research article aerodynamic optimal shape design based on body-fitted grid...

Research ArticleAerodynamic Optimal Shape Design Based onBody-Fitted Grid Generation

Farzad Mohebbi and Mathieu Sellier

Department of Mechanical Engineering The University of Canterbury Private Bag 4800 Christchurch 8140 New Zealand

Correspondence should be addressed to Farzad Mohebbi farzadmohebbiyahoocom

Received 9 April 2014 Accepted 11 June 2014 Published 27 August 2014

Academic Editor Caner Ozdemir

Copyright copy 2014 F Mohebbi and M Sellier This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper is concernedwith an optimal shape design problem in aerodynamicsThe inverse problem in question consists in findingthe optimal shape an airfoil placed in a potential flow at a given angle of attack should have such that the pressure distribution onits surface matches a desired one The numerical method to achieve this aim is based on a body-fitted grid generation technique(elliptic O-type) to generate a mesh over the airfoil surface and solve for the flow equation The O-type scheme is used due to itsability to generate a high quality (fine and orthogonal) grid around the airfoil surfaceThis paper describes a novel and very efficientsensitivity analysis scheme to compute the sensitivity of the pressure distribution to variation of grid node positions and both theconjugate gradient method (CGM) and a version of the quasi-Newton method (ie BFGS) are used as optimization algorithmsto minimize the difference between the computed pressure distribution on the airfoil surface and desired one The elliptic gridgeneration technique allows us to map the physical domain (body) onto a fixed computational domain and to discretize the flowequation using the finite difference method (FDM)

1 Introduction

Thanks to the advent of modern high speed computers overthe last few decades computational fluid dynamics (CFD)has been extensively employed as an analysis and as a designoptimization tool Among themethodologies often employedin shape optimization are gradient-based techniques Thesetechniques may be applied to minimize a specified objectivefunction In airfoil shape optimization the objective functioncan be for example a measure of difference between thepressure distribution on the airfoil surface and a desired oneand it would be desirable tominimize this objective functionIn this paper we consider the 2D shape optimization of anairfoil in an irrotational and incompressible flow governed bythe Laplace equationTheprocedure employed is based on theelliptic grid generation a novel sensitivity analysis (based onfinite difference method) and an optimization method Theconjugate gradient method and an efficient version of quasi-Newton method (BFGS) will be used as the optimizationalgorithmsThe airfoil surface is parameterized using the gridpoints and the Bezier curve Three different types of design

variables were considered the grid points the Bezier curvecontrol points and the maximum thickness of NACA00xxairfoils It will be represented that the use of the Beziercurve significantly improves the optimization performanceto reach the optimal shape Furthermore it will be shownthat the proposed sensitivity analysis method reduces thecomputation cost significantly even for large number of thedesign variables

Some of the earliest studies using a combination ofCFD with numerical optimization in aerodynamic weremade by Hicks et al [1] and Hicks and Henne [2] In[1] a procedure for optimal design of symmetric low-dragnonlifting transonic airfoils in inviscid flow is proposed Theproposed procedure uses an optimization program based onthemethod of the feasible directions coupled with an analysisprogram that utilizes a relaxation method to solve the partialdifferential equation that governs the inviscid transonic andsmall disturbance fluid flow The drag minimization withgeometric constraints is considered in this reference In fluiddynamics Pironneau was the first one to use the adjointequations for design [3]This is the first application of control

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 505372 22 pageshttpdxdoiorg1011552014505372

2 Mathematical Problems in Engineering

theory to design optimization However within the field ofaeronautical computational fluid dynamics Jameson was thefirst researcher who used the continuous adjoint formulationfor aerodynamic shape optimization in transonic potentialflows andflows governed byEuler equations [4ndash7] Giles et almade considerable contributions to the development of thediscrete adjoint approach [8ndash11] In [10] the adjoint equationsare formulated for the transonic design applications forwhich there are shocks The adjoint equations were alreadyformulated for the incompressible or subsonic flows in whichthe assumption that the original nonlinear flow solution issmooth is valid In [11] a number of algorithm develop-ments are presented for adjoint methods using the ldquodiscreterdquoapproach In continuous adjoint method the original par-tial differential equations are linearized the adjoint partialdifferential equation and appropriate boundary conditionsare formulated and finally the equations are discretizedUnlike the continuous adjoint approach in discrete adjointapproach the partial differential equations are discretizedthe discrete equations are linearized and then the transposeof the linear operator is used to form the adjoint problemThe adjoint equations have also been used by Baysal andEleshaky to infer the optimal design for a scramjet-afterbodyconfigurationwhich yields themaximumaxial thrust [12] andby Tarsquoasan et al to obtain an optimal airfoil shape [13] Baysaland Eleshakyrsquos work was based on a computational fluiddynamics-sensitivity analysis algorithm (two different quasi-analytical approaches the direct method and the adjointvariable method) to solve Euler equations for the inviscidanalysis of the flow Adjoint methods have been applied toincompressible viscous flowproblems byCabuk et al [14] andDesai and Ito [15] Cabuk et al worked on the problem ofdetermining the profile of a channel or duct that provides themaximum static pressure rise by solving the incompressiblelaminar flow governed by the steady state Navier-Stokesequations Early applications of discrete adjoint methods onunstructured meshes can be found in works by Elliott andPeraire in inviscid [16] and viscous flows [17] for 2D and3D [18] configurations In [16] an inverse design procedurefor single- andmultielement airfoils using unstructured gridsand based on the Euler equations is presented The discreteadjoint method is used to compute the sensitivities and theresults are compared with corresponding finite differencevalues It is shown that the use of the adjoint method prac-tically eliminates the dependence of the objective functiongradient computation on the number of design variablesThecontinuous adjoint approach for unstructured grids has beendeveloped by Anderson and Venkatakrishnan [19] In [19]aerodynamic shape optimization on unstructured grids usinga continuous adjoint approach is developed and analyzed forinviscid and viscous flows B spline and Bezier curves areemployed to parameterize the airfoil surface The objectivefunctions considered include drag minimization lift max-imization and matching a specified pressure distributionThe quasi-Newton optimization method is used to obtainthe optimal design Evolutionary algorithms as methods thatdo not need the computation of the gradient have recentlygained much attention in the context of aerodynamic shapeoptimization [20ndash24] Although they are of extremely high

computational cost they have the advantage that they canescape from a ldquolocal minimumrdquo (a major issue in using gra-dient based methods) and have the ability of finding globallyoptimum solutions amongst many local optima [25 26] Adetailed study of many methods in shape optimization influid mechanics is given byMohammadi and Pironneau [27]

The adjoint approach as an alternative to the finitedifference method to compute the gradient of functionalwith respect to the design variables is computationallyvery efficient Therefore as far as the computational cost isconcerned it is the appropriate choice This is the case whenthere are a large number of design variables which makes useof the finite difference method impractical The differencesbetween the adjoint method and the finite difference one (tocompute the gradient of functional with respect to the designvariables) can be summarized as follows

Adjoint Method 119873 design variables 1 flow solution and 1

adjoint calculation

Finite Difference Method 119873 design variables 119873 flow solu-tions Because

120597J

120597120572119895

=

[J (120572119895 + 120575120572119895) minusJ (120572119895)]

120575120572119895

(1)

where J is the objective function and 120572119895 are the designvariables [28] As can be seen aerodynamic shape optimiza-tion with large number of design variables is computationallypractical only when the adjoint method is used However aswill be shown in this paper a novel sensitivity analysis willbe presented which makes use of the finite difference methodcomparable (from computation cost viewpoint) to the use ofadjoint method The numerical algorithm used in this paperis already employed in shape optimization problems in heatconduction [29 30] The numerical algorithm consists ofthree steps namely grid generation and flow equation solverto find the pressure on the airfoil surface sensitivity analysisto compute the gradient of the objective functionwith respectto the design variables and an optimization method tominimize the functional and reach optimum solution

2 Governing Equation

For a two-dimensional incompressible flow a stream func-tion 120595 can be defined such that

119906 =120597120595

120597119910

V = minus120597120595

120597119909

(2)

where 119906 and V are the components of the velocity vector Vthat is V = 119906i + Vj (i and j are the unit vectors in 119909 and119910 directions resp) Combining the above definitions withthe irrotationality condition leads to the following Laplaceequation for the stream function

1205972120595

1205971199092+

1205972120595

1205971199102= 0 (3)

Mathematical Problems in Engineering 3

Vprop

prop

prop

prop

prop

y

x

VpropVprop

Vprop

120595 = constant

Figure 1 Boundary conditions at infinity and on the airfoil surface(no-penetration)

Consider an irrotational incompressible flow over anairfoil (Figure 1) The boundary conditions are as shown inFigure 1

Conditions at Infinity Far away from the airfoil surface(toward infinity) in all directions the flow approaches theuniform freestream conditions If the angle of attack (AOA)is 120572 the free stream velocity 119881infin the components of the flowvelocity can be written as

119906 =120597120595

120597119910= 119881infin cos120572

V = minus120597120595

120597119909= 119881infin sin120572

(4)

Condition on the Airfoil Surface The relevant boundarycondition at the airfoil surface for this inviscid flow is theno-penetration boundary conditionThus the velocity vectormust be tangent to the surface This wall boundary conditioncan be expressed by

120597120595

120597119904= 0 or 120595 = constant (5)

where 119904 is tangent to the surface In the problem of the flowover an airfoil if the free stream velocity and the angle ofattack are known from the boundary conditions at infinity(see (4)) and the wall boundary condition (see (5)) one cancompute the stream function 120595 at any point of the physicaldomain (flow region)Then by knowing 120595 one can computethe velocity of all points Since for an incompressible flow thepressure coefficient is a function of the velocity only one canobtain the pressure of any point in the flow region as will beshown

Pressure CoefficientThe pressure coefficient 119862119901 is defined as

119862119901 =119901 minus 119901infin

(12) 120588infin1198812infin

= 1 minus (119881

119881infin

)

2

(6)

(MN1 + N2 + 3) N2

N2

N1

N1

N3

(MN1 + 2)

D

F

C

E

(M 1)

(MN)

N = 2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4) (MN1 + 2N2 + N3 + 5)

A (1 1)

H (1 N)

M

M

B

G

yx

Figure 2 Physical domain showing the discretization of the bound-aries used for O-type elliptic grid generation technique

where 119881 is the velocity of fluid at the point at which thepressure coefficient 119862119901 is being evaluated At standard sealevel conditions

120588infin = 123 kgm3

119901infin = 101 times 105Nm2

(7)

where 120588infin and 119901infin are the freestream density and pressurerespectively From (6)

119862119901 = 0 indicates that the point at which the pressurecoefficient 119862119901 is being evaluated is located at infinity

119862119901 = 1 indicates that the point at which the pressurecoefficient 119862119901 is being evaluated is a stagnation point (where119881 = 0) For an incompressible flow this is the maximumallowable value of 119862119901 anywhere in the flow field

And in regions of the flow where 119881 gt 119881infin 119862119901 value willbe negative

3 Grid Generation and Flow Solver

To calculate the pressure at any point in the flow region agrid should be generated over the regionThe grid generationmethod considered in this study is the elliptic grid generationwhich was proposed by Thompson et al [31] and is basedon solving a system of elliptic partial differential equationsto distribute nodes in the interior of the physical domainby mapping the irregular physical domain from the 119909 and119910 physical plane (Figure 2) onto the 120585 and 120578 computationalplane (Figure 3) which is a regular region

TheO-type elliptic grid generation technique is employedhere which results in a smooth and orthogonal grid over theairfoil surfaceThediscretization of the physical domain (flowregion) and the corresponding computational domain areshown in Figures 2 and 3 respectively In the computationaldomain 119872 and 119873 = 21198731 + 21198732 + 1198733 + 6 are the numberof nodes in the 120585 and 120578 directions respectively The resultingO-type gird scheme over an airfoil for the case 1198732 = 1198731 and1198733 = 21198731 minus 1 or119873 = 61198731 + 5 is shown in Figure 4


N = 2N1 + 2N2 + N3 + 6

H M G

F

E

D

C

BMA

N

(MN)

