def test_find_n_roots():
"""test for find_n_roots"""
#find_n_roots(func, args=(), n=1, x0=0.001, dx=0.001, p=1.0, fsolve_kwargs={}):
assert_allclose(find_n_roots(math.sin, n=3, x0=0.1, dx=0.1, p=1.01),
np.arange(1,4)*np.pi, atol=1e-5)
#bessel roots from http://mathworld.wolfram.com/BesselFunctionZeros.html
assert_allclose(find_n_roots(j0, n=3, x0=0.1, dx=0.1, p=1.01),
np.array([2.4048, 5.5201, 8.6537]), atol=1e-4)
def test_find_n_roots():
"""test for find_n_roots"""
#find_n_roots(func, args=(), n=1, x0=0.001, dx=0.001, p=1.0, fsolve_kwargs={}):
assert_allclose(find_n_roots(math.sin, n=3, x0=0.1, dx=0.1, p=1.01),
np.arange(1, 4) * np.pi,
atol=1e-5)
#bessel roots from http://mathworld.wolfram.com/BesselFunctionZeros.html
assert_allclose(find_n_roots(j0, n=3, x0=0.1, dx=0.1, p=1.01),
np.array([2.4048, 5.5201, 8.6537]),
atol=1e-4)
def _find_sn(self):
"""Find the radial eigenvalues"""
if self.kh is None:
self._sn=np.array([0])
else:
self._sn = find_n_roots(self._radial_characteristic_curve,
n=self.nh,x0=self.radial_roots_x0,
dx=self.radial_roots_dx,
p = self.radial_roots_p, max_iter=self.max_iter)
def _find_sn(self):
"""Find the radial eigenvalues"""
if self.kh is None:
self._sn = np.array([0])
else:
self._sn = find_n_roots(self._radial_characteristic_curve,
n=self.nh,
x0=self.radial_roots_x0,
dx=self.radial_roots_dx,
p=self.radial_roots_p,
max_iter=self.max_iter)
def _find_beta(self):
"""find the eigenvalues of the solution
"""
H = self.zlayer[-1]
x0 = 0.1 / H**2
self._beta0 = np.empty(self.n, dtype=float)
self._beta0[:] = find_n_roots(self._characteristic_eqn, n=self.n,
x0=x0, dx=x0, p=1.01)
return
def _find_beta(self):
"""find the eigenvalues of the solution
"""
H = self.zlayer[-1]
x0 = 0.1 / H**2
self._beta0 = np.empty(self.n, dtype=float)
self._beta0[:] = find_n_roots(self._characteristic_eqn,
n=self.n,
x0=x0,
dx=x0,
p=1.01)
return
def _find_alp(self):
"""find alp by matrix transfer method"""
self._alp = np.zeros((self.nh, self.nv), dtype=float)
for n, s in enumerate(self._sn):
if s==0:
alp_start_offset = min(1e-7, self.vertical_roots_dx)
else:
alp_start_offset = 0
if n==0:
alp=self.vertical_roots_x0
else:
alp = self._alp[n-1,0]
self._alp[n,:] = find_n_roots(self._vertical_characteristic_curve,
args=(s,),n= self.nv, x0 = alp+alp_start_offset,
dx = self.vertical_roots_dx, p = self.vertical_roots_p,
max_iter=self.max_iter, fsolve_kwargs={})
def _find_alp(self):
"""find alp by matrix transfer method"""
self._alp = np.zeros((self.nh, self.nv), dtype=float)
for n, s in enumerate(self._sn):
if s == 0:
alp_start_offset = min(1e-7, self.vertical_roots_dx)
else:
alp_start_offset = 0
if n == 0:
alp = self.vertical_roots_x0
else:
alp = self._alp[n - 1, 0]
self._alp[n, :] = find_n_roots(self._vertical_characteristic_curve,
args=(s, ),
n=self.nv,
x0=alp + alp_start_offset,
dx=self.vertical_roots_dx,
p=self.vertical_roots_p,
max_iter=self.max_iter,
fsolve_kwargs={})
def __init__(self,
BC="SS",
nterms=50,
E=None,
I=None,
rho=None,
A=None,
L=None,
k1=None,
k3=None,
mu=None,
Fz=None,
v=None,
v_norm=None,
kf=None,
Fz_norm=None,
mu_norm=None,
k1_norm=None,
k3_norm=None,
nquad=30):
self.BC = BC
self.nterms = nterms
self.E = E
self.I = I
self.rho = rho
self.I = I
self.A = A
self.L = L
self.k1 = k1
self.k3 = k3
