Dialami et al 2016a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 179315710 to Dialami et al 2016a

2018-05-23T13:10:51Z

Cinmemj moved page Draft Samper 179315710 to Dialami et al 2016a

Cinmemj at 13:07, 23 May 2018

2018-05-23T13:07:26Z

Cinmemj at 13:05, 23 May 2018

2018-05-23T13:05:54Z

Cinmemj at 13:00, 23 May 2018

2018-05-23T13:00:13Z

Cinmemj at 12:59, 23 May 2018

2018-05-23T12:59:08Z

Cinmemj at 12:58, 23 May 2018

2018-05-23T12:58:18Z

Cinmemj at 12:48, 23 May 2018

2018-05-23T12:48:39Z

Cinmemj at 12:47, 23 May 2018

2018-05-23T12:47:14Z

Cinmemj at 12:42, 23 May 2018

2018-05-23T12:42:07Z

Cinmemj at 12:38, 23 May 2018

2018-05-23T12:38:07Z

Show changes

@@ Line 1,571: / Line 1,571: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> \dfrac{dT\left(\mathbf{\chi },t\right)}{dt} &\cong &\dfrac{\partial  T\left(\mathbf{\chi },t\right)}{\partial t}+\dfrac{\partial T\left( \mathbf{\chi },t\right)}{\partial \mathbf{\chi }}\dfrac{d\mathbf{\chi }}{dt}=  \dfrac{\partial T\left(\mathbf{\chi },t\right)}{\partial t}+\mathbf{c}\cdot \nabla _{\mathbf{\chi }}T\left(\mathbf{\chi },t\right) </math>
+| style="text-align: center;" | <math> \dfrac{dT\left(\mathbf{\chi },t\right)}{dt} \cong \dfrac{\partial  T\left(\mathbf{\chi },t\right)}{\partial t}+\dfrac{\partial T\left( \mathbf{\chi },t\right)}{\partial \mathbf{\chi }}\dfrac{d\mathbf{\chi }}{dt}=  \dfrac{\partial T\left(\mathbf{\chi },t\right)}{\partial t}+\mathbf{c}\cdot \nabla _{\mathbf{\chi }}T\left(\mathbf{\chi },t\right) </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (77)

@@ Line 1,571: / Line 1,571: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{array}{l} \dfrac{dT\left(\mathbf{\chi },t\right)}{dt} &\cong &\dfrac{\partial  T\left(\mathbf{\chi },t\right)}{\partial t}+\dfrac{\partial T\left( \mathbf{\chi },t\right)}{\partial \mathbf{\chi }}\dfrac{d\mathbf{\chi }}{dt}=  \\ &&\dfrac{\partial T\left(\mathbf{\chi },t\right)}{\partial t}+\mathbf{c}\cdot \nabla _{\mathbf{\chi }}T\left(\mathbf{\chi },t\right) \end{array}</math>
+| style="text-align: center;" | <math> \dfrac{dT\left(\mathbf{\chi },t\right)}{dt} &\cong &\dfrac{\partial  T\left(\mathbf{\chi },t\right)}{\partial t}+\dfrac{\partial T\left( \mathbf{\chi },t\right)}{\partial \mathbf{\chi }}\dfrac{d\mathbf{\chi }}{dt}=  \dfrac{\partial T\left(\mathbf{\chi },t\right)}{\partial t}+\mathbf{c}\cdot \nabla _{\mathbf{\chi }}T\left(\mathbf{\chi },t\right) </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (77)

@@ Line 2,155: / Line 2,155: @@
 * '''Residual stress computation''': The paper proposes a local-global two level strategy for the prediction of residual stresses in FSW processes. At the first step, the heat power is calculated at local level from both viscous dissipation and friction. Then the heat power is transferred into the global level simulation in order to obtain the residual stresses.
-===BIBLIOGRAPHY===
+==BIBLIOGRAPHY==
 <div id="cite-1"></div>

← Older revision		Revision as of 12:59, 23 May 2018
Line 2,108:		Line 2,108:

