optimization of sailplane wing parameter

MAJOR ASSIGNMENT REPORT

OPTIMIZATION OF SAILPLANE WINGS TO FIT INDONESIA’S

ATMOSPHERE PROPERTIES

Submitted as Requirement for Completing Engineering Optimization Course

Hendi Aji Pratama

23613300

Advisor:

Dr. Rais Zain

AERONAUTICS AND ASTRONAUTICS ENGINEERING

FACULTY OF MECHANICAL AND AEROSPACE ENGINEERING

INSTITUTE TECHNOLOGY OF BANDUNG

2014

CHAPTER 1

INTRODUCTION

1.1 Background

One of Indonesia Aerosport activity is gliding or soaring with a sailplane. The fact that

Indonesia only has one type of sailplane which is Schweizer SGS 1-26 that are

manufactured at 1950 make the achievement of its community is very limited. The

main problem of SGS 1-26 is not because this sailplane is bad, but it is not quite

compatible with Indonesia climate and atmospheric condition. SGS 1-26 may perform

greatly in another side of the world, but in Indonesia, it’s just not good enough.

This is the list the specification and performance of the SGS 1-26:

The rate of descent of SGS 1-26 may still climb the thermal updraft of Indonesia,but it

is not low enough. Thermal updraft only occurs in a few minutes, so climbing the

thermal as fast and as high as possible is quite a mandatory. SGS 1-26 performance

cannot help the sailplane pilot to reach the peak of Indonesia’s thermal. Based on this

problem, a sailplane need to be designed, to fit the atmosphere and thermal updraft

properties of Indonesia.

Figure 1.1 Schweizer SGS 1-26

The design need to have Rate of Descent / Sink as low as possible, so it can climb

thermal effectively. The Glide Ratio needs to as high as possible so it can fly further.

The requirement of the design is able to climb Indonesia’s weak and narrow thermal .

Indonesia’s thermal mostly have 200 – 300 meter diameter and 5000 m in height. The

strength of the thermal is varied from 0.5-3.0 m/s. That makes the glider needs to

have rate of sink less than 0.5 m/s , just to barely climb the thermal.

To answer the requirement, optimum design need to be found. The optimization are

targeted to find the optimum physical properties that related to the requirements.

Properties those related to rate of sink and glide ratio are wing Aspect Ratio, wing

area,lift coefficient CL,drag coefficient CD,and weight of the aircraft. To simplify the

problem, the parameter those will be optimized are wing area, aspect ratio, and

weight.

1.2 Purpose and Goals

The purpose of this project is to perform an optimization to sailplane design

parameter that will result in good performance of the sailplane,which is indicated by

minimum value of Rate of Descent. The goal of this project is to achive optimum design

parameter and minimum value of Rate of Descent that way lower than Indonesia’s

thermal strength.

CHAPTER 2

PROBLEM FORMULATION

2.1 Objective Function and Optimization Parameter

The objective of this optimization is to found the minimum value of Sinking Speed /

Rate of Descent which can formulated as following equation:

𝑉𝑠𝑖𝑛𝑘 =𝐶𝐷

𝐶𝐿

√2𝑊

𝜌𝑆𝐶𝐿𝑐𝑜𝑠∅ … (1)

The fact that to achieve rate of sink, The value of W must be greater than L, it can

become a constraint:

𝐿 < 𝑊 … (2)

To simplify the optimization, constraint (2) can be turn into some equation that help

completing the objective function:

𝐿 + 𝛿 = 𝑊 . . (3)

And then using the equation of lift:

𝐿 =1

2𝜌𝑣2𝑆𝐶𝐿 … (4)

Equation (3) become:

𝐶𝐿 =2(𝑊 − 𝛿)

𝜌𝑉2𝑆 … (5)

Another equation that may helping this optimization is the drag coefficient equation:

𝐶𝐷 = 𝐶𝐷0 +𝐶𝐿

2

𝜋 𝐴 𝑒… (6)

In which, the value of CL can be calculated using equation (5). The value of can be

calculated with the function of A as shown in this equation:

𝑒 = 1.78(1 − 0.045 𝐴0.68) − 0.64) … (7)

To find the value of V in equation (5), equation of turning radius as shown below,will

be also put into the objective function:

𝑅 =𝑉2

𝑔 𝑡𝑎𝑛𝜑… (8)

The Objective function then can be completed using substitution of equation (5) ,

(6), (7) and (8) into equation 1. From that substitution, then using assumption:

- Air density, ρ = 1.225 kg / m3

- Drag Coefficient, CD0 = 0.0125

- Banking Angle, Φ = 30⁰

The equation then left 5 variables that will used as the optimization parameter:

𝑋 = [ 𝑥1 𝑥2 𝑥3 𝑥4 𝑥5] = [𝑆 𝐴 𝑊 𝑅 𝛿]

2.2 Goal and Constraint

The main goal of this optimization is parameter of glider’s wing that proven can be

used to climb a thermal updraft that has vertical speed of 0.5 m/s - 3.0 m/s. Then the

optimization is success if at least:

