aerodynamic shape optimization nose cone of f-35 lightning ii
TRANSCRIPT
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/325312945
Aerodynamic shape optimization Nose Cone of F-35 Lightning II
Research · May 2018
DOI: 10.13140/RG.2.2.29953.15200
CITATIONS
0READS
1,582
1 author:
Some of the authors of this publication are also working on these related projects:
Double Multiple Stream Tube Model and Matlab Analysis of Vertical Axis Wind Turbine View project
Flapped wing aerodynamic shape optimization for minimize drag to lift ratio View project
Ege Konuk
Old Dominion University
5 PUBLICATIONS 0 CITATIONS
SEE PROFILE
All content following this page was uploaded by Ege Konuk on 23 May 2018.
The user has requested enhancement of the downloaded file.
Page 1 of 13
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.
Page 2 of 13
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.
Page 3 of 13
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
Page 4 of 13
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
Page 5 of 13
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).
Page 6 of 13
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
Page 8 of 13
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)
Page 12 of 13
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
Page 13 of 13
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)?
View publication statsView publication stats