Wing shape optimization - Revision history

Jpk254: /* Numerical example */

2021-12-16T04:59:39Z

Numerical example

← Older revision		Revision as of 00:59, 16 December 2021
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	</math>		</math>

	<math>(R_1, R_2, R_3)^T = ( \boldsymbol{v} \cdot \nabla ) \boldsymbol{v} + \~~nable\~~p - \nu \nabla^2 (\boldsymbol{v})</math>		<math>(R_1, R_2, R_3)^T = ( \boldsymbol{v} \cdot \nabla ) \boldsymbol{v} + \nabla p - \nu \nabla^2 (\boldsymbol{v})</math>

	<math>R_4 = - \nabla\\boldsymbol{u}</math><ref name=":1" />		<math>R_4 = - \nabla \boldsymbol{u}</math><ref name=":1" />

Jpk254: /* Numerical example */

2021-12-16T04:58:09Z

Numerical example

← Older revision		Revision as of 00:58, 16 December 2021
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	<math>J = J(\boldsymbol{v}, p, \boldsymbol{\beta}) = - \oint p \hat{n} \sin(\theta) dS </math>		<math>J = J(\boldsymbol{v}, p, \boldsymbol{\beta}) = - \oint p \hat{n} \sin(\theta) dS </math>

	Where '''<math>\boldsymbol{v} </math>''' is the vector of velocity magnitudes in the x and y directions, <math>p </math> is pressure, and <math>\boldsymbol{\beta} </math> is the vector containing the y coordinates that parameterize the airfoil. To determine the lift produced by the airfoil, a computational mesh must be created surrounding the airfoil. In each of the cells of the mesh, the governing equations of fluid mechanics, conservation of mass (Navier-Stokes) and conservation of mass (continuity), must be solved. This means that each variable in the objective equations (and the constraints), '''<math>\boldsymbol{v} </math>''', <math>p </math>, and <math>\boldsymbol{\beta} </math>, are vectors of the length equal to the number of cells in the computational domain (often in the range from 1e3 to 1e7). This is done in a CFD package, specifically OpenFOAM in this case, and is not the focus of this numerical example. Conservation of momentum and conservation of mass form the constraints. Additionally, in this case, a maximum value for coefficient of drag is chosen. (Note that the simulation has boundary conditions that must also be satisfied, i.e., specified inlet velocity, but these are typically not listed as optimization constraints.)		Where '''<math>\boldsymbol{v} </math>''' is the vector of velocity magnitudes in the x and y directions, <math>p </math> is pressure, and <math>\boldsymbol{\beta} </math> is the vector containing the y coordinates that parameterize the airfoil. To determine the lift produced by the airfoil, a computational mesh must be created surrounding the airfoil. In each of the cells of the mesh, the governing equations of fluid mechanics, conservation of mass (Navier-Stokes) and conservation of mass (continuity), must be solved. This means that each variable in the objective equations (and the constraints), '''<math>\boldsymbol{v} </math>''', <math>p </math>, and <math>\boldsymbol{\beta} </math>, are vectors of the length equal to the number of cells in the computational domain (about 3e6 cells in this case; often in the range from 1e3 to 1e7). This is done in a CFD package, specifically OpenFOAM in this case, and is not the focus of this numerical example. Conservation of momentum and conservation of mass form the constraints. Additionally, in this case, a maximum value for coefficient of drag is chosen. (Note that the simulation has boundary conditions that must also be satisfied, i.e., specified inlet velocity, but these are typically not listed as optimization constraints.)

