Examples¶
iPython Notebook Examples¶
More examples, including some in-depth and experimental models, can be seen on nbviewer.
A Trivial GP¶
The most trivial GP we can think of: minimize \(x\) subject to the constraint \(x \ge 1\).
from gpkit import Variable, GP
# Decision variable
x = Variable('x')
# Constraint
constraints = [x >= 1]
# Objective (to minimize)
objective = x
# Formulate the GP
gp = GP(objective, constraints)
# Solve the GP
sol = gp.solve(printing=False)
# print selected results
print "Optimal cost: %s" % sol['cost']
print "Optimal x val: %s" % sol(x)
Of course, the optimal value is 1. Output:
Optimal cost: 1.0
Optimal x val: 1.0
Maximizing the Volume of a Box¶
This example comes from Section 2.4 of the GP tutorial, by S. Boyd et. al.
from gpkit import Variable, GP
# Parameters
alpha = Variable("alpha", 2, "-", "lower limit, wall aspect ratio")
beta = Variable("beta", 10, "-", "upper limit, wall aspect ratio")
gamma = Variable("gamma", 2, "-", "lower limit, floor aspect ratio")
delta = Variable("delta", 10, "-", "upper limit, floor aspect ratio")
A_wall = Variable("A_{wall}", 200, "m^2", "upper limit, wall area")
A_floor = Variable("A_{floor}", 50, "m^2", "upper limit, floor area")
# Decision variables
h = Variable("h", "m", "height")
w = Variable("w", "m", "width")
d = Variable("d", "m", "depth")
#Constraints
constraints = [A_wall >= 2*h*w + 2*h*d,
A_floor >= w*d,
h/w >= alpha,
h/w <= beta,
d/w >= gamma,
d/w <= delta]
#Objective function
V = h*w*d
objective = 1/V #To maximize V, we minimize its reciprocal
# Formulate the GP
gp = GP(objective, constraints)
# Solve the GP
sol = gp.solve(printing=False)
# Print results table
print sol.table()
The output is
Cost
----
0.003674 [1/m**3]
Free variables
--------------
d : 8.17 [m] depth
h : 8.163 [m] height
w : 4.081 [m] width
Swept variables
---------------
Constants
---------
A_{floor} : 50 [m**2] upper limit, floor area
A_{wall} : 200 [m**2] upper limit, wall area
alpha : 2 lower limit, wall aspect ratio
beta : 10 upper limit, wall aspect ratio
delta : 10 upper limit, floor aspect ratio
gamma : 2 lower limit, floor aspect ratio
Constant and swept variable sensitivities
-----------------------------------------
A_{wall} : -1.5 upper limit, wall area
alpha : 0.5 lower limit, wall aspect ratio
Water Tank¶
Say we had a fixed mass of water we wanted to contain within a tank, but also wanted to minimize the cost of the material we had to purchase (i.e. the surface area of the tank):
from gpkit import Variable, VectorVariable, GP
M = Variable("M", 100, "kg", "Mass of Water in the Tank")
rho = Variable("\\rho", 1000, "kg/m^3", "Density of Water in the Tank")
A = Variable("A", "m^2", "Surface Area of the Tank")
V = Variable("V", "m^3", "Volume of the Tank")
d = VectorVariable(3, "d", "m", "Dimension Vector")
constraints = (A >= 2*(d[0]*d[1] + d[0]*d[2] + d[1]*d[2]),
V == d[0]*d[1]*d[2],
M == V*rho
)
gp = GP(A, constraints)
sol = gp.solve(printing=False)
print sol.table()
The output is
Cost
----
1.293 [m**2]
Free variables
--------------
A : 1.293 [m**2] Surface Area of the Tank
V : 0.1 [m**3] Volume of the Tank
d : [ 0.464 0.464 0.464 ] [m] Dimension Vector
Swept variables
---------------
Constants
---------
M : 100 [kg] Mass of Water in the Tank
\rho : 1000 [kg/m**3] Density of Water in the Tank
Constant and swept variable sensitivities
-----------------------------------------
M : 0.6667 Mass of Water in the Tank
\rho : -0.6667 Density of Water in the Tank
Simple Wing¶
This example comes from Section 3 of Geometric Programming for Aircraft Design Optimization, by W. Hoburg and P. Abbeel.
