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coupling-vhm-moving.py
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coupling-vhm-moving.py
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#!/usr/bin/env python3
# Coupling using a variable horizon (VHCM)
# @author [email protected]
# @author [email protected]
# @date 02/05/2021
import numpy as np
import sys
import matplotlib.pyplot as plt
import matplotlib
from matplotlib.ticker import FormatStrFormatter
pgf_with_latex = {"text.usetex": True, "font.size" : 12, "pgf.preamble" : [r'\usepackage{xfrac}'] }
example = sys.argv[1]
solution = sys.argv[2]
g = -1
eps = 0.01
condition = False
#############################################################################
# Solve the system
#############################################################################
def solve(M,f):
return np.linalg.solve(M,f)
#############################################################################
# Loading
#############################################################################
def f(x):
global g
if example == "Cubic":
g = 27
return -6*x
elif example == "Quartic":
g = 108
return -12 * x*x
elif example == "Quadratic":
g = 6
return -2
elif example == "Linear":
g = 1
return 0
elif example == "Linear-cubic":
g = 31./4.
if x < 1.5:
return 0
else:
return 9-6*x
elif example == "Sin":
g = 2*np.pi*np.cos(2*np.pi*3)
return 4*np.pi*np.pi*np.sin(2*np.pi*x)
elif example == "Cos":
g = -2*np.pi*np.sin(2*np.pi*3+np.pi/2)
return 4*np.pi*np.pi*np.cos(2*np.pi*x+np.pi/2)
elif example == "Steep" :
g = 1-np.exp((3/3-1)/eps)/(eps*(1-np.exp(-1/eps)))
return np.exp((x/3-1)/eps)/(3*eps*eps*(1-np.exp(-1/eps)))
elif example == "Tan":
g = (np.pi * np.power(np.sin(np.pi * 3. /8.),2)) / ( 8. * np.power(np.cos( np.pi * 3. / 8.),2)) + np.pi / 8.
return -((np.pi * np.pi * np.power(np.sin(np.pi * x / 8.),3)) / (32. * np.power(np.cos( np.pi * x / 8.),3)) + (np.pi * np.pi * np.power(np.sin(np.pi * x / 8.),1)) / (32. * np.cos( np.pi * x /8.)))
else:
print("Error: Either provide Linear, Quadratic, Quartic, or Cubic")
sys.exit()
def forceFull(n,h):
force = np.zeros(n)
for i in range(1,n-1):
force[i] = f(i * h)
force[n-1] = g
return force
def forceCoupling(n,x):
force = np.zeros(n)
for i in range(1,n-1):
force[i] = f(x[i])
force[n-1] = g
return force
#############################################################################
# Exact solution
#############################################################################
def exactSolution(x):
if example == "Cubic":
return x * x * x
elif example == "Quartic":
return x * x * x * x
elif example == "Quadratic":
return x * x
elif example == "Linear":
return x
elif example == "Linear-cubic":
return np.where(x < 1.5, x, x + (x-1.5) * (x-1.5) * (x-1.5) )
elif example == "Sin":
return np.sin(2*np.pi*x)
elif example == "Cos":
return np.cos(2*np.pi*x+np.pi/2)
elif example == "Steep":
return x-3*(np.exp((x/3-1)/eps)-np.exp(-1/eps))/(1-np.exp(-1/eps))
elif example == "Tan":
return np.tan(3*x/4*np.pi/6)
else:
print("Error: Either provide Linear, Quadratic, Quartic, or Cubic")
sys.exit()
#############################################################################
# Assemble the stiffness matrix for the finite difference model (FD)
#############################################################################
def FDM(n,h):
M = np.zeros([n,n])
M[0][0] = 1
for i in range(1,n-1):
M[i][i-1] = -2
M[i][i] = 4
M[i][i+1] = -2
#M[n-1][n-1] = 1
M[n-1][n-1] = 11*h / 3
