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![Introduction
Optimization Algorithms
Shape Optimization
Data Gathering
Black-Box Optimization
Simulation-based Optimization
OptiFlow
Optimization Examples
Final Remarks
Future Research
Acknowledgments
References
M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT)
Parameter File
# Variables Definitions:
[begin{variables}]
define(h, @) # Height of Wind Screen
[end{variables}]
define(L, 3.00) # Wind Screen Length
define(t, 0.15) # Wind Screen Width
# Wind Screen Coordinates
define(x1, calc(0−13.00))
define(x2, calc(x1−t))
define(y1, calc(h−0.25))
define(y2, calc(y1+L))
David Gutierrez Rivera Aerodynamic Shape Optimization 33 / 54](https://image.slidesharecdn.com/mscp-aerodynamicshapeoptimization-141114115259-conversion-gate02/75/Mscp-aerodynamic-shape-optimization-51-2048.jpg)
![Introduction
Optimization Algorithms
Shape Optimization
Data Gathering
Black-Box Optimization
Simulation-based Optimization
OptiFlow
Optimization Examples
Final Remarks
Future Research
Acknowledgments
References
M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT)
Parameter File
# Variables Definitions:
[begin{variables}]
define(h, @) # Height of Wind Screen
[end{variables}]
define(L, 3.00) # Wind Screen Length
define(t, 0.15) # Wind Screen Width
# Wind Screen Coordinates
define(x1, calc(0−13.00))
define(x2, calc(x1−t))
define(y1, calc(h−0.25))
define(y2, calc(y1+L))
4 //num cornerpoints**SCREEN3(SCHEME3)
0.3 0.0 //release distance*spacing hull
0.02 0.001 0.0 −0.1 //merg1*merg2*merg3*merg4
−4 −3 −1 −3 −1 //section color coding:drag*lift*moment*displ*rotation
1 x2 y1 n1
1 x1 y1 n2
1 x1 y2 n1
1 x2 y2 n2
David Gutierrez Rivera Aerodynamic Shape Optimization 33 / 54](https://image.slidesharecdn.com/mscp-aerodynamicshapeoptimization-141114115259-conversion-gate02/75/Mscp-aerodynamic-shape-optimization-52-2048.jpg)
![Introduction
Optimization Algorithms
Shape Optimization
Data Gathering
Black-Box Optimization
Simulation-based Optimization
OptiFlow
Optimization Examples
Final Remarks
Future Research
Acknowledgments
References
M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT)
Objective Function
function [Drag] = M4 Neath ViaductWS(X)
%% M4 Neath ViaductWS Objective Function
steps = 500;
Drag(1) = VXF FORCES(X,'DRAG','MAX',steps,'SECTION',11);
Drag(2) = VXF FORCES(X,'DRAG','MAX',steps,'eval',false,'SECTION',12);
Drag(3) = VXF FORCES(X,'DRAG','MAX',steps,'eval',false,'SECTION',13);
Drag(4) = VXF FORCES(X,'DRAG','MAX',steps,'eval',false,'SECTION',14);
[Drag,Lane] = max(Drag(:))
% Call optiOUT to generate Plots and Output Data
optiOUT(X, [Drag]);
end
David Gutierrez Rivera Aerodynamic Shape Optimization 34 / 54](https://image.slidesharecdn.com/mscp-aerodynamicshapeoptimization-141114115259-conversion-gate02/75/Mscp-aerodynamic-shape-optimization-53-2048.jpg)
![Introduction
Optimization Algorithms
Shape Optimization
Data Gathering
Black-Box Optimization
Simulation-based Optimization
OptiFlow
Optimization Examples
Final Remarks
Future Research
Acknowledgments
References
M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT)
Parameter File
# Variables Definitions:
[begin{variables}]
define(ex, calc(0.1+Imper)) # Eccentricity in X−Dir
define(ey, calc(0.1+Imper)) # Eccentricity in Y−Dir
[end{variables}]
# Rotor Dimensions:
define(D1, 0.572) # Rotor Inside Diameter
define(D2, 0.584) # Rotor Outside Diameter
define(t, calc(D2−D1)) # Rotor Thickness
David Gutierrez Rivera Aerodynamic Shape Optimization 40 / 54](https://image.slidesharecdn.com/mscp-aerodynamicshapeoptimization-141114115259-conversion-gate02/75/Mscp-aerodynamic-shape-optimization-60-2048.jpg)
![Introduction
Optimization Algorithms
Shape Optimization
Data Gathering
Black-Box Optimization
Simulation-based Optimization
OptiFlow
Optimization Examples
Final Remarks
Future Research
Acknowledgments
References
M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT)
Objective Function
f u n c t i o n [ Power ] = r o t o r d y n a (X)
%% r o t o r d y n a O b j e c t i v e Function
