Thanks a lot @dokken now it works 
For the sake of completeness, here is the edited version of the tutorial on Hyperelasticty with enforced average displacement condition.
I only started using FEniCS and FEniCSx yesterday, so this code is very open to improvements.
from dolfinx import log, default_scalar_type
from dolfinx.fem.petsc import NonlinearProblem
#from dolfinx.nls.petsc import NewtonSolver
from scifem import NewtonSolver, assemble_scalar, BlockedNewtonSolver
import pyvista
import numpy as np
import ufl
import scifem
from mpi4py import MPI
from dolfinx import fem, mesh, plot
from petsc4py import PETSc
ftype = PETSc.ScalarType
NX, NY = 20, 10 # number of elements in horizontal direction and vertical direction
LX, LY = 10., 5. # length in repective directions
# Create a 2D rectangular mesh
domain = mesh.create_rectangle(MPI.COMM_WORLD, [[0.0, 0.0], [LX, LY]], [NX, NY], mesh.CellType.quadrilateral)
# Define the function space for 2D
V = fem.functionspace(domain, ("Lagrange", 2, (domain.geometry.dim, )))
R = scifem.create_real_functionspace(domain, (1,)) # Adding a single Lagrange multiplier to the global system
def left(x):
return np.isclose(x[0], 0)
def right(x):
return np.isclose(x[1], LY) * (x[0]>0.8*LX-1e-5) # The boundary where the displacement is prescribed
fdim = domain.topology.dim - 1
left_facets = mesh.locate_entities_boundary(domain, fdim, left)
right_facets = mesh.locate_entities_boundary(domain, fdim, right)
# 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])
# Fixed boundary condition on the left edge
u_bc = np.array((0,) * domain.geometry.dim, dtype=default_scalar_type)
left_dofs = fem.locate_dofs_topological(V, facet_tag.dim, facet_tag.find(1))
bcs = [fem.dirichletbc(u_bc, left_dofs, V)]
# Enforved displacement
disp = fem.Constant(domain, default_scalar_type((0)))
# Define function space including the lagrange multiplier
W = ufl.MixedFunctionSpace(V, R)
v, test_q = ufl.TestFunctions(W)
du, dq = ufl.TrialFunctions(W)
# Set zero-valued initial guess
u = fem.Function(V, dtype=ftype)
#u.x.array[:] = 0.0 # edit this for non-zero initial guess
q = fem.Function(R, dtype=ftype)
# 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))
# Elasticity parameters
E = default_scalar_type(1.0)
nu = default_scalar_type(0.4)
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)
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)
F0 = ufl.inner(ufl.grad(v), P) * dx + ufl.inner(q[0],v[1]) * ds(2)
F1 = ufl.inner(test_q[0],(u[1]-disp)) * ds(2)
F_ = F0 + F1
F = list(ufl.extract_blocks(F_))
J = ufl.extract_blocks(ufl.derivative(F_, u, du) + ufl.derivative(F_, q, dq))
petsc_options = {"ksp_type": "preonly", "pc_type": "lu", "pc_factor_mat_solver_type": "mumps"}
solver = BlockedNewtonSolver(F, [u, q], bcs=bcs, J=None, petsc_options=petsc_options)
#solver = NewtonSolver(F, J, [u, q], bcs=bcs, max_iterations=25, petsc_options=petsc_options)
# Set Newton solver options
solver.atol = 1e-8
solver.rtol = 1e-8
solver.convergence_criterion = "incremental"
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
log.set_log_level(log.LogLevel.INFO)
tval0 = -1.0
for n in range(1, 10):
disp.value = n * tval0
num_its = solver.solve()
u.x.scatter_forward()
q.x.scatter_forward()
print(f"Time step {n}, Number of iterations {num_its}, Disp {disp.value}, Load {q.x.array[:]}")
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, 1])
plotter.write_frame()
plotter.close()