Franci et al 2014b - Revision history

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Cinmemj at 13:34, 22 March 2018

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Cinmemj at 13:04, 22 March 2018

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Cinmemj at 12:56, 22 March 2018

2018-03-22T12:56:21Z

@@ Line 1,558: / Line 1,558: @@
 <div id="cite-27"></div>
-'''[[#citeF-27|[27]]]'''  E.&nbsp;Walhorn, A.&nbsp;Kolke&nbsp;B. Hubner, and D.Dinkler. Fluid-structure coupling within a monolithic model involving free   surface flows. ''Computer <math>{&}</math> Structures Methods in Applied Mechanics and   Engineering'', 83 (25-26):2100&#8211;2111, 2005.
+'''[[#citeF-27|[27]]]'''  E.&nbsp;Walhorn, A.&nbsp;Kolke&nbsp;B. Hubner, and D.Dinkler. Fluid-structure coupling within a monolithic model involving free   surface flows. ''Computer & Structures Methods in Applied Mechanics and   Engineering'', 83 (25-26):2100&#8211;2111, 2005.
 <div id="cite-28"></div>
 '''[[#citeF-28|[28]]]'''  C.&nbsp;Wood, A.J. Gil, O.Hassan, and J.Bonet. Partitioned block gauss-seidel coupling for dynamic fluid-strucure   interaction. ''Computers and Structures'', 88:1367&#8211;1382, 2010.

@@ Line 1,477: / Line 1,477: @@
 Finally, the unified formulation has been tested solving different benchmark problems. In particular, the collapse of a water column on a deformable membrane and the water entry of a nylon sphere have been numerically studied in 2D and 3D. The comparison of the numerical results with the ones published in the literature and with scientific tests is fully satisfactory and evidences the efficiency and the accuracy of the proposed method.
-===BIBLIOGRAPHY===
+==Bibliography==
 <div id="cite-1"></div>

@@ Line 1,250: / Line 1,250: @@
 <div class="center">
-<math display="inline">{{\bf C}^{\mu }}=\mu \left[                      \begin{array}{cccccc}                        2/3 & -1/3 & -1/3 & 0 & 0 & 0 \\                         &   2/3 & -1/3 &0  &0  & 0 \\                         &  &  2/3 & 0 & 0 & 0 \\                         &  &  & 1 & 0 & 0 \\                        {Sym.} &  &  &  & 1 & 0 \\                         &  &  &  &  & 1 \\                      \end{array} \right] }</math></div>
+<math display="inline">{{\bf C}^{\mu }}=\mu \left[                      \begin{array}{cccccc}                        2/3 & -1/3 & -1/3 & 0 & 0 & 0 \\                         &   2/3 & -1/3 &0  &0  & 0 \\                         &  &  2/3 & 0 & 0 & 0 \\                         &  &  & 1 & 0 & 0 \\                        {Sym.} &  &  &  & 1 & 0 \\                         &  &  &  &  & 1 \\                      \end{array} \right] </math></div>
 The volumetric part of the tangent matrix can compromise the conditioning of the linear system because its terms are orders of magnitude larger than the viscous part. In order to prevent the numerical instabilities originated by the ill-conditioning of the tangent matrix, a reduced pseudo-bulk modulus can be used in the expression of <math display="inline">{K^{\kappa }}</math> without altering the numerical results <span id='citeF-12'></span>[[#cite-12|[12]]].