(MN1 + 2N2 + N3 + 5)

(MN1 + N2 + N3 + 4)

(MN1 + N2 + 3)

(MN1 + 2)

(M 1)120578

120585

(1 N)

(1 1)

Figure 3 Computational domain showing the discretization of thephysical domain boundaries

The initial guess for the elliptic grid generation is per-formed using the transfinite interpolation (TFI) methodSince TFI method is an algebraic technique and does notrequire much computational time it will be an appropriateinitial guess for the elliptic grid generation method andaccelerates convergence time for the ellipticmethod Anotheradvantage of using the TFImethod as an initial guess is that itprevents the grids generated by the elliptic (O-type) methodfrom folding

If 119881infin and 120572 are known then from (4) one can obtainthe stream function 120595 at any point on the boundaries of thephysical domain as follows

1205952 = 1205951 + (1199102 minus 1199101) 119881infin cos120572 (8)

1205952 = 1205951 minus (1199092 minus 1199091) 119881infin sin120572 (9)

where subscripts 1 and 2 refer to any two arbitrary grid pointson the boundaries of the physical domain Equations (8) and(9) are applied to vertical and horizontal boundaries of thephysical domain respectively By knowing the values of thestream function 120595 on the boundaries of the physical domainas well as on the airfoil surface we can obtain the values of 120595over the physical domain by applying the Kutta condition [3233] and using the following formula (bymapping the physicaldomain onto the computational domain [29])

120572120595120585120585 minus 2120573120595120585120578 + 120574120595120578120578

= minus1198692(119875 (120585 120578) 120595120585 + 119876 (120585 120578) 120595120578)

(10)

where

120572 = 1199092120578 + 1199102120578

120573 = 119909120585119909120578 + 119910120585119910120578

120574 = 1199092120585 + 1199102120585

119869 = 119909120585119910120578 minus 119909120578119910120585 (Jacobian of transformation)

(11)

119875 and 119876 are grid control functions which control the densityof grids towards a specified coordinate line or about a specific

grid point Equations (10) and (11) are discretized using thefinite differencemethod Formore details please refer to [29]

Velocity Calculation There are three sections where thevelocity must be known

(1) the outer boundaries (four sides CD DE EF and FCof the rectangle shown in Figure 2)

(2) the airfoil surface (AH in Figure 2)(3) the inside of the physical domain

The velocity values on the outer boundaries are knownfrom the conditions at infinity (using (4)) In other words119909-component of the velocity vector (119906) on all the outerboundaries is equal to 119881infin cos120572 and 119910-component of thevelocity vector (V) on all the outer boundaries is equal to119881infin sin120572 For the inside of the physical domain and the airfoilsurface we can use the flowing relationships to evaluatethe velocity These relationships are obtained by using thetransformation relationships and chain rule in mapping thephysical domain onto the computational one Consider

119906119894119895 =120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895

(120595120585)119894119895+ (119909120585)119894119895

(120595120578)119894119895] (12)

V119894119895 = minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895

(120595120585)119894119895minus (119910120585)119894119895

(120595120578)119894119895] (13)

The central and forward difference schemes are used for theinside of the physical domain and the airfoil surface respec-tively After obtaining the components of the velocity vectorthe total velocity (velocity distribution) can be computed by

119881119894119895 = radic1199062119894119895 + V2119894119895 (14)

As stated before for an incompressible flow the pressurecoefficient can be expressed in terms of the velocity onlyThus(6) can be used to determine the pressure at any grid point inthe domain Therefore

119901119894119895 =1

2120588 (1198812infin minus 119881

2119894119895) + 119901infin (15)

Validation of the Results for the Pressure Distribution Theresults obtained here are compared with the results given in[34] which are obtained both analytically and by using thepanel method (see Figure 7)

Validation Case The pressure coefficient distribution (119862119901)over the NACA 0012 airfoil at an angle of attack 120572 = 9

∘ isplotted The results are compared with the results from [34]TheO-type grid size used in the computation is 155 times 155Thecomputation time is 53 seconds

4 Airfoil Parameterization

So far the airfoil surface is parameterized by grid pointswhich result in accurate pressure distribution on the airfoil


0 05 1

0

02

04

06

0 01 02

0

005

01

015

09 1 11

0

005

01

minus02

minus04

minus06

YY

Y

X

X X

minus005

minus01

minus015

minus005

minus01

minus015minus01

Close-up view of O-type grid around the airfoil Magnifed view of grid around the leading edge

Magnifed view of grid around the trailing edge

Figure 4O-type grid (elliptic) around an airfoilThis close-up view of the grid shows orthogonality and smoothness of the gridlines especiallynear airfoil surface

surface (see Figures 14 19 and 25) However a large numberof grid points are needed to obtain such accurate resultswhich in turn lead to high (see Figures 5 and 6) computationcost The design variables are the coordinate (usually 119910-coordinate) of grid points Therefore the optimization pro-cessmay be inappropriate if there are a large number of designvariables since it is difficult to maintain a smooth geometrythe optimization problem will be difficult to solve and theoptimization strategy is likely to fail or be impractical [35]Thus alternative methods of airfoil surface parameterizationare needed These methods should represent great flexibilityin defining the airfoil surfacewithminimumdesign variablesIn this paper in addition to the grid points to represent theairfoil surface Bezier curves (a special subset of B-spline) are

employed due to their ability to produce airfoil surfaces easilyand precisely with only a few control points

Bezier Curve A Bezier curve is a special case of a B-splinecurve and is mathematically defined by

119875 (119905) =

119899

sum

119894=0

119861119894119869119899119894 (119905) (16)

where

119869119899119894 (119905) =119899

119894 (119899 minus 119894)119905119894(1 minus 119905)

119899minus119894 (17)

is Bernstein basis polynomial of degree 119899 By convention00

equiv 1 and 0 equiv 1 Here 119899 the degree of the Bernstein


0 02 04 06 08 1

0

1

minus5

minus4

minus3

minus2

minus1

Cp

xc

Figure 5 The pressure coefficient distribution over an NACA 0012airfoil at a 9∘ angle of attack obtained numerically

minus04

minus03

minus02

minus01

0

01

05 10

Cptimes10

minus1

NACA 0012 airfoil120572 = 9∘

Upper surfaceLower surfaceClassical solution

2 orderpanel

method

Vinfin

9∘

xc

Figure 6 The pressure coefficient distribution over an NACA 0012airfoil at a 9∘ angle of attack [34]

basis polynomial is one less than the number of points in theBezier polygon In other words the number of control pointsis 119899 + 1 The points 119861119894 are the vertices of a Bezier polygonor the control points of a Bezier curve The curve begins at1198610 and ends at 119861119899 The order of a Bezier curve 119896 is equal to119899 + 1 In other words the order of a Bezier curve is equal tothe number of the control points [36]

In this paper two different Bezier curves of order 7(degree = 6) and of order 11 (degree = 10) will be consideredAs it will be shown the Bezier curve of order 7 representsthe better optimization performance due to its less designvariables However this kind of Bezier curve is not able toproduce very accurate airfoil shapes Indeed it is appropriateto NACA 00xx airfoils only On the other hand the Bezier

0 02 04 06 08 1

0

1

Results from the referenceResults from our method

Airfoil NACA 0012Angle of attack 9

minus6

minus5

minus4

minus3

minus2

minus1

Cp

xc

Figure 7 Comparison between the results from [34] and the resultsfrom our method for validation case The figure shows an excellentagreement between the results

curve of order 11 can successfully generate any airfoil shapewith a high degree of accuracy Therefore the formulationfor the Bezier curve of order 11 only will be given here Theformulation for the Bezier curve of order 7 can be written ina similar fashion

The parametric Bezier curve of order 11 is as follows

119899 = 10 997904rArr number of control points = 11

119875 (119905) =

10

sum

119894=0

11986111989411986910119894 (119905)

= 1198610119869100 (119905) + sdot sdot sdot + 119861101198691010 (119905)

= 1198610

10

0 (10 minus 0)1199050(1 minus 119905)

10minus0

+ sdot sdot sdot + 11986110

10

10 (10 minus 10)11990510(1 minus 119905)

10minus10

(18)

Therefore

119875 (119905) = 1198610 (1 minus 119905)10

+ 119861110119905 (1 minus 119905)9+ 119861245119905

2(1 minus 119905)

8

+ 11986131201199053(1 minus 119905)

7+ 1198614210119905

4(1 minus 119905)

6

+ 11986152521199055(1 minus 119905)

5+ 1198616210119905

6(1 minus 119905)

4

+ 11986171201199057(1 minus 119905)

3+ 119861845119905

8(1 minus 119905)

2

+ 1198619101199059(1 minus 119905)

1+ 11986110119905

10

(19)

In order to construct the airfoil surface two Bezier curve willbe considered corresponding to the upper and lower surfaces


respectively Here there are 11 control points (vertices) foreach surface Since the coordinates of the airfoil surface areknown the problem is to determine values for the controlpoints 119861119894 (119894 = 0 10) In other words our problem is tospecify the coordinates of the control points 119861119894 so that thecurve passes through the predetermined data points on theairfoil surface Equation (16) can be written in matrix form asfollows

[119875 (119905)] = [119869 (119905)] [119861] (20)

If the number of the chosen data points on the airfoil surfaceis119898 and the degree of Bezier curve is 119899 then [119875(119905)] is a119898times2

matrix [119869(119905)] is a 119898 times (119899 + 1)matrix and [119861] is a (119899 + 1) times 2

matrix Two columns of the matrix [119875(119905)] pertain to the 119909-and 119910-coordinates of the predetermined data on the airfoilsurface Equation (20) can be rewritten as

[119875 (119905)]119898times2 = [119869 (119905)]119898times(119899+1)[119861](119899+1)times2 (21)

If 119898 = 119899 + 1 the matrix [119869(119905)]119898times(119899+1) will be a square matrixand it can be inverted In such a case (21) can be written asfollows to find the matrix [119861]

[119861](119899+1)times2 = [119869 (119905)]minus1119898times(119899+1)[119875 (119905)]119898times2 (22)

However the number of the airfoil surface data points isusually more than the number of control points In such acase there aremore equations than unknowns and thematrix[119869(119905)]119898times(119899+1) is no longer a square matrix Hence it is requiredto convert it to a square matrix by multiplying both sides of(21) by the transpose of [119869(119905)]119898times(119899+1) as follows

[119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2

= [119869 (119905)]119879(119899+1)times119898[119869 (119905)]119898times(119899+1)[119861](119899+1)times2

(23)

Thus

[119861](119899+1)times2 = [[119869 (119905)]119879(119899+1)times119898[119869 (119905)]119898times(119899+1)]

minus1

sdot [119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2

(24)

NACA 0015 and TsAGI ldquoBrdquo 12 airfoils produced by Beziercurve with 119899 = 10 and 119898 = 51 and their comparisonwith conventional NACA 0015 and TsAGI ldquoBrdquo 12 airfoils areshown in Figures 8 and 9 respectively There is an excellentagreement between two airfoils in each figure

The predetermined data for the NACA airfoils can beextracted from for example the software JavaFoil [37] whichis based on the analytical NACA formulations

NACA 00xx Symmetric Airfoils Since the maximum thick-ness of a NACA 00xx symmetric airfoil will be considered asa design variable the equation for generating such airfoils isgiven as follows

plusmn119910119905 =119905

02[02969radic119909 minus 01260119909

minus035161199092+ 02843119909

3minus 01015119909

4]

(25)

0 02 04 06 08 1

0

01

02

03

04

NACA 0015 (standard)NACA 0015 (Bezier curve)

0 005 01 015

0

005

09 095 1

0001002003

minus01

minus02

minus03

minus04

minus05

minus005

minus001

minus002

minus003

X

X

X

Y

Y

Y

Figure 8 Comparison between the standard airfoil and the Beziercurve for a NACA0015

0 02 04 06 08 1

0

01

02

03

04

05

0 005 01 015 02

0

005

0985 099 0995 1

0

0002

0004minus01

minus02

minus03

minus04

minus05

minus005

minus0002

minus0004

X

X

X

Y

Y

Y

TsAGI ldquoBrdquo 12 (Bezier curve)TsAGI ldquoBrdquo 12 (standard)

