Boroomand Mossaiby 2004b - Revision history

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Cinmemj at 11:17, 25 June 2019

2019-06-25T11:17:33Z

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 It may be noticed that existence of superconvergence is independent of the average of minimum angles since there are some cases in which the average minimum angle are quite small while the results are still superconvergent.   This means that, regarding error estimation part in an adaptive procedure, a mesh generation program should face more limitations with respect to mesh pattern.  On the other hand, if the results of recovery methods are taken as approximate gradients, the restrictions regarding minimum angle, i.e. skewness of elements, may be relaxed.
-Since we have so far studied structured mesh configurations, a question regarding the structure of the output mesh from a 3D mesh generation program, structured/unstructured mesh, may now be raised. Obviously, it is not possible to consider all possible unstructured mesh configurations and thus the answer needs an engineering judgment for generalization of the present results to unstructured meshes though the need of more studies, in this regard, is felt. Nevertheless, in this study, we examine the performance of the recovery methods in some artificially produced unstructured patterns.  For this we select the pattern of RRR and change the location of central node to produce skew elements inside the patch.   The results obtained, for some specific locations of the inside node with respect to the center of the patch are shown in Figure 14.  In this figure the patch is considered to be at interior parts of a fictitious domain. It can be seen that the robustness values are still in an acceptable range when the mesh becomes very skew though SPR exhibits better robustness for highly distorted meshes.
+Since we have so far studied structured mesh configurations, a question regarding the structure of the output mesh from a 3D mesh generation program, structured/unstructured mesh, may now be raised. Obviously, it is not possible to consider all possible unstructured mesh configurations and thus the answer needs an engineering judgment for generalization of the present results to unstructured meshes though the need of more studies, in this regard, is felt. Nevertheless, in this study, we examine the performance of the recovery methods in some artificially produced unstructured patterns.  For this we select the pattern of RRR and change the location of central node to produce skew elements inside the patch.   The results obtained, for some specific locations of the inside node with respect to the center of the patch are shown in Figure [[#img-14|14]].  In this figure the patch is considered to be at interior parts of a fictitious domain. It can be seen that the robustness values are still in an acceptable range when the mesh becomes very skew though SPR exhibits better robustness for highly distorted meshes.
 <div id='img-14'></div>

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 ====Enhanced gradients at boundaries====
-Because in this paper assessment of the recoveries at boundaries of the domain is aimed, it is necessary to specify the way that nodal values of recovered gradients are calculated.  The reader may note that there are several ways for gradient calculation at boundaries.  An efficient way which was suggested in <span id="citeF-23"></span>[[#cite-23|[23]],<span id="citeF-24"></span>[[#cite-24|24]]] is using the information from patches nearby the boundary node.  This means that one can directly use the smooth field obtained in ([[#eq-2|2]]) for patches near boundary and substitute the coordinates (or local normalized coordinates) of the boundary node in the polynomial terms.  Since a node at the boundary may generally fall in several patches, especially in three dimensional domains, it is preferred to obtain the information from patch of closest internal nodal point.    However, this makes some ambiguity when unstructured meshes are used since the distances between a boundary node and the nodes around it may not be so different.  Therefore, here, we employ an averaging approach.
+Because in this paper assessment of the recoveries at boundaries of the domain is aimed, it is necessary to specify the way that nodal values of recovered gradients are calculated.  The reader may note that there are several ways for gradient calculation at boundaries.  An efficient way which was suggested in <span id="citeF-23"></span>[[#cite-23|[23]],<span id="citeF-24"></span>[[#cite-24|24]]] is using the information from patches nearby the boundary node.  This means that one can directly use the smooth field obtained in (2) for patches near boundary and substitute the coordinates (or local normalized coordinates) of the boundary node in the polynomial terms.  Since a node at the boundary may generally fall in several patches, especially in three dimensional domains, it is preferred to obtain the information from patch of closest internal nodal point.    However, this makes some ambiguity when unstructured meshes are used since the distances between a boundary node and the nodes around it may not be so different.  Therefore, here, we employ an averaging approach.
 ==4. TEST RESULTS==

← Older revision		Revision as of 11:30, 25 June 2019
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−	In above subscript <math display="inline">p</math> denotes patch-wise view of notations, <math display="inline">{\sigma }^\ast </math> represent the matrix of stresses/gradients each of its components is expressed as (~~[[#eq-~~2~~\|2]]~~) and <math display="inline">\boldsymbol{F}_p</math> is a vector of forces induced by finite element stress/gradient field maintaining the patch in equilibrium state. The procedure resembles to the solution of a system, as small as a patch, using a set of piecewise continuous weight functions and a set of generalized coordinates as the polynomial terms. The formulation can be written for strain or even displacement field.	+	In above subscript <math display="inline">p</math> denotes patch-wise view of notations, <math display="inline">{\sigma }^\ast </math> represent the matrix of stresses/gradients each of its components is expressed as (2) and <math display="inline">\boldsymbol{F}_p</math> is a vector of forces induced by finite element stress/gradient field maintaining the patch in equilibrium state. The procedure resembles to the solution of a system, as small as a patch, using a set of piecewise continuous weight functions and a set of generalized coordinates as the polynomial terms. The formulation can be written for strain or even displacement field.