@@ Line 1,119: / Line 1,119: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>\sigma =\sigma ' - pI =2 \mu D^{dev} - pI </math>
+| style="text-align: center;" | <math>\boldsymbol{\sigma} =\boldsymbol{\sigma}' - p{\bf I} =2 \mu {\bf D}^{dev} - p{\bf I} </math>
 |}
 | style="width: 5px;text-align: right;" | (75)
@@ Line 1,136: / Line 1,136: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>^{n+1}\sigma = \left(2 \mu I^{dev} - \Delta t \kappa I\otimes I  \right): D - ^{n}p I  </math>
+| style="text-align: center;" | <math>^{n+1}\boldsymbol{\sigma} = \left(2 \mu {\bf I}^{dev} - \Delta t \kappa {\bf I}\otimes {\bf I}  \right): {\bf D} - ^{n}p {\bf I}  </math>
 |}
 | style="width: 5px;text-align: right;" | (76)
@@ Line 1,148: / Line 1,148: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>I^{dev}=I-\frac{1}{3}I\otimes I </math>
+| style="text-align: center;" | <math>{\bf I}^{dev}={\bf I}-\frac{1}{3}{\bf I}\otimes {\bf I} </math>
 |}
 | style="width: 5px;text-align: right;" | (77)
@@ Line 1,161: / Line 1,161: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>^{n+1}\Delta \sigma =^{n+1}\sigma  - ^{n}\sigma =\left( C^{d} +  C^{\kappa } \right): D </math>
+| style="text-align: center;" | <math>^{n+1}\Delta \boldsymbol{\sigma} =^{n+1}\boldsymbol{\sigma}  - ^{n}\boldsymbol{\sigma} =\left( {\bf C}^{d} +  {\bf C}^{\kappa } \right): {\bf D} </math>
 |}
 | style="width: 5px;text-align: right;" | (78)
@@ Line 1,173: / Line 1,173: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>^{n}\sigma _{ij}= - ^{n}p I </math>
+| style="text-align: center;" | <math>^{n}\boldsymbol{\sigma} _{ij}= - ^{n}p {\bf I} </math>
 |}
 | style="width: 5px;text-align: right;" | (79)
@@ Line 1,183: / Line 1,183: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>C^{d}=2 \mu I^{dev} </math>
+| style="text-align: center;" | <math>{\bf C}^{d}=2 \mu {\bf I}^{dev} </math>
 |}
 | style="width: 5px;text-align: right;" | (80)
@@ Line 1,193: / Line 1,193: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>C^{\kappa }= - \Delta t \kappa I\otimes I </math>
+| style="text-align: center;" | <math>{\bf C}^{\kappa }= - \Delta t \kappa {\bf I}\otimes {\bf I} </math>
 |}
 | style="width: 5px;text-align: right;" | (81)
@@ Line 1,206: / Line 1,206: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>\dot \sigma =\frac{\Delta  \sigma }{\Delta t}= \left(\frac{ C^{d}}{\Delta t} + \frac{C^{\kappa }}{\Delta t}   \right): D  </math>
+| style="text-align: center;" | <math>\dot \boldsymbol{\sigma} =\frac{\Delta  \boldsymbol{\sigma}}{\Delta t}= \left(\frac{ {\bf C}^{d}}{\Delta t} + \frac{{\bf C}^{\kappa }}{\Delta t}   \right): D  </math>
 |}
 | style="width: 5px;text-align: right;" | (82)
@@ Line 1,219: / Line 1,219: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>K^{m}_{NF}=K^{\mu } +K^{\kappa }  </math>
+| style="text-align: center;" | <math>{\bf K}^{m}_{NF}={\bf K}^{\mu } +{\bf K}^{\kappa }  </math>
 |}
 | style="width: 5px;text-align: right;" | (83)
 |}
-where <math display="inline">{K^{\mu }}</math> and <math display="inline">{K^{\kappa }}</math> are defined for a generic finite element <math display="inline">{e}</math> and the pair of nodes <math display="inline">{I,J}</math> as:
