def Hessian(self,StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
mu = self.mu
lamb = self.lamb
c1 = self.c1
c2 = self.c2
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
E = ElectricFieldx
# print(E.shape, b.shape, np.dot(b,ElectricFieldx).shape)
be = np.dot(b,ElectricFieldx).reshape(self.ndim)
# Elasticity
C = lamb*(2.*J-1.)*einsum("ij,kl",I,I) +(mu/J - lamb*(J-1))*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) )
C_Voigt = Voigt(C,1)
# Coupled Tensor (e - 3rd order)
e_voigt = 2*c2/J*( einsum('ij,k',b,be) + einsum('i,jk',be,b) )
e_voigt = Voigt(e_voigt,1)
# Dielectric Tensor (Permittivity - 2nd order)
# Permittivity = -2./J*np.dot((c1*I+c2*b),b)
Permittivity = 2./J*np.dot((c1*I+c2*b),b)
factor = -1.
H1 = np.concatenate((C_Voigt,factor*e_voigt),axis=1)
H2 = np.concatenate((factor*e_voigt.T,Permittivity),axis=1)
H_Voigt = np.concatenate((H1,H2),axis=0)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=None, elem=0, gcounter=0):
"""Hessian split into isochoroic and volumetric parts"""
I = StrainTensors['I']
b = StrainTensors['b'][gcounter]
J = StrainTensors['J'][gcounter]
mu = self.mu
kappa = self.kappa
# ISOCHORIC
H_Voigt = 2*mu*J**(-5./3.)*(1./9.*trace(b)*einsum('ij,kl',I,I) - \
1./3.*(einsum('ij,kl',b,I) + einsum('ij,kl',I,b)) +\
1./6.*trace(b)*(einsum('ik,jl',I,I) + einsum('il,jk',I,I)) )
# VOLUMETRIC CHANGES
if self.is_nearly_incompressible:
H_Voigt += self.pressure * (
einsum('ij,kl', I, I) -
(einsum('ik,jl', I, I) + einsum('il,jk', I, I)))
else:
H_Voigt += kappa * (
(2. * J - 1.) * einsum('ij,kl', I, I) - (J - 1.) *
(einsum('ik,jl', I, I) + einsum('il,jk', I, I)))
H_Voigt = Voigt(H_Voigt, 1)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def CauchyStress(self,StrainTensors,ElectricFieldx,elem=0,gcounter=0):
strain = StrainTensors['strain'][gcounter]
strain_Voigt = Voigt(strain)
E = self.E
E_A = self.E_A
G_A = self.G_A
v = self.nu
H_Voigt = np.array([
[ -(E*(- E*v**2 + E_A))/((v + 1)*(2*E*v**2 + E_A*v - E_A)), -(E*v*(E_A + E*v))/((v + 1)*(2*E*v**2 + E_A*v - E_A)), -(E_A*E*v)/(2*E*v**2 + E_A*v - E_A), 0, 0, 0],
[ -(E*v*(E_A + E*v))/((v + 1)*(2*E*v**2 + E_A*v - E_A)), -(E*(- E*v**2 + E_A))/((v + 1)*(2*E*v**2 + E_A*v - E_A)), -(E_A*E*v)/(2*E*v**2 + E_A*v - E_A), 0, 0, 0],
[ -(E_A*E*v)/(2*E*v**2 + E_A*v - E_A), -(E_A*E*v)/(2*E*v**2 + E_A*v - E_A), (E_A**2*(v - 1))/(2*E*v**2 + E_A*v - E_A), 0, 0, 0],
[ 0, 0, 0, E/(2*(v + 1)), 0, 0],
[ 0, 0, 0, 0, G_A, 0],
[ 0, 0, 0, 0, 0, G_A]
])
if self.ndim == 2:
# CAREFUL WITH THIS SLICING AS SOME MATERIAL CONSTANTS WOULD BE REMOVED.
# ESSENTIALLY IN PLANE STRAIN ANISOTROPY THE BEHAVIOUR OF MATERIAL
# PERPENDICULAR TO THE PLANE IS LOST
H_Voigt = H_Voigt[np.array([2,1,-1])[:,None],[2,1,-1]]
stress = UnVoigt(np.dot(H_Voigt,strain_Voigt))
return stress
def Hessian(self, StrainTensors, ElectricFieldx=None, elem=0, gcounter=0):
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
F = StrainTensors['F'][gcounter]
C = np.dot(F.T, F)
if np.isclose(J, 0) or J < 0:
delta = np.sqrt(0.04 * J * J + 1e-8)
J = 0.5 * (J + np.sqrt(J**2 + 4 * delta**2))
mu = self.mu
lamb = self.lamb
E = 0.5 * (C - I)
C_Voigt = lamb * np.einsum("ij,kl", I, I) + mu * (
np.einsum("ik,jl", I, I) + np.einsum("il,jk", I, I))
C_Voigt = 1. / J * np.einsum("iI,jJ,kK,lL,IJKL", F, F, F, F, C_Voigt)
# same here
# C_Voigt = lamb/J * np.einsum("ij,kl",b,b) + mu/J * (np.einsum("ik,jl",b,b) + np.einsum("il,jk",b,b))
C_Voigt = Voigt(C_Voigt, 1)
self.H_VoigtSize = C_Voigt.shape[0]
return C_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
mu1 = self.mu1
mu2 = self.mu2
mu3 = self.mu3
lamb = self.lamb
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
F = StrainTensors['F'][gcounter]
H = J * np.linalg.inv(F).T
N = self.anisotropic_orientations[elem][:, None]
HN = np.dot(H, N)[:, 0]
innerHN = einsum('i,i', HN, HN)
outerHN = einsum('i,j', HN, HN)
H_Voigt = 2.*mu2/J* ( 2.0*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b) ) + \
2.*(mu1+2.*mu2+mu3)/J * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) ) + \
lamb*(2.*J-1.)*einsum('ij,kl',I,I) - lamb*(J-1.) * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) ) - \
4.*mu3/J * ( einsum('ij,kl',I,outerHN) + einsum('ij,kl',outerHN,I) ) + \
2.*mu3/J*innerHN*(2.0*einsum('ij,kl',I,I) - einsum('ik,jl',I,I) - einsum('il,jk',I,I) ) + \
2.*mu3/J * ( einsum('ik,jl',I,outerHN) + einsum('il,jk',I,outerHN) + \
einsum('ik,jl',outerHN,I) + einsum('il,jk',outerHN,I) )
# 2.*mu3/J * ( einsum('il,j,k',I,HN,HN) + einsum('jl,i,k',I,HN,HN) + \
# einsum('ik,j,l',I,HN,HN) + einsum('jk,i,l',I,HN,HN) )
H_Voigt = Voigt(H_Voigt, 1)
return H_Voigt
def Hessian(self,
StrainTensors,
ElectricDisplacementx,
elem=0,
gcounter=0):
d = self.ndim
I = StrainTensors['I']
F = StrainTensors['F'][gcounter]
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
if np.isclose(J, 0) or J < 0:
delta = np.sqrt(0.04 * J * J + 1e-8)
