Ω + = +

( )1 ig KU CU MU F+ + + =& &&

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• Statics– Stiffness Matrix;

– Analysis; KU = P• U and P may have Multiple Columns

– We Need

– By Implicit Differentiation

– where

– We have Already Decomposed K

Sensitivity AnalysisSensitivity Analysis

1

NE

ei

K k=

=∑

i

UX∂∂

i i i

U P KK UX X X∂ ∂ ∂

= −∂ ∂ ∂

1

NEe

i ii

kKX X

=

∂∂=

∂ ∂∑

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TOPOLOGY OPTIMIZATION (SHAPE SYNTHESIS)

• Microstructure definition• Sensitivity• Problem solution

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• Consider material being porous• Void of regular geometry determines

porosity• Derive homogenious material properties

TOPOLOGICAL OPTIMIZATION: TOPOLOGICAL OPTIMIZATION: GENESIS GENESIS

HOMOGENEISATION THEORYHOMOGENEISATION THEORY

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SimTech 2005 All rights reserved Topology Optimization

Optimization

Optimization

Initial Design

Initial Design

Isodensity Results

Density Distribution Results

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SimTech 2005 All rights reserved Topology Optimization

with Manufacturing Variables

Traditional

Casting in X direction

Initial Design

Casting in Z direction

• Objective:– Minimize Strain

Energy

• Constraints:– Mass constraints

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With No Manufacturing Constraints

Casting with Fix Parting Plane

Casting with variable Parting Plane

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SimTech 2005 All rights reserved Topology Optimization with Casting and

Extrusion Variables

Traditional480 design variables

Casting320 design variables

Extrusion160 design variables

Min Global Strain Energys.t Mass Fraction <= 0.2

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Minimum Size Control

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Michell Truss-like Problem

Number of Design variables=60,704

Number of elements = 60,704


Energy


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Michell Truss-like Problem

Number of Design variables=60,704Number of design domains = 2Number of poles = 1984*2Number of design variables = 7936


Energy


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Mitchell Truss-like Problem

Number of Elements=60,704Number of design domains = 2Number of poles = 1984*2Number of design variables = 7936

Number of Elements=60,704

Number of Design Variables = 60,704

Traditional Results Casting Results

Design variables reduced by 87%

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SimTech 2005 All rights reserved Topology Problem

Load and Boundary Conditions

Minimize Strain Energy

S.t. MASSFR <= 0.1

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SimTech 2005 All rights reserved ANALYS PROBLEM SUMMARY

NUMBER OF GRID POINTS: 188053

NUMBER OF LOCAL COORDINATE SYSTEMS: 1

NUMBER OF CTETRA ELEMENTS: 1003520

TOTAL NUMBER OF NON RIGID ELEMENTS: 1003520

NUMBER OF DEGREES OF FREEDOM: 564147

TOPOLOGY OPTIMIZATION PROBLEM SUMMARY

OBJECTIVE FUNCTION: MINIMIZE STRAIN ENERGY

NUMBER OF INDEPENDENT DESIGNABLE ELEMENTS: 1,003,520

NUMBER OF DESIGNABLE ELEMENTS: 1003520

NUMBER OF TOPOLOGY RESPONSES: 2

NUMBER OF TOPOLOGY CONSTRAINTS: 1

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SimTech 2005 All rights reserved Standard Topology optimization Results

Initial Design

Final DesignDesign Variables= 1,003,520

Number of Elements= 1,003,520

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SimTech 2005 All rights reserved Castable Topology optimization Results

Initial Design

Final DesignDesign Variables= 13,440


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SimTech 2005 All rights reserved Topology optimization Results

Initial Design

Final Design


Design Variables= 2,400

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SimTech 2005 All rights reserved Topology optimization Results

Initial Design

Final Design



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Tetra Results




Design Variables= 1,003,520

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SimTech 2005 All rights reserved Autorib Example

Automatically Generated Candidate Rib Stiffeners

Best 5% of Ribs for IncreasedTorsional Natural Frequency

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SimTech 2005 All rights reserved Application and examples

Design of rib pattern for the plate with hole subject to torsional load

Mass constraint : 10.25%

Max initial disp : 3.61

Max final disp :2.82

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SupportSupport

Concentrated load

• Objective–Minimize strain energy (improve rigidity)

• Constraint–25% mass constraint

Hinge Design

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SimTech 2005 All rights reserved Topometry Optimization Example:

• Objective:– Maximize Strain Energy

• Constraints:– Mass

• Design Variables: 324– Each Element thickness

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Topometry Is Flexible

PSOLIDPBUSHPELASPVECTOR

Topometry Designable Properties

PMASSPCONM3PDAMPPVISC

PHBDYPELASH

PRODPSHEARPSHELLPAXISPBAR

PCOMP

Topology Designable Properties

Topometry Results Topology Results

•Works with almost all elements

•Can produce topology answers

•Can produce continuous answers

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SimTech 2005 All rights reserved Topometry Is Versatile

Topometry Responses

TopologyResponses

Natural Frequencies Statics Special

Buckling Heat Transfer

Frequency Response

Frequency

Mode Shape

DisplacementStrain Energy

StressStrain Forces

DisplacementVelocity

AccelerationStressStrain Force

Backling Load Factor Temperature

Mass GeometricEquationSubroutines

•Works with most load cases

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SimTech 2005 All rights reserved Topometry work with Other Types of Optimization

Topometry + Topography

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Topometry + Shape

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Topometry + Topography + Shape

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SimTech 2005 All rights reserved With Topometry you can solve new problems

Buckling load factor is increased from 5.63 to 10.05 for a constant volume

Reinforcement pattern for buckling of a thin panel

actual thicknesses !

