4. Zastosowania Optymalizacja wielokryterialna

4. Zastosowania Optymalizacja wielokryterialna Tadeusz Burczyński 1,2) 1), Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland 2) Cracow University of Technology, Institute of Computer Modelling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracow, Poland 1

Outline of the presentation 1. Multi-objective problem 2. Thermoelasticity problem 3. Evolutionary algorithms 4. Numerical examples 5. Conclusions 2

Multi-objective problem general information 3

Multi-objective problem general information Find the vector x = [ x1, x2,... x n ] T inequality constrains g i ( x) ł 0 i = 1, 2,..., m equality constrains hi (x) = 0 i 1, 2,..., = p which minimizes the vector of k objective functions f (x) = [ f 1 (x ), f 2 (x ),..., f k (x ) ] T 4

Multicriteria optimization selected methods Min-max method Pure weighting method Weighting min-max method Global criterion method Constraint Method VEGA: Vector Evaluated Genetic Algorithm (Schaffer 1985). HLGA: Hajela's and Lin's Weighting-based Genetic Algorithm (1992). FFGA: Fonseca's and Fleming's Multiobjective Genetic Algorithm (1993). NPGA: The Niched Pareto Genetic Algorithm (Horn, Nafpliotis, Goldberg 1994). NSGA: The Nondominated Sorting Genetic Algorithm (Srinivas, Deb 1994). SPEA: The Strength Pareto Evolutionary Algorithm (Zitzler, Thiele 1999). 5

Pure weighting method f ( x) = ĺ k i= 1 wi f i ( x) (f1, f2,..., fk,) f where: k number of objective functions; x solutions vector wi appropriate weights Selection of the weights? wi [ 0, 1] ĺ k i= 1 wi = 1 6

Pareto frontier concept For the minimization problem the set of k objective function: f(x)=(f1(x),f2(x),...,fk(x)); Solution x is dominated, if exists admissible solution y not worse of x for each objective function fi: fi(y) fi (x) (i=1,... k) otherwise x is non-dominated solution 7

Pareto frontier concept An example of the bi-objective problem 8

VEGA (Vector Evaluated Genetic Algorithm) Population is divided into k-th subpopulation Selection is performed independently (each population correspond to different criterion) Evolutionary operators exceed boundaries of the subpopulations (covers whole populations) 9

Types of rank individuals Population ranking according to Goldberg (1989) Ranks after four steps Ranks after one step 10

Types of rank individuals Population ranking according to Fonseca and Fleming (1998) Ranking equals the number of individuals by which is dominated, plus one after ranking 11

SPEA (Strength Pareto Evolutionary Algorithm) external set P contains only non-dominated solutions rank of the individual depends on dominations by external set (domination within population P is negleced) size of the external set is reduced by clustering 12

SPEA (Strength Pareto Evolutionary Algorithm) 13

Heat exchangers (radiators) 14

Direct problem Γ = t ČΓ Γt ΓT qi = Γ c : qi α (T= i T Ą ) q r ΓĆ q Γ λ T,ii + Q Ti = Γ q : qi = q ΓČ Ć Γ = c ΓČ = c heat conduction problem thermal boundary conditions Γ T : Ti u ΓT u + λ 0= - thermal conductivity 15

Direct problem Γ = t ČΓ Γt ΓT u ΓT = q ΓČ Ć Γ = u ΓĆ q Γ ΓČ c Γ = c radiation problem ů 1 ε ( p) r ( ξ) ň=ęeb ( pε ) ξ q (p ) +K (,p )d ε ( p ) ű Γc q (ξ ) + ( ε)eb (ξ ) r blackbody emissive power eb = σ T 4 kernel function K ( ξ, p) = cosφ ξ cos 2ξ p p φ (, p) σ c (p ),,p c - Stefan Boltzman constant ě 1 if ξ can be seen from p otherwise 0 β βξ (,ξ p ) = 16