# J = 0.5 * (J + np.sqrt(J**2 + 4 *delta**2))
# J = 1.
trb = trace(b) + 0
C_Voigt = 4./d * J**(-1) * einsum("ij,kl", (-1./d * trb * I + b), (-1./d) * J**(-2./d) * I ) +\
4./d**2 * J**(-1) * J**(-2./d) * (-einsum("ij,kl", I, b) + 0.5 * trb * (einsum("ik,jl",I,I)+einsum("il,jk",I,I)) )
# factor = np.exp(1./d * J**(-2./d) * trb - 1.)
# sigma = 2./d * J**(-2./d - 1.) * (-1./d * trb * I + b)
# C_Voigt = factor * (np.einsum("ij,kl", sigma, sigma) + C_Voigt)
C_Voigt = Voigt(C_Voigt, 1)
# s = np.linalg.svd(C_Voigt)[1]
# if np.any(s < 0):
# print(s)
return C_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
lamb = self.lamb
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
# Actual version
# d = np.einsum
# H_Voigt = Voigt( 4.0*beta/J*d('ij,kl',b,b) - 2.0*beta/J*( d('ik,jl',b,b) + d('il,jk',b,b) ) +\
# (lamb+4.0*beta+4.0*alpha/J)*d('ij,kl',I,I) + 2.0*(lamb*(J-1.0) -4.0*beta -2.0*alpha/J)*d('ij,kl',I,I) -\
# 1.0*(lamb*(J-1.0) -4.0*beta -2.0*alpha/J)*(d('ik,jl',I,I)+d('il,jk',I,I)) ,1)
# # Simplified version
# H_Voigt = 2.0*beta/J*( 2.0*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b) ) + \
# (lamb*(2.0*J-1.0) -4.0*beta)*einsum('ij,kl',I,I) - \
# (lamb*(J-1.0) -4.0*beta -2.0*alpha/J)*( einsum('ik,jl',I,I) + einsum('il,jk',I,I) )
# Further simplified version
H_Voigt = 2.0*self.beta/J*( 2.0*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b) ) + \
(lamb*(2.0*J-1.0) -4.0*self.beta)*self.Iijkl - \
(lamb*(J-1.0) -4.0*self.beta -2.0*self.alpha/J)*self.Iikjl
H_Voigt = Voigt(H_Voigt, 1)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=None, elem=0, gcounter=0):
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
F = StrainTensors['F'][gcounter]
b = StrainTensors['b'][gcounter]
if np.isclose(J, 0) or J < 0:
delta = np.sqrt(0.04 * J * J + 1e-8)
else:
delta = 1e-4
J = 0.5 * (J + np.sqrt(J**2 + 4 * delta**2))
# cr = J**2
mu = self.mu
lamb = self.lamb
C_Voigt = 2 * (lamb * (1. - 1./J) - mu / J) * self.d2CrdCdC(StrainTensors, gcounter) +\
(mu+lamb)/J**3 * J * np.einsum("ij,kl", self.dCrdC(StrainTensors, gcounter), self.dCrdC(StrainTensors, gcounter))
C_Voigt = Voigt(C_Voigt, 1)
# print(C_Voigt)
# exit()
self.H_VoigtSize = C_Voigt.shape[0]
return C_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
"""Hessian split into isochoroic and volumetric parts"""
I = StrainTensors['I']
b = StrainTensors['b'][gcounter]
J = StrainTensors['J'][gcounter]
if np.isclose(J, 0) or J < 0:
delta = np.sqrt(0.04 * J * J + 1e-8)
# J = 0.5 * (J + np.sqrt(J**2 + 4 *delta**2))
mu = self.mu
trb = trace(b)
if self.ndim == 2:
trb += 1.
# ISOCHORIC
H_Voigt = 2*mu*J**(-5./3.)*(1./9.*trb*einsum('ij,kl',I,I) - \
1./3.*(einsum('ij,kl',b,I) + einsum('ij,kl',I,b)) +\
1./6.*trb*(einsum('ik,jl',I,I) + einsum('il,jk',I,I)) )
# VOLUMETRIC
# H_Voigt += self.pressure[elem]*(einsum('ij,kl',I,I) - (einsum('ik,jl',I,I) + einsum('il,jk',I,I)))
H_Voigt += self.lamb * (2 * J - 1) * einsum(
'ij,kl', I, I) - self.lamb * (J - 1) * (einsum('ik,jl', I, I) +
einsum('il,jk', I, I))
H_Voigt = Voigt(H_Voigt, 1)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
# Get material constants (5 in this case)
E = self.E
E_A = self.E_A
v = self.nu
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
F = StrainTensors['F'][gcounter]
H = J * np.linalg.inv(F).T
N = self.anisotropic_orientations[elem][:, None]
FN = np.dot(F, N)[:, 0]
HN = np.dot(H, N)[:, 0]
innerFN = einsum('i,i', FN, FN)
innerHN = einsum('i,i', HN, HN)
outerHN = einsum('i,j', HN, HN)
gamma = self.gamma
lamb = -(E_A * E * v) / (2. * E * v**2 + E_A * v - E_A)
ut = (E**2 * v**2 + E_A * E * v**2 + E_A * E * v -
E_A * E) / (2 * (v + 1) * (2 * E * v**2 + E_A * v - E_A))
beta = 0.