Figure 9 Comparison between the standard airfoil and the Beziercurve for a TsAGI ldquoBrdquo 12

where 119909 is coordinates along the chord of the airfoil from 0

to 119888 (119888 is the chord length and is assumed equal to 1) 119910119905 isthe thickness coordinates above and below the line extendingalong the length of the airfoil and 119905 is maximum thicknessof the airfoil in percentage of chord (ie 119905 in a 15 thickairfoil would be 015) Equation (25) can be used to find


the 119910-coordinates of a NACA 00xx symmetric airfoil byknowing the values for 119909 and 119905 As will be shown themaximum thickness of such airfoils will also be consideredas a design variable By optimizing the thickness the optimalshape for such airfoils will be obtainedThis kind of optimiza-tion problem however is not comprehensive and producesthe optimalNACA00xx symmetric airfoils only In summarythree kinds of design variable will be considered in this paperfor airfoil shape optimizationwhich are grid points on a givenairfoil surface extracted from say the software JavaFoil theBezier curve control points and the maximum thickness ofNACA 00xx symmetric airfoils

5 Shape Optimization

Different objective functions may be considered for theaerodynamics shape optimization including maximizing thelift-drag ratio maximizing the lift and minimizing the dragIn the framework of this paper the shape optimizationproblem will be to infer the shape an airfoil should have sothat the pressure distribution on the airfoil surface matchesa prescribed one (an inverse problem) In inverse designproblem the desired pressure distribution of the target designmay be specified a priori

Design Variable (DV) Here the airfoil grid points theBezier curve control points and the maximum thicknessof NACA00xx airfoils are considered as design variablesTherefore one has the following

Case 1 the airfoil grid points as design variable (seeFigure 13)Case 2 the Bezier curve control points as designvariableCase 3 the maximum thickness of NACA00xx air-foils

Case 1 The mathematical expression for the objective func-tion considered for Case 1 can be stated as

J =

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))2 (26)

where 119875(1119895) is the pressure at grid points 1198651119895 on the airfoilsurface and 119875119889(1119895) is the desirable pressure at grid points 1198651119895on the airfoil surface (Figure 10) The aim is to minimize Jand to reach the desirable pressure distribution by changingthe position of the grid points on the airfoil surface Sincethe 119909-coordinates of the grid points can be constant duringthe optimization process only the 119910-coordinates of the gridpoints are considered as design variables Two end points ofairfoil namely leading edge (119895 = (119873 + 1) 2) and trailing edge(119895 = 1119873) are fixed Thus they are not considered as designvariables


J =

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)2 (27)

0 02 04 06 08 1

0

02

04

06

Airfoil

Leading edge Trailing edge

minus02

minus04

(1 N + 1

2)

(1 N)

Y

X

F(1j)

(1 1)

Figure 10 Illustration of the airfoil surface points to be optimizedso that the objective function reaches a minimum

0 02 04 06 08 1

0

02

Control pointsAirfoil grid points

NACA 0028

09998 1000265

00027000275

00028000285

00029000295

0003

minus02

minus04

minus06

B10 U

B10 U

B0 U

B1 UB2 U

B8 U

B9 U

B10 LB0 L

B1 L

B2 L

B8 L

Y

Y

X

X

(B) Control point(L) Lower surface(U) Upper surface

Figure 11 Illustration of the Bezier control points (119861119894) to be opti-mized so that the objective function (see (27)) reaches a minimum

where 119898 is the number of the predetermined data on eachof the upper and the lower surfaces of the airfoil 119875119894119861 is thepressure at point 119894 of the airfoil surface generated by theBeziercurve and 119875119894119861119889

is the desirable pressure at point 119894 Why does2119898minus4119898 data points for the upper surface119898 data points forthe lower surface and the leading and the trailing edges fortwo surfaces are considered fixed The aim is to minimizeJand to reach the desirable pressure distribution by changingthe 119910-position of the control points 119861119894 (119894 = 1 9) oneach of the upper and the lower surfaces of the airfoil (seeFigure 11) 1198610 and 11986110 which are concerned with the leading


0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X

Figure 12 The location for the maximum thickness on the upperand lower airfoil surfaces 119910-coordinates of the points 1 and 2 areconsidered as the design variables

edge and the trailing edge respectively are considered fixedfor both upper and lower surfaces Therefore for the shapeoptimization problemwith a Bezier curve of order 11 we have2 times (11 minus 2) = 18 design variables For the shape optimizationproblem with a Bezier curve of order 7 we have 2 times (7 minus 2)= 10 design variables The reason for considering these twokinds of the Bezier curve is twofold

(1) to show that the optimization problem will be moresuccessful if we have less number of design variable

(2) to have a very accurate and flexible representation ofthe airfoil shapes a degree of at least 10 should beused

Case 3 The airfoil surface is generated by the analyticalNACA formula (25) and the maximum thickness is con-sidered as the design variable To show the accuracy of thesensitivity scheme the upper and lower airfoil surfaces aregenerated separately and hence the design variables will betwo maximum thicknesses in the upper and lower airfoilsurfaces As shown in Figure 12 if the indices 1 and 2denote the location of maximum thickness on the upperand lower airfoil surfaces respectively then themathematicalexpression for the objective function considered for Case 3 isas follows

J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)

6 Sensitivity Analysis

Suppose we wish to calculate the sensitivity of pressureof nodes on the airfoil surface (see Figure 10) 1198751119895 (119895 =

2 119873 minus 1 119895 = (119873 + 1) 2) to the 119910-position of the nodeson the airfoil surface 11991011198951015840 (119895

1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X

Figure 13 Grid used in Test Case 1 (around initial shape)

0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X

Figure 14 Pressure distribution around the airfoil surface (initialshape)

The sensitivity analysis can be performed by introducingsmall perturbations to the 119910-coordinate of each point onthe airfoil surface individually The grid generation and flowproblem may be solved for this perturbed shape to obtainthe new values for the pressure 1198751119895 Using these values forthe pressure the dependency of the pressure 1198751119895 to theperturbation of the 119910-position of points of coordinates (1 1198951015840)11991011198951015840 can be evaluated The finite difference method may beused to formulate these sensitivities as follows

1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)

where 120576 may be say 10minus6 The term 12057611991011198951015840 is the perturba-tion in the 119910-position of points of coordinates (1 119895

1015840) 11991011198951015840


Since the sensitivity of each pressure 1198751119895 (119895 = 2 119873 minus

1 119895 = (119873 + 1) 2) to each 119910-position of points of coordinates(1 1198951015840) (1198951015840

= 2 119873 minus 1 1198951015840

= (119873 + 1) 2) is required thecomputation of the sensitivity coefficients using this methodrequires (119873 minus 3) additional solutions of the flow problemTherefore this method is only suitable when the number ofpoints on the airfoil surface is smallThus for the airfoil shapeoptimization problem which demand a fine grid to obtainaccurate results the perturbation method using the finitedifference method will be of high computation cost In thispaper wewill expand the novelmethodused in evaluating thesensitivity matrix in the shape optimization of heat transferproblems As will be shown it requires only one solution ofthe flow problem (at each iteration) to compute all sensitivitycoefficients

With regard to (26) the following equation can bewrittenin order to calculate the Jacobian matrix

120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)

where (119895 = 2 119873 minus 1 119895 = (119873 + 1) 2) and (119897 = 2 119873 minus

1 119897 = (119873 + 1) 2) The expression 12059711987511198951205971199101119897 in the aboverelation is called the Jacobian coefficient In this case thesensitivity matrix can be expanded as

Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014

sdot sdot sdot12059711987512

1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014

sdot sdot sdot1205971198751119873minus1

1205971199101119873minus1

]]]]]]]]]]]

]

(31)

Since the physical domain is mapped onto the computa-tional one the chain rule may be used to correlate variablesin the two domains Therefore

1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)

As pointed out before the 119909-coordinate of the grid pointsare considered fixed and they are not included in the designvariables Thus (32) is written here to derive the requiredrelations for the sensitivity coefficients By interchanging 119909

and 120585 and 119910 and 120578 and solving the derived equations for120597119875120597119909 and 120597119875120597119910 we finally obtain

1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)

where 119869 = (119909120585119910120578 minus 119909120578119910120585)1119897 is the Jacobian of the transforma-tionUsing the finite differencemethod to discretize the equa-tions in the computational domain we can write appropriate

algebraic approximations for all partial derivatives involvedin the above equation Therefore

(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)

which are based on the central and the forward differencesEquations (35) through (38) are employed to calculate thesensitivity coefficients in (31)

Bezier Control Points as Design VariablesWith regard to (27)and considering the control points of the Bezier curve asdesign variable we can write

120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)

Using the chain rule we can write

120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)

where 1199101198941015840119861 (1198941015840= 1 2119898 minus 4) are the 119910-coordinate of the

predetermined grid points to be passed by the Bezier curveand 119861119910119897

(119897 = 1 18) are the 119910-coordinate of Bezier controlpoints whose number is equal to 18 (9 for each of the upperand lower surfaces) The term 1205971198751198941198611205971199101198941015840119861 can be computedby the expressions derived for Case 1 (see (31)) The size ofthe matrix formed by the arrays 1205971198751198941198611205971199101198941015840119861 is (2119898 minus 4) times

(2119898 minus 4) The term 1205971199101198941015840119861120597119861119910119897can be easily evaluated by

taking derivative of (19) with respect to the control points119861 (noting that [119875(119905)] equiv [119909(119905) 119910(119905)]) The control pointsmay be renumbered so that 1198611199101

= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871

The indices 119880 and 119871 denote the upper and lower surfacesrespectively The direction of numbering is from right to leftfor the upper surface and from left to right for the lowersurface The reason for this renumbering is the compatibilitywith the grid point data reading (most of the airfoil data arein this format) as well as the pressure reading to compute theobjective function (see (27)) However we should note thatthe Bezier curve evaluation is from left to right for both theupper and lower surfaces The size of the matrix formed bythe arrays 1205971199101198941015840119861120597119861119910119897 is (2119898 minus 4) times 18 Because the upper andlower surfaces are constructed separately the variation of 119910of the upper surface with respect to the change in position ofthe lower surface control points as well as the variation of 119910


of the lower surface with respect to the change in position ofthe upper surface control points is zero

MaximumThickness as Design Variables In a similar deriva-tion to Case 1 the sensitivity matrix for Case 3 will be

Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)

7 Optimization Method

In this paper two powerful optimization methods namelythe conjugate gradient method and the quasi-Newtonmethod will be used For the airfoil grid points as adesign variable (Case 1) both optimization methods will beemployed However for the Bezier curve control points as adesign variable (Case 2) only the quasi-Newton method willbe used For Case 3 (the maximum thickness of NACA00xxairfoils as a design variable) only the conjugate gradientmethod will be employed

Conjugate Gradient Method The conjugate gradient algo-rithm to obtain the optimal shape for the airfoil is as follows

(1) Specify the physical domain the boundary condi-tions the problem conditions such as Mach numberand the angle of attack and the desired airfoil surfacepressure distribution

(2) Generate the boundary fitted grids using the gridgeneration methods described earlier

(3) Solve the direct flow problem of finding the pressurevalues at any grid points of the physical domain andhence the airfoil surface

(4) Using (26) compute the objective function (J(119896))(5) If the value of the objective function obtained in step

(4) is less than the specified stopping criterion theoptimization is finished Otherwise go to step (6)

(6) Compute the sensitivity matrix (Ja) from (31)(7) Compute the gradient direction nablaJ(119896) from (30)(8) Compute the conjugation coefficient 120574

(119896) from thefollowing equation (the Polak-Ribiere formula)

120574(119896)

=

[nablaJ(119896)]119879(nablaJ(119896) minus nablaJ(119896minus1))

1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=


[nablaJ(119896minus1)]119879nablaJ(119896minus1)

(42)

For 119896 = 0 set 120574(0) = 0(9) Compute the direction of descent d(119896) from the

following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)

(10) Compute the search step size 120573(119896) from the following

120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)

