# Cantilever exact solution

Hi,

I’m trying to verify the cantilever beam problem described in Implementation — FEniCSx tutorial against the exact solution for displacement y = \frac{Fx^2}{24EI} (x^2 + 6l^2 -4lx) , where F is force, l length in x-direction, E Young’s modulus and I the second moment of area.

My exact and numerical solutions don’t match. I’m dividing the force by the number of nodes for the exact solution, which gets things closer to the numerical solution, but it’s still not great. Does anyone have any thoughts on what I might be doing wrong?

This is how I am defining my exact solution:

# Define the exact solution:
I = fem.Constant(domain, (W*W)**3/12) #second moment of area
E = fem.Constant(domain, (mu*(3*lambda_ + 2*mu)) / (1+mu)) #Youngs mod based on lame params
l = fem.Constant(domain, 1.0) #length of domain, L
f_ex = fem.Constant(domain, ScalarType((0, 0, -rho*g/1029))) #diving force by nb of mesh nodes else it's huge

x = ufl.SpatialCoordinate(domain)
u_exact = (f_ex*x[0]**2 /(24*E*I)) * (x[0]+6*l**2 - 4*l*x[0])

# Interpolate u_exact onto higher dimensional space:
V2 = fem.VectorFunctionSpace(domain, ("CG", 2))
expr = fem.Expression(u_exact, V2.element.interpolation_points())
uex = fem.Function(V2)
uex.interpolate(expr)


and these are my solutions (numerical above, “exact” below):

The entire code (including the LE tutorial for the numerical solution) is below for completeness. Any pointers would be greatly appreciated

Thanks,
Rose

import scipy
import dolfinx
import numpy as np
import pyvista
import ufl
import scipy.io
from dolfinx import fem, io, mesh, plot, nls, log
from dolfinx.io import gmshio
from mpi4py import MPI
from petsc4py import PETSc
import matplotlib as mpl
import matplotlib.pyplot as plt
from matplotlib import cm
from scipy.interpolate import griddata
from scipy.interpolate import LinearNDInterpolator
from scipy.io import savemat

# Scaled variable
L = 1
W = 0.2
mu = 1
rho = 1
delta = W/L
gamma = 0.4*delta**2
beta = 1.25
lambda_ = beta
g = gamma

import numpy as np
import ufl
from mpi4py import MPI
from petsc4py.PETSc import ScalarType
from dolfinx import mesh, fem, plot, io

# Mesh and function space:
domain = mesh.create_box(MPI.COMM_WORLD, [np.array([0,0,0]), np.array([L, W, W])],
[20,6,6], cell_type=mesh.CellType.hexahedron)
V = fem.VectorFunctionSpace(domain, ("CG", 1))

# BCs:
def clamped_boundary(x):
return np.isclose(x[0], 0)
fdim = domain.topology.dim - 1
boundary_facets = mesh.locate_entities_boundary(domain, fdim, clamped_boundary)
u_D = np.array([0,0,0], dtype=ScalarType)
bc = fem.dirichletbc(u_D, fem.locate_dofs_topological(V, fdim, boundary_facets), V)

# Variational fomulation
T = fem.Constant(domain, ScalarType((0, 0, 0)))
ds = ufl.Measure("ds", domain=domain)
def epsilon(u):
def sigma(u):
return lambda_ * ufl.nabla_div(u) * ufl.Identity(len(u)) + 2*mu*epsilon(u)
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
f = fem.Constant(domain, ScalarType((0, 0, -rho*g)))
a = ufl.inner(sigma(u), epsilon(v)) * ufl.dx
L = ufl.dot(f, v) * ufl.dx + ufl.dot(T, v) * ds

# Solve numerically
problem = fem.petsc.LinearProblem(a, L, bcs=[bc], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
uh = problem.solve()

# Define the exact solution:
I = fem.Constant(domain, (W*W)**3/12) #second moment of area
E = fem.Constant(domain, (mu*(3*lambda_ + 2*mu)) / (1+mu)) #Youngs mod based on lame params
l = fem.Constant(domain, 1.0) #length of domain, L
f_ex = fem.Constant(domain, ScalarType((0, 0, -rho*g/1029))) #diving force by nb of mesh nodes else it's huge

x = ufl.SpatialCoordinate(domain)
u_exact = (f_ex*x[0]**2 /(24*E*I)) * (x[0]+6*l**2 - 4*l*x[0])

# Interpolate u_exact onto higher dimensional space:
V2 = fem.VectorFunctionSpace(domain, ("CG", 2))
expr = fem.Expression(u_exact, V2.element.interpolation_points())
uex = fem.Function(V2)
uex.interpolate(expr)

# Numerical solution plot
import pyvista
pyvista.start_xvfb()
# Create plotter and pyvista grid
p = pyvista.Plotter()
topology, cell_types, geometry = plot.create_vtk_mesh(V)
grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)
# Attach vector values to grid and warp grid by vector
grid["u"] = uh.x.array.reshape((geometry.shape[0], 3))
warped = grid.warp_by_vector("u", factor=1.5)
p.show_axes()
if not pyvista.OFF_SCREEN:
p.show()
else:
figure_as_array = p.screenshot("deflection.png")

# Exact solution plot
pyvista.start_xvfb()
# Create plotter and pyvista grid
p = pyvista.Plotter()
topology, cell_types, geometry = plot.create_vtk_mesh(V)
grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)
# Attach vector values to grid and warp grid by vector
grid["uex_1"] = uex_1.x.array.reshape((geometry.shape[0], 3))