self.mu = mu
self.Fz = Fz
self.v = v
self.v_norm = v_norm
self.kf = kf
self.Fz_norm = Fz_norm
self.mu_norm = mu_norm
self.k1_norm = k1_norm
self.k3_norm = k3_norm
self.nquad = nquad
# normalised parameters
if kf is None:
self.kf = 1 / self.L * np.sqrt(self.I / self.A)
if v_norm is None:
self.v_norm = self.v * np.sqrt(self.rho / self.E)
if self.kf is None:
self.kf = 1 / self.L * np.sqrt(self.I / self.A)
if Fz_norm is None:
self.Fz_norm = self.Fz / (self.E * self.A)
if self.mu_norm is None:
self.mu_norm = self.mu / self.A * np.sqrt(self.L**2 / (self.rho * self.E))
if self.k1_norm is None:
self.k1_norm = self.k1 * self.L**2 / (self.E * self.A)
if self.k3_norm is None:
self.k3_norm = self.k3 * self.L**4 / (self.E* self.A)
# phi, beta, and quadrature points
if self.BC == "SS":
self.phi = self.phiSS
self.beta = np.pi * (np.arange(self.nterms) + 1)
if not self.nquad is None:
self.xj = 0.5 * (1 - np.cos(np.pi * np.arange(self.nquad) / (self.nquad - 1)))
self.BC_coeff = 2
elif self.BC == "CC" or self.BC == "FF":
def _f(beta):
return 1 - np.cos(beta) * np.cosh(beta)
self.beta = find_n_roots(_f, n=self.nterms, x0=0.001, dx=3.14159 / 10, p=1.1)
if not self.nquad is None:
self.xj = 0.5*(1 - np.cos(np.pi * np.arange(self.nquad)/(self.nquad - 3)))
self.xj[0] = 0
self.xj[1] = 0.0001
self.xj[-2] = 0.0001
self.xj[-1] = 0
self.BC_coeff = 1 # Coefficeint to multiply Fz(vt) and k3*Integral terms by
if self.BC == "CC":
self.phi = self.phiCC
if self.BC == "FF":
self.phi = self.phiFF
self.beta[1:] = self.beta[0:-1]
self.beta[0] = 0.0
else:
raise ValueError("only BC='SS', 'CC' and 'FF' have been implemented, you have {}".format(self.BC))
# quadrature weighting
if not self.nquad is None:
rhs = np.reciprocal(np.arange(self.nquad, dtype=float) + 1)
lhs = self.xj[np.newaxis, :] ** np.arange(self.nquad)[:, np.newaxis]
self.Ij = np.linalg.solve(lhs, rhs)
self.vsvdot = np.zeros(2 * self.nterms) #vector of state values for odeint
def zhuandyin2012(z, t, alpha, p, q, drn=0, tpor=None, H = 1, kv0 = 1, mv0 = 0.1, gamw = 10,
ui = 1, nterms = 20, plot_eigs=False):
"""Analysis and mathematical solutions for consolidation of a soil layer
with depth-dependent parameters under confined compression.
An implementation of Zhu and Yin (2012) [1]_.
Features:
- Single layer.
- Permeability and volume compressibility can vary with depth with a
power lay relationship.
- Instant load uniform with depth.
- PTIB ansd PTPB drainage conditions.
- Pore pressure vs depth at variaous times
- Degree of consolidation based on surface settlement vs time.
- Settlement vs time.
Parameters
----------
z : float or 1d array/list of float
Depth.
t : float or 1d array/list of float
Time for degree of consolidation calcs.
tpor : float or 1d array/list of float
Time for pore pressure vs depth calcs.
alpha, p, q : float
Exponent in depth dependence of permeability and compressibility
respectively. e.g. mv = mv0 * (1+alpha*z/H)**q. Note p/q cannot be
2; alpha cannot be zero.
drn : [0,1], optional
Drainage. drn=0 is pervious top pervious bottom. drn=1 is pervious
bottom, impervious bottom. default drn=0.
tpor : float or 1d array/list of float, optional
Time values for pore pressure vs depth calcs. Default tpor=None i.e.
time values will be taken from `t`.
