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.
import numpy as np
import ufl
from petsc4py import PETSc
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.VectorFunctionSpace(domain, ("CG", 2))
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=PETSc.ScalarType)
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, PETSc.ScalarType((0, 0, 0)))
T = fem.Constant(domain, PETSc.ScalarType((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 = PETSc.ScalarType(1.0e4)
nu = PETSc.ScalarType(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 = fem.petsc.NonlinearProblem(F, u, bcs)
and then create and customize the Newton solver
from dolfinx import nls
solver = nls.petsc.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.
import pyvista
import matplotlib.pyplot as plt
pyvista.start_xvfb()
plotter = pyvista.Plotter()
plotter.open_gif("deformation.gif", fps=3)
topology, cells, geometry = plot.create_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 y-component of the traction
from dolfinx import log
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)
plotter.update_coordinates(warped_n.points.copy(), render=False)
plotter.update_scalar_bar_range([0, 10])
plotter.update_scalars(magnitude.x.array)
plotter.write_frame()
plotter.close()
2023-05-01 08:15:16.899 ( 6.720s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:17.388 ( 7.208s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.488564, 0.460000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:17.604 ( 7.425s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:17.968 ( 7.788s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.363549, 0.330000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:17.979 ( 7.799s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 22.2455 (tol = 1e-08) r (rel) = 0.134278(tol = 1e-08)
2023-05-01 08:15:18.186 ( 8.006s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:18.535 ( 8.355s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.348530, 0.320000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:18.546 ( 8.366s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 2.43261 (tol = 1e-08) r (rel) = 0.0146837(tol = 1e-08)
2023-05-01 08:15:18.751 ( 8.571s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:19.100 ( 8.920s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.349096, 0.300000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:19.111 ( 8.931s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 4.43158 (tol = 1e-08) r (rel) = 0.0267498(tol = 1e-08)
2023-05-01 08:15:19.319 ( 9.139s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:19.649 ( 9.469s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.330337, 0.300000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:19.661 ( 9.481s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 0.144189 (tol = 1e-08) r (rel) = 0.000870353(tol = 1e-08)
2023-05-01 08:15:19.869 ( 9.689s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:20.204 ( 10.024s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.335163, 0.300000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:20.216 ( 10.036s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 0.021424 (tol = 1e-08) r (rel) = 0.000129319(tol = 1e-08)
2023-05-01 08:15:20.424 ( 10.244s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:20.763 ( 10.584s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.339770, 0.330000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:20.777 ( 10.598s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 7: r (abs) = 4.80067e-06 (tol = 1e-08) r (rel) = 2.89777e-08(tol = 1e-08)
2023-05-01 08:15:20.986 ( 10.806s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:21.321 ( 11.142s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.335571, 0.310000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:21.333 ( 11.153s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 8: r (abs) = 2.64464e-11 (tol = 1e-08) r (rel) = 1.59635e-13(tol = 1e-08)
2023-05-01 08:15:21.333 ( 11.153s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 8 iterations and 8 linear solver iterations.