0.5 ≤ 𝐹(𝑥) ≤ 3.0

To narrow the optimization, these are the constraint that will be used in this

optimization:

- Glide Ratio: 10 ≤ 𝐶𝐿/𝐶𝐷 ≤ 150

- Wing Area : 10 ≤ S ≤ 30

- Aspect Ratio: 10 ≤ A ≤ 25

- Glider’s weight, 220 kg ≤ W ≤ 350 kg

- Turn Radius, 30 m ≤ R ≤ 150 m

- −1000 ≤ 𝛿 ≤ −0.01

CHAPTER 3

OPTIMIZATION

3.1 Optimization Process

Optimization process are done using 3 solver that available in Matlab R2013a: Genetic

Algorithm, Simulated Annealing, and Constrained Minimization (fmincon). All of the

solver will have 3 initial point, which are:

- Outside Boundary : X = [100 100 10000 400 -7000]

- Inside Boundary 1 : X = [20 20 3000 100 -500]

- Inside Boundary 2 : X = [15 11 2500 50 -10]

Total there will be 5 computation that will be done in this optimization. Genetic

Algorithm and fmincon will be executed twice,once using Glide Ratio constraint (see

section 2), and another one will be without Glide Ratio constraint. Simulated

Annealing will executed without Glide Ratio Constraint, because currently, Matlab’s

simulated annealing only support bound constrainted optimization. These are the

optimization step of the 3 solver:

3.1.1 Constrained Optimization (fmincon)

The computation using fmincon are using following code:

a. With Glide Ratio constraint:

function [history,searchdir] = runfmincon

history.x = [];

history.fval = [];

searchdir = [];

UB=[30 25 3500 150 -0.01];

LB=[10 10 2200 30 -1000];

x0 = [100 100 10000 400 -7000];

options =

optimoptions(@fmincon,'OutputFcn',@outfun,...

'Display','iter','Algorithm','active-set');

xsol =

fmincon(@obfun,x0,[],[],[],[],LB,UB,@confun,options);

function stop = outfun(x,optimValues,state)

stop = false;

switch state

case 'init'

hold on

case 'iter'

history.fval = [history.fval;

optimValues.fval];

history.x = [history.x; x];

searchdir = [searchdir;...

optimValues.searchdirection'];

iter=optimValues.iteration+1;

n(iter)=iter;

x1(iter)=x(1);

x2(iter)=x(2);

x3(iter)=x(3)/100;

x4(iter)=x(4);

x5(iter)=x(5)/100;

fx(iter)=optimValues.fval;

title('Sequence of Points Computed by

fmincon');

case 'done'

hold off

otherwise

end

end

function f = obfun(x)

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1));

e=(1.78*(1-0.045*x(2)^0.68)-0.64);

CD=0.0125+(CL^2/(3.14*x(2)*e));

f=(CD/CL)*sqrt(2*x(3)/(1.061*x(1)*CL));

end

function [c, ceq] = confun(x)

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1));

e=(1.78*(1-0.045*x(2)^0.68)-0.64);

CD=0.0125+(CL^2/(3.14*x(2)*e));

c =[10-CL/CD;

CL/CD-150];

ceq = [];

end

[ax,h1,h2]=plotyy(n,[x1' x2' x3' x4'

x5'],n,fx,'plot','plot');

legend([h1;h2],'x1 - S(m2)','x2 - AR','x3 - W(x 100

N)','x4 - R(m)','x5 - delta (x 100 N)','fx -

Vsink(m/s)','Location','southwest','Orientation','ver

tical');

end

b. Without glide ratio constraint

%=========================

% FMINCON

% Without Constraints

%=========================

function [history,searchdir] = runfmincon

history.x = [];

history.fval = [];

searchdir = [];

%BOUNDS

UB=[30 25 3500 150 -0.01];

LB=[10 10 2200 30 -1000];

%INITIAL VALUES

x0 = [15 11 2500 50 -10];

options =

optimoptions(@fmincon,'OutputFcn',@outfun,...

'Display','iter','Algorithm','active-set');

[x,fval]=

fmincon(@obfun,x0,[],[],[],[],LB,UB,[],options)

function stop = outfun(x,optimValues,state)

stop = false;

switch state

case 'init'

hold on

case 'iter'

history.fval = [history.fval;

optimValues.fval];

history.x = [history.x; x];

searchdir = [searchdir;...

optimValues.searchdirection'];

iter=optimValues.iteration+1;

n(iter)=iter;

x1(iter)=x(1);

x2(iter)=x(2);

x3(iter)=x(3)/100;

x4(iter)=x(4);

x5(iter)=x(5)/100;

fx(iter)=optimValues.fval;

title('Sequence of Points Computed by

fmincon');

case 'done'

hold off

otherwise

end

end

%OBJECTIVE FUNCTION

function f = obfun(x)

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1));

e=(1.78*(1-0.045*x(2)^0.68)-0.64);

CD=0.0125+(CL^2/(3.14*x(2)*e));

f=(CD/CL)*sqrt(2*x(3)/(1.061*x(1)*CL));

end

%CONSTRAINT FUNCTION

function [c, ceq] = confun(x)

