October ENSEEIHT-IRIT, Team APO collaboration with GREM 3 -LAPLACE, Toulouse. Design of Electrical Rotating Machines using

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1 using IBBA using ENSEEIHT-IRIT, Team APO collaboration with GREM 3 -LAPLACE, Toulouse October 2007

2 Collaborations with the GREM 3 Team (LAPLACE-ENSEEIHT) using IBBA Bertrand Nogarede, Professor : Director of GREM 3, collaboration since 1995 (2 PhD-thesis). Yvan Lefèvre, CR, HDR : collaboration since 2002, (1 PhD student). François Pigache, MC : collaboration since Jean-François Rouchon, MC : specialist in mechanic, Carole Hénaux, MC : specialist in electrotechnic, Eric Duhayon, MC : specialist of materials, Dominique Harribey, IR.

3 Outline using IBBA Actuators: an Direct and and Formulations Mathematical Formulations = Optimization Problems Classification and Principle of Optimization Methods Principle of Branch Interval Analysis Mixed and Constrained Problems Some Designs of with Magnetic Effects Combinatorial Models for Numerical Examples of Design with a Black-Box Constraint Some Realizations and

4 Outline using IBBA Actuators: an Direct and and Formulations Mathematical Formulations = Optimization Problems Classification and Principle of Optimization Methods Principle of Branch Interval Analysis Mixed and Constrained Problems Some Designs of with Magnetic Effects Combinatorial Models for Numerical Examples of Design with a Black-Box Constraint Some Realizations and

5 Outline using IBBA Actuators: an Direct and and Formulations Mathematical Formulations = Optimization Problems Classification and Principle of Optimization Methods Principle of Branch Interval Analysis Mixed and Constrained Problems Some Designs of with Magnetic Effects Combinatorial Models for Numerical Examples of Design with a Black-Box Constraint Some Realizations and

6 Outline using IBBA Actuators: an Direct and and Formulations Mathematical Formulations = Optimization Problems Classification and Principle of Optimization Methods Principle of Branch Interval Analysis Mixed and Constrained Problems Some Designs of with Magnetic Effects Combinatorial Models for Numerical Examples of Design with a Black-Box Constraint Some Realizations and

7 Direct and using Direct and Inverse Problem Mathematical Formulations Optimization IBBA

8 Direct and using Direct and Inverse Problem Mathematical Formulations Optimization IBBA

9 Direct and using Direct and Inverse Problem Mathematical Formulations Optimization IBBA

10 Direct and using Direct and Inverse Problem Mathematical Formulations Optimization IBBA

11 Direct and using Direct and Inverse Problem Mathematical Formulations Optimization IBBA

12 Mathematical Formulations using Direct and Inverse Problem Mathematical Formulations Optimization IBBA Dimensioning : min f (x) x R n g i (x) 0 i {1,..., n g } h j (x) = 0 j {1,..., n h } More General : min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h }

13 Mathematical Formulations using Direct and Inverse Problem Mathematical Formulations Optimization IBBA Dimensioning : min f (x) x R n g i (x) 0 i {1,..., n g } h j (x) = 0 j {1,..., n h } More General : min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h }

14 Classification and Principle of Optimization Methods using Direct and Inverse Problem Mathematical Formulations Optimization IBBA Multistart Method Metaheuristic Methods Taboo Research, (Glover and Hansen), VNS, (Mladenovitch and Hansen), Kangourou Method... Stochastic Global Optimization Methods Simulated Annealing, Genetic, Evolutionary... Deterministic Global Optimization Methods Particular structure of problems: Convex functions + Theory, Linear programs: Simplex Algorithm (Danzig) Quadratic programs: (Sherali, Audet, Hansen et al.)..., More General Problems = Branch Techniques Difference of convex or monotonic functions, (Horst and Tuy), Interval analysis (Ratsheck, Rokne, E. Hansen)...