V a r i a b l e s = getappdata (0 , ' V a r i a b l e s ' ) ; % V a r i a b l e s Data
modelPATH = getappdata (0 , 'modelPATH ' ) ; % Model Path
% Savonious Rotor Data
D = 0 . 5 7 8 ; % Rotor Avg Diameter [m]
m = 10; % Mass [ kg ]
% C a l c u l a t e Arm v e c t o r
Arm = [−X(1)/2 , −X( 2 ) / 2 ] + [(4/6)∗D/ pi , D/ 2 ] ;
Arm = norm (Arm ) ;
% C a l c u l a t e I n e r t i a l Mass
Mass = 2∗m∗Armˆ2;
m4Mod( 'MASS22 ' , Mass , f u l l f i l e (modelPATH , V a r i a b l e s . FILE ) ) ;
% Angular V e l o c i t y
Omega = VXF DERIV(X, 'RDISPL ' , 'TIME ' ,50 , 'SECTION ' , 1 ) ;
Omega = mean(Omega ) ;
% T o r s i o n a l Force
Torque = VXF FORCES(X, 'MOMENT' , 'MEAN' ,50 , ' e v a l ' , f a l s e ) ;
Power = Torque ∗ Omega ;
% C a l l optiOUT to generate P l o t s and Output Data
optiOUT (X, [ Power ] ) ;
end
David Gutierrez Rivera Aerodynamic Shape Optimization 43 / 54](https://image.slidesharecdn.com/mscp-aerodynamicshapeoptimization-141114115259-conversion-gate02/75/Mscp-aerodynamic-shape-optimization-63-2048.jpg)
Uploaded byDavid Gutierrez Rivera
Mscp - aerodynamic shape optimization
The document discusses optimization algorithms for aerodynamic shape optimization. It covers topics such as what optimization is, computational fluid dynamics and vortex particle methods for simulations, local and global optimization algorithms, shape parametrization, objective functions, and examples of optimizing aerodynamic shapes like wind shields and wind turbines.
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Mscp - aerodynamic shape optimization
- 1. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Aerodynamic Shape Optimization using Vortex Particle Simulations Master’s Presentation David Gutierrez Rivera Bauhaus Universit¨at Weimar April 4, 2014 David Gutierrez Rivera Aerodynamic Shape Optimization 1 / 54
- 2. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Overview 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 2 / 54
- 3. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics Introduction 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 3 / 54
- 4. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics What is Optimization? Optimization is to find the optimum value(s) to achieve certain goal(s) David Gutierrez Rivera Aerodynamic Shape Optimization 4 / 54
- 5. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics What is Optimization? Optimization is to find the optimum value(s) to achieve certain goal(s) David Gutierrez Rivera Aerodynamic Shape Optimization 4 / 54
- 6. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics What is Optimization? Optimization is to find the optimum value(s) to achieve certain goal(s) David Gutierrez Rivera Aerodynamic Shape Optimization 4 / 54
- 7. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics What is Optimization? Optimization is to find the optimum value(s) to achieve certain goal(s) David Gutierrez Rivera Aerodynamic Shape Optimization 4 / 54
- 8. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics What is Optimization? Optimization is to find the optimum value(s) to achieve certain goal(s) David Gutierrez Rivera Aerodynamic Shape Optimization 4 / 54
- 9. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics What is Optimization? Optimization is to find the optimum value(s) to achieve certain goal(s) David Gutierrez Rivera Aerodynamic Shape Optimization 4 / 54
- 10. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics A Mathematical View Mathematically, optimization is formulated as: minimize x f (x) subject to gi (x) ≤ 0, i = 1, . . . , m hi (x) = 0, i = 1, . . . , n (1) David Gutierrez Rivera Aerodynamic Shape Optimization 5 / 54
- 11. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics A Mathematical View Mathematically, optimization is formulated as: minimize x f (x) subject to gi (x) ≤ 0, i = 1, . . . , m hi (x) = 0, i = 1, . . . , n (1) where, f (x) : Rn → R, is the objective function to be minimized gi (x) ≤ 0, are inequality constraints hi (x) = 0, are equality constraints David Gutierrez Rivera Aerodynamic Shape Optimization 5 / 54