M[n-1][n-2] = -18*h / 3
M[n-1][n-3] = 9* h / 3
M[n-1][n-4] = -2* h / 3
M *= 1./(2.*h*h)
return M
#############################################################################
# Assemble the stiffness matrix for the coupling of FDM - FDM - FDM
#############################################################################
def CouplingFDFD(n,h):
M = np.zeros([3*n,3*n])
M[0][0] = 1
for i in range(1,n-1):
M[i][i-1] = -2
M[i][i] = 4
M[i][i+1] = -2
M[n-1][n-1] = -1
M[n-1][n] = 1
M[n][n-1] = 3*h
M[n][n-2] = -4*h
M[n][n-3] = 1*h
M[n][n] = 3*h
M[n][n+1] = -4*h
M[n][n+2] = 1*h
for i in range(n+1,2*n-1):
M[i][i-1] = -2
M[i][i] = 4
M[i][i+1] = -2
M[2*n-1][2*n-1] = -1
M[2*n-1][2*n] = 1
M[2*n][2*n-1] = 3*h
M[2*n][2*n-2] = -4*h
M[2*n][2*n-3] = h
M[2*n][2*n] = 3*h
M[2*n][2*n+1] = -4*h
M[2*n][2*n+2] = h
for i in range(2*n+1,3*n-1):
M[i][i-1] = -2
M[i][i] = 4
M[i][i+1] = -2
M[3*n-1][3*n-1] = 11*h / 3
M[3*n-1][3*n-2] = -18*h / 3
M[3*n-1][3*n-3] = 9 * h / 3
M[3*n-1][3*n-4] = -2 * h / 3
M *= 1./(2.*h*h)
return M
#############################################################################
# Assemble the stiffness matrix for the varibale horizon model (VHM)
#############################################################################
def VHM(n,h):
MVHM = np.zeros([n,n])
MVHM[0][0] = 1.
MVHM[1][0] = -8.
MVHM[1][1] = 16.
MVHM[1][2] = -8.
for i in range(2,n-2):
MVHM[i][i-2] = -1.
MVHM[i][i-1] = -4.
MVHM[i][i] = 10.
MVHM[i][i+1] = -4.
MVHM[i][i+2] = -1.
MVHM[n-2][n-1] = -8.
MVHM[n-2][n-2] = 16.
MVHM[n-2][n-3] = -8.
MVHM[n-1][n-1] = 12.*h
MVHM[n-1][n-2] = -16.*h
MVHM[n-1][n-3] = 4.*h
MVHM *= 1./(8.*h*h)
return MVHM
#############################################################################
# Assemble the stiffness matrix for the coupling of FDM - VHM - FDM
#############################################################################
def CouplingFDVHM(nodes1,nodes2,nodes3,h):
total = nodes1 + nodes2 + nodes3
fVHM = 1./(8.*h*h)
fFDM = 1./(2.*h*h)
M = np.zeros([total,total])
M[0][0] = 1 * fFDM
for i in range(1,nodes1-1):
M[i][i-1] = -2 * fFDM
M[i][i] = 4 * fFDM
M[i][i+1] = -2 * fFDM
M[nodes1-1][nodes1-1] = -1
M[nodes1-1][nodes1] = 1
M[nodes1][nodes1-1] = 11*h * fFDM / 3
M[nodes1][nodes1-2] = -18*h * fFDM / 3
M[nodes1][nodes1-3] = 9*h * fFDM / 3
M[nodes1][nodes1-4] = -2*h * fFDM / 3
M[nodes1][nodes1] = 11 / 6 / h
M[nodes1][nodes1+1] = -18 / 6 / h
M[nodes1][nodes1+2] = 9 / 6 / h
M[nodes1][nodes1+3] = -2 / 6 / h
M[nodes1+1][nodes1] = -8 * fVHM
M[nodes1+1][nodes1+1] = 16 * fVHM
M[nodes1+1][nodes1+2] = -8 * fVHM
for i in range(nodes1+2,nodes1+nodes2-2):
M[i][i-2] = -1. * fVHM
M[i][i-1] = -4. * fVHM
M[i][i] = 10. * fVHM
M[i][i+1] = -4. * fVHM
M[i][i+2] = -1. * fVHM
n = nodes1 + nodes2
M[n-2][n-3] = -8 * fVHM
M[n-2][n-2] = 16 * fVHM
M[n-2][n-1] = -8 * fVHM
M[n-1][n-1] = -1
M[n-1][n] = 1
M[n][n-1] = 11 / 6 / h
M[n][n-2] = -18 / 6 / h
M[n][n-3] = 9 / 6 / h
M[n][n-4] = -2 / 6 / h
M[n][n] = 11*h * fFDM / 3
M[n][n+1] = -18*h * fFDM / 3
M[n][n+2] = 9*h * fFDM / 3
M[n][n+3] = -2*h * fFDM / 3
for i in range(n+1,n+nodes3-1):
M[i][i-1] = -2 * fFDM
M[i][i] = 4 * fFDM
M[i][i+1] = -2 * fFDM
n += nodes3
#M[n-1][n-1] = 1
M[n-1][n-1] = 11*h * fFDM / 3
M[n-1][n-2] = -18*h * fFDM / 3
M[n-1][n-3] = 9 * h * fFDM / 3
M[n-1][n-4] = -2 * h * fFDM / 3
if condition :
print(np.linalg.cond(M))
with open("con-vhm-neumann.txt", "a") as f:
f.write(str(np.linalg.cond(M))+"\n")
f.close()
np.savetxt("vhm-matrix.csv", M, delimiter=",")
return M
markers = ['s','o','x','.']
