Minimize and maximize these variables and these functions?

 

This is a genetic multi-objective algorithm of a turbojet two spool afterburner engine. The idea is to see in 3D Pareto front maximizing and minimizing the functions in relation to the constraints and to opt for the best overall design.

 Establish realistic values for the variables and equations .

 

Variables and equations:

%variables

f=1; % f= fuel to air ratio

Fn=2; % Fn= net thurst

m0=3; % m0= air mass flow rate

p3=4; % p3=inlet pressure

Tpz=5; % Tpz=primary zone temperature

o=6; % o=equivalence ratio

t=7; % t=residence time

aa=8; % aa=pressure term

WTO=9; % Wto=take off weight

WLanding=10; % Wlanding=landing weight

WF=11; % Wf=fuel weight

WE=12; % WE=empty weight

WPL=13; % Wpl=payload weight

WC=14; % Wc=crew weight

M=15; % Fuel load mass

Fab=16; % afterburner fuel-to-air ratio

V9=17; % exhaust velocity

V0=18; % cruising speed

f=19; % fuel to air ratio

QR=20; % fuel heating value

NB=21; % burner efficiency

T1=22; % inlet temperature

T0=23; % ambient static temperature

L=24; % lift

p=25; % density

V=26; % velocity

A=27; % wing area

bw=28; % wing span

sw=29; % Trapezoidal wing area in ft

WZF=30; % zero fuel weight

b=31; % upper dimension of wing

cr=32; % right dimension side of wing

ct=33; % left dimension side of wing

WS=34; % wing span

RC=35; % root chord

l=36; % lenght

m=37; % moment

W0=38; % gross weight of aicraft

VH=39; % maximum demonstrated level airpeed

VS=40; % stalling speed

VS1=41; % stalling speed with flaps retracted

KG=42; % Gust alleviation factor

UREF=43; % reference gust velocity

VC=44; % design cruise speed

pi=3.14; % pi number

Ude=45; % vertical gust velocity

x=46; % distance in ft penetrated into the gust

CMGC=47; % Mean geometric chord

UG=48; % i dont know the name

D=49; % Drag

Y=Wi/Wf; % fuel weight fraction for the cruise segment

Y1=49;

CL=51; % lift coefficient

CD=52; % drag coefficient

A=53; % Area

%Maximize this ones

NP=((Fn/m0)*V0)/((1+f+Fab)*((V9^2)/2)-((V0^2)/2); % propulsive efficiency

NO=((Fn/m0)*V0)/(f*QR*NB); % overall efficiency

TE=1-(1/(T1/T0)); % thermal efficiency

T=m0*V0; % Thurst

%and Minimize this ones :

TSFC =(f/(Fn/m0)); % TSFC=specific fuel consumption

Nox=(10^13)*(p3/(1.4*(10^6)))*exp(-71442/Tpz)*(7.56*(o^7.2)-1.6)*(t^0.64); % NOx Emissions (LTO)

WF=WTO-WLanding; % Fuel Weight

WTO=WT0*(1-(WF/WTO)-(WE/WTO)); % take off weight

WTO=WPL/WC; % take off weight

maxFT=M/(TSFC/Fn); % max flight time

%constraints to apply

CLmin=L/(p*(V^2)*(A/2)); % maximum lift coefficient

CLmax=L/(p*(V^2)*(A/2));% maximum lift coefficient % maximum lift coefficient

WW=(17*bw*sw*sqrt(WZF/WTO)); % wing weight

S=b*((cr+ct)/2); % wing planform area

WV=(5*WS*RC*l)/0.5; % wing volume

PM=m/(0.5*p*V^2)*(A^2); % Pitching moment

LF=2.1+(24000/(W0+10000)); % Load factor

MaximumDCS=33*sqrt((W0)/S)); % minimum design cruising speed

MinimumDCS=0.9*VH; % maximum design cruising speed

DDS=VD>1.4*MinimumDCS; % design dive speed

nplus=(0.003388*(V^2)*S*CLmax)/W0; % Positive stall line:

nless=(-0.003388*(V^2)*S*CLmin)/W0; % Negative stall line:

VA=VS*sqrt(nplus); % maneuvering speeds

VG=sqrt((2*nless*W)/(p*S*CLmin)); % maneuvering speeds

DSMGI=VS1*((KG*UREF*VC*a)/498*W0)^1/3; % Design speed for maximum Gust Intensity

GLF=(Ude/2)*(1-cos(2*pi*x/(25*CMGC)) % Gust load factor

GAF=0.88*UG/(5.3+UG); % Gust alleviation factor

R=(V/TSFC)*(L/D)*ln(Wi/WF); % Range

E=((1/TSFC)*(CL/CD)*ln(Wo/W1)); % Endurance

M=p*A*V; % mass flow rate

MODEL CODE:

% Define the optimization variables

vars = optimvar('vars', 4, 'LowerBound', [0 0 0 0], 'UpperBound', [2000 800 30 10]);

% Define the objective functions

f(1) = specific_fuel_consumption(vars(1), vars(2), vars(3), vars(4));

f(2) = NOx_emissions(vars(1), vars(2), vars(3), vars(4));

f(3) = fuel_weight(vars(1), vars(2), vars(3), vars(4));

f(4) = Wto(vars(1), vars(2), vars(3), vars(4));

f(5) = flight_time(vars(1), vars(2), vars(3), vars(4));

g(1) = propulsive_efficiency(vars(1), vars(2), vars(3), vars(4));

g(2) = overall_efficiency(vars(1), vars(2), vars(3), vars(4));

g(3) = thermal_efficiency(vars(1), vars(2), vars(3), vars(4));

g(4) = thurst(vars(1), vars(2), vars(3), vars(4));

% Define the optimization problem

prob = optimproblem('Objective', [f, -g]);

% Define the constraints

prob.Constraints.con1 = vars(1) <= 2000;

prob.Constraints.con2 = vars(2) <= 800;

prob.Constraints.con3 = vars(3) <= 30;

prob.Constraints.con4 = vars(4) <= 10;

% Define the solver options

options = optimoptions('gamultiobj', 'PlotFcn', 'gaplotpareto');

% Solve the problem

[x, fval, exitflag, output] = gamultiobj(@(x) evaluate_objective(x), 4, [], [], [], [], [0 0 0 0], [2000 800 30 10], @(x) evaluate_constraints(x), options);

% Plot the 3D Pareto front

pareto(fval(:,1), -fval(:,2), -fval(:,3));

xlabel('Structures');

ylabel('Aerodynamics');

zlabel('Propulsion');

King Regards