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poroelectric.py
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# Poro-electro-elasticity Equations
# epsilon dE/dt + sigma E - curl H - L grad p = J
# mu dH/dt + curl E = 0
# -lambda grad div u - G div grad u + alpha p = 0
# d/dt(c p + alpha div u) + L div E - kappa laplacian p = 0
from fenics import *
from dolfin import *
from ufl import nabla_div
import numpy as np
import matplotlib.pyplot as plt
#from mpl_toolkits.mplot3d import Axes3D
# Definition of constants and parameters
epsilon0 = 1
sigma0 = 2*pi*pi
L0 = 0.5
mu0 = 1
lambda0 = 2
G0 = 1
alpha0 = 1
c0 = 1
kappa0 = 1/(3*pi*pi)
#kappa0 = 1
T = 1.0 # final time
num_steps = 10 # number of time steps
dt = T / num_steps # time step size
# Create mesh and define function space
# Load mesh
mesh = UnitCubeMesh(6, 6, 6)
#mesh = UnitCubeMesh(10, 10, 10)
# Build function space
D1 = FiniteElement("N1curl", mesh.ufl_cell(), 1)
B1 = FiniteElement("RT", mesh.ufl_cell(), 1)
V = VectorElement("Lagrange", mesh.ufl_cell(), 1)
Q = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
element = MixedElement([D1, B1, V, Q])
W = FunctionSpace(mesh, element)
# Exact solutions, boundary conditions and known functions
E_ex = Expression(('sin(pi*x[1])*sin(pi*x[2])*exp(-t)', 'sin(pi*x[2])*sin(pi*x[0])*exp(-t)', \
'sin(pi*x[0])*sin(pi*x[1])*exp(-t)'), degree = 2, t = 0)
#E_ex = Expression(('sin(pi*x[1])*sin(pi*x[2])*(1+t)', 'sin(pi*x[2])*sin(pi*x[0])*(1+t)', \
# 'sin(pi*x[0])*sin(pi*x[1])*(1+t)'), degree = 2, t = 0)
#E_ex = Expression(('0', '0', '0'), degree = 2, t = 0)
H_ex = Expression(('pi*sin(pi*x[0])*(cos(pi*x[1])-cos(pi*x[2]))*exp(-t)', \
'pi*sin(pi*x[1])*(cos(pi*x[2])-cos(pi*x[0]))*exp(-t)', \
'pi*sin(pi*x[2])*(cos(pi*x[0])-cos(pi*x[1]))*exp(-t)'), degree = 2, t = 0)
#H_ex = Expression(('-pi*sin(pi*x[0])*(cos(pi*x[1])-cos(pi*x[2]))*(t+t*t/2)', \
# '-pi*sin(pi*x[1])*(cos(pi*x[2])-cos(pi*x[0]))*(t+t*t/2)', \
# '-pi*sin(pi*x[2])*(cos(pi*x[0])-cos(pi*x[1]))*(t+t*t/2)'), degree = 2, t = 0)
#H_ex = Expression(('0', '0', '0'), degree = 2, t = 0)
J = Expression(('(-1-0.5*pi*cos(pi*x[0]))*sin(pi*x[1])*sin(pi*x[2])*exp(-t)', \
'(-1-0.5*pi*cos(pi*x[1]))*sin(pi*x[2])*sin(pi*x[0])*exp(-t)', \
'(-1-0.5*pi*cos(pi*x[2]))*sin(pi*x[0])*sin(pi*x[1])*exp(-t)'), degree = 2, t = 0)
#J = Expression(('0', '0', '0'), degree = 2, t = 0)
u_D = Expression(('sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])*exp(-t)', 'sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])*exp(-t)', '0'), degree = 2, t = 0)
#p_D = Expression('0', degree = 2, t = 0)
#u_D = Expression(('0', '0', '0'), degree = 2, t = 0)
