### Need help with a model

Hi!

I'm a MSc-student at University of Bergen (Norway) and I'm writing my thesis with use of FEniCS. I'm using it to study PDEs for fluid flow in poreous media.

My mesh is 2D, with (0, 1) x (0, 0.1) with 100 cells in the x-direction and 10 cells in the y-direction. As this is a time-dependent problem, I'm first using an initial condition to solve Darcy's equation with an initial condition for pressure and viscosity (dependent of concentration) and a fixed permeability tensor ((2,0),(0,1)). Darcy's equation:

\[

\begin{equation}

u = - K \nabla p

\end{equation}

\]

After the flux is calculated from this, the flux is used in convection–diffusion equation:

\[

\begin{equation}

\frac{\delta c}{\delta t} - \nabla \cdot \left( D \nabla c \right) + \nabla \cdot \left( u c \right) = f_c

\end{equation}

\]

Here \( p \) is pressure, \( u \) is flux (vector), \( K \) permability tensor and \( c \) consentration (number between 0 and 1, defined in each cell). For these calculations, CG-3 elements is used for the pressure, DRT-2 elements for flux and P-2 elements for consentration.

After these calculations is done, everything looks okey and the flow seams to be correct compared to my constructed solutions. But for our use, I want to project this 2D solution onto a 1D plot, where the y-axis is an averaged value for each x-point. Is there any examples on this or does anyone know how to do this most efficiently?

### 2 Answers

`p = f(x,y)`

Otherwise, you can perform the integration across the domain numerically by solving a variational problem :

\[

v(x,y) = \int_a^x u(s,y) ds = U(x,y) - U(a,y)

\]

where \(U\) is the anti-derivative of \(v\):

\[

\frac{\partial U}{\partial x} = u(x,y).

\]

The integral at the start \(x = a\) will be zero, and so \(v(x,y) = U(x,y)\) and the BVP is therefore

\[

\frac{\partial v}{\partial x} = u(x,y), \hspace{5mm} v(a,y) = 0.

\]

with associated variational problem consisting of finding \(v\) such that

\[

\int_{\Omega} \frac{\partial v}{\partial x} \phi \mathrm{d}\Omega = \int_{\Omega} u \phi \mathrm{d}\Omega

\]

for all test functions \(\phi\).

An example in one dimension with \(u = \cos(x)\), analytical solution \(v = \sin(x)\) :

```
from fenics import *
mesh = IntervalMesh(1000,0,2*pi)
Q = FunctionSpace(mesh, 'CG', 1)
x = SpatialCoordinate(mesh)[0]
t = mesh.coordinates()[:,0]
u = interpolate(Expression('cos(x[0])', degree=2), Q)
s = interpolate(Expression('sin(x[0])', degree=2), Q)
v = TrialFunction(Q)
phi = TestFunction(Q)
def left(x, on_boundary):
return on_boundary and abs(x[0]) < 1e-14
# integral is zero on the left
bcs = DirichletBC(Q, 0.0, left)
a = v.dx(0) * phi * dx
L = u * phi * dx
v = Function(Q)
solve(a == L, v, bcs)
uf = u.vector().array()[::-1]
vf = v.vector().array()[::-1]
sf = s.vector().array()[::-1]
r = vf - sf
#===============================================================================
# plot :
from pylab import *
from mpl_toolkits.axes_grid1.inset_locator import zoomed_inset_axes
from mpl_toolkits.axes_grid1.inset_locator import mark_inset
mpl.rcParams['font.family'] = 'serif'
mpl.rcParams['text.usetex'] = True
mpl.rcParams['text.latex.preamble'] = ['\usepackage{fouriernc}']
fig = figure(figsize=(5,3.5))
ax = fig.add_subplot(111)
ax.plot(t, uf, 'k', lw=2.0, label=r"$u$")
ax.plot(t, vf, 'r--', lw=2.0, label=r"$v$")
ax.plot(t, r, '#880cbc', lw=2.0, label=r"$\epsilon$")
ax.set_xlim(0,2*pi)
leg = ax.legend(loc='upper center', fontsize='medium')
leg.get_frame().set_alpha(0.0)
ax.set_xlabel(r'$x$')
ax.grid()
tight_layout()
savefig("integration.pdf")
```

This is easily translatable to 2- or 3-dimensional space.

You want the average, so you'll have to divide the integral afterwards by the width of the dimension you integrated -- which may vary with the other dimension -- depending on your mesh.**edit:** fixed the script above and description of example problem solved.

I use trig functions so that I can verify the numerical integration.

In 3D, why do you think a tensor function space should be used, and for what? I do not follow.