Problem with weak form formulation

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6 months ago by
Hello everybody,

I have this PDE for the transversal vibration of a tensioned beam with variable axial tension:

EJ*v*uzzz - To*v*uzz  - w*z*v*uzz - w*v*uz + m*v*utt = f(z)*v

where E,J,To,w are constant, z is the space variable, t is the time variable,u(z,t) is the unknown displacement function and v is the test function.

I am trying to integrate by parts but I do not know how to integrate the term:   w*z*v*uzz ?
In particular, how should I handle the integration by parts if I have also the space variable 'z' which multiplies the trial and test function ?

Thank you very much for your help!

Diego

Community: FEniCS Project

1
6 months ago by
Let $\phi(z) = w z v(z)$ and $\varGamma = \partial \varOmega$.  Then

\begin{align*} \int_{\varOmega} \phi \frac{\partial^2 u}{\partial z^2} \mathrm{d}\varOmega &= - \int_{\varOmega} \frac{\partial \phi}{\partial z} \frac{\partial u}{\partial z} \mathrm{d}\varOmega + \int_{\varGamma} \phi \frac{\partial u}{\partial z} n_z \mathrm{d}\varGamma \\ &= - \int_{\varOmega} w \frac{\partial (z v)}{\partial z} \frac{\partial u}{\partial z} \mathrm{d}\varOmega + \int_{\varGamma} w z v \frac{\partial u}{\partial z} n_z \mathrm{d}\varGamma \\ &= - \int_{\varOmega} w \left[ v \frac{\partial z}{\partial z} + z \frac{\partial v}{\partial z} \right] \frac{\partial u}{\partial z} \mathrm{d}\varOmega + \int_{\varGamma} w z v \frac{\partial u}{\partial z} n_z \mathrm{d}\varGamma \\ &= - \int_{\varOmega} w \left[ v + z \frac{\partial v}{\partial z} \right] \frac{\partial u}{\partial z} \mathrm{d}\varOmega + \int_{\varGamma} w z v \frac{\partial u}{\partial z} n_z \mathrm{d}\varGamma, \end{align*}
where $n_z$ is the $z$ component of the normal vector $\underline{n}$.

Thanks to Adam's good eye below.
1
I think there should also be the term
$- w \int_\Omega \frac{\partial u}{\partial z} v \,\mathrm{d} \Omega,$
arising from the derivative of $z$.
written 6 months ago by Adam Janecka
I suppose you integrate by parts this way, using $\phi(z) = w z v$:

\begin{align*} \int_{\Omega} \phi \frac{\partial^2 u}{\partial z^2} d\Omega &= - \int_{\Omega} \frac{\partial \phi}{\partial z} \frac{\partial u}{\partial z} d\Omega + \int_{\partial \Omega} \phi \frac{\partial u}{\partial z} n_z d\partial \Omega \\ &= - \int_{\Omega} w \frac{\partial (z v)}{\partial z} \frac{\partial u}{\partial z} d\Omega + \int_{\partial \Omega} w z v \frac{\partial u}{\partial z} n_z d\partial \Omega \\ &= - \int_{\Omega} w v \frac{\partial z}{\partial z} \frac{\partial u}{\partial z} d\Omega - \int_{\Omega} w z \frac{\partial v}{\partial z} \frac{\partial u}{\partial z} d\Omega + \int_{\partial \Omega} w z v \frac{\partial u}{\partial z} n_z d\partial \Omega \\ &= - \int_{\Omega} w v \frac{\partial u}{\partial z} d\Omega - \int_{\Omega} w z \frac{\partial v}{\partial z} \frac{\partial u}{\partial z} d\Omega + \int_{\partial \Omega} w z v \frac{\partial u}{\partial z} n_z d\partial \Omega, \end{align*}
but...  nevermind, you're right.
written 6 months ago by pf4d
Exactly, omitting $w$,
$\int_{\partial \Omega} z \frac{\partial u}{\partial z} n_z v \,\mathrm{d} a = \int_{\Omega} \frac{\partial}{\partial z} \left( z \frac{\partial u}{\partial z} v \right) \,\mathrm{d} v = \int_\Omega \frac{\partial u}{\partial z} v \,\mathrm{d}v + \int_{\Omega} z \frac{\partial^2 u}{\partial z^2} v \,\mathrm{d} v + \int_{\Omega} z \frac{\partial u}{\partial z} \frac{\partial v}{\partial z} \,\mathrm{d} v.$
I do not see, how you can 'move' the derivative just to the test function a leave $z$ untouched.
written 6 months ago by Adam Janecka
Yes, you are correct.
written 6 months ago by pf4d
I like how you use mathrm on those d's, but I dislike your choice of "dv" and "da"  that is confusing.
written 6 months ago by pf4d
Thanks. 'dv' should indicate volume, while 'da' area -- maybe 'ds' as for surface would be better.
written 6 months ago by Adam Janecka
personally, I prefer the explicit or the use of $\varGamma = \partial \varOmega$.
written 6 months ago by pf4d
I'm fine with it (or anything that makes sense) ;-)
written 6 months ago by Adam Janecka
I edited my answer once again after you kindled my obsessive-math-symbol-notation disorder.
written 6 months ago by pf4d
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6 months ago by