% Nonlinear inequality constraints

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1));

e=(1.78*(1-0.045*x(2)^0.68)-0.64);

CD=0.0125+(CL^2/(3.14*x(2)*e));

c =[10-CL/CD;

CL/CD-150];

% Nonlinear equality constraints

ceq = [];

end

subplot (3,2,1)

plot(n,x1)

title('Wing Area - x1 (m2)')

ylabel('x1 (m2)')

subplot (3,2,2)

plot(n,x2)

title('Aspect Ratio - x2 ')

ylabel('x2')

subplot (3,2,3)

plot(n,x3)

title('Weight - x3 (N)')

ylabel('x3 (N)')

subplot (3,2,4)

plot(n,x4)

title('Turn Radius - x4 (m)')

ylabel('x4 (m)')

subplot (3,2,5)

plot(n,x5)

title('Delta [W-L]- x5 (N)')

xlabel ('Number of iteration')

ylabel('x5 (N)')

subplot (3,2,6)

plot(n,fx)

title('Rate of Descent - f(x) (m/s)')


ylabel('f(x) (m/s)')

ha=axes('Position',[0 0 1 1],'Xlim',[0 1],'Ylim',[0

1],'Box','off','Visible','off','Units','normalized','

clipping','off');

text(0.5,1,'\bf Fmincon Result (WITHOUT Glide Ratio

Constraint, Initial point inside boundary - 2

)','HorizontalAlignment','center','VerticalAlignment'

,'top')

end

c.

The output of the solver and the plot of parameter change in every iteration,at every

cases can be seen in Appendix A at the back of this report.

3.1.2 Genetic Algorithm

The computation using Genetic Algorithm are using following code:

a. With Glide Ratio constraint:

%==================================

% GENETIC ALGORITHM

% (with glide ratio constraint)

%==================================

function [history,searchdir] = ga_ma2

ObjectiveFunction = @ma_objfun_ga;

nvars = 5; % Number of variables

% BOUND

UB=[30 25 4000 150 -0.01];

LB=[10 10 2200 30 -1000];

% INITIAL VALUES

X0=[15 11 2500 50 -10];

ConstraintFunction = @ma_constraint;

opt = gaoptimset('OutputFcns',@ga_outfun, ...

'Display','iter','Generations',50,'TolFun',1E-

20,'TolCon',1E-

20,'Mutation',@mutationadaptfeasible,'InitialPopulati

on',X0);

[x,fval,exitflag] =

ga(ObjectiveFunction,nvars,[],[],[],[],LB,UB, ...

ConstraintFunction,opt)

function [state, options,optchanged] =

ga_outfun(options,state,flag)

optchanged = false;

switch flag

case 'init'

hold on

case 'iter'

iter=state.Generation;

n(iter)=iter;

x1(iter)=state.Population(1,1);





fx(iter)=state.Best(iter);

case 'done'

hold off

end

end

subplot (3,2,1)

plot(n,x1)


ylabel('x1 (m2)')

ylim([25 32])

subplot (3,2,2)

plot(n,x2)


ylabel('x2')

subplot (3,2,3)

plot(n,x3)


ylabel('x3 (N)')

subplot (3,2,4)

plot(n,x4)


ylabel('x4 (m)')

subplot (3,2,5)

plot(n,x5)


xlabel ('Number of generation')

ylabel('x5 (N)')

subplot (3,2,6)

plot(n,fx)


xlabel ('Number of generation')



1],'Box','off','Visible','off','Units','normalized','

clipping','off');

text(0.5,1,'\bf Genetic Algorithm Result (WITH Glide

Ratio Constraint, Initial point inside boundary - 2

)','HorizontalAlignment','center','VerticalAlignment'

,'top')

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1))

e=(1.78*(1-0.045*(x(2)^0.68))-0.64)

CD=0.0125+(CL^2/(3.14*x(2)*e))

CL_CD=CL/CD

V=sqrt(x(4)*6.3)

%OBJECTIVE/FITNESS FUNCTION

function f=ma_objfun_ga(x)

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1));

e=(1.78*(1-0.045*x(2)^0.68)-0.64);

CD=0.0125+(CL^2/(3.14*x(2)*e));

f=(CD/CL)*sqrt(2*x(3)/(1.061*x(1)*CL));

end

%CONSTRAINTS FUNCTION

function [c, ceq] = ma_constraint(x)

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1));

e=(1.78*(1-0.045*x(2)^0.68)-0.64);

CD=0.0125+(CL^2/(3.14*x(2)*e));

c =[10-CL/CD;

CL/CD-150];

ceq = [];

end

end

b. Without Glide Ratio constraint:

%================================== % GENETIC ALGORITHM % (without glide ratio constraint) %==================================

function [history,searchdir] = ga_ma2 ObjectiveFunction = @ma_objfun_ga; nvars = 5; % Number of variables % BOUND UB=[30 25 4000 150 -0.01]; LB=[10 10 2200 30 -1000];

