Colom Cobb et al 2019b - Revision history

Julio at 17:31, 6 August 2019

2019-08-06T17:31:19Z

Julio at 12:43, 25 July 2019

2019-07-25T12:43:35Z

Julio at 12:40, 25 July 2019

2019-07-25T12:40:00Z

Julio at 12:25, 25 July 2019

2019-07-25T12:25:34Z

Julio: Julio moved page Draft Garcia-Espinosa 450194514 to Colom Cobb et al 2019b

2019-07-13T13:08:09Z

Julio moved page Draft Garcia-Espinosa 450194514 to Colom Cobb et al 2019b

Julio at 13:07, 13 July 2019

2019-07-13T13:07:42Z

Julio at 13:06, 13 July 2019

2019-07-13T13:06:58Z

Julio at 13:05, 13 July 2019

2019-07-13T13:05:05Z

Julio at 18:32, 5 July 2019

2019-07-05T18:32:32Z

Scipediacontent at 10:51, 5 July 2019

2019-07-05T10:51:52Z

@@ Line 238: / Line 238: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| <math>{\mathit{\boldsymbol{U}}}_{\lambda }^{n+1,i+1}={\mathit{\boldsymbol{U}}}_{\lambda }^{n+1/2}+</math><math>\frac{\Delta t}{2}{\mathit{\boldsymbol{a}}}^{n+1}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{n+1}\right)</math>
+| <math>{\mathit{\boldsymbol{U}}}_{\lambda }^{n+1}={\mathit{\boldsymbol{U}}}_{\lambda }^{n+1/2}+</math><math>\frac{\Delta t}{2}{\mathit{\boldsymbol{a}}}^{n+1}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{n+1}\right)</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref3545805'></span><span id='_Ref527454686'></span>(17)

@@ Line 649: / Line 649: @@
 {| style="margin:auto;width: 100%;"
 |-
-| <math>{\tilde{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}=</math><math>\underbrace{{{\mathit{\boldsymbol{X}}}_{\lambda }^{0}\mathit{\boldsymbol{+}}\Delta t{\mathit{\boldsymbol{u}}}^{0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) +\frac{\Delta {t}^{2}}{2}{\mathit{\boldsymbol{a}}}^{0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{{\hat{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}\mathit{\boldsymbol{=}}{\mathit{\boldsymbol{X}}}_{\lambda }^{1}\mathit{\boldsymbol{+}}O(\Delta {t}^{3})\mathit{\boldsymbol{\, }}}+</math><math>\underbrace{{{\Delta t\epsilon }^{\mathit{\boldsymbol{u}},0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta t{h}_{e}^{2})}+</math><math>\underbrace{{\Delta t{\boldsymbol{\epsilon }}_{h}^{P,0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta t{h}_{e}^{2})}+</math><math>\underbrace{{{\frac{\Delta {t}^{2}}{2}\epsilon }^{\mathit{\boldsymbol{a}},0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta {t}^{2}{h}_{e}^{2})}</math>
+| <math>{\tilde{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}=</math><math>\underbrace{{{\mathit{\boldsymbol{X}}}_{\lambda }^{0}\mathit{\boldsymbol{+}}\Delta t{\mathit{\boldsymbol{u}}}^{0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) +\frac{\Delta {t}^{2}}{2}{\mathit{\boldsymbol{a}}}^{0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{{\hat{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}\mathit{\boldsymbol{=}}{\mathit{\boldsymbol{X}}}_{\lambda }^{1}\mathit{\boldsymbol{+}}O(\Delta {t}^{3})\mathit{\boldsymbol{\, }}}+</math><math>\underbrace{{{\Delta t\epsilon }^{\mathit{\boldsymbol{u}},0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta t{h}_{e}^{2})}+</math><math>\underbrace{{\Delta t{\boldsymbol{\epsilon }}_{h}^{P,0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta t{h}_{e}^{2})}+</math><math>\underbrace{{{\frac{\Delta {t}^{2}}{2}}{\epsilon}^{\mathit{\boldsymbol{a}},0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta {t}^{2}{h}_{e}^{2})}</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|(49)