- 12. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics A Programmatic View Programmatically, optimization can be viewed as a loop: David Gutierrez Rivera Aerodynamic Shape Optimization 6 / 54
- 13. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics Basics of Aerodynamics The Navier-Stokes Partial Differential Equation: David Gutierrez Rivera Aerodynamic Shape Optimization 7 / 54
- 14. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics Computational Fluid Dynamics (CFD) Is the branch of fluid mechanics that uses numerical methods to solve fluid flow problems. David Gutierrez Rivera Aerodynamic Shape Optimization 8 / 54
- 15. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics Computational Fluid Dynamics (CFD) Is the branch of fluid mechanics that uses numerical methods to solve fluid flow problems. Discretization Methods Finite Volume Method Boundary Element Method High-Resolution Schemes Turbulence Models Reynolds-Averaged NavierStokes (RANS) Large eddy simulation (LES) Vortex methods David Gutierrez Rivera Aerodynamic Shape Optimization 8 / 54
- 16. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References What is Optimization? Basics of Aerodynamics The Vortex Particle Method (VPM) Characteristics: Is a grid-free technique for simulation of turbulent flows. It uses vortices as the computational elements. David Gutierrez Rivera Aerodynamic Shape Optimization 9 / 54
- 17. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Local Global Optimization Algorithms 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 10 / 54
- 18. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Local Global Optimization Algorithms Classification Linearity Linear NonLinear Constraints Unconstrained Constrained Objectives Single-Objective Multi-Objective Modality uni-modal (Local) multi-modal (Global) David Gutierrez Rivera Aerodynamic Shape Optimization 11 / 54
- 19. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Local Global Optimization Algorithms Classification Linearity Linear NonLinear Constraints Unconstrained Constrained Objectives Single-Objective Multi-Objective Modality uni-modal (Local) multi-modal (Global) David Gutierrez Rivera Aerodynamic Shape Optimization 11 / 54
- 20. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Local Global Global vs. Local David Gutierrez Rivera Aerodynamic Shape Optimization 12 / 54
- 21. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Local Global Local Gradient-free Methods Golden Section Simplex Gradient-based Methods Gradient-Descent Newton David Gutierrez Rivera Aerodynamic Shape Optimization 13 / 54
- 22. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Local Global Global non-Heuristic Methods Deterministic Stochastic Heuristic Methods Evolutionary Genetic Algorithm Swarm Intelligence Swarm Intelligence Particle Swarm Ant Colony David Gutierrez Rivera Aerodynamic Shape Optimization 14 / 54
- 23. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Shape Optimization 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 15 / 54
- 24. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Shape Optimization Shape optimization is formulated as: minimize Ω f (Ω) subject to gi (Ω) ≤ 0, i = 1, . . . , m hi (Ω) = 0, i = 1, . . . , n (2) David Gutierrez Rivera Aerodynamic Shape Optimization 16 / 54
- 25. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Shape Optimization Shape optimization is formulated as: minimize Ω f (Ω) subject to gi (Ω) ≤ 0, i = 1, . . . , m hi (Ω) = 0, i = 1, . . . , n (2) where, Ω is a set of variable parameters that make up the geometry that we want to optimize. David Gutierrez Rivera Aerodynamic Shape Optimization 16 / 54
- 26. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Parametrization David Gutierrez Rivera Aerodynamic Shape Optimization 17 / 54
- 27. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Parametrization David Gutierrez Rivera Aerodynamic Shape Optimization 17 / 54