(11) Evaluate the new 119910-coordinates of the airfoil surfacegrid nodes as follows

y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)

(12) Set the next iteration (119896 = 119896 + 1) and return to step(2)

The above algorithm is for the airfoil grid points as a designvariable (Case 1) only The algorithm for Case 3 can beexpressed in a similar way

Quasi-Newton Method Quasi-Newton method is anotherpowerful optimization method used in this paper In quasi-Newton method the Hessian matrix (which is composedof the second partial derivatives) is replaced by an approx-imation of it The approximation uses only the first partialderivatives The Broyden-Fletcher-Goldfarb-Shanno (BFGS)method is a quasi-Newtonmethod for solving unconstrainednonlinear optimization In the BFGS method the Hessianmatrix approximation B(119896) is updated iteratively The stepsof BFGS method can be summarized as follows




(4) Using (26) compute the objective function (J(119896))(5) If value of the objective function obtained in step


(6) Compute the sensitivity matrix (Ja) from (31)

(7) Compute the gradient direction nablaJ(119896) from (30)

(8) The initial Hessian matrix approximation B(1) istaken as the identity matrix namely B(1) = I

(9) Set S(119896) = minusB(119896)nablaJ(119896) and the iteration number as119896 = 1

(10) Compute the search step size 120573(119896) (from (44)) in the

direction S(119896) and set

y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)

(11) Repeat the steps (2) to (7) with these new values of yfor the grid points119910-coordinates to calculatenablaJ(119896+1)


Table 1 Data used for Test Case 1

Airfoil Grid sizeAngle ofattack

120572

Free streamvelocity119881infin

Initial TsAGIldquoBrdquo 10 300 times 305 1∘ 70msDesired NACA 0015 350 times 365 1∘ 70ms

(12) Update the Hessian matrix approximation as

B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)

minus nablaJ(119896)

(48)

(13) Set the new iteration number as 119896 = 119896 + 1 and go tostep (9)

8 Results

In this section the results obtained for the shape optimizationof an airfoil in the incompressible irrotational and inviscidflow under given boundary conditions are presented Threekinds of the design variable (the airfoil grid points the Beziercurve control points and the maximum thickness of NACA00xx airfoils) as well as two optimization methods (CG andBFGS) are considered In all test cases in this paper whichemploy the Bezier curve the number of predetermined airfoildata119898 is set equal to the Bezier curve order 119899 + 1

Test Case 1 In this test case the airfoil surface is parame-terized by a Bezier curve of order 11 The total number ofthe design variable is 18 namely 9 design variables for eachof the upper and lower surfaces At first two parametriccurves for two surfaces (upper and lower) are obtained using11 grid points and then a fine grid is generated to obtainaccurate results The data for Test Case 1 is given in Table 1The comparison of the initial and optimal airfoil shapes andsomemagnified parts of them are shown in Figures 15 20 and26 In this test case BFGS optimization method is employed

The convergence of the objective function is shown inFigures 17 23 29 37 and 39 The initial and minimumvalues for the objective function are approximately 3517433and 2877116 respectively which shows 182 reduction inobjective function (see Figures 16 21 and 27) The minimumvalue for the objective function takes place in iteration 14The optimization time spent on the 1st iteration (which isequivalent to one direct flow solution) is 11 minutes and 43seconds and the total optimization time for 30 iterations is 11minutes and 46 secondswhich shows the proposed sensitivity

0 02 04 06 08 1

0

02

04

Initial shapeOptimal shape

0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04

minus001minus002minus003

minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X

Figure 15 Comparison of the initial and optimal shapes and somemagnified parts including the leading edge middle parts and thetrailing edge

0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X

Figure 16 Comparison of the initial and optimal shapes The 119910-axis has been greatly exaggerated to highlight difference in the airfoilshapes

analysis efficiency 30 iterations take only 3 seconds Thereason for the difference between the 1st iteration and thefollowing ones is that the solution after the 1st iteration is avery good initial guess for the 2nd iteration and the directsolution converges quickly In other words what is a bittime consuming for the 1st iteration is the grid generationand stream function loops not the pressure calculationsensitivity analysis and optimization stages Moreover a fine


Iteration number0 5 10 15 20 25 30

5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n

Figure 17 Objective function value versus the iteration number

0 1

04

02

0

minus02

minus04

Y

X

05


grid (300 times 305) and a tolerance of 10minus8 are used in theiterative loopswhich increase the computation timeThe codeis programmed in Fortran 77 using a Fortran compiler (Force20) and the computations are run on a PCwith Intel PentiumDual 173 and 1G RAM All the computations in the testcases in this paper are performed using the above mentionedcompiler and PC Therefore there is no need to repeat it inthe following test cases

Test Case 2 Test Case 2 is similar to Test Case 1 but withdifferent specifications (see Figure 18) The data for this testcase is given in Table 2

The explanation is similar to Test Case 1 Thus only theresults will be given in Table 3

Now an optimal shape design problem using a Beziercurve of order 7 is given As it will be shown it decreasesthe objective function value much bigger than when using

06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04

Optimal shapeInitial shape

004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X

Figure 20 Comparison of the initial and optimal shapes and somemagnified parts including the leading and trailing edges



120572


Initial NACA 0028 400 times 425 2∘ 70ms

Desired NACA 0016 350 times 365 2∘ 70ms

a Bezier curve of order 11 as there is less number of designvariables (10 design variables for a Bezier curve of order 7


Table 3 Results for Test Case 2

Initial objectivefunction

Minimum objectivefunction (iteration 3)

Computational time1st iteration

Total computationaltime 30 iterations

Percentage ofreduction inobjectivefunction

10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X


versus 18 design variables for a Bezier curve of order 11)However as it will be shown using a Bezier curve of order7 is not comprehensive for all airfoil shapes and is suitable toNACA 00XX or similar airfoils only In other words it is notable to produce all airfoil shapes precisely

Test Case 3 (using a Bezier curve of order 7 see Figure 24 andTable 4) The results are given in Table 5

Although there is a decrease of 59 in objective functionthe 59 approach from the initial shape to desired one isnot seen (see Figure 28) This indicates that the solution ofthe inverse problem is not unique The reason for this canbe found in the trailing edge configuration for the initialoptimal and desired airfoil shapes (Figure 28)

Although the results of using the Bezier curve of order7 is very promising its drawback is that it is restricted tothe simple and symmetric airfoil shapes such as NACA 00xx

For other airfoil shapes there can be seen some oscillationsaround the trailing edge (see Figure 30)

Moreover the Bezier curve of order 7 fails to representthe NACA 00xx airfoils accurately In other words the Beziercurve of order 11 is the appropriate option to produce veryaccurate airfoils (Figure 31)

Test Case 4 In this test case the airfoil surface is parame-terized by grid points obtained from the analytical NACAformula (eg software JavaFoil) In this case the number ofthe design variables is equal to the number of grid pointsminus three (one for leading edge and two for trailingedge)Therefore we have an aerodynamic shape optimizationproblem with a high number of the design variables as weshould have a fine grid to obtain sufficiently accurate resultsIt is known that the optimization process becomes morechallenging by increasing the number of design variablesHence we have a difficult shape optimization problem in TestCase 4 The data used for Test Case 4 is given in Table 6A very fine grid (400 times 425) is used for the initial airfoilshape (NACA 0012) The number of the design variables is119873 minus 3 which is 425 ndash 3 = 422 Therefore a time consumingoptimization problem is expected However by using thesensitivity analysis scheme proposed in this paper the totaltime for the optimization problem in Test Case 4 using bothCG and BFGS optimization methods is about 46 minutes for8000 iterations The computation time for the 1st iterationis about 25 minutes The comparison of the computationtimes for the 1st iteration and 8000 ones indicates again theefficiency of the sensitivity analysis scheme The summaryof the results is presented in Table 7 The comparison ofthe initial optimal and desired airfoil shapes is given inFigure 32 As can be seen in the figure the variation of theshape is minuteThe convergence of the objective function toa local minimum when both the CG and BFGS optimizationmethods are used as well as a comparison of them is shownin Figures 33 34 and 35 respectively The plots reveal thebetter performance of the BFGS method in minimizing theobjective function

In Test Cases 5 and 6 the maximum thickness of theNACA 00xx is considered as a design variable As mentionedpreviously the conjugate gradient method is employed as theoptimization method

Test Case 5 The data for the problem including the condi-tions for the initial and desired airfoils are given in Table 8

Figure 36 represents the comparison of the initial opti-mal and desired shapes for airfoils (also see Figures 22 and38) The desired airfoil shape is a NACA0018 at conditionsstated in Table 8 As can be seen this shape is shown by








898639 368148 9m and 32 s 9m and 43 s 59



120572




Table 7 Summary of the results of Test Case 4

InitialJ FinalJ Number ofiterations

Computationtime (total)

ReductioninJ

CG 6351997 6053996 8000 sim46mins 47BFGS 6351997 6036943 8000 sim46mins 5



120572




a black color line The optimization process is started bya NACA0011 as an initial shape which is shown by a redcolor line The optimal shape (shown by a blue color line)which is obtained by the conjugate gradient method is in anexcellent agreement (full matching) with the desired oneTheobjective function variation is shown in Figure 37The initialand final values for the objective function are about 77359745and 1548 respectively which reveals an approximately 100reduction in the objective function within 31 iterations Thetotal time for the optimization (for 31 iterations) is 2 minutesand 14 seconds The tolerance used in iterative steps in theprogram is 10minus8 Although such a tolerance value increasesthe computation time it enhances the accuracy of the resultsIf the 119910-components of the maximum thickness in upperand lower airfoil surfaces are denoted by 119910thick119880 and 119910thick119871respectively the value of the pressure for these two locationsfor initial optimal and desired shapes are reported in Tables10 and 13The difference values show the validity of the shapeoptimization process

Test Case 6 The data for the problem is given in Table 11Theexplanation for the results is similar to Test Case 5 (see Tables9 and 12)

0 05 1

0

02

04

06

08

Optimal shapeInitial shapeDesired shape

097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X

Figure 22 Comparison of the initial optimal and desired shapes




ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method

As pointed out previously for the aerodynamic shape opti-mization problems requiring a large number of design vari-ables the use of finite difference method to evaluate thegradient by introducing a small perturbation to each designvariable separately and then solving the flow problem is ofvery high computational cost because it requires a numberof additional flow solutions equal to the number of designvariables For optimal shape design problems with a highnumber of design variables the adjoint method [4] cancompute the gradients of objective functionmuch faster thanthe finite difference method

The aerodynamic shape optimization problem of interesthere can be expressed as

minimization of objective function J

subject to constraint R = 0 (the governing equation) (49)


Table 10 Comparison of the pressure at the maximum thicknesses of the airfoil surface (upper and lower surfaces) for the initial optimaland desired shapes

Initial shape Optimal shape Desired shape Difference (Pa)

Pressure at 119910thickU (Pa) 10008004 9941137 9941006

Difference in initial and desired= 66998

Difference in optimal and desired= 131

Pressure at 119910thickL (Pa) 10081439 10024825 10024454





120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X

Figure 25 Pressure distribution around the airfoil surface (initial shape)

The objective function J and the governing equation R =

0 depend on the flow variables W and the geometry designvariable X119863

J = J (WX119863)

R = R (WX119863) = 0

(50)

The derivative of the objective functionJwith respect to thedesign variables X119863 can be expressed as

119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)

which states that a change in the objective function is dueto a combination of a variation in the flow solution 120597W and

a variation in the design variable (change in geometry) 120597X119863In a similar way we have

119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)

If the sensitivity analysis is performed using (51) and (52)the problem is referred to as the ldquoprimal problemrdquo Solvingthe primal problem comes with the same difficulties as weencounter with use of the finite differencemethod It requiresthe additional flow solutions proportional to the number ofthe design variablesX119863Therefore the adjointmethod comesto the picture by introducing a vector of Lagrange multipliers


