Hyperelasticity#
Author: Jørgen S. Dokken and Garth N. Wells
This section shows how to solve the hyperelasticity problem for deformation of a beam.
We will also show how to create a constant boundary condition for a vector function space.
We start by importing DOLFINx and some additional dependencies.
Then, we create a slender cantilever consisting of hexahedral elements and create the function space V for our unknown.
from dolfinx import log, default_scalar_type
from dolfinx.fem.petsc import NonlinearProblem
from dolfinx.nls.petsc import NewtonSolver
import pyvista
import numpy as np
import ufl
from mpi4py import MPI
from dolfinx import fem, mesh, plot
L = 20.0
domain = mesh.create_box(MPI.COMM_WORLD, [[0.0, 0.0, 0.0], [L, 1, 1]], [20, 5, 5], mesh.CellType.hexahedron)
V = fem.functionspace(domain, ("Lagrange", 2, (domain.geometry.dim, )))
We create two python functions for determining the facets to apply boundary conditions to
def left(x):
return np.isclose(x[0], 0)
def right(x):
return np.isclose(x[0], L)
fdim = domain.topology.dim - 1
left_facets = mesh.locate_entities_boundary(domain, fdim, left)
right_facets = mesh.locate_entities_boundary(domain, fdim, right)
Next, we create a marker based on these two functions
# Concatenate and sort the arrays based on facet indices. Left facets marked with 1, right facets with two
marked_facets = np.hstack([left_facets, right_facets])
marked_values = np.hstack([np.full_like(left_facets, 1), np.full_like(right_facets, 2)])
sorted_facets = np.argsort(marked_facets)
facet_tag = mesh.meshtags(domain, fdim, marked_facets[sorted_facets], marked_values[sorted_facets])
We then create a function for supplying the boundary condition on the left side, which is fixed.
u_bc = np.array((0,) * domain.geometry.dim, dtype=default_scalar_type)
To apply the boundary condition, we identity the dofs located on the facets marked by the MeshTag.
left_dofs = fem.locate_dofs_topological(V, facet_tag.dim, facet_tag.find(1))
bcs = [fem.dirichletbc(u_bc, left_dofs, V)]
Next, we define the body force on the reference configuration (B), and nominal (first Piola-Kirchhoff) traction (T).
B = fem.Constant(domain, default_scalar_type((0, 0, 0)))
T = fem.Constant(domain, default_scalar_type((0, 0, 0)))
Define the test and solution functions on the space \(V\)
v = ufl.TestFunction(V)
u = fem.Function(V)
Define kinematic quantities used in the problem
# Spatial dimension
d = len(u)
# Identity tensor
I = ufl.variable(ufl.Identity(d))
# Deformation gradient
F = ufl.variable(I + ufl.grad(u))
# Right Cauchy-Green tensor
C = ufl.variable(F.T * F)
# Invariants of deformation tensors
Ic = ufl.variable(ufl.tr(C))
J = ufl.variable(ufl.det(F))
Define the elasticity model via a stored strain energy density function \(\psi\), and create the expression for the first Piola-Kirchhoff stress:
# Elasticity parameters
E = default_scalar_type(1.0e4)
nu = default_scalar_type(0.3)
mu = fem.Constant(domain, E / (2 * (1 + nu)))
lmbda = fem.Constant(domain, E * nu / ((1 + nu) * (1 - 2 * nu)))
# Stored strain energy density (compressible neo-Hookean model)
psi = (mu / 2) * (Ic - 3) - mu * ufl.ln(J) + (lmbda / 2) * (ufl.ln(J))**2
# Stress
# Hyper-elasticity
P = ufl.diff(psi, F)
Comparison to linear elasticity
To illustrate the difference between linear and hyperelasticity, the following lines can be uncommented to solve the linear elasticity problem.