- 28. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Objective Functions Similar to mathematical functions. They involve 3 basic operations: David Gutierrez Rivera Aerodynamic Shape Optimization 18 / 54
- 29. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Objective Functions Similar to mathematical functions. They involve 3 basic operations: Substitution f (x) = a0 + n i=1 an · cos(n x ) + n i=1 bn · sin(n x ) David Gutierrez Rivera Aerodynamic Shape Optimization 18 / 54
- 30. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Objective Functions Similar to mathematical functions. They involve 3 basic operations: Substitution f (x) = a0 + n i=1 an · cos(n 3.1416 ) + n i=1 bn · sin(n 3.1416 ) Substitution David Gutierrez Rivera Aerodynamic Shape Optimization 18 / 54
- 31. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Objective Functions Similar to mathematical functions. They involve 3 basic operations: Evaluation f (x) = a0 + n i=1 an · cos(n 3.1416 ) + n i=1 bn · sin(n 3.1416 ) Evaluation David Gutierrez Rivera Aerodynamic Shape Optimization 18 / 54
- 32. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Parametrization Objective Functions Objective Functions Similar to mathematical functions. They involve 3 basic operations: Read Output f (x) = Value Read Output David Gutierrez Rivera Aerodynamic Shape Optimization 18 / 54
- 33. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Data Gathering 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 19 / 54
- 34. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Data Gathering David Gutierrez Rivera Aerodynamic Shape Optimization 20 / 54
- 35. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Data Gathering Data File It is a text-based tab-delimited file containing the data points from the variables and the corresponding values of the objective functions. David Gutierrez Rivera Aerodynamic Shape Optimization 21 / 54
- 36. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Data Gathering Data File It is a text-based tab-delimited file containing the data points from the variables and the corresponding values of the objective functions. Variable#1 Variable#2 ... Variable#n Objective#1 ... Objective#m Variable#1 Variable#2 ... Variable#n Objective#1 ... Objective#m ... ... ... ... ... ... ... Variable#1 Variable#2 ... Variable#n Objective#1 ... Objective#m David Gutierrez Rivera Aerodynamic Shape Optimization 21 / 54
- 37. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Black-Box Optimization 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 22 / 54
- 38. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Black-Box Optimization Black-Box Process They are known from their inputs and outputs. Little is known of how it works internally. David Gutierrez Rivera Aerodynamic Shape Optimization 23 / 54
- 39. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Black-Box Optimization David Gutierrez Rivera Aerodynamic Shape Optimization 23 / 54
- 40. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Black-Box Optimization David Gutierrez Rivera Aerodynamic Shape Optimization 24 / 54
- 41. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Noisy Functions David Gutierrez Rivera Aerodynamic Shape Optimization 25 / 54
- 42. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Simulation-based Optimization 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 26 / 54
- 43. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Simulation-based Optimization David Gutierrez Rivera Aerodynamic Shape Optimization 27 / 54
- 44. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Moving Boundary Problem David Gutierrez Rivera Aerodynamic Shape Optimization 28 / 54
- 45. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References OptiFlow 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 29 / 54
- 46. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Optimization Examples 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 30 / 54
