# Component-wise Dirichlet BC#

Author: Jørgen S. Dokken

In this section, we will learn how to prescribe Dirichlet boundary conditions on a component of your unknown $$u_h$$. We will illustrate the problem using a VectorElement. However, the method generalizes to any MixedElement.

We will use a slightly modified version of the linear elasticity demo, namely $$$-\nabla \cdot \sigma (u) = f\quad \text{in } \Omega,$$$

$\sigma \cdot n = 0 \quad \text{on } \partial \Omega_N,$
$u= 0\quad \text{at } \partial\Omega_{D},$
$u_x=0 \quad \text{at } \partial\Omega_{Dx},$
$\sigma(u)= \lambda \mathrm{tr}(\epsilon(u))I + 2 \mu \epsilon(u), \qquad \epsilon(u) = \frac{1}{2}\left(\nabla u + (\nabla u )^T\right).$

We will consider a two dimensional box spanning $$[0,L]\times[0,H]$$, where $$\partial\Omega_N$$ is the left and right side of the beam, $$\partial\Omega_D$$ the bottom of the beam, while $$\partial\Omega_{Dx}$$ is the right side of the beam. We will prescribe a displacement $$u_x=0$$ on the right side of the beam, while the beam is being deformed under its own weight. The sides of the box is traction free.

L = 1
H = 1.3
lambda_ = 1.25
mu = 1
rho = 1
g = 1


As in the previous demos, we define our mesh and function space. We will create a ufl.VectorElement to create a two dimensional vector space.

from dolfinx.fem import (Constant, dirichletbc, Function, FunctionSpace, locate_dofs_geometrical,
locate_dofs_topological)
from dolfinx.fem.petsc import LinearProblem
from dolfinx.mesh import CellType, create_rectangle, locate_entities_boundary
from ufl import Identity, Measure, TestFunction, TrialFunction, VectorElement, dot, dx, inner, grad, nabla_div, sym
from mpi4py import MPI
from petsc4py.PETSc import ScalarType

import numpy as np
import pyvista

mesh = create_rectangle(MPI.COMM_WORLD, np.array([[0,0],[L, H]]), [30,30], cell_type=CellType.triangle)
element = VectorElement("CG", mesh.ufl_cell(), 1)
V = FunctionSpace(mesh, element)


## Boundary conditions#

As we would like to clamp the boundary at $$x=0$$, we do this by using a marker function, we use dolfinx.fem.locate_dofs_geometrical to identify the relevant degrees of freedom.

def clamped_boundary(x):
return np.isclose(x[1], 0)

u_zero = np.array((0,)*mesh.geometry.dim, dtype=ScalarType)
bc = dirichletbc(u_zero, locate_dofs_geometrical(V, clamped_boundary), V)


Next we would like to constrain the $$x$$-component of our solution at $$x=L$$ to $$0$$. We start by creating the sub space only containing the $$x$$ -component.

Next, we locate the degrees of freedom on the top boundary. However, as the boundary condition is in a sub space of our solution, we need to supply both the parent space $$V$$ and the sub space $$V_0$$ to dolfinx.locate_dofs_topological.

def right(x):
return np.logical_and(np.isclose(x[0], L), x[1] < H)
boundary_facets = locate_entities_boundary(mesh, mesh.topology.dim-1, right)
boundary_dofs_x = locate_dofs_topological(V.sub(0), mesh.topology.dim-1, boundary_facets)


We can now create our Dirichlet condition

bcx = dirichletbc(ScalarType(0), boundary_dofs_x, V.sub(0))
bcs = [bc, bcx]


As we want the traction $$T$$ over the remaining boundary to be $$0$$, we create a dolfinx.Constant

T = Constant(mesh, ScalarType((0, 0)))


We also want to specify the integration measure $$\mathrm{d}s$$, which should be the integral over the boundary of our domain. We do this by using ufl, and its built in integration measures

ds = Measure("ds", domain=mesh)


## Variational formulation#

We are now ready to create our variational formulation in close to mathematical syntax, as for the previous problems.

def epsilon(u):
def sigma(u):
return lambda_ * nabla_div(u) * Identity(len(u)) + 2*mu*epsilon(u)

u = TrialFunction(V)
v = TestFunction(V)
f = Constant(mesh, ScalarType((0, -rho*g)))
a = inner(sigma(u), epsilon(v)) * dx
L = dot(f, v) * dx + dot(T, v) * ds


## Solve the linear variational problem#

As in the previous demos, we assemble the matrix and right hand side vector and use PETSc to solve our variational problem

problem = LinearProblem(a, L, bcs=bcs, petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
uh = problem.solve()


## Visualization#

pyvista.start_xvfb()
from dolfinx.plot import create_vtk_mesh

# Create plotter and pyvista grid
p = pyvista.Plotter()
topology, cell_types, x = create_vtk_mesh(V)
grid = pyvista.UnstructuredGrid(topology, cell_types, x)

# Attach vector values to grid and warp grid by vector

vals = np.zeros((x.shape[0], 3))
vals[:,:len(uh)] = uh.x.array.reshape((x.shape[0], len(uh)))
grid["u"] = vals