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Arbitrary polyhedral volume

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== Arbitrary Polyhedral Volume ==
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The volume of arbitrary polyhedral can be calculated by using [[Greens theorem | Green-Gauss Theorem]].
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<p>
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The volume of arbitrary polyhedral can be calculated by using Green-Gauss Theorem. <br>
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:<math>\int\limits_\Omega  {div(\vec F)d\Omega  = } \oint\limits_S {\vec F \bullet d\vec S}  
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<math>\int\limits_\Omega  {div(\vec F)d\Omega  = } \oint\limits_S {\vec F \bullet d\vec S}  
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</math>
</math>
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<br>
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By choosing the function<br>
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By choosing the function
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<math>
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:<math>
\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}
\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}
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</math><br>
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</math>
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Where (x,y,z) are centroid of the surface enclosing the volume under consideration.  
Where (x,y,z) are centroid of the surface enclosing the volume under consideration.  
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As we have, <br>
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As we have,
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<math>
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:<math>
div(\vec F) = 1
div(\vec F) = 1
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</math><br>
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</math>
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Hence the volume can be calculated as: <br>
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<math>
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Hence the volume can be calculated as:
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:<math>
volume = \oint\limits_S {\vec F \bullet \hat ndS}
volume = \oint\limits_S {\vec F \bullet \hat ndS}
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</math><br>
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</math>
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where the normal of the surface pointing outwards is given by: <br>
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<math>
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where the normal of the surface pointing outwards is given by:
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:<math>
\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)
\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)
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</math><br>
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</math>
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Final expression could be written as <br>
Final expression could be written as <br>
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<math>
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:<math>
volume = \frac{1}{3}\sum\limits_{faces} {\left[ {\left( {x \times n_x  + y \times n_y  + z \times n_z } \right) \bullet S} \right]}
volume = \frac{1}{3}\sum\limits_{faces} {\left[ {\left( {x \times n_x  + y \times n_y  + z \times n_z } \right) \bullet S} \right]}
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</math><br>
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</math>
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where S is magnitude of Surface Area.
where S is magnitude of Surface Area.
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</p>
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----
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<i> Return to [[Numerical methods | Numerical Methods]] </i>

Latest revision as of 06:18, 3 October 2005

The volume of arbitrary polyhedral can be calculated by using Green-Gauss Theorem.

\int\limits_\Omega  {div(\vec F)d\Omega  = } \oint\limits_S {\vec F \bullet d\vec S}

By choosing the function


\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}

Where (x,y,z) are centroid of the surface enclosing the volume under consideration. As we have,


div(\vec F) = 1

Hence the volume can be calculated as:


volume = \oint\limits_S {\vec F \bullet \hat ndS}

where the normal of the surface pointing outwards is given by:


\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)

Final expression could be written as


volume = \frac{1}{3}\sum\limits_{faces} {\left[ {\left( {x \times n_x  + y \times n_y  + z \times n_z } \right) \bullet S} \right]}

where S is magnitude of Surface Area.



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