Time step 1, Number of iterations 8, Load [ 0. 0. -1.5]
2023-05-01 08:15:21.784 ( 11.605s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:22.141 ( 11.962s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.357014, 0.310000, 0.050000 (PETSc Krylov solver)
2023-05-01 08:15:22.358 ( 12.179s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:22.698 ( 12.518s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.339284, 0.310000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:22.710 ( 12.530s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 17.3254 (tol = 1e-08) r (rel) = 0.117842(tol = 1e-08)
2023-05-01 08:15:22.917 ( 12.738s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:23.269 ( 13.089s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.351818, 0.310000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:23.280 ( 13.101s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 5.14882 (tol = 1e-08) r (rel) = 0.0350207(tol = 1e-08)
2023-05-01 08:15:23.485 ( 13.306s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:23.835 ( 13.656s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.349881, 0.330000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:23.846 ( 13.667s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 7.24003 (tol = 1e-08) r (rel) = 0.0492445(tol = 1e-08)
2023-05-01 08:15:24.054 ( 13.875s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:24.395 ( 14.215s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.340651, 0.300000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:24.408 ( 14.228s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 0.777889 (tol = 1e-08) r (rel) = 0.00529096(tol = 1e-08)
2023-05-01 08:15:24.614 ( 14.434s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:24.952 ( 14.773s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.338947, 0.330000, 0.010000 (PETSc Krylov solver)
2023-05-01 08:15:24.964 ( 14.785s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 1.25525 (tol = 1e-08) r (rel) = 0.00853785(tol = 1e-08)
2023-05-01 08:15:25.170 ( 14.991s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:25.508 ( 15.328s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.337341, 0.310000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:25.520 ( 15.340s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 7: r (abs) = 0.00849512 (tol = 1e-08) r (rel) = 5.77813e-05(tol = 1e-08)
2023-05-01 08:15:25.727 ( 15.548s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:26.067 ( 15.887s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.339699, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:26.080 ( 15.901s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 8: r (abs) = 0.000192107 (tol = 1e-08) r (rel) = 1.30665e-06(tol = 1e-08)
2023-05-01 08:15:26.287 ( 16.107s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:26.632 ( 16.452s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.344593, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:26.643 ( 16.463s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 9: r (abs) = 1.70757e-10 (tol = 1e-08) r (rel) = 1.16144e-12(tol = 1e-08)
2023-05-01 08:15:26.643 ( 16.463s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 9 iterations and 9 linear solver iterations.
Time step 2, Number of iterations 9, Load [ 0. 0. -3.]
2023-05-01 08:15:26.977 ( 16.798s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:27.322 ( 17.142s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.344678, 0.310000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:27.540 ( 17.361s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:27.884 ( 17.704s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.343635, 0.340000, 0.010000 (PETSc Krylov solver)
2023-05-01 08:15:27.897 ( 17.718s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 10.0011 (tol = 1e-08) r (rel) = 0.0887471(tol = 1e-08)
2023-05-01 08:15:28.103 ( 17.923s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:28.444 ( 18.264s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.341099, 0.310000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:28.455 ( 18.276s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 5.33026 (tol = 1e-08) r (rel) = 0.0472992(tol = 1e-08)
2023-05-01 08:15:28.661 ( 18.482s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:28.998 ( 18.818s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.336554, 0.300000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:29.009 ( 18.829s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 11.9901 (tol = 1e-08) r (rel) = 0.106397(tol = 1e-08)
2023-05-01 08:15:29.214 ( 19.034s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:29.549 ( 19.370s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.335408, 0.310000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:29.560 ( 19.380s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 2.29702 (tol = 1e-08) r (rel) = 0.0203831(tol = 1e-08)
2023-05-01 08:15:29.766 ( 19.587s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:30.121 ( 19.941s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.354027, 0.320000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:30.132 ( 19.953s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 3.90234 (tol = 1e-08) r (rel) = 0.0346282(tol = 1e-08)
2023-05-01 08:15:30.340 ( 20.160s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:30.683 ( 20.504s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.343931, 0.310000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:30.694 ( 20.515s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 7: r (abs) = 0.236535 (tol = 1e-08) r (rel) = 0.00209895(tol = 1e-08)
2023-05-01 08:15:30.901 ( 20.721s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:31.246 ( 21.066s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.345437, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:31.257 ( 21.078s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 8: r (abs) = 0.0427142 (tol = 1e-08) r (rel) = 0.000379034(tol = 1e-08)
2023-05-01 08:15:31.463 ( 21.283s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:31.817 ( 21.638s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.354546, 0.340000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:31.828 ( 21.648s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 9: r (abs) = 2.87798e-05 (tol = 1e-08) r (rel) = 2.55384e-07(tol = 1e-08)
2023-05-01 08:15:32.033 ( 21.853s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:32.372 ( 22.192s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.338649, 0.310000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:32.386 ( 22.206s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 10: r (abs) = 6.09481e-10 (tol = 1e-08) r (rel) = 5.40836e-12(tol = 1e-08)
2023-05-01 08:15:32.386 ( 22.206s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 10 iterations and 10 linear solver iterations.
