class IsotropicElectroMechanics_109(Material):
""" Electromechanical model in terms of internal energy
W(C,D) = W_mn(C) + 1/2/eps_2/J (FD0*FD0)
W_mn(C) = u1*C:I+u2*G:I - 2*(u1+2*u2)*lnJ + lamb/2*(J-1)**2
"""
def __init__(self, ndim, **kwargs):
mtype = type(self).__name__
super(IsotropicElectroMechanics_109, self).__init__(mtype, ndim, **kwargs)
# REQUIRES SEPARATELY
self.nvar = self.ndim+1
self.energy_type = "internal_energy"
self.legendre_transform = LegendreTransform()
self.nature = "nonlinear"
self.fields = "electro_mechanics"
if self.ndim == 2:
self.H_VoigtSize = 5
elif self.ndim == 3:
self.H_VoigtSize = 9
# LOW LEVEL DISPATCHER
self.has_low_level_dispatcher = True
# self.has_low_level_dispatcher = False
def KineticMeasures(self,F,ElectricFieldx, elem=0):
from Florence.MaterialLibrary.LLDispatch._IsotropicElectroMechanics_109_ import KineticMeasures
return KineticMeasures(self,np.ascontiguousarray(F), ElectricFieldx)
def Hessian(self,StrainTensors,ElectricDisplacementx,elem=0,gcounter=0):
mu_v = self.mu_v
mu1 = self.mu1
mu2 = self.mu2
lamb = self.lamb
eps_2 = self.eps_2
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
D = ElectricDisplacementx.reshape(self.ndim,1)
Dx = D.reshape(self.ndim)
DD = np.dot(D.T,D)[0,0]
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 = 1./eps_2*(0.5*DD*(einsum('ik,jl',I,I) + einsum('il,jk',I,I) + einsum('ij,kl',I,I) ) - \
einsum('ij,k,l',I,Dx,Dx) - einsum('i,j,kl',Dx,Dx,I))
# C_linear = self.H_Voigt
C_linear = H_Voigt = lamb*einsum('ij,kl',I,I)+mu_v*(einsum('ik,jl',I,I)+einsum('il,jk',I,I))
self.elasticity_tensor = C_mech + C_elect + C_linear
self.coupling_tensor = 1./eps_2*(einsum('ik,j',I,Dx) + einsum('i,jk',Dx,I) - einsum('ij,k',I,Dx))
self.dielectric_tensor = 1./eps_2*I
# TRANSFORM TENSORS TO THEIR ENTHALPY COUNTERPART
E_Voigt, P_Voigt, C_Voigt = self.legendre_transform.InternalEnergyToEnthalpy(self.dielectric_tensor,
self.coupling_tensor, self.elasticity_tensor)
# BUILD 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 CauchyStress(self,StrainTensors,ElectricDisplacementx,elem=0,gcounter=0):
mu_v = self.mu_v
mu1 = self.mu1
mu2 = self.mu2
lamb = self.lamb
eps_2 = self.eps_2
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
strain = StrainTensors['strain'][gcounter]
D = ElectricDisplacementx.reshape(self.ndim,1)
DD = np.dot(D.T,D)[0,0]
trb = trace(b)
if self.ndim==2:
trb += 1.
simga_mech = 2.0*mu1/J*b + \
2.0*mu2/J*(trb*b - np.dot(b,b)) -\
2.0*(mu1+2*mu2)/J*I +\
lamb*(J-1.)*I
sigma_electric = 1./eps_2*(np.dot(D,D.T) - 0.5*DD*I)
# CHECK IF THIS IS NECESSARY
if self.ndim == 3:
tre = trace(strain)
elif self.ndim == 2:
tre = trace(strain) + 1
# USE FASTER TRACE FUNCTION
sigma_linear = 2.*mu_v*strain + lamb*tre*I
sigma = simga_mech + sigma_electric + sigma_linear
return sigma
def ElectricDisplacementx(self,StrainTensors,ElectricFieldx,elem=0,gcounter=0):
D = self.legendre_transform.GetElectricDisplacement(self, StrainTensors, ElectricFieldx, elem, gcounter)
# SANITY CHECK FOR IMPLICIT COMPUTATUTAION OF D
# I = StrainTensors['I']
# J = StrainTensors['J'][gcounter]
# b = StrainTensors['b'][gcounter]
# E = ElectricFieldx.reshape(self.ndim,1)
# eps_1 = self.eps_1
# eps_2 = self.eps_2
# inverse = np.linalg.inv(J/eps_1*np.linalg.inv(b) + 1./eps_2*I)
# D_exact = np.dot(inverse,E)
# print np.linalg.norm(D - D_exact)
# return D_exact
return D
