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Aerodynamic shape optimization Nose Cone of F-35 Lightning II

Research · May 2018

DOI: 10.13140/RG.2.2.29953.15200

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Ege Konuk

Old Dominion University

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Aerodynamic shape optimization Nose Cone of F-35

Lightning II

Ege Konuk, UIN: 01090996

Mechanical and Aerospace Engineering

Old Dominion University

Abstract

This project aims to maps out a potentially better design points for the Nose cone structure of the

Lockheed Martin F-35 Lightning II aircraft. The optimization is performed by using the Built-In

functions of MATLAB optimization. Computational Fluid Dynamics (CFD) analysis has

conducted to predict the flow behavior mainly drag values around the particular shape of the nose

cone. QUICKERSIM CFD software has been used to obtain the aerodynamics of the nose cone.

Optimized shape is obtained at Mach 1.61 which is the certified max speed for the F-35 fighter jet.

CFD solves the compressible Euler equations for the domain. The PARSEC curve approach has

been applied for the physical two-dimensional shape of the nose. Two-dimensional shape of the

nose seen from bird’s eye view has selected for the obtain necessary simplicity for the projected

deadline of the project and assumptions made accordingly. Optimization loop has been developed

and implemented as to function automatically when the user initiated the main optimization. The

necessary relation has been established inside of each script. Analysis has provided a solution close

to nine percent decrease in drag values with out manipulating the constraints given.

I. Introduction

Nose cone is the term that used to refer to the foremost part of the rocket, missile or aircraft. The

significance of this section throughout the aircraft structure is intensifies when the vehicle flying

at the supersonic flight regime. This condition of flight has dramatic effects on every part of the

air vehicle that exposed to the air flow. One of the most important factor when it comes the efficient

and safe supersonic aircraft is to design of the nose cone. This cone has to offer minimum amount

of drag without sacrificing too much on the performance of the fighter and the volume that it stores

inside. It generally houses the radome (radar system for the aircraft). This put inherent constraint

of the design of the cone. Especially in very high-speed flow where extreme temperatures involved,

it has to be made by temperature resistant material even dough those materials there still some

room for safety has put by limiting the temperature created by shock effect happening on the high-

speed fluid flowing through the nose cone to ensure safe flight. There many parameters are to fine

tune in order to get an optimal one with the given restrictions and resources to design and build a

nose cone for the particular air vehicle.

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a) b)

Figure 1. Lockheed Martin F-35 Lightning II (a) Bow Shock waves around the similar fighter

jet (b)

a) b)

Figure 2. Top view of Symmetric Nose cone of Lockheed Martin F-35 Lightning II (a) Front

view of the CAD drawing of the Lockheed Martin F-35 Lightning II (b)

In the present work optimization algorithm will be implemented to a CFD code and optimization

will be performed conjunction with the analysis that given from CFD. MATLAB algorithm

interior-method considered to this optimization. CFD based optimizations has been used by

decades now however for the recent years there is a big growth in the research areas because of

the advent of the fast computers and more accurate and efficient mathematical models designers

able to achieve the better solution that might not be obtainable by early of the development stages

of the design.

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II. Formulation Methods

Shape Representation

There are various ways to represent a shape like nose cone. Specially, for this project there are

couple of condition are considered while settle on a method to represent the shape of a nose cone

a. It should be able to explore and capture the unconventional shapes.

b. The method for shape representation must refrain on creating the sharp discontinues and

must create a somewhat smooth curvature all around.

c. It must be correctly manipulated by constraint imposed by the optimization

d. Accuracy of initial shape prediction should be adjustable by the user

Once the method has been selected for the shape model design variables can be obtained and

implemented into the optimization code. For this project PARSEC representation has selected

because it meets with the desired requirements.

The PARSEC representation has been most promising way to obtain a curve such as the nose cone.

This has been provided by the small number of design variable in spite of the promising accuracy.

The profile defined for the objective basically a 6th degree polynomial fit around shape and Y and

X coordinates are defined as shown

𝑌 = ∑ 𝑎𝑛𝑋𝑛

7

𝑛=1

Where the 𝑎𝑛 are the polynomial coefficient which will the design variables of the problem. For

this problem Length of the cone has been fixed by using the same X values at each iteration of the

optimization. However thickest width of nose cone has been given a little relaxation constraint.

Finally, Coefficients are obtained from initial shape(F35) are given;

𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7

Upper Bounds -0.1469 1.0392 -0.7655 0.8827 -0.3473 0.8962 -1.1879e-05

Initial Coefficients -0.2099 0.7994 -1.0935 0.6790 -0.4962 0.6894 -1.697e-05

Lower Bounds -0.2728 0.5596 -1.4216 0.4753 -0.6451 0.4826 -2.206e-05

Table 1. Upper-lower bounds and Initial polynomial Coefficients

Hence, the initial shape is plotted as shown

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Figure 3. Initial shape of the nose curvature of the F-35 fighter jet

Optimization

Optimization has been performed in the MATLAB. Hence the built-in MATLAB function has

used for this problem. MATLAB “fmincon” function has used to utilized for this problem.