- 47. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) M4 Neath Viaduct Wind Shield Description This bridge is located in South Wales and crosses the River Neath. A section of the bridge is very exposed in flat topography, therefore a wind shielding system is desired for reducing the overturning forces on vehicles. David Gutierrez Rivera Aerodynamic Shape Optimization 31 / 54
- 48. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) M4 Neath Viaduct Wind Shield Description The CFD model is the same as the one used on the VXFlow tutorial. David Gutierrez Rivera Aerodynamic Shape Optimization 31 / 54
- 49. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) M4 Neath Viaduct Wind Shield Description The CFD model is the same as the one used on the VXFlow tutorial. Four small sections were used to obtain the interior drag forces. David Gutierrez Rivera Aerodynamic Shape Optimization 31 / 54
- 50. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Optimization Model Find optimum height (h) for the Wind Shield. David Gutierrez Rivera Aerodynamic Shape Optimization 32 / 54
- 51. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Parameter File # Variables Definitions: [begin{variables}] define(h, @) # Height of Wind Screen [end{variables}] define(L, 3.00) # Wind Screen Length define(t, 0.15) # Wind Screen Width # Wind Screen Coordinates define(x1, calc(0−13.00)) define(x2, calc(x1−t)) define(y1, calc(h−0.25)) define(y2, calc(y1+L)) David Gutierrez Rivera Aerodynamic Shape Optimization 33 / 54
- 52. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Parameter File # Variables Definitions: [begin{variables}] define(h, @) # Height of Wind Screen [end{variables}] define(L, 3.00) # Wind Screen Length define(t, 0.15) # Wind Screen Width # Wind Screen Coordinates define(x1, calc(0−13.00)) define(x2, calc(x1−t)) define(y1, calc(h−0.25)) define(y2, calc(y1+L)) 4 //num cornerpoints**SCREEN3(SCHEME3) 0.3 0.0 //release distance*spacing hull 0.02 0.001 0.0 −0.1 //merg1*merg2*merg3*merg4 −4 −3 −1 −3 −1 //section color coding:drag*lift*moment*displ*rotation 1 x2 y1 n1 1 x1 y1 n2 1 x1 y2 n1 1 x2 y2 n2 David Gutierrez Rivera Aerodynamic Shape Optimization 33 / 54
- 53. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Objective Function function [Drag] = M4 Neath ViaductWS(X) %% M4 Neath ViaductWS Objective Function steps = 500; Drag(1) = VXF FORCES(X,'DRAG','MAX',steps,'SECTION',11); Drag(2) = VXF FORCES(X,'DRAG','MAX',steps,'eval',false,'SECTION',12); Drag(3) = VXF FORCES(X,'DRAG','MAX',steps,'eval',false,'SECTION',13); Drag(4) = VXF FORCES(X,'DRAG','MAX',steps,'eval',false,'SECTION',14); [Drag,Lane] = max(Drag(:)) % Call optiOUT to generate Plots and Output Data optiOUT(X, [Drag]); end David Gutierrez Rivera Aerodynamic Shape Optimization 34 / 54
- 54. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Data Points David Gutierrez Rivera Aerodynamic Shape Optimization 35 / 54
- 55. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Run-Time Optimization David Gutierrez Rivera Aerodynamic Shape Optimization 36 / 54
- 56. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Vertical-Axis Wind Turbine (VAWT) Description The Savonius Wind Turbine is a Vertical-Axis Wind Turbine (VAWT) which main advantages are its reliability and practicality. David Gutierrez Rivera Aerodynamic Shape Optimization 37 / 54
- 57. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Optimization Model Description Find optimum values of ex and ey to maximize Torque or Power. David Gutierrez Rivera Aerodynamic Shape Optimization 38 / 54
- 58. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Optimization Model Description Find optimum values of ex and ey to maximize Torque or Power. Constrained to the shaded area. David Gutierrez Rivera Aerodynamic Shape Optimization 38 / 54
- 59. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Dynamic Model David Gutierrez Rivera Aerodynamic Shape Optimization 39 / 54
- 60. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Parameter File # Variables Definitions: [begin{variables}] define(ex, calc(0.1+Imper)) # Eccentricity in X−Dir define(ey, calc(0.1+Imper)) # Eccentricity in Y−Dir [end{variables}] # Rotor Dimensions: define(D1, 0.572) # Rotor Inside Diameter define(D2, 0.584) # Rotor Outside Diameter define(t, calc(D2−D1)) # Rotor Thickness David Gutierrez Rivera Aerodynamic Shape Optimization 40 / 54