# P = 2.0 * mu * ufl.sym(ufl.grad(u)) + lmbda * ufl.tr(ufl.sym(ufl.grad(u))) * I
Define the variational form with traction integral over all facets with value 2. We set the quadrature degree for the integrals to 4.
metadata = {"quadrature_degree": 4}
ds = ufl.Measure('ds', domain=domain, subdomain_data=facet_tag, metadata=metadata)
dx = ufl.Measure("dx", domain=domain, metadata=metadata)
# Define form F (we want to find u such that F(u) = 0)
F = ufl.inner(ufl.grad(v), P) * dx - ufl.inner(v, B) * dx - ufl.inner(v, T) * ds(2)
As the varitional form is non-linear and written on residual form, we use the non-linear problem class from DOLFINx to set up required structures to use a Newton solver.
problem = NonlinearProblem(F, u, bcs)
and then create and customize the Newton solver
solver = NewtonSolver(domain.comm, problem)
# Set Newton solver options
solver.atol = 1e-8
solver.rtol = 1e-8
solver.convergence_criterion = "incremental"
We create a function to plot the solution at each time step.
pyvista.start_xvfb()
plotter = pyvista.Plotter()
plotter.open_gif("deformation.gif", fps=3)
topology, cells, geometry = plot.vtk_mesh(u.function_space)
function_grid = pyvista.UnstructuredGrid(topology, cells, geometry)
values = np.zeros((geometry.shape[0], 3))
values[:, :len(u)] = u.x.array.reshape(geometry.shape[0], len(u))
function_grid["u"] = values
function_grid.set_active_vectors("u")
# Warp mesh by deformation
warped = function_grid.warp_by_vector("u", factor=1)
warped.set_active_vectors("u")
# Add mesh to plotter and visualize
actor = plotter.add_mesh(warped, show_edges=True, lighting=False, clim=[0, 10])
# Compute magnitude of displacement to visualize in GIF
Vs = fem.functionspace(domain, ("Lagrange", 2))
magnitude = fem.Function(Vs)
us = fem.Expression(ufl.sqrt(sum([u[i]**2 for i in range(len(u))])), Vs.element.interpolation_points())
magnitude.interpolate(us)
warped["mag"] = magnitude.x.array
Finally, we solve the problem over several time steps, updating the z-component of the traction
log.set_log_level(log.LogLevel.INFO)
tval0 = -1.5
for n in range(1, 10):
T.value[2] = n * tval0
num_its, converged = solver.solve(u)
assert (converged)
u.x.scatter_forward()
print(f"Time step {n}, Number of iterations {num_its}, Load {T.value}")
function_grid["u"][:, :len(u)] = u.x.array.reshape(geometry.shape[0], len(u))
magnitude.interpolate(us)
warped.set_active_scalars("mag")
warped_n = function_grid.warp_by_vector(factor=1)
warped.points[:, :] = warped_n.points
warped.point_data["mag"][:] = magnitude.x.array
plotter.update_scalar_bar_range([0, 10])
plotter.write_frame()
plotter.close()
2024-07-22 08:05:13.390 ( 6.572s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:15.898 ( 9.080s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:17.907 ( 11.090s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 22.2455 (tol = 1e-08) r (rel) = 0.134278(tol = 1e-08)
2024-07-22 08:05:18.236 ( 11.418s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:20.233 ( 13.415s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 2.43261 (tol = 1e-08) r (rel) = 0.0146837(tol = 1e-08)
2024-07-22 08:05:20.568 ( 13.750s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:22.561 ( 15.743s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 4.43158 (tol = 1e-08) r (rel) = 0.0267498(tol = 1e-08)
2024-07-22 08:05:22.886 ( 16.068s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:24.873 ( 18.055s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 0.144189 (tol = 1e-08) r (rel) = 0.000870353(tol = 1e-08)
2024-07-22 08:05:25.197 ( 18.380s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:27.189 ( 20.371s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 0.021424 (tol = 1e-08) r (rel) = 0.000129319(tol = 1e-08)
2024-07-22 08:05:27.524 ( 20.706s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:29.525 ( 22.707s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 7: r (abs) = 4.80065e-06 (tol = 1e-08) r (rel) = 2.89776e-08(tol = 1e-08)
2024-07-22 08:05:29.850 ( 23.032s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:31.864 ( 25.047s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 8: r (abs) = 2.65767e-11 (tol = 1e-08) r (rel) = 1.60422e-13(tol = 1e-08)
2024-07-22 08:05:31.864 ( 25.047s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 8 iterations and 8 linear solver iterations.