% INITIAL VALUES X0=[15 11 2500 50 -10]; ConstraintFunction = @ma_constraint; opt = gaoptimset('OutputFcns',@ga_outfun, ... 'Display','iter','Generations',50,'TolFun',1E-

20,'TolCon',1E-

20,'Mutation',@mutationadaptfeasible,'InitialPopulation',X

0);

[x,fval,exitflag] =

ga(ObjectiveFunction,nvars,[],[],[],[],LB,UB, ... [],opt)

function [state, options,optchanged] =

ga_outfun(options,state,flag) optchanged = false; switch flag case 'init' hold on case 'iter'

iter=state.Generation; n(iter)=iter;

x1(iter)=state.Population(1,1); x2(iter)=state.Population(1,2); x3(iter)=state.Population(1,3); x4(iter)=state.Population(1,4); x5(iter)=state.Population(1,5); fx(iter)=state.Best(iter);

case 'done' hold off end end

subplot (3,2,1) plot(n,x1) title('Wing Area - x1 (m2)') ylabel('x1 (m2)') ylim([25 32])

subplot (3,2,2) plot(n,x2) title('Aspect Ratio - x2 ') ylabel('x2')

subplot (3,2,3) plot(n,x3) title('Weight - x3 (N)') ylabel('x3 (N)')

subplot (3,2,4) plot(n,x4) title('Turn Radius - x4 (m)') ylabel('x4 (m)')

subplot (3,2,5) plot(n,x5) title('Delta [W-L]- x5 (N)') xlabel ('Number of generation') ylabel('x5 (N)')

subplot (3,2,6) plot(n,fx) title('Rate of Descent - f(x) (m/s)') xlabel ('Number of generation') ylabel('f(x) (m/s)')


1],'Box','off','Visible','off','Units','normalized','clipp

ing','off'); text(0.5,1,'\bf Genetic Algorithm Result (WITHOUT Glide


)','HorizontalAlignment','center','VerticalAlignment','top

')

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1)) e=(1.78*(1-0.045*(x(2)^0.68))-0.64) CD=0.0125+(CL^2/(3.14*x(2)*e)) CL_CD=CL/CD V=sqrt(x(4)*6.3)

%OBJECTIVE/FITNESS FUNCTION function f=ma_objfun_ga(x) CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1)); e=(1.78*(1-0.045*x(2)^0.68)-0.64); CD=0.0125+(CL^2/(3.14*x(2)*e)); f=(CD/CL)*sqrt(2*x(3)/(1.061*x(1)*CL)); end

%CONSTRAINTS FUNCTION function [c, ceq] = ma_constraint(x) CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1)); e=(1.78*(1-0.045*x(2)^0.68)-0.64); CD=0.0125+(CL^2/(3.14*x(2)*e)); c =[10-CL/CD; CL/CD-150]; ceq = []; end end

The output of the solver and the plot of parameter change in every generation, at

every cases, can be seen in Appendix A at the back of this report.

3.1.3 Simulated Annealing

The computation using Simulated Annealing are using following code:

%==========================

% SIMULATED ANNEALING

%==========================

function [history,searchdir] = simulated_annealing

Objfun=@ma_objfunc_sa;

%INITIAL VALUES

x0=[15 11 2500 50 -10];

%BOUNDS

UB=[30 25 4000 150 -0.01];

LB=[10 10 2200 30 -1000];

options = saoptimset('OutputFcns',@sa_output);

[x,fval,exitflag,output]=simulannealbnd(Objfun,x0,LB,UB,op

tions)

function [stop,options,optchanged] =

sa_output(options,optimvalues,flag)

stop = false;

optchanged = false;

switch flag

case 'init'

hold on;

case 'iter'

iter=optimvalues.iteration;

n(iter)=iter;

x1(iter)=optimvalues.bestx(1);


x3(iter)=optimvalues.bestx(3)/100;


x5(iter)=optimvalues.bestx(5)/100;

fx(iter)=optimvalues.bestfval;

case 'done'

hold off;

end

end

subplot (3,2,1)

plot(n,x1)


ylabel('x1 (m2)')

ylim([5 35])

subplot (3,2,2)

plot(n,x2)


ylabel('x2')

subplot (3,2,3)

plot(n,x3)


ylabel('x3 (N)')

subplot (3,2,4)

plot(n,x4)


ylabel('x4 (m)')

subplot (3,2,5)

plot(n,x5)



ylabel('x5 (N)')

subplot (3,2,6)

plot(n,fx)





1],'Box','off','Visible','off','Units','normalized','clipp

ing','off');

text(0.5,1,'\bf Simulated Annealing Result (WITHOUT Glide


)','HorizontalAlignment','center','VerticalAlignment','top

')

%OBJECTIVE/FITNESS FUNCTION

function f=ma_objfunc_sa(x)

CL= 2*(x(3)-x(5))/(6.93*x(4)*x(1));

e=(1.78*(1-0.045*x(2)^0.68)-0.64);

CD=0.0125+(CL^2/(3.14*x(2)*e));

f=(CD/CL)*sqrt(2*x(3)/(1.061*x(1)*CL));

end

end

The output of the solver and the plot of parameter change in every generation, at

every cases, can be seen in Appendix A at the back of this report.