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Where to Reinforce?

• Objective:– Maximize Sum of 12

Lowest Natural frequencies


• Design Variables: 34,560– Each Element thickness

• Objective:– Maximize Sum of 12

Lowest Natural frequencies


• Design Variables: 64– Each Element thickness

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SimTech 2005 All rights reserved Where to Reinforce?

+15 kg => 10 HZ Gains +15 kg => 18 HZ GainsOptimal answer:

117 Kg: 17 HzOptimal answer:

70 Kg: 22 Hz

Sizing Topometry

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SimTech 2005 All rights reserved Where to Reinforce?

AVG HZ Gains Vs Added Mass

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

0 20 40 60 80 100 120 140

Added Mass Kg

AvG

Sum

of 1

2 Fr

eque

ncie

s

SIZINGTOPOMETR

There are natural limits. Once reached no more progress can be made Need to change technology or methodology etc

Is this really needed?

Sizing Topometry?

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Where to Locate Welds?• Objective:

– Maximize Natural frequencies


• Design Variables: 4316– Weld stiffness Retained Welds: 70%

Removed Welds: 30%Car Model with Welds

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There is more to engineering than structures ...

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ASSEMBLY DESIGN

Montecarlo analysis

16 design variables

Results:

Prototypes built according to SimTech specifications and successfully tested

Shrink-fitting. Provide nominal size and tolerances for:

•mandrel

•shaft

•cam

•cam groove

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φ cercle base came

φ extérieur tube

φ intérieur tube

φ dudgeon

φ intérieur came

σσmaxmax

CCglissgliss

plage deplage de pputilutillargelarge

contraintes moins contraintes moins éélevlevéées et stableses et stables

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AUTOMOTIVE CRASH DESIGN

DOE on:

thickness/material variables

geometry variables (domain elements)

assembly variables

Results:

Failure mode: identification of mode shapes and controlling variables

Simultaneous reduction of:

mass (12-20%)

intrusion(s) (16-30%)

deformable barrier energy (5-8%)

Multi-objective (Danner + BFD+Side+Rear...)

Distributed computing environment(16 procs LINUX server)

Dedicated tools developed

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• 1975

OPTIMIZATION WORKSOPTIMIZATION WORKS

COMBAT

MISSION

INITIAL OPTIMUM

SUPERSONIC CRUISE AIRCRAFT

SOLVED BY THE ACSYNT PROGRAM

5 DESIGN VARIABLES, 2 PERFORMANCE CONSTRAINTS

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• 1976: A Two Hour Optimization Study

OPTIMIZATION WORKSOPTIMIZATION WORKS

STOL AIRCRAFT TAKEOFF

CONVENTIONAL: W = WG0

VARIATIONAL CALCULUS: W = 2.5WG0

NUMERICAL OPTIMIZATION: W = 1.2WG0

20gFlightSpeed

2g

500 ft/minClimb

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Shape Optimization of a Pin

• Pin must carry a specified load

• Non-linear contact problem solved using ABAQUS

• Three materials: pin, adhesive,solid base

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Shape Optimization of a Pin

• Minimize maximum stress inthe solid base

• Constraints: displacement, stress

• Nine shape design variables

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Optimize Heat Sink Shape (for PC processor)

Minimize: MassSubject To:– min heat dissipation into the air– max tO in thyristor– max tO in chassis

Analysis:Flux2D - FE based package for the analysis ofelectromagnetic and thermaldevices and processes

VisualDOC/FLUX2D

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Initial design Final design

Design variables:height of the baseheight and width of fins

Result:Result:47% mass reduction47% mass reductionall constraints satisfiedall constraints satisfied

Initial design was choseninfeasible for demo

Final design looks likenormal heat sink in PC

VisualDOC/FLUX2D

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CORE+

CO

IL

-CO

IL

GAP

C-Shaped Magnetic Circuit FLUX2D Model

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CORE

GAP

X

Y

Flux Density in GAP of Initial Design

- Initial geometry gives a non-uniform magnetic field (flux) in the air gap- Optimize the geometry of the gap to give a prescribed point flux, or uniform flux along the length of the gap

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Pt. 1

Pt. 2Pt. 3 Pt. 4

Pt. 5

Pt. 6GAP

Change the Y coordinates of points 1-6 and X coordinates of points 2-5 in order to produce a uniform flux density of 0.6Tesla within the gap. Note: Symmetry Imposed

Case 3: Optimum Flux Density in GAP

Minimize the sum of the squares of the error (SSE) at 200 points

CORE

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Flux Density Variation in GAP of Optimized Designs

Designing all X and Y coordinates produces the flattest flux density as shown in case 3 above

M agnetic Flux Density in G ap

0.57

0.62

0.67

0.72

0.77

0 2 4 6 8 10

X (mm)

Flux

(Tes

la)

Case 1Case 2Case 3Orig

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Airfoil Optimization• NACA 4-digit airfoil• Design variables:

– maximum mean line camber as fraction of chord (m)

– chordwise position of maximum camber (p)

– maximum thickness as fraction of chord (t)

– Angle of attack (α)• Maximize ratio of Lift/Drag.• Use GAMBIT/FLUENT for geometric/flow

modelling.

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SimTech 2005 All rights reserved Optimization Results

Pressure Distribution

Initial design

Final design

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SimTech 2005 All rights reserved CONCLUSIONS

• Shape optimization is a reality for engineering structures

• It can be applied to design in non-linear environment with much greater care and effort

Documents

OPTIMISATION DE FORME DANS LA MECANIQUE DES ...MECANIQUE DES STRUCTURES •Présentation de SimTech •Concepts généraux dans l ’optimisation de structures •Optimisation topologique