- 61. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Parameter File # Coord. Calculations: # Section 1 define(x1, calc(−(ex/2))) define(y11, calc(D1−(ey/2))) define(y12, calc(y11−D1)) define(y13, calc(y12−t)) define(y14, calc(y11+t)) define(xc1, calc(−(ex/2))) define(yc1, calc(y11−D1/2)) # Section 2 define(x2, calc(ex/2)) define(y21, calc(−D1+(ey/2))) define(y22, calc(y21+D1)) define(y23, calc(y22+t)) define(y24, calc(y21−t)) define(xc2, calc(ex/2)) define(yc2, calc(y21+D1/2)) David Gutierrez Rivera Aerodynamic Shape Optimization 41 / 54
- 62. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Parameter File 4 //num cornerpoints 0.2 0.0 //release distance*spacing hull 0.0 0.002 0.0 −0.02 //merg1*merg2*merg3*merg4 −4 4 −1 −3 −1 //section color coding:drag*lift*moment*displ*rotation 3 x1 y11 nd1 xc1 yc1 1 x1 y12 nt 2 x1 y13 nd2 xc1 yc1 1 x1 y14 nt 4 //num cornerpoints 0.2 0.0 //release distance*spacing hull 0.0 0.002 0.0 −0.02 //merg1*merg2*merg3*merg4 −4 4 −1 −3 −1 //section color coding:drag*lift*moment*displ*rotation 3 x2 y21 nd1 xc2 yc2 1 x2 y22 nt 2 x2 y23 nd2 xc2 yc2 1 x2 y24 nt David Gutierrez Rivera Aerodynamic Shape Optimization 42 / 54
- 63. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Objective Function f u n c t i o n [ Power ] = r o t o r d y n a (X) %% r o t o r d y n a O b j e c t i v e Function V a r i a b l e s = getappdata (0 , ' V a r i a b l e s ' ) ; % V a r i a b l e s Data modelPATH = getappdata (0 , 'modelPATH ' ) ; % Model Path % Savonious Rotor Data D = 0 . 5 7 8 ; % Rotor Avg Diameter [m] m = 10; % Mass [ kg ] % C a l c u l a t e Arm v e c t o r Arm = [−X(1)/2 , −X( 2 ) / 2 ] + [(4/6)∗D/ pi , D/ 2 ] ; Arm = norm (Arm ) ; % C a l c u l a t e I n e r t i a l Mass Mass = 2∗m∗Armˆ2; m4Mod( 'MASS22 ' , Mass , f u l l f i l e (modelPATH , V a r i a b l e s . FILE ) ) ; % Angular V e l o c i t y Omega = VXF DERIV(X, 'RDISPL ' , 'TIME ' ,50 , 'SECTION ' , 1 ) ; Omega = mean(Omega ) ; % T o r s i o n a l Force Torque = VXF FORCES(X, 'MOMENT' , 'MEAN' ,50 , ' e v a l ' , f a l s e ) ; Power = Torque ∗ Omega ; % C a l l optiOUT to generate P l o t s and Output Data optiOUT (X, [ Power ] ) ; end David Gutierrez Rivera Aerodynamic Shape Optimization 43 / 54
- 64. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Data Points David Gutierrez Rivera Aerodynamic Shape Optimization 44 / 54
- 65. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) Final Shape David Gutierrez Rivera Aerodynamic Shape Optimization 45 / 54
- 66. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 46 / 54
- 67. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks The Vortex Particle Method is a David Gutierrez Rivera Aerodynamic Shape Optimization 47 / 54
- 68. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks The Vortex Particle Method is a Relatively low computational cost and highly accurate simulation. David Gutierrez Rivera Aerodynamic Shape Optimization 47 / 54
- 69. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks The Vortex Particle Method is a Relatively low computational cost and highly accurate simulation. A mesh-free numerical method. David Gutierrez Rivera Aerodynamic Shape Optimization 47 / 54
- 70. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks Some Recommendations: David Gutierrez Rivera Aerodynamic Shape Optimization 48 / 54
- 71. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks Some Recommendations: Tweak Local Optimization Algorithms David Gutierrez Rivera Aerodynamic Shape Optimization 48 / 54
- 72. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks Some Recommendations: Tweak Local Optimization Algorithms Run-Time Smoothing David Gutierrez Rivera Aerodynamic Shape Optimization 48 / 54
- 73. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks Some Recommendations: Tweak Local Optimization Algorithms Run-Time Smoothing Global Optimization Algorithms David Gutierrez Rivera Aerodynamic Shape Optimization 48 / 54