Time step 1, Number of iterations 8, Load [ 0. 0. -1.5]
2024-07-22 08:05:32.382 ( 25.565s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:34.702 ( 27.884s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:36.693 ( 29.875s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 17.3254 (tol = 1e-08) r (rel) = 0.117842(tol = 1e-08)
2024-07-22 08:05:37.021 ( 30.203s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:39.014 ( 32.197s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 5.14882 (tol = 1e-08) r (rel) = 0.0350207(tol = 1e-08)
2024-07-22 08:05:39.348 ( 32.530s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:41.343 ( 34.525s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 7.24003 (tol = 1e-08) r (rel) = 0.0492445(tol = 1e-08)
2024-07-22 08:05:41.678 ( 34.860s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:43.663 ( 36.845s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 0.777889 (tol = 1e-08) r (rel) = 0.00529096(tol = 1e-08)
2024-07-22 08:05:43.987 ( 37.169s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:46.026 ( 39.208s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 1.25525 (tol = 1e-08) r (rel) = 0.00853785(tol = 1e-08)
2024-07-22 08:05:46.355 ( 39.537s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:48.357 ( 41.539s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 7: r (abs) = 0.00849512 (tol = 1e-08) r (rel) = 5.77812e-05(tol = 1e-08)
2024-07-22 08:05:48.690 ( 41.872s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:50.687 ( 43.869s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 8: r (abs) = 0.000192107 (tol = 1e-08) r (rel) = 1.30665e-06(tol = 1e-08)
2024-07-22 08:05:51.022 ( 44.205s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:53.031 ( 46.213s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 9: r (abs) = 1.70902e-10 (tol = 1e-08) r (rel) = 1.16242e-12(tol = 1e-08)
2024-07-22 08:05:53.031 ( 46.213s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 9 iterations and 9 linear solver iterations.
Time step 2, Number of iterations 9, Load [ 0. 0. -3.]
2024-07-22 08:05:53.424 ( 46.606s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:55.735 ( 48.917s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:05:57.726 ( 50.908s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 10.0011 (tol = 1e-08) r (rel) = 0.0887471(tol = 1e-08)
2024-07-22 08:05:58.061 ( 51.243s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:00.057 ( 53.239s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 5.33026 (tol = 1e-08) r (rel) = 0.0472992(tol = 1e-08)
2024-07-22 08:06:00.392 ( 53.574s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:02.406 ( 55.589s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 11.9901 (tol = 1e-08) r (rel) = 0.106397(tol = 1e-08)
2024-07-22 08:06:02.734 ( 55.916s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:04.720 ( 57.902s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 2.29702 (tol = 1e-08) r (rel) = 0.0203831(tol = 1e-08)
2024-07-22 08:06:05.058 ( 58.240s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:07.048 ( 60.230s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 3.90234 (tol = 1e-08) r (rel) = 0.0346282(tol = 1e-08)
2024-07-22 08:06:07.374 ( 60.556s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:09.365 ( 62.547s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 7: r (abs) = 0.236535 (tol = 1e-08) r (rel) = 0.00209895(tol = 1e-08)
2024-07-22 08:06:09.699 ( 62.882s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:11.685 ( 64.868s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 8: r (abs) = 0.0427142 (tol = 1e-08) r (rel) = 0.000379034(tol = 1e-08)
2024-07-22 08:06:12.020 ( 65.202s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:14.011 ( 67.193s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 9: r (abs) = 2.87798e-05 (tol = 1e-08) r (rel) = 2.55384e-07(tol = 1e-08)
2024-07-22 08:06:14.345 ( 67.527s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:16.335 ( 69.517s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 10: r (abs) = 6.09146e-10 (tol = 1e-08) r (rel) = 5.40539e-12(tol = 1e-08)
2024-07-22 08:06:16.335 ( 69.517s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 10 iterations and 10 linear solver iterations.