eta_1 = (E_A**2 * v**2 - E_A**2 - 2 * E_A * E * v**2 + E_A * E +
E**2 * v**2) / (4 * (gamma - 1) * (v + 1) *
(2 * E * v**2 + E_A * v - E_A))
eta_2 = -(E_A**2 * v - E_A**2 + E_A * E - E_A * E * v) / (
4 * (gamma - 1) * (2 * E * v**2 + E_A * v - E_A))
alpha = ut - 4 * gamma * beta - 2 * (1 - gamma) * eta_1 - 2 * (
1 - gamma) * eta_2
alpha = alpha / 2. / gamma
eta = [eta_1, eta_2]
H_Voigt = 2.*gamma*beta/J* ( 2.0*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b) ) - \
(- lamb*(2.*J-1.) ) *einsum('ij,kl',I,I) + \
(ut - lamb*(J-1.) ) * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) )
for m in range(2, 4):
H_Voigt += self.TransverseHessianNCN(StrainTensors, m, eta[m - 2],
gamma, FN, innerFN, elem,
gcounter)
H_Voigt += self.TransverseHessianNGN(StrainTensors, 1., eta_1, gamma,
HN, innerHN, elem, gcounter)
H_Voigt += self.TransverseHessianNGN(StrainTensors, 1., eta_2, gamma,
HN, innerHN, elem, gcounter)
H_Voigt = Voigt(H_Voigt, 1)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self,StrainTensors,ElectricFieldx=0,elem=0,gcounter=0):
mu1 = self.mu1
mu2 = self.mu2
lamb = self.lamb
eps_1 = self.eps_1
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
E = ElectricFieldx.reshape(self.ndim,1)
Ex = E.reshape(E.shape[0])
EE = np.dot(E,E.T)
be = np.dot(b,ElectricFieldx).reshape(self.ndim,1)
C_mech = 2.*mu2/J*(2*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b)) +\
2.*(mu1+2.*mu2)/J*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) ) + \
lamb*(2.*J-1.)*einsum("ij,kl",I,I) - lamb*(J-1)*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) )
C_elect = -eps_1/J* (einsum("ik,j,l",I,Ex,Ex) + einsum("il,j,k",I,Ex,Ex) + einsum("i,k,jl",Ex,Ex,I) + einsum("i,l,jk",Ex,Ex,I))
C_Voigt = Voigt(C_mech + C_elect, 1)
P_Voigt = eps_1/J*(einsum('ik,j',I,Ex)+einsum('i,jk',Ex,I))
P_Voigt = Voigt(P_Voigt,1)
E_Voigt = -eps_1/J*I
# Build the Hessian
factor = -1.
H1 = np.concatenate((C_Voigt,factor*P_Voigt),axis=1)
H2 = np.concatenate((factor*P_Voigt.T,E_Voigt),axis=1)
H_Voigt = np.concatenate((H1,H2),axis=0)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
mu = self.mu
lamb = self.lamb
c1 = self.c1
c2 = self.c2
eps_1 = self.eps_1
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
E = 1.0 * ElectricFieldx.reshape(self.ndim, 1)
Ex = E.reshape(E.shape[0])
EE = np.dot(E, E.T)
be = np.dot(b, ElectricFieldx).reshape(self.ndim)
C_Voigt = lamb/J*einsum('ij,kl',I,I) - (lamb*np.log(J) - mu)/J*( einsum('ik,jl',I,I) + einsum('il,jk',I,I) ) + \
eps_1*( einsum('ij,kl',I,EE) + einsum('ij,kl',EE,I) - einsum('ik,jl',EE,I) - einsum('il,jk',EE,I) - \
einsum('ik,jl',I,EE) - einsum('il,jk',I,EE) ) + \
eps_1*(np.dot(E.T,E)[0,0])*0.5*( einsum('ik,jl',I,I) + einsum('il,jk',I,I) - einsum('ij,kl',I,I) )
C_Voigt = Voigt(C_Voigt, 1)
P_Voigt = eps_1*( einsum('ik,j',I,Ex) + einsum('jk,i',I,Ex) - einsum('ij,k',I,Ex)) +\
2.0*c2/J*( einsum('ik,j',b,be) + einsum('i,jk',be,b) )
P_Voigt = Voigt(P_Voigt, 1)
E_Voigt = -eps_1 * I + 2.0 * c1 / J * b + 2.0 * c2 / J * np.dot(b, b)
# Build the Hessian
factor = -1.
H1 = np.concatenate((C_Voigt, factor * P_Voigt), axis=1)
H2 = np.concatenate((factor * P_Voigt.T, E_Voigt), axis=1)
H_Voigt = np.concatenate((H1, H2), axis=0)
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
# CHECK IF FIBRE ORIENTATION IS SET
if self.anisotropic_orientations is None:
raise ValueError(
"Fibre orientation for non-isotropic material is not available"
)
# Get material constants (5 in this case)
E = self.E
E_A = self.E_A
G_A = self.G_A
v = self.nu
mu = self.mu
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
F = StrainTensors['F'][gcounter]
H = J * np.linalg.inv(F).T
N = self.anisotropic_orientations[elem, :, None]
FN = np.dot(F, N)[:, 0]
alpha = E / 8.0 / (1. + v)
beta = E / 8.0 / (1. + v)
eta_1 = 4. * alpha - G_A
lamb = -(3 * E) / (2 * (v + 1)) - (E * (-E * v**2 + E_A)) / (
(v + 1) * (2 * E * v**2 + E_A * v - E_A))
eta_2 = E/(4*(v + 1)) - (E_A*E*v)/(4*(2*E*v**2 + E_A*v - E_A)) + \
(E*(- E*v**2 + E_A))/(4*(v + 1)*(2*E*v**2 + E_A*v - E_A))
gamma = (E_A**2*(v - 1))/(8*(2*E*v**2 + E_A*v - E_A)) - G_A/2 + \
(E_A*E*v)/(4*(2*E*v**2 + E_A*v - E_A)) - \
(E*(- E*v**2 + E_A))/(8*(v + 1)*(2*E*v**2 + E_A*v - E_A))
ut = 2 * alpha + 4 * beta
H_Voigt = 2.*beta/J* ( 2.0*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b) ) + \
lamb*(2.*J-1.) *einsum('ij,kl',I,I) + \
