# Cantilever Beam Function

The Cantilever Beam function is defined as: $f(w,t) = \frac{4L^3}{Ewt}*\sqrt{ (\frac{Y}{t^2})^2 + (\frac{X}{w^2})^2 }$ With parameters L,E,X and Y given.

Let's import Surrogates and Plots:

using Surrogates
using Plots
default()

Define the objective function:

function f(x)
t = x
w = x
L = 100.0
E = 2.770674127819261e7
X = 530.8038576066307
Y = 997.8714938733949
return (4*L^3)/(E*w*t)*sqrt( (Y/t^2)^2 + (X/w^2)^2)
end
f (generic function with 1 method)

Let's plot it:

n = 100
lb = [1.0,1.0]
ub = [8.0,8.0]
xys = sample(n,lb,ub,SobolSample());
zs = f.(xys);
x, y = 0:8, 0:8
p1 = surface(x, y, (x1,x2) -> f((x1,x2)))
xs = [xy for xy in xys]
ys = [xy for xy in xys]
p2 = contour(x, y, (x1,x2) -> f((x1,x2)))
scatter!(xs, ys)
plot(p1, p2, title="True function")

Fitting different Surrogates:

mypoly = PolynomialChaosSurrogate(xys, zs,  lb, ub)
loba = PolynomialChaosSurrogate(xys, zs,  lb, ub)
rad = RadialBasis(xys,zs,lb,ub)

Plotting:

p1 = surface(x, y, (x, y) -> mypoly([x y]))
scatter!(xs, ys, zs, marker_z=zs)
p2 = contour(x, y, (x, y) -> mypoly([x y]))
scatter!(xs, ys, marker_z=zs)
plot(p1, p2, title="Polynomial expansion")
p1 = surface(x, y, (x, y) -> loba([x y]))
scatter!(xs, ys, zs, marker_z=zs)
p2 = contour(x, y, (x, y) -> loba([x y]))
scatter!(xs, ys, marker_z=zs)
plot(p1, p2, title="Lobachevsky")
p1 = surface(x, y, (x, y) -> rad([x y]))
scatter!(xs, ys, zs, marker_z=zs)
p2 = contour(x, y, (x, y) -> rad([x y]))
scatter!(xs, ys, marker_z=zs)
plot(p1, p2, title="Inverse distance surrogate")