H : float, optional
Drainage path length. default H=1.
kv0 : float, optional
Vertical coefficient of permeability. Default kv=1.
mv0 : float, optional
Volume compressibility. Default mv=0.1.
gamw : float, optional
Unit weight of water. Default gamw=10.
ui : float, optional
Initial uniform pore water pressure. Default ui=1.
nterms : int, optional
maximum number of series terms. default nterms=20.
plot_eigs : True/False, optional
If True then a plot of the characteristic curve and associated
eigenvalues will be created. Use plt.show after runnning the program
to display the curve. Use this to assess if the eigenvalues are
correct. Default plot_eigs=False.
Returns
-------
por : 2d array of float
Pore pressure at depth and time. ppress is an array of size
(len(z), len(t)).
doc : 1d array of float
Degree of consolidation based on surface settlement.
settlement : 1d array of float
Surface settlement at time values
Notes
-----
kv = kv0 * (1+alp*z/H)**p
mv = mv0 * (1+alp*z/H)**q
References
----------
Code based on theroy in [1]_
..[1] Zhu, G., and J. Yin. 2012. 'Analysis and Mathematical Solutions
for Consolidation of a Soil Layer with Depth-Dependent Parameters
under Confined Compression'. International Journal of Geomechanics
12 (4): 451-61.
"""
def Zmu(eta, mu, nu, b):
"""Depth fn"""
return (bessely(nu, eta) * besselj(mu, eta * b) -
besselj(nu, eta) * bessely(mu, eta * b))
def Zmu_(eta, mu, nu, b):
"""derivative of Zmu"""
return ((besselj(mu - 1.0, eta * b)/2.0 -
besselj(mu + 1.0, eta * b)/2.0)*bessely(nu, eta) -
(bessely(mu - 1.0, eta * b)/2.0 -
bessely(mu + 1.0, eta * b)/2.0)*besselj(nu, eta))
def drn1root(eta, mu, nu, b, p, n):
"""eigenvalue function"""
return (1-p)/(2-n)*Zmu(eta, nu, nu, b) + eta * b * Zmu_(eta, nu, nu, b)
z = np.atleast_1d(z)
t = np.atleast_1d(t)
if tpor is None:
tpor = t
else:
tpor = np.atleast_1d(tpor)
#derived parameters
n = p-q
b = (1 + alpha)**(1 - n/2)
nu = abs((1 - p) / (2 - n))
# print('drn', drn)
# print('p',p)
# print('q', q)
# print('n', n)
# print('b', b)
# print('nu', nu)
# print('(1 - p) / (2 - n)',(1 - p) / (2 - n))
# plot_eigs=True
if drn==0:
etam = find_n_roots(Zmu, args=(nu, nu, b), n = nterms, p=1.01)
if plot_eigs:
fig=plt.figure(figsize=(20,5))
ax = fig.add_subplot('111')
x = np.linspace(0.01,etam[-1],1000)
y = Zmu(x, nu, nu, b)
ax.plot(x, y, marker='.', markersize=3, ls='-')
ax.plot(etam,np.zeros_like(etam), 'o', markersize=6, )
ax.set_ylim(-0.3,0.3)
ax.set_xlabel('$\eta$')
ax.set_ylabel('characteristic curve')
ax.grid()
fig.tight_layout()
elif drn==1:
etam = find_n_roots(drn1root, args=(nu, nu, b, p, n), n = nterms, p=1.01)
if plot_eigs:
fig=plt.figure(figsize=(20,5))
ax = fig.add_subplot('111')
x = np.linspace(0.01,etam[-1],1000)
y = drn1root(x, nu, nu, b, p, n)
ax.plot(x, y, marker='.', markersize=3, ls='-')