(ut/J - lamb*(J-1.) ) * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) ) + \
4.*eta_2/J*( einsum('ij,k,l',b,FN,FN) + einsum('i,j,kl',FN,FN,b) ) + \
8.*gamma/J*( einsum('i,j,k,l',FN,FN,FN,FN) ) - \
eta_1/J*( einsum('jk,i,l',b,FN,FN) + einsum('ik,j,l',b,FN,FN) +
einsum('jl,i,k',b,FN,FN) + einsum('il,j,k',b,FN,FN) )
H_Voigt = Voigt(H_Voigt, 1)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
alpha = self.alpha
beta = self.beta
kappa = self.kappa
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
F = StrainTensors['F'][gcounter]
# b=np.dot(F,F.T)
H = J * np.linalg.inv(F).T
g = np.dot(H, H.T)
if self.ndim == 2:
trb = trace(b) + 1
trg = trace(g) + J**2
elif self.ndim == 3:
trb = trace(b)
trg = trace(g)
H_Voigt = -4/3.*alpha*J**(-5/3.)*( einsum('ij,kl',b,I) + einsum('ij,kl',I,b) ) + \
4.*alpha/9.*J**(-5/3.)*trb*einsum('ij,kl',I,I) + \
2/3.*alpha*J**(-5/3.)*trb*( einsum('il,jk',I,I) + einsum('ik,jl',I,I) ) + \
beta*J**(-3)*trg**(3./2.)* ( einsum('ij,kl',I,I) - einsum('ik,jl',I,I) - einsum('il,jk',I,I) ) - \
3.*beta*J**(-3)*trg**(1./2.)*( einsum('ij,kl',I,g) + einsum('ij,kl',g,I) ) + \
6.*beta*J**(-3)*trg**(1./2.)*( einsum('ik,jl',I,g) + einsum('il,jk',g,I) ) + \
3.*beta*J**(-3)*trg**(-1./2.)*( einsum('ij,kl',g,g) ) + \
kappa*(2.0*J-1)*einsum('ij,kl',I,I) - kappa*(J-1)*(einsum('ik,jl',I,I)+einsum('il,jk',I,I)) # #
# # WITH PRE-COMPUTED IDENTITY TENSORS
# H_Voigt = -4/3.*alpha*J**(-5/3.)*( einsum('ij,kl',b,I) + einsum('ij,kl',I,b) ) + \
# 4.*alpha/9.*J**(-5/3.)*trb*self.Iijkl + \
# 2/3.*alpha*J**(-5/3.)*trb*self.Iikjl + \
# beta*J**(-3)*trg**(3./2.)*( self.Iijkl - self.Iikjl ) - \
# 3.*beta*J**(-3)*trg**(1./2.)*( einsum('ij,kl',I,g) + einsum('ij,kl',g,I) ) + \
# 6.*beta*J**(-3)*trg**(1./2.)*( einsum('ik,jl',I,g) + einsum('il,jk',g,I) ) + \
# 3.*beta*J**(-3)*trg**(-1./2.)*( einsum('ij,kl',g,g) ) + \
# kappa*(2.0*J-1)*self.Iijkl - kappa*(J-1)*self.Iikjl
H_Voigt = Voigt(H_Voigt, 1)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self,StrainTensors,ElectricFieldx,elem=0,gcounter=0):
mu = self.mu
lamb = self.lamb
eta = self.eta
I = StrainTensors['I']
self.elasticity_tensor = Voigt(lamb*einsum('ij,kl',I,I)+mu*(einsum('ik,jl',I,I)+einsum('il,jk',I,I)),1)
self.gradient_elasticity_tensor = 2.*eta*I
self.coupling_tensor0 = np.zeros((self.elasticity_tensor.shape[0],self.gradient_elasticity_tensor.shape[0]))
# # BUILD HESSIAN
# factor = 1.
# H1 = np.concatenate((self.elasticity_tensor,factor*self.coupling_tensor0),axis=1)
# H2 = np.concatenate((factor*self.coupling_tensor0.T,self.gradient_elasticity_tensor),axis=1)
# H_Voigt = np.concatenate((H1,H2),axis=0)
# return H_Voigt
return None
def CauchyStress(self,StrainTensors,ElectricFieldx,elem=0,gcounter=0):
strain = StrainTensors['strain'][gcounter]
strain_Voigt = Voigt(strain)
E = self.E
E_A = self.E_A
G_A = self.G_A
v = self.nu
H_Voigt = np.array([
[ -(E*(- E*v**2 + E_A))/((v + 1)*(2*E*v**2 + E_A*v - E_A)), -(E*v*(E_A + E*v))/((v + 1)*(2*E*v**2 + E_A*v - E_A)), -(E_A*E*v)/(2*E*v**2 + E_A*v - E_A), 0, 0, 0],
[ -(E*v*(E_A + E*v))/((v + 1)*(2*E*v**2 + E_A*v - E_A)), -(E*(- E*v**2 + E_A))/((v + 1)*(2*E*v**2 + E_A*v - E_A)), -(E_A*E*v)/(2*E*v**2 + E_A*v - E_A), 0, 0, 0],
[ -(E_A*E*v)/(2*E*v**2 + E_A*v - E_A), -(E_A*E*v)/(2*E*v**2 + E_A*v - E_A), (E_A**2*(v - 1))/(2*E*v**2 + E_A*v - E_A), 0, 0, 0],
[ 0, 0, 0, E/(2*(v + 1)), 0, 0],
[ 0, 0, 0, 0, G_A, 0],
[ 0, 0, 0, 0, 0, G_A]
])
if self.ndim == 2:
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
"""Hessian split into isochoroic and volumetric parts"""
I = StrainTensors['I']
b = StrainTensors['b'][gcounter]
J = StrainTensors['J'][gcounter]
mu = self.mu
# ISOCHORIC
H_Voigt = 2*mu*J**(-5./3.)*(1./9.*trace(b)*einsum('ij,kl',I,I) - \
1./3.*(einsum('ij,kl',b,I) + einsum('ij,kl',I,b)) +\
1./6.*trace(b)*(einsum('ik,jl',I,I) + einsum('il,jk',I,I)) )
# VOLUMETRIC
H_Voigt += self.pressure[elem]*(einsum('ij,kl',I,I) - (einsum('ik,jl',I,I) + einsum('il,jk',I,I)))
H_Voigt = Voigt(H_Voigt,1)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
mu = self.mu
lamb = self.lamb
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
F = StrainTensors['F'][gcounter]
H = J * np.linalg.inv(F).T
N = self.anisotropic_orientations[elem][:, None]
FN = np.dot(F, N)[:, 0]
HN = np.dot(H, N)[:, 0]
innerHN = einsum('i,i', HN, HN)
outerHN = einsum('i,j', HN, HN)
# FIX GAMMA
gamma = 0.5
# gamma = 1.0
alpha = mu / 2. / gamma
beta = mu / 2. / gamma
eta = mu / 3.
