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

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We then define the matrices
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We define the added mass matrix by
 
<center><math>
 
<center><math>
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A_{\mu\nu} = \mathrm{Re} \left\{ -\frac{\mathrm{i}}{\omega}\rho\iint_{\partial\Omega_{B}}
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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>
 
</math></center>
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which is called the added mass matrix and
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and the damping matrix by
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We then define the matrices
 
 
<center><math>
 
<center><math>
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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>
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which is called the damping matrix and the forcing vector is
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and the forcing vector by
 
<center><math>
 
<center><math>
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f_{\mu} = -\mathrm{i}\omega\rho\iint_{\partial\Omega_{B}}
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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>

Latest revision as of 12:57, 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\} }[/math]

and the damping matrix by

[math]\displaystyle{ 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]
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