15 Principle of a Branch Algorithm for a problem with constraints using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Choice and Subdivision of the box X, (in 2 parts for each iteration) = list of possible solutions, Reduction of the sub-boxes, by using a constraint propagation technique, Computation of bounds of the functions F, G j, H j on the sub-boxes, - inclusion functions - Elimination of the sub-boxes which cannot contain the global optimum: F L (X ) > f or Gi L (X ) > 0 or 0 H(X ), where f denotes the current solution, STOP when accurate enclosures of the optimum are obtained.

16 Principle of a Branch Algorithm for a problem with constraints using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Choice and Subdivision of the box X, (in 2 parts for each iteration) = list of possible solutions, Reduction of the sub-boxes, by using a constraint propagation technique, Computation of bounds of the functions F, G j, H j on the sub-boxes, - inclusion functions - Elimination of the sub-boxes which cannot contain the global optimum: F L (X ) > f or Gi L (X ) > 0 or 0 H(X ), where f denotes the current solution, STOP when accurate enclosures of the optimum are obtained.

17 Principle of a Branch Algorithm for a problem with constraints using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Choice and Subdivision of the box X, (in 2 parts for each iteration) = list of possible solutions, Reduction of the sub-boxes, by using a constraint propagation technique, Computation of bounds of the functions F, G j, H j on the sub-boxes, - inclusion functions - Elimination of the sub-boxes which cannot contain the global optimum: F L (X ) > f or Gi L (X ) > 0 or 0 H(X ), where f denotes the current solution, STOP when accurate enclosures of the optimum are obtained.

18 Principle of a Branch Algorithm for a problem with constraints using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Choice and Subdivision of the box X, (in 2 parts for each iteration) = list of possible solutions, Reduction of the sub-boxes, by using a constraint propagation technique, Computation of bounds of the functions F, G j, H j on the sub-boxes, - inclusion functions - Elimination of the sub-boxes which cannot contain the global optimum: F L (X ) > f or Gi L (X ) > 0 or 0 H(X ), where f denotes the current solution, STOP when accurate enclosures of the optimum are obtained.

19 Principle of a Branch Algorithm for a problem with constraints using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Choice and Subdivision of the box X, (in 2 parts for each iteration) = list of possible solutions, Reduction of the sub-boxes, by using a constraint propagation technique, Computation of bounds of the functions F, G j, H j on the sub-boxes, - inclusion functions - Elimination of the sub-boxes which cannot contain the global optimum: F L (X ) > f or Gi L (X ) > 0 or 0 H(X ), where f denotes the current solution, STOP when accurate enclosures of the optimum are obtained.

20 Interval Analysis using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems 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] [ 1 d, 1 ] if 0 [c, d]. c Remark Subtraction and division are not the inverse operations of addition and respectively multiplication. 0 = extended interval arithmetic, (E. Hansen). Numerical errors = rounded interval analysis, (Moore).

21 Interval Analysis using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems 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] [ 1 d, 1 ] if 0 [c, d]. c Remark Subtraction and division are not the inverse operations of addition and respectively multiplication. 0 = extended interval arithmetic, (E. Hansen). Numerical errors = rounded interval analysis, (Moore).

22 Some Properties of Interval Analysis and Inclusion Functions using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Property For all x X and y Y, one has: x y X Y, where is +,,,. Property Let A, B, C 3 intervals, therefore A (B + C) A B + A C. Property Let Y 1, Y 2, Z 1, Z 2 4 intervals, if Y 1 Z 1 and if Y 2 Z 2 then Y 1 Y 2 Z 1 Z 2 where is +,,,. Definition An inclusion function F (X ) of f over a box X is such that f (X ) := [min f (x), max f (x)] F (X ) = [F L (X ), F U (X )] x X x X

23 Natural Extension: an Inclusion Function using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Theorem The natural extension into interval of an expression of f over a box X is an inclusion function. Example Let f (x) = x 2 x + 1 and x X = [0, 1] Inclusion functions: F 1 (X ) = X 2 X + 1 = [0, 1] 2 [0, 1] + [1, 1] = [0, 2], F 2 (X ) = X (X 1) + 1 = [0, 1]([0, 1] 1) + [1, 1] = [0, 1] [ 1, 0] + [1, 1] = [0, 1], ( F 3 (X ) = X 1 ) [ 2 4 = 1 2, 1 ] [ ] = 4, 1,