0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X


Ψ By adding (52) as a constraint to the sensitivity equation(51) we obtain

119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06

Initial shapeOptimal shapeDesired shape

094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001


Iteration number0 20 40 60 80 100

0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n


Rearranging the terms inside (53) we get

119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08

Airfoil TsAGI ldquoBrdquo 12

09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001

Bezier curve (degree = 6)Bezier curve (degree = 10)

minus02

minus00002

minus00004

minus00006

X

XY

Y

Figure 30 Oscillations around the trailing edge

0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128

00013000132 Standard

Bezier (degree = 6)Bezier (degree = 10)

Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y

NACA 0012 (upper surface) 28 points

Figure 31 Comparison of an analytical NACA 0012 airfoil (uppersurface only) with one obtained by using the Bezier curves of orders7 and 11 The plots represent an excellent agreement between theanalytical NACA and the Bezier of order 11

then (54) reduces to

119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)

Equation (55) is the adjoint equation and the vector Ψ is theadjoint variables Equations (55) and (56) are referred to asthe ldquodual problemrdquo The adjoint equation is a linear system

0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY

Initial shape (IS)Optimal shape (OS)

minus0059954

minus0059956

minus0059958

minus000123minus000124minus000125minus000126minus000127minus000128minus000129minus00013

minus000131

minus0013796

minus0013798

minus00138

minus0013802

Figure 32 Comparison of the initial and optimal airfoil shapes withthe magnified sections of them to show the variation of the shapeBoth BFGS and CG are used in optimization process

Iteration number0 2000 4000 6000 8000

Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n

Figure 33 Convergence of the objective function CG is used as theoptimization method

and can be solved to obtainΨ Then the determinedΨ can besubstituted into (56) to obtain the gradient of the objectivefunction It can be seen that the gradient of the objectivefunction can be determined without the need for additionalflow solutions The computational cost of solving the adjointequation is comparable to that of solving the flow equationTherefore the computational cost of evaluating the objectivefunction gradient is roughly equal to the computational costof two flow equation solutions independent of the number ofdesign variables [38ndash41]



Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n

Figure 34 Convergence of the objective function BFGS is used asthe optimization method


CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n

Figure 35 Comparison of the CG and BFGSmethods in decreasingthe objective function

From the accuracy of the derivatives view point the finitedifference method (based on the perturbation scheme) iscompared to the adjoint method [42ndash44] The comparisonshows a very good agreement between two methods There-fore our aim here is to compare our novel shape sensitivitymethod to the adjoint method from the efficiency view pointonly As mentioned above the computational cost of solvingthe adjoint equation is comparable to that of solving theflow equation whereas the computational cost of our novelmethod is comparable to that of computation of an algebraicexpression for arrays of a matrix As seen in Test Case 4the computation time for iterations 2 to 8000 is 46 ndash 25 =

0 02 04 06 08 1

0

02

04

Desired airfoil (NACA 0018)Initial airfoil (NACA 0011)Optimal airfoil

0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y

Figure 36 The initial optimal and desired shapes for the airfoilThere is an excellent agreement between the optimal and desiredairfoil shapes


0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n


21 minutes (about 7 iterations per second) which reveals theefficiency of the proposed sensitivity analysis

10 Conclusion

This paper addressed the aerodynamic shape optimization foran airfoil in an irrotational and incompressible flow governed


0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06


by Laplace equation using a type of the elliptic grid genera-tion (O-type) a novel and very efficient sensitivity analysismethod and the conjugate gradient and BFGS optimizationmethodsThe airfoil was parameterized using the grid pointsand the Bezier curve Three different types of design variablewere considered the grid points the Bezier curve controlpoints and the maximum thickness of NACA00xx airfoils Itwas represented that the use of the Bezier curve significantlyimproves the optimization performance to reach the optimal

shape The results obtained in test cases presented in thispaper show that the proposed sensitivity analysis methodreduces the computation cost even for large number of thedesign variables (Test Case 4) and confirm accuracy andefficiency of the proposed shape optimization algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R M Hicks E M Murman and G N Vanderplaats AnAssessment of Airfoil Design by Numerical Optimization 1974

[2] R M Hicks and P A Henne ldquoWing design by numericaloptimizationrdquo Journal of Aircraft vol 15 no 7 pp 407ndash412 1978

[3] O Pironneau ldquoOn optimumdesign in fluidmechanicsrdquo Journalof Fluid Mechanics vol 64 pp 97ndash110 1974

[4] A Jameson ldquoAerodynamic design via control theoryrdquo Journal ofScientific Computing vol 3 pp 233ndash260 1988

[5] A Jameson ldquoComputational aerodynamics for aircraft designrdquoScience vol 245 no 4916 pp 361ndash371 1989

[6] A Jameson ldquoOptimum aerodynamic design using CFD andcontrol theoryrdquo AIAA Paper 95-1729 1995

[7] A Jameson and J Reuther Control Theory Based Airfoil DesignUsing the Euler Equations Research Institute for AdvancedComputer Science NASA Ames Research Center 1994

[8] M B Giles andN A Pierce ldquoAdjoint equations in CFD dualityboundary conditions and solution behaviourrdquoAIAA Paper vol97 p 1850 1997

[9] M B Giles and N A Pierce ldquoOn the properties of solutionsof the adjoint Euler equationsrdquo Numerical Methods for FluidDynamics pp 1ndash16 1998

[10] M B Giles Discrete Adjoint Approximations with ShocksSpringer New York NY USA 2003

[11] M B Giles M C Duta J Muller and N A Pierce ldquoAlgorithmdevelopments for discrete adjoint methodsrdquo AIAA Journal vol41 no 2 pp 198ndash205 2003

[12] O Baysal and M E Eleshaky ldquoAerodynamic design opti-mization using sensitivity analysis and computational fluiddynamicsrdquo AIAA Journal vol 30 no 3 pp 718ndash725 1992

[13] S Tarsquoasan G Kuruvila and M Salas ldquoAerodynamic designand optimization in one shotrdquo in Proceedings of the 30th AIAAAerospace Sciences Meeting and Exhibit Reno Nev USA 1992

[14] H Cabuk C-H Sung andVModi ldquoAdjoint operator approachto shape design for internal incompressible flowsrdquo in Pro-ceedings of the 3rd International Conference on Inverse DesignConcepts and Optimization in Engineering Sciences (ICIDES-3rsquo91) pp 391ndash404 1991

[15] M Desai and K Ito ldquoOptimal controls of Navier-Stokes equa-tionsrdquo SIAM Journal on Control and Optimization vol 32 no 5pp 1428ndash1446 1994

[16] J Elliott and J Peraire ldquoAerodynamic design using unstructuredmeshesrdquo AIAA Paper 1996

[17] J Elliott and J Peraire ldquoAerodynamic optimization on unstruc-tured meshes with viscous effectsrdquo AIAA Paper vol 97 p 18491997

[18] J Elliott and J Peraire ldquoPractical three-dimensional aerody-namic design and optimization using unstructured meshesrdquoAIAA Journal vol 35 no 9 pp 1479ndash1485 1997


[19] W K Anderson and V Venkatakrishnan ldquoAerodynamic designoptimization on unstructured grids with a continuous adjointformulationrdquo Computers and Fluids vol 28 no 4-5 pp 443ndash480 1999

[20] L Gonzalez E Whitney K Srinivas and J Periaux ldquoMul-tidisciplinary aircraft design and optimisation using a robustevolutionary technique with variable Fidelity modelsrdquo in Pro-ceedings of the 10th AIAAISSMOMultidisciplinary Analysis andOptimization Conference vol 4625 2004

[21] I C Parmee and A H Watson ldquoPreliminary airframe designusing co-evolutionary multiobjective genetic algorithmsrdquo inProceedings of the Genetic and Evolutionary Computation Con-ference pp 1657ndash1665 1999

[22] S Obayashi ldquoMultidisciplinary design optimization of aircraftwing planform based on evolutionary algorithmsrdquo in Proceed-ings of the IEEE International Conference on Systems Man andCybernetics pp 3148ndash3153 October 1998

[23] A Oyama M-S Liou and S Obayashi ldquoTransonic axial-flowblade shape optimization using evolutionary algorithm andthree-dimensional Navier-Stokes solverrdquo in Proceedings of the9th AIAAISSMO Symposium and Exhibit on MultidisciplinaryAnalysis and Optimization Atlanta Ga USA 2002

[24] H-S Chung S Choi and J J Alonso ldquoSupersonic business jetdesign using knowledge-based genetic algorithmwith adaptiveunstructured grid methodologyrdquo in Proceedings of the 21stApplied Aerodynamics Conference 2003

[25] A Jameson and K Ou ldquoOptimization methods in computa-tional fluid dynamicsrdquo inEncyclopedia of Aerospace EngineeringJohn Wiley amp Sons New York NY USA 2010

[26] D Thevenin and G Janiga Optimization and ComputationalFluid Dynamics Springer 2008

[27] B Mohammadi and O Pironneau Applied Shape Optimizationfor Fluids Oxford University Press Oxford UK 2009

[28] P Castonguay and S K Nadarajah ldquoEffect of shape parameter-ization on aerodynamic shape optimizationrdquo in Proceedings ofthe 45th AIAA Aerospace Sciences Meeting and Exhibit pp 8ndash11January 2007

[29] F Mohebbi and M Sellier ldquoOptimal shape design in heattransfer based on body-fitted grid generationrdquo InternationalJournal for Computational Methods in Engineering Science andMechanics vol 14 no 3 pp 227ndash243 2013

[30] F Mohebbi and M Sellier ldquoThree-dimensional optimal shapedesign in heat transfer based on body-fitted grid generationrdquoInternational Journal for Computational Methods in EngineeringScience and Mechanics vol 14 no 6 pp 473ndash490 2013

[31] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinatesystem for field containing any number of arbitrary two-dimensional bodiesrdquo Journal of Computational Physics vol 15no 3 pp 299ndash319 1974

[32] M W Kutta Lifting Forces in Flowing Fluids 1902[33] F Mohebbi and M Sellier ldquoOn the Kutta condition in potential

flow over airfoilrdquo Journal of Aerodynamics vol 2014 Article ID676912 10 pages 2014

[34] J D Anderson Fundamentals of Aerodynamics McGraw-Hill2001

[35] J A Samareh ldquoA survey of shape parameterization techniquesrdquoin NASA Conference Publication pp 333ndash344 Citeseer 1999

[36] D F Rogers An Introduction to NURBS With HistoricalPerspective Morgan Kaufmann Publications 2001

[37] M Hepperle ldquoJavafoilmdashanalysis of airfoilsrdquo 2008 httpwwwmh-aerotoolsdeairfoilsjavafoilhtm

[38] A Jameson ldquoAerodynamic shape optimization using the adjointmethodrdquo Lectures at the Von Karman Institute Brussels Bel-gium 2003

[39] A Jameson L Martinelli and N A Pierce ldquoOptimum aero-dynamic design using the Navier-Stokes equationsrdquoTheoreticaland Computational Fluid Dynamics vol 10 no 1ndash4 pp 213ndash2371998

[40] M H Straathof Shape parameterization in aircraft design aNovelmethod based on B-splines [Dissertation] DelftUniversityof Technology 2012

[41] M B Giles and N A Pierce ldquoAn introduction to the adjointapproach to designrdquo Flow Turbulence and Combustion vol 65no 3-4 pp 393ndash415 2000

[42] W K Anderson and D L Bonhaus ldquoAirfoil design on unstruc-tured grids for turbulent flowsrdquo AIAA Journal vol 37 no 2 pp185ndash191 1999

[43] T D Economon P Francisco and J J Alonso ldquoOptimal shapedesign for open rotor bladesrdquo in Proceedings of the 30th AIAAApplied Aerodynamics Conference pp 1414ndash1436 June 2012

