def phi_r(r, X, d):
"""
Return local basis function phi_r at local point X in
a 1D element with d+1 nodes.
"""
nodes = np.linspace(-1, 1, d+1)
return Lagrange_polynomial(X, r, nodes)
def basis(d, point_distribution='uniform', symbolic=True):
"""
Return all local basis function phi as functions of the
local point X in a 1D element with d+1 nodes.
If symbolic=True, return symbolic expressions, else
return Python functions of X.
point_distribution can be 'uniform' or 'Chebyshev'.
"""
X = sym.symbols('X')
if d == 0:
phi_sym = [1]
else:
if point_distribution == 'uniform':
if symbolic:
h = sym.Rational(1, d) # node spacing
nodes = [2 * i * h - 1 for i in range(d + 1)]
else:
nodes = np.linspace(-1, 1, d + 1)
elif point_distribution == 'Chebyshev':
# Just numeric nodes
nodes = Chebyshev_nodes(-1, 1, d)
phi_sym = [Lagrange_polynomial(X, r, nodes) for r in range(d + 1)]
# Transform to Python functions
phi_num = [
sym.lambdify([X], phi_sym[r], modules='numpy') for r in range(d + 1)
]
return phi_sym if symbolic else phi_num
def phi_r(r, X, d, point_distribution='uniform'):
"""
Return local basis function phi_r at local point X in
a 1D element with d+1 nodes.
point_distribution can be 'uniform' or 'Chebyshev'.
"""
if point_distribution == 'uniform':
nodes = np.linspace(-1, 1, d+1)
elif point_distribution == 'Chebyshev':
nodes = Chebyshev_nodes(-1, 1, d)
return Lagrange_polynomial(X, r, nodes)
def phi_r(r, X, d):
"""
Return local basis function phi_r at local point X in
a 1D element with d+1 nodes.
"""
if isinstance(X, sm.Symbol):
# Use sm.Rational and integers for nodes
# (to maximize nice-looking output)
h = sm.Rational(1, d)
nodes = [2 * i * h - 1 for i in range(d + 1)]
else:
# X is numeric: use floats for nodes
nodes = np.linspace(-1, 1, d + 1)
return Lagrange_polynomial(X, r, nodes)
def phi_r(r, X, d, point_distribution='uniform'):
"""
Return local basis function phi_r at local point X in
a 1D element with d+1 nodes.
point_distribution can be 'uniform' or 'Chebyshev'.
"""
if point_distribution == 'uniform':
if isinstance(X, sm.Symbol):
# Use sm.Rational and integers for nodes
# (to maximize nice-looking output)
h = sm.Rational(1, d)
nodes = [2 * i * h - 1 for i in range(d + 1)]
else:
# X is numeric: use floats for nodes
nodes = np.linspace(-1, 1, d + 1)
elif point_distribution == 'Chebyshev':
nodes = Chebyshev_nodes(-1, 1, d)
return Lagrange_polynomial(X, r, nodes)
