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#
# Author: Martin Sandve Alnes
# Date: 2008-12-22
#
# Modified by Garth N. Wells, 2009
# Cell and its properties
cell = tetrahedron
d = cell.d
n = cell.n
x = cell.x
# Elements
u_element = VectorElement("CG", cell, 2)
p_element = FiniteElement("CG", cell, 1)
A_element = TensorElement("CG", cell, 1)
# Test and trial functions
v = TestFunction(u_element)
w = TrialFunction(u_element)
# Displacement at current and two previous timesteps
u = Coefficient(u_element)
up = Coefficient(u_element)
upp = Coefficient(u_element)
# Time parameters
dt = Constant(cell)
# Fiber field
A = Coefficient(A_element)
# External forces
T = Coefficient(u_element)
p0 = Coefficient(p_element)
N = cell.n
# Material parameters FIXME
rho = Constant(cell)
K = Constant(cell)
c00 = Constant(cell)
c11 = Constant(cell)
c22 = Constant(cell)
# Deformation gradient
I = Identity(d)
F = I + grad(u)
F = variable(F)
Finv = inv(F)
J = det(F)
# Left Cauchy-Green deformation tensor
B = F*F.T
I1_B = tr(B)
I2_B = (I1_B**2 - tr(B*B))/2
I3_B = J**2
# Right Cauchy-Green deformation tensor
C = F.T*F
I1_C = tr(C)
I2_C = (I1_C**2 - tr(C*C))/2
I3_C = J**2
# Green strain tensor
E = (C-I)/2
# Mapping of strain in fiber directions
Ef = A*E*A.T
# Strain energy function W(Q(Ef))
Q = c00*Ef[0,0]**2 + c11*Ef[1,1]**2 + c22*Ef[2,2]**2 # FIXME: insert some simple law here
W = (K/2)*(exp(Q) - 1) # + p stuff
# First Piola-Kirchoff stress tensor
P = diff(W, F)
# Acceleration term discretized with finite differences
k = dt/rho
acc = (u - 2*up + upp)
# Residual equation # FIXME: Can contain errors, not tested!
a_F = inner(acc, v)*dx \
+ k*inner(P, grad(v))*dx \
- k*dot(J*Finv*T, v)*ds(0) \
- k*dot(J*Finv*p0*N, v)*ds(1)
# Jacobi matrix of residual equation
a_J = derivative(a_F, u, w)
# Export forms
forms = [a_F, a_J]
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