Direct problem Γ = t ČΓ Γt ΓT u ΓT = q ΓČ Ć Γ = u ΓĆ q Γ ΓČ c Γ = c heat conduction problem ST = R thermoelasticity problem G ui, jj mechanical boundary conditions Γ t : ti Γ u : ui _ ti = _ ui = G + u j, ji 1 2v 2G (1 v ) + α t T, i 1 2v G - shear modulus ν - Poisson s ratio α t - thermal expansion coefficient dis c re tiza t io n 0 = distribution of the temperature KU = F 17

Fitness function evaluation 18

Formulation of the optimization problem single-objective optimization (constraint method) The minimum volume of the structure: min V (X) X Constrains: maximal value of temperature T T ad Ł 0 maximal value of equivalent stress σ eq ad eq σł 0 The minimization of the maximal value of the equivalent stress: min σ X max eq (X) V V ad Ł 0 Constrains: maximal value of volume of the structure The minimization of the maximal value of the temperature: min T max (X) X V V ad Ł 0 Constrains: maximal value of volume of the structure The maximization of the total dissipated heat flux: max q(x) X Constrains: maximal value of equivalent stress σ cost of the radiator eq ad eq σł 0 c c ad Ł 0 19

Formulation of the optimization problem multiobjective optimization (Pareto approach) min V (X) X Minimization / Maximization min σ X 2 or 3 functionals simultaneously max eq (X) min T max (X) X max q (X) X 20

Design vector = chromosome x = < x1, x2,... xi,... x N > xil xi xir where: xi, i = 1, N genes which represent the geometry of the boundary Geometric constrains: admissible values of design variables positions of the control points of the Bezier curve 21

Evolutionary algorithm operators Uniform mutation Gaussian mutation Simple crossover Ranking selection 22

Evolutionary algorithm operators Uniform mutation Gaussian mutation Simple crossover Ranking selection 23

Evolutionary algorithm operators Uniform mutation Gaussian mutation Simple crossover Ranking selection 24

Evolutionary algorithm operators Uniform mutation Gaussian mutation Simple crossover Ranking selection single-objective optimization rank is calcualted on the base of a position of the chromosome after sorting 25

Calculation of the dominance rank (multi-objective optimization) # of dominators ED( xi ; x j ) = popsize ĺ ( x (n ) n= 1 i x j (n )) 2 rank is calcualted on the base of the number of individuals by which is dominated and scaled value of the Euclidian distance (ED) 26

Evolutionary algorithm (single-objective optimization) 27

Evolutionary algorithm (multiobjective optimization) 28

Geometry modeling Bezier curve C ( u) = ĺ p i= 0 Bi, p ( u ) Pi 0 Ł u 1Ł P control points of the Bezier curve i p! p 1 Bi, p ( u ) = ui ( 1 u ) i!( p i )! Bi,p basis function of the Bezier curve 29

Numerical examples Material properties Parameter Young modulus Poisson ratio Thermal expansion coef. Heat conductivity Emissivity Value 120 000 MPa 0.3 16.5 10-6 1/K 400 W/mK 0.8 The fitness function is created by the method of penalty function taking into account: the volume of the structure, the equivalent stress, the temperature or heat flux and imposed constrains. For the multi-objective optimization only geometrical constrains are applied. 30

Example 1 The admissible values of the design parameters Design variable Range Z1 Z2 Z3, Z4, Z5 20mm 100mm 2mm 10mm 4mm 10mm Boundary conditions values Boundary conditions Value Dissipated heat P 80W 10N Ambient temperature Heat convection coefficient Emissivity 25ºC 2W/m2K constrains min V (x) x The maximal value of equivalent stress The maximal value of the temperature 0.8 σ ad eq = 20 MPa T ad = 70, 80, 90 C 31

