Cardiac mechanics Benchmark - Problem 1¶

This code implements problem 1 in the Cardiac Mechanic Benchmark paper

Land S, Gurev V, Arens S, Augustin CM, Baron L, Blake R, Bradley C, Castro S, Crozier A, Favino M, Fastl TE. Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour. Proc. R. Soc. A. 2015 Dec 8;471(2184):20150641.

import dolfin
import numpy as np

try:
    from dolfin_adjoint import (
        BoxMesh,
        Constant,
        DirichletBC,
        Expression,
        Mesh,
        interpolate,
    )
except ImportError:
    from dolfin import BoxMesh, Expression, DirichletBC, Constant, interpolate, Mesh

import pulse
from fenics_plotly import plot

Create the Beam geometry

# Length
L = 10
# Width
W = 1

# Create mesh
mesh = BoxMesh(dolfin.Point(0, 0, 0), dolfin.Point(L, W, W), 30, 3, 3)

# Mark boundary subdomians
left = dolfin.CompiledSubDomain("near(x[0], side) && on_boundary", side=0)
bottom = dolfin.CompiledSubDomain("near(x[2], side) && on_boundary", side=0)

boundary_markers = dolfin.MeshFunction("size_t", mesh, mesh.topology().dim() - 1)
boundary_markers.set_all(0)

left_marker = 1
left.mark(boundary_markers, 1)
bottom_marker = 2
bottom.mark(boundary_markers, 2)

marker_functions = pulse.MarkerFunctions(ffun=boundary_markers)

# Create mictrotructure
f0 = Expression(("1.0", "0.0", "0.0"), degree=1, cell=mesh.ufl_cell())
s0 = Expression(("0.0", "1.0", "0.0"), degree=1, cell=mesh.ufl_cell())
n0 = Expression(("0.0", "0.0", "1.0"), degree=1, cell=mesh.ufl_cell())

# Collect the mictrotructure
microstructure = pulse.Microstructure(f0=f0, s0=s0, n0=n0)

# Create the geometry
geometry = pulse.Geometry(
    mesh=mesh,
    marker_functions=marker_functions,
    microstructure=microstructure,
)

# Create the material
material_parameters = pulse.Guccione.default_parameters()
material_parameters["CC"] = 2.0
material_parameters["bf"] = 8.0
material_parameters["bfs"] = 4.0
material_parameters["bt"] = 2.0

material = pulse.Guccione(parameters=material_parameters)

# Define Dirichlet boundary. Fix at the left boundary
def dirichlet_bc(W):
    V = W if W.sub(0).num_sub_spaces() == 0 else W.sub(0)
    return DirichletBC(V, Constant((0.0, 0.0, 0.0)), left)

# Traction at the bottom of the beam
p_bottom = Constant(0.004)
neumann_bc = pulse.NeumannBC(traction=p_bottom, marker=bottom_marker)

# Collect Boundary Conditions
bcs = pulse.BoundaryConditions(dirichlet=(dirichlet_bc,), neumann=(neumann_bc,))

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

# Solve problem
problem.solve()

(6, True)

# Get displacement and hydrostatic pressure
u, p = problem.state.split(deepcopy=True)

point = np.array([10.0, 0.5, 1.0])
disp = np.zeros(3)
u.eval(disp, point)

print(
    ("Get z-position of point ({}): {:.4f} mm" "").format(
        ", ".join(["{:.1f}".format(p) for p in point]),
        point[2] + disp[2],
    ),
)

Get z-position of point (10.0, 0.5, 1.0): 4.1552 mm

V = dolfin.VectorFunctionSpace(geometry.mesh, "CG", 1)
u_int = interpolate(u, V)
mesh = Mesh(geometry.mesh)
dolfin.ALE.move(mesh, u_int)

fig = plot(geometry.mesh, show=False)
fig.add_plot(plot(mesh, color="red", show=False))
fig.show()