	In this section, firstly, the comparison between the temperature fields obtained at local and global levels are shown to validate the local-global coupling strategy. Then, the residual stresses computed using the strategy are illustrated. The meshes at local and global levels are shown in figure [[#img-41\|41]]. The total dissipated power is obtained at the steady state and used as the heat input energy for the global level analysis. It is applied to the control volume representing the TMAZ at each time-step of the analysis. The temperature fields at local and global levels at the end of the process is shown in figure [[#img-42\|42]]. The comparison between the temperature fields obtained from both local and global level analyses for different lines parallel to the weld line at the bottom surface or thermocouples placed at identical locations with those obtained from experiments are presented in figure [[#img-43\|43]]. The global level analysis provides the residual stress field. The comparison between the variations of the residual stress (normal stresses in x direction) along the transverse direction (along the line perpendicular to the weld line) and experimental results are plotted in figure [[#img-44\|44]] at the same location. Longitudinal stresses after fixture release and cooling are depicted in this figure.		In this section, firstly, the comparison between the temperature fields obtained at local and global levels are shown to validate the local-global coupling strategy. Then, the residual stresses computed using the strategy are illustrated. The meshes at local and global levels are shown in figure [[#img-41\|41]]. The total dissipated power is obtained at the steady state and used as the heat input energy for the global level analysis. It is applied to the control volume representing the TMAZ at each time-step of the analysis. The temperature fields at local and global levels at the end of the process is shown in figure [[#img-42\|42]]. The comparison between the temperature fields obtained from both local and global level analyses for different lines parallel to the weld line at the bottom surface or thermocouples placed at identical locations with those obtained from experiments are presented in figure [[#img-43\|43]]. The global level analysis provides the residual stress field. The comparison between the variations of the residual stress (normal stresses in x direction) along the transverse direction (along the line perpendicular to the weld line) and experimental results are plotted in figure [[#img-44\|44]] at the same location. Longitudinal stresses after fixture release and cooling are depicted in this figure.
		+
		+	<div id='img-41'></div>
		+	{\| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
		+	\|-
		+	\|[[Image:draft_Samper_179315710-41.png\|600px\|Mesh resolution at local (left) and global (right) levels.]]
		+	\|- style="text-align: center; font-size: 75%;"
		+	\| colspan="1" \| '''Figure 41:''' Mesh resolution at local (left) and global (right) levels.
		+	\|}
		+
		+	<div id='img-42'></div>
		+	{\| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
		+	\|-
		+	\|[[Image:draft_Samper_179315710-42.png\|600px\|Temperature field at local (left) and global (right) levels.]]
		+	\|- style="text-align: center; font-size: 75%;"
		+	\| colspan="1" \| '''Figure 42:''' Temperature field at local (left) and global (right) levels.
		+	\|}
		+
		+	<div id='img-43'></div>
		+	{\| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
		+	\|-
		+	\|[[Image:draft_Samper_179315710-43.png\|600px\|Temperature evolution obtained from local and global level analyses compared with experiment]]
		+	\|- style="text-align: center; font-size: 75%;"
		+	\| colspan="1" \| '''Figure 43:''' Temperature evolution obtained from local and global level analyses compared with experiment
		+	\|}
		+
		+
		+	<div id='img-44'></div>
		+	{\| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
		+	\|-
		+	\|[[Image:draft_Samper_179315710-44.png\|600px\|Comparison between the stresses obtained from global level analysis and experiment at a line orthogonal to the weld line.]]
		+	\|- style="text-align: center; font-size: 75%;"
		+	\| colspan="1" \| '''Figure 44:''' Comparison between the stresses obtained from global level analysis and experiment at a line orthogonal to the weld line.
		+	\|}

	==9 Conclusion==		==9 Conclusion==

@@ Line 2,103: / Line 2,103: @@
 |}
-This power re-distribution preserves the total amount of energy input independently of the mesh used (<span id='citeF-39'></span>[[#cite-39|[39]]]).<div id='img-41'></div>
+This power re-distribution preserves the total amount of energy input independently of the mesh used (<span id='citeF-39'></span>[[#cite-39|[39]]]).
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:draft_Samper_179315710-41.png|600px|Mesh resolution at local (left) and global (right) levels.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 41:''' Mesh resolution at local (left) and global (right) levels.
-|}
-<div id='img-42'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:draft_Samper_179315710-42.png|600px|Temperature field at local (left) and global (right) levels.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 42:''' Temperature field at local (left) and global (right) levels.
-|}
-<div id='img-43'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:draft_Samper_179315710-43.png|600px|Temperature evolution obtained from local and global level analyses compared with experiment]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 43:''' Temperature evolution obtained from local and global level analyses compared with experiment
-|}
-<div id='img-44'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:draft_Samper_179315710-44.png|600px|Comparison between the stresses obtained from global level analysis and experiment at a line orthogonal to the weld line.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 44:''' Comparison between the stresses obtained from global level analysis and experiment at a line orthogonal to the weld line.
-|}
 ===8.3 Experimental validation===

← Older revision	Revision as of 13:10, 23 May 2018
(No difference)