Results of the optimization (constrained method) T ad = 90 C T ad = 80 C T ad = 70 C Z1 Z2 Z3 Z4 Z5 Volume 24.19mm 31.42mm 41.46mm 10mm 10mm 9.817mm 4.129mm 4mm 4mm 4mm 4.844mm 5.698mm 4mm 4mm 4mm 10367mm3 14107mm3 19649mm3 T ad = 90 C T ad = 80 C T ad = 70 C 32

Results of the optimization (Pareto approach) f1 - volume f2 - equivalent stress 33

Results of the optimization (Pareto approach) f1 max. temperature f2 - equivalent stress 34

Example 2 boundary conditions value 1000W/m2 heat flux heat convection coeficient ambitne temperature emissivity pressure min V (x) x min T max (X) X min σ X max eq (X) 2W/m2K 25ºC 0.8 5000Pa σ ad eq = 15MPa T ad = 70 C V ad = 150000mm 3 35

Geometry modelling P 0 ľsym ľ P 5 N 0 ľsym ľ N 5 P1 ľsym ľ P 4 N1 ľsym ľ N 4 P 2 ľsym ľ P 3 N 2 ľsym ľ N 3 Number of design parameters = 7 Design variable P, P1, P2, P3, P4, P5 N0, N1, N2, N3, N4, N5 H 0 Range 30mm 200mm 4mm 12mm 7mm 15mm 36

Results of the optimization (constrained method) 0 5 P =P 1 4 P =P 2 3 P =P min T max (X) 200mm min V (X) 110,6mm 30mm 30mm max min σ (X) eq 80,5mm 51,7mm 71,3mm X X X 0 5 N =N 1 2 3 N =N N =N H Fitness function value 4mm 4mm 7mm 49,48ºC 4,2mm 4mm 4mm 7mm 0.0073 11,4mm 5,6mm 99,13mm 138,9mm 4,49mm minimization of the maximal value of the temperature 4 minimization of the value of the radiator 10,3mm 8,85mm 0,97MPa minimization of the maximal value of the equivalent stress 37

Results of the optimization (Pareto approach) f1 - volume f2 max. temperature 38

Example 3 The problem of the optimal distribution of the material Fitness function max q(x) X Constrains: σ eq ad eq σł 0 σ ad eq = 20 MPa c c ad Ł 0 The cost of the radiator c is sum of factors for all fins Parameter Young modulus Poisson ratio Thermal expansion coef. Heat conductivity Material cost aluminum 68 000 MPa 0.34 24 10-6 1/K 210 W/mK 0.1 copper 110 000 MPa 0.35 16.5 10-6 1/K 380 W/mK 0.2 silver 76 000 MPa 0.39 19.5 10-6 1/K 420 W/mK 1 39

The boundary conditions values boundary condition fixed temperature heat convection coefficient ambient temperature pressure value 80 C 40 W/m2K 25ºC 1000Pa 40

Results of the optimization constraints: the maximal cost cad cad=4 cad=2.5 cad=9 41

Example 4 D Z e s i g n 1 2, Z Z Z v, 5 6 a Mr i a i bn [ m Z 3, 0Z. 0. 0 0. 0 l ev am l u a e x v a l u ] [ m ] 04 1 0. 0 5 0 2 5 0. 0 0 6 0 2 5 0. 0 0 8 e 42

Results of the optimization (Pareto approach) f1 - volume f2 - equivalent stress 43

Results of the optimization (Pareto approach) f1 heat flux f2 - equivalent stress 44

Results of the optimization (Pareto approach) f1 volume f2 - equivalent stress f3 heat flux 45

Conclusions An effective intelligent technique of evolutionary design based on constraint and Pareto approach has been presented. The important feature of this approach is its great flexibility and the strong probability of finding the global optimal solution. The preparation of the model may be aided by parametric curves Different types of fitness function can be easy formulated The choice of one objective and incorporate the other objectives as constrains requires performing optimization many times with different values of the constrains Approach based on Pareto frontier is considerably faster and more convinient comparing to constraint or weighting method The radiative transfer of heat between surfaces plays a significant role especially for higher values of the temperature 46