Nadukandi et al 2016a - Revision history

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Julio at 17:36, 11 April 2019

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Julio: Julio moved page Draft Garcia-Espinosa 557861078 to Nadukandi et al 2016a

2019-04-11T17:11:05Z

Julio moved page Draft Garcia-Espinosa 557861078 to Nadukandi et al 2016a

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 The application of the semi-Lagrangian particle finite element method (SL–PFEM) for the seakeeping simulation of the wave adaptive modular vehicle under spray generating conditions is presented. The time integration of the Lagrangian advection is done using the explicit integration of the velocity and acceleration along the streamlines (X-IVAS). Despite the suitability of the SL–PFEM for the considered seakeeping application, small time steps were needed in the X-IVAS scheme to control the solution accuracy. A preliminary proposal to overcome this limitation of the X-IVAS scheme for seakeeping simulations is presented.
 <!--
 == Full document ==

@@ Line 7: / Line 7: @@
 -->
-==2 Introduction==
+==1 Introduction==
 The Particle Finite Element Method (PFEM)&nbsp;<span id='citeF-1'></span>[[#cite-1|[1]]] is a versatile framework for the analysis of fluid&#8211;structure interaction problems. The PFEM combines Lagrangian particle-based techniques with the advantage of the integral formulation of the Finite Element Method (FEM). It has been shown&nbsp;<span id='citeF-2'></span>[[#cite-2|[2]]] to successfully simulate a wide variety of complex engineering problems, e.g., free-surface/multi-fluid flows with violent interface motions, multi-fluid mixing and buoyancy-driven segregation problems etc.  The latest development within the framework of the PFEM is the X-IVAS (eXplicit Integration along the Velocity and Acceleration Streamlines) scheme&nbsp;<span id='citeF-3'></span>[[#cite-3|[3]]]. It is a semi-implicit scheme built over a semi-Lagrangian (SL) formulation of the PFEM.
@@ Line 20: / Line 20: @@
 <span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]]) The SL&#8211;PFEM method using the X-IVAS scheme has been called the Particle Finite Element Method second generation (PFEM-2) by the original authors&nbsp;<span id='citeF-3'></span>[[#cite-3|[3]]]. However, we choose the former nomenclature (SL&#8211;PFEM + X-IVAS) to acknowledge its connection (see Sect. [[#3 Semi-Lagrangian formulation|3]]) with semi-Lagrangian schemes.</span>
-==3 Semi-Lagrangian formulation==
+==2 Semi-Lagrangian formulation==
 Semi-Lagrangian schemes can be classified into two groups based on the way the problem variables are stored in the formulation: the Eulerian storage (ES) group and the Lagrangian storage (LS) group. However, the keyword ''semi-Lagrangian'' almost always points to ES&#8211;SL schemes in the literature.  The ES&#8211;SL schemes are extensions of the Courant&#8211;Isaacson&#8211;Rees&nbsp;<span id='citeF-5'></span>[[#cite-5|[5]]] method for hyperbolic equations. John Sawyer&nbsp;<span id='citeF-6'></span>[[#cite-6|[6]]] and Andre Robert&nbsp;<span id='citeF-7'></span>[[#cite-7|[7]]] made seminal contributions to ES&#8211;SL schemes. ES&#8211;SL schemes are unconditionally stable and therefore allow for large time steps.  The solution is defined on a mesh and the nodal values are stored.  Lagrangian particles are used as an auxiliary tool to compute the advective time evolution of the variables.  For each time step (say <math display="inline">t^{n}</math> to <math display="inline">t^{n+1}</math>) particles are initially placed at the mesh nodes and are transported backwards in time (for <math display="inline">t^{n+1}-t^{n}</math>s) along the pathlines passing through these nodes at time <math display="inline">t^{n+1}</math>.  The <math display="inline">t^{n}</math> solution at the terminal positions are then assigned to the <math display="inline">t^{n+1}</math> nodal values.  The backward trajectories seldom end at existing nodes.  The ES&#8211;SL schemes provide models for the a priori unknown pathlines and to determine the <math display="inline">t^{n}</math> solution at the terminal positions.
@@ Line 34: / Line 34: @@
 Celledoni et&nbsp;al. showed&nbsp;<span id='citeF-19'></span>[[#cite-19|[19]]] that second-order ES&#8211;SL schemes are effectively second-order approximations to standard exponential integrators.  A similar connection between the X-IVAS scheme (a LS&#8211;SL scheme) and exponential integrators was pointed out&nbsp;<span id='citeF-20'></span>[[#cite-20|[20]]] by the first author, albeit using heuristic arguments.  Exponential integrators are designed to integrate stiff initial value problems with large time-steps.  This connection explains why the SL schemes admit very large time-steps without deteriorating the solution accuracy.
