def weak_residuals(U0, U1, U, w, kn, problem):
"Return weak residuals"
# Extract variables
U_F0, P_F0, U_S0, P_S0, U_M0 = U0
U_F1, P_F1, U_S1, P_S1, U_M1 = U1
U_F, P_F, U_S, P_S, U_M = U
v_F, q_F, s_F, v_S, q_S, v_M, q_M = w
# Get problem parameters
Omega = problem.mesh()
rho_F = problem.fluid_density()
mu_F = problem.fluid_viscosity()
rho_S = problem.structure_density()
mu_S = problem.structure_mu()
lmbda_S = problem.structure_lmbda()
alpha_M = problem.mesh_alpha()
mu_M = problem.mesh_mu()
lmbda_M = problem.mesh_lmbda()
# MER: added these
B = problem.structure_body_force()
G_0 = problem.structure_boundary_traction_extra()
F_M = problem.mesh_right_hand_side()
# Define normals
N = FacetNormal(Omega)
N_F = N
N_S = -N
# Define cell integrals
dx_F = dx(0)
dx_S = dx(1)
# Define facet integrals
dS_F = dS(0)
dS_S = dS(1)
d_FSI = dS(2)
# Define time derivatives
dt_U_F = (1/kn) * (U_F1 - U_F0)
dt_U_M = (1/kn) * (U_M1 - U_M0)
Dt_U_F = rho_F * J(U_M) * (dt_U_F + dot(grad(U_F), dot(inv(F(U_M)), U_F - dt_U_M)))
Dt_U_S = (1/kn) * (U_S1 - U_S0)
Dt_P_S = rho_S * (1/kn) * (P_S1 - P_S0)
Dt_U_M = alpha_M * (1/kn) * (U_M1 - U_M0)
# Define stresses
Sigma_F = PiolaTransform(_Sigma_F(U_F, P_F, U_M, mu_F), U_M)
Sigma_S = _Sigma_S(U_S, mu_S, lmbda_S)
Sigma_M = _Sigma_M(U_M, mu_M, lmbda_M)
# Fluid residual
Id = I(U_F)
R_F = inner(v_F, Dt_U_F)*dx_F + inner(grad(v_F), Sigma_F)*dx_F \
- inner(v_F, mu_F*J(U_M)*dot(dot(inv(F(U_M)).T, grad(U_F).T), dot(inv(F(U_M)).T, N_F)))*ds \
+ inner(v_F, J(U_M)*P_F*dot(Id, dot(inv(F(U_M)).T, N_F)))*ds \
+ inner(q_F, div(J(U_M)*dot(inv(F(U_M)), U_F)))*dx_F
# Structure residual
# MER: What about the body force? (Added, makes a difference, good)
# MER And the extra stress? (Added, sign dubious, b/c evaluates to
# zero since it is orthogonal to test case dual solution)
R_S = inner(v_S, Dt_P_S)*dx_S + inner(grad(v_S), Sigma_S)*dx_S \
- inner(v_S('-'), dot(Sigma_F('+'), N_S('+')))*d_FSI \
+ inner(q_S, Dt_U_S - P_S)*dx_S \
- inner(v_S, B)*dx_S \
+ inner(v_S('-'), G_0('-'))*d_FSI \
# Mesh residual contributions
R_M = inner(v_M, Dt_U_M)*dx_F + inner(sym(grad(v_M)), Sigma_M)*dx_F \
+ inner(q_M, U_M - U_S)('+')*d_FSI \
- inner(v_M, F_M)*dx_F
return R_F, R_S, R_M
def weak_residuals(U0, U1, U, w, kn, problem):
"Return weak residuals"
# Extract variables
U_F0, P_F0, U_S0, P_S0, U_M0 = U0
U_F1, P_F1, U_S1, P_S1, U_M1 = U1
U_F, P_F, U_S, P_S, U_M = U
v_F, q_F, v_S, q_S, v_M, q_M = w
# Get problem parameters
Omega = problem.mesh()
rho_F = problem.fluid_density()
mu_F = problem.fluid_viscosity()
rho_S = problem.structure_density()
mu_S = problem.structure_mu()
lmbda_S = problem.structure_lmbda()
alpha_M = problem.mesh_alpha()
mu_M = problem.mesh_mu()
lmbda_M = problem.mesh_lmbda()