@@ Line 313: / Line 313: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| <math>{\hat{\mathit{\boldsymbol{u}}}}^{n+1,i+1}={\mathit{\boldsymbol{u}}}^{n+1/2}+\frac{\Delta t}{2}\left( -\right. </math><math>\left. \nabla \left( \frac{{P}^{n+1,i}}{\rho }\right) +\nu \Delta {\hat{\mathit{\boldsymbol{u}}}}^{n+1,i+1}+\right. </math><math>\left. {\mathit{\boldsymbol{f}}}^{n+1}\right)</math>
+| <math>{\hat{\mathit{\boldsymbol{u}}}}^{n+1,i+1}={\mathit{\boldsymbol{u}}}^{n+1/2}+\frac{\Delta t}{2}\left(-\nabla \left( \frac{{P}^{n+1,i}}{\rho }\right) +\nu \Delta {\hat{\mathit{\boldsymbol{u}}}}^{n+1,i+1}+{\mathit{\boldsymbol{f}}}^{n+1}\right)</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref527460900'></span>(22)
@@ Line 320: / Line 320: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| <math>{\mathit{\boldsymbol{u}}}^{n+1,i+1}-{\hat{\mathit{\boldsymbol{u}}}}^{n+1,i+1}=\frac{\Delta t}{2}\left( -\right. </math><math>\left. \nabla \left( \frac{{P}^{n+1,i+1}}{\rho }\right) +\nabla \left( \frac{{P}^{n+1,i}}{\rho }\right) \right)</math>
+| <math>{\mathit{\boldsymbol{u}}}^{n+1,i+1}-{\hat{\mathit{\boldsymbol{u}}}}^{n+1,i+1}=\frac{\Delta t}{2}\left(-\nabla \left( \frac{{P}^{n+1,i+1}}{\rho }\right) +\nabla \left( \frac{{P}^{n+1,i}}{\rho }\right) \right)</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref527458693'></span>(23)

@@ Line 225: / Line 225: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| <math>\frac{{\mathit{\boldsymbol{U}}}_{\lambda }^{n+1}-{\mathit{\boldsymbol{U}}}_{\lambda }^{n}}{\Delta t}=</math><math>\frac{1}{2}\left( {\mathit{\boldsymbol{a}}}^{n}({\mathit{\boldsymbol{X}}}_{\lambda }^{n})+\right. </math><math>\left. {\mathit{\boldsymbol{a}}}^{n+1}({\mathit{\boldsymbol{X}}}_{\lambda }^{n+1})\right)</math>
+| <math>\frac{{\mathit{\boldsymbol{U}}}_{\lambda }^{n+1}-{\mathit{\boldsymbol{U}}}_{\lambda }^{n}}{\Delta t}=\frac{1}{2}\left( {\mathit{\boldsymbol{a}}}^{n}({\mathit{\boldsymbol{X}}}_{\lambda }^{n})+{\mathit{\boldsymbol{a}}}^{n+1}({\mathit{\boldsymbol{X}}}_{\lambda }^{n+1})\right)</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref527451071'></span>(16)
@@ Line 350: / Line 350: @@
 |  style="width: 5px;text-align: right;white-space: nowrap;"|(25)
 |}
 ====1.3 The SL-PFEM method for solving the incompressible Navier-Stokes equations====

← Older revision		Revision as of 13:07, 13 July 2019
Line 24:		Line 24:

	In this paper, a second order SL-PFEM scheme for solving the incompressible Navier-Stokes equations is presented. This scheme is based on the second order velocity Verlet algorithm, which uses an explicit integration for the particle’s trajectory and an implicit integration for the velocity. The algorithm is completed with a predictor-multicorrector scheme for the integration of the velocity correction using the Finite Element Method. A second order projector based on least squares is used to transfer the intrinsic variables information from the particles onto the background mesh, while a second order interpolation scheme is used to transfer the accelerations from the mesh to the particles. Convergence analyses are carried out to assess the second order convergence.		In this paper, a second order SL-PFEM scheme for solving the incompressible Navier-Stokes equations is presented. This scheme is based on the second order velocity Verlet algorithm, which uses an explicit integration for the particle’s trajectory and an implicit integration for the velocity. The algorithm is completed with a predictor-multicorrector scheme for the integration of the velocity correction using the Finite Element Method. A second order projector based on least squares is used to transfer the intrinsic variables information from the particles onto the background mesh, while a second order interpolation scheme is used to transfer the accelerations from the mesh to the particles. Convergence analyses are carried out to assess the second order convergence.
		+
		+	The [https://link.springer.com/article/10.1007%2Fs40571-019-00258-9 final publication] is available at link.springer.com.

