García-Espinosa 2016a - Revision history

Scipediacontent at 09:34, 26 August 2019

2019-08-26T09:34:09Z

Scipediacontent at 09:33, 26 August 2019

2019-08-26T09:33:04Z

Scipediacontent at 09:31, 26 August 2019

2019-08-26T09:31:46Z

Scipediacontent at 09:30, 26 August 2019

2019-08-26T09:30:00Z

Scipediacontent at 09:29, 26 August 2019

2019-08-26T09:29:19Z

Julio at 10:18, 4 April 2017

2017-04-04T10:18:09Z

Scipediacontent: Abstract added

2016-07-13T16:27:10Z

Abstract added

Julio: Julio moved page Draft García-Espinosa 882755931 to García-Espinosa 2016a

2016-06-03T18:04:35Z

Julio moved page Draft García-Espinosa 882755931 to García-Espinosa 2016a

Julio at 15:15, 23 May 2016

2016-05-23T15:15:34Z

Julio at 15:11, 23 May 2016

2016-05-23T15:11:11Z

@@ Line 279: / Line 279: @@
 The fluid-structure coupling is performed by an implicit interfield iteration algorithm [<span id='cite-23'></span>[[#23|23]]], based on a Block Gauss-Seidel method [<span id='cite-24'></span>[[#24|24]]]. For the ''i-''th iteration of the current time step ''n'', let <math display="inline">{\boldsymbol{\phi }}^{n,i}</math>, <math display="inline">{\boldsymbol{\eta }}^{n,i}</math>, <math display="inline">{\boldsymbol{p}}^{n,i}</math> be the vectors of the velocity potential, free surface elevation and pressure in the corresponding nodes of the fluid, <math display="inline">{\boldsymbol{x}}^{n,i}</math> the vector containing the three components of the displacement of the nodes of the seals structure, with <math display="inline">{\boldsymbol{x}}_{k}^{n,i}</math> the vector of the ''k''-th component of the displacement field, and <math display="inline">{\boldsymbol{H}}_{seal}^{n,i-1}</math> the vector of the elevation field of the seal nodes, then the fluid-structure interaction algorithm, can be written as:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="text-align: center;"| <math>\begin{matrix}\left({\boldsymbol{p}}^{n,i},\, {\boldsymbol{\phi }}^{n,i}\right)={\boldsymbol{F}}^{p}\left({\boldsymbol{\phi }}^{n,i-1},{\boldsymbol{\eta }}^{n,i-1},{\boldsymbol{H}}_{seal}^{n,i-1},{{\boldsymbol{\eta }}^{n-1,i-1}\boldsymbol{,\phi }}^{n-1}\right)\\{\boldsymbol{x}}^{n,i}={\boldsymbol{S}}^{q}\left({\boldsymbol{x}}^{n,i-1},{\boldsymbol{p}}^{n,i},{\boldsymbol{x}}^{n-1},{\boldsymbol{x}}^{n-2}\right)\\{\boldsymbol{H}}_{seal}^{n,i-1}={\boldsymbol{H}}_{seal}^{0}+{\boldsymbol{x}}_{k}^{n,i}\end{matrix}</math>
-| style="text-align: right;"|(11)
+{| style="text-align: center;margin:auto;width: 100%;"
 |}
@@ Line 301: / Line 305: @@
 Let <math display="inline">{\boldsymbol{I}}^{FS}{\boldsymbol{,\, I}}^{SF}</math> be the interpolator operators from the fluid boundary mesh to the solid interface, and vice-versa, then the fluid-structure interaction algorithm reads
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
-| <math>\begin{matrix}\left({\boldsymbol{p}}^{n,i},\, {\boldsymbol{\phi }}^{n,i}\right)={\boldsymbol{F}}^{p}\left({\boldsymbol{\phi }}^{n,i-1},{\boldsymbol{\eta }}^{n,i-1},{\boldsymbol{H}}_{seal}^{n,i-1},{{\boldsymbol{\eta }}^{n-1,i-1}\boldsymbol{,\phi }}^{n-1}\right)\\{\check{\boldsymbol{p}}}^{n,i}={\boldsymbol{I}}^{FS}\left({\boldsymbol{p}}^{n,i}\right)\\{\boldsymbol{x}}^{n,i}={\boldsymbol{S}}^{q}\left({\boldsymbol{x}}^{n,i-1},{\check{\boldsymbol{p}}}^{n,i},{\boldsymbol{x}}^{n-1},{\boldsymbol{x}}^{n-2}\right)\\{\check{\boldsymbol{x}}}^{n,i}={\boldsymbol{I}}^{SF}\left({\boldsymbol{x}}^{n,i}\right)\\{\boldsymbol{H}}_{seal}^{n,i-1}={\boldsymbol{H}}_{seal}^{0}+{\check{\boldsymbol{x}}}_{k}^{n,i}\end{matrix}</math>
