def exo5():
valeur = np.array([0.5, 1., 1.5])
F = np.array([])
G = np.array([])
n = 400
t = np.linspace(-2 * pi, 2 * pi, n)
for a in valeur:
f = np.array([])
g = np.array([])
for i in range(0, n):
f = np.append(f, t[i] - a * sin(t[i]))
g = np.append(g, 1 - a * cos(t[i]))
F = np.append(F, f)
G = np.append(G, g)
F = F.reshape((3, n))
G = G.reshape((3, n))
# Question a
Graph2D(t, F, "t", "f", "f(t)", ["0.5", "1.", "1.5"]).show()
Graph2D(t, G, "t", "g", "g(t)", ["0.5", "1.", "1.5"]).show()
# Question b
Graph2D(F, G, "f", "g", "question b", legend=["0.5", "1.", "1.5"]).show()
# Question c
for i in range(0, n):
if (abs(G[2, i]) < 0.05): # intersection with the x-axis
print("abscissa :", F[2, i], "i :", i, "t :", t[i])
if (abs(F[2, i]) < 1e-3): # double point
print("i :", i, "t :", t[i], "g :", G[2, i])
def well():
# potential well number
nw = 4
yd = 1
yg = 1
y0 = np.array([yg, yd])
zd = 0
zg = 0
z0 = np.array([zg, zd])
n = 300
a = getA(nw, n)
#testPotential(nw,n)
global E
E = -266.7709 / Eh
E1 = E * Eh
x, y1, z1 = tirEuler(n, a, y0, z0, yDer, zDer)
y1 = y1 / np.linalg.norm(y1)
E = -3.98506 / Eh
E2 = E * Eh
x, y2, z2 = tirEuler(n, a, y0, z0, yDer, zDer)
y2 = y2 / np.linalg.norm(y2)
legend = ["E = " + str(E1), "E = " + str(E2)]
graph = Graph2D(x, [y1, y2], "x [a0]", "Psi(x)", "Potential well", legend)
graph.show()
def exo2():
"""
"physicists" Hermite polynomials computation
"""
# question b :
print("H5(0.5) :", H(5, 0.5))
min = -3
max = 3
N = 200
x = np.linspace(min, max, N)
#tracer
y = [
formHn(0, min, max, N),
formHn(1, min, max, N),
formHn(2, min, max, N),
formHn(3, min, max, N),
formHn(4, min, max, N),
formHn(5, min, max, N)
]
legend = ['H0', 'H1', 'H2', 'H3', 'H4', 'H5']
graph = Graph2D(x, y, "x", "Hn(x)", "polynômes d'Hermite", legend)
graph.ylimit(-50, 50)
graph.show()
def testPotential(nw, n):
a = getA(nw, n)
x = np.linspace(0, a, n + 1)
v = [V(x[i], nw) for i in range(0, n + 1)]
graph = Graph2D(x, [v], "x [a0]", "V(x)", "Kronig Penney potential")
graph.show()
def testVoltageStep(nw, n):
a = getA(nw, n)
x = np.linspace(0, a, n + 1)
v = [V(x[i], nw) + V1(x[i], a) for i in range(0, n + 1)]
graph = Graph2D(x, [v], "x [a0]", "V(x)", "Potential + voltage step")
graph.show()
def questionC2():
T = 5000
x, y, z = approx(T, I)
ye = [exact(xi, I) for xi in x]
legend = ["T = 0 N", "T = 5000 N"]
graph = Graph2D(x, [ye, y], "x", "y", "Déformation poutre", legend)
graph.show()
def questionC1():
x, y, z = approx(T, I)
#Exact Solution
ye = [exact(xi, I) for xi in x]
legend = ["exacte", "approchée"]
graph = Graph2D(x, [ye, y], "x", "y", "Déformation poutre", legend)
graph.show()
def testImpurity(nw, n):
a = getA(nw, n)
x = np.linspace(0, a, n + 1)
iw = randint(1, nw)
v = [V(x[i], nw) + impurity(x[i], iw) for i in range(0, n + 1)]
#v = [impurity(x[i],iw) for i in range(0,n+1)]
graph = Graph2D(x, [v], "x [a0]", "V(x)", "adding Impurity")
graph.show()
def testAmorphous(nw, n):
a = getA(nw, n)
x = np.linspace(0, a, n + 1)
sizes = [1.5 * a0 * random() for i in range(0, nw)]
#print("sizes : {}".format(sizes))
v = [V(x[i], nw) + amorphous(x[i], nw, sizes) for i in range(0, n + 1)]
graph = Graph2D(x, [v], "x [a0]", "V(x)", "amorphous solid")
graph.show()
def testTransport():
x = np.linspace(0,4,81)
c = 1
Tfin = 2
r = 0.9
opt = "imp"
U = transport1D(x,c,Tfin,r,opt="imp")
graph = Graph2D(x,[U],"x","u","Transport equation {}".format(opt))
graph.show()
def exo3():
L = 2
M = 4
N = 3
n = 200
t = np.linspace(0, 2 * pi, n)
x = np.array([])
y = np.array([])
for i in range(0, n):
x = np.append(x, sin(L * t[i]) * cos(M * t[i]))
y = np.append(y, sin(L * t[i]) * sin(N * t[i]))
yt = Graph2D(t, [x], "t", "x")
yx = Graph2D(x, [y], "x", "y")
tx = Graph2D(x, [t], "x", "t")
yt.show()
yx.show()
tx.show()
def interpolExp():
x0 = np.array([-1, 0.5, 1.5, 2.])