	The original form of REP, proposed in <span id="citeF-15"></span>[[#cite-15\|[15]]], directly employs Equation ([[#eq-5\|5]]) to construct enhanced field of gradients in a coupled form, while the improved version uses spectral form of ([[#eq-5\|5]]) to construct the recovered gradient field for each gradient component separately.		The original form of REP, proposed in <span id="citeF-15"></span>[[#cite-15\|[15]]], directly employs Equation ([[#eq-5\|5]]) to construct enhanced field of gradients in a coupled form, while the improved version uses spectral form of ([[#eq-5\|5]]) to construct the recovered gradient field for each gradient component separately.

@@ Line 741: / Line 741: @@
 ====Superconvergent Patch Recovery (SPR)====
-One of the most robust recovery methods was proposed by Zienkiewicz and Zhu in 1992 <span id="citeF-9"></span>[[#cite-9|[9]].  Although at the time of its proposal some benchmark problems were used to demonstrate its performance, later comprehensive studies by Babuška and co-workers <span id="citeF-9"></span>[[#cite-9|[9]]-<span id="citeF-11"></span>[[#cite-11|11]]] proved its superiority over the other methods in terms of robustness and asymptotic convergence rate in two dimensional problems. It is noteworthy to mention that the studies in <span id="citeF-9"></span>[[#cite-9|[9]]-<span id="citeF-11"></span>[[#cite-11|11]]] show that none of the augmented forms of SPR, as some researches suggest <span id="citeF-6"></span>[[#cite-6|[6]]-<span id="citeF-8"></span>[[#cite-8|8]]] can compete with the original formulation.  In this paper we shall perform a comprehensive study on the performance of SPR when applied to three dimensional problems.
+One of the most robust recovery methods was proposed by Zienkiewicz and Zhu in 1992 <span id="citeF-4"></span>[[#cite-4|[4]]].  Although at the time of its proposal some benchmark problems were used to demonstrate its performance, later comprehensive studies by Babuška and co-workers <span id="citeF-9"></span>[[#cite-9|[9]]-<span id="citeF-11"></span>[[#cite-11|11]]] proved its superiority over the other methods in terms of robustness and asymptotic convergence rate in two dimensional problems. It is noteworthy to mention that the studies in <span id="citeF-9"></span>[[#cite-9|[9]]-<span id="citeF-11"></span>[[#cite-11|11]]] show that none of the augmented forms of SPR, as some researches suggest <span id="citeF-6"></span>[[#cite-6|[6]]-<span id="citeF-8"></span>[[#cite-8|8]]] can compete with the original formulation.  In this paper we shall perform a comprehensive study on the performance of SPR when applied to three dimensional problems.
 The procedure of SPR may be found in many texts and papers <span id="citeF-4"></span>[[#cite-4|[4]],<span id="citeF-5"></span>[[#cite-5|5]],<span id="citeF-21"></span>[[#cite-21|21]]-<span id="citeF-24"></span>[[#cite-24|24]]].  The procedure starts with assuming a continuous gradient field, in the form of a polynomial, over a patch of elements built on a vertex node

@@ Line 741: / Line 741: @@
 ====Superconvergent Patch Recovery (SPR)====
-One of the most robust recovery methods was proposed by Zienkiewicz and Zhu in 1992.  Although at the time of its proposal some benchmark problems were used to demonstrate its performance, later comprehensive studies by Babuška and co-workers <span id="citeF-9"></span>[[#cite-9|[9]]-<span id="citeF-11"></span>[[#cite-11|11]]] proved its superiority over the other methods in terms of robustness and asymptotic convergence rate in two dimensional problems. It is noteworthy to mention that the studies in <span id="citeF-9"></span>[[#cite-9|[9]]-<span id="citeF-11"></span>[[#cite-11|11]]] show that none of the augmented forms of SPR, as some researches suggest <span id="citeF-6"></span>[[#cite-6|[6]]-<span id="citeF-8"></span>[[#cite-8|8]]] can compete with the original formulation.  In this paper we shall perform a comprehensive study on the performance of SPR when applied to three dimensional problems.
+One of the most robust recovery methods was proposed by Zienkiewicz and Zhu in 1992 <span id="citeF-9"></span>[[#cite-9|[9]].  Although at the time of its proposal some benchmark problems were used to demonstrate its performance, later comprehensive studies by Babuška and co-workers <span id="citeF-9"></span>[[#cite-9|[9]]-<span id="citeF-11"></span>[[#cite-11|11]]] proved its superiority over the other methods in terms of robustness and asymptotic convergence rate in two dimensional problems. It is noteworthy to mention that the studies in <span id="citeF-9"></span>[[#cite-9|[9]]-<span id="citeF-11"></span>[[#cite-11|11]]] show that none of the augmented forms of SPR, as some researches suggest <span id="citeF-6"></span>[[#cite-6|[6]]-<span id="citeF-8"></span>[[#cite-8|8]]] can compete with the original formulation.  In this paper we shall perform a comprehensive study on the performance of SPR when applied to three dimensional problems.
 The procedure of SPR may be found in many texts and papers <span id="citeF-4"></span>[[#cite-4|[4]],<span id="citeF-5"></span>[[#cite-5|5]],<span id="citeF-21"></span>[[#cite-21|21]]-<span id="citeF-24"></span>[[#cite-24|24]]].  The procedure starts with assuming a continuous gradient field, in the form of a polynomial, over a patch of elements built on a vertex node

