Design of Electrical Rotating Machines using Interval Branch and Bound based Algorithms

Design ofElectricalRotating

Machines usingInterval Branch

and Boundbased

Algorithms

Frederic Messine

Inverse Problemof Design

IBBA Algorithms

Design ofMachines

Realizations &Conclusion

Design of Electrical Rotating Machines usingInterval Branch and Bound based Algorithms

Frederic Messine

ENSEEIHT-IRIT, Team APO collaboration with GREM3-LAPLACE,Toulouse

October 2007

and Boundbased

Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

Collaborations with the GREM3 Team(LAPLACE-ENSEEIHT)

I Bertrand Nogarede, Professor : Director of GREM3,collaboration since 1995 (2 PhD-thesis).

I Yvan Lefevre, CR, HDR : collaboration since 2002, (1PhD student).

I Francois Pigache, MC : collaboration since 2005.

I Jean-Francois Rouchon, MC : specialist in mechanic,

I Carole Henaux, MC : specialist in electrotechnic,

I Eric Duhayon, MC : specialist of materials,

I Dominique Harribey, IR.

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Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

Outline

Design of Actuators: an Inverse ProblemDirect and Inverse Problem of Design and FormulationsMathematical Formulations =⇒ Optimization ProblemsClassification and Principle of Optimization Methods

Interval Branch and Bound AlgorithmsPrinciple of Branch and Bound AlgorithmsInterval AnalysisMixed and Constrained Problems

Some Designs of Electrical Rotating MachinesRotating Machines with Magnetic EffectsCombinatorial Models for Electrical MachinesNumerical Examples of Electrical MachinesDesign with a Black-Box Constraint

Some Realizations and Conclusion

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Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

Outline

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Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

Outline

and Boundbased

Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

Outline

and Boundbased

Algorithms

Frederic Messine

Direct and InverseProblem

MathematicalFormulations

OptimizationAlgorithms

IBBA Algorithms

Design ofMachines

Direct and Inverse Problem of Design

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Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

and Boundbased

Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

and Boundbased

Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

and Boundbased

Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

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Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

Mathematical Formulations

I Dimensioning Inverse Problem:minx∈Rn

gi (x) ≤ 0 ∀i ∈ {1, . . . , ng}hj(x) = 0 ∀j ∈ {1, . . . , nh}

I More General Inverse Problem of Design:min

x∈Rnr ,z∈Nne ,

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

gi (x , z , σ, b) ≤ 0 ∀i ∈ {1, . . . , ng}hj(x , z , σ, b) = 0 ∀j ∈ {1, . . . , nh}

and Boundbased

Algorithms

Frederic Messine

IBBA Algorithms

Design ofMachines

Mathematical Formulations

I Dimensioning Inverse Problem:minx∈Rn

gi (x) ≤ 0 ∀i ∈ {1, . . . , ng}hj(x) = 0 ∀j ∈ {1, . . . , nh}

I More General Inverse Problem of Design:min

x∈Rnr ,z∈Nne ,

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

gi (x , z , σ, b) ≤ 0 ∀i ∈ {1, . . . , ng}hj(x , z , σ, b) = 0 ∀j ∈ {1, . . . , nh}

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Frederic Messine

IBBA Algorithms

Design ofMachines

Classification and Principle of OptimizationMethods

I Multistart MethodI Metaheuristic Methods

I Taboo Research, (Glover and Hansen),I VNS, (Mladenovitch and Hansen),I Kangourou Method...

I Stochastic Global Optimization MethodsI Simulated Annealing,I Genetic Algorithms,I Evolutionary Algorithms...

I Deterministic Global Optimization MethodsI Particular structure of problems:

I Convex functions + Theory,I Linear programs: Simplex Algorithm (Danzig)I Quadratic programs: (Sherali, Audet, Hansen et al.)...,

I More General Problems =⇒ Branch and BoundTechniques

I Difference of convex or monotonic functions, (Horst andTuy),

I Interval analysis (Ratsheck, Rokne, E. Hansen)...

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Frederic Messine

IBBA Algorithms

Branch and BoundPrinciple

Interval Analysis

Mixed andConstrainedProblems

Design ofMachines

Principle of a Branch and Bound Algorithm for aproblem with constraints

I Choice and Subdivision of the box X , (in 2 parts for eachiteration) =⇒ list of possible solutions,

I Reduction of the sub-boxes, by using a constraintpropagation technique,

I Computation of bounds of the functions F , Gj , Hj on thesub-boxes, - inclusion functions -

I Elimination of the sub-boxes which cannot contain theglobal optimum: F L(X ) > f or GL

i (X ) > 0 or 0 6∈ H(X ),where f denotes the current solution,

I STOP when accurate enclosures of the optimum areobtained.

