def __init__(self, deg, basis_type):
deg = int(deg)
if basis_type == 'L': #1D Lagrange basis of degree deg
z = Symbol('z')
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch
elif basis_type == 'M': #1D Legendre Polynomials (Fischer calls this spectral)
z = Symbol('z')
x = Symbol('x')
def gen_legendre_basis(n):
if n == -2:
return (1 - 1 * z) / 2
elif n == -1:
return (1 + 1 * z) / 2
else:
return ((2 * (n + 3) - 3) / 2.)**0.5 * integrate(
legendre((n + 1), x), (x, -1, z))
N = Array([gen_legendre_basis(i - 2) for i in range(deg + 1)])
N2 = N.tolist()
N2[-1] = N[1]
N2[1] = N[-1]
N = Array(N2)
dfN = diff(N, z) + 1.e-25 * N
# print(N)
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch
elif basis_type == 'G': #1D GFEM Functions (degree denotes the total approximation space)
z = Symbol('z')
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch #Enrichment order corresponding to GFEM shape functions
else:
raise Exception('Element type not implemented yet')
def __init__(self,deg,basis_type):
deg = int(deg)
from sympy import Symbol,diff,Array,lambdify,legendre,integrate,simplify,permutedims
if basis_type == 'L': #1D Lagrange basis of degree deg
z=Symbol('z')
Xi=np.linspace(-1,1,deg+1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z-Xi[i])/(Xi[k]-Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg+1)])
dfN = diff(N,z)+1.e-25*N
self.Ns=lambdify(z,N,'numpy')
self.dN=lambdify(z,dfN,'numpy')
# if deg==2.: # denotes the number of nodes
# N=1/2*Array([1-z,1+z])
# dfN=diff(N,z)
# self.Ns=lambdify(z,N,'numpy')
# self.dN=lambdify(z,dfN,'numpy')
# elif deg==3.:
# N=1/2*Array([z*(z-1),2*(1+z)*(1-z),z*(1+z)])
# dfN=diff(N,z)
# self.Ns=lambdify(z,N,'numpy')
# self.dN=lambdify(z,dfN,'numpy')
# elif deg==4.: #not implemented yet. the no. of nodes change. define a general geom.nNnodes (?)
# N=1/2*Array([z*(z-1),2*(1+z)*(1-z),z*(1+z)])
# dfN=diff(N,z)
# self.Ns=lambdify(z,N,'numpy')
# self.dN=lambdify(z,dfN,'numpy')
# else:
# raise Exception('Element type not implemented yet')
elif basis_type == 'M': #1D Legendre Polynomials (Fischer calls this spectral)
z = Symbol('z')
x=Symbol('x')
def gen_legendre_basis(n):
if n==-2:
return (1-1*z)/2
elif n==-1:
return (1+1*z)/2
else:
return ((2*(n+3)-3)/2.)**0.5*integrate(legendre((n+1),x),(x,-1,z))
N=Array([gen_legendre_basis(i-2) for i in range(deg+1)])
# print(N)
N2 = N.tolist()
N2[-1] = N[1]
N2[1] = N[-1]
#N[-1] = N[-1] - N[1]
N=Array(N2)
dfN=diff(N,z)+1.e-25*N
# print(N)
self.Ns=lambdify(z,N,'numpy')
self.dN=lambdify(z,dfN,'numpy')
else:
raise Exception('Element type not implemented yet')
def __init__(self, deg, basis_type):
deg = int(deg)
from sympy import Symbol, diff, Array, lambdify, legendre, integrate, simplify
if basis_type == 'L': #1D Lagrange basis of degree deg
z = Symbol('z')
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
elif basis_type == 'M': #1D Legendre Polynomials (Fischer calls this spectral)
z = Symbol('z')
x = Symbol('x')
def gen_legendre_basis(n):
if n == -2:
return (1 - 1 * z) / 2
elif n == -1:
return (1 + 1 * z) / 2
else:
return ((2 * (n + 3) - 3) / 2.)**0.5 * integrate(
legendre((n + 1), x), (x, -1, z))