Time step 3, Number of iterations 10, Load [ 0. 0. -4.5]
2023-05-01 08:15:32.719 ( 22.540s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:33.057 ( 22.877s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.337536, 0.330000, 0.010000 (PETSc Krylov solver)
2023-05-01 08:15:33.273 ( 23.094s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:33.615 ( 23.435s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.341698, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:33.626 ( 23.447s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 5.50693 (tol = 1e-08) r (rel) = 0.0653918(tol = 1e-08)
2023-05-01 08:15:33.832 ( 23.652s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:34.171 ( 23.991s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.338977, 0.300000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:34.183 ( 24.003s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 26.2489 (tol = 1e-08) r (rel) = 0.311692(tol = 1e-08)
2023-05-01 08:15:34.390 ( 24.210s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:34.724 ( 24.545s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.334859, 0.300000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:34.738 ( 24.558s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 2.30927 (tol = 1e-08) r (rel) = 0.0274213(tol = 1e-08)
2023-05-01 08:15:34.944 ( 24.764s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:35.285 ( 25.106s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.341763, 0.320000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:35.298 ( 25.119s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 14.0562 (tol = 1e-08) r (rel) = 0.16691(tol = 1e-08)
2023-05-01 08:15:35.506 ( 25.327s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:35.852 ( 25.673s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.346272, 0.330000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:35.864 ( 25.684s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 0.222774 (tol = 1e-08) r (rel) = 0.00264532(tol = 1e-08)
2023-05-01 08:15:36.071 ( 25.891s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:36.424 ( 26.244s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.352958, 0.320000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:36.435 ( 26.256s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 7: r (abs) = 0.286671 (tol = 1e-08) r (rel) = 0.00340406(tol = 1e-08)
2023-05-01 08:15:36.642 ( 26.462s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:36.983 ( 26.804s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.341577, 0.330000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:36.995 ( 26.815s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 8: r (abs) = 0.000321869 (tol = 1e-08) r (rel) = 3.82203e-06(tol = 1e-08)
2023-05-01 08:15:37.201 ( 27.021s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:37.547 ( 27.367s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.345877, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:37.558 ( 27.378s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 9: r (abs) = 2.63796e-07 (tol = 1e-08) r (rel) = 3.13244e-09(tol = 1e-08)
2023-05-01 08:15:37.558 ( 27.378s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 9 iterations and 9 linear solver iterations.
Time step 4, Number of iterations 9, Load [ 0. 0. -6.]
2023-05-01 08:15:37.910 ( 27.731s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:38.258 ( 28.078s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.347165, 0.330000, 0.010000 (PETSc Krylov solver)
2023-05-01 08:15:38.476 ( 28.296s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:38.832 ( 28.652s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.355631, 0.340000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:38.844 ( 28.664s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 3.19462 (tol = 1e-08) r (rel) = 0.0496479(tol = 1e-08)
2023-05-01 08:15:39.052 ( 28.873s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:39.399 ( 29.219s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.346019, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:39.410 ( 29.230s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 7.71429 (tol = 1e-08) r (rel) = 0.119888(tol = 1e-08)
2023-05-01 08:15:39.616 ( 29.436s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:39.968 ( 29.789s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.352422, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:39.980 ( 29.801s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 0.850873 (tol = 1e-08) r (rel) = 0.0132235(tol = 1e-08)
2023-05-01 08:15:40.186 ( 30.006s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:40.530 ( 30.350s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.343922, 0.320000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:40.541 ( 30.361s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 0.371434 (tol = 1e-08) r (rel) = 0.0057725(tol = 1e-08)
2023-05-01 08:15:40.746 ( 30.567s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:41.094 ( 30.914s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.347678, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:41.106 ( 30.926s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 0.00215066 (tol = 1e-08) r (rel) = 3.34236e-05(tol = 1e-08)
2023-05-01 08:15:41.311 ( 31.132s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:41.673 ( 31.494s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.361820, 0.350000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:41.685 ( 31.505s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 7: r (abs) = 2.54607e-06 (tol = 1e-08) r (rel) = 3.95687e-08(tol = 1e-08)
2023-05-01 08:15:41.892 ( 31.712s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:42.254 ( 32.074s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.361755, 0.320000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:42.265 ( 32.086s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 8: r (abs) = 1.51485e-13 (tol = 1e-08) r (rel) = 2.35425e-15(tol = 1e-08)
2023-05-01 08:15:42.265 ( 32.086s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 8 iterations and 8 linear solver iterations.