f = np.exp(x0)
x = np.linspace(-2.3, 50)
poly = pDivise(x, x0, f)
xExp = np.exp(x)
graph = Graph2D(x, [poly, xExp], 'x', 'f(x)', 'Exp interpolation',
[r'$p_{0123}}$', 'exp'])
graph.xlimit(-2, 3)
graph.ylimit(-5, 11)
graph.show()
def testHeat():
x = np.array([
0, 0.025, 0.05, 0.125, 0.175, 0.2, 0.25, 0.3, 0.4, 0.425, 0.5, 0.575,
0.6, 0.675, 0.725, 0.75, 0.8, 0.85, 0.95, 0.975, 1
])
r = 0.09
Tfin = 1
U = heat1D(x, r, Tfin)
graph = Graph2D(x, U, "x", "u", "1D Heat equation solution")
graph.show()
def exo1():
a = 0.6
b = 7.
min = 0.
max = 5.
n = 150
t = np.linspace(min, max, n)
x = np.array([])
y = np.array([])
for i in range(0, n):
x = np.append(x, exp(-a * t[i]) * cos(b * t[i]))
y = np.append(y, exp(-a * t[i]) * sin(b * t[i]))
yt = Graph2D(t, [x], "t", "x")
yx = Graph2D(x, [y], "x", "y")
tx = Graph2D(x, [t], "x", "t")
yt.show()
yx.show()
tx.show()
def questionE():
# Effet des forces de compressions
T = -5000
b = 1e-1
a = 5e-2
I = b * a**3
x, y, z = approx(T, I)
ye = [exact(xi, I) for xi in x]
legend = ["T = 0 N", "T = -5000 N"]
graph = Graph2D(x, [ye, y], "x", "y", "Déformation poutre", legend)
graph.show()
def testPoisson():
def f(x):
return x * (1 - x)
x = np.array([
0, 0.025, 0.05, 0.125, 0.175, 0.2, 0.25, 0.3, 0.4, 0.425, 0.5, 0.575,
0.6, 0.675, 0.725, 0.75, 0.8, 0.85, 0.95, 0.975, 1
])
u = poisson1D(x, f)
graph = Graph2D(x, [u], "x", "u", "1D Poisson equation solution")
graph.show()
def test():
# Plot of a Bezier curve :
xPoints = np.array([0, 2, 4, 6])
yPoints = np.array([1, 4, 1, 4])
u = np.linspace(0, 1, 50)
x, y = curve(u, xPoints, yPoints)
print("P0 : ({},{})".format(x[0], y[0]))
print("P3 : ({},{})".format(x[-1], y[-1]))
graph = Graph2D([xPoints, x], [yPoints, y], 'x', 'y', 'Bézier curve',
['points', 'curve'])
graph.show()
def pinpointK(n,x0,s0,t0,k1,k2):
global k
k = k1; k1 = k
r,s1,t1 = tirRK(n,1,s0,t0,sDer,tDer,x0)
print("S(1) : {}".format(s1[-1]))
k = k2; k2 = k
r,s2,t2 = tirRK(n,1,s0,t0,sDer,tDer,x0)
print("S(1) : {}".format(s2[-1]))
legend = ["k = "+str(k1),"k = "+str(k2)]
graph = Graph2D(r,[s1,s2],"rho","S(rho)","Méthode du tir",legend)
graph.show()
def exo7():
"""
Plot the approximation versus the actual function
"""
n = 100
x = np.linspace(-pi, pi, n)
approx = np.array([])
TAN = np.tan(x)
for i in range(0, n):
approx = np.append(approx, tan(x[i]))
Graph2D(x, [TAN, approx], 'x', 'tan(x)', 'exo7',
['tan', 'fraction']).show()
def exo4():
min = -2 * pi
max = 2 * pi
n = 400
t = np.linspace(min, max, n)
x = np.array([])
y = np.array([])
for i in range(0, n):
x = np.append(x, cos(t[i]) * cos(t[i] / 2.))
