This file is part of https://github.com/diehlpk/reusommer21.
Copyright (c) 2021 Patrick Diehl
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, version 3.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.
In [1]:
from sympy import Symbol, simplify, lambdify
import numpy as np
import matplotlib.pyplot as plt
from functools import reduce
import operator
import numpy as np
import matplotlib.pyplot as plt
import sys
from matplotlib.ticker import FormatStrFormatter
In [2]:
def interpolate_lagrange(x, x_values, y_values):
"""
x : value at which to evaluate y, should be between min and max x_values
x_values: list or numpy array containing values of x
y_values: list or numpy array contaning values of y
"""
def _basis(j):
p = [(x - x_values[m])/(x_values[j] - x_values[m]) for m in range(k) if m != j]
return reduce(operator.mul, p)
assert len(x_values) != 0 and (len(x_values) == len(y_values)), 'x and y cannot be empty and must have the same length'
k = len(x_values)
basis = []
for j in range(k):
basis.append(_basis(j))
#return sum(_basis(j)*y_values[j] for j in range(k))
return basis
In [3]:
example = "Quartic"
save_results = False
g = -1
con = []
has_condition = True
In [4]:
#############################################################################
# Solve the system
#############################################################################
def solve(M,f):
return np.linalg.solve(M,f)
In [5]:
#############################################################################
# Loading
#############################################################################
def f(x):
global g
if example == "Cubic":
g = 3/27*3*3
return -6/27*x
elif example == "Quartic":
g = 4/81 * 3 * 3 * 3
return -12/81 * x*x
elif example == "Quadratic":
g = 6/9
return -2/9
elif example == "Linear":
g = 1/3
return 0
elif example == "Linear-cubic":
g = 31./4.
if x < 1.5:
return 0
else:
return 9-6*x
else:
print("Error: Either provide Quadratic, Quartic, or Cubic")
sys.exit()
def forceFull(n,x):
force = np.zeros(n)
for i in range(1,n-1):
force[i] = f(x[i])
force[n-1] = g
return force
def forceCoupling(n,x):
dim = n
force = np.zeros(dim)
for i in range(1,dim-1):
force[i] = f(x[i]) / exactSolution(x[dim-1])
force[dim-1] = g / exactSolution(x[dim-1])
return force
In [6]:
#############################################################################
# Exact solution
#############################################################################
def exactSolution(x):
if example == "Cubic":
return 1/27 * x * x * x
elif example == "Quartic":
return x * x * x * x / 81
elif example == "Quadratic":
return 1/9 * x * x
elif example == "Linear":
return x/3
elif example == "Linear-cubic":
return np.where(x < 1.5, x, x + (x-1.5) * (x-1.5) * (x-1.5) )
else:
print("Error: Either provide Linear, Quadratic, Quartic, or Cubic")
sys.exit()
In [7]:
#############################################################################
# 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] = 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
In [8]:
#############################################################################
# Assemble the stiffness matrix for the coupling of FDM - VHM - FDM
#############################################################################
def CouplingMDCM(n,h,nFD,hFD,xAll):
fVHM = 1./(8.*h*h)
fFDM = 1./(2.*hFD*hFD)
dim = 2*nFD + n + 3
M = np.zeros([dim,dim])
M[0][0] = 1
for i in range(1,nFD-1):
M[i][i-1] = -2 * fFDM
M[i][i] = 4 * fFDM
M[i][i+1] = -2 * fFDM
# Interpolate the last FD node
weights = interpolate_lagrange(xAll[nFD-1], [xAll[nFD],xAll[nFD+1],xAll[nFD+2],xAll[nFD+3]], ["u1","u2","u3","u4"])
M[nFD-1][nFD-1] = -1 # FM node at 1
M[nFD-1][nFD] = weights[0]
M[nFD-1][nFD+1] = weights[1]
M[nFD-1][nFD+2] = weights[2]
M[nFD-1][nFD+3] = weights[3]
# Interpolate the first PD node
weights = interpolate_lagrange(xAll[nFD], [xAll[nFD-4],xAll[nFD-3],xAll[nFD-2],xAll[nFD-1]], ["u1","u2","u3","u4"])
M[nFD][nFD] = -1
M[nFD][nFD-1] = weights[3]
M[nFD][nFD-2] = weights[2]