+| <math>\begin{matrix}\left( {\boldsymbol{p}}^{n,i},\, {\boldsymbol{\phi }}^{n,i}\right) ={\boldsymbol{F}}^{p}\left( {\boldsymbol{\phi }}^{n,i-1},{\boldsymbol{\eta }}^{n,i-1},{\boldsymbol{H}}_{seal}^{n,i-1},{{\boldsymbol{\eta }}^{n-1,i-1}\boldsymbol{,\phi }}^{n-1}\right) \\{\check{\boldsymbol{p}}}^{n,i}={\boldsymbol{I}}^{FS}\left( {\boldsymbol{p}}^{n,i}\right) \\{\boldsymbol{x}}^{n,i}={\boldsymbol{S}}^{q}\left( {\boldsymbol{x}}^{n,i-1},{\check{\boldsymbol{p}}}^{n,i},{\boldsymbol{x}}^{n-1},{\boldsymbol{x}}^{n-2}\right) \\{\check{\boldsymbol{x}}}^{n,i}={\boldsymbol{I}}^{SF}\left( {\boldsymbol{x}}^{n,i}\right) \\{\boldsymbol{H}}_{seal}^{n,i-1}={\boldsymbol{H}}_{seal}^{0}+{\check{\boldsymbol{x}}}_{k}^{n,i}\end{matrix}</math>
-| style="text-align: right;"|(12)
+|}
 |}
 Even though the iterations within each time step lead to the monolithic solution of the coupled problem, sometimes relaxation techniques are combined with this method to ensure the convergence of the coupling scheme. In this work, a relaxation method based on Aitken’s acceleration has been adopted [<span id='cite-25'></span>[[#25|25]]]. Let <math display="inline">{\omega }_{i}</math> be the optimal relaxation parameter, obtained via Aitken’s method, the resulting algorithm can be written as:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
-| <math>\begin{matrix}\left({\boldsymbol{p}}^{n,i},\, {\boldsymbol{\phi }}^{n,i}\right)={\boldsymbol{F}}^{p}\left({\boldsymbol{\phi }}^{n,i-1},{\boldsymbol{\eta }}^{n,i-1},{\boldsymbol{H}}_{seal}^{n,i-1},{{\boldsymbol{\eta }}^{n-1,i-1}\boldsymbol{,\phi }}^{n-1}\right)\\{\hat{\boldsymbol{p}}}^{n,i}=\left(1-{\omega }_{i}\right){\hat{\boldsymbol{p}}}^{n,i-1}+{\omega }_{i}{\boldsymbol{p}}^{n,i}\\{\check{\boldsymbol{p}}}^{n,i}={\boldsymbol{I}}^{FS}\left({\hat{\boldsymbol{p}}}^{n,i}\right)\\{\boldsymbol{x}}^{n,i}={\boldsymbol{S}}^{q}\left({\boldsymbol{x}}^{n,i-1},{\check{\boldsymbol{p}}}^{n,i},{\boldsymbol{x}}^{n-1},{\boldsymbol{x}}^{n-2}\right)\\{\check{\boldsymbol{x}}}^{n,i}={\boldsymbol{I}}^{SF}\left({\boldsymbol{x}}^{n,i}\right)\\{\boldsymbol{H}}_{seal}^{n,i-1}={\boldsymbol{H}}_{seal}^{0}+{\check{\boldsymbol{x}}}_{k}^{n,i}\end{matrix}</math>
+| <math>\begin{matrix}\left( {\boldsymbol{p}}^{n,i},\, {\boldsymbol{\phi }}^{n,i}\right) ={\boldsymbol{F}}^{p}\left( {\boldsymbol{\phi }}^{n,i-1},{\boldsymbol{\eta }}^{n,i-1},{\boldsymbol{H}}_{seal}^{n,i-1},{{\boldsymbol{\eta }}^{n-1,i-1}\boldsymbol{,\phi }}^{n-1}\right) \\{\hat{\boldsymbol{p}}}^{n,i}=\left( 1-{\omega }_{i}\right) {\hat{\boldsymbol{p}}}^{n,i-1}+{\omega }_{i}{\boldsymbol{p}}^{n,i}\\{\check{\boldsymbol{p}}}^{n,i}={\boldsymbol{I}}^{FS}\left( {\hat{\boldsymbol{p}}}^{n,i}\right) \\{\boldsymbol{x}}^{n,i}={\boldsymbol{S}}^{q}\left( {\boldsymbol{x}}^{n,i-1},{\check{\boldsymbol{p}}}^{n,i},{\boldsymbol{x}}^{n-1},{\boldsymbol{x}}^{n-2}\right) \\{\check{\boldsymbol{x}}}^{n,i}={\boldsymbol{I}}^{SF}\left( {\boldsymbol{x}}^{n,i}\right) \\{\boldsymbol{H}}_{seal}^{n,i-1}={\boldsymbol{H}}_{seal}^{0}+{\check{\boldsymbol{x}}}_{k}^{n,i}\end{matrix}</math>
-| style="text-align: right;"|(13)
+|}
 |}
@@ Line 322: / Line 334: @@
 <span id='_Toc337282452'></span>
 ==5. Validation of FSSI algorithm with flexible bow seals ==

