# Defining function space for free boundary problem

**URL:** <https://fenicsproject.discourse.group/t/defining-function-space-for-free-boundary-problem/6194>\
**Category:** mathematics\
**Created:** [July 8, 2021, 2:03pm UTC](https://fenicsproject.discourse.group/t/defining-function-space-for-free-boundary-problem/6194 "2021-07-08T14:03:47Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Niall\_hanevy](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/niall_hanevy/32/2608_2.png) [@Niall\_hanevy](https://fenicsproject.discourse.group/u/Niall_hanevy)\
**Post date:** [July 8, 2021, 2:03pm UTC](https://fenicsproject.discourse.group/t/defining-function-space-for-free-boundary-problem/6194/1 "2021-07-08T14:03:47Z")

</div>

Hi,

I am trying to solve a fiber drawing model with a free boundary.  
After making a change of variables I have the weak form

\int\_{d\eta d\zeta} R^2 \eta q \left( \frac{1}{R \eta} \frac{\partial }{\partial \eta} \left(\eta u \right) + \left[\frac{\partial}{\partial \zeta} - \eta \frac{R'}{R}\frac{\partial}{\partial \eta}\right]w \right) = 0 \\ \int\_{d\eta d\zeta} - \frac{\partial \phi\_1}{\partial \eta} R \eta \left(-p + \frac{2}{R}\frac{\partial u}{\partial \eta} \right) - \textcolor{red}{\epsilon} \left(\eta \frac{\partial }{\partial \zeta} \left(\phi\_1 R^2 \right) + R' R \frac{\partial }{\partial \eta} \left( \phi\_1 \eta^2 \right) \right) \left[\epsilon \left(\frac{\partial u }{\partial \zeta} + \eta \frac{R'}{R}\frac{\partial u }{\partial \eta} \right) + \epsilon^{-1} \frac{1}{R} \frac{\partial w }{\partial \eta} \right] \\ - \phi\_1 R \left( - p + 2 \frac{u}{R\eta }\right) = 0 \\ \int\_{d\eta d\zeta} \frac{\partial \phi\_2}{\partial \eta} \left[\epsilon \left(\frac{\partial u }{\partial \zeta} + \eta \frac{R'}{R}\frac{\partial u }{\partial \eta} \right) + \epsilon^{-1} \frac{1}{R} \frac{\partial w }{\partial \eta} \right] \\ + \textcolor{red}{\epsilon} \left[\eta \frac{\partial}{\partial \zeta} \left(R^2 \phi\_2 \right) + R R' \frac{\partial}{\partial \eta } \left( \eta^2 \phi\_2 \right)\right] \left[- p + 2 \frac{\partial w}{\partial \zeta} - 2 \eta \frac{R'}{R}\frac{\partial w}{\partial \eta} \right] \\ = 0 \\ \int\_{d \eta d \zeta} \frac{\partial R}{\partial \eta} + \int\_{dS} u - w \frac{\partial R}{\partial \eta} = 0

However when I am creating my function space R, the free boundary variable is a vector valued function rather than a scalar function like the pressure.  
`

# Define Taylor–Hood function space W

V = VectorElement(“Lagrange”, triangle, 2)  
Q = FiniteElement(“Lagrange”, triangle, 1)  
S = VectorElement(“Lagrange”, triangle, 1)

W = FunctionSpace(mesh, MixedElement([V, Q, S]))

# Define Function and TestFunction(s)

w = Function(W)  
(u, p, r) = split(w)  
(v, q, s) = split(TestFunction(W))

`

Any ideas what I am doing wrong?

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<div class="post-metadata">

**Author:** ![nate](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/nate/32/17_2.png) [@nate](https://fenicsproject.discourse.group/u/nate)\
**Post date:** [July 8, 2021, 2:07pm UTC](https://fenicsproject.discourse.group/t/defining-function-space-for-free-boundary-problem/6194/2 "2021-07-08T14:07:19Z")

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Do you just mean you want your `s` and `r` functions to be scalar?

If so, simply change:

```auto
S = VectorElement(“Lagrange”, triangle, 1)

```

to

```auto
S = FiniteElement(“Lagrange”, triangle, 1)

```

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<div class="post-metadata">

**Author:** ![Niall\_hanevy](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/niall_hanevy/32/2608_2.png) [@Niall\_hanevy](https://fenicsproject.discourse.group/u/Niall_hanevy)\
**Post date:** [July 8, 2021, 2:09pm UTC](https://fenicsproject.discourse.group/t/defining-function-space-for-free-boundary-problem/6194/3 "2021-07-08T14:09:30Z")

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God I really should have seen that.

cheers nate!