Optimization algorithm has kept at default. Since there is no gradient provided the default

algorithm selected as ‘Interior-Point’.

Optimization formulation defined as following

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓(𝐷(𝑢), 𝑢(𝑎)) = 𝐶𝑑

Where 𝑢(𝑎) = ∑ 𝑎𝑛𝑋𝑛7𝑛=1

𝑎𝑛 ∈ 𝑅7

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 − 𝑠𝑙𝑜𝑝𝑒(𝑢(𝑎)) ≤ 0 𝑔1

𝑠𝑙𝑜𝑝𝑒 (𝑢(𝑎𝑦+1 − 𝑎𝑦)) ≤ 0 𝑔2

|∑ 𝑎𝑛

7

𝑛=1

− 0.368 − 10−3| ≤ 0 𝑔3

CFD Analysis of Nose Cone

QUICKERSIM, a commercial code is implemented as a analysis of the flow. It is a Finite element

CFD code and solves Euler equation in compressible regimes. There is no turbulence modelling

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or viscous term involved. However, for the simplicity of this problem these are necessary

assumption. This way the optimization was able to utilized as an iterative manner with using any

techniques to provide a gradient or collecting data for DOE analysis.

CFD code was initialized by using the following parameters

• Speed of the flow= 1,61 Mach

• Density: 1.225 kg/m3

• Atmospheric Pressure: 101325 Pa

• Convergence Criteria: Residuals < 1E − 3

Mesh creation

Mesh has been generated using the finite element mesh generator software called “GMSH”. The

choice of the software was purely on the requirement based. Since the CFD that is used in the

analysis only sufficiently tested and properly working with GMSH the decision made totally upon

not to upset the accuracy of the solution and avoid compatibility problems.

Initially, the prototype mesh file was tested to verify the MATLAB’s internals are working in order

then the steps were taken to implement a mesh generation cycle into the optimization cycle. Hence

following steps have carried out;

1. Script written that generates compatible Geometry file (.geo).

2. The script then structured to work as function that can be called from the objective function

script.

3. Parameter has been established to achieved the desired parameters.

Element size factor:0.5

Nose cone mesh factor: 0.15

Rest of the domain mesh factor: 1.5

𝑴𝒆𝒔𝒉 𝑺𝒊𝒛𝒆 𝒂𝒕 𝑵𝒐𝒔𝒆 𝑪𝒐𝒏𝒆: 𝐸𝑙𝑒𝑚𝑒𝑛𝑡 𝑠𝑖𝑧𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 × 𝑁𝑜𝑠𝑒 𝑐𝑜𝑛𝑒 𝑚𝑒𝑠ℎ 𝑓𝑎𝑐𝑡𝑜𝑟 = 0.5 × 0.15

= 0.075

𝑴𝒆𝒔𝒉 𝑺𝒊𝒛𝒆 𝒂𝒕 𝒐𝒖𝒕𝒆𝒓 𝒘𝒂𝒍𝒍𝒔: 𝐸𝑙𝑒𝑚𝑒𝑛𝑡 𝑠𝑖𝑧𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 × 𝑅𝑒𝑠𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑚𝑒𝑠ℎ 𝑓𝑎𝑐𝑡𝑜𝑟

= 0.5 × 1.5 = 0.75

4. Python used to call the GMSH and obtain a Mesh file (.msh) from geometry file (.geo).

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a) b)

Figure 4. Initial mesh created with parameters defined in the geometry file (a) Adapted Mesh

that has optimized and refined with the pressure distribution (b)

As it shown from figure domain has been selected to be close at recommend sizes for this type of

analysis in the CFD.

In y direction ≈ 14 times of the half width of the Nose cone on both directions

In x direction ≈ 2 times of the length of the Nose cone on both directions

Coding Cycle

At last the cycle obtained can be schematized as it seen on the following figure.

Page 7 of 13

Fmincon (Main Script)

Geometry Generator

CFD-SolverDrag Obtained

Results feed into

optimization

Figure 5. Schematics of the optimization cycle

III. Results

Optimization Code tested the following total 204 Cases

Mesh created

by Python call

Geometric

parameters inputted

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Figure 6. Total cases for curvature of Nose Cone

Last convergence plot has shown in below

Figure 7. Residual vs number of iteration plot for optimal shape

a) b)

Page 9 of 13

Figure 8. Pressure Contours of the Initial Shape (a) Pressure Contours of the Optimal Shape(b)

a) b)

Figure 9. Mach Contours of the Initial Shape (a) Mach Number Contours of the Optimal

Shape(b)

Finally, the optimal shape has been determined by the solver

Figure 10. Comparison of initial Nose Cone Curvature and Optimal Nose Cone Curvature

Page 10 of 13

Polynomial coefficients are obtained as follows;

𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7

Initial Coefficients -0.2099 0.7994 -1.0935 0.6790 -0.4962 0.6894 -1.697e-05

Optimal Coefficients -0.2170 0.8554 -1.2569 0.7182 -0.4911 0.7593 -1.6969e-05

Table 2. Initial and Optimal polynomial Coefficients

MATLAB FMINCON Founds the optimization parameter as such

Iteration Function

count

Function

Value

Feasibility Normal of the

Step

First-Order

Optimality

20 248 6.178 × 101 0 0 1.291 × 10−5

Table 3. Optimization Parameters

Drag coefficients for initial and Optimal shape of the Nose cone

(𝐶𝑑)𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 0.6777 (𝐶𝑑)𝑜𝑝𝑡𝑖𝑚𝑎𝑙 = 0.6178

IV. Conclusion and comments

After applying the nonlinear optimization in MATLAB and by using the Polynomial(Parsec) curve

approach to create the geometry and utilizing the meshing software into the optimization cycle the

following conclusion can be made. Total seven design variables considered for the optimization

study for Nose cone shape of the F-35 fighter jet which purely consist the polynomial coefficients

if the Parsec curve. Minimizing drag coefficients was the objective of my study and from the result

it seemed to be a successful study. Drag coefficient was definitely decreased by almost 9 percent

which is a considered significant decrease for designers and aerodynamicists whose main research

and development areas are focus on improving the product that will serve in the most critical parts

of the national defense industry. We must realize that this study is purely test case and may not

reflect the real values on what actual performance might yield. However, I believe it’s a good

compromise on the computational resources and the accuracy of the results.

V. References

1. https://quickersim.com/cfdtoolbox/

2. https://grabcad.com/library/f35-3

3. N.R Deepak, T. Ray, R. Boyce, Nose Cone design Optimization for a Hypersonic Flight

Experimental Trajectory, AIAA 2007-7998

4. https://www.mathworks.com/help/optim/ug/fmincon.html

5. GMSH Reference Manual

Page 11 of 13

VI. Appendix

Main Optimization Script

clc;close all;clear

% addpath('./Release');

addpath('C:\Users\Ege''s Tablet\Documents\MATLAB\Add-

Ons\Apps\QuickerSimCFDToolbox\code\Release');

load x

coor

load ycoor

a=polyfit(x,y,6);

y=polyval(a,x);

figure(5) % initial curve of nose

plot(x,y,'-*')

grid minor

xlabel('x')

ylabel('y')

hold on

gridPts=length(x);

x0 = [a(1),a(2),a(3),a(4),a(5),a(6),a(7)]; %initial values

tic;

lb = [a(1)-abs(0.3*a(1)),a(2)-abs(0.3*a(2)),a(3)-

abs(0.3*a(3)),a(4)-abs(0.3*a(4)),...

a(5)-abs(0.3*a(5)),a(6)-abs(0.3*a(6)),a(7)-abs(0.3*a(7))];

disp(x0)

ub =

[a(1)+abs(0.3*a(1)),a(2)+abs(0.3*a(2)),a(3)+abs(0.3*a(3)),a(4)+a

bs(0.3*a(4)),...

a(5)+abs(0.3*a(5)),a(6)+abs(0.3*a(6)),a(7)+abs(0.3*a(7))];

options =

optimoptions(@fmincon,'Display','iter');%,'PlotFcn',{@optimplotx

,...

% @optimplotfval});

fun=@(a)objfun(a);

[a,fval,lambda] =

fmincon(fun,x0,[],[],[],[],lb,ub,@nonlcon,options);

elapsed = toc;

y=polyval(a,x);

fprintf('TIC TOC: %g\n', elapsed);

figure(5) % final curve of nose

plot(x,y,'-+')

fprintf('%.f',a)

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function f = objfun(a)

load xcoor

disp('a=')

disp(a)

y=polyval(a,x);

gridPts=length(x);

conemshsize=0.15; %0.15

wallmshsize=1.5; %0.75

aa=5; %4 % x domain

bb=10; %8 % y domain

Mach=1.61; % mach number

% create geometry

[out,in,walls,folder]=geo_get(gridPts,conemshsize,wallmshsize,aa

,bb,x,y);

% cfd solver

% disp(y)

[D,Cd]=compressiblesolver(out,in,walls,Mach,folder);

% Cd=a(5)*5;

% disp(Cd)

f= Cd; %obj. func.

function [c,ceq] = nonlcon(a)

load xcoor

y=polyval(a,x);

figure(5)

plot(x,y)

hold on

% ainit=polyfit(x,y,6);

q=polyder(a);

slope=polyval(q,x); % slope of each node

c(1)=slope(1);

for i=1:length(slope)-1

c(2)=slope(i+1)-slope(i); %negative slope constraint

if c(2)>0

break;

end

if c(1)<0

continue

else

c(1)=slope(i+1);

end

end

%if there is no negative slope till end clear first consraint

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if c(2)==slope(end)-slope(end-1)

c(1)=0;

end

c(3)=abs(sum(a)-0.368020305)-1e-03; % fix the width of cone at

end.

ceq = []; %non eq. cons. a(end)?

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