@@ Line 225: / Line 225: @@
 The main challenge for an algorithm like this is to be capable of capturing when a node goes from dry to wet and vice versa, as well as estimating the pressure field on the wet nodes. <span id='cite-_Ref326165139'></span>[[#_Ref326165139|Figure 3]] shows the wetting and drying iterative algorithm. For a dry node, the implementation of both, the kinetic and dynamic boundary condition is the same as for any other node not interacting with the seal. However, for wet nodes, the free surface boundary condition is imposed via imposing that the free surface elevation matches the seal elevation, <math display="inline">\, {H}_{seal}</math>, and ensuring that there is no flow across the seal. These two conditions are represented by the following equations:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="vertical-align: top;"| <math>{\eta }_{wet}^{n+1}={H}_{seal}^{n+1}</math>
-<math>{\phi }_{z}^{n+1}=2\left(\frac{{\eta }^{n+1}{-\eta }^{n}}{\Delta t}+\, \left|{\mathit{\boldsymbol{U}}}^{n+1/2}\right|{\left({\delta }_{L}\eta \right)}^{n+1/2}\right)-{\phi }_{z}^{n}</math>
+<math>{\phi }_{z}^{n+1}=2\left( \frac{{\eta }^{n+1}{-\eta }^{n}}{\Delta t}+\, \left| {\mathit{\boldsymbol{U}}}^{n+1/2}\right| {\left( {\delta }_{L}\eta \right) }^{n+1/2}\right) -</math><math>{\phi }_{z}^{n}</math>
-| style="text-align: right;"|(9)
+|}
 |}
@@ Line 247: / Line 251: @@
 The change from being a dry node to become a wet node is identified via the kinematic BC though the condition  <math display="inline">{\eta }_{wet}^{n+1}>{H}_{seal}^{n+1}\,</math>  . On the other hand, the switch from being a wet node to become a dry one is carried out by comparing the dynamic pressure with the reference pressure. The dynamic pressure on wet nodes is obtained by the dynamic BC:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
-| style="vertical-align: top;"| <math>{P}_{wet}^{n+1}=-\rho \left[\frac{{\phi }^{n+1}{-\phi }^{n}}{\Delta t}+\, \left|{\boldsymbol{U}}^{n+1}\right|\cdot {\left({\delta }_{L}\phi \right)}^{n+1}-{\frac{1}{2}}{\left({\nabla }_{h}\phi \right)}^{n+1}\cdot {\left({\nabla }_{h}\phi \right)}^{n+1}+{\frac{1}{2}}{\phi }_{z}^{n+1}{\phi }_{z}^{n+1}+\, g{\eta }^{n+1}\right]</math>
+| <math>{P}_{wet}^{n+1}=-\rho \left[ \frac{{\phi }^{n+1}{-\phi }^{n}}{\Delta t}+\, \left| {\boldsymbol{U}}^{n+1}\right| \cdot {\left( {\delta }_{L}\phi \right) }^{n+1}-\right. </math><math>\left. {\frac{1}{2}}{\left( {\nabla }_{h}\phi \right) }^{n+1}\cdot {\left( {\nabla }_{h}\phi \right) }^{n+1}+\right. </math><math>\left. {\frac{1}{2}}{\phi }_{z}^{n+1}{\phi }_{z}^{n+1}+\, g{\eta }^{n+1}\right]</math>
-| style="text-align: right;"|(10)
+|}
 |}
@@ Line 265: / Line 273: @@
 <span id='_Toc337282430'></span>
 ===4.2  Free surface-flexible seals coupling algorithm===

