# Modelling with Boundary Conditions¶

We use the preceding example (Poisson equation on the unit square) but want to specify different boundary conditions on the four sides.

Again, we first import numpy and pygimli, the solver and post processing functionality.

import numpy as np
import pygimli as pg

from pygimli.solver import solve
from pygimli.viewer import show
from pygimli.mplviewer import drawStreams


We create 21 x 21 node grid to solve on.

grid = pg.createGrid(x=np.linspace(-1.0, 1.0, 21),
y=np.linspace(-1.0, 1.0, 21))


We start considering inhomogeneous Dirichlet boundary conditions (BC).

There are different ways of specifying BCs. They can be maps from markers to values, explicit functions or implicit (lambda) functions.

The boundary 1 (top) and 2 (left) are directly mapped to the values 1 and 2. On side 3 (bottom) a lambda function 3+x is used (p is the boundary position and p its x coordinate. On side 4 (right) a function uDirichlet is used that simply returns 4 in this example but can compute anything as a function of the individual boundaries b.

Short test: setting single node dirichlet BC

u = solve(grid, f=1., bc={'Dirichlet': [grid.node(2), 0.]})

ax, _ = pg.show(grid, u, label='Solution $u$',)
show(grid, ax=ax)

def uDirichlet(boundary):
"""Return a solution value for coordinate p."""
return 4.0

dirichletBC = [[1, 1.0],                                     # left
[grid.findBoundaryByMarker(2), 2.0],          # right
[grid.findBoundaryByMarker(3),
lambda boundary: 3.0 + boundary.center()], # top
[grid.findBoundaryByMarker(4), uDirichlet]]   # bottom The BC are passed using the bc keyword dictionary. Note that showMesh returns the created figure ax ax while drawMesh plots on it and it can also be used as a class with plotting or decoration methods.

u = solve(grid, f=1., bc={'Dirichlet': dirichletBC})

ax = show(grid, data=u, colorBar=True,
orientation='vertical', label='Solution $u$',
levels=np.linspace(1.0, 4.0, 17), hold=1)

show(grid, ax=ax)

ax.text(0, 1.01, '$u=3+x$', ha='center')
ax.text(-1.01, 0, '$u=1$', va='center', ha='right', rotation='vertical')
ax.text(0, -1.01, '$u=4$', ha='center', va='top')
ax.text(1.02, 0, '$u=2$', va='center', ha='left',  rotation='vertical')

ax.set_title('$\\nabla\cdot(1\\nabla u)=1$')

ax.set_xlim([-1.1, 1.1])  # some boundary for the text
ax.set_ylim([-1.1, 1.1]) Alternatively we can define the gradients of the solution on the boundary, i.e., Neumann type BC. This is done with another map (marker, value) and passed by bc Dictionary.

neumannBC = [[1, -0.5],  # left
[grid.findBoundaryByMarker(4), 2.5]]  # bottom

dirichletBC = [3, 1.0]  # top

u = solve(grid, f=0., bc={'Dirichlet': dirichletBC, 'Neumann': neumannBC})


Note that on boundary 4 (right) no BC is explicitly applied leading to default (natural) BC that are of homogeneous Neumann type $$\frac{\partial u}{\partial n}=0$$

ax = show(grid, data=u, filled=True, colorBar=True,
orientation='vertical', label='Solution $u$',
levels=np.linspace(min(u), max(u), 14), hold=True)

# the solution (i.e., the flow direction).

drawStreams(ax, grid, u)

ax.text(0.0, 1.01, '$u=1$',
horizontalalignment='center')  # top -- 3
ax.text(-1.0, 0.0, '$\partial u/\partial n=-0.5$',
va='center', ha='right', rotation='vertical')  # left -- 1
ax.text(0.0, -1.01, '$\partial u/\partial n=2.5$',
ha='center', va='top')  # bot -- 4
ax.text(1.01, 0.0, '$\partial u/\partial n=0$',
va='center', ha='left', rotation='vertical')  # right -- 2

ax.set_title('$\\nabla\cdot(1\\nabla u)=0$')

ax.set_xlim([-1.1, 1.1])
ax.set_ylim([-1.1, 1.1])

pg.wait() Gallery generated by Sphinx-Gallery

Created using Bootstrap, Sphinx and pyGIMLi 1.0.12+15.g7e48e344 on Dec 31, 2019.