M[nFD][nFD-3] = weights[1]
M[nFD][nFD-4] = weights[0]
# Interpolate the second PD node
weights = interpolate_lagrange(xAll[nFD+1], [xAll[nFD-4],xAll[nFD-3],xAll[nFD-2],xAll[nFD-1]], ["u1","u2","u3","u4"])
M[nFD+1][nFD+1] = -1
M[nFD+1][nFD-1] = weights[3]
M[nFD+1][nFD-2] = weights[2]
M[nFD+1][nFD-3] = weights[1]
M[nFD+1][nFD-4] = weights[0]
mid = nFD+n+1
for i in range(nFD+2,mid):
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
# Interpolate the first PD node
weights = interpolate_lagrange(xAll[mid], [xAll[mid+2],xAll[mid+3],xAll[mid+4],xAll[mid+5]], ["u1","u2","u3","u4"])
M[mid][mid] = -1
M[mid][mid+2] = weights[0]
M[mid][mid+3] = weights[1]
M[mid][mid+4] = weights[2]
M[mid][mid+5] = weights[3]
# Interpolate the second PD node
weights = interpolate_lagrange(xAll[mid+1], [xAll[mid+2],xAll[mid+3],xAll[mid+4],xAll[mid+5]], ["u1","u2","u3","u4"])
M[mid+1][mid+1] = -1
M[mid+1][mid+2] = weights[0]
M[mid+1][mid+3] = weights[1]
M[mid+1][mid+4] = weights[2]
M[mid+1][mid+5] = weights[3]
# Same end node and start node
weights = interpolate_lagrange(xAll[mid+2], [xAll[mid-2],xAll[mid-1],xAll[mid],xAll[mid+1]], ["u1","u2","u3","u4"])
M[mid+2][mid+2] = -1
M[mid+2][mid-2] = weights[0]
M[mid+2][mid-1] = weights[1]
M[mid+2][mid] = weights[2]
M[mid+2][mid+1] = weights[3]
for i in range(mid+3,dim-1):
M[i][i-1] = -2 * fFDM
M[i][i] = 4 * fFDM
M[i][i+1] = -2 * fFDM
M[dim-1][dim-1] = 11 * hFD * fFDM / 3
M[dim-1][dim-2] = -18 * hFD * fFDM / 3
M[dim-1][dim-3] = 9 * hFD * fFDM / 3
M[dim-1][dim-4] = -2 * hFD * fFDM / 3
if has_condition:
con.append(np.linalg.cond(M))
return M
In [9]:
def compute(amount):
hFD = 1./ amount
h = hFD / 5
nFD = int(1 / hFD)+1
n = int(1 / h)+1
x1 = np.linspace(0,1,nFD)
x2 = np.linspace(1-1.5*h,2+1.5*h,n+3)
x3 = np.linspace(2*1,3*1,nFD)
xFull = np.linspace(0,3*1,3*nFD-2)
xAll = np.concatenate([x1,x2,x3])
forceFD = forceFull(len(xFull),xFull)
MFD = FDM(len(xFull),hFD)
uFDM = solve(MFD,forceFD)
forceCoupled = forceCoupling(len(xAll),xAll)
forceCoupled[nFD-1] = 0
forceCoupled[nFD] = 0
forceCoupled[nFD+1] = 0
forceCoupled[nFD+n+1] = 0
forceCoupled[nFD+n+2] = 0
forceCoupled[nFD+n+3] = 0
MCoupling = CouplingMDCM(n,h,nFD,hFD,xAll)
np.savetxt("m.csv", MCoupling, delimiter=",")
uCoupled = solve(MCoupling,forceCoupled)
return xAll, xFull, uCoupled, uFDM , nFD, n
In [10]:
xAll = []
xFull = []
uCoupled = []
uFDM = []
nFD = []
n = []
for i in range(5, 9):
res = compute(i)
xAll.append(res[0])
xFull.append(res[1])
uCoupled.append(res[2])
uFDM.append(res[3])
nFD.append(res[4])
n.append(res[5])
In [11]:
xSlice = []
uSlice = []
for i in range(0, len(xAll)):
xSlice.append(np.concatenate([xAll[i][0:nFD[i]-1],xAll[i][nFD[i]:nFD[i]+2],[xAll[i][nFD[i]-1]],xAll[i][nFD[i]+2:nFD[i]+n[i]+1],[xAll[i][nFD[i]+n[i]+3]],xAll[i][nFD[i]+n[i]+1:nFD[i]+n[i]+3],xAll[i][nFD[i]+n[i]+4:]]))
uSlice.append(np.concatenate([uCoupled[i][0:nFD[i]-1],uCoupled[i][nFD[i]:nFD[i]+2],[uCoupled[i][nFD[i]-1]],uCoupled[i][nFD[i]+2:nFD[i]+n[i]+1],[uCoupled[i][nFD[i]+n[i]+3]],uCoupled[i][nFD[i]+n[i]+1:nFD[i]+n[i]+3],uCoupled[i][nFD[i]+n[i]+4:]]))
plt.plot(xSlice[i],uSlice[i],label="m="+str(i+4))
plt.plot(xFull[0],uFDM[0],label="FDM")
plt.grid()
plt.legend()
plt.axvline(x=1,c="#536872")
plt.axvline(x=2,c="#536872")
Out[11]:
<matplotlib.lines.Line2D at 0x10e9888e0>
In [12]:
plt.clf()
markers = ['s','o','x','.']
level = [2,3,4,5]
for i in range(0, len(xAll)):
plt.plot(xSlice[i],uSlice[i]-exactSolution(xSlice[i]),color="black",marker=markers[i],markevery=level[i],label="n="+str(i+4))
plt.gca().yaxis.set_major_formatter(FormatStrFormatter('%0.3f'))
plt.xlabel(r"$x$")
plt.ylabel(r"Error in displacement w.r.t exact solution")
plt.title(r"Example with " + str(example).lower() + " solution for MDCM with m = 2")
plt.grid()
plt.axvline(x=1,c="#536872")
plt.axvline(x=2,c="#536872")
plt.legend()
plt.savefig("MDCM-adaptive-endpoint-non-matching-cubic-"+str(example).lower()+".pdf")
In [14]:
if has_condition :
np.savetxt("con_mdcm_n-cubic-non-matching.csv", con, delimiter=",")