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IBBA Algorithms

Interval Analysis

Design ofMachines

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

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Algorithms

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

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Algorithms

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

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Algorithms

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

Interval Analysis

Let X = [a, b] and Y = [c , d ] 2 intervals.Moore (1966) defines the interval arithmetic as follows:

[a, b] + [c , d ] = [a + c , b + d ][a, b]− [c , d ] = [a− d , b − c][a, b]× [c , d ] = [min{ac, ad , bc , bd},

max{ac, ad , bc , bd}][a, b]÷ [c , d ] = [a, b]× [

c] if 0 6∈ [c , d ].

RemarkSubtraction and division are not the inverse operations ofaddition and respectively multiplication.

÷0 =⇒ extended interval arithmetic, (E. Hansen).Numerical errors =⇒ rounded interval analysis, (Moore).

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

Interval Analysis

Let X = [a, b] and Y = [c , d ] 2 intervals.Moore (1966) defines the interval arithmetic as follows:

[a, b] + [c , d ] = [a + c , b + d ][a, b]− [c , d ] = [a− d , b − c][a, b]× [c , d ] = [min{ac, ad , bc , bd},

max{ac, ad , bc , bd}][a, b]÷ [c , d ] = [a, b]× [

c] if 0 6∈ [c , d ].

RemarkSubtraction and division are not the inverse operations ofaddition and respectively multiplication.

÷0 =⇒ extended interval arithmetic, (E. Hansen).Numerical errors =⇒ rounded interval analysis, (Moore).

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

Some Properties of Interval Analysis and InclusionFunctions

Property

For all x ∈ X and y ∈ Y , one has: x ? y ∈ X ? Y , where ? is+,−,×,÷.

Property

Let A,B,C 3 intervals, thereforeA× (B + C ) ⊆ A× B + A× C .

Property

Let Y1,Y2,Z1,Z2 4 intervals, if Y1 ⊆ Z1 and if Y2 ⊆ Z2 thenY1 ? Y2 ⊆ Z1 ? Z2 where ? is +,−,×,÷.

DefinitionAn inclusion function F (X ) of f over a box X is such that

f (X ) := [minx∈X

f (x),maxx∈X

f (x)] ⊆ F (X ) = [F L(X ),FU(X )]

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IBBA Algorithms

Interval Analysis

Design ofMachines

Natural Extension: an Inclusion Function

TheoremThe natural extension into interval of an expression of f overa box X is an inclusion function.

Example

Let f (x) = x2 − x + 1 and x ∈ X = [0, 1]

Inclusion functions:

I F1(X ) = X 2 − X + 1 = [0, 1]2 − [0, 1] + [1, 1] = [0, 2],

I F2(X ) = X (X − 1) + 1 = [0, 1]([0, 1]− 1) + [1, 1] =[0, 1]× [−1, 0] + [1, 1] = [0, 1],

I F3(X ) =

(X − 1

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

Example

Let f (x) = x2 − x + 1 and x ∈ X = [0, 1]

I F1(X ) = X 2 − X + 1 = [0, 1]2 − [0, 1] + [1, 1] = [0, 2],

I F2(X ) = X (X − 1) + 1 = [0, 1]([0, 1]− 1) + [1, 1] =[0, 1]× [−1, 0] + [1, 1] = [0, 1],

I F3(X ) =

(X − 1

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IBBA Algorithms

Interval Analysis

Design ofMachines

Example

Let f (x) = x2 − x + 1 and x ∈ X = [0, 1]

I F1(X ) = X 2 − X + 1 = [0, 1]2 − [0, 1] + [1, 1] = [0, 2],

I F2(X ) = X (X − 1) + 1 = [0, 1]([0, 1]− 1) + [1, 1] =[0, 1]× [−1, 0] + [1, 1] = [0, 1],

I F3(X ) =

(X − 1

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

Example

Let f (x) = x2 − x + 1 and x ∈ X = [0, 1]

I F1(X ) = X 2 − X + 1 = [0, 1]2 − [0, 1] + [1, 1] = [0, 2],

I F2(X ) = X (X − 1) + 1 = [0, 1]([0, 1]− 1) + [1, 1] =[0, 1]× [−1, 0] + [1, 1] = [0, 1],

I F3(X ) =

(X − 1

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

Propagation Techniques

c(x) ∈ [a, b] is a contraint =⇒ implicit (or explicit) relationsbetween the variables of the problem.

Idea: use some deduction steps for reducing the box X .

Linear case: if c(x) =n∑

aixi then:

[a, b]−

n∑i=1,i 6=k

∩ Xk , if ak 6= 0. (1)

where k is in {1, · · · , n} and Xi is the ith component of X .