plt.axvline(x=0.5,c="#536872")
plt.axvline(x=1.5,c="#536872")
start = 4
for i in range(start,start+4):
n = np.power(2,i)
h = 1./n
print(h)
nodes1 = int(0.5/h)+1
nodes2 = int(1./h)+1
nodes3 = int(1.5/h)+1
nodesFull = 3 * n + 1
print(nodesFull,h)
x1 = np.linspace(0,0.5,nodes1)
#print(x1)
x2 = np.linspace(0.5,1.5,nodes2)
#print(x2)
x3 = np.linspace(1.5,3.,nodes3)
#print(x3)
x = np.array(np.concatenate((x1,x2,x3)))
#print(x)
xFull = np.linspace(0,3.,nodesFull)
print(xFull)
forceCoupled = forceCoupling(nodes1+nodes2+nodes3,x)
forceCoupled[nodes1-1] = 0
forceCoupled[nodes1] = 0
forceCoupled[nodes1+nodes2-1] = 0
forceCoupled[nodes1+nodes2] = 0
np.savetxt("vhm-force.csv", forceCoupled, delimiter=",")
uFDMVHM = solve(CouplingFDVHM(nodes1,nodes2,nodes3,h),forceCoupled)
uSlice = np.array(np.concatenate((uFDMVHM[0:nodes1],uFDMVHM[nodes1+1:nodes1+nodes2],uFDMVHM[nodes1+nodes2+1:nodes1+nodes2+nodes3])))
print(xFull-np.array(np.concatenate((x[0:nodes1],x[nodes1+1:nodes1+nodes2],x[nodes1+nodes2+1:nodes1+nodes2+nodes3]))))
uFD = solve(FDM(nodesFull,h),forceFull(nodesFull,h))
print("------")
print(uFDMVHM[nodes1-1])
print(uFDMVHM[nodes1])
#print(uFD[nodes1-1])
#print("PD")
#upfd = (-2 * uFDMVHM[nodes1-4] + 9 * uFDMVHM[nodes1-3] - 18 * uFDMVHM[nodes1-2] + 11 * uFDMVHM[nodes1-1] ) / 6
#uppd = (-2 * uFDMVHM[nodes1+3] + 9 * uFDMVHM[nodes1+2] - 18 * uFDMVHM[nodes1+1] + 11 * uFDMVHM[nodes1] ) / 6
#print(upfd)
#print(uppd)
#print(abs(uppd)-abs(upfd))
#upfdl = (-2 * uFD[nodes1-4] + 9 * uFD[nodes1-3] - 18 * uFD[nodes1-2] + 11 * uFD[nodes1-1] ) / 6
#upfdr = (-2 * uFD[nodes1+2] + 9 * uFD[nodes1+1] - 18 * uFD[nodes1] + 11 * uFD[nodes1-1] ) / 6
#print("FDM ")
#print(upfdl)
#print(upfdr)
#print(abs(upfdl)-abs(upfdr))
#print("-- Second interface")
#upfd = (-2 * uFDMVHM[nodes1+nodes2-4] + 9 * uFDMVHM[nodes1+nodes2-3] - 18 * uFDMVHM[nodes1+nodes2-2] + 11 * uFDMVHM[nodes1+nodes2-1] ) / 6
#uppd = (-2 * uFDMVHM[nodes1+nodes2+3] + 9 * uFDMVHM[nodes1+nodes2+2] - 18 * uFDMVHM[nodes1+nodes2+1] + 11 * uFDMVHM[nodes1+nodes2] ) / 6
#print(upfd)
#print(uppd)
#print(abs(uppd)-abs(upfd))
#print("FDM")
#upfdl = (-2 * uFD[nodes1+nodes2-4] + 9 * uFD[nodes1+nodes2-3] - 18 * uFD[nodes1+nodes2-2] + 11 * uFD[nodes1+nodes2-1] ) / 6
#upfdr = (-2 * uFD[nodes1+nodes2+2] + 9 * uFD[nodes1+nodes2+1] - 18 * uFD[nodes1+nodes2] + 11 * uFD[nodes1+nodes2-1] ) / 6
