https://www.cfd-online.com/W/index.php?title=Arbitrary_polyhedral_volume&feed=atom&action=history
Arbitrary polyhedral volume - Revision history
2024-03-28T11:56:32Z
Revision history for this page on the wiki
MediaWiki 1.16.5
https://www.cfd-online.com/W/index.php?title=Arbitrary_polyhedral_volume&diff=3084&oldid=prev
Zxaar at 06:18, 3 October 2005
2005-10-03T06:18:13Z
<p></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 06:18, 3 October 2005</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 36:</td>
<td colspan="2" class="diff-lineno">Line 36:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where S is magnitude of Surface Area.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where S is magnitude of Surface Area.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">----</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><i> Return to [[Numerical methods | Numerical Methods]] </i></ins></div></td></tr>
</table>
Zxaar
https://www.cfd-online.com/W/index.php?title=Arbitrary_polyhedral_volume&diff=1774&oldid=prev
Jola at 14:54, 12 September 2005
2005-09-12T14:54:25Z
<p></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 14:54, 12 September 2005</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">== Arbitrary Polyhedral Volume ==</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The volume of arbitrary polyhedral can be calculated by using [[Greens theorem | Green-Gauss Theorem]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The volume of arbitrary polyhedral can be calculated by using [[Greens theorem | Green-Gauss Theorem]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
</table>
Jola
https://www.cfd-online.com/W/index.php?title=Arbitrary_polyhedral_volume&diff=1772&oldid=prev
Jola: Arbitrary Polyhedral Volume moved to Arbitrary polyhedral volume
2005-09-12T14:53:57Z
<p>Arbitrary Polyhedral Volume moved to Arbitrary polyhedral volume</p>
<table style="background-color: white; color:black;">
<tr valign='top'>
<td colspan='1' style="background-color: white; color:black;">← Older revision</td>
<td colspan='1' style="background-color: white; color:black;">Revision as of 14:53, 12 September 2005</td>
</tr></table>
Jola
https://www.cfd-online.com/W/index.php?title=Arbitrary_polyhedral_volume&diff=1770&oldid=prev
Jola at 14:03, 12 September 2005
2005-09-12T14:03:03Z
<p></p>
<table style="background-color: white; color:black;">
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 14:03, 12 September 2005</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Arbitrary Polyhedral Volume ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Arbitrary Polyhedral Volume ==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"><p></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The volume of arbitrary polyhedral can be calculated by using Green-Gauss Theorem. <del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The volume of arbitrary polyhedral can be calculated by using <ins class="diffchange diffchange-inline">[[Greens theorem | </ins>Green-Gauss Theorem<ins class="diffchange diffchange-inline">]]</ins>.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>\int\limits_\Omega {div(\vec F)d\Omega = } \oint\limits_S {\vec F \bullet d\vec S} </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:</ins><math>\int\limits_\Omega {div(\vec F)d\Omega = } \oint\limits_S {\vec F \bullet d\vec S} </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>By choosing the function<del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>By choosing the function</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:</ins><math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math><del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Where (x,y,z) are centroid of the surface enclosing the volume under consideration. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Where (x,y,z) are centroid of the surface enclosing the volume under consideration. </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>As we have, <del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>As we have,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:</ins><math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>div(\vec F) = 1</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>div(\vec F) = 1</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math<del class="diffchange diffchange-inline">><br</del>></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Hence the volume can be calculated as: <del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Hence the volume can be calculated as:</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:</ins><math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>volume = \oint\limits_S {\vec F \bullet \hat ndS}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>volume = \oint\limits_S {\vec F \bullet \hat ndS}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math<del class="diffchange diffchange-inline">><br</del>></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>where the normal of the surface pointing outwards is given by: <del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>where the normal of the surface pointing outwards is given by:</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:</ins><math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math><del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Final expression could be written as <br></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Final expression could be written as <br></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:</ins><math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>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]}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>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]}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math><del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where S is magnitude of Surface Area.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where S is magnitude of Surface Area.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></p></del></div></td><td colspan="2"> </td></tr>
</table>
Jola
https://www.cfd-online.com/W/index.php?title=Arbitrary_polyhedral_volume&diff=1767&oldid=prev
Zxaar at 12:26, 12 September 2005
2005-09-12T12:26:41Z
<p></p>
<p><b>New page</b></p><div>== Arbitrary Polyhedral Volume ==<br />
<p><br />
The volume of arbitrary polyhedral can be calculated by using Green-Gauss Theorem. <br><br />
<math>\int\limits_\Omega {div(\vec F)d\Omega = } \oint\limits_S {\vec F \bullet d\vec S} <br />
</math><br />
<br><br />
By choosing the function<br><br />
<math><br />
\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}<br />
</math><br><br />
Where (x,y,z) are centroid of the surface enclosing the volume under consideration. <br />
As we have, <br><br />
<math><br />
div(\vec F) = 1<br />
</math><br><br />
Hence the volume can be calculated as: <br><br />
<math><br />
volume = \oint\limits_S {\vec F \bullet \hat ndS}<br />
</math><br><br />
where the normal of the surface pointing outwards is given by: <br><br />
<math><br />
\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)<br />
</math><br><br />
Final expression could be written as <br><br />
<math><br />
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]}<br />
</math><br><br />
where S is magnitude of Surface Area.<br />
</p></div>
Zxaar