[44] S K NadarajahThe Discrete Adjoint Approach to AerodynamicShape Optimization Citeseer 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


theory to design optimization However within the field ofaeronautical computational fluid dynamics Jameson was thefirst researcher who used the continuous adjoint formulationfor aerodynamic shape optimization in transonic potentialflows andflows governed byEuler equations [4ndash7] Giles et almade considerable contributions to the development of thediscrete adjoint approach [8ndash11] In [10] the adjoint equationsare formulated for the transonic design applications forwhich there are shocks The adjoint equations were alreadyformulated for the incompressible or subsonic flows in whichthe assumption that the original nonlinear flow solution issmooth is valid In [11] a number of algorithm develop-ments are presented for adjoint methods using the ldquodiscreterdquoapproach In continuous adjoint method the original par-tial differential equations are linearized the adjoint partialdifferential equation and appropriate boundary conditionsare formulated and finally the equations are discretizedUnlike the continuous adjoint approach in discrete adjointapproach the partial differential equations are discretizedthe discrete equations are linearized and then the transposeof the linear operator is used to form the adjoint problemThe adjoint equations have also been used by Baysal andEleshaky to infer the optimal design for a scramjet-afterbodyconfigurationwhich yields themaximumaxial thrust [12] andby Tarsquoasan et al to obtain an optimal airfoil shape [13] Baysaland Eleshakyrsquos work was based on a computational fluiddynamics-sensitivity analysis algorithm (two different quasi-analytical approaches the direct method and the adjointvariable method) to solve Euler equations for the inviscidanalysis of the flow Adjoint methods have been applied toincompressible viscous flowproblems byCabuk et al [14] andDesai and Ito [15] Cabuk et al worked on the problem ofdetermining the profile of a channel or duct that provides themaximum static pressure rise by solving the incompressiblelaminar flow governed by the steady state Navier-Stokesequations Early applications of discrete adjoint methods onunstructured meshes can be found in works by Elliott andPeraire in inviscid [16] and viscous flows [17] for 2D and3D [18] configurations In [16] an inverse design procedurefor single- andmultielement airfoils using unstructured gridsand based on the Euler equations is presented The discreteadjoint method is used to compute the sensitivities and theresults are compared with corresponding finite differencevalues It is shown that the use of the adjoint method prac-tically eliminates the dependence of the objective functiongradient computation on the number of design variablesThecontinuous adjoint approach for unstructured grids has beendeveloped by Anderson and Venkatakrishnan [19] In [19]aerodynamic shape optimization on unstructured grids usinga continuous adjoint approach is developed and analyzed forinviscid and viscous flows B spline and Bezier curves areemployed to parameterize the airfoil surface The objectivefunctions considered include drag minimization lift max-imization and matching a specified pressure distributionThe quasi-Newton optimization method is used to obtainthe optimal design Evolutionary algorithms as methods thatdo not need the computation of the gradient have recentlygained much attention in the context of aerodynamic shapeoptimization [20ndash24] Although they are of extremely high

computational cost they have the advantage that they canescape from a ldquolocal minimumrdquo (a major issue in using gra-dient based methods) and have the ability of finding globallyoptimum solutions amongst many local optima [25 26] Adetailed study of many methods in shape optimization influid mechanics is given byMohammadi and Pironneau [27]

The adjoint approach as an alternative to the finitedifference method to compute the gradient of functionalwith respect to the design variables is computationallyvery efficient Therefore as far as the computational cost isconcerned it is the appropriate choice This is the case whenthere are a large number of design variables which makes useof the finite difference method impractical The differencesbetween the adjoint method and the finite difference one (tocompute the gradient of functional with respect to the designvariables) can be summarized as follows

Adjoint Method 119873 design variables 1 flow solution and 1

adjoint calculation

Finite Difference Method 119873 design variables 119873 flow solu-tions Because

120597J

120597120572119895

=

[J (120572119895 + 120575120572119895) minusJ (120572119895)]

120575120572119895

(1)

where J is the objective function and 120572119895 are the designvariables [28] As can be seen aerodynamic shape optimiza-tion with large number of design variables is computationallypractical only when the adjoint method is used However aswill be shown in this paper a novel sensitivity analysis willbe presented which makes use of the finite difference methodcomparable (from computation cost viewpoint) to the use ofadjoint method The numerical algorithm used in this paperis already employed in shape optimization problems in heatconduction [29 30] The numerical algorithm consists ofthree steps namely grid generation and flow equation solverto find the pressure on the airfoil surface sensitivity analysisto compute the gradient of the objective functionwith respectto the design variables and an optimization method tominimize the functional and reach optimum solution

2 Governing Equation

For a two-dimensional incompressible flow a stream func-tion 120595 can be defined such that

119906 =120597120595

120597119910

V = minus120597120595

120597119909

(2)

where 119906 and V are the components of the velocity vector Vthat is V = 119906i + Vj (i and j are the unit vectors in 119909 and119910 directions resp) Combining the above definitions withthe irrotationality condition leads to the following Laplaceequation for the stream function

1205972120595

1205971199092+

1205972120595

1205971199102= 0 (3)


Vprop

prop

prop

prop

prop

y

x

VpropVprop

Vprop

120595 = constant




119906 =120597120595

120597119910= 119881infin cos120572

V = minus120597120595

120597119909= 119881infin sin120572

(4)


120597120595

120597119904= 0 or 120595 = constant (5)



119862119901 =119901 minus 119901infin

(12) 120588infin1198812infin

= 1 minus (119881

119881infin

)

2

(6)

(MN1 + N2 + 3) N2

N2

N1

N1

N3

(MN1 + 2)

D

F

C

E

(M 1)

(MN)

N = 2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4) (MN1 + 2N2 + N3 + 5)

A (1 1)

H (1 N)

M

M

B

G

yx



120588infin = 123 kgm3


(7)









N = 2N1 + 2N2 + N3 + 6

H M G

F

E

D

C

BMA

N

(MN)

(MN1 + 2N2 + N3 + 5)

(MN1 + N2 + N3 + 4)

(MN1 + N2 + 3)

(MN1 + 2)

(M 1)120578

120585

(1 N)

(1 1)




1205952 = 1205951 + (1199102 minus 1199101) 119881infin cos120572 (8)



120572120595120585120585 minus 2120573120595120585120578 + 120574120595120578120578

= minus1198692(119875 (120585 120578) 120595120585 + 119876 (120585 120578) 120595120578)

(10)

where

120572 = 1199092120578 + 1199102120578

120573 = 119909120585119909120578 + 119910120585119910120578

120574 = 1199092120585 + 1199102120585


(11)







119906119894119895 =120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895

(120595120585)119894119895+ (119909120585)119894119895

(120595120578)119894119895] (12)

V119894119895 = minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895

(120595120585)119894119895minus (119910120585)119894119895

(120595120578)119894119895] (13)


119881119894119895 = radic1199062119894119895 + V2119894119895 (14)


119901119894119895 =1

2120588 (1198812infin minus 119881

2119894119895) + 119901infin (15)







0 05 1

0

02

04

06

0 01 02

0

005

01

015

09 1 11

0

005

01

minus02

minus04

minus06

YY

Y

X

X X

minus005

minus01

minus015

minus005

minus01

minus015minus01







119875 (119905) =

119899

sum

119894=0

119861119894119869119899119894 (119905) (16)

where

119869119899119894 (119905) =119899

119894 (119899 minus 119894)119905119894(1 minus 119905)

119899minus119894 (17)




0 02 04 06 08 1

0

1

minus5

minus4

minus3

minus2

minus1

Cp

xc


minus04

minus03

minus02

minus01

0

01

05 10

Cptimes10

minus1

NACA 0012 airfoil120572 = 9∘


2 orderpanel

method

Vinfin

9∘

xc




0 02 04 06 08 1

0

1



minus6

minus5

minus4

minus3

minus2

minus1

Cp

xc





119875 (119905) =

10

sum

119894=0

11986111989411986910119894 (119905)

= 1198610119869100 (119905) + sdot sdot sdot + 119861101198691010 (119905)

= 1198610

10

0 (10 minus 0)1199050(1 minus 119905)

10minus0


10

10 (10 minus 10)11990510(1 minus 119905)

10minus10

(18)

Therefore

119875 (119905) = 1198610 (1 minus 119905)10

+ 119861110119905 (1 minus 119905)9+ 119861245119905

2(1 minus 119905)

8

+ 11986131201199053(1 minus 119905)

7+ 1198614210119905

4(1 minus 119905)

6

+ 11986152521199055(1 minus 119905)

5+ 1198616210119905

6(1 minus 119905)

4

+ 11986171201199057(1 minus 119905)

3+ 119861845119905

8(1 minus 119905)

2

+ 1198619101199059(1 minus 119905)

1+ 11986110119905

10

(19)




[119875 (119905)] = [119869 (119905)] [119861] (20)








[119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2


(23)

Thus


minus1

sdot [119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2

(24)




plusmn119910119905 =119905

02[02969radic119909 minus 01260119909

minus035161199092+ 02843119909

3minus 01015119909

4]

(25)

0 02 04 06 08 1

0

01

02

03

04


0 005 01 015

0

005

09 095 1

0001002003

minus01

minus02

minus03

minus04

minus05

minus005

minus001

minus002

minus003

X

X

X

Y

Y

Y


0 02 04 06 08 1

0

01

02

03

04

05

0 005 01 015 02

0

005

0985 099 0995 1

0

0002

0004minus01

minus02

minus03

minus04

minus05

minus005

minus0002

minus0004

X

X

X

Y

Y

Y












J =

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))2 (26)



J =

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)2 (27)

0 02 04 06 08 1

0

02

04

06

Airfoil


minus02

minus04

(1 N + 1

2)

(1 N)

Y

X

F(1j)

(1 1)


0 02 04 06 08 1

0

02


NACA 0028

09998 1000265

00027000275

00028000285

00029000295

0003

minus02

minus04

minus06

B10 U

B10 U

B0 U

B1 UB2 U

B8 U

B9 U

B10 LB0 L

B1 L

B2 L

B8 L

Y

Y

X

X






0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X






J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)




1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X


0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X



1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)


1015840) 11991011198951015840




= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Vprop

prop

prop

prop

prop

y

x

VpropVprop

Vprop

120595 = constant




119906 =120597120595

120597119910= 119881infin cos120572

V = minus120597120595

120597119909= 119881infin sin120572

(4)


120597120595

120597119904= 0 or 120595 = constant (5)



119862119901 =119901 minus 119901infin

(12) 120588infin1198812infin

= 1 minus (119881

119881infin

)

2

(6)

(MN1 + N2 + 3) N2

N2

N1

N1

N3

(MN1 + 2)

D

F

C

E

(M 1)

(MN)

N = 2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4) (MN1 + 2N2 + N3 + 5)

A (1 1)

H (1 N)

M

M

B

G

yx



120588infin = 123 kgm3


(7)









N = 2N1 + 2N2 + N3 + 6

H M G

F

E

D

C

BMA

N

(MN)

(MN1 + 2N2 + N3 + 5)

(MN1 + N2 + N3 + 4)

(MN1 + N2 + 3)

(MN1 + 2)

(M 1)120578

120585

(1 N)

(1 1)




1205952 = 1205951 + (1199102 minus 1199101) 119881infin cos120572 (8)



120572120595120585120585 minus 2120573120595120585120578 + 120574120595120578120578

= minus1198692(119875 (120585 120578) 120595120585 + 119876 (120585 120578) 120595120578)

(10)

where

120572 = 1199092120578 + 1199102120578

120573 = 119909120585119909120578 + 119910120585119910120578

120574 = 1199092120585 + 1199102120585


(11)







119906119894119895 =120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895

(120595120585)119894119895+ (119909120585)119894119895

(120595120578)119894119895] (12)

V119894119895 = minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895

(120595120585)119894119895minus (119910120585)119894119895

(120595120578)119894119895] (13)


119881119894119895 = radic1199062119894119895 + V2119894119895 (14)


119901119894119895 =1

2120588 (1198812infin minus 119881

2119894119895) + 119901infin (15)







0 05 1

0

02

04

06

0 01 02

0

005

01

015

09 1 11

0

005

01

minus02

minus04

minus06

YY

Y

X

X X

minus005

minus01

minus015

minus005

minus01

minus015minus01







119875 (119905) =

119899

sum

119894=0

119861119894119869119899119894 (119905) (16)

where

119869119899119894 (119905) =119899

119894 (119899 minus 119894)119905119894(1 minus 119905)

119899minus119894 (17)