@@ Line 120: / Line 120: @@
 Solving the free surface boundary condition efficiently is essential when dealing with free surface-structure interaction (FSSI) problems. The free surface conditions can be rewritten as:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="vertical-align: top;"| <math>{\partial }_{t}\phi +\mathit{\boldsymbol{U}}\cdot {\nabla }_{h}\phi -{{\frac{1}{2}}\nabla }_{h}\phi \cdot {\nabla }_{h}\phi +\, \frac{P}{\rho }+g\eta =0</math>
-<math>{\partial }_{t}\eta +\mathit{\boldsymbol{U}}\cdot {\nabla }_{h}\eta -{\partial }_{z}\phi =0</math>
+<math>{\partial }_{t}\eta +\mathit{\boldsymbol{U}}\cdot {\nabla }_{h}\eta -{\partial }_{z}\phi =</math><math>0</math>
-| style="text-align: right;"|(4)
+|}
 |}
@@ Line 132: / Line 136: @@
 The numerical schemes adopted for solving the kinematic-dynamic free surface boundary conditions are based on an implicit midpoint rule for the kinematic condition, and backward Euler for the dynamic condition. Then <math display="inline">{\phi }^{n+1}</math> can be imposed as a Dirichlet boundary condition in the system <span id='cite-_Ref382924951'></span>[[#_Ref382924951|(2)]]. The schemes read as follows:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="vertical-align: top;"| <math>{\phi }^{n+1}={\phi }^{n}-\Delta t\, {\mathit{\boldsymbol{U}}}^{n+1}\cdot {\left({\nabla }_{h}\phi \right)}^{n+1}-\Delta t\cdot {\left({\nabla }_{h}\phi \right)}^{n+1}\cdot {\left({\nabla }_{h}\phi \right)}^{n+1}-\Delta t\, \left({P}^{n+1}/\rho +g{\eta }^{n+1}\right)=0</math>
-<math>{\eta }^{n+1}={\eta }^{n}-\Delta t\, {\mathit{\boldsymbol{U}}}^{n+1/2}\cdot {\left({\nabla }_{h}\eta \right)}^{n+1/2}+\Delta t{\left({\partial }_{z}\phi \right)}^{n+1/2}=0</math>
+<math>{\eta }^{n+1}={\eta }^{n}-\Delta t\, {\mathit{\boldsymbol{U}}}^{n+1/2}\cdot {\left( {\nabla }_{h}\eta \right) }^{n+1/2}+</math><math>\Delta t{\left( {\partial }_{z}\phi \right) }^{n+1/2}=0</math>
-| style="text-align: right;"|(5)
+|}
 |}
 It is well known that the convective terms appearing in the above equations can result in solutions corrupted by spurious node-to-node oscillations, if no special care is taken when evaluating them. In this work, the convective term is obtained by differentiating along streamlines:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="vertical-align: top;"| <math>{\mathit{\boldsymbol{U}}}^{n+1}\cdot {\left({\nabla }_{h}\phi \right)}^{n+1}=\left|{\mathit{\boldsymbol{U}}}^{n+1}\right|{\left({\partial }_{L}\phi \right)}^{n+1}</math>
-<math>{\mathit{\boldsymbol{U}}}^{n}\cdot {\left({\nabla }_{h}\eta \right)}^{n}=\left|{\mathit{\boldsymbol{U}}}^{n}\right|{\left({\partial }_{L}\eta \right)}^{n}</math>
+<math>{\mathit{\boldsymbol{U}}}^{n}\cdot {\left( {\nabla }_{h}\eta \right) }^{n}=\left| {\mathit{\boldsymbol{U}}}^{n}\right| {\left( {\partial }_{L}\eta \right) }^{n}</math>
-| style="text-align: right;"|(6)
+|}
 |}
@@ Line 163: / Line 175: @@
 <span id='cite-_Ref326165045'></span>[[#_Ref326165045|Figure 1]] shows the tracing of the streamline at node C. The left (L) and forward left (FL) points are the upstream points, while the right (R) point corresponds to the downstream point. The values of the velocity potential <math display="inline">\phi</math>  and free surface elevation <math display="inline">\eta</math>  at L, FL and R points are obtained by linear interpolation between the nodes of the edges where they lie on. The streamline differential operator reads as:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="vertical-align: top;"| <math>{\partial }_{L}\phi \approx {\delta }_{L}\phi ={\alpha }_{R}{\phi }_{R}+{\alpha }_{C}{\phi }_{C}+{\alpha }_{L}{\phi }_{L}+{\alpha }_{FL}{\phi }_{FL}</math>
-<math>{\partial }_{L}\eta \approx {\delta }_{L}\eta ={\alpha }_{R}{\eta }_{R}+{\alpha }_{C}{\eta }_{C}+{\alpha }_{L}{\eta }_{L}+{\alpha }_{FL}{\eta }_{FL}</math>
+<math>{\partial }_{L}\eta \approx {\delta }_{L}\eta ={\alpha }_{R}{\eta }_{R}+{\alpha }_{C}{\eta }_{C}+</math><math>{\alpha }_{L}{\eta }_{L}+{\alpha }_{FL}{\eta }_{FL}</math>
-| style="text-align: right;"|(7)
+|}
 |}
 where the stencils are obtained as a one dimensional third order finite difference scheme:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="vertical-align: top;"| <math>\left[\begin{matrix}\begin{matrix}1&\quad \, \, 1\\{\Delta x}_{R}&\quad \, \, 0\end{matrix}&\begin{matrix}1&1\\{-\Delta x}_{L}&\quad \, \, -{\Delta x}_{L}-{\Delta x}_{FL}\end{matrix}\\\begin{matrix}{\left({\Delta x}_{R}\right)}^{2}&0\\{\left({\Delta x}_{R}\right)}^{3}&0\end{matrix}&\quad \begin{matrix}{\left({\Delta x}_{L}\right)}^{2}&\, {\left({\Delta x}_{L}+{\Delta x}_{FL}\right)}^{2}\\{-\left({\Delta x}_{L}\right)}^{3}&-{\left({\Delta x}_{L}+{\Delta x}_{FL}\right)}^{3}\end{matrix}\end{matrix}\right]\left[\begin{matrix}\begin{matrix}{\alpha }_{R}\\{\alpha }_{C}\end{matrix}\\\begin{matrix}{\alpha }_{L}\\{\alpha }_{FL}\end{matrix}\end{matrix}\right]=\left[\begin{matrix}\begin{matrix}0\\1\end{matrix}\\\begin{matrix}0\\0\end{matrix}\end{matrix}\right]</math>
-| style="text-align: right;"|(8)
+{| style="vertical-align: top;margin:auto;width: 100%;"
 |}

