Difference between revisions of "Template:Added mass damping and force matrices definition"

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We then define the matrices
+
We define the added mass matrix by
 
<center><math>
 
<center><math>
A_{\mu\nu} = \mathrm{Re} \left\{ -\frac{\mathrm{i}}{\omega}\rho\iint_{\partial\Omega_{B}}
+
A_{\mu\nu} = \mathrm{Re} \left\{\rho\iint_{\partial\Omega_{B}}
 
  \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\}
 
  \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\}
</math></center>
+
and the damping matrix by
which is called the added mass matrix and
 
We then define the matrices
 
 
<center><math>
 
<center><math>
B_{\mu\nu} = \mathrm{Im} \left\{ \rho\iint_{\partial\Omega_{B}}
+
B_{\mu\nu} = \mathrm{Im} \left\{ \omega \rho\iint_{\partial\Omega_{B}}
 
  \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\}
 
  \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\}
 
</math></center>
 
</math></center>
which is called the damping matrix and the forcing vector is
+
and the forcing vector by
 
<center><math>
 
<center><math>
 
f_{\mu} = \mathrm{i}\omega\rho\iint_{\partial\Omega_{B}}
 
f_{\mu} = \mathrm{i}\omega\rho\iint_{\partial\Omega_{B}}
 
\left(\phi^{\mathrm{I}} +  \phi^{\mathrm{D}} \right) \mathbf{n}_{\mu}\, dS
 
\left(\phi^{\mathrm{I}} +  \phi^{\mathrm{D}} \right) \mathbf{n}_{\mu}\, dS
 
</math></center>
 
</math></center>

Revision as of 12:56, 26 April 2011

We define the added mass matrix by

[math]\displaystyle{ A_{\mu\nu} = \mathrm{Re} \left\{\rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\} and the damping matrix by \lt center\gt \lt math\gt B_{\mu\nu} = \mathrm{Im} \left\{ \omega \rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\} }[/math]

and the forcing vector by

[math]\displaystyle{ f_{\mu} = \mathrm{i}\omega\rho\iint_{\partial\Omega_{B}} \left(\phi^{\mathrm{I}} + \phi^{\mathrm{D}} \right) \mathbf{n}_{\mu}\, dS }[/math]