Want to define some value on interior nodes

General
1

Hello everyone

If I marked all interior nodes with point 1 using syntax

from dolfin import *
import numpy as np

mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, "Lagrange", 1)

u = TrialFunction(V)
v = TestFunction(V)
NodesMarker = MeshFunction('size_t', mesh, 0)
NodesMarker.set_all(0) 

a = inner(grad(u), grad(v)) * dx 
L = inner(f, v) * dx

Now I want to assign u = 10 on all interior nodes(i.e. to point 0). How I can do this thing.
There is any way to define in weak formulation or we can do in any other way.

2

You can assign this directly to your function using vertex_to_dof_map

from dolfin import *
import numpy as np

mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, "Lagrange", 1)

class MarkedArea(SubDomain):
    def inside(self, x, on_boundary):
        return x[0] < 0.5
NodesMarker = MeshFunction('size_t', mesh, 0)
NodesMarker.set_all(0) 
MarkedArea().mark(NodesMarker, 1) 

v_to_d = vertex_to_dof_map(V)
dofs = v_to_d[np.flatnonzero(NodesMarker.array()==1)]
values = np.full_like(dofs, 10, dtype=np.float64)
uh = Function(V)
uh.vector().vec().setValuesLocal(dofs, values) 
uh.vector().update_ghost_values()
print(assemble(uh*dx)) 
File("u.pvd") << uh
3 
from dolfin import *
import numpy as np

# define mesh
N = 1
mesh = UnitSquareMesh(N, N)

V = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
R = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
me = MixedElement([V, R])
W = FunctionSpace(mesh, me)

(u, lmda) = TrialFunctions(W)
(v, tau) = TestFunctions(W)
g = Constant(0)

# Here i marked these two set of nodes with some condition
ActiveNodes = NodesMarker.where_equal(1)
InActiveNodes = NodesMarker.where_equal(2)

a = inner(grad(u), grad(v)) * dx - lmda * v * dx 
L = inner(f, v) * dx

Now i have to impose u = g on ActiveNodes and lmda = 0 on InActiveNodes. I have to impose in such a way so that i can change set of active nodes and Inactive nodes in a iterative process.
Please suggest something. Hope to hear from you soon. THere is any way to do these things via weak formulation.
shall we impose something in a weak formulation using dS(1) but it doesn’t work. suggest something in such a way.

4

You could impose this weakly using Nitsche’s method for weak enforcement of boundary condition, i.e.
create a g where you assign the values on the boundary, then integrate over your boundary ds with this value. The variational formulation for Nitsche’s method has been shown for instance: Weak imposition of Dirichlet conditions for the Poisson problem — FEniCSx tutorial

1 Like
5

Sir, i have to impose u=g on interior nodes not on boundary nodes…dS is used for interior edges

What we can use in our weak formulation in order to impose u=g on interior nodes.

6

Wouldn’t his mean that your problem can be split into two sections:
u=\begin{cases} u=g \text{ on } \Omega_0\\ -\Delta u = f \text{ on } \Omega_1\\ u=g \text{ on } \partial\Omega_0\cap\partial\Omega_1 \end{cases}
Given that you mark all cells that satisfies being in \Omega_0 with 0, all cells in \Omega_1 with one, you can write:

while some_condition:
  # Update cell marker based on some criterion
  dx = Measure("dx", domain=mesh, subdomain_data=cell_marker)
  a = inner(u, v)*dx(0) + inner(grad(u), grad(v))*dx(1)
  L = inner(g, v)*dx(0) + inner(f, v)*dx(1)
  solve(a==L, uh)
1 Like
7

sir, you didn’t get my points May be there is a case in which two nodes of a cell belongs to Ω0 and one node in Ω1.
so, i have to impose condition on a node.
Here i foam two points 1 and 2, some nodes are in point 1 and some nodes in point 2 i.e. these are discrete points.
On the nodes which are in point 1, I have to impose u = g there.
Tell me if i will do like that given below, Is it works?

from dolfin import *
import numpy as np

# define mesh
N = 1
mesh = UnitSquareMesh(N, N)

V = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
R = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
me = MixedElement([V, R])
W = FunctionSpace(mesh, me)

(u, lmda) = TrialFunctions(W)
(v, tau) = TestFunctions(W)
g = Constant(0)

# Here i marked these two set of nodes with some condition
ActiveNodes = NodesMarker.where_equal(1)
InActiveNodes = NodesMarker.where_equal(2)

a = inner(grad(u), grad(v)) * dx - lmda * v * dx 
L = inner(f, v) * dx 
v_to_d = vertex_to_dof_map(W)
dofs1 = v_to_d[np.flatnonzero(ActiveNodes)]
values1 = np.full_like(dofs1, g2 , dtype=np.float64)
W.sub(0).vector().vec().setValuesLocal(dofs1, values1) 
W.sub(0).vector().update_ghost_values()
dofs2 = v_to_d[np.flatnonzero(InActiveNodes)]
values2 = np.full_like(dofs2, Constant(0) , dtype=np.float64)
W.sub(1).vector().vec().setValuesLocal(dofs2, values2) 
W.sub(1).vector().update_ghost_values()
solve(a == L, w,bc)
8

How about something along the lines of:

from IPython import embed
from dolfin import *
import numpy as np

# define mesh
N = 23
mesh = UnitSquareMesh(N, N)

V = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
R = FiniteElement("Real", mesh.ufl_cell(), 0)
me = MixedElement([V, R])
W = FunctionSpace(mesh, me)

(u, lmda) = TrialFunctions(W)
(v, tau) = TestFunctions(W)
g = Constant(-0.01)
x = SpatialCoordinate(mesh)
f = sin(x[0])+3*cos(x[1])


class MarkedArea(SubDomain):
    def inside(self, x, on_boundary):
        return x[0] < 0.5


a = inner(grad(u), grad(v)) * dx - lmda * v * dx + u*tau*dx
L = inner(f, v) * dx
bc = DirichletBC(W.sub(0), g, MarkedArea(), method="pointwise")
w = Function(W)

solve(a == L, w, bc)

u, r = w.split(deepcopy=True)
File("w.pvd").write(u)

u
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