import sys, os
sys.path.insert(0, os.path.join(os.pardir, 'src'))
from fe_approx1D_numint import approximate, u_glob
from sympy import tanh, Symbol, lambdify
x = Symbol('x')
steepness = 20
arg = steepness * (x - 0.5)
approximate(tanh(arg),
symbolic=False,
numint='GaussLegendre4',
d=3,
N_e=1,
filename='fe_p3_tanh_1e')
approximate(tanh(arg),
symbolic=False,
numint='GaussLegendre4',
d=3,
N_e=2,
filename='fe_p3_tanh_2e')
approximate(tanh(arg),
symbolic=False,
numint='GaussLegendre4',
d=3,
N_e=4,
filename='fe_p3_tanh_4e')
approximate(tanh(arg),
symbolic=False,
numint='GaussLegendre5',
d=4,
results = {}
d_values = [1, 2, 3, 4]
for case in cases:
f = cases[case]['f']
f_func = lambdify([x], f, modules='numpy')
Omega = cases[case]['Omega']
results[case] = {}
for d in d_values:
results[case][d] = {'E': [], 'h': [], 'r': []}
for N_e in [4, 8, 16, 32, 64, 128]:
try:
c = approximate(
f, symbolic=False,
numint='GaussLegendre%d' % (d+1),
d=d, N_e=N_e, Omega=Omega,
filename='tmp_%s_d%d_e%d' % (case, d, N_e))
except np.linalg.linalg.LinAlgError as e:
print str(e)
continue
vertices, cells, dof_map = mesh_uniform(
N_e, d, Omega, symbolic=False)
xc, u, _ = u_glob(c, vertices, cells, dof_map, 51)
e = f_func(xc) - u
# Trapezoidal integration of the L2 error over the
# xc/u patches
e2 = e**2
L2_error = 0
for i in range(len(xc)-1):
L2_error += 0.5*(e2[i+1] + e2[i])*(xc[i+1] - xc[i])
import sys, os
sys.path.insert(0, os.path.join(os.pardir, 'src'))
from fe_approx1D_numint import approximate
from sympy import tanh, Symbol
x = Symbol('x')
steepness = 20
arg = steepness * (x - 0.5)
approximate(tanh(arg),
symbolic=False,
numint='GaussLegendre2',
d=1,
N_e=4,
filename='fe_p1_tanh_4e')
approximate(tanh(arg),
symbolic=False,
numint='GaussLegendre2',
d=1,
N_e=8,
filename='fe_p1_tanh_8e')
approximate(tanh(arg),
symbolic=False,
numint='GaussLegendre2',
d=1,
N_e=16,
filename='fe_p1_tanh_16e')
approximate(tanh(arg),
symbolic=False,
numint='GaussLegendre3',
d=2,
N_e=2,
def approx(functionclass='intro'):
"""
Exemplify approximating various functions by various choices
of finite element functions
"""
import os
if functionclass == 'intro':
# Note: no plot if symbol=True
approximate(x * (1 - x),
symbolic=False,
d=1,
N_e=2,
filename='fe_p1_x2_2e')
#approximate(x*(1-x), symbolic=True, numint='Trapezoidal',
# d=1, N_e=2, filename='fe_p1_x2_2eT')
approximate(x * (1 - x),
symbolic=False,
d=1,
N_e=4,
filename='fe_p1_x2_4e')
approximate(x * (1 - x),
symbolic=False,
d=2,
N_e=1,
filename='fe_p2_x2_1e')
#approximate(x*(1-x), symbolic=True, numint='Simpson',
# d=2, N_e=4, filename='fe_p1_x2_2eS')
approximate(x * (1 - x),
symbolic=False,
d=1,
N_e=4,
filename='fe_p1_x2_4e')
# Only the extended fe_approx1D_numint can do P0 elements
import fe_approx1D_numint
fe_approx1D_numint.approximate(x * (1 - x),
symbolic=False,
d=0,
N_e=4,
filename='fe_p0_x2_4e')
fe_approx1D_numint.approximate(x * (1 - x),
symbolic=False,
d=0,
N_e=8,
filename='fe_p0_x2_8e')
for ext in 'pdf', 'png':