N = Array([gen_legendre_basis(i - 2) for i in range(deg + 1)])
N2 = N.tolist()
N2[-1] = N[1]
N2[1] = N[-1]
N = Array(N2)
dfN = diff(N, z) + 1.e-25 * N
# print(N)
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
# elif basis_type == 'GFEM1D1': #1D GFEM Linear Enrichment Functions
else:
raise Exception('Element type not implemented yet')
def __init__(self, deg, basis_type):
# print(basis_type)
deg = int(deg)
if basis_type == 'L': # 1D Lagrange basis of degree deg
z = Symbol('z')
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = 1
elif basis_type == 'M': # 1D Legendre Polynomials
z = Symbol('z')
x = Symbol('x')
def gen_legendre_basis(n):
if n == -2:
return (1 - 1 * z) / 2
elif n == -1:
return (1 + 1 * z) / 2
else:
return ((2 * (n + 3) - 3) / 2.)**0.5 * integrate(
legendre((n + 1), x), (x, -1, z))
N = Array([gen_legendre_basis(i - 2) for i in range(deg + 1)])
N2 = N.tolist()
N2[-1] = N[1]
N2[1] = N[-1]
N = Array(N2)
dfN = diff(N, z) + 1.e-25 * N
# print(N)
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = 1
elif basis_type == 'G': #1D GFEM Functions (degree denotes the total approximation space)
z = Symbol('z', real=True)
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch #Enrichment order corresponding to GFEM shape functions
xalph = Symbol('xalph', real=True)
halph = Symbol('halph', real=True)
N1 = Array([((z - xalph) / halph)**n + 1.e-26 * (z - xalph) / halph
for n in range(self.enrich + 1)
]) #shape consistency take E_o = 1
Nf1 = lambdify((z, xalph, halph), N1, 'numpy')
dfN1 = lambdify((z, xalph, halph),
diff(N1, z) + 1.e-26 * N1, 'numpy')
self.Np = Nf1
self.dNp = dfN1
elif basis_type == 'IF':
# print('here')
z = Symbol('z', real=True)
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch # Enrichment order corresponding to GFEM shape functions
xalph = Symbol('xalph', real=True)
halph = Symbol('halph', real=True)
xgam = Symbol('xgam', real=True)
Nf2 = simplify(
Array([(abs(z - xgam))**m * ((z - xalph) / halph)**n + 1.e-26 *
(z - xalph) / halph for m in range(2)
for n in range(enrch + 1)
])) # Enrichment by non-polynomials (for interfaces)
dfN2 = simplify(Nf2.diff(z) + 1.e-26 * Nf2)
self.NIF = lambdify(
(z, xgam, xalph, halph), Nf2, 'numpy'
) # Final enrichment functions including both polynomial and level set
self.DNIF = lambdify((z, xgam, xalph, halph), dfN2, 'numpy')
elif basis_type == 'IFM':
# print('here')
z = Symbol('z', real=True)
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch # Enrichment order corresponding to GFEM shape functions
xalph = Symbol('xalph', real=True)
halph = Symbol('halph', real=True)
xgam = Symbol('xgam', real=True)
Nf2 = simplify(
Array([(abs(z - xalph))**m * ((z - xalph) / halph)**n +
1.e-26 * (z - xalph) / halph for m in range(2)
for n in range(enrch + 1)
])) # Enrichment by non-polynomials (for interfaces)
dfN2 = simplify(Nf2.diff(z) + 1.e-26 * Nf2)
Nf3 = simplify(
Array([((z - xalph) / halph)**n + 1.e-26 * (z - xalph) / halph
for n in range(enrch + 1)
])) # Enrichment by non-polynomials (for interfaces)
dfN3 = simplify(Nf3.diff(z) + 1.e-26 * Nf3)
self.NIF = lambdify(