+where <math display="inline">{{\bf K}^{\mu }}</math> and <math display="inline">{{\bf K}^{\kappa }}</math> are defined for a generic finite element <math display="inline">{e}</math> and the pair of nodes <math display="inline">{I,J}</math> as:
 <span id="eq-84"></span>
@@ Line 1,232: / Line 1,232: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>K^{\mu }_{IJ}=\int _{\Omega ^e} B^{eT}_I  C^{\mu }  B^e_J    d\Omega  </math>
+| style="text-align: center;" | <math>{\bf K}^{\mu }_{IJ}=\int _{\Omega ^e} {\bf B}^{eT}_I  {\bf C}^{\mu }  {\bf B}^e_J    d\Omega  </math>
 |}
 | style="width: 5px;text-align: right;" | (84)
@@ Line 1,243: / Line 1,243: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>K^{\kappa }_{IJ}=\int _{\Omega ^e} B^{eT}_I  m \kappa  m^T B^e_J    d\Omega  </math>
+| style="text-align: center;" | <math>{\bf K}^{\kappa }_{IJ}=\int _{\Omega ^e} {\bf B}^{eT}_I  {\bf m} \kappa  {\bf m}^T {\bf B}^e_J    d\Omega  </math>
 |}
 | style="width: 5px;text-align: right;" | (85)
 |}
-<math display="inline">\begin{array}{l} \\   \hbox{with}       {C^{\mu }}=\mu \left[                      \begin{array}{cccccc}                        2/3 & -1/3 & -1/3 & 0 & 0 & 0 \\                         &   2/3 & -1/3 &0  &0  & 0 \\                         &  &  2/3 & 0 & 0 & 0 \\                         &  &  & 1 & 0 & 0 \\                        {Sym.} &  &  &  & 1 & 0 \\                         &  &  &  &  & 1 \\                      \end{array} \right] \\ \\ \end{array}</math>
+<div class="center">
 The volumetric part of the tangent matrix can compromise the conditioning of the linear system because its terms are orders of magnitude larger than the viscous part. In order to prevent the numerical instabilities originated by the ill-conditioning of the tangent matrix, a reduced pseudo-bulk modulus can be used in the expression of <math display="inline">{K^{\kappa }}</math> without altering the numerical results <span id='citeF-12'></span>[[#cite-12|[12]]].
@@ Line 1,266: / Line 1,268: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>\left(\frac{1}{ \Delta t} \hbox{M}_{1} +   \frac{1}{ {\Delta t}^2} \hbox{M}_{2} + {L}+{M}_b \right){^{n+1}\bar {p}} = {Q}^T {^{n+1}\bar{v}}+{^ng}^S </math>
+| style="text-align: center;" | <math>\left(\frac{1}{ \Delta t} {\bf M}_{1} +   \frac{1}{ {\Delta t}^2} \mathbf{M}_{2} + {\bf L}+{\bf M}_b \right){^{n+1}\bar {\bf p}} = {\bf Q}^T {^{n+1}\bar{\bf v}}+{^ng}^S </math>
 |}
 | style="width: 5px;text-align: right;" | (86)
@@ Line 1,279: / Line 1,281: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>^n{g}^S = {f}_p + \frac{1}{ \Delta t} \hbox{M}_{1} {^{n} \bar {p}} + \frac{1}{ \Delta t} \hbox{M}_{2}  \left(\frac{ ^{n}\bar {p}}{\Delta t} + {^n \dot{\bar{p}}} \right)  </math>
+| style="text-align: center;" | <math>^n{\bf g}^S = {\bf f}_p + \frac{1}{ \Delta t} {\bf M}_{1} {^{n} \bar {\bf p}} + \frac{1}{ \Delta t} {\bf M}_{2}  \left(\frac{ ^{n}\bar {\bf p}}{\Delta t} + {^n \dot{\bar{p}}} \right)  </math>
 |}
 | style="width: 5px;text-align: right;" | (87)