cmd = 'doconce combine_images fe_p1_x2_2e.%(ext)s fe_p1_x2_4e.%(ext)s fe_p1_x2_2e_4e.%(ext)s' % vars(
)
os.system(cmd)
cmd = 'doconce combine_images fe_p0_x2_4e.%(ext)s fe_p0_x2_8e.%(ext)s fe_p1_x2_4e_8e.%(ext)s' % vars(
)
os.system(cmd)
elif functionclass == 'special':
# Does not work well because Heaviside cannot be analytically
# integrated (not important) and numpy cannot evaluate
# Heaviside (serious for plotting)
from sympy import Heaviside, Rational
approximate(Heaviside(x - Rational(1, 2)),
symbolic=False,
d=1,
N_e=2,
filename='fe_p1_H_2e')
approximate(Heaviside(x - Rational(1, 2)),
symbolic=False,
d=1,
N_e=4,
filename='fe_p1_H_4e')
approximate(Heaviside(x - Rational(1, 2)),
symbolic=False,
d=1,
N_e=8,
filename='fe_p1_H_8e')
elif functionclass == 'easy':
approximate(1 - x, symbolic=False, d=1, N_e=4, filename='fe_p1_x_4e')
approximate(x * (1 - x),
symbolic=False,
d=2,
N_e=4,
filename='fe_p2_x2_4e')
approximate(x * (1 - x)**8,
symbolic=False,
d=1,
N_e=4,
filename='fe_p1_x9_4e')
approximate(x * (1 - x)**8,
symbolic=False,
d=1,
N_e=8,
filename='fe_p1_x9_8e')
approximate(x * (1 - x)**8,
symbolic=False,
d=2,
N_e=2,
filename='fe_p2_x9_2e')
approximate(x * (1 - x)**8,
symbolic=False,
d=2,
N_e=4,
filename='fe_p2_x9_4e')
for ext in 'pdf', 'png':
cmd = 'doconce combine_images fe_p1_x9_4e.%(ext)s fe_p1_x9_8e.%(ext)s fe_p1_x9_4e_8e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p2_x9_2e.%(ext)s fe_p2_x9_4e.%(ext)s fe_p2_x9_2e_4e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p1_x9_4e.%(ext)s fe_p2_x9_2e.%(ext)s fe_p1_x9_8e.%(ext)s fe_p2_x9_4e.%(ext)s fe_p1_p2_x9_248e.%(ext)s'
os.system(cmd)
elif functionclass == 'hard':
approximate(sin(pi * x),
symbolic=False,
d=1,
N_e=4,
filename='fe_p1_sin_4e')
approximate(sin(pi * x),
symbolic=False,
d=1,
N_e=8,
filename='fe_p1_sin_8e')
approximate(sin(pi * x),
symbolic=False,
d=2,
N_e=2,
filename='fe_p2_sin_2e')
approximate(sin(pi * x),
symbolic=False,
d=2,
N_e=4,
filename='fe_p2_sin_4e')
approximate(sin(pi * x)**2,
symbolic=False,
d=1,
N_e=4,
filename='fe_p1_sin2_4e')
approximate(sin(pi * x)**2,
symbolic=False,
d=1,
N_e=4,
filename='fe_p1_sin2_8e')
approximate(sin(pi * x)**2,
symbolic=False,
d=2,
N_e=2,
filename='fe_p2_sin2_2e')
approximate(sin(pi * x)**2,
symbolic=False,
d=2,
N_e=4,
filename='fe_p2_sin2_4e')
for ext in 'pdf', 'png':
cmd = 'doconce combine_images fe_p1_sin_4e.%(ext)s fe_p1_sin_8e.%(ext)s fe_p1_sin_4e_8e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p2_sin_2e.%(ext)s fe_p2_sin_4e.%(ext)s fe_p2_sin_2e_4e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p1_sin2_4e.%(ext)s fe_p1_sin2_8e.%(ext)s fe_p1_sin2_4e_8e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p2_sin2_2e.%(ext)s fe_p2_sin2_4e.%(ext)s fe_p2_sin2_2e_4e.%(ext)s'
os.system(cmd)
import sys, os
sys.path.insert(0, os.path.join(os.pardir, 'src-approx'))
from fe_approx1D_numint import approximate
from sympy import tanh, Symbol
x = Symbol('x')
steepness = 20
arg = steepness*(x-0.5)
approximate(tanh(arg), symbolic=False, numint='GaussLegendre2',
