Compute stress and strain¶

In this demo we show how to compute stress and strain

import dolfin

try:
    from dolfin_adjoint import Constant, DirichletBC, Function, project
except ImportError:
    from dolfin import DirichletBC, Constant, project, Function

import pulse
from fenics_plotly import plot

gamma_space = "R_0"
geometry = pulse.Geometry.from_file(pulse.mesh_paths["prolate_ellipsoid"])
# geometry = pulse.geometries.prolate_ellipsoid_geometry(mesh_size_factor=3.0)
geometry.mesh.coordinates()[:] *= 0.1
# breakpoint()
activation = Function(dolfin.FunctionSpace(geometry.mesh, "R", 0))

2022-06-05 12:06:52,805 [1066] INFO     pulse.geometry_utils: 
Load mesh from h5

matparams = pulse.HolzapfelOgden.default_parameters()
material = pulse.HolzapfelOgden(
    active_model=pulse.ActiveModels.active_stress,
    T_ref=1.0,  # Total active stress should be activation * T_ref
    eta=0.2,  # Fraction of transverse stress
    activation=activation,
    parameters=matparams,
    f0=geometry.f0,
    s0=geometry.s0,
    n0=geometry.n0,
)

# LV Pressure
lvp = Constant(0.0)
lv_marker = geometry.markers["ENDO"][0]
lv_pressure = pulse.NeumannBC(traction=lvp, marker=lv_marker, name="lv")
neumann_bc = [lv_pressure]

# Add spring term at the base with stiffness 1.0 kPa/cm^2
base_spring = 1.0
robin_bc = [
    pulse.RobinBC(value=Constant(base_spring), marker=geometry.markers["BASE"][0]),
]

def fix_basal_plane(W):
    V = W if W.sub(0).num_sub_spaces() == 0 else W.sub(0)
    bc = DirichletBC(
        V.sub(0),
        Constant(0.0),
        geometry.ffun,
        geometry.markers["BASE"][0],
    )
    return bc

dirichlet_bc = [fix_basal_plane]

# Collect boundary conditions
bcs = pulse.BoundaryConditions(
    dirichlet=dirichlet_bc,
    neumann=neumann_bc,
    robin=robin_bc,
)

# Create the problem
problem = pulse.MechanicsProblem(geometry, material, bcs)

# Solve the problem
pulse.iterate.iterate(problem, lvp, 15.0, max_nr_crash=100, max_iters=100)
pulse.iterate.iterate(problem, activation, 60.0)

2022-06-05 12:06:52,893 [1066] INFO     pulse.iterate: Iterating to target control (f_35)...

2022-06-05 12:06:52,893 [1066] INFO     pulse.iterate: Current control: f_35 = 0.000

2022-06-05 12:06:52,895 [1066] INFO     pulse.iterate: Target: 15.000

*** Warning: PETSc SNES solver diverged in 0 iterations with divergence reason DIVERGED_FNORM_NAN.
*** Warning: PETSc SNES solver diverged in 0 iterations with divergence reason DIVERGED_FNORM_NAN.

*** Warning: PETSc SNES solver diverged in 0 iterations with divergence reason DIVERGED_FNORM_NAN.

*** Warning: PETSc SNES solver diverged in 0 iterations with divergence reason DIVERGED_FNORM_NAN.

*** Warning: PETSc SNES solver diverged in 0 iterations with divergence reason DIVERGED_FNORM_NAN.
*** Warning: PETSc SNES solver diverged in 0 iterations with divergence reason DIVERGED_FNORM_NAN.

*** Warning: PETSc SNES solver diverged in 0 iterations with divergence reason DIVERGED_FNORM_NAN.
*** Warning: PETSc SNES solver diverged in 1 iterations with divergence reason DIVERGED_DTOL.

*** Warning: PETSc SNES solver diverged in 2 iterations with divergence reason DIVERGED_FNORM_NAN.

2022-06-05 12:08:59,886 [1066] INFO     pulse.iterate: Iterating to target control (f_22)...

2022-06-05 12:08:59,887 [1066] INFO     pulse.iterate: Current control: f_22 = 0.000

2022-06-05 12:08:59,887 [1066] INFO     pulse.iterate: Target: 60.000

*** Warning: PETSc SNES solver diverged in 0 iterations with divergence reason DIVERGED_FNORM_NAN.

([Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), MixedElement(VectorElement(FiniteElement('Lagrange', tetrahedron, 2), dim=3), FiniteElement('Lagrange', tetrahedron, 1))), 1077),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), MixedElement(VectorElement(FiniteElement('Lagrange', tetrahedron, 2), dim=3), FiniteElement('Lagrange', tetrahedron, 1))), 1141),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), MixedElement(VectorElement(FiniteElement('Lagrange', tetrahedron, 2), dim=3), FiniteElement('Lagrange', tetrahedron, 1))), 1266),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), MixedElement(VectorElement(FiniteElement('Lagrange', tetrahedron, 2), dim=3), FiniteElement('Lagrange', tetrahedron, 1))), 1329),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), MixedElement(VectorElement(FiniteElement('Lagrange', tetrahedron, 2), dim=3), FiniteElement('Lagrange', tetrahedron, 1))), 1408),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), MixedElement(VectorElement(FiniteElement('Lagrange', tetrahedron, 2), dim=3), FiniteElement('Lagrange', tetrahedron, 1))), 1472)],
 [Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), FiniteElement('Real', tetrahedron, 0)), 1075),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), FiniteElement('Real', tetrahedron, 0)), 1139),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), FiniteElement('Real', tetrahedron, 0)), 1264),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), FiniteElement('Real', tetrahedron, 0)), 1327),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), FiniteElement('Real', tetrahedron, 0)), 1406),
  Coefficient(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', tetrahedron, 1), dim=3), 1), FiniteElement('Real', tetrahedron, 0)), 1470)])

V = dolfin.VectorFunctionSpace(geometry.mesh, "CG", 1)
u, p = problem.state.split(deepcopy=True)
# u_int = interpolate(u, V)
# mesh = Mesh(geometry.mesh)
# dolfin.ALE.move(mesh, u_int)

# Get the solution
# u, p = problem.state.split(deepcopy=True)
# dolfin.File("u.pvd") << u

W = dolfin.FunctionSpace(geometry.mesh, "CG", 1)
F = pulse.kinematics.DeformationGradient(u)
E = pulse.kinematics.GreenLagrangeStrain(F)
# Green strain normal to fiber direction
Ef = project(dolfin.inner(E * geometry.f0, geometry.f0), W)
# Save to file for visualization in paraview
# dolfin.File("Ef.pvd") << Ef
plot(Ef)

<fenics_plotly.fenics_plotly.FEniCSPlotFig at 0x7f8be7a03b80>

P = material.FirstPiolaStress(F, p)
# First piola stress normal to fiber direction
Pf = project(dolfin.inner(P * geometry.f0, geometry.f0), W)
# Save to file for visualization in paraview
# dolfin.File("Pf.pvd") << Pf
plot(Pf, colorscale="viridis")

<fenics_plotly.fenics_plotly.FEniCSPlotFig at 0x7f8beb7ce6b0>

T = material.CauchyStress(F, p)
f = F * geometry.f0
# Cauchy fiber stress
Tf = dolfin.project(dolfin.inner(T * f, f), W)
# Save to file for visualization in paraview
# dolfin.File("Tf.pvd") << Tf
plot(Tf, colorscale="jet")

<fenics_plotly.fenics_plotly.FEniCSPlotFig at 0x7f8beb7ce320>