24 Natural Extension: an Inclusion Function using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Theorem The natural extension into interval of an expression of f over a box X is an inclusion function. Example Let f (x) = x 2 x + 1 and x X = [0, 1] Inclusion functions: F 1 (X ) = X 2 X + 1 = [0, 1] 2 [0, 1] + [1, 1] = [0, 2], F 2 (X ) = X (X 1) + 1 = [0, 1]([0, 1] 1) + [1, 1] = [0, 1] [ 1, 0] + [1, 1] = [0, 1], ( F 3 (X ) = X 1 ) [ 2 4 = 1 2, 1 ] [ ] = 4, 1,

25 Natural Extension: an Inclusion Function using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Theorem The natural extension into interval of an expression of f over a box X is an inclusion function. Example Let f (x) = x 2 x + 1 and x X = [0, 1] Inclusion functions: F 1 (X ) = X 2 X + 1 = [0, 1] 2 [0, 1] + [1, 1] = [0, 2], F 2 (X ) = X (X 1) + 1 = [0, 1]([0, 1] 1) + [1, 1] = [0, 1] [ 1, 0] + [1, 1] = [0, 1], ( F 3 (X ) = X 1 ) [ 2 4 = 1 2, 1 ] [ ] = 4, 1,

26 Natural Extension: an Inclusion Function using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Theorem The natural extension into interval of an expression of f over a box X is an inclusion function. Example Let f (x) = x 2 x + 1 and x X = [0, 1] Inclusion functions: F 1 (X ) = X 2 X + 1 = [0, 1] 2 [0, 1] + [1, 1] = [0, 2], F 2 (X ) = X (X 1) + 1 = [0, 1]([0, 1] 1) + [1, 1] = [0, 1] [ 1, 0] + [1, 1] = [0, 1], ( F 3 (X ) = X 1 ) [ 2 4 = 1 2, 1 ] [ ] = 4, 1,

27 Propagation Techniques using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems c(x) [a, b] is a contraint = implicit (or explicit) relations between the variables of the problem. Idea: use some deduction steps for reducing the box X. n Linear case: if c(x) = a i x i then: [a, b] X k := i=1 n i=1,i k a k a i X i X k, if a k 0. (1) where k is in {1,, n} and X i is the i th component of X. Non-linear case: Idea (E. Hansen): one linearizes using T 1 (or T 2 ). Then one solves a linear system with interval coefficients. Other Idea: construction of the calculus tree and propagation.

28 Propagation Techniques using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems c(x) [a, b] is a contraint = implicit (or explicit) relations between the variables of the problem. Idea: use some deduction steps for reducing the box X. n Linear case: if c(x) = a i x i then: [a, b] X k := i=1 n i=1,i k a k a i X i X k, if a k 0. (1) where k is in {1,, n} and X i is the i th component of X. Non-linear case: Idea (E. Hansen): one linearizes using T 1 (or T 2 ). Then one solves a linear system with interval coefficients. Other Idea: construction of the calculus tree and propagation.

29 Propagation Techniques using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems c(x) [a, b] is a contraint = implicit (or explicit) relations between the variables of the problem. Idea: use some deduction steps for reducing the box X. n Linear case: if c(x) = a i x i then: [a, b] X k := i=1 n i=1,i k a k a i X i X k, if a k 0. (1) where k is in {1,, n} and X i is the i th component of X. Non-linear case: Idea (E. Hansen): one linearizes using T 1 (or T 2 ). Then one solves a linear system with interval coefficients. Other Idea: construction of the calculus tree and propagation.

30 Example of Propagation Technique on the Calculus Tree using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Let c(x) = 2x 3 x 2 + x 1 and c(x) = 3 where x i [1, 3] for all i {1, 2, 3}.