@@ Line 88: / Line 88: @@
 Find <math display="inline">{\, [{\phi }_{h}]\epsilon Q}_{h}</math>, by solving the discrete variational problem:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
-| style="vertical-align: top;"| <math>\int_{\Omega }^{}\nabla {\upsilon }_{h}\cdot \nabla {\phi }_{h}d\Omega =</math>
+{| style="vertical-align: top;margin:auto;width: 100%;"
-<math>=\int_{{\Gamma }^{B}}^{}{\upsilon }_{h}\cdot {\hat{\phi }}_{n}^{B}d\Gamma +\int_{{\Gamma }^{{Z}_{0}}}^{}{\upsilon }_{h}\cdot {\hat{\phi }}_{n}^{{Z}_{0}}d\Gamma +\int_{{\Gamma }^{{Z}_{-H}}}^{}{\upsilon }_{h}\cdot {\hat{\phi }}_{n}^{{Z}_{-H}}d\Gamma \quad \, \forall {\upsilon }_{h}\in {Q}_{h}</math>
+|-
+<math>=\int_{{\Gamma }^{B}}^{}{\upsilon }_{h}\cdot {\hat{\phi }}_{n}^{B}d\Gamma +\int_{{\Gamma }^{{Z}_{0}}}^{}{\upsilon }_{h}\cdot {\hat{\phi }}_{n}^{{Z}_{0}}d\Gamma +\int_{{\Gamma }^{{Z}_{-H}}}^{}{\upsilon }_{h}\cdot {\hat{\phi }}_{n}^{{Z}_{-H}}d\Gamma \quad \, \forall {\upsilon }_{h}\in {Q}_{h}</math>
-| style="text-align: right;vertical-align: top;"|
+|}
-| style="text-align: right;"|<span id='_Ref382924947'></span>
+|  style="text-align: right;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref382924947'></span><span id='_Ref382924951'></span>(2)
 (2)
 |}
 Where <math display="inline">{\hat{\phi }}_{n}^{B},\, {\hat{\phi }}_{n}^{{Z}_{0}}\,</math> and <math display="inline">{\hat{\phi }}_{n}^{{Z}_{-H}}</math> are the potential normal gradients corresponding to the Neumann boundary conditions on body, radiation boundary, free surface and bottom, respectively. At this point, it is useful to introduce the associated matrix form
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="vertical-align: top;"| <math>\overline{\overline{\mathit{\boldsymbol{L}}}}\mathit{\boldsymbol{\phi }}={\mathit{\boldsymbol{b}}}^{B}+{\mathit{\boldsymbol{b}}}^{\Gamma o}+{\mathit{\boldsymbol{b}}}^{{\Gamma }_{-H}}</math>
-| style="text-align: right;vertical-align: top;"|
+{| style="vertical-align: top;margin:auto;width: 100%;"
-| style="text-align: right;"|<span id='_Ref382927175'></span>
+|-
-(3)
+| <math>\overline{\overline{\mathit{\boldsymbol{L}}}}\mathit{\boldsymbol{\phi }}={\mathit{\boldsymbol{b}}}^{B}+</math><math>{\mathit{\boldsymbol{b}}}^{\Gamma o}+{\mathit{\boldsymbol{b}}}^{{\Gamma }_{-H}}</math>
 |}
@@ Line 113: / Line 115: @@
 <span id='_Toc337282425'></span>
 ===2.3  Free surface boundary condition===