# Define normals
N = FacetNormal(Omega)
N_F = N
N_S = -N
# Define cell integrals
dx_F = dx(0)
dx_S = dx(1)
# Define facet integrals
dS_F = dS(0)
dS_S = dS(1)
d_FSI = dS(2)
# Define time derivatives
dt_U_F = (1 / kn) * (U_F1 - U_F0)
dt_U_M = (1 / kn) * (U_M1 - U_M0)
Dt_U_F = rho_F * J(U_M) * (dt_U_F +
dot(grad(U_F), dot(inv(F(U_M)), U_F - dt_U_M)))
Dt_U_S = (1 / kn) * (U_S1 - U_S0)
Dt_P_S = rho_S * (1 / kn) * (P_S1 - P_S0)
Dt_U_M = alpha_M * (1 / kn) * (U_M1 - U_M0)
# Define stresses
Sigma_F = PiolaTransform(_Sigma_F(U_F, P_F, U_M, mu_F), U_M)
Sigma_S = _Sigma_S(U_S, mu_S, lmbda_S)
Sigma_M = _Sigma_M(U_M, mu_M, lmbda_M)
# Fluid residual
R_F = inner(v_F, Dt_U_F)*dx_F + inner(grad(v_F), Sigma_F)*dx_F \
- inner(v_F, mu_F*J(U_M)*dot(dot(inv(F(U_M)).T, grad(U_F).T), dot(inv(F(U_M)).T, N_F)))*ds \
+ inner(v_F, J(U_M)*P_F*dot(I, dot(inv(F(U_M)).T, N_F)))*ds \
+ inner(q_F, div(J(U_M)*dot(inv(F(U_M)), U_F)))*dx_F
# Structure residual
R_S = inner(v_S, Dt_P_S)*dx_S + inner(grad(v_S), Sigma_S)*dx_S \
- inner(v_S('-'), dot(Sigma_F('+'), N_S('+')))*d_FSI \
+ inner(q_S, Dt_U_S - P_S)*dx_S
# Mesh residual contributions
R_M = inner(v_M, Dt_U_M)*dx_F + inner(sym(grad(v_M)), Sigma_M)*dx_F \
+ inner(q_M, U_M - U_S)('+')*d_FSI
return R_F, R_S, R_M
def strong_residuals(U0, U1, U, Z, EZ, w, kn, problem):
"Return strong residuals (integrated by parts)"
# Extract variables
U_F0, P_F0, U_S0, P_S0, U_M0 = U0
U_F1, P_F1, U_S1, P_S1, U_M1 = U1
U_F, P_F, U_S, P_S, U_M = U
Z_F, Y_F, X_F, Z_S, Y_S, Z_M, Y_M = Z
EZ_F, EY_F, EX_F ,EZ_S, EY_S, EZ_M, EY_M = EZ
# Get problem parameters
Omega = problem.mesh()
Omega_F = problem.fluid_mesh()
Omega_S = problem.structure_mesh()
rho_F = problem.fluid_density()
mu_F = problem.fluid_viscosity()
rho_S = problem.structure_density()
mu_S = problem.structure_mu()
lmbda_S = problem.structure_lmbda()
alpha_M = problem.mesh_alpha()
mu_M = problem.mesh_mu()
lmbda_M = problem.mesh_lmbda()
# Define normals
N = FacetNormal(Omega)
N_F = N
N_S = N
# FIXME: Check sign of N_S, should it be -N?
# Define inner products
dx_F = dx(0)
dx_S = dx(1)
# Define "facet" products
dS_F = dS(0)
dS_S = dS(1)
d_FSI = dS(2)
# Define midpoint values
U_F = 0.5 * (U_F0 + U_F1)
P_F = 0.5 * (P_F0 + P_F1)
U_S = 0.5 * (U_S0 + U_S1)
P_S = 0.5 * (P_S0 + P_S1)
U_M = 0.5 * (U_M0 + U_M1)
# Define time derivatives
dt_U_F = (1/kn) * (U_F1 - U_F0)
dt_U_M = (1/kn) * (U_M1 - U_M0)
Dt_U_F = rho_F * J(U_M) * (dt_U_F + dot(grad(U_F), dot(inv(F(U_M)), U_F - dt_U_M)))
Dt_U_S = (1/kn) * (U_S1 - U_S0)
Dt_P_S = rho_S * (1/kn) * (P_S1 - P_S0)
Dt_U_M = alpha_M * (1/kn) * (U_M1 - U_M0)
# Define stresses
Sigma_F = PiolaTransform(_Sigma_F(U_F, P_F, U_M, mu_F), U_M)
Sigma_S = _Sigma_S(U_S, mu_S, lmbda_S)
Sigma_M = _Sigma_M(U_M, mu_M, lmbda_M)