-==4 Semi-Lagrangian Particle Finite Element Method==
+==3 Semi-Lagrangian Particle Finite Element Method==
 '''Notation'''  Vectors are written using bold italic font and matrices are written using bold upright font. The independent variables in Lagrangian kinematics are <math display="inline">\{ \lambda , t\} </math>, where <math display="inline">\lambda </math> represents a label to identify particles and <math display="inline">t</math> represents the time elapsed after labeling. The primary dependent variable is the fluid particle trajectory denoted as <math display="inline">\boldsymbol{X}(\lambda ,t)</math>. The independent variables in Eulerian kinematics are <math display="inline">\{ \boldsymbol{x}, t\} </math>, where <math display="inline">\boldsymbol{x}</math> denotes the spatial coordinate. The primary dependent variable is the fluid velocity <math display="inline">\boldsymbol{u}(\boldsymbol{x},t)</math>.
@@ Line 153: / Line 153: @@
 |}
-==5 Seakeeping simulation of the WAM-V using the SL&#8211;PFEM==
+==4 Seakeeping simulation of the WAM-V using the SL&#8211;PFEM==
 Experimental testing of the WAM-V prototype revealed an operability problem&#8211;-at high speeds the hull generates an intense water spray (see Figure&nbsp;[[#img-1|1]]).  Beyond a critical speed the spray leads to reduced visibility and the operator (or the reconnaissance equipment) will get drenched.  Both represent operational and also safety hazards, especially in cold climates where there is a heightened risk of marine ice formation.
@@ Line 270: / Line 270: @@
 '''Remark'''  For a wide class of engineering applications the X-IVAS scheme usually admits very large time steps in the Lagrangain advection stage, e.g. <math display="inline">10</math>&#8211;<math display="inline">15</math> times the one given by the CFL condition.  However in the simulations presented here we had to use small time steps (of the same order as the CFL limit) to control the accuracy of the solution.  The reason for this limitation is discussed in the next section.
-==6 Trajectory approximation for Stokes waves==
+==5 Trajectory approximation for Stokes waves==
 Consider a second-order Stokes wave in deep sea and position the 2D coordinate system at the height of the mean sea level. The solution of the free surface <math display="inline">\eta (x,t)</math> is
@@ Line 370: / Line 370: @@
 <span style="text-align: center; font-size: 75%;">([[#fnc-2|<sup>1</sup>]]) http://cordis.europa.eu/project/rcn/200435_en.html</span>
-==7 Conclusions==
+==6 Conclusions==
 In the seakeeping simulation of the WAM-V, we have seen that the semi-Lagrangian Particle Finite Element method is a suitable CFD tool for the analysis of incompressible flows subjected to challenging physical conditions, e.g. violent interface motions, spray generating conditions etc. Due to the Lagrangian treatment of the advective processes and the Lagrangian data storage strategy in the SL&#8211;PFEM, the interfaces are accurately tracked. Further, the computational task associated to advective transport is mutually exclusive and hence scalable on parallel computers.
@@ Line 376: / Line 376: @@
 However, seakeeping simulations often involve water wave motion. Examining a second-order Stokes wave propagating on the surface of a deep sea, we have seen that the streamlines are not a good approximation to the pathlines. In such situations the use of large time steps in the X-IVAS scheme will result in significant errors in the fluid configuration. The particle trajectories driven by the acceleration in the initial configuration of the second-order Stokes wave are shown to be a better approximation to the pathlines. For seakeeping simulations with the SL&#8211;PFEM, this latter time-integration scheme seems to be a promising alternative to mitigate the small time-step limitation of the X-IVAS scheme.
-==8 Compliance with Ethical Standards==
+==7 Compliance with Ethical Standards==
 The authors declare that they have no conflict of interest. This study was partially supported by the WAM-V project funded under the Navy Grant N62909-12-1-7101 issued by Office of Naval Research Global, the SAFECON project (ref. 267521, FP7-IDEAS-ERC) and the FORECAST project (ref. 664910, H2020-ERC-2014-PoC) of the European Research Council (European Commission). The United States Government has a royalty-free license throughout the world in all copyrightable material contained herein.