			<math>\boldsymbol{R} (\boldsymbol{v} , p , \boldsymbol{\beta} ) = 0
			</math>

			<math>(R_1, R_2, R_3)^T = ( \boldsymbol{v} \cdot \nabla ) \boldsymbol{v} + \nable\p - \nu \nabla^2 (\boldsymbol{v})</math>

			<math>R_4 = - \nabla\\boldsymbol{u}</math><ref name=":1" />


	Besides solving the physics of the fluid surrounding the airfoil, the airfoil design producing the most lift must also be solved using an optimization method. In this case, the steepest descent method is used because that is the method used by the solver adjointOptimisationFoam in OpenFOAM and the initial NACA 0012 airfoil is assumed to be close to the optimal design.		Besides solving the physics of the fluid surrounding the airfoil, the airfoil design producing the most lift must also be solved using an optimization method. In this case, the steepest descent method is used because that is the method used by the solver adjointOptimisationFoam in OpenFOAM and the initial NACA 0012 airfoil is assumed to be close to the optimal design.

	<math>\boldsymbol{\beta}_{N+1} = \boldsymbol{\beta}_{N} + \alpha \centerdot \delta L </math>		<math>\boldsymbol{\beta}_{N+1} = \boldsymbol{\beta}_{N} + \alpha \centerdot \delta L </math><ref name=":1" />

	Where <math>\boldsymbol{\beta}_{N} </math> are the current coordinates, <math>\boldsymbol{\beta}_{N+1} </math> are the next coordinates, and <math>\alpha </math> is the time step. The <math>\delta L </math> term is the variation in the Lagrange function <math>L </math> which combines the objective and constraints multiped by Lagrange multipliers called adjoint variables. This is not discussed in detail here, but essentially, by creating the Lagrange function, it can be manipulated so that the objective only depends on the airfoil shape, the entire flow domain does not need to be solved every optimization iteration.		Where <math>\boldsymbol{\beta}_{N} </math> are the current coordinates, <math>\boldsymbol{\beta}_{N+1} </math> are the next coordinates, and <math>\alpha </math> is the time step. The <math>\delta L </math> term is the variation in the Lagrange function <math>L </math> which combines the objective and constraints multiped by Lagrange multipliers called adjoint variables. This is not discussed in detail here, but essentially, by creating the Lagrange function, it can be manipulated so that the objective only depends on the airfoil shape, the entire flow domain does not need to be solved every optimization iteration.


	Although changes in the position of airfoil coordinates are not visible after just one iteration, after multiple iterations, you can see the y coordinates of the airfoil start to “move”.		Although changes in the position of airfoil coordinates are not visible after just one iteration, after multiple iterations, you can see the y coordinates of the airfoil start to “move”.
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	After nearly fifteen thousand optimization iterations in OpenFOAM, the solution converged.		After nearly fifteen thousand optimization iterations in OpenFOAM, the solution converged.


	[[File:PressureContours-5.png\|thumb\|right\|upright=1.45\|'''Figure 5. Pressure contours of optimized airfoil''']]		[[File:PressureContours-5.png\|thumb\|right\|upright=1.45\|'''Figure 5. Pressure contours of optimized airfoil''']]

Santi Correa: /* Conclusion */

2021-12-16T04:57:57Z

Conclusion

← Older revision		Revision as of 00:57, 16 December 2021
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	==Conclusion==		==Conclusion==