@@ Line 682: / Line 682: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>{^{n+1}\dot{p}}= \frac{ ^{n+1}{p} - {^{n}p} }{\Delta t}  </math>
+| style="text-align: center;" | <math>{^{n+1}\dot{\bf p}}= \frac{ ^{n+1}{\bf p} - {^{n}{\bf p}} }{\Delta t}  </math>
 |}
 | style="width: 5px;text-align: right;" | (45)
@@ Line 693: / Line 693: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>{^{n+1}\ddot{p}}= \frac{^{n+1} {p} - {^{n}p} }{{\Delta t}^2}-\frac{^{n}\dot{p} }{\Delta t}   </math>
+| style="text-align: center;" | <math>{^{n+1}\ddot{\bf p}}= \frac{^{n+1} {\bf p} - {^{n}{\bf p}} }{{\Delta t}^2}-\frac{^{n}\dot{\bf p} }{\Delta t}   </math>
 |}
 | style="width: 5px;text-align: right;" | (46)

@@ Line 657: / Line 657: @@
 |}
-where <math display="inline">\boldsymbol{\sigma}' ^{\bigtriangledown }</math>  is the deviatoric part of the Cauchy stress rate tensor, <math display="inline">{p^{\bigtriangledown }}</math> is the rate of  the pressure, <math display="inline">{\bf c}^{\sigma '} }</math> is the tangent moduli for <math display="inline">\boldsymbol{\sigma}'^{\bigtriangledown}</math> and <math display="inline">{{\bf D}^{dev} }</math> is the deviatoric part of the deformation rate tensor. The pressure is related to the volumetric part of the rate of deformation by the following relation:
+where <math display="inline">\boldsymbol{\sigma}'^{\bigtriangledown }</math>  is the deviatoric part of the Cauchy stress rate tensor, <math display="inline">{p^{\bigtriangledown }}</math> is the rate of  the pressure, <math display="inline">{\bf c}^{\sigma '}</math> is the tangent moduli for <math display="inline">\boldsymbol{\sigma}'^{\bigtriangledown}</math> and <math display="inline">{{\bf D}^{dev} }</math> is the deviatoric part of the deformation rate tensor. The pressure is related to the volumetric part of the rate of deformation by the following relation:
 <span id="eq-44"></span>

@@ Line 624: / Line 624: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>\sigma =\sigma ' - pI </math>
+| style="text-align: center;" | <math>\boldsymbol{\sigma} =\boldsymbol{\sigma}' - p{\bf I} </math>
 |}
 | style="width: 5px;text-align: right;" | (41)
 |}
-where <math display="inline">{\sigma '}</math> is the deviatoric part of the Cauchy stress, <math display="inline">{p}</math> is the pressure  and <math display="inline">{I}</math> is the identity tensor.
+where <math display="inline">\boldsymbol{\sigma}'</math> is the deviatoric part of the Cauchy stress, <math display="inline">{p}</math> is the pressure  and <math display="inline">{I}</math> is the identity tensor.
 In terms of stress rate, that would be:
@@ Line 639: / Line 639: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>\sigma ^{\bigtriangledown }=  \sigma ' ^{\bigtriangledown } - p^{\bigtriangledown }I    </math>
+| style="text-align: center;" | <math>\boldsymbol{\sigma} ^{\bigtriangledown }=  \boldsymbol{\sigma}' ^{\bigtriangledown } - p^{\bigtriangledown }{\bf I}   </math>
 |}
 | style="width: 5px;text-align: right;" | (42)
@@ Line 652: / Line 652: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>\sigma ' ^{\bigtriangledown } =  c^{\sigma '} : D^{dev}  </math>
+| style="text-align: center;" | <math>\boldsymbol{\sigma}' ^{\bigtriangledown } =  {\bf c}^{\sigma '} : {\bf D}^{dev}  </math>
 |}
 | style="width: 5px;text-align: right;" | (43)
 |}
-where <math display="inline">{{{\sigma } ' ^{\bigtriangledown }}}</math>  is the deviatoric part of the Cauchy stress rate tensor, <math display="inline">{p^{\bigtriangledown }}</math> is the rate of  the pressure, <math display="inline">{c^{\sigma '} }</math> is the tangent moduli for <math display="inline">{{{\sigma } ' ^{\bigtriangledown }}}</math> and <math display="inline">{D^{dev} }</math> is the deviatoric part of the deformation rate tensor. The pressure is related to the volumetric part of the rate of deformation by the following relation:
+where <math display="inline">\boldsymbol{\sigma}' ^{\bigtriangledown }</math>  is the deviatoric part of the Cauchy stress rate tensor, <math display="inline">{p^{\bigtriangledown }}</math> is the rate of  the pressure, <math display="inline">{\bf c}^{\sigma '} }</math> is the tangent moduli for <math display="inline">\boldsymbol{\sigma}'^{\bigtriangledown}</math> and <math display="inline">{{\bf D}^{dev} }</math> is the deviatoric part of the deformation rate tensor. The pressure is related to the volumetric part of the rate of deformation by the following relation:
 <span id="eq-44"></span>
@@ Line 670: / Line 670: @@
 |}
-where <math display="inline">{\kappa }</math> is the bulk modulus of the material and <math display="inline">{{D}^{v}= trace(D)}</math> is the volumetric strain rate.   Eq.([[#eq-44|44]]) is the closure equation for the mixed velocity-pressure formulation.
+where <math display="inline">{\kappa }</math> is the bulk modulus of the material and <math display="inline">{{\bf D}^{v}= trace(D)}</math> is the volumetric strain rate.   Eq.([[#eq-44|44]]) is the closure equation for the mixed velocity-pressure formulation.
 ===3.2 Time integration===