Time step 3, Number of iterations 10, Load [ 0. 0. -4.5]
2024-07-22 08:06:16.721 ( 69.903s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:19.050 ( 72.232s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:21.043 ( 74.225s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 5.50693 (tol = 1e-08) r (rel) = 0.0653918(tol = 1e-08)
2024-07-22 08:06:21.381 ( 74.564s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:23.395 ( 76.577s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 26.2489 (tol = 1e-08) r (rel) = 0.311692(tol = 1e-08)
2024-07-22 08:06:23.725 ( 76.907s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:25.718 ( 78.901s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 2.30927 (tol = 1e-08) r (rel) = 0.0274213(tol = 1e-08)
2024-07-22 08:06:26.044 ( 79.226s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:28.097 ( 81.279s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 14.0562 (tol = 1e-08) r (rel) = 0.16691(tol = 1e-08)
2024-07-22 08:06:28.422 ( 81.604s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:30.412 ( 83.595s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 0.222774 (tol = 1e-08) r (rel) = 0.00264532(tol = 1e-08)
2024-07-22 08:06:30.742 ( 83.925s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:32.743 ( 85.925s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 7: r (abs) = 0.286671 (tol = 1e-08) r (rel) = 0.00340406(tol = 1e-08)
2024-07-22 08:06:33.079 ( 86.261s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:35.090 ( 88.273s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 8: r (abs) = 0.000321869 (tol = 1e-08) r (rel) = 3.82203e-06(tol = 1e-08)
2024-07-22 08:06:35.426 ( 88.608s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:37.418 ( 90.600s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 9: r (abs) = 2.63796e-07 (tol = 1e-08) r (rel) = 3.13244e-09(tol = 1e-08)
2024-07-22 08:06:37.418 ( 90.600s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 9 iterations and 9 linear solver iterations.
Time step 4, Number of iterations 9, Load [ 0. 0. -6.]
2024-07-22 08:06:37.803 ( 90.985s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:40.132 ( 93.314s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:42.121 ( 95.303s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 3.19462 (tol = 1e-08) r (rel) = 0.0496479(tol = 1e-08)
2024-07-22 08:06:42.456 ( 95.639s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:44.451 ( 97.633s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 7.71429 (tol = 1e-08) r (rel) = 0.119888(tol = 1e-08)
2024-07-22 08:06:44.787 ( 97.969s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:46.780 ( 99.962s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 0.850873 (tol = 1e-08) r (rel) = 0.0132235(tol = 1e-08)
2024-07-22 08:06:47.114 ( 100.296s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:49.122 ( 102.304s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 0.371434 (tol = 1e-08) r (rel) = 0.0057725(tol = 1e-08)
2024-07-22 08:06:49.454 ( 102.637s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:51.447 ( 104.629s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 0.00215066 (tol = 1e-08) r (rel) = 3.34236e-05(tol = 1e-08)
2024-07-22 08:06:51.782 ( 104.964s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:53.781 ( 106.963s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 7: r (abs) = 2.54607e-06 (tol = 1e-08) r (rel) = 3.95687e-08(tol = 1e-08)
2024-07-22 08:06:54.109 ( 107.291s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:56.112 ( 109.294s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 8: r (abs) = 1.73699e-13 (tol = 1e-08) r (rel) = 2.69948e-15(tol = 1e-08)
2024-07-22 08:06:56.112 ( 109.294s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 8 iterations and 8 linear solver iterations.
Time step 5, Number of iterations 8, Load [ 0. 0. -7.5]
2024-07-22 08:06:56.497 ( 109.679s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:06:58.813 ( 111.996s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:00.803 ( 113.985s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 2.00649 (tol = 1e-08) r (rel) = 0.0395622(tol = 1e-08)
2024-07-22 08:07:01.131 ( 114.313s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:03.123 ( 116.305s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 4.60977 (tol = 1e-08) r (rel) = 0.0908914(tol = 1e-08)
2024-07-22 08:07:03.452 ( 116.635s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:05.441 ( 118.624s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 0.185372 (tol = 1e-08) r (rel) = 0.00365501(tol = 1e-08)
2024-07-22 08:07:05.770 ( 118.952s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:07.767 ( 120.949s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 0.024688 (tol = 1e-08) r (rel) = 0.000486777(tol = 1e-08)
Time step 6, Number of iterations 7, Load [ 0. 0. -9.]