	With the various uses of wing-shape optimization, it is abundantly clear that this optimization method is robust enough to withstand most problems that carry real world constraints/parameters. The shape of a blade or a wing is subjected to a wide variety of dimensional and performance constraints, magnifying the need to correctly set up one’s optimization problem. These constraints then directly affect the outcome of the objective, such as optimizing aerodynamic performance through drag and lift coefficient constraining. By reducing the drag coefficient of the solution, one also reduces the amount of generated lift, expectedly lowering an aircraft in altitude<ref>ZADEH, Parviz MOHAMMAD, and Mohsen Sayadi. "An efficient aerodynamic shape optimization of blended wing body UAV using multi-fidelity models." Chinese Journal of Aeronautics 31.6 (2018): 1165-1180.</ref>. Another constraint that factors into this optimization is the wing length, which can be constrained due to a physical or monetary/material constraint. As seen, certain regulatory factors have to be followed for the solution to be feasible, but there are other driving factors that are out of engineering control.		With the various uses of wing-shape optimization, it is abundantly clear that this optimization method is robust enough to withstand most problems that carry real world constraints/parameters. The shape of a blade or a wing is subjected to a wide variety of dimensional and performance constraints, magnifying the need to correctly set up one’s optimization problem. These constraints then directly affect the outcome of the objective, such as optimizing aerodynamic performance through drag and lift coefficient constraining. By reducing the drag coefficient of the solution, one also reduces the amount of generated lift, expectedly lowering an aircraft in altitude<ref>ZADEH, Parviz MOHAMMAD, and Mohsen Sayadi. "An efficient aerodynamic shape optimization of blended wing body UAV using multi-fidelity models." Chinese Journal of Aeronautics 31.6 (2018): 1165-1180.</ref>. Another constraint that factors into this optimization is the wing length, which can be constrained due to a physical or monetary/material constraint. Using the example provided in the Applications Section regarding airwing optimization, across-sectional areas of wings also greatly affect the constraints needed to be satisfied. As seen, certain regulatory factors have to be followed for the solution to be feasible, but there are other driving factors that are out of engineering control.

	<math display="inline">\begin{array}{lcl} C_D & = & {C_L^2 \over \piARe} \\ C_D & = & Drag \ Coefficient \\ C_L & = & Lift \ Coefficient \\ AR & = & Aspect \ Ratio \\ e & = & Efficiency \ Factor \end{array}</math><ref>“Induced Drag Coefficient.” NASA, NASA, <nowiki>https://www.grc.nasa.gov/www/k-12/airplane/induced.html</nowiki>.</ref>		<math display="inline">\begin{array}{lcl} C_D & = & {C_L^2 \over \piARe} \\ C_D & = & Drag \ Coefficient \\ C_L & = & Lift \ Coefficient \\ AR & = & Aspect \ Ratio \\ e & = & Efficiency \ Factor \end{array}</math><ref>“Induced Drag Coefficient.” NASA, NASA, <nowiki>https://www.grc.nasa.gov/www/k-12/airplane/induced.html</nowiki>.</ref>

Jpk254: /* Numerical example */

2021-12-16T04:37:11Z

Numerical example

← Older revision		Revision as of 00:37, 16 December 2021
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	One problem is presented at two different levels of complexity. The objective is to maximum the coefficient of lift of a 2-D airfoil starting from the NACA 0012 profile (a common profile developed by the National Advisory Committee for Aeronautics). The actual problem being solved is developed in the open-source CFD package OpenFOAM (Open Source Field Operation and Manipulation) with results presented. The simplified version is presented here in depth to illustrate the process of wing shape optimization.[[File:Naca0012-1.jpg\|thumb\|upright\|'''Figure 2.''' Initial parameterized airfoil]]The problem formulation begins with the initial airfoil design being parameterized with points defined by x and y coordinates – these coordinates are contained in the <math>\boldsymbol{\beta} </math> vector. In this case, the x coordinates are fixed, so <math>\boldsymbol{\beta} </math> only contains y values. The objective of this problem is to maximize the lift produced by the airfoil.		One problem is presented at two different levels of complexity. The objective is to maximum the coefficient of lift of a 2-D airfoil starting from the NACA 0012 profile (a common profile developed by the National Advisory Committee for Aeronautics). The actual problem being solved is developed in the open-source CFD package OpenFOAM (Open Source Field Operation and Manipulation) with results presented. The simplified version is presented here in depth to illustrate the process of wing shape optimization.[[File:Naca0012-1.jpg\|thumb\|upright=1.2\|'''Figure 2.''' Initial parameterized airfoil]]The problem formulation begins with the initial airfoil design being parameterized with points defined by x and y coordinates – these coordinates are contained in the <math>\boldsymbol{\beta} </math> vector. In this case, the x coordinates are fixed, so <math>\boldsymbol{\beta} </math> only contains y values. The objective of this problem is to maximize the lift produced by the airfoil.