CHAPTER 4

ANALYSIS AND CONCLUSION

4.1 Solver Analysis & Conclusion

Table 4.1 Result Summary

Table 4.1 above shows the summary of the result of the 3 solvers. Genetic Algorithm and

Simulated Annealing show unconsistent result from every execution. It mostly caused by the

concept of these optimization is creating random number and calculate the fitness/objective

function until the optimum result is found. However, from the iteration plot, GA result is more

stable when SA result is more likely to oscilate randomly.

Fmincon yield consistent result , event in constraint-unconstrainted case. But the result is still

not as good as GA. With different initial point it seems that fmincon end up in different

minima. It should be because the solver found the local minima and assume it as the global

minima,so the algortihm stopped.

Different initial point also affect the result of GA and SA, but it’s unclear if it’s because

of one of the initial point closer to the global minima, or because of the random

algorithm of SA and FA . But from the plot of the parameter change in every iteration,

initial point inside the constraint are more likely to make some parameter reach

convergence faster. This may be the proof that the initial point is closer to the global

minima.

4.2 Optimization Conclusion

The optimum parameters obtained from this optimization, taken from the GA best

result are:

Wing Area : 30 m2

Aspect Ratio : 23

Weight : 2279 N

Turn Radius : 30 m

That will yield optimum Rate of Descent 0.51 m/s. However, the set of result of the

optimization are only the ideal numbers. In practical, it’s very difficult to design a

sailplane with those set of parameter. For example with Aspect Ratio of 23 the wing

and the aircraft will more likely need tougher structure and become very heavy. It may

be practically unfeasible to have the aircraft weighted 2279 N.

To understand the behavior of every parameter in the optimization, variations of

boundary / side constraints are applied. Each time, one parameter are varied when

the others still the same as the constraints explained at the early part of this report.

Figure 4.1 Aspect Ratio vs Rate Of Descent

0

10

20

30

40

50

60

70

0.4 0.45 0.5 0.55

Asp

ect

Rat

io

Rate Of Descent (m/s)

Aspect RatioBounds

OptimumAspect Ratio

Table 4.1 shows the behavior of rate of descent based on aspect ratio. Rate of descent

tend to become lower when Aspect Ratio is bigger. However, at some point of

increasing Aspect Ratio, the rate of descent are not decreasing anymore, and the

optimum aspect ratio chosen by solver are not the upper bound of the aspect ratio.

Figure 4.2 Wing Area vs Rate of Descent

Table 4.2 shows the behavior of rate of descent based on wing area. Rate of descent

tend to become lower when wing area is bigger. However, at some point of increasing

wing area, the rate of descent are not decreasing significantly anymore, and the

optimum aspect ratio chosen by solver are not the upper bound of the aspect ratio.

Figure 4.3 Weight vs Rate Of Descent

0

500

1000

1500

2000

2500

3000

0.44 0.46 0.48 0.5 0.52 0.54 0.56

Wei

ght

(N)


Weight LowerBounds

Optimum Weight

0

5

10

15

20

25

30

0.51 0.52 0.53 0.54 0.55 0.56 0.57

Win

g A

rea

(m2

)


Wing Area Bounds

Optimum Wing Area

Table 4.3 shows the behavior of rate of descent based on weight. Rate of descent tend

to become lower at lower weight. However, based on the result trend of the

optimization the weight seems not significantly affect the optimization. The optimum

value barely moving close to upper or lower bound. It may be caused by that weight

are not the critical parameter that define rate of descent.

Figure 4.4 Turn Radius vs Rate Of Descent

Table 4.4 shows the behavior of rate of descent based on turn radius. Rate of descent

tend to become lower at lower turn radius. The optimum value of turn radius is more

likely very close to the lower bound.

0

10

20

30

40

50

60

0.48 0.5 0.52 0.54 0.56 0.58 0.6

Turn

Rad

ius

(m)


Turn Radius Lower Bound

Optimum Turn Radius

CHAPTER 5

RECOMMENDATION

Based on the result and conclusion of this optimization project, future works that can be done

related to this optimization project should discover and use another solver. Matlab still have

limitation that need to be fixed. Trial using another solver or maybe the improved version of

Matlab solver needs to be done to validate which are the real best solver for this

optimization.

Optimization topic in this project also still need to be explored deeper. In this project, basicly

the parameter that are used are only four, Where the rest of parameter related to rate of

descent are simplified using assumed value. The case in this project is also still the simplified

version, because the optimization are only centered in sailplane wing. In the real case, rate of

descent should be affected not only by the wing, but also by the fuselage and the tail. In future

works, many parameters are need to be added, such as wing twist, taper, incidence, fuselage

drag, tail drag, etc. to find the real optimum design of sailplane that have minimum rate of

descent.

The algorithm applied to the solver also need many improvements. One necessary

improvement is control to the constraints, so the combination optimum value that come out

from the solver are really make sense in the real world. So the optimization result might

applied to the the real sailplane design project.