ax.plot(etam,np.zeros_like(etam), 'o', markersize=6, )
ax.set_ylim(-0.3,0.3)
ax.set_xlabel('$\eta$')
ax.set_ylabel('characteristic curve')
ax.grid()
fig.tight_layout()
eta0=etam[0]
etam = etam[np.newaxis, np.newaxis, :]
cv0 = kv0 / (mv0 * gamw)
dTv = cv0 * (2 - n)**2 * alpha**2 * eta0**2 / (np.pi**2 * H**2)
y = (1 + alpha * z / H)**(1-n/2)
y = y[:, np.newaxis, np.newaxis]
T = dTv * tpor
T = T[np.newaxis, :, np.newaxis]
Tm = np.exp(-np.pi**2 * etam**2 / (4 * eta0**2) * T)
# if np.allclose(tpor, t):
# Tm_ = Tm.copy()
Zm = Zmu(etam, nu, nu, y)
if drn == 0:
# if -nu == (p - 1) / (2 - n):
if np.allclose(-nu, (p - 1) / (2 - n)):
numer = 2*np.pi * (2 + np.pi * etam * b**(1-nu) * Zmu(etam, nu-1, nu, b))
denom = 4-(b * np.pi * etam * Zmu(etam, 1 + nu, nu, b))**2
Cm = numer/denom
else:
numer = 2*np.pi * (2 - np.pi * etam * b**(1+nu) * Zmu(etam, 1+nu, nu, b))
denom = 4-(b * np.pi * etam * Zmu(etam, 1 + nu, nu, b))**2
Cm = numer/denom
elif drn == 1:
numer = 4 * np.pi
denom = 4 - (b * np.pi * etam * Zmu(etam, nu, nu, b))**2
Cm = numer / denom
por = Cm*Zm*Tm
por *= ui * y**((1 - p)/(2 - n))
por = np.sum(por, axis=2)
#degree of consolidation
etam = etam[0, :, :]
# if np.allclose(tpor, t):
# Tm=Tm_[0,:,:]
# else:
T = dTv * t
T = T[:, np.newaxis]
Tm = np.exp(-np.pi**2 * etam**2 / (4 * eta0**2) * T)
if drn == 0:
# if -nu == (p - 1) / (2 - n):
if np.allclose(-nu, (p - 1) / (2 - n)):
numer = (2 + np.pi * etam * b**(1-nu) * Zmu(etam, nu-1, nu, b))**2
denom = etam**2 * ((b * np.pi * etam * Zmu(etam, 1 + nu, nu, b))**2-4)
Cm = numer/denom
else:
print('got here')
numer = (2 - np.pi * etam * b**(1+nu) * Zmu(etam, 1 + nu, nu, b))**2
denom = etam**2 * ((b * np.pi * etam * Zmu(etam, 1 + nu, nu, b))**2-4)
Cm = numer/denom
elif drn == 1:
numer = 1.0
denom = etam**2*((b * np.pi * etam * Zmu(etam, nu, nu, b))**2-4)
Cm = numer / denom
doc = np.sum(Cm*Tm,axis=1)
if drn==0:
numer = 4*(1+q)
# numer_settle = 4
elif drn==1:
numer = 16*(1+q)
# numer_settle = 16
denom = (2 - n) * ((1+alpha)**(1 + q) - 1)
# denom_settle = (2 - n)
# settle = mv0*ui*(1 - (H/alpha*numer_settle / denom_settle) * doc)
fr = numer / denom
doc*= -fr
doc += 1
settle = doc * ui* mv0*((1+alpha)**(1 + q) - 1)/(1+q)*H/alpha
# print(",".join(['{:.3g}']*4).format((1 - p) / (2 - n), nu, t[0], doc[0]))
return por, doc, settle
def zhuandyin2012(z, t, alpha, p, q, drn=0, tpor=None, H = 1, kv0 = 1, mv0 = 0.1, gamw = 10,
ui = 1, nterms = 20, plot_eigs=False):
"""Analysis and mathematical solutions for consolidation of a soil layer
with depth-dependent parameters under confined compression.
An implementation of Zhu and Yin (2012) [1]_.
Features:
- Single layer.
- Permeability and volume compressibility can vary with depth with a
power lay relationship.
- Instant load uniform with depth.