	<span id='_GoBack'></span>'''Keywords: '''Semi-Lagrangian, Particles, PFEM, FEM, Second-order		<span id='_GoBack'></span>'''Keywords: '''Semi-Lagrangian, Particles, PFEM, FEM, Second-order

← Older revision	Revision as of 13:08, 13 July 2019
(No difference)

← Older revision		Revision as of 13:06, 13 July 2019
Line 1,457:		Line 1,457:
	<div id="12"></div>		<div id="12"></div>
	[[#cite-12\|[12]]] H. Jiang and L. Cheng. Strouhal Reynolds number relationship for flow past a circular cylinder. J. Fluid Mech. (2017), vol. 832, pp. 170-188.		[[#cite-12\|[12]]] H. Jiang and L. Cheng. Strouhal Reynolds number relationship for flow past a circular cylinder. J. Fluid Mech. (2017), vol. 832, pp. 170-188.
−
−
−	~~The [https://link.springer.com/article/10.1007%2Fs40571-019-00258-9 final publication] is available at link.springer.com.~~

@@ Line 34: / Line 34: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-  [[Image:Draft_Garcia-Espinosa_450194514-image1.png|384px]] </div>
+  [[Image:Draft_Garcia-Espinosa_450194514-image1.png|600px]] </div>
 <div id="_Ref532897315" class="center" style="width: auto; margin-left: auto; margin-right: auto;">

@@ Line 968: / Line 968: @@
-{| style="width: 63%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
+{| style="text-align: center;width: 63%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
 |-
 |  style="vertical-align: top;width: 0%;"|[[Image:Draft_Garcia-Espinosa_450194514-image5.jpeg|366px]]
@@ Line 992: / Line 992: @@
 <span style="text-align: center; font-size: 75%;">Figure 6: Streamlines comparison. Up: Second order Verlet-SLPFEM. Down: First Order Euler SL-PFEM</span></div>
-{| style="width: 57%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
+{| style="text-align: center;width: 57%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
 |-
 |  style="border: 1pt solid black;vertical-align: top;width: 0%;"|[[Image:Draft_Garcia-Espinosa_450194514-image10.png|336px]]
@@ Line 1,003: / Line 1,003: @@
 <span style="text-align: center; font-size: 75%;">Figure 7: Horizontal velocity and pressure profiles at the mid-section x=0.5. Red dots: spectral method [</span><span id='cite-_Ref6225974'></span>[[#_Ref6225974|<span style="text-align: center; font-size: 75%;">11</span>]]<span style="text-align: center; font-size: 75%;">]. Solid line: Second Order Verlet-SLPFEM. Dash line: First Order Euler-SLPFEM.</span></div>
-{| style="width: 58%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
+{| style="text-align: center;width: 58%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
 |-
 |  style="vertical-align: top;width: 0%;"|[[Image:Draft_Garcia-Espinosa_450194514-image12.png|336px]]

