def u_glob(U, cells, vertices, dof_map, resolution_per_element=51):
"""
Compute (x, y) coordinates of a curve y = u(x), where u is a
finite element function: u(x) = sum_i of U_i*phi_i(x).
(The solution of the linear system is in U.)
Method: Run through each element and compute curve coordinates
over the element.
This function works with cells, vertices, and dof_map.
"""
x_patches = []
u_patches = []
nodes = {} # node coordinates (use dict to avoid multiple values)
for e in range(len(cells)):
Omega_e = (vertices[cells[e][0]], vertices[cells[e][-1]])
d = len(dof_map[e]) - 1
phi = basis(d)
X = np.linspace(-1, 1, resolution_per_element)
x = affine_mapping(X, Omega_e)
x_patches.append(x)
u_cell = 0 # u(x) over this cell
for r in range(d+1):
i = dof_map[e][r] # global dof number
u_cell += U[i]*phi[r](X)
u_patches.append(u_cell)
# Compute global coordinates of local nodes,
# assuming all dofs corresponds to values at nodes
X = np.linspace(-1, 1, d+1)
x = affine_mapping(X, Omega_e)
for r in range(d+1):
nodes[dof_map[e][r]] = x[r]
nodes = np.array([nodes[i] for i in sorted(nodes)])
x = np.concatenate(x_patches)
u = np.concatenate(u_patches)
return x, u, nodes
def approximate(f, symbolic=False, d=1, N_e=4, numint=None,
Omega=[0, 1], filename='tmp'):
"""
Compute the finite element approximation, using Lagrange
elements of degree d, to a symbolic expression f (with x
as independent variable) on a domain Omega. N_e is the
number of elements.
symbolic=True implies symbolic expressions in the
calculations, while symbolic=False means numerical
computing.
numint is the name of the numerical integration rule
(Trapezoidal, Simpson, GaussLegendre2, GaussLegendre3,
GaussLegendre4, etc.). numint=None implies exact
integration.
"""
numint_name = numint # save name
if symbolic:
if numint == 'Trapezoidal':
numint = [[sym.S(-1), sym.S(1)], [sym.S(1), sym.S(1)]] # sympy integers
elif numint == 'Simpson':
numint = [[sym.S(-1), sym.S(0), sym.S(1)],
[sym.Rational(1,3), sym.Rational(4,3), sym.Rational(1,3)]]
elif numint == 'Midpoint':
numint = [[sym.S(0)], [sym.S(2)]]
elif numint == 'GaussLegendre2':
numint = [[-1/sym.sqrt(3), 1/sym.sqrt(3)], [sym.S(1), sym.S(1)]]
elif numint == 'GaussLegendre3':
numint = [[-sym.sqrt(sym.Rational(3,5)), 0,
sym.sqrt(sym.Rational(3,5))],
[sym.Rational(5,9), sym.Rational(8,9),
sym.Rational(5,9)]]
elif numint is not None:
print 'Numerical rule %s is not supported for symbolic computing' % numint
numint = None
else:
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1./3, 4./3, 1./3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1/sqrt(3), 1/sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3./5), 0, sqrt(3./5)],
[5./9, 8./9, 5./9]]
elif numint == 'GaussLegendre4':
numint = [[-0.86113631, -0.33998104, 0.33998104, 0.86113631],
[ 0.34785485, 0.65214515, 0.65214515, 0.34785485]]
elif numint == 'GaussLegendre5':
numint = [[-0.90617985, -0.53846931, -0. , 0.53846931, 0.90617985],
[ 0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689]]
elif numint is not None:
print 'Numerical rule %s is not supported for numerical computing' % numint
numint = None
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega, symbolic)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e])-1) for e in range(N_e)]
print 'phi basis (reference element):\n', phi
A, b = assemble(vertices, cells, dof_map, phi, f,
symbolic=symbolic, numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
#print sym.latex(A, mode='plain')
#print sym.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
else:
c = np.linalg.solve(A, b)
print 'c:\n', c
if not symbolic:
print 'Plain interpolation/collocation:'
x = sym.Symbol('x')
f = sym.lambdify([x], f, modules='numpy')
try:
f_at_vertices = [f(xc) for xc in vertices]
print f_at_vertices
except Exception as e:
print 'could not evaluate f numerically:'
print e
# else: nodes are symbolic so f(nodes[i]) only makes sense
# in the non-symbolic case
if filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
if numint is None:
title += ', exact integration'
else:
title += ', integration: %s' % numint_name
x_u, u = u_glob(np.asarray(c), vertices, cells, dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
import scitools.std as plt
plt.plot(x_u, u, '-',
x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def approximate(f, d, N_e, numint, Omega=[0,1], filename='tmp'):
"""
Compute the finite element approximation, using Lagrange
elements of degree d, to a Python functionn f on a domain
Omega. N_e is the number of elements.
numint is the name of the numerical integration rule
(Trapezoidal, Simpson, GaussLegendre2, GaussLegendre3,
GaussLegendre4, etc.). numint=None implies exact
integration.