REFERENCE

[1] FAA. Glider Flying Handbook, Skyhorse Publishing,Washington DC.July 2007

[2] Thomas,Fred. Fundamentals of Sailplane Design. College Park Press, Maryland.

1999

[3] Handojo,Vega H. 2013. Sailplane Performance Estimation. ITB Internship Report,

Bandung

[4] Rujgrok, G.J.J. 1990. Element of Airplane Performance. Delft University Press,

Netherland

[5] Pratama, Hendi A. 2014. Definition of Glider Design Basic Parameter Based on

Thermal Characteristic of Indonesia. Design Project 1 Report. Bandung

APPENDIX A

TABLE A.1 Result of Constrained Minimization (fmincon)

With glide ratio constraint.initial point outside boundary x0 = [100 100 10000 400 -7000]

Without glide ratio constraint.initial point outside boundary x0 = [100 100 10000 400 -7000]

With glide ratio constraint.initial point inside boundary - 1 x0 = [20 20 3000 100 -500]

Without glide ratio constraint.initial point inside boundary - 1 x0 = [20 20 3000 100 -500]

With glide ratio constraint.initial point inside boundary -2 X0 = [15 11 2500 50 -10];


Figure A.1 Plot of Parameter Changes in Every Iteration (fmincon, With Glide Ratio, Initial Point Outside Boundary)

0 5 10 15 20 25 3010

15

20

25

30Wing Area - x1 (m2)

x1 (

m2)

0 5 10 15 20 25 3018

20

22

24

26Aspect Ratio - x2

x2

0 5 10 15 20 25 3034.985

34.99

34.995

35

35.005Weight - x3 (N)

x3 (

N)

0 5 10 15 20 25 300

50

100

150Turn Radius - x4 (m)

x4 (

m)

0 5 10 15 20 25 30-10

-10

-10Delta [W-L]- x5 (N)

Number of iteration

x5 (

N)

0 5 10 15 20 25 300.5

1

1.5Rate of Descent - f(x) (m/s)

Number of iteration

f(x)

(m/s

)

Fmincon Result (With Glide Ratio Constraint, Initial point outside boundary)

Figure A.2 Plot of Parameter Changes in Every Iteration (fmincon, With Glide Ratio, Initial Point Inside Boundary - 1)

0 5 10 15 20 2510

15

20

25


x1 (

m2)

0 5 10 15 20 2520

22

24

26Aspect Ratio - x2

x2

0 5 10 15 20 2529.996

29.998

30

30.002

30.004Weight - x3 (N)

x3 (

N)

0 5 10 15 20 2520

40

60

80


x4 (

m)

0 5 10 15 20 25-5.03

-5.02

-5.01


Number of iteration

x5 (

N)

0 5 10 15 20 250.4

0.6

0.8

1Rate of Descent - f(x) (m/s)

Number of iteration

f(x)

(m/s

)

Fmincon Result (With Glide Ratio Constraint, Initial point inside boundary - 1)

Figure A.3 Plot of Parameter Changes in Every Iteration (fmincon, With Glide Ratio, Initial Point Inside Boundary -2)

0 5 10 15 20 2515

20

25


x1 (

m2)

0 5 10 15 20 2510

15

20

25Aspect Ratio - x2

x2

0 5 10 15 20 2524.999

24.9995

25

25.0005Weight - x3 (N)

x3 (

N)

0 5 10 15 20 2530

35

40

45


x4 (

m)

0 5 10 15 20 25-0.104

-0.102

-0.1

-0.098Delta [W-L]- x5 (N)

Number of iteration

x5 (

N)

0 5 10 15 20 250.4

0.6

0.8


Number of iteration

f(x)

(m/s

)


Figure A.4 Plot of Parameter Changes in Every Iteration (fmincon, Without Glide Ratio, Initial Point Outside Boundary )

0 5 10 15 20 25 3010

15

20

25


x1 (

m2)

0 5 10 15 20 25 3018

20

22

24

26Aspect Ratio - x2

x2

0 5 10 15 20 25 3034.985

34.99

34.995

35

35.005Weight - x3 (N)

x3 (

N)

0 5 10 15 20 25 300

50

100


x4 (

m)

0 5 10 15 20 25 30-10

-10


Number of iteration

x5 (

N)

0 5 10 15 20 25 300.5

1


Number of iteration

f(x)

(m/s

)

Fmincon Result (WITHOUT Glide Ratio Constraint, Initial point ouside boundary )

Figure A.5 Plot of Parameter Changes in Every Iteration (fmincon, Without Glide Ratio, Initial Point Inside Boundary - 1)

0 5 10 15 20 2510

15

20

25


x1 (

m2)

0 5 10 15 20 2520

22

24

26Aspect Ratio - x2

x2

0 5 10 15 20 2529.996

29.998

30

30.002

30.004Weight - x3 (N)

x3 (

N)

0 5 10 15 20 2520

40

60

80


x4 (

m)

0 5 10 15 20 25-5.03

-5.02

-5.01


Number of iteration

x5 (

N)

0 5 10 15 20 250.4

0.6

0.8


Number of iterationf(

x)

