Boroomand Mossaiby 2004a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 725107104 to Boroomand Mossaiby 2004a

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Cinmemj moved page Draft Samper 725107104 to Boroomand Mossaiby 2004a

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Cinmemj at 14:31, 14 June 2018

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@@ Line 1,563: / Line 1,563: @@
 <math>f\left({\mu }_1,{\mu }_2\right)=33+\frac{1}{4}\left({\mu }_1^2+\right. </math><math>\left. {\mu }_2^2+{\mu }_1^{-2}+{\mu }_2^{-2}\right)-</math><math>8\left({\mu }_1+{\mu }_2+{\mu }_1^{-1}+{\mu }_2^{-1}\right)-</math><math>\frac{1}{2}\left({\mu }_1{\mu }_2+{\mu }_1^{-1}{\mu }_2+\right. </math><math>\left. {\mu }_1{\mu }_2^{-1}+{\mu }_1^{-1}{\mu }_2^{-1}\right)=</math><math>0</math>
-We evaluate <math display="inline">{\mu }_2</math> in terms of
+We evaluate <math display="inline">{\mu }_2</math> in terms of  <math display="inline">{\mu }_1</math>
 {|
 |-
-| <math display="inline">{\mu }_1</math>
+| <math>{\mu }_2=g_1({\mu }_1)=\frac{1}{2}\left(16+{\mu }_1^{-1}- 8{\mu }_1^{-\frac{1}{2}}-8{\mu }_1^{\frac{1}{2}}+ {\mu }_1-{\mu }_1^{-1}\left(-1+5{\mu }_1^{\frac{1}{2}}- 5{\mu }_1+{\mu }_1^{\frac{3}{2}}\right){\left(1-6{\mu }_1^{\frac{1}{2}}+{\mu }_1\right)}^{\frac{1}{2}}\right)</math>
-| <math display="inline">{\mu }_2=g_1({\mu }_1)=\frac{1}{2}\left(16+{\mu }_1^{-1}-\right. </math><math>\left. 8{\mu }_1^{-\frac{1}{2}}-8{\mu }_1^{\frac{1}{2}}+\right. </math><math>\left. {\mu }_1-{\mu }_1^{-1}\left(-1+5{\mu }_1^{\frac{1}{2}}-\right. \right. </math><math>\left. \left. 5{\mu }_1+{\mu }_1^{\frac{3}{2}}\right){\left(1-6{\mu }_1^{\frac{1}{2}}+{\mu }_1\right)}^{\frac{1}{2}}\right)</math>
 |}
-<math display="inline">{\mu }_2=g_2\left({\mu }_1\right)=\frac{1}{2}\left(16+\right. </math><math>\left. {\mu }_1^{-1}+8{\mu }_1^{-\frac{1}{2}}+8{\mu }_1^{\frac{1}{2}}+\right. </math><math>\left. {\mu }_1-{\mu }_1^{-1}\left(1+5{\mu }_1^{\frac{1}{2}}+\right. \right. </math><math>\left. \left. 5{\mu }_1+{\mu }_1^{\frac{3}{2}}\right){\left(1+6{\mu }_1^{\frac{1}{2}}+{\mu }_1\right)}^{\frac{1}{2}}\right)</math>
+<math>{\mu }_2=g_2\left({\mu }_1\right)=\frac{1}{2}\left(16+ {\mu }_1^{-1}+8{\mu }_1^{-\frac{1}{2}}+8{\mu }_1^{\frac{1}{2}}+{\mu }_1-{\mu }_1^{-1}\left(1+5{\mu }_1^{\frac{1}{2}}+ 5{\mu }_1+{\mu }_1^{\frac{3}{2}}\right){\left(1+6{\mu }_1^{\frac{1}{2}}+{\mu }_1\right)}^{\frac{1}{2}}\right)</math>
 Similar expressions may be derived for  <math display="inline">{\mu }_1</math> in terms of  <math display="inline">{\mu }_2</math> . Generally, such explicit relations are not necessary and what needed is just choosing a value for  <math display="inline">{\mu }_1</math> , as an integration point in the feasible domain, and solving an algebraic equation for its roots. Nevertheless, to give some insight to the problem the explicit relations are given here. Now we define
-−
 {|
 |-
-| <math display="inline">{\left({\mu }_2\right)}_1=h_1({\mu }_1)</math>
+| <math>{\left({\mu }_2\right)}_1=h_1({\mu }_1)</math>
-| <math display="inline">h_1=\left\{g\mbox{ }\vert \mbox{ }\mbox{ }g\in \left\{g_1,g_3\right\}\mbox{ },\mbox{ }\vert g\vert \leq 1\right\}</math>
+| <math>h_1=\left\{g\mbox{ }\vert \mbox{ }\mbox{ }g\in \left\{g_1,g_3\right\}\mbox{ },\mbox{ }\vert g\vert \leq 1\right\}</math>
 |}
+,
 {|
 |-
-| <math display="inline">{\left({\mu }_2\right)}_2=h_2({\mu }_1)</math>
+| <math>{\left({\mu }_2\right)}_2=h_2({\mu }_1)</math>
-| <math display="inline">h_2=\left\{g\mbox{ }\vert \mbox{ }\mbox{ }g\in \left\{g_2,g_4\right\}\mbox{ },\mbox{ }\vert g\vert \leq 1\right\}</math>
+| <math>h_2=\left\{g\mbox{ }\vert \mbox{ }\mbox{ }g\in \left\{g_2,g_4\right\}\mbox{ },\mbox{ }\vert g\vert \leq 1\right\}</math>
 |}