31 using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Example of Propagation Technique on the Calculus Tree Let c(x) = 2x 3 x 2 + x 1 and c(x) = 3 where x i [1, 3] for all i {1, 2, 3}. The propagation is: * [2,6] * [2,18] + [3,21] X =[1,3] 2 X =[1,3] 1 2 X 3 =[1,3]

32 using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Example of Propagation Technique on the Calculus Tree Let c(x) = 2x 3 x 2 + x 1 and c(x) = 3 where x i [1, 3] for all i {1, 2, 3}. The propagation is: * [2,6] * [2,18] + [3,21] X =[1,3] 2 X =[1,3] * 3-[ 1,3] 1 =[0,2] X 3-[2,18] =[-16,1] 1 [1,1] * [0,2] =[0,2] [1,3] + 3 X [0,2] 2 [2,6] = [0,1] [1,1] 2 X 3 =[1,3] 2 X [0,2] =[0,1] 3 2 [1,1]

33 for Mixed Problems using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Continuous variables: real variables (dimensions of an electrical machines such as diameter). Discrete variables: integer (the number of pair of poles of a machine), boolean (machine with or without slot), categorical variable (which kind of magnet is used). For integer and boolean variables = relaxation for computing bounds + particular bisection technique and propagation. For categorical variables = we introduce 4 particular algorithms with propagation and retro-propagation + properties about the bisection techniques.

34 for Mixed Problems using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Continuous variables: real variables (dimensions of an electrical machines such as diameter). Discrete variables: integer (the number of pair of poles of a machine), boolean (machine with or without slot), categorical variable (which kind of magnet is used). For integer and boolean variables = relaxation for computing bounds + particular bisection technique and propagation. For categorical variables = we introduce 4 particular algorithms with propagation and retro-propagation + properties about the bisection techniques.

35 for Mixed Problems using IBBA Branch Principle Interval Analysis Mixed and Constrained Problems Continuous variables: real variables (dimensions of an electrical machines such as diameter). Discrete variables: integer (the number of pair of poles of a machine), boolean (machine with or without slot), categorical variable (which kind of magnet is used). For integer and boolean variables = relaxation for computing bounds + particular bisection technique and propagation. For categorical variables = we introduce 4 particular algorithms with propagation and retro-propagation + properties about the bisection techniques.

36 with Magnetic Effects using IBBA MAPSE Combinatorial Models Numerical Exemples Extension

37 Example for the Dimensioning of an Motor using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Slotless Machine with Permanent Magnet: βπ p D 2 C l a E IBBA standard (defined by Ratschek and Rokne 1988) 1h35, IBBA + propagation due to E. Hansen 41.5s, IBBA + propagation with the calculus tree 0.5s. e C

38 Example for the Dimensioning of an Motor using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Slotless Machine with Permanent Magnet: βπ p D 2 C l a E IBBA standard (defined by Ratschek and Rokne 1988) 1h35, IBBA + propagation due to E. Hansen 41.5s, IBBA + propagation with the calculus tree 0.5s. e C

39 Example for the Dimensioning of an Motor using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Slotless Machine with Permanent Magnet: βπ p D 2 C l a E IBBA standard (defined by Ratschek and Rokne 1988) 1h35, IBBA + propagation due to E. Hansen 41.5s, IBBA + propagation with the calculus tree 0.5s. e C

40 Combination of Different Figure: 4 structures possible machines 2 modes (rectangular or sinusoidal waveform). using b b r e W it h S lo t s Inverse Rotor C C D 2 D 2 d Internal Rotor a la E IBBA MAPSE Combinatorial Models Numerical Exemples Extension W it h o u t S lo t βπ p g bf Rectangular or Sinusoidal Waveform

41 Discrete Variables for Modeling using IBBA MAPSE Combinatorial Models Numerical Exemples Extension 1. b r = 1 for machines with an internal rotoric configuration and b r = 0 for an external one, 2. b e = 1 for machines with slots or b e = 0 slotless machines, 3. b f = 1 represents rectangular waveform or b f = 0 for a sinusoïdale one. 3 boolean variables to represent 8 possible structures + 2 categorical variables.