2024-07-22 08:07:08.103 ( 121.285s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:10.102 ( 123.285s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 5.69255e-06 (tol = 1e-08) r (rel) = 1.12241e-07(tol = 1e-08)
2024-07-22 08:07:10.437 ( 123.619s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:12.432 ( 125.614s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 7: r (abs) = 2.62006e-11 (tol = 1e-08) r (rel) = 5.166e-13(tol = 1e-08)
2024-07-22 08:07:12.432 ( 125.614s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 7 iterations and 7 linear solver iterations.
2024-07-22 08:07:12.816 ( 125.998s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:15.123 ( 128.305s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:17.112 ( 130.295s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 1.38506 (tol = 1e-08) r (rel) = 0.0336622(tol = 1e-08)
2024-07-22 08:07:17.447 ( 130.629s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:19.437 ( 132.619s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 3.03739 (tol = 1e-08) r (rel) = 0.07382(tol = 1e-08)
2024-07-22 08:07:19.769 ( 132.951s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:21.758 ( 134.940s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 0.0412386 (tol = 1e-08) r (rel) = 0.00100225(tol = 1e-08)
2024-07-22 08:07:22.093 ( 135.275s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:24.082 ( 137.264s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 0.00205057 (tol = 1e-08) r (rel) = 4.98364e-05(tol = 1e-08)
2024-07-22 08:07:24.408 ( 137.590s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:26.406 ( 139.588s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 1.78867e-08 (tol = 1e-08) r (rel) = 4.34714e-10(tol = 1e-08)
2024-07-22 08:07:26.406 ( 139.588s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 6 iterations and 6 linear solver iterations.
Time step 7, Number of iterations 6, Load [ 0. 0. -10.5]
2024-07-22 08:07:26.791 ( 139.973s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:29.110 ( 142.292s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:31.100 ( 144.283s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 1.06336 (tol = 1e-08) r (rel) = 0.031085(tol = 1e-08)
2024-07-22 08:07:31.435 ( 144.617s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:33.426 ( 146.608s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 2.0477 (tol = 1e-08) r (rel) = 0.0598598(tol = 1e-08)
2024-07-22 08:07:33.750 ( 146.932s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:35.740 ( 148.922s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 0.00897719 (tol = 1e-08) r (rel) = 0.000262427(tol = 1e-08)
2024-07-22 08:07:36.073 ( 149.255s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:38.067 ( 151.250s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 0.000167422 (tol = 1e-08) r (rel) = 4.89419e-06(tol = 1e-08)
2024-07-22 08:07:38.400 ( 151.583s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:40.393 ( 153.575s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 3.24488e-11 (tol = 1e-08) r (rel) = 9.48564e-13(tol = 1e-08)
2024-07-22 08:07:40.393 ( 153.575s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 6 iterations and 6 linear solver iterations.
Time step 8, Number of iterations 6, Load [ 0. 0. -12.]
2024-07-22 08:07:40.777 ( 153.959s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:43.095 ( 156.277s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:45.093 ( 158.275s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 2: r (abs) = 0.898789 (tol = 1e-08) r (rel) = 0.0309666(tol = 1e-08)
2024-07-22 08:07:45.427 ( 158.610s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:47.427 ( 160.610s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 3: r (abs) = 1.38354 (tol = 1e-08) r (rel) = 0.0476679(tol = 1e-08)
2024-07-22 08:07:47.754 ( 160.936s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:49.750 ( 162.932s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 4: r (abs) = 0.00185096 (tol = 1e-08) r (rel) = 6.37724e-05(tol = 1e-08)
2024-07-22 08:07:50.075 ( 163.257s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:52.093 ( 165.275s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 5: r (abs) = 7.87183e-06 (tol = 1e-08) r (rel) = 2.71213e-07(tol = 1e-08)
2024-07-22 08:07:52.423 ( 165.606s) [main ] petsc.cpp:700 INFO| PETSc Krylov solver starting to solve system.
2024-07-22 08:07:54.501 ( 167.683s) [main ] NewtonSolver.cpp:38 INFO| Newton iteration 6: r (abs) = 2.05992e-13 (tol = 1e-08) r (rel) = 7.09718e-15(tol = 1e-08)
2024-07-22 08:07:54.501 ( 167.683s) [main ] NewtonSolver.cpp:252 INFO| Newton solver finished in 6 iterations and 6 linear solver iterations.
Time step 9, Number of iterations 6, Load [ 0. 0. -13.5]