ut = 2. * gamma * (alpha + 2.0 * beta) + 2. * (1. - gamma) * eta
lamb = lamb + 2. * gamma * alpha - 2 * (1. - gamma) * eta
H_Voigt = 2.*gamma*beta/J* ( 2.0*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b) ) + \
4.*(1-gamma)*eta/J * einsum('i,j,k,l',FN,FN,FN,FN) + \
4.*(1-gamma)*eta/J * ( innerHN * einsum('ij,kl',I,I) - \
0.5*innerHN * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) ) - \
einsum('ij,k,l',I,HN,HN) - einsum('i,j,kl',HN,HN,I) ) + \
2.*(1-gamma)*eta/J * ( einsum('il,j,k',I,HN,HN) + einsum('jl,i,k',I,HN,HN) + \
einsum('ik,j,l',I,HN,HN) + einsum('jk,i,l',I,HN,HN) ) - \
ut*einsum('ij,kl',I,I) + ut * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) ) + \
lamb*(2.*J-1.)*einsum('ij,kl',I,I) - lamb*(J-1.) * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) )
H_Voigt = Voigt(H_Voigt, 1)
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def Hessian(self,
StrainTensors,
ElectricDisplacementx,
elem=0,
gcounter=0):
mu1 = self.mu1
mu2 = self.mu2
lamb = self.lamb
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
C_Voigt = 2.*mu2/J*(2*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b)) +\
2.*(mu1+2.*mu2)/J*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) ) + \
lamb*(2.*J-1.)*einsum("ij,kl",I,I) - lamb*(J-1)*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) )
C_Voigt = Voigt(C_Voigt, 1)
return C_Voigt
def Hessian(self,StrainTensors,ElectricFieldx=None,elem=0,gcounter=0):
kappa = self.kappa
mu = self.mu
k1 = self.k1
k2 = self.k2
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
F = StrainTensors['F'][gcounter]
if self.ndim == 3:
trb = trace(b)
elif self.ndim == 2:
trb = trace(b) + 1
H_Voigt = 2.*mu*J**(-5./3.) * (1./9.*trb*einsum('ij,kl',I,I) - \
1./3.*einsum('ij,kl',I,b) - 1./3.*einsum('ij,kl',b,I) + \
1./6.*trb*(einsum('il,jk',I,I) + einsum('ik,jl',I,I)) )
if self.is_nearly_incompressible:
H_Voigt += self.pressure*(einsum('ij,kl',I,I) - (einsum('ik,jl',I,I) + einsum('il,jk',I,I)))
else:
H_Voigt += kappa*((2.*J-1.)*einsum('ij,kl',I,I) - (J-1.)*(einsum('ik,jl',I,I) + einsum('il,jk',I,I)))
# Anisotropic contibution
nfibres = self.anisotropic_orientations.shape[1]
for i_fibre in range(nfibres):
N = self.anisotropic_orientations[elem,i_fibre,:,None]
FN = np.dot(F,N)[:,0]
innerFN = einsum('i,i',FN,FN)
outerFN = einsum('i,j',FN,FN)
expo = np.exp(k2*(innerFN-1.)**2)
H_Voigt += 4.*k1/J*(1.+2.*k2*(innerFN-1.)**2)*expo*einsum('ij,kl',outerFN,outerFN)
H_Voigt = Voigt(H_Voigt ,1)
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=None, elem=0, gcounter=0):
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
# Jb = 1.27168554933
Jb = self.Jbar[elem]
mu2 = self.mu / J - self.lamb * (J - Jb)
lamb2 = self.lamb * (2 * J - Jb)
H_Voigt_n = lamb2 * self.vIijIkl + mu2 * self.vIikIjl
# return H_Voigt
H_Voigt_l = Voigt(
self.lamb * einsum('ij,kl', I, I) + self.mu *
(einsum('ik,jl', I, I) + einsum('il,jk', I, I)), 1)
alp = 0.1
# H_Voigt = (1-alp)*H_Voigt_l + alp*H_Voigt_n
H_Voigt = H_Voigt_n
# return H_Voigt_l + H_Voigt_n
return H_Voigt
def Hessian(self,StrainTensors,ElectricFieldx,elem=0,gcounter=0):
mu = self.mu
lamb = self.lamb
eta = self.eta
eps = self.eps
I = StrainTensors['I']
self.elasticity_tensor = Voigt(lamb*einsum('ij,kl',I,I)+mu*(einsum('ik,jl',I,I)+einsum('il,jk',I,I)),1)
self.gradient_elasticity_tensor = 2.*eta*I
self.coupling_tensor0 = np.zeros((self.elasticity_tensor.shape[0],self.gradient_elasticity_tensor.shape[0]))
# Piezoelectric tensor must be 6x3 [3D] or 3x2 [2D]
self.piezoelectric_tensor = self.P
# Dielectric tensor
self.dielectric_tensor = -self.eps*I
# # BUILD HESSIAN
factor = -1.
H1 = np.concatenate((self.elasticity_tensor,factor*self.piezoelectric_tensor),axis=1)
H2 = np.concatenate((factor*self.piezoelectric_tensor.T,self.dielectric_tensor),axis=1)
H_Voigt = np.concatenate((H1,H2),axis=0)
return H_Voigt
def Hessian(self, StrainTensors, ElectricFieldx=None, elem=0, gcounter=0):
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
F = StrainTensors['F'][gcounter]
b = StrainTensors['b'][gcounter]
if np.isclose(J, 0) or J < 0:
delta = np.sqrt(0.04 * J * J + 1e-8)
# J = 0.5 * (J + np.sqrt(J**2 + 4 *delta**2))
mu = self.mu
lamb = self.lamb
alpha = self.alpha
C_Voigt = (lamb * (2 * J - alpha) - mu) * np.einsum(
"ij,kl", I, I) + (mu - lamb *
(J - alpha)) * (np.einsum("ik,jl", I, I) +
np.einsum("il,jk", I, I))
C_Voigt = Voigt(C_Voigt, 1)
self.H_VoigtSize = C_Voigt.shape[0]
return C_Voigt
def Hessian(self,
StrainTensors,
ElectricDisplacementx,
elem=0,
gcounter=0):
mu = self.mu
mu1 = self.mu1
mu2 = self.mu2
lamb = self.lamb
d = self.ndim
I = StrainTensors['I']
F = StrainTensors['F'][gcounter]
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
trb = trace(b) + 0
C_Voigt0 = 4./d * J**(-1) * einsum("ij,kl", (-1./d * trb * I + b), (-1./d) * J**(-2./d) * I ) +\
4./d**2 * J**(-1) * J**(-2./d) * (-einsum("ij,kl", I, b) + 0.5 * trb * (einsum("ik,jl",I,I)+einsum("il,jk",I,I)) )
C_Voigt0 *= (1. / d * J**(-2 / d) * trb - 1)
sigma = 2. / d * J**(-2. / d - 1.) * (-1. / d * trb * I + b)
C_Voigt1 = einsum("ij,kl", sigma, sigma)
C_Voigt = C_Voigt0 + C_Voigt1
# C_Voigt = C_Voigt0
C_Voigt = mu * C_Voigt
C_Voigt += 2.*mu2/J*(2*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b)) +\
2.*(mu1+2.*mu2)/J*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) ) + \
lamb*(2.*J-1.)*einsum("ij,kl",I,I) - lamb*(J-1)*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) )
C_Voigt = Voigt(C_Voigt, 1)
return C_Voigt
eta = mu/3.