42 Combinatorial Models for using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Γ em = k Γ D [D + (1 b e )(2b r 1)E] LB e K S, ( ) a K S = k r Ej b e a + d + (1 b e), [ k Γ = π b f [1 K f ] ] 2 β + (1 b f ) 2 2 sin(β π 2 ), [ ] E + g K f = 1.5pβ (1 b e ).b f, D 2J(σ m )l a B e = ] 1, (2b r 1)D ln k c k c =... 1 b e [ 1 N ea 2 5πD.g+πD.a [ D+2E(2br 1)(1 b e) D 2(2b r 1)[l a+g] Generally, the torque Γ em is fixed = a strong equality ],

43 Examples of 4 optimal machines with magnetical effects using IBBA MAPSE Combinatorial Models Numerical Exemples Extension

44 Multicriteria Optimization using IBBA MAPSE Combinatorial Models Numerical Exemples Extension

45 Analytical versus Numerical Models using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Schedule of conditions = a fixed torque. Optimal solutions satisfy the equality constraint about 1%. Computation of the torque using finite elements methods (by a numerical model); software EFCAD (from the LEEI) or ANSYS. EFCAD and ANSYS are too general = NUMT

46 Analytical versus Numerical Models using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Schedule of conditions = a fixed torque. Optimal solutions satisfy the equality constraint about 1%. Computation of the torque using finite elements methods (by a numerical model); software EFCAD (from the LEEI) or ANSYS. EFCAD and ANSYS are too general = NUMT

47 Analytical versus Numerical Models using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Schedule of conditions = a fixed torque. Optimal solutions satisfy the equality constraint about 1%. Computation of the torque using finite elements methods (by a numerical model); software EFCAD (from the LEEI) or ANSYS. EFCAD and ANSYS are too general = NUMT

48 Analytical versus Numerical Models using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Schedule of conditions = a fixed torque. Optimal solutions satisfy the equality constraint about 1%. Computation of the torque using finite elements methods (by a numerical model); software EFCAD (from the LEEI) or ANSYS. EFCAD and ANSYS are too general = NUMT

49 Direct and using IBBA MAPSE Combinatorial Models Numerical Exemples Extension

50 Numerical Validations: NUMT Algorithm using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Figure: Draw 2 optimal solutions (min mass and min multicriteria).

51 Numerical Validations: NUMT Algorithm using Couple (N.m) Min Masse Min Vol Min Multi IBBA MAPSE Combinatorial Models Numerical Exemples Extension pi 3*pi/4 pi/2 pi/4 0 pi/4 pi/2 3*pi/4 pi Angle de calage : π Ψ π Figure: Torque of 3 solutions and design of teeth of the slot. Using Triangle and EFCAD. Name of the Software: NUMT.

52 Discussions using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Generally about 10%! Analytical Value Numerical Value If the gap is less than 3% our solution is validated. Else = modifications of the solution until the numerical value is correct. Optimize with a black-box constraint: min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b Extension of IBBA? f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} NUMT (x, z, σ, b) = Γ em

53 Discussions using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Generally about 10%! Analytical Value Numerical Value If the gap is less than 3% our solution is validated. Else = modifications of the solution until the numerical value is correct. Optimize with a black-box constraint: min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b Extension of IBBA? f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} NUMT (x, z, σ, b) = Γ em

54 Discussions using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Generally about 10%! Analytical Value Numerical Value If the gap is less than 3% our solution is validated. Else = modifications of the solution until the numerical value is correct. Optimize with a black-box constraint: min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b Extension of IBBA? f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} NUMT (x, z, σ, b) = Γ em

55 Discussions using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Generally about 10%! Analytical Value Numerical Value If the gap is less than 3% our solution is validated. Else = modifications of the solution until the numerical value is correct. Optimize with a black-box constraint: min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b Extension of IBBA? f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} NUMT (x, z, σ, b) = Γ em

56 Discussions using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Generally about 10%! Analytical Value Numerical Value If the gap is less than 3% our solution is validated. Else = modifications of the solution until the numerical value is correct. Optimize with a black-box constraint: min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b Extension of IBBA? f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} NUMT (x, z, σ, b) = Γ em

57 IBBA+NUMT using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Idea: Define a zone where the numerical solution is searched. Analytical model + tolerance about 10% = zone. In this zone, all the solutions satisfies that NUMT (x, z, σ, b ) = Γ em about 1%. min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} 0.9 Γ em Γ(x, z, σ, b) 1.1 Γ em NUMT (x, z, σ, b) = Γ em

