def GetLookupTable(fluid):
"""Retrieve a lookup table for the interpolated velocity generated by a point force.
If table does not already exist it will be created and stored."""
path = join(dirname(dirname(__file__)), "GreensFunction")
path = join(path, '2D' if fluid.Dim == 2 else '3D')
name = join(path, fluid.Params_ToString() + '.npy')
try:
# Load the table from file
Table = numpy.load(name)
except:
# File didn't exist, create table from scratch
print "Greens function at\n ", name, "\ndoesn't exist, creating it...",
Table = CalculateLookupTable(fluid)
# Save the new lookup table
ensure_filedir(name)
numpy.save(name, Table)
print "Done."
print
return Table
def GetLookupTable (fluid): """Retrieve a lookup table for the interpolated velocity generated by a point force. If table does not already exist it will be created and stored.""" path = join(dirname(dirname(__file__)), "GreensFunction") path = join(path, '2D' if fluid.Dim == 2 else '3D') name = join(path, fluid.Params_ToString() + '.npy') try: # Load the table from file Table = numpy.load(name); except: # File didn't exist, create table from scratch print "Greens function at\n ", name, "\ndoesn't exist, creating it...", Table = CalculateLookupTable(fluid) # Save the new lookup table ensure_filedir(name) numpy.save(name, Table) print "Done." print return Table
def CreateDecomposition(u, NumTerms = 50, MinPanelWidth = 1): N, dt = u.N[0], u.dt # Get the path for where to save the decomposition DecompPath = GetDecompPath() # Get the lookup table for the Greens function of the Spread/Interpolated Stokes flow Greens = GetLookupTable(u) # The Greens function is a 3x3 matrix field, hence we must calculate 9 expansions, one for each component. # By symmetry only 3 of these expansions are unique. for __j in range(1): for __k in range(2): # Get the (__j,__k) component of the Greens function and call it Phi, and its Fourier transform PhiHat Phi = Greens[...,__j,__k] PhiHat = real(fft.fftn(Phi)) # The largest sized panel we care about has width N / 4, because larger panels are never well separated from anything. PanelWidth = N / 4 # We loop through all panel sizes, starting with N / 4, then N / 8, then N / 16, and so on, until a preset stopping size is passed. while PanelWidth > MinPanelWidth: print "__j ==", __j print "__k ==", __k print "PanelWidth ==", PanelWidth # Two lists of expansion coefficients, one for the incoming and one for the outgoing. U, V = [], [] # Fill the lists with arbitrary initial guesses. for l in range(NumTerms): U.append(ones((N,N,N))) V.append(ones((N,N,N))) for j1 in range(N): for j2 in range(N): for j3 in range(N): V[l][j1,j2,j3] = 1 # Define the incoming and outgoing regions. # Outgoing Region (Omega_Out) # [U_x1,U_x2]x[U_y1,U_y2]x[U_z1,U_z2] U_x1 = N / 2 - PanelWidth / 2 U_x2 = N / 2 + PanelWidth / 2 U_y1 = N / 2 - PanelWidth / 2 U_y2 = N / 2 + PanelWidth / 2 U_z1 = N / 2 - PanelWidth / 2 U_z2 = N / 2 + PanelWidth / 2 U_x1 -= 1 U_x2 += 1 U_y1 -= 1 U_y2 += 1 U_z1 -= 1 U_z2 += 1 # Incoming Region (Omega_In) # [V_x1,V_x2]x[V_y1,V_y2]x[V_z1,V_z2] V_width = 3 * PanelWidth V_x1 = N / 2 - V_width / 2 V_x2 = N / 2 + V_width / 2 V_y1 = N / 2 - V_width / 2 V_y2 = N / 2 + V_width / 2 V_z1 = N / 2 - V_width / 2 V_z2 = N / 2 + V_width / 2 V_x1 += 1 V_x2 -= 1 V_y1 += 1 V_y2 -= 1 V_z1 += 1 V_z2 -= 1 # Define the indicator functions for the incoming and outgoing regions. def Enforce_U_Restriction(U): """Ensure that the tensor field U is zero outside the Outgoing Region""" hold = U[U_x1:U_x2+1, U_y1:U_y2+1, U_z1:U_z2+1].copy() U *= 0 U[U_x1:U_x2 + 1,U_y1:U_y2 + 1,U_z1:U_z2 + 1] = hold return U def Enforce_V_Restriction(V): """Ensure that the tensor field V is zero outside the Incoming Region""" V[V_x1:V_x2 + 1,V_y1:V_y2 + 1,V_z1:V_z2 + 1] *= 0 V[1:N/2,:,:] *= 0 V[:,1:N/2,:] *= 0 V[:,:,1:N/2] *= 0 return V U[0] = Enforce_U_Restriction(U[0]) # A list to store the singular values _s = [] # We wish to calculate the first few singular values, from 1 to NumTerms for l in range(NumTerms): V[l] = Enforce_V_Restriction(V[l]) # We perform the Power Iteration a fixed number of times to approximate each singular value. for i in range(PowerIterations): # Calculate the product Phi' * U less = 0 for _l in range(l): less += U[_l]*(V[_l]*V[l]).sum() U[l] = (Convolve(PhiHat, V[l]) - less) / (V[l]**2).sum() U[l] = Enforce_U_Restriction(U[l]) # Renormalize norm1 = (U[l]**2).sum()**.5 norm2 = (V[l]**2).sum()**.5 U[l] *= (norm2 / norm1)**.5 V[l] *= (norm1 / norm2)**.5 # Calculate the product Phi' * V less = 0 for _l in range(l): less += U[_l]*(V[_l]*U[l]).sum() V[l] = (Convolve(PhiHat, U[l]) - less) / (U[l]**2).sum() V[l] = Enforce_V_Restriction(V[l]) # Renormalize norm1 = (U[l]**2).sum()**.5 norm2 = (V[l]**2).sum()**.5 U[l] *= (norm2 / norm1)**.5 V[l] *= (norm1 / norm2)**.5 print l, _s.append( ((U[l]**2).sum() * (V[l]**2).sum())**.5 ) # Save the decomposition for l in range(NumTerms): name = path.join(DecompPath, u.Params_ToString()) name = path.join(name, 'Panel_Width_' + str(PanelWidth)) name = path.join(name, str(l) + '_' + str(__j) + str(__k) + '.npy') ensure_filedir(name) # We store both components U and V together in the same scalar field. Composite = U[l] + V[l] Composite = Shift(Composite, -2, -2, -2) numpy.save(name, Composite[N/2 - PanelWidth/2 - 2:, N/2 - PanelWidth/2 - 2:, N/2 - PanelWidth/2 - 2:]) # Sort and print the singular values _s.sort() _s.reverse() print "\n\nSingular Values:\n" + str(_s) + "\n\n\n" # Reduce the width of Omega_Out by half PanelWidth /= 2