ut = 2.*gamma*(alpha+2.0*beta) + 2.*(1. - gamma)*eta
lamb = lamb + 2.*gamma*alpha - 2*(1.- gamma)*eta
H_Voigt = 2.*gamma*beta/J* ( 2.0*einsum('ij,kl',b,b) - einsum('ik,jl',b,b) - einsum('il,jk',b,b) ) + \
4.*(1-gamma)*eta/J * einsum('i,j,k,l',FN,FN,FN,FN) + \
4.*(1-gamma)*eta/J * ( innerHN * einsum('ij,kl',I,I) - \
0.5*innerHN * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) ) - \
einsum('ij,k,l',I,HN,HN) - einsum('i,j,kl',HN,HN,I) ) + \
2.*(1-gamma)*eta/J * ( einsum('il,j,k',I,HN,HN) + einsum('jl,i,k',I,HN,HN) + \
einsum('ik,j,l',I,HN,HN) + einsum('jk,i,l',I,HN,HN) ) - \
ut*einsum('ij,kl',I,I) + ut * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) ) + \
lamb*(2.*J-1.)*einsum('ij,kl',I,I) - lamb*(J-1.) * ( einsum('ik,jl',I,I) + einsum('il,jk',I,I) )
H_Voigt = Voigt(H_Voigt ,1)
<<<<<<< HEAD
=======
>>>>>>> upstream/master
self.H_VoigtSize = H_Voigt.shape[0]
return H_Voigt
def CauchyStress(self,StrainTensors,ElectricFieldx,elem=0,gcounter=0):
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
J = StrainTensors['J'][gcounter]
# Jb = 1.27168554933
Jb = self.Jbar[elem]
mu2 = self.mu/J- self.lamb*(J-Jb)
<<<<<<< HEAD
lamb2 = self.lamb*(2*J-Jb)
=======
lamb2 = self.lamb*(2*J-Jb)
>>>>>>> upstream/master
H_Voigt_n = lamb2*self.vIijIkl+mu2*self.vIikIjl
# return H_Voigt
H_Voigt_l = Voigt(self.lamb*einsum('ij,kl',I,I)+self.mu*(einsum('ik,jl',I,I)+einsum('il,jk',I,I)) ,1)
<<<<<<< HEAD
=======
>>>>>>> upstream/master
alp = 0.1
# H_Voigt = (1-alp)*H_Voigt_l + alp*H_Voigt_n
H_Voigt = H_Voigt_n
# return H_Voigt_l + H_Voigt_n
return H_Voigt
def CauchyStress(self,StrainTensors,ElectricFieldx=None,elem=0,gcounter=0):
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
def __checkdata__(self, material, boundary_condition,
formulation, mesh, function_spaces, solver, contact_formulation=None):
## IMPORTANT
self.requires_geometry_update = False
self.analysis_nature = "linear"
function_spaces, solver = super(CoupleStressSolver,self).__checkdata__(material, boundary_condition,
formulation, mesh, function_spaces, solver, contact_formulation=contact_formulation)
# BUILD THE TANGENT OPERATORS BEFORE HAND
if material.mtype == "CoupleStressModel":
I = np.eye(material.ndim,material.ndim)
material.elasticity_tensor = Voigt(material.lamb*np.einsum('ij,kl',I,I)+material.mu*(np.einsum('ik,jl',I,I)+np.einsum('il,jk',I,I)),1)
material.gradient_elasticity_tensor = 2.*material.eta*I
material.coupling_tensor0 = np.zeros((material.elasticity_tensor.shape[0],
material.gradient_elasticity_tensor.shape[0]))
# print material.elasticity_tensor
ngauss = function_spaces[0].AllGauss.shape[0]
d0 = material.elasticity_tensor.shape[0]
d1 = material.elasticity_tensor.shape[1]
d2 = material.gradient_elasticity_tensor.shape[0]
d3 = material.gradient_elasticity_tensor.shape[1]
material.elasticity_tensors = np.tile(material.elasticity_tensor.ravel(),ngauss).reshape(ngauss,d0,d1)
material.gradient_elasticity_tensors = np.tile(material.gradient_elasticity_tensor.ravel(),ngauss).reshape(ngauss,d2,d3)
elif material.mtype == "IsotropicLinearFlexoelectricModel":
I = np.eye(material.ndim,material.ndim)
material.elasticity_tensor = Voigt(material.lamb*np.einsum('ij,kl',I,I)+material.mu*(np.einsum('ik,jl',I,I)+np.einsum('il,jk',I,I)),1)
material.gradient_elasticity_tensor = 2.*material.eta*I
material.coupling_tensor0 = np.zeros((material.elasticity_tensor.shape[0],
material.gradient_elasticity_tensor.shape[0]))
material.piezoelectric_tensor = material.P
material.flexoelectric_tensor = material.f
material.dielectric_tensor = -material.eps*np.eye(material.ndim,material.ndim)
# print material.elasticity_tensor
ngauss = function_spaces[0].AllGauss.shape[0]
d0 = material.elasticity_tensor.shape[0]
d1 = material.elasticity_tensor.shape[1]
d2 = material.gradient_elasticity_tensor.shape[0]
d3 = material.gradient_elasticity_tensor.shape[1]
material.elasticity_tensors = np.tile(material.elasticity_tensor.ravel(),ngauss).reshape(ngauss,d0,d1)
material.gradient_elasticity_tensors = np.tile(material.gradient_elasticity_tensor.ravel(),ngauss).reshape(ngauss,d2,d3)
material.piezoelectric_tensors = np.tile(material.P.ravel(),ngauss).reshape(ngauss,material.P.shape[0],material.P.shape[1])
material.flexoelectric_tensors = np.tile(material.f.ravel(),ngauss).reshape(ngauss,material.f.shape[0],material.f.shape[1])
material.dielectric_tensors = np.tile(material.dielectric_tensor.ravel(),ngauss).reshape(ngauss,material.ndim,material.ndim)
factor = -1.
