Boroomand et al 2004b - Revision history

Cinmemj: Cinmemj moved page Draft Samper 287811841 to Boroomand et al 2004b

2018-07-13T10:48:21Z

Cinmemj moved page Draft Samper 287811841 to Boroomand et al 2004b

Cinmemj at 10:46, 13 July 2018

2018-07-13T10:46:55Z

Cinmemj at 10:44, 13 July 2018

2018-07-13T10:44:43Z

Cinmemj at 10:43, 13 July 2018

2018-07-13T10:43:25Z

Cinmemj at 10:42, 13 July 2018

2018-07-13T10:42:13Z

Show changes

Cinmemj at 10:23, 13 July 2018

2018-07-13T10:23:57Z

Cinmemj at 10:20, 13 July 2018

2018-07-13T10:20:55Z

Cinmemj at 10:15, 13 July 2018

2018-07-13T10:15:47Z

Cinmemj at 14:58, 10 July 2018

2018-07-10T14:58:30Z

Cinmemj at 14:36, 10 July 2018

2018-07-10T14:36:49Z

← Older revision		Revision as of 10:46, 13 July 2018
Line 1,150:		Line 1,150:
	Series of irregular quadrilateral and triangular meshes are chosen as Figure 25 and 26 to give a view of mesh independency of the solution. Figures 27 and 28 depict the velocity obtained along <math display="inline">x=0.5</math> and pressure distribution along <math display="inline">y=0.5</math> . It can be seen that the finite element solution is not dependent on the mesh used.		Series of irregular quadrilateral and triangular meshes are chosen as Figure 25 and 26 to give a view of mesh independency of the solution. Figures 27 and 28 depict the velocity obtained along <math display="inline">x=0.5</math> and pressure distribution along <math display="inline">y=0.5</math> . It can be seen that the finite element solution is not dependent on the mesh used.

−	~~==edit==~~

	{\|style="width: 100%;border-collapse: collapse;"		{\|style="width: 100%;border-collapse: collapse;"

@@ Line 1,229: / Line 1,229: @@
 |-
 |  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 12%;"| <math display="inline">\nu =0.3</math>
-|  style="border: 1pt solid black;vertical-align: top;width: 12%;"| <math display="inline">\nu =0.499</math>
+|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 12%;"| <math display="inline">\nu =0.499</math>
 |  style="border: 1pt solid black;text-align: center;vertical-align: top;"|'''Iteration 5'''
 |  style="border: 1pt solid black;text-align: center;vertical-align: top;"|'''Iteration 10'''

@@ Line 1,260: / Line 1,260: @@
 |}
-=6. CONCLUSIONS=
+==6. CONCLUSIONS==
 A simple iterative method has been presented for finite element solution of fully incompressible problems using mesh of linear elements.  For linear quadrilateral elements the control volumes are taken as elements volumes and for linear triangular elements the control volumes are built on partial contributions of elements adjacent to edges.   In the case that elements are considered as control volumes, the procedure may be interpreted as a special case of general iterative methods available for mixed formulation though the pressure correction part is different.  However, the most advantage of the proposed method is its capability to be extended to more general cases of control volumes as described for the case of triangular elements.
@@ Line 1,270: / Line 1,270: @@
 It has been demonstrated that although the formulation is based on displacement/velocity field, the method eliminates the locking/instability effects generally observed in linear elements and the answers obtained are similar to those obtained from mixed formulation.  The iteration procedure is consistent with non-linear problems and can be implemented in any finite element code without need of much effort.
-==Acknowledgment ==
+==Acknowledgments==
 The authors wish to thank Dr. A.R. Pishevar in Mechanical Engineering Department of Isfahan University of Technology for useful discussions.

@@ Line 1,154: / Line 1,154: @@
 {|style="width: 100%;border-collapse: collapse;"
 |-
-| style="padding-right:15px;"|[[Image:draft_Samper_287811841-image245.png|center|138px]]
+| style="padding-right:15px;"|[[Image:draft_Samper_287811841-image245.png|center|150px]]
-| style="padding-right:15px;"|[[Image:draft_Samper_287811841-image246.png|center|138px]]
+| style="padding-right:15px;"|[[Image:draft_Samper_287811841-image246.png|center|150px]]
-|[[Image:draft_Samper_287811841-image247.png|center|138px]]
+|[[Image:draft_Samper_287811841-image247.png|center|150px]]
 |-
 |style="text-align: center;"| Mesh Q1
@@ Line 1,190: / Line 1,190: @@
 As a final example we consider the cantilever beam of Figure 29.  Plane strain conditions are assumed and two types of elements are employed in the solution.  In order to compare the results with an exact solution we replace the one dimensional problem of Figure 29-a with its equivalent form as a two dimensional problem of Figure 29-b.  Considering a fully incompressible material with shear modulus  <math display="inline">G</math> , the exact solution is given as
-<math display="inline">u_{Exact}=\frac{-\mbox{ }\mbox{ }6\mbox{ }M}{G\mbox{ }b\mbox{ }h^3}\mbox{ }\mbox{ }x\mbox{ }y</math>,       <math display="inline">v_{Exact}=\mbox{ }\mbox{ }\frac{3\mbox{ }M}{G\mbox{ }b\mbox{ }h^3}\mbox{ }\mbox{ }(y^2+</math><math>x^2)</math>
+<math>u_{Exact}=\frac{-\mbox{ }\mbox{ }6\mbox{ }M}{G\mbox{ }b\mbox{ }h^3}\mbox{ }\mbox{ }x\mbox{ }y</math> ,
 Where  <math display="inline">u</math> and  <math display="inline">v</math> are displacements in  <math display="inline">x</math> and  <math display="inline">y</math> directions and also  <math display="inline">b</math> and  <math display="inline">h</math> are the width and height of the beam.  The body force and traction of the problem are found from the exact solution and applied to the meshes shown in Figures 29-c and 29-d. The reader may notice that the body force has just vertical component,  <math display="inline">b_y</math> , and has a relation with vertical component of tractions at  <math display="inline">y=\pm \mbox{ }h/2</math> as  <math display="inline">h\mbox{ }\mbox{ }b_y=2\mbox{ }t_y</math> .  The problem is solved using  <math display="inline">G=100</math> ,  <math display="inline">b=1</math> ,  <math display="inline">h=2</math> and <math display="inline">L=8</math> .  The results for vertical displacement of the beam tip, i.e. at  <math display="inline">x=8</math> and <math display="inline">y=0</math> , are shown in Table 1 and are normalized with respect to the corresponding exact value.  The results are obtained from both conventional displacement approach, with  <math display="inline">\nu =0.499</math> , and the proposed method for meshes shown in the figure and some finer meshes with similar patterns.  It can be clearly seen that the proposed method does not exhibit locking behavior for both linear rectangular and triangular types of elements.  To compare the results with compressible cases we have included results obtained from  <math display="inline">\nu =0.3</math> (the values are normalized with respect to exact solution at fully incompressible limit).

