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

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(Created page with 'We then define the matrices <center><math> A_{\mu\nu} = \mathrm{Re} \left\{ -\frac{\mathrm{i}}{\omega}\rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} n_{\mu}\, dS \right\…')
 
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<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\{ -\frac{\mathrm{i}}{\omega}\rho\iint_{\partial\Omega_{B}}
−
  \phi_{\nu}^{\mathrm{R}} n_{\mu}\, dS \right\}
+
  \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\}
 
</math></center>
 
</math></center>
 
which is called the added mass matrix and  
 
which is called the added mass matrix and  
Line 8: Line 8:
 
<center><math>
 
<center><math>
 
B_{\mu\nu} = \mathrm{Im} \left\{ \rho\iint_{\partial\Omega_{B}}
 
B_{\mu\nu} = \mathrm{Im} \left\{ \rho\iint_{\partial\Omega_{B}}
−
  \phi_{\nu}^{\mathrm{R}} 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
 
which is called the damping matrix and the forcing vector is
 
<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) n_{\mu}\, dS
+
\left(\phi^{\mathrm{I}} +  \phi^{\mathrm{D}} \right) \mathbf{n}_{\mu}\, dS
 
</math></center>
 
</math></center>

Revision as of 22:26, 28 April 2010

We then define the matrices

[math]\displaystyle{ A_{\mu\nu} = \mathrm{Re} \left\{ -\frac{\mathrm{i}}{\omega}\rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\} }[/math]

which is called the added mass matrix and We then define the matrices

[math]\displaystyle{ B_{\mu\nu} = \mathrm{Im} \left\{ \rho\iint_{\partial\Omega_{B}} \phi_{\nu}^{\mathrm{R}} \mathbf{n}_{\mu}\, dS \right\} }[/math]

which is called the damping matrix and the forcing vector is

[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]
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