@@ Line 1: / Line 1: @@
 == Abstract ==
 The application of the semi-Lagrangian particle finite element method (SL–PFEM) for the seakeeping simulation of the wave adaptive modular vehicle under spray generating conditions is presented. The time integration of the Lagrangian advection is done using the explicit integration of the velocity and acceleration along the streamlines (X-IVAS). Despite the suitability of the SL–PFEM for the considered seakeeping application, small time steps were needed in the X-IVAS scheme to control the solution accuracy. A preliminary proposal to overcome this limitation of the X-IVAS scheme for seakeeping simulations is presented.
 <!--
 == Full document ==

@@ Line 307: / Line 307: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{X}^{h}(\lambda ,t) =  \boldsymbol{X}^{h}(\lambda ,t^{n}) + \int ^{t}_{t^{n}} \boldsymbol{u}^{h}(\boldsymbol{X}^{h}(\lambda ,\xi ), t^{n}) \diff \xi </math>
+| style="text-align: center;" | <math>\boldsymbol{X}^{h}(\lambda ,t) =  \boldsymbol{X}^{h}(\lambda ,t^{n}) + \int ^{t}_{t^{n}} \boldsymbol{u}^{h}(\boldsymbol{X}^{h}(\lambda ,\xi ), t^{n}) \xi </math>
 |-
 | style="text-align: center;" |
@@ Line 322: / Line 322: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{U}^{h}(\lambda ,t) =  \boldsymbol{U}^{h}(\lambda ,t^{n}) + \int ^{t}_{t^{n}} \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,\tau ), t^{n}) \diff \tau  </math>
+| style="text-align: center;" | <math>\boldsymbol{U}^{h}(\lambda ,t) =  \boldsymbol{U}^{h}(\lambda ,t^{n}) + \int ^{t}_{t^{n}} \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,\tau ), t^{n}) {d} \tau  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (18)
@@ Line 335: / Line 335: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\diff ^{2}}{\diff ^{2} t}\boldsymbol{X}^{h}(\lambda ,t) = \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,t),t^{n}) = \mathbf{C}^{n} \boldsymbol{X}^{h}(\lambda ,t) + \boldsymbol{d}^{n}  </math>
+| style="text-align: center;" | <math>\frac{{d} ^{2}}{{d} ^{2} t}\boldsymbol{X}^{h}(\lambda ,t) = \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,t),t^{n}) = \mathbf{C}^{n} \boldsymbol{X}^{h}(\lambda ,t) + \boldsymbol{d}^{n}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (19)
@@ Line 349: / Line 349: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{U}^{h}(\lambda ,t) =  \boldsymbol{U}^{h}(\lambda ,t^{n}) + \int ^{t}_{t^{n}} \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,\xi ), t^{n}) \diff \xi </math>
+| style="text-align: center;" | <math>\boldsymbol{U}^{h}(\lambda ,t) =  \boldsymbol{U}^{h}(\lambda ,t^{n}) + \int ^{t}_{t^{n}} \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,\xi ), t^{n}) {d} \xi </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (20)
 |-
 | style="text-align: center;" | <math>  \boldsymbol{X}^{h}(\lambda ,t) =  \boldsymbol{X}^{h}(\lambda ,t^{n}) + (t-t^{n}) \boldsymbol{U}^{h}(\lambda ,t^{n}) </math>
 |-
-| style="text-align: center;" | <math>                       + \int ^{t}_{t^{n}} \int ^{\tau }_{t^{n}} \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,\xi ), t^{n}) \diff \xi \diff \tau  </math>
+| style="text-align: center;" | <math>                       + \int ^{t}_{t^{n}} \int ^{\tau }_{t^{n}} \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,\xi ), t^{n}) {d} \xi {d} \tau  </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (21)
 |}

@@ Line 294: / Line 294: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Garcia-Espinosa_570669631-trajectories_following_streamlines.png|600px|Trajectory approximation in the X-IVAS scheme. The velocity field in the initial configuration is shown in the background as a vector plot.]]
+|[[File:Nadukandi_et_al_2016a_2567_streamlinesA.png|600px|Trajectory approximation in the X-IVAS scheme. The velocity field in the initial configuration is shown in the background as a vector plot.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 6:''' Trajectory approximation in the X-IVAS scheme. The velocity field in the initial configuration is shown in the background as a vector plot.
@@ Line 362: / Line 362: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Garcia-Espinosa_570669631-trajectories_drivenby_accelerations.png|600px|Trajectory driven by the acceleration field (shown in the background as a vector plot) in the initial configuration.]]
+|[[File:Nadukandi_et_al_2016a_2051_streamlinesB.png|600px|Trajectory driven by the acceleration field (shown in the background as a vector plot) in the initial configuration.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 7:''' Trajectory driven by the acceleration field (shown in the background as a vector plot) in the initial configuration.