(m/s

)


Figure A.6 Plot of Parameter Changes in Every Iteration (fmincon, Without Glide Ratio, Initial Point Inside Boundary - 2)

0 5 10 15 20 2515

20

25


x1 (

m2)

0 5 10 15 20 2510

15

20

25Aspect Ratio - x2

x2

0 5 10 15 20 2524.999

24.9995

25

25.0005Weight - x3 (N)

x3 (

N)

0 5 10 15 20 2530

35

40

45


x4 (

m)

0 5 10 15 20 25-0.104

-0.102

-0.1

-0.098Delta [W-L]- x5 (N)

Number of iteration

x5 (

N)

0 5 10 15 20 250.4

0.6

0.8


Number of iterationf(

x)

(m/s

)

Fmincon Result (WITHOUT Glide Ratio Constraint, Initial point inside boundary - 2 )

TABLE A.2 Result of Genetic Algorithm

With glide ratio constrain, initial point outside boundary x0 = [100 100 10000 400 -7000]

Withoulide ratio constraint.initial point outside boundary x0 = [100 100 10000 400 -7000]

With glide ratio constraint.initial point inside boundary – 1 x0 = [20 20 3000 100 -500]

Without glide ratio constraint.initial point inside boundary - 1 x0 = [20 20 3000 100 -500]



Figure A.7 Plot of Parameter Changes in Every Iteration (Genetic Algorithm, With Glide Ratio, Initial Point Outside Boundary)

0 2 4 6 8 10 12 14 16 18 20

26

28

30


x1 (

m2)

0 2 4 6 8 10 12 14 16 18 2022.9

23

23.1

23.2

23.3Aspect Ratio - x2

x2

0 2 4 6 8 10 12 14 16 18 202220

2240

2260

2280Weight - x3 (N)

x3 (

N)

0 2 4 6 8 10 12 14 16 18 2030

40

50


x4 (

m)

0 2 4 6 8 10 12 14 16 18 20-700

-680

-660

-640Delta [W-L]- x5 (N)

Number of generation

x5 (

N)

0 2 4 6 8 10 12 14 16 18 200.5

0.55

0.6

0.65



f(x)

(m/s

)

Genetic Algorithm Result (WITH Glide Ratio Constraint, Initial point outside boundary )

Figure A.8 Plot of Parameter Changes in Every Iteration (Genetic Algorithm, With Glide Ratio, Initial Point Inside Boundary - 1)

0 2 4 6 8 10 12 14 16 18 20

26

28

30


x1 (

m2)

0 2 4 6 8 10 12 14 16 18 2023.1

23.2

23.3

23.4


x2

0 2 4 6 8 10 12 14 16 18 202400

2420

2440

2460Weight - x3 (N)

x3 (

N)

0 2 4 6 8 10 12 14 16 18 2030

32

34


x4 (

m)

0 2 4 6 8 10 12 14 16 18 20-520

-510

-500

-490Delta [W-L]- x5 (N)


x5 (

N)

0 2 4 6 8 10 12 14 16 18 200.52

0.53

0.54



f(x)

(m/s

)

Genetic Algorithm Result (WITH Glide Ratio Constraint, Initial point inside boundary - 1 )

Figure A.9 Plot of Parameter Changes in Every Iteration (Genetic Algorithm, With Glide Ratio, Initial Point Inside Boundary - 2)

0 2 4 6 8 10 12 14 16 18 20

26

28

30


x1 (

m2)

0 2 4 6 8 10 12 14 16 18 2023.12

23.14

23.16

23.18


x2

0 2 4 6 8 10 12 14 16 18 202260

2280

2300

2320Weight - x3 (N)

x3 (

N)

0 2 4 6 8 10 12 14 16 18 2030

30.05

30.1

30.15Turn Radius - x4 (m)

x4 (

m)

0 2 4 6 8 10 12 14 16 18 20-500

-490

-480

-470Delta [W-L]- x5 (N)


x5 (

N)

0 2 4 6 8 10 12 14 16 18 200.514

0.516

0.518

0.52



f(x)

(m/s

)

Genetic Algorithm Result (WITH Glide Ratio Constraint, Initial point inside boundary - 2 )

Figure A.10 Plot of Parameter Changes in Every Iteration (Genetic Algorithm, Without Glide Ratio, Initial Point Outside Boundary)

0 5 10 15 20 25 30 35 40 45 50

26

28

30


x1 (

m2)

0 5 10 15 20 25 30 35 40 45 5022.5

23

23.5

24


x2

0 5 10 15 20 25 30 35 40 45 502800

2900

3000

3100

3200Weight - x3 (N)

x3 (

N)

0 5 10 15 20 25 30 35 40 45 5030

35

40


x4 (

m)

0 5 10 15 20 25 30 35 40 45 50-800

-600

-400

-200Delta [W-L]- x5 (N)


x5 (

N)

0 5 10 15 20 25 30 35 40 45 50

0.58

0.6



f(x)

(m/s

)

Genetic Algorithm Result (WITHOUT Glide Ratio Constraint, Initial point outside boundary )