	<math>J = J(\boldsymbol{v}, p, \boldsymbol{\beta}) = - \oint p \hat{n} \sin(\theta) dS </math>		<math>J = J(\boldsymbol{v}, p, \boldsymbol{\beta}) = - \oint p \hat{n} \sin(\theta) dS </math>
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	Although changes in the position of airfoil coordinates are not visible after just one iteration, after multiple iterations, you can see the y coordinates of the airfoil start to “move”.		Although changes in the position of airfoil coordinates are not visible after just one iteration, after multiple iterations, you can see the y coordinates of the airfoil start to “move”.

	[[File:Naca0012-2.jpg\|thumb\|center\|upright=1\|'''Figure 3.''' Airfoil parameterization after multiple iterations]]		[[File:Naca0012-2.jpg\|thumb\|center\|upright=1.2\|'''Figure 3.''' Airfoil parameterization after multiple iterations]]

	After nearly fifteen thousand optimization iterations in OpenFOAM, the solution converged.		After nearly fifteen thousand optimization iterations in OpenFOAM, the solution converged.

Jpk254: /* Numerical example */

2021-12-16T04:36:31Z

Numerical example

← Older revision		Revision as of 00:36, 16 December 2021
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	One problem is presented at two different levels of complexity. The objective is to maximum the coefficient of lift of a 2-D airfoil starting from the NACA 0012 profile (a common profile developed by the National Advisory Committee for Aeronautics). The actual problem being solved is developed in the open-source CFD package OpenFOAM (Open Source Field Operation and Manipulation) with results presented. The simplified version is presented here in depth to illustrate the process of wing shape optimization.[[File:Naca0012-1.jpg\|thumb\|'''Figure 2.''' Initial parameterized airfoil]]The problem formulation begins with the initial airfoil design being parameterized with points defined by x and y coordinates – these coordinates are contained in the <math>\boldsymbol{\beta} </math> vector. In this case, the x coordinates are fixed, so <math>\boldsymbol{\beta} </math> only contains y values. The objective of this problem is to maximize the lift produced by the airfoil.		One problem is presented at two different levels of complexity. The objective is to maximum the coefficient of lift of a 2-D airfoil starting from the NACA 0012 profile (a common profile developed by the National Advisory Committee for Aeronautics). The actual problem being solved is developed in the open-source CFD package OpenFOAM (Open Source Field Operation and Manipulation) with results presented. The simplified version is presented here in depth to illustrate the process of wing shape optimization.[[File:Naca0012-1.jpg\|thumb\|upright\|'''Figure 2.''' Initial parameterized airfoil]]The problem formulation begins with the initial airfoil design being parameterized with points defined by x and y coordinates – these coordinates are contained in the <math>\boldsymbol{\beta} </math> vector. In this case, the x coordinates are fixed, so <math>\boldsymbol{\beta} </math> only contains y values. The objective of this problem is to maximize the lift produced by the airfoil.

	<math>J = J(\boldsymbol{v}, p, \boldsymbol{\beta}) = - \oint p \hat{n} \sin(\theta) dS </math>		<math>J = J(\boldsymbol{v}, p, \boldsymbol{\beta}) = - \oint p \hat{n} \sin(\theta) dS </math>
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	Although changes in the position of airfoil coordinates are not visible after just one iteration, after multiple iterations, you can see the y coordinates of the airfoil start to “move”.		Although changes in the position of airfoil coordinates are not visible after just one iteration, after multiple iterations, you can see the y coordinates of the airfoil start to “move”.

	[[File:Naca0012-2.jpg\|thumb\|center\|'''Figure 3.''' Airfoil parameterization after multiple iterations]]		[[File:Naca0012-2.jpg\|thumb\|center\|upright=1\|'''Figure 3.''' Airfoil parameterization after multiple iterations]]

	After nearly fifteen thousand optimization iterations in OpenFOAM, the solution converged.		After nearly fifteen thousand optimization iterations in OpenFOAM, the solution converged.