from gpkit.shortcuts import *
import numpy as np
# Define the constants in the problem
CDA0 = Var('(CDA_0)', 0.0306, 'm^2', label='Fuselage drag area')
C_Lmax = Var('C_{L,max}', 2.0, label='Maximum C_L, flaps down')
e = Var('e', 0.96, label='Oswald efficiency factor')
k = Var('k', 1.2, label='Form factor')
mu = Var('\\mu', 1.78E-5, 'kg/m/s', label='Viscosity of air')
N_lift = Var('N_{lift}', 2.5, label='Ultimate load factor')
rho = Var('\\rho', 1.23, 'kg/m^3', label='Density of air')
Sw_S = Var('\\frac{S_{wet}}{S}', 2.05, label='Wetted area ratio')
tau = Var('\\tau', 0.12, label='Airfoil thickness-to-chord ratio')
V_min = Var('V_{min}', 22, 'm/s', label='Desired landing speed')
W_0 = Var('W_0', 4940, 'N', label='Aircraft weight excluding wing')
cww1 = Var('cww1', 45.42, 'N/m^2', 'Wing weight area factor')
cww2 = Var('cww2', 8.71E-5, '1/m', 'Wing weight bending factor')
# Define decision variables
A = Var('A', label='Aspect ratio')
C_D = Var('C_D', label='Drag coefficient')
C_L = Var('C_L', label='Lift coefficient')
C_f = Var('C_f', label='Skin friction coefficient')
S = Var('S', 'm^2', label='Wing planform area')
Re = Var('Re', label='Reynolds number')
W = Var('Re', 'N', label='Total aircraft weight')
W_w = Var('W_w', 'N', label='Wing weight')
V = Var('V', 'm/s', label='Cruise velocity')
# Define objective function
objective = 0.5 * rho * V**2 * C_D * S
# Define constraints
constraints = [C_f * Re**0.2 >= 0.074,
C_D >= CDA0 / S + k * C_f * Sw_S + C_L**2 / (np.pi * A * e),
0.5 * rho * V**2 * C_L * S >= W,
W >= W_0 + W_w,
W_w >= (cww1*S +
cww2 * N_lift * A**1.5 * (W_0 * W * S)**0.5 / tau),
0.5 * rho * V_min**2 * C_Lmax * S >= W,
Re == rho * V * (S/A)**0.5 / mu]
# Formulate the GP
gp = GP(objective, constraints)
# Solve the GP
sol = gp.solve()
# Print the solution table
print sol.table()
The output is
Using solver 'cvxopt'
Solving for 9 variables.
Solving took 0.0629 seconds.
Cost
----
255 [kg*m/s**2]
Free variables
--------------
A : 12.7 Aspect ratio
C_D : 0.0231 Drag coefficient
C_L : 0.6512 Lift coefficient
C_f : 0.003857 Skin friction coefficient
Re : 7189 [N] Total aircraft weight
Re : 2.598e+06 Reynolds number
S : 12.08 [m**2] Wing planform area
V : 38.55 [m/s] Cruise velocity
W_w : 2249 [N] Wing weight
Swept variables
---------------
Constants
---------
(CDA_0) : 0.0306 [m**2] Fuselage drag area
C_{L,max} : 2 Maximum C_L, flaps down
N_{lift} : 2.5 Ultimate load factor
V_{min} : 22 [m/s] Desired landing speed
W_0 : 4940 [N] Aircraft weight excluding wing
\frac{S_{wet}}{S} : 2.05 Wetted area ratio
\mu : 1.78e-05 [kg/m/s] Viscosity of air
\rho : 1.23 [kg/m**3] Density of air
\tau : 0.12 Airfoil thickness-to-chord ratio
cww1 : 45.42 [N/m**2] Wing weight area factor
cww2 : 8.71e-05 [1/m] Wing weight bending factor
e : 0.96 Oswald efficiency factor
k : 1.2 Form factor
Constant and swept variable sensitivities
-----------------------------------------
(CDA_0) : 0.1097 Fuselage drag area
C_{L,max} : -0.1307 Maximum C_L, flaps down
N_{lift} : 0.2923 Ultimate load factor
V_{min} : -0.2614 Desired landing speed
W_0 : 0.9953 Aircraft weight excluding wing
\frac{S_{wet}}{S} : 0.4108 Wetted area ratio
\mu : 0.08217 Viscosity of air
\rho : -0.1718 Density of air
\tau : -0.2923 Airfoil thickness-to-chord ratio
cww1 : 0.09428 Wing weight area factor
cww2 : 0.2923 Wing weight bending factor
e : -0.4795 Oswald efficiency factor
k : 0.4108 Form factor