Figure A.11 Plot of Parameter Changes in Every Iteration (Genetic Algorithm, Without Glide Ratio, Initial Point Inside Boundary - 1)

0 5 10 15 20 25 30 35 40 45 50

26

28

30


x1 (

m2)

0 5 10 15 20 25 30 35 40 45 5015

20

25Aspect Ratio - x2

x2

0 5 10 15 20 25 30 35 40 45 502000

2500

3000

3500Weight - x3 (N)

x3 (

N)

0 5 10 15 20 25 30 35 40 45 5030

35

40


x4 (

m)

0 5 10 15 20 25 30 35 40 45 50-900

-800

-700

-600

-500Delta [W-L]- x5 (N)


x5 (

N)

0 5 10 15 20 25 30 35 40 45 500.5

0.55

0.6

0.65



f(x)

(m/s

)

Genetic Algorithm Result (WITHOUT Glide Ratio Constraint, Initial point inside boundary - 1 )

Figure A.12 Plot of Parameter Changes in Every Iteration (Genetic Algorithm, Without Glide Ratio, Initial Point Inside Boundary - 2)

15 20 25 30 35 40 45 50

26

28

30


x1 (

m2)

0 5 10 15 20 25 30 35 40 45 5021

22

23

24

25Aspect Ratio - x2

x2

0 5 10 15 20 25 30 35 40 45 502200

2400

2600

2800Weight - x3 (N)

x3 (

N)

0 5 10 15 20 25 30 35 40 45 5035

40

45


x4 (

m)

0 5 10 15 20 25 30 35 40 45 50-600

-400

-200

0Delta [W-L]- x5 (N)


x5 (

N)

0 5 10 15 20 25 30 35 40 45 500.55

0.6



f(x)

(m/s

)

Genetic Algorithm Result (WITHOUT Glide Ratio Constraint, Initial point inside boundary - 2 )

TABLE A.3 Result of Simulated Annealing

Without glide ratio constraint.initial point outside boundary x0 = [100 100 10000 400 -7000]

Without glide ratio constraint.initial point inside boundary - 1

x0 = [20 20 3000 100 -500]

With glide ratio constraint.initial point inside boundary -2

X0 = [15 11 2500 50 -10];

Figure A.13 Plot of Parameter Changes in Every Iteration (Simulated Annealing, Without Glide Ratio, Initial Point Outside Boundary)

0 1000 2000 3000 4000 5000 6000 7000

10

20

30

Wing Area - x1 (m2)

x1 (

m2)

0 1000 2000 3000 4000 5000 6000 700010

15

20

25Aspect Ratio - x2

x2

0 1000 2000 3000 4000 5000 6000 700034

36

38

40Weight - x3 (N)

x3 (

N)

0 1000 2000 3000 4000 5000 6000 70000

50

100


x4 (

m)

0 1000 2000 3000 4000 5000 6000 7000-10

-9

-8

-7


Number of iteration

x5 (

N)

0 1000 2000 3000 4000 5000 6000 7000

0.8

1

1.2


Number of iteration

f(x)

(m/s

)

Simulated Annealing Result (WITHOUT Glide Ratio Constraint, Initial point outside boundary )

Figure A.14 Plot of Parameter Changes in Every Iteration (Simulated Annealing, Without Glide Ratio, Initial Point Inside Boundary - 1)

0 1000 2000 3000 4000 5000 6000

10

20

30

Wing Area - x1 (m2)

x1 (

m2)

0 1000 2000 3000 4000 5000 600010

15

20

25Aspect Ratio - x2

x2

0 1000 2000 3000 4000 5000 600024

26

28

30

32Weight - x3 (N)

x3 (

N)

0 1000 2000 3000 4000 5000 600020

40

60

80


x4 (

m)

0 1000 2000 3000 4000 5000 6000-7

-6

-5

-4


Number of iteration

x5 (

N)

0 1000 2000 3000 4000 5000 60000.4

0.6

0.8


Number of iteration

f(x)

(m/s

)

Simulated Annealing Result (WITHOUT Glide Ratio Constraint, Initial point inside boundary - 1 )

Figure A.15 Plot of Parameter Changes in Every Iteration (Simulated Annealing, Without Glide Ratio, Initial Point Inside Boundary – 2)

0 500 1000 1500 2000 2500 3000

10

20

30

Wing Area - x1 (m2)x1 (

m2)

0 500 1000 1500 2000 2500 300010

15

20

25Aspect Ratio - x2

x2

0 500 1000 1500 2000 2500 300024.5

25

25.5Weight - x3 (N)

x3 (

N)

0 500 1000 1500 2000 2500 300030

40

50


x4 (

m)

0 500 1000 1500 2000 2500 3000-1

-0.8

-0.6

-0.4

-0.2Delta [W-L]- x5 (N)

Number of iteration

x5 (

N)

0 500 1000 1500 2000 2500 30000.5

0.6

0.7

0.8


Number of iteration

f(x)

(m/s

)

Simulated Annealing Result (WITHOUT Glide Ratio Constraint, Initial point inside boundary - 2 )

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