d=1, N_e=4, filename='fe_p1_tanh_4e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre2',
d=1, N_e=8, filename='fe_p1_tanh_8e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre2',
d=1, N_e=16, filename='fe_p1_tanh_16e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre3',
d=2, N_e=2, filename='fe_p2_tanh_2e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre3',
d=2, N_e=4, filename='fe_p2_tanh_4e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre3',
d=2, N_e=8, filename='fe_p2_tanh_8e')
results = {}
d_values = [1, 2, 3, 4]
for case in cases:
f = cases[case]['f']
f_func = lambdify([x], f, modules='numpy')
Omega = cases[case]['Omega']
results[case] = {}
for d in d_values:
results[case][d] = {'E': [], 'h': [], 'r': []}
for N_e in [4, 8, 16, 32, 64, 128]:
try:
c = approximate(
f, symbolic=False,
numint='GaussLegendre%d' % (d+1),
d=d, N_e=N_e, Omega=Omega,
filename='tmp_%s_d%d_e%d' % (case, d, N_e))
except np.linalg.linalg.LinAlgError as e:
print(str(e))
continue
vertices, cells, dof_map = mesh_uniform(
N_e, d, Omega, symbolic=False)
xc, u, _ = u_glob(c, vertices, cells, dof_map, 51)
e = f_func(xc) - u
# Trapezoidal integration of the L2 error over the
# xc/u patches
e2 = e**2
L2_error = 0
for i in range(len(xc)-1):
L2_error += 0.5*(e2[i+1] + e2[i])*(xc[i+1] - xc[i])
def approx(functionclass='intro'):
"""
Exemplify approximating various functions by various choices
of finite element functions
"""
import os
if functionclass == 'intro':
# Note: no plot if symbol=True
approximate(x*(1-x), symbolic=False,
d=1, N_e=2, filename='fe_p1_x2_2e')
#approximate(x*(1-x), symbolic=True, numint='Trapezoidal',
# d=1, N_e=2, filename='fe_p1_x2_2eT')
approximate(x*(1-x), symbolic=False,
d=1, N_e=4, filename='fe_p1_x2_4e')
approximate(x*(1-x), symbolic=False,
d=2, N_e=1, filename='fe_p2_x2_1e')
#approximate(x*(1-x), symbolic=True, numint='Simpson',
# d=2, N_e=4, filename='fe_p1_x2_2eS')
approximate(x*(1-x), symbolic=False,
d=1, N_e=4, filename='fe_p1_x2_4e')
# Only the extended fe_approx1D_numint can do P0 elements
import fe_approx1D_numint
fe_approx1D_numint.approximate(x*(1-x), symbolic=False,
d=0, N_e=4, filename='fe_p0_x2_4e')
fe_approx1D_numint.approximate(x*(1-x), symbolic=False,
d=0, N_e=8, filename='fe_p0_x2_8e')
for ext in 'pdf', 'png':
cmd = 'doconce combine_images fe_p1_x2_2e.%(ext)s fe_p1_x2_4e.%(ext)s fe_p1_x2_2e_4e.%(ext)s' % vars()
os.system(cmd)
cmd = 'doconce combine_images fe_p0_x2_4e.%(ext)s fe_p0_x2_8e.%(ext)s fe_p1_x2_4e_8e.%(ext)s' % vars()
os.system(cmd)
elif functionclass == 'special':
# Does not work well because Heaviside cannot be analytically
# integrated (not important) and numpy cannot evaluate
# Heaviside (serious for plotting)
from sympy import Heaviside, Rational
approximate(Heaviside(x - Rational(1,2)), symbolic=False,
d=1, N_e=2, filename='fe_p1_H_2e')
approximate(Heaviside(x - Rational(1,2)), symbolic=False,
d=1, N_e=4, filename='fe_p1_H_4e')
approximate(Heaviside(x - Rational(1,2)), symbolic=False,
d=1, N_e=8, filename='fe_p1_H_8e')