y = np.append(y, sin(t[i]) * cos(3. * t[i]))
# period of x : 4*pi, period of y : pi
a = Graph2D(t, [x, y], "t", "x", "question a", ["x", "y"])
a.show()
# plot parametric curve
b = Graph2D(x, [y], "x", "y", "question b")
b.show()
for i in range(0, n):
# loop on the intervals satisfying
if (y[i] <= 1e-8 and abs(x[i]) <= 1 and abs(x[i]) >= 0.8):
print("i :", i, "t :", t[i])
def exo6():
min = -pi
max = 2 * pi
n = 400
t = np.linspace(min, max, n)
# Question a
terme0 = np.cos(t) # 2*pi periodic
terme1 = -1 / 3. * np.cos(3 * t) # 2*pi/3 periodic
terme2 = 1 / 5. * np.cos(5 * t) # 2*pi/5 periodic
Graph2D(t, [terme0, terme1, terme2], "t", "f", "question a").show()
# Question b
def Sn(n, t):
terme = np.cos(t)
somme = terme
for i in range(1, n + 1):
terme = -1 / (2 * i + 1) * np.cos((2 * i + 1) * t)
somme += terme
return somme
y = [Sn(3, t), Sn(4, t), Sn(6, t), Sn(10, t)]
legend = ["s3", "s4", "s6", "s10"]
Graph2D(t, y, "t", "Sn", "Partial sums", legend).show()
def exo5():
n = 200
min = -10.
max = 2.
X = np.linspace(-10., 2., n)
A = np.array([])
B = np.array([])
x = -10.
pas = (max - min) / n
for i in range(0, n):
A = np.append(A, Ai(x))
B = np.append(B, Bi(x))
x += pas
graph = Graph2D(X, [A, B], 'x', 'f(x)', "Airy Functions", ['Ai', 'Bi'])
graph.ylimit(-.5, 1)
graph.show()
def questionD():
x = np.linspace(0, L, 100)
# a < b :
b = 1e-1
a = 5e-2
I = b * a**3
y1 = [exact(xi, I) for xi in x]
# a > b :
a = 1e-1
b = 5e-2
I = b * a**3
y2 = [exact(xi, I) for xi in x]
legend = ["a < b", "a > b"]
graph = Graph2D(x, [y1, y2], "x", "y", "Déformation poutre", legend)
graph.show()
def exo1():
def f(x):
return x-0.2*sin(x)-0.5
def derf(x):
return 1-0.2*cos(x)
def g(x):
return 0.2*sin(x)+0.5
tol = 1e-5
n = 20
a = 0
b = 1
# with the bissection method
xa = bissection(a,b,f,tol)
print("Bissection : x* = {}".format(xa))
# with the regulaFalsi method
xb = regulaFalsi(a,b,f,tol,n)
print("Regula Falsi : x* = {}".format(xb))
# with the NewtonRaphson method
xc = newtonRaphson(b,tol,f,derf)
print("Newton Raphson : x* = {}".format(xc))
# with the fixed point method
xd,x,y = iterations(g,0,tol,n)
xf = np.linspace(a,b,50)
yf = [g(x) for x in xf]
print("Fixed Points : x* = {}".format(xd))
legend = ['x','f','iteration']
graph = Graph2D([xf,xf,x],[xf,yf,y],'x','f(x)','f(x)=x',legend)
graph.show()
# with the secant method
xe = secant(a,b,f,tol,n)
print("Secant : x* = {}".format(xe))
def exo3():
a = 0.5
b = 1.5
tol = 1e-1
def f(x):
return 1 / tan(x) - x
def derf(x):
return -1 / sin(x)**2 - 1
x = np.linspace(a, b, 50)
y = np.divide(1, np.tan(x))
Graph2D(x, [x, y], 'x', 'f(x)', 'Fixed point ?',
['y = x', 'cotan(x)']).show()
# question b
xb = bissection(a, b, f, tol)
print("Bissection : x* = {}".format(xb))