#print(upfdl)
#print(upfdr)
#print(abs(upfdl)-abs(upfdr))
#print(uSlice[36])
print(uFD[nodes1])
#print((uSlice-uFD)[35])
print((uSlice-uFD)[nodes1])
print((uSlice-uFD)[nodes2])
print(uSlice-uFD)
#print((uSlice-uFD)[37])
#print((uSlice-uFD)[38])
#print(uFD)
#print(uFDMVHM)
print("-------")
if example == "Steep":
plt.plot(xFull,uSlice,label=r"$\delta$=1/"+str(int(n/2))+"",c="black",marker=markers[i-start],markevery=n)
if i == start+3:
plt.plot(xFull,exactSolution(xFull),label="Exact solution",c="black")
if example == "Quartic" or "Linear-cubic" or example == "Sin" or example == "Cos" or example == "Tan":
if solution == "FDM" :
uFD = solve(FDM(nodesFull,h),forceFull(nodesFull,h))
plt.plot(xFull,uSlice-uFD,label=r"$\delta$=1/"+str(int(n/2))+"",c="black",marker=markers[i-start],markevery=n)
#plt.plot(xFull,uFD,label=r"$\delta$=1/"+str(int(n/2))+"",c="black",marker=markers[i-start],markevery=n)
#plt.errorbar(xFull,uSlice-uFD,yerr=uSlice-uFD)
plt.ylabel("Error in displacement w.r.t. FDM")
elif solution == "Exact":
uFD = solve(FDM(nodesFull,h),forceFull(nodesFull,h))
plt.plot(xFull,uFD-exactSolution(xFull),label=r"LLEM-VHM ($\delta$=1/"+str(int(n/2))+")",c="black",marker=markers[i-start],markevery=n)
plt.plot(xFull,uSlice-exactSolution(xFull),label=r"LLEM-VHM ($\delta$=1/"+str(int(n/2))+")",c="blue",marker=markers[i-start],markevery=n)
plt.ylabel("Error in displacement w.r.t. exact solution")
elif i == 4:
plt.plot(xFull,exactSolution(xFull),label="Exact solution",c="black")
plt.plot(xFull,uSlice,label=r"LLEM-VHM ($\delta$=1/"+str(int(n/2))+")",c="black",marker=markers[i-4],markevery=n)
plt.ylabel("Displacement")
np.savetxt("coupling-"+example.lower()+"-vhm.csv",uSlice)
plt.gca().yaxis.set_major_formatter(FormatStrFormatter('%0.5f'))
plt.title("Example with "+example.lower()+" solution for VHCM with $m=2$")
plt.legend()
plt.grid()
plt.xlabel("$x$")
if solution == "FDM" :
if example == "Steep":
plt.savefig("coupling-"+example.lower()+"-"+str(start)+"-"+str(eps)+"-vhm-moving.pdf",bbox_inches='tight')
else:
plt.savefig("coupling-"+example.lower()+"-"+str(start)+"-vhm-moving.pdf",bbox_inches='tight')
else:
plt.savefig("coupling-"+example.lower()+"-"+str(start)+"-vhm-exact-moving.pdf",bbox_inches='tight')