- 74. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Final Remarks Some Recommendations: Tweak Local Optimization Algorithms Run-Time Smoothing Global Optimization Algorithms Parallelization David Gutierrez Rivera Aerodynamic Shape Optimization 48 / 54
- 75. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Future Research 1 Introduction What is Optimization? Basics of Aerodynamics 2 Optimization Algorithms Local Global 3 Shape Optimization Parametrization Objective Functions 4 Data Gathering 5 Black-Box Optimization 6 Simulation-based Optimization 7 OptiFlow 8 Optimization Examples M4 Neath Viaduct Wind Shield Vertical-Axis Wind Turbine (VAWT) 9 Final Remarks 10 Future Research David Gutierrez Rivera Aerodynamic Shape Optimization 49 / 54
- 76. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Future Research Some ideas: David Gutierrez Rivera Aerodynamic Shape Optimization 50 / 54
- 77. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Future Research Some ideas: Multi-Disciplinary Optimization (MDO) David Gutierrez Rivera Aerodynamic Shape Optimization 50 / 54
- 78. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Future Research Some ideas: Multi-Disciplinary Optimization (MDO) Life-Cycle Design David Gutierrez Rivera Aerodynamic Shape Optimization 50 / 54
- 79. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Future Research Some ideas: Multi-Disciplinary Optimization (MDO) Life-Cycle Design Development of Optimization Algorithms David Gutierrez Rivera Aerodynamic Shape Optimization 50 / 54
- 80. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References Acknowledgments Many Thanks to: Prof. Guido Morgenthal M.Sc. Khaled Ibrahim M.Sc. Benjamin Bendig M.Sc. Samir Chawdhury Shanmugam Narayanan David Gutierrez Rivera Aerodynamic Shape Optimization 51 / 54
- 81. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References References Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, The Edinburgh Building, Cambridge, CB2 8RU, UK, 2004. Pages: 1-11, 455-496. http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf. Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer, 175 Fifth Avenue, New York, NY 10010, USA, 1999. Pages: 2-3, 4-7, 10-30. Igor Griva, Stephen G. Nash, and Ariela Sofer. Linear and Nonlinear Optimization. Siam, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA, 2nd edition, 2009. Pages: 35-40, 54-58, 355-450. David G. Luenberger and Yinyu Ye. Linear and Nonlinear Programming. Springer, 233 Spring Street, New York,NY 10013, USA, 3rd edition, 2008. Pages: 2-7, 183-257. David Gutierrez Rivera Aerodynamic Shape Optimization 52 / 54
- 82. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References References Florent Brunet. Contributions to Parametric Image Registration and 3D Surface Reconstruction. Universite d’Auvergne, Ph.D. Thesis, November 2010. Chapter 2, Pages: 27-44. http://www.brnt.eu/publications/brunet2010phd.pdf. Prof. DK Chaturvedi. Advances in Evolutionary Optimization Techniques. YouTube, November 2013. https://www.youtube.com/watch?v=ftxiiaHNweQ. David Gutierrez Rivera. OptiFlow v0.6.1a. Weimar, Germany, March 2014. OptiFlow Userguide. Guido Morgenthal. Aerodynamic Analysis of Structures Using High-resolution Vortex Particle Methods. University of Cambridge, Ph.D. Thesis, October 2002. Pages: 21-31, 121-142. Wikipedia. Savonius wind turbine. The Wikimedia Foundation, March 2014. http://en.wikipedia.org/wiki/Savonius_wind_turbine. David Gutierrez Rivera Aerodynamic Shape Optimization 53 / 54
- 83. Introduction Optimization Algorithms Shape Optimization DataGathering Black-Box Optimization Simulation-based Optimization OptiFlow Optimization Examples Final Remarks Future Research Acknowledgments References References John James Tomick. On Convergence of the Nelder-Mead Simplex Algorithm for Unconstrained Stochastic Optimization. The Pennsylvania State University, Ph.D. Thesis, May 1995. http://www.dtic.mil/dtic/tr/fulltext/u2/a289453.pdf. David Gutierrez Rivera Aerodynamic Shape Optimization 54 / 54