"""
from math import sqrt
numint_name = numint # save name
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1./3, 4./3, 1./3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1/sqrt(3), 1/sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3./5), 0, sqrt(3./5)],
[5./9, 8./9, 5./9]]
elif numint == 'GaussLegendre4':
numint = [[-0.86113631, -0.33998104, 0.33998104,
0.86113631],
[ 0.34785485, 0.65214515, 0.65214515,
0.34785485]]
elif numint == 'GaussLegendre5':
numint = [[-0.90617985, -0.53846931, -0. ,
0.53846931, 0.90617985],
[ 0.23692689, 0.47862867, 0.56888889,
0.47862867, 0.23692689]]
elif numint is not None:
print 'Numerical rule %s is not supported for numerical computing' % numint
sys.exit(1)
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e])-1) for e in range(N_e)]
A, b = assemble(vertices, cells, dof_map, phi, f,
numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
c = np.linalg.solve(A, b)
print 'c:\n', c
if filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
title += ', integration: %s' % numint_name
x_u, u, _ = u_glob(np.asarray(c), vertices, cells, dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
import scitools.std as plt
plt.plot(x_u, u, '-',
x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def approximate(f, d, N_e, numint, Omega=[0, 1], filename='tmp'):
"""
Compute the finite element approximation, using Lagrange
elements of degree d, to a Python functionn f on a domain
Omega. N_e is the number of elements.
numint is the name of the numerical integration rule
(Trapezoidal, Simpson, GaussLegendre2, GaussLegendre3,
GaussLegendre4, etc.). numint=None implies exact
integration.
"""
from math import sqrt
numint_name = numint # save name
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1. / 3, 4. / 3, 1. / 3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1 / sqrt(3), 1 / sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3. / 5), 0, sqrt(3. / 5)], [5. / 9, 8. / 9, 5. / 9]]
elif numint == 'GaussLegendre4':
numint = [[-0.86113631, -0.33998104, 0.33998104, 0.86113631],
[0.34785485, 0.65214515, 0.65214515, 0.34785485]]
elif numint == 'GaussLegendre5':
numint = [[-0.90617985, -0.53846931, -0., 0.53846931, 0.90617985],
[0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689]]
elif numint is not None:
print 'Numerical rule %s is not supported '\
'for numerical computing' % numint
sys.exit(1)
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e]) - 1) for e in range(N_e)]
A, b = assemble(vertices, cells, dof_map, phi, f, numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
c = np.linalg.solve(A, b)
print 'c:\n', c
if filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
title += ', integration: %s' % numint_name
x_u, u, _ = u_glob(np.asarray(c),
vertices,
cells,
dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
import scitools.std as plt
plt.plot(x_u, u, '-', x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def approximate(f, symbolic=False, d=1, N_e=4, numint=None,
Omega=[0, 1], collocation=False, filename='tmp'):
"""
Compute the finite element approximation, using Lagrange
elements of degree d, to a symbolic expression f (with x
as independent variable) on a domain Omega. N_e is the
number of elements.
symbolic=True implies symbolic expressions in the
calculations, while symbolic=False means numerical
computing.
numint is the name of the numerical integration rule
(Trapezoidal, Simpson, GaussLegendre2, GaussLegendre3,
GaussLegendre4, etc.). numint=None implies exact
integration.
"""
numint_name = numint # save name
if symbolic:
if numint == 'Trapezoidal':
numint = [[sym.S(-1), sym.S(1)], [sym.S(1), sym.S(1)]] # sympy integers
elif numint == 'Simpson':
numint = [[sym.S(-1), sym.S(0), sym.S(1)],
[sym.Rational(1,3), sym.Rational(4,3), sym.Rational(1,3)]]
elif numint == 'Midpoint':
numint = [[sym.S(0)], [sym.S(2)]]
elif numint == 'GaussLegendre2':
numint = [[-1/sym.sqrt(3), 1/sym.sqrt(3)], [sym.S(1), sym.S(1)]]
elif numint == 'GaussLegendre3':
numint = [[-sym.sqrt(sym.Rational(3,5)), 0,
sym.sqrt(sym.Rational(3,5))],
[sym.Rational(5,9), sym.Rational(8,9),
sym.Rational(5,9)]]
elif numint is not None:
print 'Numerical rule %s is not supported for symbolic computing' % numint
numint = None
else:
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1./3, 4./3, 1./3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1/sqrt(3), 1/sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3./5), 0, sqrt(3./5)],
[5./9, 8./9, 5./9]]
elif numint == 'GaussLegendre4':
numint = [[-0.86113631, -0.33998104, 0.33998104, 0.86113631],
[ 0.34785485, 0.65214515, 0.65214515, 0.34785485]]
elif numint == 'GaussLegendre5':
numint = [[-0.90617985, -0.53846931, -0. , 0.53846931, 0.90617985],
[ 0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689]]
elif numint is not None:
print 'Numerical rule %s is not supported for numerical computing' % numint
numint = None
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega, symbolic)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e])-1) for e in range(N_e)]
A, b = assemble(vertices, cells, dof_map, phi, f,
symbolic=symbolic, numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
#print sym.latex(A, mode='plain')
#print sym.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
c = np.asarray([c[i,0] for i in range(c.shape[0])])
else:
c = np.linalg.solve(A, b)
print 'c:\n', c
x = sym.Symbol('x')
f = sym.lambdify([x], f, modules='numpy')
if collocation and not symbolic:
print 'Plain interpolation/collocation:'
# Should use vertices, but compute all nodes!
f_at_vertices = [f(xc) for xc in vertices]
print f_at_vertices
if filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
if numint is None:
title += ', exact integration'
else:
title += ', integration: %s' % numint_name
x_u, u, _ = u_glob(c, vertices, cells, dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
import scitools.std as plt
plt.plot(x_u, u, '-',
x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c