Time step 5, Number of iterations 8, Load [ 0. 0. -7.5]
2023-05-01 08:15:42.604 ( 32.424s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:42.956 ( 32.777s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.352284, 0.310000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:43.173 ( 32.993s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:43.525 ( 33.346s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.352173, 0.330000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:43.536 ( 33.356s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 2.00649 (tol = 1e-08) r (rel) = 0.0395622(tol = 1e-08)
2023-05-01 08:15:43.742 ( 33.562s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:44.105 ( 33.925s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.362470, 0.350000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:44.116 ( 33.936s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 4.60977 (tol = 1e-08) r (rel) = 0.0908914(tol = 1e-08)
2023-05-01 08:15:44.322 ( 34.142s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:44.674 ( 34.494s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.352249, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:44.686 ( 34.506s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 0.185372 (tol = 1e-08) r (rel) = 0.00365501(tol = 1e-08)
2023-05-01 08:15:44.893 ( 34.713s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:45.245 ( 35.066s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.352834, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:45.258 ( 35.078s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 0.024688 (tol = 1e-08) r (rel) = 0.000486777(tol = 1e-08)
2023-05-01 08:15:45.465 ( 35.285s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:45.815 ( 35.635s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.349954, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:45.826 ( 35.646s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 5.69254e-06 (tol = 1e-08) r (rel) = 1.12241e-07(tol = 1e-08)
2023-05-01 08:15:46.033 ( 35.853s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:46.388 ( 36.208s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.355105, 0.330000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:46.400 ( 36.220s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 7: r (abs) = 2.67626e-11 (tol = 1e-08) r (rel) = 5.27682e-13(tol = 1e-08)
2023-05-01 08:15:46.400 ( 36.220s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 7 iterations and 7 linear solver iterations.
Time step 6, Number of iterations 7, Load [ 0. 0. -9.]
2023-05-01 08:15:46.738 ( 36.558s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:47.091 ( 36.911s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.352828, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:47.309 ( 37.130s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:47.663 ( 37.484s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.354094, 0.330000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:47.675 ( 37.495s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 1.38506 (tol = 1e-08) r (rel) = 0.0336622(tol = 1e-08)
2023-05-01 08:15:47.880 ( 37.700s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:48.230 ( 38.050s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.350240, 0.320000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:48.241 ( 38.061s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 3.03739 (tol = 1e-08) r (rel) = 0.07382(tol = 1e-08)
2023-05-01 08:15:48.446 ( 38.267s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:48.804 ( 38.624s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.357498, 0.330000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:48.817 ( 38.637s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 0.0412386 (tol = 1e-08) r (rel) = 0.00100225(tol = 1e-08)
2023-05-01 08:15:49.024 ( 38.845s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:49.388 ( 39.208s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.363553, 0.340000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:49.399 ( 39.220s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 0.00205057 (tol = 1e-08) r (rel) = 4.98364e-05(tol = 1e-08)
2023-05-01 08:15:49.606 ( 39.426s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:49.970 ( 39.790s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.363638, 0.340000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:49.983 ( 39.803s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 1.78865e-08 (tol = 1e-08) r (rel) = 4.34709e-10(tol = 1e-08)
2023-05-01 08:15:49.983 ( 39.803s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 6 iterations and 6 linear solver iterations.
