def Hessian(self, StrainTensors, elem=0, gcounter=0):
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
# ISOCHORIC PART
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)) )
# VOLUMETRIC PART
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 += self.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, 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 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 += self.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, elem=0, gcounter=0):
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
# ISOCHORIC PART
stress = mu * J**(-5. / 3.) * (b - 1. / 3. * trb * I)
# VOLUMETRIC PART
if self.is_nearly_incompressible:
stress += self.pressure * I
else:
stress += self.kappa * (J - 1.) * 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)
stress += 2. * k1 / J * (innerFN - 1.) * expo * outerFN
return stress
def CauchyStress(self,StrainTensors,elem=0,gcounter=0):
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
mu = self.mu
# ISOCHORIC PART
stress = mu*J**(-5./3.)*(b - 1./3.*trace(b)*I)
# VOLUMETRIC PART
if self.is_nearly_incompressible:
stress += self.pressure*I
else:
stress += self.kappa*(J-1.)*I
return stress
def Hessian(self, StrainTensors, elem=0, gcounter=0):
#print(np.finfo(np.float64).eps)
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
F = StrainTensors['F'][gcounter]
# Directional vector for element
Normal = self.anisotropic_orientations[elem][0][:, None]
Normal = np.dot(I, Normal)[:, 0]
Tangential = self.anisotropic_orientations[elem][1][:, None]
Tangential = np.dot(I, Tangential)[:, 0]
Axial = self.anisotropic_orientations[elem][2][:, None]
Axial = np.dot(I, Axial)[:, 0]
Rotation = np.eye(3, 3, dtype=np.float64)
for i in range(3):
Rotation[0, i] = Normal[i]
Rotation[1, i] = Tangential[i]
Rotation[2, i] = Axial[i]
# Growth tensor (same for all constituent)
outerNormal = einsum('i,j', Normal, Normal)
outerTangential = I - outerNormal
F_g = self.StateVariables[gcounter][20] * outerNormal + outerTangential
# Remodeling tensor for elastin
F_r = I + (self.StateVariables[gcounter][0:9].reshape(3, 3) -
I) * self.factor_increment
F_r = np.dot(Rotation.T, np.dot(F_r, Rotation))
# Elastin Growth and Remodeling tensor
F_gr = np.dot(F_r, F_g)
F_gr_inv = np.linalg.inv(F_gr)
# ELASTIN
F_e = np.dot(F, F_gr_inv)
J_e = np.linalg.det(F_e)
b_e = np.dot(F_e, F_e.T)
if self.ndim == 3:
trb = trace(b_e)
elif self.ndim == 2:
trb = trace(b_e) + 1.
H_Voigt = 2.*self.mu*J_e**(-2./3.)*(1./9.*trb*einsum('ij,kl',I,I) - \
1./3.*(einsum('ij,kl',I,b_e) + einsum('ij,kl',b_e,I)) + \
1./6.*trb*(einsum('ik,jl',I,I) + einsum('il,jk',I,I)) )*self.StateVariables[gcounter][14]/J
if self.is_nearly_incompressible:
H_Voigt += self.pressure*J_e*(einsum('ij,kl',I,I) - \
(einsum('ik,jl',I,I) + einsum('il,jk',I,I)))*self.StateVariables[gcounter][14]/J
else:
H_Voigt += self.kappa*J_e*((2.*J_e-1.)*einsum('ij,kl',I,I) - \
(J_e-1.)*(einsum('ik,jl',I,I) + einsum('il,jk',I,I)))*self.StateVariables[gcounter][14]/J
#SMC AND COLLAGEN FIBRES
for fibre_i in [1, 2, 3, 4, 5]:
# Fibre direction
N = self.anisotropic_orientations[elem][fibre_i][:, None]
N = np.dot(I, N)[:, 0]
FN = np.dot(F, N)
# Remodeling stretch in fibre direction
lambda_r = 1. + (self.StateVariables[gcounter][fibre_i + 8] -
1.) * self.factor_increment
# TOTAL deformation
innerFN = einsum('i,i', FN, FN)
outerFN = einsum('i,j', FN, FN)
# ELASTIC deformation
innerFN_e = innerFN / lambda_r**2
outerFN_e = outerFN / lambda_r**2
if fibre_i is 1:
k1 = self.k1m * self.StateVariables[gcounter][15] / J
k2 = self.k2m
# Anisotropic Stiffness for this key
H_Voigt += 4. * k1 * (1. + 2. * k2 *
(innerFN_e - 1.)**2) * np.exp(
k2 * (innerFN_e - 1.)**2) * einsum(
'ij,kl', outerFN_e, outerFN_e)
# Active stress for SMC
s_act = self.maxi_active_stress * self.StateVariables[
gcounter][15] / J
stretch_m = self.maxi_active_stretch
stretch_0 = self.zero_active_stretch
stretch_a = self.active_stretch
H_Voigt += -2.*(s_act/(self.rho0*innerFN**2))*\
(1.-((stretch_m-stretch_a)/(stretch_m-stretch_0))**2)*einsum('ij,kl',outerFN,outerFN)
elif fibre_i is not 1:
k1 = self.k1c if (innerFN_e - 1.0) >= 0.0 else 0.075 * self.k1c
k1 = k1 * self.StateVariables[gcounter][14 + fibre_i] / J
k2 = self.k2c
# Anisotropic Stiffness for this key
H_Voigt += 4. * k1 * (1. + 2. * k2 *
(innerFN_e - 1.)**2) * np.exp(
k2 * (innerFN_e - 1.)**2) * einsum(
'ij,kl', outerFN_e, outerFN_e)
H_Voigt = Voigt(H_Voigt, 1)
return H_Voigt