beta = 1.0 # Boundary value at y = L
# Guessed values
s = [1.0, 1.5]
z0 = np.zeros(2)
dpdx_list = [-5.0, -2.5, -1.0, 0.0, 1.0, 2.5, 5.0]
legends = []
for dpdx in dpdx_list:
phi = []
for svalue in s:
z0[1] = svalue
z = rk4(f, z0, y)
phi.append(z[-1, 0] - beta)
# Compute correct initial guess
s_star = (s[0] * phi[1] - s[1] * phi[0]) / (phi[1] - phi[0])
z0[1] = s_star
# Solve the initial value problem which is a solution to the boundary value problem
z = rk4(f, z0, y)
plot(z[:, 0], y, '-.')
legends.append('rk4: dp=' + str(dpdx))
# Plot the analytical solution
plot(u_a(y, dpdx), y, ':')
legends.append('exa: dp=' + str(dpdx))
"""
def func(Theta, t):
theta = Theta[0]
dtheta = Theta[1]
if dtheta >=0:
out = np.array([dtheta, math.cos(theta) - mu*math.sin(theta)])
else:
out = np.array([dtheta, math.cos(theta) + mu*math.sin(theta)])
return out
mu = 0.4
Tau = np.linspace(0, 10, 101)
Theta_0 = np.array([0, 0])
THETA = rk4(func, Theta_0, Tau)
animatePendulum(THETA[:,0], Tau, l=1)
from ODEschemes import euler, heun, rk4
import numpy as np
N=20
L = 1.0
y = np.linspace(0,L,N+1)
def f(z, t, dpdx=0.0):
"""RHS for Couette-Posieulle flow"""
zout = np.zeros_like(z)
zout[:] = [z[1], -dpdx]
return zout
s0=1.0
z0=np.zeros(2)
z0[1] = s0
z = rk4(f, z0, y)
phi0 =z[]
figure(figsize=(20, 8))
# hold('on')
LNWDT = 2
FNT = 18
rcParams['lines.linewidth'] = LNWDT
rcParams['font.size'] = FNT
# computing and plotting
# smooth ball with drag
for i in range(0, N2):
z0 = np.zeros(4)
z0[2] = v0 * np.cos(angle[i])
z0[3] = v0 * np.sin(angle[i])
z = rk4(f, z0, time)
plot(z[:, 0], z[:, 1], ':', color=line_color[i])
legends.append('angle=' + str(alfa[i]) + ', smooth ball')
# golf ball with drag
for i in range(0, N2):
z0 = np.zeros(4)
z0[2] = v0 * np.cos(angle[i])
z0[3] = v0 * np.sin(angle[i])
z = rk4(f2, z0, time)
plot(z[:, 0], z[:, 1], '-.', color=line_color[i])
legends.append('angle=' + str(alfa[i]) + ', golf ball')
# golf ball with drag and lift
for i in range(0, N2):
# main program starts here
T = 10 # end of simulation
N = 20 # no of time steps
time = np.linspace(0, T, N+1)
z0=np.zeros(2)
z0[0] = 2.0
ze = euler(f, z0, time) # compute response with constant CD using Euler's method
ze2 = euler(f2, z0, time) # compute response with varying CD using Euler's method
zh = heun(f, z0, time) # compute response with constant CD using Heun's method
zh2 = heun(f2, z0, time) # compute response with varying CD using Heun's method
zrk4 = rk4(f, z0, time) # compute response with constant CD using RK4
zrk4_2 = rk4(f2, z0, time) # compute response with varying CD using RK4
k1 = np.sqrt(g*4*rho_s*d/(3*rho_f*CD))
k2 = np.sqrt(3*rho_f*g*CD/(4*rho_s*d))
v_a = k1*np.tanh(k2*time) # compute response with constant CD using analytical solution
# plotting
legends=[]
line_type=['-',':','.','-.',':','.','-.']
plot(time, v_a, line_type[0])
legends.append('Analytical (constant CD)')
plot(time, ze[:,1], line_type[1])
a semicircle:
theta'' = - mu*(theta')^2 +g/r(cos(theta) - mu*sin(theta))
"""
def func(Theta, t):
theta = Theta[0]
dtheta = Theta[1]
if dtheta >= 0:
out = np.array([dtheta, math.cos(theta) - mu * math.sin(theta)])
else:
out = np.array([dtheta, math.cos(theta) + mu * math.sin(theta)])
return out
mu = 0.4
Tau = np.linspace(0, 10, 101)
Theta_0 = np.array([0, 0])
THETA = rk4(func, Theta_0, Tau)
animatePendulum(THETA[:, 0], Tau, l=1)