class Piezoelectric_100(Material):
"""
Piezoelectric model in terms of internal energy
W(C,D) = W_mn(C) + 1/2/eps_1 (D0*D0) + 1/2/eps_2/J (FD0*FD0)
+ u3*(FD0/sqrt(u3*eps_3)+FN)*(FD0/sqrt(u3*eps_3)+FN) + u3*HN*HN - 2*sqrt(u3/eps_3)*D0*N
W_mn(C) = u1*C:I+u2*G:I - 2*(u1+2*u2+u3)*lnJ + lamb/2*(J-1)**2
"""
def __init__(self, ndim, **kwargs):
mtype = type(self).__name__
super(Piezoelectric_100, self).__init__(mtype, ndim, **kwargs)
# REQUIRES SEPARATELY
self.nvar = self.ndim+1
self.energy_type = "internal_energy"
self.legendre_transform = LegendreTransform()
self.nature = "nonlinear"
self.fields = "electro_mechanics"
self.is_transversely_isotropic = True
if self.ndim==3:
self.H_VoigtSize = 9
else:
self.H_VoigtSize = 5
# LOW LEVEL DISPATCHER
self.has_low_level_dispatcher = True
# self.has_low_level_dispatcher = False
def KineticMeasures(self,F,ElectricFieldx, elem=0):
from Florence.MaterialLibrary.LLDispatch._Piezoelectric_100_ import KineticMeasures
return KineticMeasures(self,np.ascontiguousarray(F), ElectricFieldx, self.anisotropic_orientations[elem][:,None])
# return KineticMeasures(self,np.ascontiguousarray(F), np.ascontiguousarray(ElectricFieldx),
# np.ascontiguousarray(self.anisotropic_orientations[elem][:,None]))
def Hessian(self,StrainTensors,ElectricDisplacementx,elem=0,gcounter=0):
mu1 = self.mu1
mu2 = self.mu2
mu3 = self.mu3
lamb = self.lamb
eps_1 = self.eps_1
eps_2 = self.eps_2
eps_3 = self.eps_3
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]
D = ElectricDisplacementx.reshape(self.ndim,1)
FN = np.dot(F,N)[:,0]
HN = np.dot(H,N)[:,0]
innerHN = einsum('i,i',HN,HN)
outerHN = einsum('i,j',HN,HN)
Dx = D.reshape(self.ndim)
DD = np.dot(D.T,D)[0,0]
# Iso + Aniso
C_mech = 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('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) )
C_elect = 1./eps_2*(0.5*DD*(einsum('ik,jl',I,I) + einsum('il,jk',I,I) + einsum('ij,kl',I,I) ) - \
einsum('ij,k,l',I,Dx,Dx) - einsum('i,j,kl',Dx,Dx,I))
self.elasticity_tensor = C_mech + C_elect
self.coupling_tensor = 1./eps_2*(einsum('ik,j',I,Dx) + einsum('i,jk',Dx,I) - einsum('ij,k',I,Dx)) + \
2.*J*sqrt(mu3/eps_3)*(einsum('ik,j',I,Dx) + einsum('i,jk',Dx,I)) + \
2.*sqrt(mu3/eps_3)*(einsum('ik,j',I,FN) + einsum('i,jk',FN,I))
self.dielectric_tensor = J/eps_1*np.linalg.inv(b) + 1./eps_2*I + 2.*J*sqrt(mu3/eps_3)*I
# TRANSFORM TENSORS TO THEIR ENTHALPY COUNTERPART
E_Voigt, P_Voigt, C_Voigt = self.legendre_transform.InternalEnergyToEnthalpy(self.dielectric_tensor,
self.coupling_tensor, self.elasticity_tensor)
# BUILD 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 CauchyStress(self,StrainTensors,ElectricDisplacementx,elem=0,gcounter=0):
mu1 = self.mu1
mu2 = self.mu2
mu3 = self.mu3
lamb = self.lamb
eps_1 = self.eps_1
eps_2 = self.eps_2
eps_3 = self.eps_3
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]
outerFN = einsum('i,j',FN,FN)
innerHN = einsum('i,i',HN,HN)
outerHN = einsum('i,j',HN,HN)
D = ElectricDisplacementx.reshape(self.ndim,1)
Dx = D.reshape(self.ndim)
DD = np.dot(D.T,D)[0,0]
D0D = np.dot(D,D.T)
if self.ndim == 3:
trb = trace(b)
elif self.ndim == 2:
trb = trace(b) + 1.