p_D = Expression('sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])*exp(-t)', degree = 2, t = 0)
f = Expression(('(5*pi*pi*sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2]) - 2*pi*pi*cos(pi*x[0])*cos(pi*x[1])*sin(pi*x[2]) + pi*cos(pi*x[0])*sin(pi*x[1])*sin(pi*x[2]))*exp(-t)', \
'(5*pi*pi*sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2]) - 2*pi*pi*cos(pi*x[0])*cos(pi*x[1])*sin(pi*x[2]) + pi*cos(pi*x[1])*sin(pi*x[2])*sin(pi*x[0]))*exp(-t)', \
'(-2*pi*pi*cos(pi*x[0])*sin(pi*x[1])*cos(pi*x[2]) - 2*pi*pi*sin(pi*x[0])*cos(pi*x[1])*cos(pi*x[2]) + pi*cos(pi*x[2])*sin(pi*x[0])*sin(pi*x[1]))*exp(-t)'), degree = 2, t = 0)
g = Expression('(-pi*cos(pi*x[0])*sin(pi*x[1])*sin(pi*x[2]) - pi*sin(pi*x[0])*cos(pi*x[1])*sin(pi*x[2]))*exp(-t)', degree = 2, t = 0)
#f = Expression(('5*pi*pi*sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])*exp(-t) + pi*cos(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])*exp(-t)', \
# '-2*pi*pi*cos(pi*x[0])*cos(pi*x[1])*sin(pi*x[2])*exp(-t) + pi*cos(pi*x[1])*sin(pi*x[2])*sin(pi*x[0])*exp(-t)', \
# '-2*pi*pi*cos(pi*x[0])*sin(pi*x[1])*cos(pi*x[2])*exp(-t) + pi*cos(pi*x[2])*sin(pi*x[0])*sin(pi*x[1])*exp(-t)'), degree = 2, t = 0)
#g = Expression('-pi*cos(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])*exp(-t)', degree = 2, t = 0)
#f = Expression(('0', '0', '0'), degree = 2, t = 0)
#g = Expression('0', degree = 2, t = 0)
U_0 = Expression(('sin(pi*x[1])*sin(pi*x[2])', 'sin(pi*x[2])*sin(pi*x[0])', 'sin(pi*x[0])*sin(pi*x[1])',\
'pi*sin(pi*x[0])*(cos(pi*x[1])-cos(pi*x[2]))', 'pi*sin(pi*x[1])*(cos(pi*x[2])-cos(pi*x[0]))', 'pi*sin(pi*x[2])*(cos(pi*x[0])-cos(pi*x[1]))',\
#U_0 = Expression(('0', '0', '0',\
# '0', '0', '0',\
'sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])', 'sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])', '0', \
#'0', '0', '0', \
'sin(pi*x[0])*sin(pi*x[1])*sin(pi*x[2])'), degree = 2)
n = FacetNormal(mesh)
T_d = Constant((0, 0, 0))
tol = 1E-14
def boundary(x, on_boundary):
return on_boundary
def clamped_boundary(x, on_boundary):
return on_boundary and x[0] < tol
bcD = DirichletBC(W.sub(0), Constant((0.0, 0.0, 0.0)), DomainBoundary())
bcB = DirichletBC(W.sub(1), Constant((0.0, 0.0, 0.0)), DomainBoundary())
bcu = DirichletBC(W.sub(2), Constant((0.0, 0.0, 0.0)), DomainBoundary())
bcp = DirichletBC(W.sub(3), Constant(0.0), boundary)
bc = [bcD, bcB, bcu, bcp]
# Define strain and stress
def epsilon(u):
return 0.5*(nabla_grad(u) + nabla_grad(u).T)
#return sym(nabla_grad(u))
def sigma(u):
return lambda0*nabla_div(u)*Identity(3) + 2*mu0*epsilon(u)
# Define expressions used in variational forms
dt = Constant(dt)
epsilon0 = Constant(epsilon0)
sigma0 = Constant(sigma0)
L0 = Constant(L0)