@@ Line 47: / Line 47: @@
 Based on the previous assumption, the governing equations for the first order diffraction-radiation wave problem on a frame of reference fixed on the ship are:
-{| style="width: 100%;"
+{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 | style="vertical-align: top;"| <math>{\nabla }^{2}\phi =0</math>
-<math>{\partial }_{t}\phi -\mathit{\boldsymbol{u}}\cdot {\nabla }_{h}\phi +{{\frac{1}{2}}\nabla }_{h}\phi \cdot {\nabla }_{h}\phi +\, \frac{P}{\rho }+g\eta =0</math>
+<math>{\partial }_{t}\phi -\mathit{\boldsymbol{u}}\cdot {\nabla }_{h}\phi +{{\frac{1}{2}}\nabla }_{h}\phi \cdot {\nabla }_{h}\phi +</math><math>\, \frac{P}{\rho }+g\eta =0</math>
-<math>{\partial }_{t}\eta +\left(\mathit{\boldsymbol{-u}}+{\nabla }_{h}\phi \right)\cdot {\nabla }_{h}\eta -{\partial }_{z}\phi =0</math>
+<math>{\partial }_{t}\eta +\left( \mathit{\boldsymbol{-u}}+{\nabla }_{h}\phi \right) \cdot {\nabla }_{h}\eta -</math><math>{\partial }_{z}\phi =0</math>
 <math>\nabla \phi \cdot {\mathit{\boldsymbol{n}}}_{b}=-\mathit{\boldsymbol{u}}\cdot {\mathit{\boldsymbol{n}}}_{b}</math>
 <math>{\partial }_{z}\phi =0</math>
-| style="vertical-align: top;"|in &#x03a9;
+|}
 in z = 0
@@ Line 67: / Line 74: @@
 in ''z = -H''
-| style="text-align: right;"|(1)
+|}
 |}

@@ Line 590: / Line 590: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 Figure 12: Numerical results: free surface elevation (a) and top view of pressure distribution on fingers (b)</div>
 As can be seen, the bow seals deformation obtained in this analysis agrees with the qualitative experimental information available in reference [<span id='cite-_Ref382813140'></span>[[#_Ref382813140|28]]]. In fact, the deformation mode obtained in the computational model is quite similar to that found experimentally.