← Older revision	Revision as of 11:34, 25 June 2019
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 |- style="text-align: center; font-size: 75%;"
 | colspan="2" | '''Figure 2:'''Boundary layer solution obtained from the proposed method for the sample problem No. 1 using <math display="inline">10x10</math>  patches with pattern of Fig. [[#img-1|1]] and Dirichlet boundary condition at <math display="inline">x=0</math> and <math display="inline">y=0</math> obtained from monomial <math display="inline">X^2</math>  as the exact solution; (a) Contour line presentation, (b) Three dimensional plot presentation.
-|}}
+|}
 <div id='img-3'></div>

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 |}
-We choose vectors associated with  <math>\vert {\mu }_2\vert \leq 1</math> , i.e. <math display="inline">{\phi }_{\mbox{ }\mbox{ }1\mbox{ }}</math> ,  <math display="inline">{\phi }_{\mbox{ }\mbox{ }3\mbox{ }}</math> and  <math display="inline">{\phi }_{\mbox{ }\mbox{ }4\mbox{ }}</math> , reflecting decay condition.   For each case we define the transformation matrix  <math display="inline">\boldsymbol{T}_0</math> in order to construct  <math display="inline">\boldsymbol{v}</math> , for instance for two patches along each axis (see Figure [[#img1|1-b]])
+We choose vectors associated with  <math>\vert {\mu }_2\vert \leq 1</math> , i.e. <math display="inline">{\phi }_{\mbox{ }\mbox{ }1\mbox{ }}</math> ,  <math display="inline">{\phi }_{\mbox{ }\mbox{ }3\mbox{ }}</math> and  <math display="inline">{\phi }_{\mbox{ }\mbox{ }4\mbox{ }}</math> , reflecting decay condition.   For each case we define the transformation matrix  <math display="inline">\boldsymbol{T}_0</math> in order to construct  <math display="inline">\boldsymbol{v}</math> , for instance for two patches along each axis (see Figure [[#img-1|1-b]])
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 1,158: / Line 1,158: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_152569619_4671_10.PNG|500px|XXXXXXXX]]
+|[[File:Draft_Samper_152569619_4671_10.PNG|400px|XXXXXXXX]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 10:'''Robustness index obtained from application of SPR to heat conduction type of problems versus average minimum angle between faces of the elements in the patch.
@@ Line 1,166: / Line 1,166: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_152569619_7397_11.PNG|500px|XXXXXXXX]]
+|[[File:Draft_Samper_152569619_7397_11.PNG|400px|XXXXXXXX]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 11:'''Robustness index obtained from application of SPR to elasticity type of problems versus average minimum angle between faces of the elements in the patch.
@@ Line 1,176: / Line 1,176: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_152569619_5263_12.PNG|500px|XXXXXXXX]]
+|[[File:Draft_Samper_152569619_5263_12.PNG|400px|XXXXXXXX]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 12:'''Robustness index obtained from application of REP (improved) to heat conduction type of problems versus average minimum angle between faces of the elements in the patch.
@@ Line 1,184: / Line 1,184: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_152569619_8727_13.PNG|500px|XXXXXXXX]]
+|[[File:Draft_Samper_152569619_8727_13.PNG|400px|XXXXXXXX]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 13:'''Robustness index obtained from application of REP (improved) to elasticity type of problems versus average minimum angle between faces of the elements in the patch.
@@ Line 1,196: / Line 1,196: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_152569619_8727_13.PNG|500px|XXXXXXXX]]
+|[[File:Draft_Samper_152569619_3832_14.PNG|400px|XXXXXXXX]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 14:'''Variation of robustness index from application of SPR and REP (improved) versus eccentricity of the interior node, as a measure of skewness of the mesh, for RRR pattern along three directions, major direction <math display="inline>x</math>, minor diagonal <math display="inline>y=x</math> and major diagonal <math display="inline>z=y=x</math>.