@@ Line 576: / Line 576: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>K= K^{m}+ K^{g} +K^{k}   </math>
+| style="text-align: center;" | <math>{\bf K}= {\bf K}^{m}+ {\bf K}^{g} +{\bf K}^{k}   </math>
 |}
 | style="width: 5px;text-align: right;" | (39)

@@ Line 286: / Line 286: @@
 |}
-The first Piola-Kirchhoff stress tensor <math display="inline">{P}</math> is not typically used because it is not symmetric and its rate is a non-objective measure. For these reasons, the second Piola-Kirchhoff stress tensor <math display="inline">{S}</math> and its rate are  preferred quantities in the TL framework. The mentioned stress rate measures are related each other through the following relation:
+The first Piola-Kirchhoff stress tensor <math display="inline">{\bf P}</math> is not typically used because it is not symmetric and its rate is a non-objective measure. For these reasons, the second Piola-Kirchhoff stress tensor <math display="inline">{\bf S}</math> and its rate are  preferred quantities in the TL framework. The mentioned stress rate measures are related each other through the following relation:
 <span id="eq-19"></span>
@@ Line 355: / Line 355: @@
 |}
-One may note that in literature (<span id='citeF-3'></span>[[#cite-3|[3]]],<span id='citeF-2'></span>[[#cite-2|[2]]]), often the term <math display="inline">{\frac{\partial N_I}{\partial X}  F}</math> is grouped in the matrix <math display="inline">{B^I_{0}}</math> defined in two dimension as:
+One may note that in literature (<span id='citeF-3'></span>[[#cite-3|[3]]],<span id='citeF-2'></span>[[#cite-2|[2]]]), often the term <math display="inline">{\frac{\partial N_I}{\partial X}  F}</math> is grouped in the matrix <math display="inline">{{\bf B}^I_{0}}</math> defined in two dimension as:
@@ Line 361: / Line 361: @@
 |-
 | style="text-align: center;" |
-<math>\begin{array}{l} \\       {B^I_{0}}= \left[                      \begin{array}{cccccc}                        \frac{\partial N_I}{\partial {X}}\frac{\partial x}{\partial {X}} & \frac{\partial N_I}{\partial {X}}\frac{\partial y}{\partial {X}} \\  \frac{\partial N_I}{\partial {Y}}\frac{\partial x}{\partial {Y}} & \frac{\partial N_I}{\partial {Y}}\frac{\partial y}{\partial {Y}} \\  \frac{\partial N_I}{\partial {X}}\frac{\partial x}{\partial {Y}}+\frac{\partial N_I}{\partial {Y}}\frac{\partial x}{\partial {X}} &  \frac{\partial N_I}{\partial {X}}\frac{\partial y}{\partial {Y}}+\frac{\partial N_I}{\partial {Y}}\frac{\partial y}{\partial {X}} \\                      \end{array} \right] \\ \\ \end{array}</math>