Non-linear case: Idea (E. Hansen): one linearizes using T1 (orT2). Then one solves a linear system with interval coefficients.Other Idea: construction of the calculus tree and propagation.

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IBBA Algorithms

Interval Analysis

Design ofMachines

aixi then:

[a, b]−

n∑i=1,i 6=k

∩ Xk , if ak 6= 0. (1)

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IBBA Algorithms

Interval Analysis

Design ofMachines

aixi then:

[a, b]−

n∑i=1,i 6=k

∩ Xk , if ak 6= 0. (1)

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IBBA Algorithms

Interval Analysis

Design ofMachines

Example of Propagation Technique based on theCalculus Tree

Let c(x) = 2x3x2 + x1 and

c(x) = 3

where xi ∈ [1, 3] for all i ∈ {1, 2, 3}.

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

c(x) = 3

where xi ∈ [1, 3] for all i ∈ {1, 2, 3}.The propagation is:

=[1,3]

[2,18]

[3,21]

=[1,3]

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Algorithms

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

c(x) = 3

where xi ∈ [1, 3] for all i ∈ {1, 2, 3}.The propagation is:

=[1,3]

[2,18]

[3,21]

=[1,3]

=[1,3] X

3-[ 1,3] =[0,2]

[0,2] =[0,2] [1,3]

3-[2,18] =[-16,1]

[0,2] = [0,1] [2,6]

[0,2] =[0,1] 2

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

Algorithms for Mixed Problems

Continuous variables: real variables (dimensions of anelectrical machines such as diameter).

Discrete variables: integer (the number of pair of poles of amachine), boolean (machine with or without slot), categoricalvariable (which kind of magnet is used).

For integer and boolean variables =⇒ relaxation forcomputing bounds + particular bisection technique andpropagation.

For categorical variables =⇒ we introduce 4 particularalgorithms with propagation and retro-propagation +properties about the bisection techniques.

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IBBA Algorithms

Interval Analysis

Design ofMachines

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Algorithms

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Frederic Messine

IBBA Algorithms

Interval Analysis

Design ofMachines

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Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Rotating Machines with Magnetic Effects

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Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Example for the Dimensioning of an ElectricalMotor

Electrical Slotless Rotating Machine with Permanent Magnet:

I IBBA standard (defined by Ratschek and Rokne 1988)−→ 1h35,

I IBBA + propagation due to E. Hansen −→ 41.5s,

I IBBA + propagation with the calculus tree −→ 0.5s.

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

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Algorithms

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

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Algorithms

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Combination of Different Rotating ElectricalMachines

C D__2

βπ__p

Internal RotorInverse Rotor

Rectangular or Sinusoidal Waveform

Without

Figure: 4 structures possible machines ×2 modes (rectangular orsinusoidal waveform).

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Discrete Variables for Modeling ElectricalMachines

1. br = 1 for machines with an internal rotoric configurationand br = 0 for an external one,

2. be = 1 for machines with slots or be = 0 slotlessmachines,

3. bf = 1 represents rectangular waveform or bf = 0 for asinusoıdale one.

3 boolean variables to represent 8 possible structures + 2categorical variables.

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Combinatorial Models for Electrical Machines

Γem = kΓD [D + (1− be)(2br − 1)E ] LBeKS,

KS = krEj

a + d+ (1− be)

kΓ =π

[bf [1−Kf ]

√β + (1− bf )

2sin(β

Kf = 1.5pβ

[E + g

](1− be).bf ,

Be =2J(σm)la

(2br − 1)D ln[

D+2E(2br−1)(1−be)D−2(2br−1)[la+g ]

1− be

5πD.g+πD.a

Generally, the torque Γem is fixed =⇒ a strong equalityconstraint.

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Examples of 4 optimal machines with magneticaleffects

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Multicriteria Optimization

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Analytical versus Numerical Models

Schedule of conditions =⇒ a fixed torque.

Optimal solutions satisfy the equality constraint about 1%.

Computation of the torque using finite elements methods (bya numerical model); software EFCAD (from the LEEI) orANSYS.

EFCAD and ANSYS are too general =⇒ NUMT

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

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Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

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Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

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Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Numerical Validations: NUMT Algorithm

Figure: Draw 2 optimal solutions (min mass and min multicriteria).

Figure: Mesh 2 above machines

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Numerical Validations: NUMT Algorithm

−pi −3*pi/4 −pi/2 −pi/4 0 pi/4 pi/2 3*pi/4 pi−10

Angle de calage : −π ≤ Ψ ≤ π

)Min MasseMin VolMin Multi

Figure: Torque of 3 solutions and design of teeth of the slot.

Using Triangle and EFCAD.Name of the Software: NUMT.