Jpk254: /* Numerical example */

2021-12-16T04:35:05Z

Numerical example

← Older revision		Revision as of 00:35, 16 December 2021
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	From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.		From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.

	[[File:Airfoil-1.png\|thumb\|~~left~~\|upright=1.49\|'''Figure 6.''' Original airfoil]]		[[File:Airfoil-1.png\|thumb\|center\|upright=1.49\|'''Figure 6.''' Original airfoil]]

	[[File:Airfoil-5.png\|thumb\|~~right~~\|upright=1.49\|'''Figure 7.''' Optimized airfoil]]		[[File:Airfoil-5.png\|thumb\|center\|upright=1.49\|'''Figure 7.''' Optimized airfoil]]

	== Applications ==		== Applications ==

Jpk254: /* Numerical example */

2021-12-16T04:34:32Z

Numerical example

← Older revision		Revision as of 00:34, 16 December 2021
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	From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.		From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.

	[[File:Airfoil-1.png\|thumb\|left\|upright=1.5\|'''Figure 6.''' Original airfoil]]		[[File:Airfoil-1.png\|thumb\|left\|upright=1.49\|'''Figure 6.''' Original airfoil]]

	[[File:Airfoil-5.png\|thumb\|right\|upright=1.5\|'''Figure 7.''' Optimized airfoil]]		[[File:Airfoil-5.png\|thumb\|right\|upright=1.49\|'''Figure 7.''' Optimized airfoil]]

	== Applications ==		== Applications ==

Jpk254: /* Numerical example */

2021-12-16T04:34:14Z

Numerical example

← Older revision		Revision as of 00:34, 16 December 2021
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	From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.		From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.

	[[File:Airfoil-1.png\|thumb\|left\|upright=1.47\|'''Figure 6.''' Original airfoil]]		[[File:Airfoil-1.png\|thumb\|left\|upright=1.5\|'''Figure 6.''' Original airfoil]]

	[[File:Airfoil-5.png\|thumb\|right\|upright=1.47\|'''Figure 7.''' Optimized airfoil]]		[[File:Airfoil-5.png\|thumb\|right\|upright=1.5\|'''Figure 7.''' Optimized airfoil]]

	== Applications ==		== Applications ==

Jpk254: /* Numerical example */

2021-12-16T04:33:54Z

Numerical example

← Older revision		Revision as of 00:33, 16 December 2021
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	From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.		From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.

	[[File:Airfoil-1.png\|thumb\|left\|upright=1.46\|'''Figure 6.''' Original airfoil]]		[[File:Airfoil-1.png\|thumb\|left\|upright=1.47\|'''Figure 6.''' Original airfoil]]

	[[File:Airfoil-5.png\|thumb\|right\|upright=1.46\|'''Figure 7.''' Optimized airfoil]]		[[File:Airfoil-5.png\|thumb\|right\|upright=1.47\|'''Figure 7.''' Optimized airfoil]]

	== Applications ==		== Applications ==

Jpk254: /* Numerical example */

2021-12-16T04:33:15Z

Numerical example

← Older revision		Revision as of 00:33, 16 December 2021
Line 50:		Line 50:
	From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.		From the contour plots, the pressure is significantly greater of the bottom of the airfoil in Figure 5 than Figure 4. This pressure difference resulted in an increase of approximately 2% in the coefficient of lift.

	[[File:Airfoil-1.png\|thumb\|left\|upright=1.45\|'''Figure 6.''' Original airfoil]]		[[File:Airfoil-1.png\|thumb\|left\|upright=1.46\|'''Figure 6.''' Original airfoil]]

	[[File:Airfoil-5.png\|thumb\|right\|upright=1.45\|'''Figure 7.''' Optimized airfoil]]		[[File:Airfoil-5.png\|thumb\|right\|upright=1.46\|'''Figure 7.''' Optimized airfoil]]

	== Applications ==		== Applications ==