# question c
xc = newtonRaphson(1, tol, f, derf)
print("Newton Raphson : x* = {}".format(xc))
def exo4():
"""
Plot the Bessel function J0(x)
"""
# Question b :
n = 1
err = 1
while (err > 1e-6):
n += 1
err = err * 10 / (n - 1)
print("n : ", n)
test = np.array([-0.1775967713, -0.2459357644, -0.0142244728])
print("J0(5)-J0(5)test : ", J0(5., n) - test[0])
print("J0(10)-J0(10)test : ", J0(10., n) - test[1])
print("J0(15)-J0(15)test : ", J0(15., n) - test[2])
x = np.linspace(0, 20, 200)
bessel0 = np.array([])
for i in range(0, 200):
bessel0 = np.append(bessel0, J0(x[i], n))
Graph2D(x, [bessel0], 'x', 'J0(x)', "Bessel function J0").show()
def drawCircles():
origin = 0.8 + 0.5j
circle1 = circle(origin, 0.75)
circle2 = circle(origin, 1)
circle3 = circle(origin, 1.25)
x = np.array([circle1.real, circle2.real, circle3.real])
y = np.array([circle1.imag, circle2.imag, circle3.imag])
#Graph2D(x,y,'x','y','Cercles',['r=0.75','r=1','r=1.25']).show()
w1 = 1 / circle1
w2 = 1 / circle2
w3 = 1 / circle3
X = np.array([w1.real, w2.real, w3.real])
Y = np.array([w1.imag, w2.imag, w3.imag])
#Graph2D(X,Y,'x','y','1/z',['r=0.75','r=1','r=1.25']).show()
b = 1
circle4 = circle(-0.25, 1.25)
wJ = circle4 + b * b / circle4
graph = Graph2D(wJ.real, [wJ.imag], 'x', 'y', 'Joukovsky')
graph.setEqual()
graph.show()
vx = [j for i,j,k,l,m,n in u]
y = [k for i,j,k,l,m,n in u]
vy = [l for i,j,k,l,m,n in u]
z = [m for i,j,k,l,m,n in u]
vz = [n for i,j,k,l,m,n in u]
return t,x,vx,y,vy,z,vz
def particuleEnergy(vx,vy,vz):
E = []
E = np.multiply(0.5*m,np.square(vx)+np.square(vy)+np.square(vz))
return E
if __name__ == "__main__":
print("T : ",T)
print("a0 :",a0/1e-10)
u0 = [5,1,0,0.1,0,-0.2]
dt = 0.01
n = 1000
t,x,vx,y,vy,z,vz = integrationRK4(u0,dt,n)
E = particuleEnergy(vx,vy,vz)
Graph2D(t,[E],"timz(s)","Energy(J)","Energy's conservation").show()
plot3DGraph(x,y,z,"x(Rt)","y(Rt)","z(Rt)","Particle's trajectory")
def testEnergy(theta,vtheta):
T = np.multiply(0.5*m*l**2,np.square(vtheta))
V = m*g*l*(1-np.cos(theta))
E = T + V
return T,V,E
if __name__ == "__main__":
# First resolution : question b
y0 = np.radians([10.,0.])
dt = 0.005
n = 500
#t,theta,vtheta = numpySoluce(y0,dt,n)
t,theta,vtheta = testRK4(y0,dt,n)
x = l*np.sin(theta)
y = -l*np.cos(theta)
Graph2D(theta,[vtheta],r'$\theta$ (rds)',r'$\dot{\theta}$ (rds/s)',
"Phase portrait").show()
Graph2D(x,[y],"x (m)","y (m)","Trajectory in Oxy").show()
# test of energy's conservation
T,V,E = testEnergy(theta,vtheta)
legend = ['V','T','E']
Graph2D(t,[V,T,E],"t","Energy","Energy's conservation",legend).show()
def exo2():
t = np.array([0, 1, 2, 3, 4, 5, 10])
N = np.array([1000, 370, 130, 50, 17, 8, 1])
decroit = Graph2D(t, [N], "t", "N(t)")
decroit.show()