# Fluid residual contributions
# FIXME: Add dual Lagrange multiplier EX_F for the fluid
R_F0 = w*inner(EZ_F - Z_F, Dt_U_F - div(Sigma_F))*dx_F
R_F1 = avg(w)*inner(EZ_F('+') - Z_F('+'), jump(Sigma_F, N_F))*dS_F
R_F2 = w*inner(EZ_F - Z_F, dot(Sigma_F, N_F))*ds
R_F3 = w*inner(EY_F - Y_F, div(J(U_M)*dot(inv(F(U_M)), U_F)))*dx_F
# Structure residual contributions (note the minus sign on N_F('+'))
R_S0 = w*inner(EZ_S - Z_S, Dt_P_S - div(Sigma_S))*dx_S
R_S1 = avg(w)*inner(EZ_S('-') - Z_S('-'), jump(Sigma_S, N_S))*dS_S
R_S2 = w('-')*inner(EZ_S('-') - Z_S('-'), dot(Sigma_S('-') - Sigma_F('+'), -N_F('+')))*d_FSI
R_S3 = w*inner(EY_S - Y_S, Dt_U_S - P_S)*dx_S
# Mesh residual contributions
R_M0 = w*inner(EZ_M - Z_M, Dt_U_M - div(Sigma_M))*dx_F
R_M1 = avg(w)*inner(EZ_M('+') - Z_M('+'), jump(Sigma_M, N_F))*dS_F
R_M2 = w('+')*inner(EY_M - Y_M, U_M - U_S)('+')*d_FSI # this should be zero
return (R_F0, R_F1, R_F2, R_F3), (R_S0, R_S1, R_S2, R_S3), (R_M0, R_M1, R_M2)
def weak_residuals(U0, U1, U, w, kn, problem):
"Return weak residuals"
# Extract variables
U_F0, P_F0, U_S0, P_S0, U_M0 = U0
U_F1, P_F1, U_S1, P_S1, U_M1 = U1
U_F, P_F, U_S, P_S, U_M = U
v_F, q_F, v_S, q_S, v_M, q_M = w
# Get problem parameters
Omega = problem.mesh()
rho_F = problem.fluid_density()
mu_F = problem.fluid_viscosity()
rho_S = problem.structure_density()
mu_S = problem.structure_mu()
lmbda_S = problem.structure_lmbda()
alpha_M = problem.mesh_alpha()
mu_M = problem.mesh_mu()
lmbda_M = problem.mesh_lmbda()
# Define normals
N = FacetNormal(Omega)
N_F = N
N_S = -N
# Define cell integrals
dx_F = dx(0)
dx_S = dx(1)
# Define facet integrals
dS_F = dS(0)
dS_S = dS(1)
d_FSI = dS(2)
# Define time derivatives
dt_U_F = (1/kn) * (U_F1 - U_F0)
dt_U_M = (1/kn) * (U_M1 - U_M0)
Dt_U_F = rho_F * J(U_M) * (dt_U_F + dot(grad(U_F), dot(inv(F(U_M)), U_F - dt_U_M)))
Dt_U_S = (1/kn) * (U_S1 - U_S0)
Dt_P_S = rho_S * (1/kn) * (P_S1 - P_S0)
Dt_U_M = alpha_M * (1/kn) * (U_M1 - U_M0)
# Define stresses
Sigma_F = PiolaTransform(_Sigma_F(U_F, P_F, U_M, mu_F), U_M)
Sigma_S = _Sigma_S(U_S, mu_S, lmbda_S)
Sigma_M = _Sigma_M(U_M, mu_M, lmbda_M)
# Fluid residual
R_F = inner(v_F, Dt_U_F)*dx_F + inner(grad(v_F), Sigma_F)*dx_F \
- inner(v_F, mu_F*J(U_M)*dot(dot(inv(F(U_M)).T, grad(U_F).T), dot(inv(F(U_M)).T, N_F)))*ds \
+ inner(v_F, J(U_M)*P_F*dot(I, dot(inv(F(U_M)).T, N_F)))*ds \
+ inner(q_F, div(J(U_M)*dot(inv(F(U_M)), U_F)))*dx_F
# Structure residual
R_S = inner(v_S, Dt_P_S)*dx_S + inner(grad(v_S), Sigma_S)*dx_S \
- inner(v_S('-'), dot(Sigma_F('+'), N_S('+')))*d_FSI \
+ inner(q_S, Dt_U_S - P_S)*dx_S
# Mesh residual contributions
R_M = inner(v_M, Dt_U_M)*dx_F + inner(sym(grad(v_M)), Sigma_M)*dx_F \
+ inner(q_M, U_M - U_S)('+')*d_FSI
return R_F, R_S, R_M