@@ Line 1,953: / Line 1,953: @@
 ===7.3 RK4 method===
 <div id='img-33'></div>
@@ Line 1,961: / Line 1,989: @@
 | colspan="1" | '''Figure 33:''' Fourth-order Runge-Kutta method. In each step the derivative is evaluated four times: once at the initial point, twice at trial midpoints, and once at a trial endpoint. From these derivatives the final value (shown as a filled dot) is calculated
 |}
 <div id='img-34'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
@@ Line 1,968: / Line 1,998: @@
 | colspan="1" | '''Figure 34:''' Zalesask's benchmark after 10 revolutions.
 |}
 <div id='img-35'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
@@ Line 1,975: / Line 2,007: @@
 | colspan="1" | '''Figure 35:''' Velocity effect on the weld quality.
 |}
 <div id='img-36'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
@@ Line 1,982: / Line 2,016: @@
 | colspan="1" | '''Figure 36:''' Material particles after the weld with a threaded pin.
 |}
 ==8 Residual stresses==

@@ Line 1,929: / Line 1,929: @@
 ===7.2 BFECC method===
-<div id='img-32'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:draft_Samper_179315710-32.png|400px|The steps of the BFECC method for a rotational velocity field]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 32:''' The steps of the BFECC method for a rotational velocity field
-|}
 According to the third-order accurate, <math>O\left(\Delta t^{3}\right)</math>, BFECC method, the error arising from backward and forward advection of the particle position is used to modify the initial position.  Denoting with <math>A\left(\mathbf{\cdot }\right)=\left(\mathbf{\cdot }\right) +\int _{t_{n}}^{t_{n+1}}\mathbf{V}\left(\mathbf{\cdot },t\right)\mathrm{d}t, </math> the first order up-winding integration operator, BFECC consists of a prediction/correction procedure: the initial position of the tracer, <math>\mathbf{X}_{n}=\mathbf{X}\left(t_{n}\right)</math>, is advected forward to <math>\mathbf{\hat{X}}_{n+1}=A\left(\mathbf{X}_{n}\right)</math>, and then backward to <math>\mathbf{\hat{X}}_{n}=A^{R}\left(\mathbf{\hat{X}}_{n+1}\right)</math>, <math>A^{R}\left(\mathbf{\cdot }\right)</math> being the backward advection operator. The difference between <math>\mathbf{\hat{X}}_{n}</math> and the initial position defines the error, <math>\mathbf{\epsilon =}\dfrac{\mathbf{X}_{n}-\mathbf{\hat{X}}_{n}}{2}</math>, of the time integration scheme.  This error is used later to correct the starting position, <math>\mathbf{\bar{X}}_{n}=\mathbf{X}_{n}+\mathbf{\epsilon }</math>, for the next advection step, achieving the final solution, <math>\mathbf{X}_{n+1}=A\left(\mathbf{\bar{X}}_{n}\right)</math>.  The steps are the following:
@@ Line 1,949: / Line 1,943: @@
 where <math>A\left(\mathbf{\circ }\right)</math> involves the computation of the velocity field of the tracer at its current position, interpolating from the nodal values (Figure [[#img-32|32]]).
 ===7.3 RK4 method===

@@ Line 1,395: / Line 1,395: @@
 |}
-The resulting system is an accurate, stable and consistent formulation for the solution of the mechanical problem subjected to the incompressibility constraint. <div id='img-22'></div>
+The resulting system is an accurate, stable and consistent formulation for the solution of the mechanical problem subjected to the incompressibility constraint.
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:draft_Samper_179315710-22.png|400px|Pressure field obtained using ASGS (top) and OSGS (bottom) stabilization method.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 22:''' Pressure field obtained using ASGS (top) and OSGS (bottom) stabilization method.
-|}
-<div id='img-23'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
-|-
-|[[Image:draft_Samper_179315710-23.png|400px|Streamlines obtained using ASGS (top) and OSGS (bottom) stabilization method.]]
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 23:''' Streamlines obtained using ASGS (top) and OSGS (bottom) stabilization method.
-|}
 The stabilization parameter <math>\tau _{v}</math>, is computed as:
@@ Line 1,422: / Line 1,410: @@
 where <math>h</math> is the element size, <math>c_{u}</math> is a constant and <math>\bar{\mu }</math> is the parameter defined according to the constitutive model (either effective viscosity or shear modulus). <math>\bar{\mu }\left(T\right)</math> is usually temperature dependent and therefore the stabilization term <math>\tau _{v}\left( T\right)</math> is a temperature dependent parameter.  Figure [[#img-22|22]] compares results of ASGS and OSGS pressure stabilization methods in a parallel flow passing through a tube. The results obtained using OSGS are nodally exact, and totally unaffected by the boundary conditions <span id='citeF-6'></span>[[#cite-6|[6]]].  Figure [[#img-23|23]] shows the streamlines. The results obtained using OSGS are totally unaffected by the boundary conditions.
 ===4.2 Convective stabilization===