@@ Line 1,537: / Line 1,537: @@
 Apart from assessment of error estimators, the method described above may be used to find a solution to a homogeneous differential equation like following Poisson’s equation
 {|
 |-
-| <math display="inline">{\nabla }^2u=0</math>
+| <math>{\nabla }^2u=0</math>  <math>x\geq 0</math> and    <math>y\geq 0</math>
-| <math display="inline">x\geq 0</math>
 |}
-and    <math display="inline">y\geq 0</math>
 with exact solution, for instance, as

@@ Line 1,389: / Line 1,389: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (85)
 |}
 or

@@ Line 1,399: / Line 1,399: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math> \bar{\bf u}=\sum _{k=1}^{2}\left(\int _{\, 0}^{\, 1}\left[\sum \textbf{v}_{i{\kern 1pt} \, \, (\mu _{k} )}^{k} \left(\textbf{v}_{i{\kern 1pt} \, (\mu _{k} )}^{'k} \right)^{T}  \right]d\mu _{k}  \right) \, \textbf{K}_{bnd}^{T} \textbf{R}'\; \textbf{F}</math>
+| <math> \bar{\bf u}=\sum _{k=1}^{2}\left(\int _{\, 0}^{\, 1}\left[\sum \textbf{v}_{i \, \, (\mu _{k} )}^{k} \left(\textbf{v}_{i \, (\mu _{k} )}^{'k} \right)^{T}  \right]d\mu _{k}  \right) \, \textbf{K}_{bnd}^{T} \textbf{R}'\; \textbf{F}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (86)

@@ Line 1,226: / Line 1,226: @@
 |}
 The rest of the procedure is construction of general asymptotic finite element solution as mentioned before.
@@ Line 1,391: / Line 1,390: @@
 |}
 or
@@ Line 1,399: / Line 1,399: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math> \bar{\bf u}=\sum _{k=1}^{2}\left(\int _{\, 0}^{\, 1}\left[\sum \textbf{v}_{i{\kern 1pt} \, \, (\mu _{k} )}^{k} \left(\textbf{v}'_{i{\kern 1pt} \, (\mu _{k} )}^{\, k} \right)^{T}  \right]d\mu _{k}  \right) \, \textbf{K}_{bnd}^{T} \textbf{R}'\; \textbf{F}</math>
+| <math> \bar{\bf u}=\sum _{k=1}^{2}\left(\int _{\, 0}^{\, 1}\left[\sum \textbf{v}_{i{\kern 1pt} \, \, (\mu _{k} )}^{k} \left(\textbf{v}_{i{\kern 1pt} \, (\mu _{k} )}^{'k} \right)^{T}  \right]d\mu _{k}  \right) \, \textbf{K}_{bnd}^{T} \textbf{R}'\; \textbf{F}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (86)

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