58 IBBA+NUMT using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Idea: Define a zone where the numerical solution is searched. Analytical model + tolerance about 10% = zone. In this zone, all the solutions satisfies that NUMT (x, z, σ, b ) = Γ em about 1%. min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} 0.9 Γ em Γ(x, z, σ, b) 1.1 Γ em NUMT (x, z, σ, b) = Γ em

59 IBBA+NUMT using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Idea: Define a zone where the numerical solution is searched. Analytical model + tolerance about 10% = zone. In this zone, all the solutions satisfies that NUMT (x, z, σ, b ) = Γ em about 1%. min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} 0.9 Γ em Γ(x, z, σ, b) 1.1 Γ em NUMT (x, z, σ, b) = Γ em

60 IBBA+NUMT using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Idea: Define a zone where the numerical solution is searched. Analytical model + tolerance about 10% = zone. In this zone, all the solutions satisfies that NUMT (x, z, σ, b ) = Γ em about 1%. min x Rnr,z Nne, σ Q nc i=1 K i,b Bn b f (x, z, σ, b) g i (x, z, σ, b) 0 i {1,..., n g } h j (x, z, σ, b) = 0 j {1,..., n h 1} 0.9 Γ em Γ(x, z, σ, b) 1.1 Γ em NUMT (x, z, σ, b) = Γ em

61 Algorithm IBBA+NUMT using IBBA MAPSE Combinatorial Models Numerical Exemples Extension 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: V 1, V For j:=1 to 2 do 5.1 Compute v j := lb(f, V j ). 5.2 Compute all the lower and upper bounds of all the analytical constraints on V j + some deduction steps. 5.3 if f v j and no analytical constraint is unsatisfied then insert (v j, V j ) in L. set m the midpoint of V j 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)). if f is changed then remove from L all (z, Z) where z > f and set ỹ := m. 6. If f min z < ɛ (where z = lb(f, Z)) then STOP. (z,z) L Else GoTo Step 4.

62 Numerical Example 1 with IBBA+NUMT using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Parameters Min V g Name Bounds Unit IBBA IBBA+NUMT D [0.01, 0.3] m L [0.01, 0.3] m l a [0.003, 0.01] m E [0.005, 0.03] m C [0.003, 0.02] m β [0.7, 0.9] k d [0.4, 0.6] p [[3, 10]] 8 8 m {1, 2} 1 1 b r {0, 1} 0 0 Volume m Mass kg Multi Analytical Torque N m Numerical Torque N m CPU - Time min 0min35s 7min15s Numerical Computations - 437

63 Numerical Example 2 with IBBA+NUMT using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Parameters Min M a Name Bounds Unit IBBA IBBA+NUMT D [0.01, 0.3] m L [0.01, 0.3] m l a [0.003, 0.01] m E [0.005, 0.03] m C [0.003, 0.02] m β [0.7, 0.9] k d [0.4, 0.6] p [[3, 10]] 8 8 m {1, 2} 1 1 b r {0, 1} 0 0 Volume m Mass kg Multi Analytical Torque N m Numerical Torque N m CPU - Time min 1min14s 8min17s Numerical Computations - 560

64 Numerical Example 3 with IBBA+NUMT using IBBA MAPSE Combinatorial Models Numerical Exemples Extension Parameters Min Multi Name Bounds Unit IBBA IBBA+NUMT D [0.01, 0.3] m L [0.01, 0.3] m l a [0.003, 0.01] m E [0.005, 0.03] m C [0.003, 0.02] m β [0.7, 0.9] k d [0.4, 0.6] p [[3, 10]] 8 8 m {1, 2} 1 1 b r {0, 1} 0 0 Volume m Mass kg Multi Analytical Torque N m Numerical Torque N m CPU - Time min 1min03s 7min37s Numerical Computations - 526

65 Some Realizations using Fre de ric Messine Figure: Motor with a strongest torque. IBBA Figure: piezoelectric bimorphs.

66 Model of a Piezoelectric Bimorph using IBBA

67 Optimal the Piezoelectric Bimorph using IBBA