← Older revision		Revision as of 16:27, 13 July 2016
Line 1:		Line 1:
		+	==Abstract==
		+
		+	This paper shows the recent work of the authors in the development of a time-domain FEM model for evaluation of the seal dynamics of a surface effect ship. The fluid solver developed for this purpose, uses a potential flow approach along with a stream-line integration of the free surface. The paper focuses on the free surface-structure algorithm that has been developed to allow the simulation of the complex and highly dynamic behavior of the seals in the interface between the air cushion, and the water.
		+
		+	The developed fluid-structure interaction solver is based, on one side, on an implicit iteration algorithm, communicating pressure forces and displacements of the seals at memory level and, on the other side, on an innovative wetting and drying scheme able to predict the water action on the seals. The stability of the iterative scheme is improved by means of relaxation, and the convergence is accelerated using Aitken’s method.
		+
		+	Several validations against experimental results have been carried out to demonstrate the developed algorithm.
		+
	==1. Introduction==		==1. Introduction==

← Older revision	Revision as of 18:04, 3 June 2016
(No difference)

@@ Line 62: / Line 62: @@
 |}
-where <math display="inline">{\nabla }_{h}\,</math> is the gradient in the horizontal plane, <math display="inline">\Omega \,</math> is the fluid domain, <math display="inline">{\Gamma }_{b}\,</math> represents the wetted surface of the ship, [[Image:draft_García-Espinosa_882755931-picture-x0000_i1025.svg|19px]] is the water depth, <math display="inline">{\mathit{\boldsymbol{n}}}_{b}</math> is the normal of the ship wetted surface pointing outwards the ship, and [[Image:draft_García-Espinosa_882755931-picture-x0000_i1026.svg|16px]] is the free surface pressure.
+where <math display="inline">{\nabla }_{h}\,</math> is the gradient in the horizontal plane, <math display="inline">\Omega \,</math> is the fluid domain, <math display="inline">{\Gamma }_{b}\,</math> represents the wetted surface of the ship, <math>H</math> is the water depth, <math display="inline">{\mathit{\boldsymbol{n}}}_{b}</math> is the normal of the ship wetted surface pointing outwards the ship, and <math>P</math> is the free surface pressure.
 ===2.2  FEM solver===

← Older revision		Revision as of 15:11, 23 May 2016
Line 1:		Line 1:
−
−
−	~~<span id='_Toc337283010'></span>~~
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<big>'''A FEM fluid-structure interaction algorithm for<br/>analysis of the seal dynamics of a Surface-Effect Ship '''</big></div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~'''Julio García-Espinosa<sup>1,2</sup>, Daniel Di Capua<sup>1,2</sup>, Borja Serván-Camas<sup>1,*</sup>,<br/>Pere-Andreu Ubach<sup>1</sup> and Eugenio Oñate<sup>1,2</sup>'''</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<sup>1</sup> Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE)</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~Gran Capitán s/n, 08034 Barcelona, Spain</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~e-mail : [mailto:julio@cimne.upc.edu julio@cimne.upc.edu], web page: [http://www.cimne.upc.edu http://www.cimne.upc.edu]</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<sup>2</sup> Universitat Politècnica de Catalunya - BarcelonaTech (UPC)</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~Campus Nàutica, Edif. NT3, C. Escar 6-8, 08039 Barcelona, Spain</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~web page: [http://www http://www].upc.edu</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<sup>*</sup> Corresponding author. Tel.: +34 93 405 4627.</div>~~
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−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~E-mail address: [mailto:bservan@cimne.upc.edu bservan@cimne.upc.edu] (B. Serván-Camas).</div>~~
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−	~~URL: [http://www.cimne.com/ http://www.cimne.com/] (B. Serván-Camas).</div>~~
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−	~~<big>'''ABSTRACT'''</big>~~
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−	This paper shows the recent work of the authors in the development of a time-domain FEM model for evaluation of the seal dynamics of a surface effect ship. The fluid solver developed for this purpose, uses a potential flow approach along with a stream-line integration of the free surface. The paper focuses on the free surface-structure algorithm that has been developed to allow the simulation of the complex and highly dynamic behavior of the seals in the interface between the air cushion, and the water.
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−	The developed fluid-structure interaction solver is based, on one side, on an implicit iteration algorithm, communicating pressure forces and displacements of the seals at memory level and, on the other side, on an innovative wetting and drying scheme able to predict the water action on the seals. The stability of the iterative scheme is improved by means of relaxation, and the convergence is accelerated using Aitken’s method.
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−	~~Several validations against experimental results have been carried out to demonstrate the developed algorithm.~~
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−	~~'''Keywords:''' surface effect ship; seal dynamics; finite element method; fluid-structure interaction.~~
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	==1. Introduction==		==1. Introduction==