def GH(p, I, J, P, n, L, B, x, y, z, dx, dy, t, a, k, w, d):
""" Computes the Green's function integral
numerically, with an increase resolution of n.
@param p Point to evaluate
@param I Point to evaluate x index
@param J Point to evaluate y index
@param P Reference point
@param n Number of divisors in each direction to
discretize each area element. Should be an even value
@param L Length of the computational domain
@param B width of the computational domain
@param x X coordinates of the area elements
@param y Y coordinates of the area elements
@param z Z coordinates of the area elements
@param dx distance between elements in the x direction
@param dy distance between elements in the y direction
@param t Simulation time (s)
@param a List of waves amplitudes
@param k List of waves k
@param w List of waves omega
@param d List of waves lags.
@return Green's function integral.
"""
# Ensure that n is an even value. If n is even
# we can grant that any point will be placed
# in p, that can be eventually the same than
# P, being G and H functions bad defined.
if n % 2:
n = n + 1
# Get the new distance between Green's points
DX = dx / n
DY = dy / n
# Get the coordinates of all the grid points
# around the evaluation one
nx = x.shape[0]
ny = y.shape[1]
X = zeros((3, 3), dtype=float32)
Y = zeros((3, 3), dtype=float32)
Z = zeros((3, 3), dtype=float32)
for i in range(0, 3):
for j in range(0, 3):
X[i, j] = p[0] + (i - 1) * dx
Y[i, j] = p[1] + (j - 1) * dy
if ((X[i, j] > -0.5 * L) and (X[i, j] < 0.5 * L)
and (Y[i, j] > -0.5 * B) and (Y[i, j] < 0.5 * B)):
Z[i, j] = z[I - 1 + i, J - 1 + j]
else:
Z[i, j] = waves.z(X[i, j], Y[i, j], t, a, k, w, d)
# Perform spline surface coeffs
K = zeros((3 * 3), dtype=float32)
K[0] = Z[0, 0] # k_{0}
K[1] = 4 * Z[1, 0] - Z[2, 0] - 3 * Z[0, 0] # k_{u}
K[2] = 4 * Z[0, 1] - Z[0, 2] - 3 * Z[0, 0] # k_{v}
K[3] = Z[2,2] - 4*Z[2,1] + 3*Z[2,0] + \
3*Z[0,2] - 12*Z[0,1] + 9*Z[0,0] + \
-4*Z[1,2] + 16*Z[1,1] - 12*Z[1,0] # k_{uv}
K[4] = 2 * Z[2, 0] + 2 * Z[0, 0] - 4 * Z[1, 0] # k_{uu}
K[5] = 2 * Z[0, 2] + 2 * Z[0, 0] - 4 * Z[0, 1] # k_{vv}
K[6] = -2*Z[2,2] + 8*Z[2,1] - 6*Z[2,0] + \
-2*Z[0,2] + 8*Z[0,1] - 6*Z[0,0] + \
4*Z[1,2] - 16*Z[1,1] + 12*Z[1,0] # k_{uuv}
K[7] = -2*Z[2,2] + 4*Z[2,1] - 2*Z[2,0] + \
-6*Z[0,2] + 12*Z[0,1] - 6*Z[0,0] + \
8*Z[1,2] - 16*Z[1,1] + 8*Z[1,0] # k_{uuv}
K[8] = 4*Z[2,2] - 8*Z[2,1] + 4*Z[2,0] + \
4*Z[0,2] - 8*Z[0,1] + 4*Z[0,0] + \
-8*Z[1,2] + 16*Z[1,1] - 8*Z[1,0] # k_{uuvv}
# Loop around the point p collecting the integral
G_tot = 0.0
H_tot = zeros((3), dtype=float32)
for i in range(0, n):
for j in range(0, n):
xx = x[I, J] - 0.5 * dx + (i + 0.5) * DX
yy = y[I, J] - 0.5 * dy + (j + 0.5) * DY
# Interpolate z
u = (xx - X[0, 0]) / (X[2, 0] - X[0, 0])
v = (yy - Y[0, 0]) / (Y[0, 2] - Y[0, 0])
zz = K[0] + K[1]*u + K[2]*v + K[3]*u*v + \
K[4]*u*u + K[5]*v*v + K[6]*u*u*v + \
K[7]*u*v*v + K[8]*u*u*v*v
p = array([xx, yy, zz])
G_tot = G_tot + G_val(p, P) * DX * DY
H_tot = H_tot + H_val(p, P) * DX * DY
return (G_tot, H_tot)
w.append(2*pi/t[i]) k.append(w[i]**2 / 9.81) c.append(w[i]/k[i]) # We can intializate the free surface x = zeros((nx,ny),dtype=float32) y = zeros((nx,ny),dtype=float32) z = zeros((nx,ny),dtype=float32) gradz = zeros((nx,ny),dtype=float32) p = zeros((nx,ny),dtype=float32) gradp = zeros((nx,ny),dtype=float32) for i in range(0,nx): for j in range(0,ny): x[i,j] = -0.5*L + (i+0.5)*drx y[i,j] = -0.5*B + (j+0.5)*dry z[i,j] = waves.z(x[i,j],y[i,j],0.0, a,k,w,d) p[i,j] = waves.phi(x[i,j],y[i,j],z[i,j],0.0, a,k,w,d) gradp[i,j] = waves.gradphi(x[i,j],y[i,j],z[i,j],0.0, a,k,w,d)[2] # We can plot starting data plot_j = int(ny/2) fig = figure() plot(x[:,plot_j], z[:,plot_j], color="blue", linewidth=2.5, linestyle="-", label="z") plot(x[:,plot_j], gradp[:,plot_j], color="red", linewidth=2.5, linestyle="-", label="vz") plot(x[:,plot_j], p[:,plot_j], color="green", linewidth=2.5, linestyle="-", label="phi") legend(loc='best') grid() show() close(fig) # Compute the error in an arbitrary point
def GH(p,I,J,P,n,L,B,x,y,z,dx,dy,t, a,k,w,d):
""" Computes the Green's function integral
numerically, with an increase resolution of n.