- PTIB ansd PTPB drainage conditions.
- Pore pressure vs depth at variaous times
- Degree of consolidation based on surface settlement vs time.
- Settlement vs time.
Parameters
----------
z : float or 1d array/list of float
Depth.
t : float or 1d array/list of float
Time for degree of consolidation calcs.
tpor : float or 1d array/list of float
Time for pore pressure vs depth calcs.
alpha, p, q : float
Exponent in depth dependence of permeability and compressibility
respectively. e.g. mv = mv0 * (1+alpha*z/H)**q. Note p/q cannot be
2; alpha cannot be zero.
drn : [0,1], optional
Drainage. drn=0 is pervious top pervious bottom. drn=1 is pervious
bottom, impervious bottom. default drn=0.
tpor : float or 1d array/list of float, optional
Time values for pore pressure vs depth calcs. Default tpor=None i.e.
time values will be taken from `t`.
H : float, optional
Drainage path length. default H=1.
kv0 : float, optional
Vertical coefficient of permeability. Default kv=1.
mv0 : float, optional
Volume compressibility. Default mv=0.1.
gamw : float, optional
Unit weight of water. Default gamw=10.
ui : float, optional
Initial uniform pore water pressure. Default ui=1.
nterms : int, optional
maximum number of series terms. default nterms=20.
plot_eigs : True/False, optional
If True then a plot of the characteristic curve and associated
eigenvalues will be created. Use plt.show after runnning the program
to display the curve. Use this to assess if the eigenvalues are
correct. Default plot_eigs=False.
Returns
-------
por : 2d array of float
Pore pressure at depth and time. ppress is an array of size
(len(z), len(t)).
doc : 1d array of float
Degree of consolidation based on surface settlement.
settlement : 1d array of float
Surface settlement at time values
Notes
-----
kv = kv0 * (1+alp*z/H)**p
mv = mv0 * (1+alp*z/H)**q
References
----------
Code based on theroy in [1]_
..[1] Zhu, G., and J. Yin. 2012. 'Analysis and Mathematical Solutions
for Consolidation of a Soil Layer with Depth-Dependent Parameters
under Confined Compression'. International Journal of Geomechanics
12 (4): 451-61.
"""
def Zmu(eta, mu, nu, b):
"""Depth fn"""
return (bessely(nu, eta) * besselj(mu, eta * b) -
besselj(nu, eta) * bessely(mu, eta * b))
def Zmu_(eta, mu, nu, b):
"""derivative of Zmu"""
return ((besselj(mu - 1.0, eta * b)/2.0 -
besselj(mu + 1.0, eta * b)/2.0)*bessely(nu, eta) -
(bessely(mu - 1.0, eta * b)/2.0 -
bessely(mu + 1.0, eta * b)/2.0)*besselj(nu, eta))
def drn1root(eta, mu, nu, b, p, n):
"""eigenvalue function"""
return (1-p)/(2-n)*Zmu(eta, nu, nu, b) + eta * b * Zmu_(eta, nu, nu, b)
z = np.atleast_1d(z)
t = np.atleast_1d(t)
if tpor is None:
tpor = t
else:
tpor = np.atleast_1d(tpor)
#derived parameters
n = p-q
b = (1 + alpha)**(1 - n/2)
nu = abs((1 - p) / (2 - n))
# print('drn', drn)
# print('p',p)
# print('q', q)
# print('n', n)
# print('b', b)
# print('nu', nu)
# print('(1 - p) / (2 - n)',(1 - p) / (2 - n))
# plot_eigs=True
if drn==0:
