# How to supply boundary condition on a surface of a sphere

Hi,

I am trying to solve Poisson equation with zero source term on a surface of a sphere. My initial starting condition is a constant value at all nodes. Which should give me the same constant value as the solution at all nodes.

Could you please let me know how I can supply constant value at each node as my initial/boundary condition?

Generally, my question is how can I supply boundary condition at all nodes for some closed surface (such as a 2D spherical surface embedded in 3D) which has no boundary?

I am copying snippet of my code below:

============================================
import numpy as np
import math
import os
from scipy import stats
import random
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import glob
import pandas as pd
###########################################################

######################################################################
######## Generating mesh begins ########
######################################################################

from mpi4py import MPI
from dolfinx import mesh
import ufl
import trimesh

# domain = mesh.create_unit_square(MPI.COMM_WORLD, 80, 80, mesh.CellType.quadrilateral)

####------------------------------####
gdim = 3
shape = “triangle”
degree = 1

cell = ufl.Cell(shape, geometric_dimension=gdim)
domain = ufl.Mesh(ufl.VectorElement(“Lagrange”, cell, degree))

x = np.array(meshUsingTrimesh.vertices.tolist())
cells = np.array(meshUsingTrimesh.faces.tolist())
mesh1 = mesh.create_mesh(MPI.COMM_WORLD, cells, x, domain)
####------------------------------####

######################################################################
######## Generating mesh ends ########
######################################################################

######################################################################
############# Interpolating function begins #############
######################################################################

from dolfinx.fem import FunctionSpace
#V = FunctionSpace(domain, (“CG”, 1))
V = FunctionSpace(mesh1, (“CG”, 1))

######################################################################
############# Interpolating function ends #############
######################################################################

######################################################################
############# Boundary conditions begins ############# gmsh tetmesh
######################################################################

from dolfinx import fem
uD = fem.Function(V)
#uD.interpolate(lambda x: 1 + x[0]**2 + 2 * x[1]**2)
uD.interpolate(lambda x: 1 + 0 * x[0] + 0 * x[1] + 0 * x[2])
#uD.interpolate(lambda x: 1 + (x[0]-1)**2)

import numpy

# Create facet to cell connectivity required to determine boundary facets

#tdim = domain.topology.dim ### **************************************************************************
tdim = mesh1.topology.dim
fdim = tdim - 1

# domain.topology.create_connectivity(fdim, tdim)

mesh1.topology.create_connectivity(fdim, tdim)

#boundary_facets = mesh.exterior_facet_indices(domain.topology)
boundary_facets = mesh.exterior_facet_indices(mesh1.topology)

boundary_dofs = fem.locate_dofs_topological(V, fdim, boundary_facets)

#dof_coordinates = V.tabulate_dof_coordinates()

bc = fem.dirichletbc(uD, boundary_dofs)
#bc = fem.dirichletbc(uD, dof_coordinates)

######################################################################
############# Boundary conditions ends #############
######################################################################

In the above I have basically taken the “Poisson equation on a square” code from FEniCSx tutorial and trying to implement it on a surface of a sphere. Now, my question is in the boundary condition, how can I supply a constant value at all nodes to start with? A 2D sphere has no such boundary. Although the above code runs without any error message but gives me infinity as output at all nodal points.

Thanks!

If you are solving it on a spherical shell, you have no exterior facets (no facet is connected to just one cell), thus there is no natural point to apply a Dirichlet condition to. Then, you get into the issue of your problem being singular. See for instance; Poisson problem with Neumann boundary condition - #6 by dokken