@@ Line 1,152: / Line 1,152: @@
 ==edit==
-{|
+{|style="width: 100%;border-collapse: collapse;"
 |-
 | style="padding-right:15px;"|[[Image:draft_Samper_287811841-image245.png|center|138px]]
@@ Line 1,162: / Line 1,162: @@
 |style="text-align: center;"|Mesh Q3
 |}
 Figure 25. Three meshes of quadrilateral elements for the problem of driven cavity in Fig. 21.
@@ Line 1,168: / Line 1,167: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-[Image:draft_Samper_287811841-image248.png|600px]] </div>
+[[File:Draft_Samper_287811841_4377_Fig26.png]]</div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

← Older revision	Revision as of 10:48, 13 July 2018
(No difference)

@@ Line 1,034: / Line 1,034: @@
-{|
+{|style="width: 100%;border-collapse: collapse;"
 |-
 | style="padding-right:15px;"|[[Image:draft_Samper_287811841-image230.png|center|180px]]

@@ Line 1,051: / Line 1,051: @@
 [[Image:draft_Samper_287811841-image233.png|468px]] </div>
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+Figure 14. Comparison of the vertical displacements at the top edge of the slotted plate problem obtained from application of the proposed method to three meshes of Fig 13.
-Figure 14. Comparison of the vertical displacements at the top edge of the slotted plate problem obtained from application of the proposed method to three meshes of Fig 13.</div>
@@ Line 1,113: / Line 1,112: @@
 [[Image:draft_Samper_287811841-image241.png|318px]] </div>
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+Figure 21.  The problem of driven cavity ([[Image:draft_Samper_287811841-image206.png|30px]]).
-Figure 21.  The problem of driven cavity ( [[Image:draft_Samper_287811841-image206.png|30px]] ).</div>
@@ Line 1,146: / Line 1,144: @@
 Figure 24.  Hourglass modes obtained from application of the penalty method to the problem of driven cavity; (a) solution with  <math display="inline">\alpha ={10}^8</math> , (b) solution with  <math display="inline">\alpha =10</math> .
@@ Line 1,153: / Line 1,150: @@
 Series of irregular quadrilateral and triangular meshes are chosen as Figure 25 and 26 to give a view of mesh independency of the solution.  Figures 27 and 28 depict the velocity obtained along  <math display="inline">x=0.5</math> and pressure distribution along  <math display="inline">y=0.5</math> .  It can be seen that the finite element solution is not dependent on the mesh used.
 {|
 |-
-| [[Image:draft_Samper_287811841-image245.png|138px]]
+| style="padding-right:15px;"|[[Image:draft_Samper_287811841-image245.png|center|138px]]
-| [[Image:draft_Samper_287811841-image246.png|center|138px]]
+| style="padding-right:15px;"|[[Image:draft_Samper_287811841-image246.png|center|138px]]
 |}
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+Figure 25. Three meshes of quadrilateral elements for the problem of driven cavity in Fig. 21.
-Figure 25. Three meshes of quadrilateral elements for the problem of driven cavity in Fig. 21.</div>
@@ Line 1,179: / Line 1,179: @@
 [[Image:draft_Samper_287811841-chart5.svg|468px]] </div>
-Figure 27. Comparison of variations of the horizontal velocity ( <math display="inline">u</math> ) along  <math display="inline">x=0.5</math> obtained from application of the proposed method to quadrilateral and triangular meshes of in Figs. 25 and 26.
+Figure 27. Comparison of variations of the horizontal velocity (<math display="inline">u</math>) along  <math display="inline">x=0.5</math> obtained from application of the proposed method to quadrilateral and triangular meshes of in Figs. 25 and 26.

@@ Line 1,033: / Line 1,033: @@
 To study the effects of mesh configuration, a series of meshes as shown in Figure 13 are selected and solved.  Figures 14 to 16, respectively, demonstrate top edge displacements, pressure distributions along  <math display="inline">x=0</math> and <math display="inline">y=0</math> , and convergence of the volume changes for the three meshes.
 {|
@@ Line 1,047: / Line 1,045: @@
 |}
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">