@@ Line 162: / Line 162: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Garcia-Espinosa_570669631-wamv_33ft_21knots.png|600px|Spray generated by the WAM-V hull at 21 knots&nbsp;<span id='citeF-25'></span>[[#cite-25|[25]]]. Courtesy of Prof. Mehdi Ahmadian, VirginiaTech, USA.]]
+|[[Image:draft_García-Espinosa_900443996-image2.png|600px|Spray generated by the WAM-V hull at 21 knots]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 1:''' Spray generated by the WAM-V hull at 21 knots&nbsp;<span id='citeF-25'></span>[[#cite-25|[25]]]. Courtesy of Prof. Mehdi Ahmadian, VirginiaTech, USA.

@@ Line 76: / Line 76: @@
 <span id="eq-4"></span>
 <span id="eq-5"></span>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{| style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" class="formulaSCP"
 |-
+|
-{| style="text-align: left; margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\diff }{\diff t}\boldsymbol{X}(\lambda ,t) = \boldsymbol{u}(\boldsymbol{X}(\lambda ,t),t) </math>
+| <math>\mathit{\boldsymbol{U}}\left( \lambda ,t\right) :=\frac{d\mathit{\boldsymbol{X}}\left( \lambda ,t\right) }{dt}=</math><math>\mathit{\boldsymbol{u}}\left( \mathit{\boldsymbol{X}}\left( \lambda ,t\right) ,t\right)</math>
-| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
+|}
 |-
-| style="text-align: center;" | <math>  \frac{\diff ^2}{\diff t^2}\boldsymbol{X}(\lambda ,t) = \boldsymbol{a}(\boldsymbol{X}(\lambda ,t),t) </math>
+| <math>\frac{d\mathit{\boldsymbol{U}}\left( \lambda ,t\right) }{dt}=\frac{{d}^{2}\mathit{\boldsymbol{X}}\left( \lambda ,t\right) }{d{t}^{2}}=</math><math>\mathit{\boldsymbol{a}}\left( \mathit{\boldsymbol{X}}\left( \lambda ,t\right) ,t\right)</math>
-| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
 |}
 |}
@@ Line 93: / Line 97: @@
 <span id="eq-6"></span>
 <span id="eq-7"></span>
-{| class="formulaSCP" style="width: 100%; text-align: left;"
+{| style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" class="formulaSCP"
 |-
+|
-{| style="text-align: left; margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{\diff }{\diff t}\boldsymbol{X}^{h}(\lambda ,t) = \boldsymbol{u}^{h}(\boldsymbol{X}^{h}(\lambda ,t),t^{n}) = \mathbf{A}^{n} \boldsymbol{X}^{h}(\lambda ,t) + \boldsymbol{b}^{n} </math>
+| <math>\frac{d{\mathit{\boldsymbol{X}}}^{h}\left( \lambda ,t\right) }{dt}={\mathit{\boldsymbol{u}}}^{h}\left( {\mathit{\boldsymbol{X}}}^{\mathit{\boldsymbol{h}}}\left( \lambda ,t\right) ,{t}^{n}\right) =</math><math>\boldsymbol{A}\mathit{\boldsymbol{\, }}{\mathit{\boldsymbol{X}}}^{h}\left( \lambda ,t\right) +</math><math>\mathit{\boldsymbol{b}}</math>
-| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
+|}
 |-
-| style="text-align: center;" | <math>  \frac{\diff }{\diff t}\boldsymbol{U}^{h}(\lambda ,t) = \boldsymbol{a}^{h}(\boldsymbol{X}^{h}(\lambda ,t),t^{n}) = \mathbf{C}^{n} \boldsymbol{X}^{h}(\lambda ,t) + \boldsymbol{d}^{n}  </math>
+| <math>\frac{d{\mathit{\boldsymbol{U}}}^{h}\left( \lambda ,t\right) }{dt}={\mathit{\boldsymbol{a}}}^{h}\left( {\mathit{\boldsymbol{X}}}^{h}\left( \lambda ,t\right) ,{t}^{n}\right) =</math><math>\boldsymbol{C}\mathit{\boldsymbol{\, }}{\mathit{\boldsymbol{X}}}^{h}\left( \lambda ,t\right) +</math><math>\mathit{\boldsymbol{d}}</math>
-| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
 |}
 |}

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