(z, xgam, xalph, halph), Nf3, 'numpy'
) # Final enrichment functions including both polynomial and level set
self.DNIF = lambdify((z, xgam, xalph, halph), dfN3, 'numpy')
else:
raise Exception('Element type not implemented yet')
def __init__(self, deg, basis_type):
deg = int(deg)
if basis_type == 'L': #1D Lagrange basis of degree deg
z = Symbol('z', real=True)
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([lag_basis(m) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch
ue = z**0.65 + sin(3 * pi * z / 2)
uep = ue.diff(z)
uepp = uep.diff(z)
self.Tf = lambdify(z, -uepp, 'numpy')
self.dE = float(integrate(0.5 * uep**2, (z, 0, 1)).evalf(20))
elif basis_type == 'M': #1D Legendre Polynomials (Fischer calls this spectral)
z = Symbol('z', real=True)
x = Symbol('x')
def gen_legendre_basis(n):
if n == -2:
return (1 - 1 * z) / 2
elif n == -1:
return (1 + 1 * z) / 2
else:
return ((2 * (n + 3) - 3) / 2.)**0.5 * integrate(
legendre((n + 1), x), (x, -1, z))
N = Array([gen_legendre_basis(i - 2) for i in range(deg + 1)])
N2 = N.tolist()
N2[-1] = N[1]
N2[1] = N[-1]
N = Array(N2)
dfN = diff(N, z) + 1.e-25 * N
# print(N)
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch
ue = z**0.65 + sin(3 * pi * z / 2)
uep = ue.diff(z)
uepp = uep.diff(z)
self.Tf = lambdify(z, -uepp, 'numpy')
self.dE = float(integrate(0.5 * uep**2, (z, 0, 1)).evalf(20))
elif basis_type == 'G': #1D GFEM Functions (degree denotes the total approximation space)
z = Symbol('z', real=True)
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-25 * N
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch #Enrichment order corresponding to GFEM shape functions
ue = z**0.65 + sin(3 * pi * z / 2)
uep = ue.diff(z)
uepp = uep.diff(z)
self.Tf = lambdify(z, -uepp, 'numpy')
self.dE = float(integrate(0.5 * uep**2, (z, 0, 1)).evalf(20))
elif basis_type == 'Singl' or basis_type == 'SGFEMSingl' or basis_type == 'GeomSingl' or basis_type == 'GeomSGFEM':
z = Symbol('z', real=True)
xalph = Symbol('xalph', real=True)
halph = Symbol('halph', real=True)
Xi = np.linspace(-1, 1, deg + 1)
def lag_basis(k):
n = 1.
for i in range(len(Xi)):
if k != i:
n *= (z - Xi[i]) / (Xi[k] - Xi[i])
return n
N = Array([simplify(lag_basis(m)) for m in range(deg + 1)])
dfN = diff(N, z) + 1.e-27 * N
# Manufactured solution
ue = z**0.65 + sin(3 * pi * z / 2)
uep = ue.diff(z)
uepp = uep.diff(z)
Nf2 = Array([
(ue)**m * ((z - xalph) / halph)**n + 1.e-27 *
(z - xalph) / halph for m in range(2)
for n in range(enrch + 1)
]) # Enrichment by non-polynomials (for interfaces)
dfN2 = Nf2.diff(z) + 1.e-26 * Nf2
# Convert to numpy functions to do array operations (vectorized)
self.Ns = lambdify(z, N, 'numpy')
self.dN = lambdify(z, dfN, 'numpy')
self.enrich = enrch
self.Tf = lambdify(z, -uepp, 'numpy')
self.dE = float(integrate(0.5 * uep**2, (z, 0, 1)).evalf(20))
self.uexact = lambdify(z, ue, 'numpy')
self.NIF = lambdify(
(z, xalph, halph), Nf2, 'numpy'
) # Final enrichment functions including both polynomial and exact soln
self.DNIF = lambdify((z, xalph, halph), dfN2, 'numpy')
self.enrich = enrch
else:
raise Exception('Element type not implemented yet')