mu0 = Constant(mu0)
lambda0 = Constant(lambda0)
G0 = Constant(G0)
alpha0 = Constant(alpha0)
c0 = Constant(c0)
kappa0 = Constant(kappa0)
# Define initial value
bfU_n = project(U_0, W)
E_n, H_n, bfu_n, p_n = split(bfU_n)
# Define variational problem
E, H, bfu, p = TrialFunctions(W)
D, B, bfv, q = TestFunctions(W)#
#+ inner(sigma(bfu), epsilon(bfv))*dx - alpha0*inner(p, nabla_div(bfv))*dx \
a = epsilon0*inner(E, D)*dx +dt*sigma0*inner(E, D)*dx - dt*inner(H, curl(D))*dx - dt*L0*inner(nabla_grad(p), D)*dx\
+ mu0*inner(H, B)*dx + dt*inner(curl(E), B)*dx\
+ lambda0*inner(nabla_div(bfu), nabla_div(bfv))*dx + G0*inner(nabla_grad(bfu), nabla_grad(bfv))*dx\
+ c0*inner(p, q)*dx + alpha0*inner(nabla_div(bfu), q)*dx\
- alpha0*inner(p, nabla_div(bfv))*dx\
- dt*L0*inner(E, nabla_grad(q))*dx + dt*kappa0*inner(nabla_grad(p), nabla_grad(q))*dx
L = dt*inner(J, D)*dx + epsilon0*inner(E_n, D)*dx + mu0*inner(H_n, B)*dx \
+ inner(f, bfv)*dx + inner((c0*p_n + dt*g), q)*dx + alpha0*inner(nabla_div(bfu_n), q)*dx #+ inner(T_d, bfv)*dx
# Time-stepping#
bfU = Function(W)
t = 0
for nn in range(num_steps):
# Update current time
t += 0.1
E_ex.t = t
H_ex.t = t
J.t = t
u_D.t = t
p_D.t = t
f.t = t
g.t = t
# Solve variational problem
solve(a == L, bfU, bc)
Eh, Hh, Uh, ph = bfU.split()
error_L21 = errornorm(E_ex, Eh, norm_type = 'l2')
error_L22 = errornorm(H_ex, Hh, norm_type = 'l2')
error_L23 = errornorm(u_D, Uh, norm_type = 'l2')
error_L24 = errornorm(p_D, ph, norm_type = 'l2')
ENorm = sqrt(assemble(inner(E_ex, E_ex)*dx(mesh)))
HNorm = sqrt(assemble(inner(H_ex, H_ex)*dx(mesh)))
UNorm = sqrt(assemble(inner(u_D, u_D)*dx(mesh)))
pNorm = sqrt(assemble(inner(p_D, p_D)*dx(mesh)))
rel_err_21 = error_L21/ENorm
rel_err_22 = error_L22/HNorm
rel_err_23 = error_L23/UNorm
rel_err_24 = error_L24/pNorm
print('At t = ', t)
print('||E - E_ex|| = ', error_L21)
print('||H - H_ex|| = ', error_L22)
print('relative error ||E - E_ex||/||E|| = ', rel_err_21)
print('relative error ||H - H_ex||/||H|| = ', rel_err_22)
print('relative error ||U - U_ex||/||U|| = ', rel_err_23)
print('relative error ||p - p_ex||/||p|| = ', rel_err_24)
# Update previous solution
bfU_n.assign(bfU)
#_un, _pn = bfU.split()
#bfu_n.assign(_un)
#p_n.assign(_pn)
plot(Eh)
plt.xlabel('x-axis')
plt.ylabel('y-axis')
#plt.zlabel('z-axis')
plt.title('plot of E')
plt.show()
plot(Hh)
plt.xlabel('x-axis')
plt.ylabel('y-axis')
#plt.zlabel('z-axis')
plt.title('plot of H')
plt.show()
plot(Uh)
plt.xlabel('x-axis')
plt.ylabel('y-axis')
#plt.zlabel('z-axis')
plt.title('plot of U')
plt.xlim([0, 1])
plt.ylim([0, 1])
plt.show()
plot(ph)
plt.xlabel('x-axis')
plt.ylabel('y-axis')
#plt.zlabel('z-axis')
plt.title('plot of p')
plt.show()