0 02 04 06 08 1

0

1

minus5

minus4

minus3

minus2

minus1

Cp

xc


minus04

minus03

minus02

minus01

0

01

05 10

Cptimes10

minus1

NACA 0012 airfoil120572 = 9∘


2 orderpanel

method

Vinfin

9∘

xc




0 02 04 06 08 1

0

1



minus6

minus5

minus4

minus3

minus2

minus1

Cp

xc





119875 (119905) =

10

sum

119894=0

11986111989411986910119894 (119905)

= 1198610119869100 (119905) + sdot sdot sdot + 119861101198691010 (119905)

= 1198610

10

0 (10 minus 0)1199050(1 minus 119905)

10minus0


10

10 (10 minus 10)11990510(1 minus 119905)

10minus10

(18)

Therefore

119875 (119905) = 1198610 (1 minus 119905)10

+ 119861110119905 (1 minus 119905)9+ 119861245119905

2(1 minus 119905)

8

+ 11986131201199053(1 minus 119905)

7+ 1198614210119905

4(1 minus 119905)

6

+ 11986152521199055(1 minus 119905)

5+ 1198616210119905

6(1 minus 119905)

4

+ 11986171201199057(1 minus 119905)

3+ 119861845119905

8(1 minus 119905)

2

+ 1198619101199059(1 minus 119905)

1+ 11986110119905

10

(19)




[119875 (119905)] = [119869 (119905)] [119861] (20)








[119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2


(23)

Thus


minus1

sdot [119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2

(24)




plusmn119910119905 =119905

02[02969radic119909 minus 01260119909

minus035161199092+ 02843119909

3minus 01015119909

4]

(25)

0 02 04 06 08 1

0

01

02

03

04


0 005 01 015

0

005

09 095 1

0001002003

minus01

minus02

minus03

minus04

minus05

minus005

minus001

minus002

minus003

X

X

X

Y

Y

Y


0 02 04 06 08 1

0

01

02

03

04

05

0 005 01 015 02

0

005

0985 099 0995 1

0

0002

0004minus01

minus02

minus03

minus04

minus05

minus005

minus0002

minus0004

X

X

X

Y

Y

Y












J =

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))2 (26)



J =

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)2 (27)

0 02 04 06 08 1

0

02

04

06

Airfoil


minus02

minus04

(1 N + 1

2)

(1 N)

Y

X

F(1j)

(1 1)


0 02 04 06 08 1

0

02


NACA 0028

09998 1000265

00027000275

00028000285

00029000295

0003

minus02

minus04

minus06

B10 U

B10 U

B0 U

B1 UB2 U

B8 U

B9 U

B10 LB0 L

B1 L

B2 L

B8 L

Y

Y

X

X






0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X






J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)




1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X


0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X



1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)


1015840) 11991011198951015840




= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










N = 2N1 + 2N2 + N3 + 6

H M G

F

E

D

C

BMA

N

(MN)

(MN1 + 2N2 + N3 + 5)

(MN1 + N2 + N3 + 4)

(MN1 + N2 + 3)

(MN1 + 2)

(M 1)120578

120585

(1 N)

(1 1)




1205952 = 1205951 + (1199102 minus 1199101) 119881infin cos120572 (8)



120572120595120585120585 minus 2120573120595120585120578 + 120574120595120578120578

= minus1198692(119875 (120585 120578) 120595120585 + 119876 (120585 120578) 120595120578)

(10)

where

120572 = 1199092120578 + 1199102120578

120573 = 119909120585119909120578 + 119910120585119910120578

120574 = 1199092120585 + 1199102120585


(11)







119906119894119895 =120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895

(120595120585)119894119895+ (119909120585)119894119895

(120595120578)119894119895] (12)

V119894119895 = minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895

(120595120585)119894119895minus (119910120585)119894119895

(120595120578)119894119895] (13)


119881119894119895 = radic1199062119894119895 + V2119894119895 (14)


119901119894119895 =1

2120588 (1198812infin minus 119881

2119894119895) + 119901infin (15)







0 05 1

0

02

04

06

0 01 02

0

005

01

015

09 1 11

0

005

01

minus02

minus04

minus06

YY

Y

X

X X

minus005

minus01

minus015

minus005

minus01

minus015minus01







119875 (119905) =

119899

sum

119894=0

119861119894119869119899119894 (119905) (16)

where

119869119899119894 (119905) =119899

119894 (119899 minus 119894)119905119894(1 minus 119905)

119899minus119894 (17)




0 02 04 06 08 1

0

1

minus5

minus4

minus3

minus2

minus1

Cp

xc


minus04

minus03

minus02

minus01

0

01

05 10

Cptimes10

minus1

NACA 0012 airfoil120572 = 9∘


2 orderpanel

method

Vinfin

9∘

xc




0 02 04 06 08 1

0

1



minus6

minus5

minus4

minus3

minus2

minus1

Cp

xc





119875 (119905) =

10

sum

119894=0

11986111989411986910119894 (119905)

= 1198610119869100 (119905) + sdot sdot sdot + 119861101198691010 (119905)

= 1198610

10

0 (10 minus 0)1199050(1 minus 119905)

10minus0


10

10 (10 minus 10)11990510(1 minus 119905)

10minus10

(18)

Therefore

119875 (119905) = 1198610 (1 minus 119905)10

+ 119861110119905 (1 minus 119905)9+ 119861245119905

2(1 minus 119905)

8

+ 11986131201199053(1 minus 119905)

7+ 1198614210119905

4(1 minus 119905)

6

+ 11986152521199055(1 minus 119905)

5+ 1198616210119905

6(1 minus 119905)

4

+ 11986171201199057(1 minus 119905)

3+ 119861845119905

8(1 minus 119905)

2

+ 1198619101199059(1 minus 119905)

1+ 11986110119905

10

(19)




[119875 (119905)] = [119869 (119905)] [119861] (20)








[119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2


(23)

Thus


minus1

sdot [119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2

(24)




plusmn119910119905 =119905

02[02969radic119909 minus 01260119909

minus035161199092+ 02843119909

3minus 01015119909

4]

(25)

0 02 04 06 08 1

0

01

02

03

04


0 005 01 015

0

005

09 095 1

0001002003

minus01

minus02

minus03

minus04

minus05

minus005

minus001

minus002

minus003

X

X

X

Y

Y

Y


0 02 04 06 08 1

0

01

02

03

04

05

0 005 01 015 02

0

005

0985 099 0995 1

0

0002

0004minus01

minus02

minus03

minus04

minus05

minus005

minus0002

minus0004

X

X

X

Y

Y

Y












J =

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))2 (26)



J =

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)2 (27)

0 02 04 06 08 1

0

02

04

06

Airfoil


minus02

minus04

(1 N + 1

2)

(1 N)

Y

X

F(1j)

(1 1)


0 02 04 06 08 1

0

02


NACA 0028

09998 1000265

00027000275

00028000285

00029000295

0003

minus02

minus04

minus06

B10 U

B10 U

B0 U

B1 UB2 U

B8 U

B9 U

B10 LB0 L

B1 L

B2 L

B8 L

Y

Y

X

X






0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X






J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)




1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X


0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X



1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)


1015840) 11991011198951015840




= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1

0

02

04

06

0 01 02

0

005

01

015

09 1 11

0

005

01

minus02

minus04

minus06

YY

Y

X

X X

minus005

minus01

minus015

minus005

minus01

minus015minus01







119875 (119905) =

119899

sum

119894=0

119861119894119869119899119894 (119905) (16)

where

119869119899119894 (119905) =119899

119894 (119899 minus 119894)119905119894(1 minus 119905)

119899minus119894 (17)




0 02 04 06 08 1

0

1

minus5

minus4

minus3

minus2

minus1

Cp

xc


minus04

minus03

minus02

minus01

0

01

05 10

Cptimes10

minus1

NACA 0012 airfoil120572 = 9∘


2 orderpanel

method

Vinfin

9∘

xc




0 02 04 06 08 1

0

1



minus6

minus5

minus4

minus3

minus2

minus1

Cp

xc





119875 (119905) =

10

sum

119894=0

11986111989411986910119894 (119905)

= 1198610119869100 (119905) + sdot sdot sdot + 119861101198691010 (119905)

= 1198610

10

0 (10 minus 0)1199050(1 minus 119905)

10minus0


10

10 (10 minus 10)11990510(1 minus 119905)

10minus10

(18)

Therefore

119875 (119905) = 1198610 (1 minus 119905)10

+ 119861110119905 (1 minus 119905)9+ 119861245119905

2(1 minus 119905)

8

+ 11986131201199053(1 minus 119905)

7+ 1198614210119905

4(1 minus 119905)

6

+ 11986152521199055(1 minus 119905)

5+ 1198616210119905

6(1 minus 119905)

4

+ 11986171201199057(1 minus 119905)

3+ 119861845119905

8(1 minus 119905)

2

+ 1198619101199059(1 minus 119905)

1+ 11986110119905

10

(19)




[119875 (119905)] = [119869 (119905)] [119861] (20)








[119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2


(23)

Thus


minus1

sdot [119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2

(24)




plusmn119910119905 =119905

02[02969radic119909 minus 01260119909

minus035161199092+ 02843119909

3minus 01015119909

4]

(25)

0 02 04 06 08 1

0

01

02

03

04


0 005 01 015

0

005

09 095 1

0001002003

minus01

minus02

minus03

minus04

minus05

minus005

minus001

minus002

minus003

X

X

X

Y

Y

Y


0 02 04 06 08 1

0

01

02

03

04

05

0 005 01 015 02

0

005

0985 099 0995 1

0

0002

0004minus01

minus02

minus03

minus04

minus05

minus005

minus0002

minus0004

X

X

X

Y

Y

Y












J =

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))2 (26)



J =

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)2 (27)

0 02 04 06 08 1

0

02

04

06

Airfoil


minus02

minus04

(1 N + 1

2)

(1 N)

Y

X

F(1j)

(1 1)


0 02 04 06 08 1

0

02


NACA 0028

09998 1000265

00027000275

00028000285

00029000295

0003

minus02

minus04

minus06

B10 U

B10 U

B0 U

B1 UB2 U

B8 U

B9 U

B10 LB0 L

B1 L

B2 L

B8 L

Y

Y

X

X






0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X






J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)




1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X


0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X



1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)


1015840) 11991011198951015840




= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1

0

1

minus5

minus4

minus3

minus2

minus1

Cp

xc


minus04

minus03

minus02

minus01

0

01

05 10

Cptimes10

minus1

NACA 0012 airfoil120572 = 9∘


2 orderpanel

method

Vinfin

9∘

xc




0 02 04 06 08 1

0

1



minus6

minus5

minus4

minus3

minus2

minus1

Cp

xc





119875 (119905) =

10

sum

119894=0

11986111989411986910119894 (119905)

= 1198610119869100 (119905) + sdot sdot sdot + 119861101198691010 (119905)

= 1198610

10

0 (10 minus 0)1199050(1 minus 119905)

10minus0


10

10 (10 minus 10)11990510(1 minus 119905)

10minus10

(18)

Therefore

119875 (119905) = 1198610 (1 minus 119905)10

+ 119861110119905 (1 minus 119905)9+ 119861245119905

2(1 minus 119905)

8

+ 11986131201199053(1 minus 119905)

7+ 1198614210119905

4(1 minus 119905)

6

+ 11986152521199055(1 minus 119905)

5+ 1198616210119905

6(1 minus 119905)

4

+ 11986171201199057(1 minus 119905)

3+ 119861845119905

8(1 minus 119905)

2

+ 1198619101199059(1 minus 119905)

1+ 11986110119905

10

(19)




[119875 (119905)] = [119869 (119905)] [119861] (20)








[119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2


(23)

Thus


minus1

sdot [119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2

(24)




plusmn119910119905 =119905

02[02969radic119909 minus 01260119909

minus035161199092+ 02843119909

3minus 01015119909

4]

(25)

0 02 04 06 08 1

0

01

02

03

04


0 005 01 015

0

005

09 095 1

0001002003

minus01

minus02

minus03

minus04

minus05

minus005

minus001

minus002

minus003

X

X

X

Y

Y

Y


0 02 04 06 08 1

0

01

02

03

04

05

0 005 01 015 02

0

005

0985 099 0995 1

0

0002

0004minus01

minus02

minus03

minus04

minus05

minus005

minus0002

minus0004

X

X

X

Y

Y

Y












J =

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))2 (26)