sigma_mech = 2.*mu1/J*b + \
2.*mu2/J*(trb*b - np.dot(b,b)) - \
2.*(mu1+2*mu2+mu3)/J*I + \
lamb*(J-1)*I +\
2*mu3/J*outerFN +\
2*mu3/J*innerHN*I - 2*mu3/J*outerHN
sigma_electric = 1./eps_2*(D0D - 0.5*DD*I) +\
2.*J*sqrt(mu3/eps_3)*D0D + 2*sqrt(mu3/eps_3)*(einsum('i,j',Dx,FN) + einsum('i,j',FN,Dx))
sigma = sigma_mech + sigma_electric
return sigma
def ElectricDisplacementx(self,StrainTensors,ElectricFieldx,elem=0,gcounter=0):
# THE ELECTRIC FIELD NEEDS TO BE MODFIED TO TAKE CARE OF CONSTANT TERMS
mu3 = self.mu3
eps_1 = self.eps_1
eps_2 = self.eps_2
eps_3 = self.eps_3
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)
HN = np.dot(H,N)
E = ElectricFieldx.reshape(self.ndim,1)
modElectricFieldx = (E - 2.*sqrt(mu3/eps_3)*FN + 2./J*sqrt(mu3/eps_3)*HN)
# D = self.legendre_transform.GetElectricDisplacement(self, StrainTensors, modElectricFieldx, elem, gcounter)
# SANITY CHECK FOR IMPLICIT COMPUTATUTAION OF D
inverse = np.linalg.inv(J/eps_1*np.linalg.inv(b) + 1./eps_2*I + 2.*J*sqrt(mu3/eps_3)*I)
D_exact = np.dot(inverse, (E - 2.*sqrt(mu3/eps_3)*FN + 2./J*sqrt(mu3/eps_3)*HN) )
# print np.linalg.norm(D - D_exact)
return D_exact
return D
class IsotropicElectroMechanics_101(Material):
"""
Simplest electromechanical model in terms of internal energy
W(C,D0) = W_neo(C) + 1/2/eps_1 (FD0*FD0)
"""
def __init__(self, ndim, **kwargs):
mtype = type(self).__name__
super(IsotropicElectroMechanics_101, self).__init__(mtype, ndim, **kwargs)
# REQUIRES SEPARATELY
self.nvar = self.ndim+1
self.energy_type = "internal_energy"
self.legendre_transform = LegendreTransform()
self.nature = "nonlinear"
self.fields = "electro_mechanics"
if self.ndim == 2:
self.H_VoigtSize = 5
elif self.ndim == 3:
self.H_VoigtSize = 9
# LOW LEVEL DISPATCHER
self.has_low_level_dispatcher = True
# self.has_low_level_dispatcher = False
def KineticMeasures(self,F,ElectricFieldx, elem=0):
from Florence.MaterialLibrary.LLDispatch._IsotropicElectroMechanics_101_ import KineticMeasures
return KineticMeasures(self,np.ascontiguousarray(F), ElectricFieldx)
def Hessian(self,StrainTensors,ElectricDisplacementx,elem=0,gcounter=0):
mu = self.mu
lamb = self.lamb
eps_1 = self.eps_1
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
# D = ElectricDisplacementx.reshape(self.ndim,1)
Dx = ElectricDisplacementx.reshape(self.ndim)
self.elasticity_tensor = lamb*(2.*J-1.)*einsum("ij,kl",I,I) + \
(mu/J - lamb*(J-1))*( einsum("ik,jl",I,I)+einsum("il,jk",I,I) )
self.coupling_tensor = J/eps_1*(einsum('ik,j',I,Dx) + einsum('i,jk',Dx,I))
self.dielectric_tensor = J/eps_1*I
# TRANSFORM TENSORS TO THEIR ENTHALPY COUNTERPART
E_Voigt, P_Voigt, C_Voigt = self.legendre_transform.InternalEnergyToEnthalpy(self.dielectric_tensor,
self.coupling_tensor, self.elasticity_tensor)
# BUILD 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 CauchyStress(self,StrainTensors,ElectricDisplacementx,elem=0,gcounter=0):
mu = self.mu
lamb = self.lamb
eps_1 = self.eps_1
I = StrainTensors['I']
J = StrainTensors['J'][gcounter]
b = StrainTensors['b'][gcounter]
D = ElectricDisplacementx.reshape(self.ndim,1)
return 1.0*mu/J*(b - I) + lamb*(J-1)*I + J/eps_1*np.dot(D,D.T)
# return 1.0*mu*(b - I) + 1./eps_1*np.dot(D,D.T)
def ElectricDisplacementx(self,StrainTensors,ElectricFieldx,elem=0,gcounter=0):
# D = self.legendre_transform.GetElectricDisplacement(self, StrainTensors, ElectricFieldx, elem, gcounter)
eps_1 = self.eps_1
J = StrainTensors['J'][gcounter]
E = ElectricFieldx.reshape(self.ndim,1)
D_exact = eps_1/J*E
# print np.linalg.norm(D - D_exact)
return D_exact
return D
def Permittivity(self,StrainTensors,ElectricDisplacementx,elem=0,gcounter=0):
eps_1 = self.eps_1
I = StrainTensors['I']
J = StrainTensors['J']
self.dielectric_tensor = J/eps_1*I
# Negative definite inverse is needed due to Legendre transform
return -eps_1/J*I
def ElectrostaticMeasures(self,F,ElectricFieldx, elem=0):
from Florence.MaterialLibrary.LLDispatch._IsotropicElectroMechanics_101_ import KineticMeasures
D, _, H_Voigt = KineticMeasures(self,np.ascontiguousarray(F), ElectricFieldx)
H_Voigt = np.ascontiguousarray(H_Voigt[:,-self.ndim:,-self.ndim:])
return D, None, H_Voigt