@param p Point to evaluate
@param I Point to evaluate x index
@param J Point to evaluate y index
@param P Reference point
@param n Number of divisors in each direction to
discretize each area element. Should be an even value
@param L Length of the computational domain
@param B width of the computational domain
@param x X coordinates of the area elements
@param y Y coordinates of the area elements
@param z Z coordinates of the area elements
@param dx distance between elements in the x direction
@param dy distance between elements in the y direction
@param t Simulation time (s)
@param a List of waves amplitudes
@param k List of waves k
@param w List of waves omega
@param d List of waves lags.
@return Green's function integral.
"""
# Ensure that n is an even value. If n is even
# we can grant that any point will be placed
# in p, that can be eventually the same than
# P, being G and H functions bad defined.
if n % 2:
n = n + 1
# Get the new distance between Green's points
DX = dx / n
DY = dy / n
# Get the coordinates of all the grid points
# around the evaluation one
nx = x.shape[0]
ny = y.shape[1]
X = zeros((3,3),dtype=float32)
Y = zeros((3,3),dtype=float32)
Z = zeros((3,3),dtype=float32)
for i in range(0,3):
for j in range(0,3):
X[i,j] = p[0] + (i-1)*dx
Y[i,j] = p[1] + (j-1)*dy
if((X[i,j] > -0.5*L) and (X[i,j] < 0.5*L) and (Y[i,j] > -0.5*B) and (Y[i,j] < 0.5*B)):
Z[i,j] = z[I-1+i,J-1+j]
else:
Z[i,j] = waves.z(X[i,j],Y[i,j],t, a,k,w,d)
# Perform spline surface coeffs
K = zeros((3*3),dtype=float32)
K[0] = Z[0,0] # k_{0}
K[1] = 4*Z[1,0] - Z[2,0] - 3*Z[0,0] # k_{u}
K[2] = 4*Z[0,1] - Z[0,2] - 3*Z[0,0] # k_{v}
K[3] = Z[2,2] - 4*Z[2,1] + 3*Z[2,0] + \
3*Z[0,2] - 12*Z[0,1] + 9*Z[0,0] + \
-4*Z[1,2] + 16*Z[1,1] - 12*Z[1,0] # k_{uv}
K[4] = 2*Z[2,0] + 2*Z[0,0] - 4*Z[1,0] # k_{uu}
K[5] = 2*Z[0,2] + 2*Z[0,0] - 4*Z[0,1] # k_{vv}
K[6] = -2*Z[2,2] + 8*Z[2,1] - 6*Z[2,0] + \
-2*Z[0,2] + 8*Z[0,1] - 6*Z[0,0] + \
4*Z[1,2] - 16*Z[1,1] + 12*Z[1,0] # k_{uuv}
K[7] = -2*Z[2,2] + 4*Z[2,1] - 2*Z[2,0] + \
-6*Z[0,2] + 12*Z[0,1] - 6*Z[0,0] + \
8*Z[1,2] - 16*Z[1,1] + 8*Z[1,0] # k_{uuv}
K[8] = 4*Z[2,2] - 8*Z[2,1] + 4*Z[2,0] + \
4*Z[0,2] - 8*Z[0,1] + 4*Z[0,0] + \
-8*Z[1,2] + 16*Z[1,1] - 8*Z[1,0] # k_{uuvv}
# Loop around the point p collecting the integral
G_tot = 0.0
H_tot = zeros((3),dtype=float32)
for i in range(0,n):
for j in range(0,n):
xx = x[I,J] - 0.5*dx + (i+0.5)*DX
yy = y[I,J] - 0.5*dy + (j+0.5)*DY
# Interpolate z
u = (xx - X[0,0]) / (X[2,0] - X[0,0])
v = (yy - Y[0,0]) / (Y[0,2] - Y[0,0])
zz = K[0] + K[1]*u + K[2]*v + K[3]*u*v + \
K[4]*u*u + K[5]*v*v + K[6]*u*u*v + \
K[7]*u*v*v + K[8]*u*u*v*v
p = array([xx,yy,zz])
G_tot = G_tot + G_val(p,P)*DX*DY
H_tot = H_tot + H_val(p,P)*DX*DY
return (G_tot,H_tot)