J =

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)2 (27)

0 02 04 06 08 1

0

02

04

06

Airfoil


minus02

minus04

(1 N + 1

2)

(1 N)

Y

X

F(1j)

(1 1)


0 02 04 06 08 1

0

02


NACA 0028

09998 1000265

00027000275

00028000285

00029000295

0003

minus02

minus04

minus06

B10 U

B10 U

B0 U

B1 UB2 U

B8 U

B9 U

B10 LB0 L

B1 L

B2 L

B8 L

Y

Y

X

X






0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X






J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)




1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X


0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X



1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)


1015840) 11991011198951015840




= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[119875 (119905)] = [119869 (119905)] [119861] (20)








[119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2


(23)

Thus


minus1

sdot [119869 (119905)]119879(119899+1)times119898[119875 (119905)]119898times2

(24)




plusmn119910119905 =119905

02[02969radic119909 minus 01260119909

minus035161199092+ 02843119909

3minus 01015119909

4]

(25)

0 02 04 06 08 1

0

01

02

03

04


0 005 01 015

0

005

09 095 1

0001002003

minus01

minus02

minus03

minus04

minus05

minus005

minus001

minus002

minus003

X

X

X

Y

Y

Y


0 02 04 06 08 1

0

01

02

03

04

05

0 005 01 015 02

0

005

0985 099 0995 1

0

0002

0004minus01

minus02

minus03

minus04

minus05

minus005

minus0002

minus0004

X

X

X

Y

Y

Y












J =

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))2 (26)



J =

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)2 (27)

0 02 04 06 08 1

0

02

04

06

Airfoil


minus02

minus04

(1 N + 1

2)

(1 N)

Y

X

F(1j)

(1 1)


0 02 04 06 08 1

0

02


NACA 0028

09998 1000265

00027000275

00028000285

00029000295

0003

minus02

minus04

minus06

B10 U

B10 U

B0 U

B1 UB2 U

B8 U

B9 U

B10 LB0 L

B1 L

B2 L

B8 L

Y

Y

X

X






0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X






J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)




1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X


0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X



1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)


1015840) 11991011198951015840




= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















J =

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))2 (26)



J =

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)2 (27)

0 02 04 06 08 1

0

02

04

06

Airfoil


minus02

minus04

(1 N + 1

2)

(1 N)

Y

X

F(1j)

(1 1)


0 02 04 06 08 1

0

02


NACA 0028

09998 1000265

00027000275

00028000285

00029000295

0003

minus02

minus04

minus06

B10 U

B10 U

B0 U

B1 UB2 U

B8 U

B9 U

B10 LB0 L

B1 L

B2 L

B8 L

Y

Y

X

X






0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X






J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)




1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X


0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X



1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)


1015840) 11991011198951015840




= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1

0

02

04

06

1

2

minus02

minus04

Y

X






J =

2

sum

119894=1

(119875119894 minus 119875119889(119894))2 (28)




1015840= 2 119873minus1 119895

1015840= (119873 + 1) 2)

0 05 1

0

02

04

minus02

minus04

Y

X


0 02 04 06 08 1 12

0

02

04

06

08

minus02minus04

minus02

minus04

minus06

Y

X



1205971198751119895

12059711991011198951015840=

1198751119895 (11991011198951015840 + 12057611991011198951015840) minus 1198751119895 (11991011198951015840)

12057611991011198951015840 (29)


1015840) 11991011198951015840




= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












= 2 119873 minus 1 1198951015840



120597J

1205971199101119897

= 2

119873minus1

sum

119895=2119895 = (119873+1)2

(1198751119895 minus 119875119889(1119895))

1205971198751119895

1205971199101119897

(30)



Ja119910 =

[[[[[[[[[[[

[

12059711987512

12059711991012

12059711987512

12059711991013

12059711987512

12059711991014


1205971199101119873minus1

12059711987513

12059711991012

12059711987513

12059711991013

12059711987513

12059711991014


1205971199101119873minus1

d

1205971198751119873minus1

12059711991012

1205971198751119873minus1

12059711991013

1205971198751119873minus1

12059711991014


1205971199101119873minus1

]]]]]]]]]]]

]

(31)


1205971198751119895

1205971199091119897

=

1205971198751119895

120597120585

120597120585

1205971199091119897

+

1205971198751119895

120597120578

120597120578

1205971199091119897

(32)

1205971198751119895

1205971199101119897

=

1205971198751119895

120597120585

120597120585

1205971199101119897

+

1205971198751119895

120597120578

120597120578

1205971199101119897

(33)



1205971198751119895

1205971199101119897

=1

119869[minus(119909120578)1119897

(119875120585)1119895+ (119909120585)1119897

(119875120578)1119895] (34)



(119875120585)1119895=

minus31198751119895 + 41198752119895 minus 1198753119895

2 (35)

(119875120578)1119895=

1198751119895+1 minus 1198751119895minus1

2 (36)

(119909120585)1119897=

minus31199091119897 + 41199092119897 minus 1199093119897

2 (37)

(119909120578)1119897=

1199091119897+1 minus 1199091119897minus1

2(38)



120597J

120597119861119910119897

= 2

2119898minus4

sum

119894=1

(119875119894119861 minus 119875119894119861119889)120597119875119894119861

120597119861119910119897

(39)


120597119875119894119861

120597119861119910119897

=120597119875119894119861

1205971199101198941015840119861

1205971199101198941015840119861

120597119861119910119897

(40)






= 1198611199109119880 1198611199102 = 1198611199108119880

1198611199109

= 1198611199101119880and 11986111991010

= 1198611199101119871 11986111991011 = 1198611199102119871

11986111991018 = 1198611199109119871





Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Ja =[[[

[

1205971198751

12059711991011205971198752

1205971199101

1205971198751

1205971199102

1205971198752

1205971199102

]]]

]

(41)











120574(119896)

=


1003817100381710038171003817nablaJ(119896minus1)1003817100381710038171003817

2

=



(42)


following

d(119896) = nablaJ(119896)

+ 120574(119896)d(119896minus1) (43)


120573(119896)

=

[Ja(119896)d(119896)]119879[1198751119895 minus 119875119889(1119895)]

[Ja(119896)d(119896)]119879 [Ja(119896)d(119896)] (44)


y(119896+1) = y(119896) minus 120573(119896)d(119896) (45)















y(119896+1) = y(119896) minus 120573(119896)S(119896) (46)





120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120572




B(119896+1) = B(119896) + (1 +

(g(119896))119879B(119896)g(119896)

(d(119896))119879g(119896))

d(119896)(d(119896))119879

(d(119896))119879g(119896)

minus

d(119896)(g(119896))119879B(119896)

(d(119896))119879g(119896)minus

B(119896)g(119896)(d(119896))119879

(d(119896))119879g(119896)

(47)

where

d(119896) = y(119896+1) minus y(119896) = minus120573(119896)S(119896)

g(119896) = nablaJ(119896+1)


(48)


8 Results




0 02 04 06 08 1

0

02

04


0 002 004 006 008 01

0001002003004005006

0748 075 0752 0754 0756

0028

003

0032

0034

0036

0165 017 0175 018 0185 098 0985 099 0995 1

0

0005minus02

minus04


minus0025

minus003

minus0035

minus004

minus0005

Y

Y Y

YY

X

XX

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X





5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5E + 06

45E + 06

4E + 06

35E + 06

3E + 06

Obj

ectiv

e fun

ctio

n


0 1

04

02

0

minus02

minus04

Y

X

05






06

04

02

0

minus02

minus04

minus06

minus08

Y

X

0 05 1


0 02 04 06 08 1

0

02

04


004 006 008 01 012 014006

007

008

009

01

011

012

092 094 096 098 1 102

0

001

minus001

minus02

minus04

Y

Y

Y

X

X

X




120572












10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















10074507 5585758 35m and 30 s 35m and 34 s 445



120572




0 05 1

0

005

01

015

Optimal shape

Initial shape

Initial shape

minus005

minus01

minus015

Y

X



















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















898639 368148 9m and 32 s 9m and 43 s 59



120572







ReductioninJ




120572






0 05 1

0

02

04

06

08


097 0975 098 0985 099 0995 1 1005

0

001

002

minus02

minus001

Y

Y

X

X





ReductioninJ

CG 77359745 1548 31 2m 14 s sim100

9 Adjoint Method
















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















120572







ReductioninJ

CG 798688034 115 21 1m 08 s sim100










105E + 07

1E + 07

95E + 06

9E + 06

85E + 06

8E + 06

75E + 06

7E + 06

65E + 06

6E + 06

55E + 06

Obj

ectiv

e fun

ctio

n



0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1

0

02

04

minus02

minus04

Y

X


0 05 1 15

0

05

1

minus05

minus1

minus05

Y

X




J = J (WX119863)

R = R (WX119863) = 0

(50)


119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

(51)



119889R

119889X119863=

120597R

120597X119863+

120597R

120597W120597W120597X119863

= 0 (52)



0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1

0

02

04

06

08


0 005 01 015

0

005

01

0985 099 0995 1

0

0005

minus005

minus0005

minus001

Y

Y Y

X

X X


0 05 1

0

005

01

Initial shape

Optimal shape

Optimal shape

minus005

minus01

Y

X



119889J

119889X119863=

120597J

120597X119863+

120597J

120597W120597W120597X119863

minus Ψ119879

=0⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597R

120597X119863+

120597R

120597W120597W120597X119863

(53)

0 02 04 06 08 1

0

02

04

06


094 095 096 097 098 099 1

0

001

Y

Y

X

X

minus02

minus001



0

8E + 06

6E + 06

4E + 06

2E + 06

Obj

ectiv

e fun

ctio

n



119889J

119889X119863= (

120597J

120597X119863minus Ψ119879 120597R

120597X119863)

+ (120597J

120597Wminus Ψ119879 120597R

120597W)

120597W120597X119863

(54)

If

120597J

120597Wminus Ψ119879 120597R

120597W= 0 (55)


0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1

0

02

04

06

08


09985 0999 09995 1 10005

0

00002

00004

00006

00008

0001


minus02

minus00002

minus00004

minus00006

X

XY

Y


0 02 04 06 08 1

0

02

04

06

08

Standard

001545 00155 001555

002055

00206

002065

00207

Standard

099985 09999 099995 1

000114000116000118

00012000122000124000126000128



Bezier degree = 6

Bezier degree = 10

Bezier degree = 6

Bezier degree = 10

X

XX

Y

Y Y




119889J

119889X119863=

120597J

120597X119863minus Ψ119879 120597R

120597X119863 (56)


0 05 1

0

02

04

0006445 000645

OS (BFGS)

OS (CG)

IS

0283652 0283656 028366

OS (CG)

OS (BFGS)

IS

099992 099996 1

OS (BFGS)

OS (CG)IS

minus02

minus04

X

X

X X

Y

Y

YY


minus0059954

minus0059956

minus0059958


minus000131

minus0013796

minus0013798

minus00138

minus0013802



Using CG645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n





Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Using BFGS64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

Obj

ectiv

e fun

ctio

n



CGBFGS

65E + 06

645E + 06

64E + 06

635E + 06

63E + 06

625E + 06

62E + 06

615E + 06

61E + 06

605E + 06

6E + 06

Obj

ectiv

e fun

ctio

n



0 02 04 06 08 1

0

02

04


0283 02835 0284 02845 028500894

00896

00898

009

00902

00904

minus02

minus04

X

X

Y

Y



0

7E + 06

6E + 06

5E + 06

4E + 06

3E + 06

2E + 06

1E + 06

Obj

ectiv

e fun

ctio

n



10 Conclusion



0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1

0

02

04

06


029995 03 030005017498017499

0175017501017502017503017504

minus02

minus04

X

X

Y

Y



0

Obj

ectiv

e fun

ctio

n

8E + 06

6E + 06

4E + 06

2E + 06






References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article aerodynamic optimal shape design based on body-fitted grid...

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