Problem with weak form formulation


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5 weeks ago by
Diego  
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


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2 Answers


1
5 weeks ago by
pf4d  
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 5 weeks 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 5 weeks 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 5 weeks ago by Adam Janecka  
Yes, you are correct.
written 5 weeks 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 5 weeks ago by pf4d  
Thanks. 'dv' should indicate volume, while 'da' area -- maybe 'ds' as for surface would be better.
written 5 weeks ago by Adam Janecka  
personally, I prefer the explicit or the use of \(\varGamma = \partial \varOmega\).  
written 5 weeks ago by pf4d  
I'm fine with it (or anything that makes sense) ;-)
written 5 weeks ago by Adam Janecka  
I edited my answer once again after you kindled my obsessive-math-symbol-notation disorder.
written 5 weeks ago by pf4d  
0
4 weeks ago by
Diego  
Thank you for your help!
I will try to implement it soon and let you know

Diego
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