@@ Line 646: / Line 646: @@
 |-
+|
-{| style="text-align: center; margin:auto;width: 100%;"
+{| style="margin:auto;width: 100%;"
 |-
-| <math>{\tilde{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}=</math><math>+++</math>
+| <math>{\tilde{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}=</math><math>\underbrace{{{\mathit{\boldsymbol{X}}}_{\lambda }^{0}\mathit{\boldsymbol{+}}\Delta t{\mathit{\boldsymbol{u}}}^{0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) +\frac{\Delta {t}^{2}}{2}{\mathit{\boldsymbol{a}}}^{0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{{\hat{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}\mathit{\boldsymbol{=}}{\mathit{\boldsymbol{X}}}_{\lambda }^{1}\mathit{\boldsymbol{+}}O(\Delta {t}^{3})\mathit{\boldsymbol{\, }}}+</math><math>\underbrace{{{\Delta t\epsilon }^{\mathit{\boldsymbol{u}},0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta t{h}_{e}^{2})}+</math><math>\underbrace{{\Delta t{\boldsymbol{\epsilon }}_{h}^{P,0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta t{h}_{e}^{2})}+</math><math>\underbrace{{{\frac{\Delta {t}^{2}}{2}\epsilon }^{\mathit{\boldsymbol{a}},0}\left( {\mathit{\boldsymbol{X}}}_{\lambda }^{0}\right) }}_{O(\Delta {t}^{2}{h}_{e}^{2})}</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|(49)
 |-
+|
-{| style="text-align: center; margin:auto;width: 100%;"
+{| style="margin:auto;width: 100%;"
 |-
-| <math>{\tilde{\mathit{\boldsymbol{U}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}=</math><math>++</math>
+| <math>{\tilde{\mathit{\boldsymbol{U}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}=</math><math>\underbrace{{{\mathit{\boldsymbol{U}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{0}}}\mathit{\boldsymbol{+}}\frac{\Delta t}{2}\left( {\mathit{\boldsymbol{a}}}^{\mathit{\boldsymbol{0}}}\left( {\mathit{\boldsymbol{X}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{0}}}\right) \mathit{\boldsymbol{+}}{\mathit{\boldsymbol{a}}}^{\mathit{\boldsymbol{1}}}\left( {\hat{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}\right) \right) }}_{{\hat{\mathit{\boldsymbol{U}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}\mathit{\boldsymbol{=}}{\mathit{\boldsymbol{U}}}_{\lambda }^{1}\mathit{\boldsymbol{+}}O(\Delta {t}^{3})}+</math><math>\underbrace{{\frac{\Delta t}{2}\left( {\mathit{\boldsymbol{a}}}^{\mathit{\boldsymbol{1}}}\left( {\tilde{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}\right) \mathit{\boldsymbol{-}}{\mathit{\boldsymbol{a}}}^{\mathit{\boldsymbol{1}}}\left( {\hat{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}\right) \right) }}_{\mathit{\boldsymbol{\sim }}\frac{\mathit{\boldsymbol{\Delta t}}}{\mathit{\boldsymbol{2}}}({h}_{e}^{2}\Delta t)\mathit{\boldsymbol{\nabla }}{\mathit{\boldsymbol{a}}}^{\mathit{\boldsymbol{1}}}}+</math><math>\underbrace{{\frac{\Delta t}{2}\left( {\mathit{\boldsymbol{\epsilon }}}^{\mathit{\boldsymbol{a,0}}}\left( {\mathit{\boldsymbol{X}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{0}}}\right) \mathit{\boldsymbol{+}}{\mathit{\boldsymbol{\epsilon }}}^{\mathit{\boldsymbol{a,1}}}\left( {\tilde{\mathit{\boldsymbol{X}}}}_{\mathit{\boldsymbol{\lambda }}}^{\mathit{\boldsymbol{1}}}\right) \right) }}_{O(\Delta t{h}_{e}^{2})}</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|(50)
@@ Line 723: / Line 723: @@
 |-
+|
-{| style="text-align: center; margin:auto;width: 100%;"
+{| style="margin:auto;width: 100%;"
 |-
-| <math>\left\{ {\hat{\mathit{\boldsymbol{u}}}}_{b}^{n+1}\right\} =\left\{ {\mathit{\boldsymbol{u}}}_{b}^{n+1}\right\} +</math><math>+</math>
+| <math>\left\{ {\hat{\mathit{\boldsymbol{u}}}}_{b}^{n+1}\right\} =\left\{ {\mathit{\boldsymbol{u}}}_{b}^{n+1}\right\} +</math><math>\underbrace{{\left\{ {\mathit{\boldsymbol{\epsilon }}}_{b}^{P,n+1}\right\} }}_{O\left( {h}_{e}^{2}\right) }+</math><math>\underbrace{{\left\{ {\mathit{\boldsymbol{e}}}_{b}^{n+1}\right\} }}_{{t}^{n+1}\left( O\left( \Delta {t}^{2}\right) +O\left( {h}_{e}^{2}\right) \right) }</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|(56)
 |}
 ===7. Convergence analyses===