elif functionclass == 'easy':
approximate(1-x, symbolic=False,
d=1, N_e=4, filename='fe_p1_x_4e')
approximate(x*(1-x), symbolic=False,
d=2, N_e=4, filename='fe_p2_x2_4e')
approximate(x*(1-x)**8, symbolic=False,
d=1, N_e=4, filename='fe_p1_x9_4e')
approximate(x*(1-x)**8, symbolic=False,
d=1, N_e=8, filename='fe_p1_x9_8e')
approximate(x*(1-x)**8, symbolic=False,
d=2, N_e=2, filename='fe_p2_x9_2e')
approximate(x*(1-x)**8, symbolic=False,
d=2, N_e=4, filename='fe_p2_x9_4e')
for ext in 'pdf', 'png':
cmd = 'doconce combine_images fe_p1_x9_4e.%(ext)s fe_p1_x9_8e.%(ext)s fe_p1_x9_4e_8e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p2_x9_2e.%(ext)s fe_p2_x9_4e.%(ext)s fe_p2_x9_2e_4e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p1_x9_4e.%(ext)s fe_p2_x9_2e.%(ext)s fe_p1_x9_8e.%(ext)s fe_p2_x9_4e.%(ext)s fe_p1_p2_x9_248e.%(ext)s'
os.system(cmd)
elif functionclass == 'hard':
approximate(sin(pi*x), symbolic=False,
d=1, N_e=4, filename='fe_p1_sin_4e')
approximate(sin(pi*x), symbolic=False,
d=1, N_e=8, filename='fe_p1_sin_8e')
approximate(sin(pi*x), symbolic=False,
d=2, N_e=2, filename='fe_p2_sin_2e')
approximate(sin(pi*x), symbolic=False,
d=2, N_e=4, filename='fe_p2_sin_4e')
approximate(sin(pi*x)**2, symbolic=False,
d=1, N_e=4, filename='fe_p1_sin2_4e')
approximate(sin(pi*x)**2, symbolic=False,
d=1, N_e=4, filename='fe_p1_sin2_8e')
approximate(sin(pi*x)**2, symbolic=False,
d=2, N_e=2, filename='fe_p2_sin2_2e')
approximate(sin(pi*x)**2, symbolic=False,
d=2, N_e=4, filename='fe_p2_sin2_4e')
for ext in 'pdf', 'png':
cmd = 'doconce combine_images fe_p1_sin_4e.%(ext)s fe_p1_sin_8e.%(ext)s fe_p1_sin_4e_8e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p2_sin_2e.%(ext)s fe_p2_sin_4e.%(ext)s fe_p2_sin_2e_4e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p1_sin2_4e.%(ext)s fe_p1_sin2_8e.%(ext)s fe_p1_sin2_4e_8e.%(ext)s'
os.system(cmd)
cmd = 'doconce combine_images fe_p2_sin2_2e.%(ext)s fe_p2_sin2_4e.%(ext)s fe_p2_sin2_2e_4e.%(ext)s'
os.system(cmd)
import sys, os
sys.path.insert(0, os.path.join(os.pardir, 'src-approx'))
from fe_approx1D_numint import approximate, u_glob
from sympy import tanh, Symbol, lambdify
x = Symbol('x')
steepness = 20
arg = steepness*(x-0.5)
approximate(tanh(arg), symbolic=False, numint='GaussLegendre4',
d=3, N_e=1, filename='fe_p3_tanh_1e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre4',
d=3, N_e=2, filename='fe_p3_tanh_2e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre4',
d=3, N_e=4, filename='fe_p3_tanh_4e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre5',
d=4, N_e=1, filename='fe_p4_tanh_1e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre5',
d=4, N_e=2, filename='fe_p4_tanh_2e')
approximate(tanh(arg), symbolic=False, numint='GaussLegendre5',
d=4, N_e=4, filename='fe_p4_tanh_4e')
# Interpolation method
import numpy as np
import matplotlib.pyplot as plt
f = lambdify([x], tanh(arg), modules='numpy')
# Compute exact f on a fine mesh
x_fine = np.linspace(0, 1, 101)
f_fine = f(x_fine)
