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)
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)
def approx(functionclass='intro'):
"""
Exemplify approximating various functions by various choices
of finite element functions
"""
if functionclass == 'intro':
approximate(x*(1-x), symbolic=True,
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=True,
d=2, N_e=4, filename='fe_p1_x2_2e')
approximate(x*(1-x), symbolic=True, numint='Simpson',
d=2, N_e=4, filename='fe_p1_x2_2eS')
#sys.exit(1)
approximate(x*(1-x), symbolic=False,
d=1, N_e=2, filename='fe_p1_x2_2e')
approximate(x*(1-x), symbolic=False,
d=1, N_e=8, filename='fe_p1_x2_8e')
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=1, N_e=4, filename='fe_p1_x2_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')
elif functionclass == 'hard':
# Note: it takes time to approximate these cases...
"""
approximate(sin(pi*x), symbolic=False,
d=1, N_e=4, filename='fe_p1_sin_4e')
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=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')