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')
def ge_singl_stiff_matrx(eflag, etypes,
znodes): # znodes = nodes of current elements
eabs = 1.e-12
limits = 10000
xi = Symbol('xi', real=True)
z = Symbol('z', real=True)
x = 0.5 * ((1 - xi) * znodes[0] + (1 + xi) * znodes[-1])
Je = znodes[-1] - znodes[0]
Je /= 2.
# print(Je)
ue = x**0.65 + sin(3 * pi * x / 2)
uez = z**0.65 + sin(3 * pi * z / 2)
Tx = -1. * uez.diff(z, 2)
Tx_xi = Tx.subs([
(z, 0.5 * ((1 - xi) * znodes[0] + (1 + xi) * znodes[-1]))
])
phi1 = 0.5 * (1 - xi)
phi2 = 0.5 * (1 + xi)
if elflag == 'nn':
Nshp = Array([phi1, phi2])
dNshp = Nshp.diff(xi)
elif elflag == 'repr' or elflag == 'bll':
if etypes == 'Singl':
Nshp = Array([phi1, phi1 * ue, phi2])
dNshp = Nshp.diff(xi)
elif etypes == 'SGFEMSingl':
ue_interp = ue - phi1 * uez.subs(
z, znodes[0]) - phi2 * uez.subs(z, znodes[-1])
Nshp = Array([phi1, phi1 * ue_interp, phi2])
dNshp = Nshp.diff(xi)
elif etypes == 'GeomSGFEM':
ue_interp = ue - phi1 * uez.subs(
z, znodes[0]) - phi2 * uez.subs(z, znodes[-1])
if elflag == 'repr':
Nshp = Array(
[phi1, phi1 * ue_interp, phi2, phi2 * ue_interp])
dNshp = Nshp.diff(xi)
elif elflag == 'bll':
Nshp = Array([phi1, phi1 * ue_interp, phi2])
dNshp = Nshp.diff(xi)
elif etypes == 'GeomSingl':
ue_interp = ue - phi1 * uez.subs(
z, znodes[0]) - phi2 * uez.subs(z, znodes[-1])
if elflag == 'repr':
Nshp = Array([phi1, phi1 * ue, phi2, phi2 * ue])
dNshp = Nshp.diff(xi)
elif elflag == 'bll':
Nshp = Array([phi1, phi1 * ue, phi2])
dNshp = Nshp.diff(xi)
stiffM = Matrix(dNshp) * Matrix(
dNshp).T * geom.E * geom.A / Je + Matrix(Nshp) * Matrix(
Nshp).T * geom.C * geom.A * Je
forvec = Nshp * Tx_xi * Je
# mat = np.array([[quad(lambdify(xi,stiffM[i,j],'numpy'),-1.,1.,points=[-1.])[0] for i in range(stiffM.shape[0])] for j in range(stiffM.shape[1])],float)
mat = np.array([[
quad(lambdify(xi, stiffM[i, j], 'numpy'),
-1.,
1.,
epsabs=eabs,
limit=limits,
points=[-1.])[0] for i in range(stiffM.shape[0])
] for j in range(stiffM.shape[0])], float)
# elemF = np.array([quad(lambdify(xi,forvec[i],'numpy'),-1.,1.,points=[-1.])[0] for i in range(len(forvec))],float)
elemF = np.array([
quad(lambdify(xi, forvec[i], 'numpy'),
-1.,
1.,
epsabs=eabs,
limit=limits,
points=[1.])[0] for i in range(len(forvec))
], float)
return mat, elemF.flatten()