Time step 7, Number of iterations 6, Load [ 0. 0. -10.5]
2023-05-01 08:15:50.315 ( 40.135s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:50.669 ( 40.489s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.353664, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:50.886 ( 40.706s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:51.244 ( 41.064s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.358309, 0.350000, 0.010000 (PETSc Krylov solver)
2023-05-01 08:15:51.255 ( 41.076s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 1.06336 (tol = 1e-08) r (rel) = 0.031085(tol = 1e-08)
2023-05-01 08:15:51.462 ( 41.282s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:51.817 ( 41.637s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.354632, 0.320000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:51.829 ( 41.649s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 2.0477 (tol = 1e-08) r (rel) = 0.0598598(tol = 1e-08)
2023-05-01 08:15:52.034 ( 41.855s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:52.407 ( 42.227s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.372294, 0.340000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:52.418 ( 42.238s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 0.00897719 (tol = 1e-08) r (rel) = 0.000262427(tol = 1e-08)
2023-05-01 08:15:52.624 ( 42.445s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:52.979 ( 42.800s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.354983, 0.320000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:52.991 ( 42.811s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 0.000167422 (tol = 1e-08) r (rel) = 4.89419e-06(tol = 1e-08)
2023-05-01 08:15:53.196 ( 43.017s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:53.551 ( 43.371s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.354459, 0.340000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:53.563 ( 43.384s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 3.23994e-11 (tol = 1e-08) r (rel) = 9.4712e-13(tol = 1e-08)
2023-05-01 08:15:53.563 ( 43.384s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 6 iterations and 6 linear solver iterations.
Time step 8, Number of iterations 6, Load [ 0. 0. -12.]
2023-05-01 08:15:53.896 ( 43.716s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:54.249 ( 44.070s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.353381, 0.310000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:54.468 ( 44.288s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:54.826 ( 44.646s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.357662, 0.330000, 0.040000 (PETSc Krylov solver)
2023-05-01 08:15:54.838 ( 44.659s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 2: r (abs) = 0.898789 (tol = 1e-08) r (rel) = 0.0309666(tol = 1e-08)
2023-05-01 08:15:55.045 ( 44.865s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:55.397 ( 45.217s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.352339, 0.310000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:55.409 ( 45.229s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 3: r (abs) = 1.38354 (tol = 1e-08) r (rel) = 0.0476679(tol = 1e-08)
2023-05-01 08:15:55.616 ( 45.436s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:55.972 ( 45.792s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.356009, 0.320000, 0.050000 (PETSc Krylov solver)
2023-05-01 08:15:55.982 ( 45.803s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 4: r (abs) = 0.00185096 (tol = 1e-08) r (rel) = 6.37724e-05(tol = 1e-08)
2023-05-01 08:15:56.188 ( 46.008s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:56.541 ( 46.362s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.353518, 0.320000, 0.030000 (PETSc Krylov solver)
2023-05-01 08:15:56.554 ( 46.374s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 5: r (abs) = 7.87183e-06 (tol = 1e-08) r (rel) = 2.71213e-07(tol = 1e-08)
Time step 9, Number of iterations 6, Load [ 0. 0. -13.5]
2023-05-01 08:15:56.760 ( 46.580s) [main ] petsc.cpp:675 INFO| PETSc Krylov solver starting to solve system.
2023-05-01 08:15:57.109 ( 46.930s) [main ] TimeLogger.cpp:28 INFO| Elapsed wall, usr, sys time: 0.349545, 0.330000, 0.020000 (PETSc Krylov solver)
2023-05-01 08:15:57.121 ( 46.941s) [main ] NewtonSolver.cpp:36 INFO| Newton iteration 6: r (abs) = 2.75782e-13 (tol = 1e-08) r (rel) = 9.50169e-15(tol = 1e-08)
2023-05-01 08:15:57.121 ( 46.941s) [main ] NewtonSolver.cpp:255 INFO| Newton solver finished in 6 iterations and 6 linear solver iterations.