+<math>\begin{array}{l} \\       {{\bf B}^I_{0}}= \left[                      \begin{array}{cccccc}                        \frac{\partial N_I}{\partial {X}}\frac{\partial x}{\partial {X}} & \frac{\partial N_I}{\partial {X}}\frac{\partial y}{\partial {X}} \\  \frac{\partial N_I}{\partial {Y}}\frac{\partial x}{\partial {Y}} & \frac{\partial N_I}{\partial {Y}}\frac{\partial y}{\partial {Y}} \\  \frac{\partial N_I}{\partial {X}}\frac{\partial x}{\partial {Y}}+\frac{\partial N_I}{\partial {Y}}\frac{\partial x}{\partial {X}} &  \frac{\partial N_I}{\partial {X}}\frac{\partial y}{\partial {Y}}+\frac{\partial N_I}{\partial {Y}}\frac{\partial y}{\partial {X}} \\                      \end{array} \right] \\ \\ \end{array}</math>
 |}
-In this work, for convenience, the matrix <math display="inline">{B^I_{0}}</math> is not used. As it will be shown in the following sections, Eq.([[#eq-22|22]]) can represent  both a Kirchhoff solid material and a quasi-incompressible fluid. If different constitutive laws are used, Eq.([[#eq-22|22]]) should be modified accordingly in order to derive the material part of the tangent matrix.
+In this work, for convenience, the matrix <math display="inline">{\bf B}^I_{0}</math> is not used. As it will be shown in the following sections, Eq.([[#eq-22|22]]) can represent  both a Kirchhoff solid material and a quasi-incompressible fluid. If different constitutive laws are used, Eq.([[#eq-22|22]]) should be modified accordingly in order to derive the material part of the tangent matrix.
 Substituting Eq.([[#eq-22|22]]) into <math display="inline">{\dot{W}^{m}}</math> of  Eq.([[#eq-17|17]]) yields
@@ Line 430: / Line 430: @@
 |}
-where <math display="inline">{c^{\tau }}</math> is the tangent moduli corresponding to the material time derivative of the Kirchhoff stress tensor and <math display="inline">{c^{ \sigma }}</math> is the tangent moduli for the rate of the Cauchy stress. The rate of the Cauchy stress tensor is related to the rate of deformation <math display="inline">{D}</math> through the fourth-order tensor <math display="inline">{C}</math> in the following way
+where <math display="inline">{c^{\tau }}</math> is the tangent moduli corresponding to the material time derivative of the Kirchhoff stress tensor and <math display="inline">{c^{ \sigma }}</math> is the tangent moduli for the rate of the Cauchy stress. The rate of the Cauchy stress tensor is related to the rate of deformation <math display="inline">{\bf D}</math> through the fourth-order tensor <math display="inline">{\bf C}</math> in the following way
 <span id="eq-29"></span>
@@ Line 438: / Line 438: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>\sigma ^{\bigtriangledown }=c^{\sigma } : D  </math>
+| style="text-align: center;" | <math>\boldsymbol{\sigma} ^{\bigtriangledown }=c^{\sigma } : {\bf D}  </math>
 |}
 | style="width: 5px;text-align: right;" | (29)
@@ Line 520: / Line 520: @@
 {| style="text-align: left; margin:auto;"
 |-
-| style="text-align: center;" | <math>{S}= {F}^{-1} {\sigma } {F}^{-T} J  </math>
+| style="text-align: center;" | <math>{\bf S}= {\bf F}^{-1} \boldsymbol{\sigma } {\bf F}^{-T} J  </math>
 |}
 | style="width: 5px;text-align: right;" | (35)