material.H_Voigt = np.zeros((ngauss,material.H_VoigtSize,material.H_VoigtSize))
for i in range(ngauss):
H1 = np.concatenate((material.elasticity_tensor,factor*material.piezoelectric_tensor),axis=1)
H2 = np.concatenate((factor*material.piezoelectric_tensor.T,material.dielectric_tensor),axis=1)
H_Voigt = np.concatenate((H1,H2),axis=0)
material.H_Voigt[i,:,:] = H_Voigt
return function_spaces, solver
def Hessian(self, StrainTensors, ElectricFieldx=0, elem=0, gcounter=0):
mu = self.mu
lamb = self.lamb
varepsilon_1 = self.eps_1
detF = StrainTensors['J'][gcounter]
mu2 = mu - lamb * (detF - 1.0)
lamb2 = lamb * (2.0 * detF - 1.0) - mu
delta = StrainTensors['I']
E = 1.0 * ElectricFieldx
Ex = E.reshape(E.shape[0])
EE = np.outer(E, E)
innerEE = np.dot(E, E.T)
I = delta
# C = lamb2*AijBkl(I,I) +mu2*(AikBjl(I,I)+AilBjk(I,I)) + varepsilon_1*(AijBkl(I,EE) + AijBkl(EE,I) - \
# 2.*AikBjl(EE,I)-2.0*AilBjk(I,EE) ) + varepsilon_1*(np.dot(E.T,E)[0,0])*(AikBjl(I,I)-0.5*AijBkl(I,I))
# ORIGINAL
# C = lamb2*AijBkl(I,I) +mu2*(AikBjl(I,I)+AilBjk(I,I)) +\
# varepsilon_1*(AijBkl(I,EE) + AijBkl(EE,I) -AikBjl(EE,I)-AilBjk(EE,I)-AilBjk(I,EE)-AikBjl(I,EE) ) +\
# varepsilon_1*(np.dot(E.T,E)[0,0])*(0.5*(AikBjl(I,I) + AilBjk(I,I))-0.5*AijBkl(I,I))
# C=0.5*(C+C.T)
# C_Voigt = C
C = lamb2*einsum("ij,kl",I,I) +mu2*(einsum("ik,jl",I,I)+einsum("il,jk",I,I)) +\
varepsilon_1*(einsum("ij,kl",I,EE) + einsum("ij,kl",EE,I) - einsum("ik,jl",EE,I)- einsum("il,jk",I,EE) -\
einsum("il,jl",I,EE)- einsum("ik,jl",I,EE) ) +\
varepsilon_1*(innerEE)*(0.5*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) )-0.5* einsum("ij,kl",I,I) )
C_Voigt = Voigt(C, 1)
# Computing the hessian
# Elasticity tensor (C - 4th order tensor)
# C[i,j,k,l] += lamb2*delta[i,j]*delta[k,l]+2.0*mu2*(delta[i,k]*delta[j,l]) #
b = StrainTensors['b'][gcounter]
be = np.dot(b, ElectricFieldx).reshape(self.ndim, 1)
# Coupled Tensor (e - 3rd order)
# e[k,i,j] += (-2.0*varepsilon_1/detF)*(be[j]*b[i,k] + be[i]*b[j,k]) #
# e[i,j,k] += 1.0*varepsilon_1*( E[i]*delta[j,k] + E[j]*delta[i,k] - delta[i,j]*E[k]) ##
# e[k,i,j] += 1.0*varepsilon_1*(E[i]*delta[j,k] + E[j]*delta[i,k] - delta[i,j]*E[k]) ##
# Note that the actual piezoelectric tensor is symmetric wrt to the last two indices
# Actual tensor is: e[k,i,j] += 1.0*varepsilon_1*(E[i]*delta[j,k] + E[j]*delta[i,k] - delta[i,j]*E[k])
# We need to make its Voigt_form symmetric with respect to (j,k) instead of (i,j)
# ORIGINAL
# e_voigt = 1.0*varepsilon_1*(AijUk(I,Ex)+AikUj(I,Ex)-UiAjk(Ex,I)).T
e_voigt = 1.0 * varepsilon_1 * (einsum('ij,k', I, Ex) + einsum(
'ik,j', I, Ex) - einsum('i,jk', Ex, I)).T
e_voigt = Voigt(np.ascontiguousarray(e_voigt), 1)
# Dielectric Tensor (Permittivity - 2nd order)
Permittivity = -varepsilon_1 * delta ##
# bb = np.dot(StrainTensors.b,StrainTensors.b) #
# Permittivity = -(2.0*varepsilon_1/detF)*bb #
factor = -1.
H1 = np.concatenate((C_Voigt, factor * e_voigt), axis=1)
H2 = np.concatenate((factor * e_voigt.T, Permittivity), axis=1)
H_Voigt = np.concatenate((H1, H2), axis=0)
self.H_VoigtSize = H_Voigt.shape[0]
# return H_Voigt, C, e, Permittivity
return H_Voigt
# C = lamb2*AijBkl(I,I) +mu2*(AikBjl(I,I)+AilBjk(I,I)) + varepsilon_1*(AijBkl(I,EE) + AijBkl(EE,I) - \
# 2.*AikBjl(EE,I)-2.0*AilBjk(I,EE) ) + varepsilon_1*(np.dot(E.T,E)[0,0])*(AikBjl(I,I)-0.5*AijBkl(I,I))
# ORIGINAL
# C = lamb2*AijBkl(I,I) +mu2*(AikBjl(I,I)+AilBjk(I,I)) +\
# varepsilon_1*(AijBkl(I,EE) + AijBkl(EE,I) -AikBjl(EE,I)-AilBjk(EE,I)-AilBjk(I,EE)-AikBjl(I,EE) ) +\
# varepsilon_1*(np.dot(E.T,E)[0,0])*(0.5*(AikBjl(I,I) + AilBjk(I,I))-0.5*AijBkl(I,I))
# C=0.5*(C+C.T)
# C_Voigt = C
C = lamb2*einsum("ij,kl",I,I) +mu2*(einsum("ik,jl",I,I)+einsum("il,jk",I,I)) +\
varepsilon_1*(einsum("ij,kl",I,EE) + einsum("ij,kl",EE,I) - einsum("ik,jl",EE,I)- einsum("il,jk",I,EE) -\
einsum("il,jl",I,EE)- einsum("ik,jl",I,EE) ) +\
<<<<<<< HEAD
varepsilon_1*(np.dot(E.T,E)[0,0])*(0.5*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) )-0.5* einsum("ij,kl",I,I) )
C_Voigt = Voigt(C,1)
# Computing the hessian
# Elasticity tensor (C - 4th order tensor)
=======
varepsilon_1*(innerEE)*(0.5*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) )-0.5* einsum("ij,kl",I,I) )
C_Voigt = Voigt(C,1)
# Computing the hessian
# Elasticity tensor (C - 4th order tensor)
>>>>>>> upstream/master
# C[i,j,k,l] += lamb2*delta[i,j]*delta[k,l]+2.0*mu2*(delta[i,k]*delta[j,l]) #
b = StrainTensors['b'][gcounter]
def Hessian(self,
StrainTensors,
ElectricDisplacementx,
elem=0,
gcounter=0):
mu = self.mu
lamb = self.lamb
d = self.ndim
I = StrainTensors['I']
F = StrainTensors['F'][gcounter]
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
det = np.linalg.det
u, s, vh = svd(F, full_matrices=True)
vh = vh.T
# print(det(u),det(vh))
# exit()
if self.ndim == 2:
s1 = s[0]
s2 = s[1]
T = np.array([[0., -1], [1, 0.]])
