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<h1>Aspect Ratio</h1>\r
\r
<p><img src="../i_blue.jpg" x-maintain-ratio="TRUE" width="30px" height="30px" border="0" class="img_whs1"> The <span style="font-weight: bold;"><B>Aspect Ratio</B></span> \r
- quality criterion for mesh elements is defined as follows:</p>\r
+ quality criterion for mesh elements reveals the degree of conformity of \r
+ a mesh element to the regular element of its type (with all edges having \r
+ the same length). The \r
+ criterion can be defined as </p>\r
+\r
+<p class="whs2"><img src="../image142.gif" width="120px" height="56px" border="0" class="img_whs3">where <span style="font-weight: bold; font-style: italic;"><I><B>К</B></I></span> \r
+ is the considered element of the mesh, <span style="font-weight: bold; font-style: italic;"><I><B>h</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>max</B></I></span> is the diameter of the element \r
+ К (i.e. the length of the longest edge), <span style="font-weight: bold; font-style: italic;"><I><B>р</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>к</B></I></span> is the radius of the sphere \r
+ inscribed in the element K, and <span style="font-weight: bold; font-style: italic;"><I><B>a</B></I></span> \r
+ is a normalisation value chosen so that <span style="font-weight: bold; font-style: italic;"><I><B>Q</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>K</B></I></span> = 1 when К is an equilateral \r
+ simplex. </p>\r
+\r
+<p class="whs2"> </p>\r
+\r
+<p>The value of <span style="font-weight: bold; font-style: italic;"><I><B>Q</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>K</B></I></span> varies between 1 and +oo and the \r
+ closer it is to 1, the better the element К is. </p>\r
+\r
+<ul type="disc" class="whs4">\r
+ \r
+ <li class=kadov-p><p>The <span style="font-weight: bold;"><B>Aspect Ratio</B></span> \r
+ of a <span style="font-weight: bold;"><B>triangle</B></span> 2D element consisting \r
+ of 3 nodes is calculated by the formula:</p></li>\r
+</ul>\r
\r
<p> </p>\r
\r
-<ul type="disc" class="whs2">\r
- \r
- <li class=kadov-p><p>For a <span style="font-weight: bold;"><B>triangle</B></span> \r
- 2D element consisting of 3 nodes the <span style="font-weight: bold;"><B>Aspect \r
- Ratio</B></span> is calculated by formula:\r
-<p> </p>\r
+<p class="whs2"><img src="../pics/image21.gif" x-maintain-ratio="TRUE" width="158px" height="110px" border="0" class="img_whs5"></p>\r
\r
-<p><img src="../image21.gif" width="158px" height="110px" border="0" class="img_whs3"></p>\r
+<p class="whs2"><img src="../pics/image23.gif" x-maintain-ratio="TRUE" width="120px" height="48px" border="0" class="img_whs6"> ,</p>\r
\r
-<p><img src="../image23.gif" width="120px" height="48px" border="0" class="img_whs4"></p>\r
+<p class="whs2"> </p>\r
\r
-<p>where :</p>\r
+<p class="whs2">where <span style="font-weight: bold; font-style: italic;"><I><B>р</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>к</B></I></span> is the half perimeter of the \r
+ element K and <span style="font-weight: bold; font-style: italic;"><I><B>S</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>K</B></I></span> is its surface.</p>\r
\r
-<p> </p>\r
+<p class="whs2"> </p>\r
\r
-<p><img src="../image27.gif" width="234px" height="96px" border="0" class="img_whs5"></p>\r
+<p class="whs2"><img src="../pics/image27.gif" x-maintain-ratio="TRUE" width="234px" height="96px" border="0" class="img_whs7"></p>\r
\r
-<p> </p>\r
+<p class="whs8"> </p>\r
\r
-<p> </p>\r
-</li>\r
-\r
- <li class=kadov-p><p class="whs5">For a <span style="font-weight: bold;"><B>quadrangle</B></span> \r
- 2D element consisting of 4 nodes the \r
- <span style="font-weight: bold;"><B>Aspect Ratio </B></span>is the\r
- quality of the worst triange which can be built taking three nodes of\r
- the quadrangle. There are four triangles to consider:</p></li>\r
+<ul type="disc" class="whs4">\r
+ \r
+ <li class=kadov-p><p class="whs8">The <span style="font-weight: bold;"><B>Aspect \r
+ Ratio</B></span> of a <span style="font-weight: bold;"><B>quadrangle</B></span> 2D \r
+ element consisting of 4 nodes is the worst (i.e. the greatest) value from \r
+ all triangles which can be built taking three nodes of the quadrangle. \r
+ There are four triangles to consider: </p></li>\r
+</ul>\r
\r
-<p><img src="../image138.gif" width="566px" border="0" class="img_whs6"></p>\r
+<p class="whs9"><img src="../pics/image138.gif" x-maintain-ratio="TRUE" width="622px" height="412px" border="0" class="img_whs10"></p>\r
\r
-</ul>\r
+<p class="whs9">Note, that the Aspect Ratio criterion is \r
+ applicable to 2D mesh elements only. For 3D elements we use the <a href="../aspect_ratio_3d.htm">Aspect \r
+ Ratio 3D</a> criterion. </p>\r
\r
+<p class="whs11"> </p>\r
\r
<p class=TODO>To apply the Aspect Ratio quality criterion to your mesh:</p>\r
\r
<p class=TODO> </p>\r
\r
-<p class="whs7">1. Display your mesh in the viewer.</p>\r
+<p class="whs2">1. Display your mesh in the viewer.</p>\r
\r
-<p class="whs7"> </p>\r
+<p class="whs2"> </p>\r
\r
-<p class="whs7">2. Choose <span style="font-weight: bold;"><B>Controls \r
- > Aspect Ratio </B></span>or click <img src="../image37.gif" width="25px" height="24px" border="0" class="img_whs8"> button in the toolbar. \r
+<p class="whs2">2. Choose <span style="font-weight: bold;"><B>Controls \r
+ > Aspect Ratio </B></span>or click <img src="../image37.gif" width="25px" height="24px" border="0" class="img_whs12"> button in the toolbar. \r
Your mesh will be displayed in the viewer with its elements colored according \r
to the applied mesh quality control criterion:</p>\r
\r
-<p class="whs7"> </p>\r
+<p class="whs2"> </p>\r
\r
-<p class="whs7"><img src="../image94.jpg" width="439px" height="313px" border="0" class="img_whs9"></p>\r
+<p class="whs2"><img src="../image94.jpg" width="439px" height="313px" border="0" class="img_whs13"></p>\r
\r
-<p class="whs7"> </p>\r
+<p class="whs2"> </p>\r
\r
-<p class="whs5"><span style="font-weight: bold;"><B>See Also</B></span> \r
+<p class="whs8"><span style="font-weight: bold;"><B>See Also</B></span> \r
a sample TUI Script of an <a href="../quality_controls.htm#bookmark8">Aspect \r
Ratio quality control</a> operation. </p>\r
\r