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Discussions

Analytical Value 6= Numerical Value

Generally about 10% !

I If the gap is less than 3% our solution is validated.

I Else =⇒ modifications of the solution until the numericalvalue is correct.

Optimize with a black-box constraint:min

x∈Rnr ,z∈Nne ,

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

gi (x , z , σ, b) ≤ 0 ∀i ∈ {1, . . . , ng}hj(x , z , σ, b) = 0 ∀j ∈ {1, . . . , nh − 1}NUMT (x , z , σ, b) = Γem

Extension of IBBA ?

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Discussions

x∈Rnr ,z∈Nne ,

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

Extension of IBBA ?

and Boundbased

Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Discussions

x∈Rnr ,z∈Nne ,

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

Extension of IBBA ?

and Boundbased

Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Discussions

x∈Rnr ,z∈Nne ,

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

Extension of IBBA ?

and Boundbased

Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Discussions

x∈Rnr ,z∈Nne ,

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

Extension of IBBA ?

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

IBBA+NUMT

Idea: Define a zone where the numerical solution is searched.Analytical model + tolerance about 10% =⇒ zone.In this zone, all the solutions satisfies thatNUMT (x∗, z∗, σ∗, b∗) = Γem about 1%.

minx∈Rnr ,z∈Nne ,

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

gi (x , z , σ, b) ≤ 0 ∀i ∈ {1, . . . , ng}hj(x , z , σ, b) = 0 ∀j ∈ {1, . . . , nh − 1}0.9× Γem ≤ Γ(x , z , σ, b) ≤ 1.1× Γem

NUMT (x , z , σ, b) = Γem

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

IBBA+NUMT

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

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Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

IBBA+NUMT

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

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Algorithms

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

IBBA+NUMT

σ∈Qnc

i=1Ki ,b∈Bnb

f (x , z , σ, b)

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Algorithms

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Algorithm IBBA+NUMT

1. Set X := the initial hypercube.

2. Set f := +∞ and set L := (+∞, X ).3. Extract from L the lowest lower bound.

4. Bisect the considered box chosen by its midpoint: V1, V2.

5. For j:=1 to 2 do

5.1 Compute vj := lb(f ,Vj).5.2 Compute all the lower and upper bounds of all

the analytical constraints on Vj + somededuction steps.

5.3 if f ≥ vj and no analytical constraint isunsatisfied then

I insert (vj , Vj) in L.I set m the midpoint of Vj

I if m satisfies all the analytical constraints

and then if the numerical constraint

NUMT(x , z , σ, b) = Γ is also satisfied then

f := min(f, f (m)).I if f is changed then remove from L all (z , Z)

where z > f and set y := m.6. If f− min

(z,Z)∈Lz < ε (where z = lb(f , Z)) then STOP.

Else GoTo Step 4.

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Numerical Example 1 with IBBA+NUMT

Parameters Min Vg

Name Bounds Unit IBBA IBBA+NUMT

D [0.01, 0.3] m 0.1331 0.1310L [0.01, 0.3] m 0.0474 0.0497la [0.003, 0.01] m 0.0047 0.0047E [0.005, 0.03] m 0.0074 0.0074C [0.003, 0.02] m 0.0049 0.0049β [0.7, 0.9] 0.89 0.89kd [0.4, 0.6] 0.5043 0.5043p [[3, 10]] 8 8m {1, 2} 1 1br {0, 1} 0 0

Volume m3 8.881 10−4 9.072 10−4

Mass kg 3.21 3.31Multi 2.09 2.15

Analytical Torque N·m 9.81 10.00Numerical Torque N·m 9.35 9.96

CPU - Time min 0min35s 7min15sNumerical Computations - 437

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Parameters Min Ma

Name Bounds Unit IBBA IBBA+NUMT

Volume m3 9.716 10−4 10.157 10−4

Mass kg 2.94 3.07Multi 2.10 2.19

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IBBA Algorithms

Design ofMachines

CombinatorialModels

NumericalExemples

Extension

Parameters Min MultiName Bounds Unit IBBA IBBA+NUMT

Volume m3 9.067 10−4 9.072 10−4

Mass kg 3.10 3.30Multi 2.07 2.14

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IBBA Algorithms

Design ofMachines

Some Realizations

Figure: Motor with a strongest torque.

Figure: Design of piezoelectric bimorphs.

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Algorithms

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IBBA Algorithms

Design ofMachines

Model of a Piezoelectric Bimorph

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Algorithms

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Frederic Messine

IBBA Algorithms

Design ofMachines

Optimal Design of the Piezoelectric Bimorph

Design of Electrical Rotating Machines using Interval Branch and Bound based Algorithms

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