s1s2 = s1 + s2
if (s1s2 < 2.0):
s1s2 = 2.0
lamb = 2. / (s1s2)
T = 1. / np.sqrt(2) * np.dot(u, np.dot(T, vh.T))
# C_Voigt = 1.0 * ( einsum("ik,jl",I,I)+einsum("il,jk",I,I) ) - 2. * lamb * np.einsum("ij,kl", T, T)
# C_Voigt = 1./ J * np.einsum("kI,lJ,iIjJ->iklj", F, F, C_Voigt)
C_Voigt = 2.0 * (einsum("ik,jl", I, I)) - 2. * lamb * np.einsum(
"ij,kl", T, T)
# C_Voigt = 2.0 * ( einsum("ij,kl",I,I)) - 2. * lamb * np.einsum("ij,kl", T, T)
C_Voigt = 1. / J * np.einsum("jJ,iJkL,lL->ijkl", F, C_Voigt, F)
# Exclude the stress term from this
R = u.dot(vh.T)
sigma = 2. * (F - R)
sigma = 1. / J * np.dot(sigma, F.T)
C_Voigt -= np.einsum("ij,kl", sigma, I)
# print(C_Voigt)
# print(np.linalg.norm(T.flatten()))
# print(Voigt(np.einsum("ij,kl", T, T),1))
# print(np.einsum("ij,kl", T, T))
# exit()
C_Voigt = np.ascontiguousarray(C_Voigt)
elif self.ndim == 3:
s1 = s[0]
s2 = s[1]
s3 = s[2]
T1 = np.array([[0., 0., 0.], [0., 0., -1], [0., 1., 0.]])
T1 = 1. / np.sqrt(2) * np.dot(u, np.dot(T1, vh.T))
T2 = np.array([[0., 0., -1], [0., 0., 0.], [1., 0., 0.]])
T2 = 1. / np.sqrt(2) * np.dot(u, np.dot(T2, vh.T))
T3 = np.array([[0., -1, 0.], [1., 0., 0.], [0., 0., 0.]])
T3 = 1. / np.sqrt(2) * np.dot(u, np.dot(T3, vh.T))
s1s2 = s1 + s2
s1s3 = s1 + s3
s2s3 = s2 + s3
if (s1s2 < 2.0):
s1s2 = 2.0
if (s1s3 < 2.0):
s1s3 = 2.0
if (s2s3 < 2.0):
s2s3 = 2.0
lamb1 = 2. / (s1s2)
lamb2 = 2. / (s1s3)
lamb3 = 2. / (s2s3)
# C_Voigt = 1.0 * ( einsum("ik,jl",I,I)+einsum("il,jk",I,I) ) - 2. * lamb3 * np.einsum("ij,kl", T1, T1) - \
# - 2. * lamb2 * np.einsum("ij,kl", T2, T2) - 2. * lamb1 * np.einsum("ij,kl", T3, T3)
# C_Voigt = 1./ J * np.einsum("kI,lJ,iIjJ->iklj", F, F, C_Voigt)
# C_Voigt = np.ascontiguousarray(C_Voigt)
C_Voigt = 2.0 * ( einsum("ik,jl",I,I)) - 2. * lamb3 * np.einsum("ij,kl", T1, T1) - \
- 2. * lamb2 * np.einsum("ij,kl", T2, T2) - 2. * lamb1 * np.einsum("ij,kl", T3, T3)
C_Voigt = 1. / J * np.einsum("jJ,iJkL,lL->ijkl", F, C_Voigt, F)
C_Voigt = np.ascontiguousarray(C_Voigt)
R = u.dot(vh.T)
sigma = 2. * (F - R)
sigma = 1. / J * np.dot(sigma, F.T)
C_Voigt -= np.einsum("ij,kl", sigma, I)
# s1 = s[0]
# s2 = s[1]
# s3 = s[2]
# T1 = np.array([[0.,-1.,0.],[1.,0.,0],[0.,0.,0.]])
# T1 = 1./np.sqrt(2) * np.dot(u, np.dot(T1, vh.T))
# T2 = np.array([[0.,0.,0.],[0.,0., 1],[0.,-1,0.]])
# T2 = 1./np.sqrt(2) * np.dot(u, np.dot(T2, vh.T))
# T3 = np.array([[0.,0.,1.],[0.,0.,0.],[-1,0.,0.]])
# T3 = 1./np.sqrt(2) * np.dot(u, np.dot(T3, vh.T))
# s1s2 = s1 + s2
# s1s3 = s1 + s3
# s2s3 = s2 + s3
# # if (s1s2 < 2.0):
# # s1s2 = 2.0
# # if (s1s3 < 2.0):
# # s1s3 = 2.0
# # if (s2s3 < 2.0):
# # s2s3 = 2.0
# lamb1 = 2. / (s1s2)
# lamb2 = 2. / (s1s3)
# lamb3 = 2. / (s2s3)
# # C_Voigt = 1.0 * ( einsum("ik,jl",I,I)+einsum("il,jk",I,I) ) - 2. * lamb1 * np.einsum("ij,kl", T1, T1) - \
# # - 2. * lamb3 * np.einsum("ij,kl", T2, T2) - 2. * lamb2 * np.einsum("ij,kl", T3, T3)
# # C_Voigt = 1./ J * np.einsum("kI,lJ,iIjJ->iklj", F, F, C_Voigt)
# # C_Voigt = np.ascontiguousarray(C_Voigt)
# C_Voigt = 2.0 * ( einsum("ik,jl",I,I)) - 2. * lamb1 * np.einsum("ij,kl", T1, T1) - \
# - 2. * lamb3 * np.einsum("ij,kl", T2, T2) - 2. * lamb2 * np.einsum("ij,kl", T3, T3)
# C_Voigt = 1./ J * np.einsum("kI,lJ,iIjJ->iklj", F, F, C_Voigt)
# C_Voigt = np.ascontiguousarray(C_Voigt)
# R = u.dot(vh.T)
# sigma = 2. * (F - R)
# sigma = 1./J * np.dot(sigma, F.T)
# C_Voigt -= np.einsum("ij,kl",sigma,I)
C_Voigt = Voigt(C_Voigt, 1)
makezero(C_Voigt)
# C_Voigt = np.eye(3,3)*2
# C_Voigt += 0.05*self.vIikIjl
# print(C_Voigt)
# s = svd(C_Voigt)[1]
# print(s)
return C_Voigt