etam = find_n_roots(Zmu, args=(nu, nu, b), n = nterms, p=1.01)
if plot_eigs:
fig=plt.figure(figsize=(20,5))
ax = fig.add_subplot('111')
x = np.linspace(0.01,etam[-1],1000)
y = Zmu(x, nu, nu, b)
ax.plot(x, y, marker='.', markersize=3, ls='-')
ax.plot(etam,np.zeros_like(etam), 'o', markersize=6, )
ax.set_ylim(-0.3,0.3)
ax.set_xlabel('$\eta$')
ax.set_ylabel('characteristic curve')
ax.grid()
fig.tight_layout()
elif drn==1:
etam = find_n_roots(drn1root, args=(nu, nu, b, p, n), n = nterms, p=1.01)
if plot_eigs:
fig=plt.figure(figsize=(20,5))
ax = fig.add_subplot('111')
x = np.linspace(0.01,etam[-1],1000)
y = drn1root(x, nu, nu, b, p, n)
ax.plot(x, y, marker='.', markersize=3, ls='-')
ax.plot(etam,np.zeros_like(etam), 'o', markersize=6, )
ax.set_ylim(-0.3,0.3)
ax.set_xlabel('$\eta$')
ax.set_ylabel('characteristic curve')
ax.grid()
fig.tight_layout()
eta0=etam[0]
etam = etam[np.newaxis, np.newaxis, :]
cv0 = kv0 / (mv0 * gamw)
dTv = cv0 * (2 - n)**2 * alpha**2 * eta0**2 / (np.pi**2 * H**2)
y = (1 + alpha * z / H)**(1-n/2)
y = y[:, np.newaxis, np.newaxis]
T = dTv * tpor
T = T[np.newaxis, :, np.newaxis]
Tm = np.exp(-np.pi**2 * etam**2 / (4 * eta0**2) * T)
# if np.allclose(tpor, t):
# Tm_ = Tm.copy()
Zm = Zmu(etam, nu, nu, y)
if drn == 0:
# if -nu == (p - 1) / (2 - n):
if np.allclose(-nu, (p - 1) / (2 - n)):
numer = 2*np.pi * (2 + np.pi * etam * b**(1-nu) * Zmu(etam, nu-1, nu, b))
denom = 4-(b * np.pi * etam * Zmu(etam, 1 + nu, nu, b))**2
Cm = numer/denom
else:
numer = 2*np.pi * (2 - np.pi * etam * b**(1+nu) * Zmu(etam, 1+nu, nu, b))
denom = 4-(b * np.pi * etam * Zmu(etam, 1 + nu, nu, b))**2
Cm = numer/denom
elif drn == 1:
numer = 4 * np.pi
denom = 4 - (b * np.pi * etam * Zmu(etam, nu, nu, b))**2
Cm = numer / denom
por = Cm*Zm*Tm
por *= ui * y**((1 - p)/(2 - n))
por = np.sum(por, axis=2)
#degree of consolidation
etam = etam[0, :, :]
# if np.allclose(tpor, t):
# Tm=Tm_[0,:,:]
# else:
T = dTv * t
T = T[:, np.newaxis]
Tm = np.exp(-np.pi**2 * etam**2 / (4 * eta0**2) * T)
if drn == 0:
# if -nu == (p - 1) / (2 - n):
if np.allclose(-nu, (p - 1) / (2 - n)):
numer = (2 + np.pi * etam * b**(1-nu) * Zmu(etam, nu-1, nu, b))**2
denom = etam**2 * ((b * np.pi * etam * Zmu(etam, 1 + nu, nu, b))**2-4)
Cm = numer/denom
else:
print('got here')
numer = (2 - np.pi * etam * b**(1+nu) * Zmu(etam, 1 + nu, nu, b))**2
denom = etam**2 * ((b * np.pi * etam * Zmu(etam, 1 + nu, nu, b))**2-4)
Cm = numer/denom
elif drn == 1:
numer = 1.0
denom = etam**2*((b * np.pi * etam * Zmu(etam, nu, nu, b))**2-4)
Cm = numer / denom
doc = np.sum(Cm*Tm,axis=1)
if drn==0:
numer = 4*(1+q)
# numer_settle = 4
elif drn==1:
numer = 16*(1+q)
# numer_settle = 16
denom = (2 - n) * ((1+alpha)**(1 + q) - 1)
# denom_settle = (2 - n)
# settle = mv0*ui*(1 - (H/alpha*numer_settle / denom_settle) * doc)
fr = numer / denom
doc*= -fr
doc += 1
settle = doc * ui* mv0*((1+alpha)**(1 + q) - 1)/(1+q)*H/alpha
# print(",".join(['{:.3g}']*4).format((1 - p) / (2 - n), nu, t[0], doc[0]))
return por, doc, settle
