Onate et al 2004b - Revision history

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 ===10.3 Wave breaking on a beach===
-A simulation of the propagation of a water wave and its breaking due to shoaling over a plane slope is presented next. This example was numerically studied in [Radovitzki and Ortiz (1998) <span id="citeF-35"></span>[[#cite-35|[35]]]] with a Lagrangian formulation using directly the standard Finite Element Method with remeshing. There is also an analytical solution for a simplified approximation that is used for  comparisons [Laitone (1960) <span id="citeF-27"></span>[[#cite-28|[28]]]] where  the geometry of the problem as well as a discussion of the analytical solution may be found. Figure [[#img-13|13]] shows the initial point distribution and Figure [[#img-11|11]] a comparison with the analytical free-surface at different time steps.
+A simulation of the propagation of a water wave and its breaking due to shoaling over a plane slope is presented next. This example was numerically studied in [Radovitzki and Ortiz (1998) <span id="citeF-35"></span>[[#cite-35|[35]]]] with a Lagrangian formulation using directly the standard Finite Element Method with remeshing. There is also an analytical solution for a simplified approximation that is used for  comparisons [Laitone (1960) <span id="citeF-26"></span>[[#cite-26|[26]]]] where  the geometry of the problem as well as a discussion of the analytical solution may be found. Figure [[#img-13|13]] shows the initial point distribution and Figure [[#img-11|11]] a comparison with the analytical free-surface at different time steps.
 <div id='img-13'></div>

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-| colspan="1" | '''Figure 7:''' Water column collapse at different time steps.
+| colspan="2" | '''Figure 7:''' Water column collapse at different time steps.
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 | colspan="1" | '''Figure 7:''' Water column collapse at different time steps.

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 ==10 Examples==
-The examples chosen show the applicability of the PFEM to solve problems involving large fluid motions and FSI situations. The fractional step algorithm of Section 5 with <math display="inline">\theta _2 =1</math> and <math display="inline">\alpha =1</math> has been used in all cases.
+The examples chosen show the applicability of the PFEM to solve problems involving large fluid motions and FSI situations. The fractional step algorithm of Section [[#5 Fractional Step Method for Fluid-Structure Interaction Analysis|5]] with <math display="inline">\theta _2 =1</math> and <math display="inline">\alpha =1</math> has been used in all cases.
 In examples [[#10.1 Collapse of a water column|10.1]]-[[#10.10 Rigid cube falling into a water container|10.10]] a value of <math display="inline">\theta _1=1</math> has  been chosen. This basically means that the final configuration <math display="inline">\Omega ^{n+1,j}</math> has been taken as the reference configuration at each iteration. In example [[#10.11 The Rayleigh-Bénard Instability|10.11]] <math display="inline">\theta _1=0</math> has been selected and, hence, the initial configuration <math display="inline">\Omega ^n</math> has been taken as a fixed reference configuration for all the iterations within a time step.

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 Despite the motion of the nodes within the iterative process, in general there is no need to regenerate the mesh at each iteration. A new mesh is typically generated  after a prescribed number of converged time steps, or when  the nodal displacements induce significant geometrical distorsions in some elements. ''In the examples presented in the paper the mesh in the fluid domain has been regenerated at each time step''.
-The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities <math display="inline">{v}^*</math> in Eq.(28a). The prescribed velocities at the boundary are applied when solving for <math display="inline">\bar{v}^{n+1,j}</math> in step 3. The prescribed pressures at the boundary are imposed by making  the pressure increments zero at the relevant boundary nodes and making <math display="inline">\bar{p}^n</math> equal to the prescribed pressure values.
+The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities <math display="inline">{v}^*</math> in Eq.([[#eq-28a|28a]]). The prescribed velocities at the boundary are applied when solving for <math display="inline">\bar{v}^{n+1,j}</math> in step 3. The prescribed pressures at the boundary are imposed by making  the pressure increments zero at the relevant boundary nodes and making <math display="inline">\bar{p}^n</math> equal to the prescribed pressure values.
 Details of the treatment of the contact conditions at the solid-fluid interface are given in Section [[#8 Identification of Boundary Surfaces|8]] [Idelsohn ''et al.'' (2004)<span id="citeF-23"></span>[[#cite-23|[23]]]].

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 As usual the deviatoric stresses <math display="inline">s_{ij}</math> are related to the strain rates <math display="inline">\dot \varepsilon _{ij}</math> by Eq.([[#eq-5|5]])
-It can be shown that the system of Eqs.([[#eq-22|22]]) leads to a stabilized numerical solution. For details see [Oñate ''et al.'' (2003)<span id="citeF-34"></span>[[#cite-34|[34]]]].
+It can be shown that the system of Eqs.([[#eq-22|22]]) leads to a stabilized numerical solution. For details see [Oñate ''et al.'' (2003)<span id="citeF-33"></span>[[#cite-33|[33]]]].
 '''Remark 1'''

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 The <math display="inline">h_i's</math> in above equations are ''characteristic lengths'' of the domain where balance of momentum and mass is enforced. In Eq.([[#eq-9|9]]) these lengths define the domain where equilibrium of boundary tractions is established. Details of the derivation of Eqs.([[#eq-7|7]])-([[#eq-10|10]]) can be found in [Oñate (1998; 2000) <span id="citeF-27"></span>[[#cite-27|[27]],<span id="citeF-28"></span>[[#cite-28|28]]]; Oñate ''et al.'' (2001) <span id="citeF-32"></span>[[#cite-32|[32]]]].
-Eqs.([[#eq-7|7]])-([[#eq-10|10]]) are the starting  point for deriving stabilized finite element methods to solve the incompressible Navier-Stokes equations in a Lagrangian frame of reference using equal order interpolation for the velocity and pressure variables [Idelsohn ''et al.'' (2002; 2003a; 2003b; 2004) <span id="citeF-19"></span>[[#cite-19|[19]]-<span id="citeF-21"></span>[[#cite-21|21]],<span id="citeF-23"></span>[[#cite-23|23]]]]; Oñate ''et al.'' (2003)<span id="citeF-33"></span>[[#cite-33|[33]]]; Aubry ''et al.'' (2004)<span id="citeF-1"></span>[[#cite-1|[1]]]]. Application of the FIC formulation to finite element and meshless analysis of fluid flow problems can be found in [García and Oñate (2003)<span id="citeF-12"></span>[[#cite-12|[12]]]; Oñate (2000; 2004)<span id="citeF-28"></span>[[#cite-28|[28]],<span id="citeF-29"></span>[[#cite-29|29]]]; Oñate ''et al.'' (2000; 2004)<span id="citeF-31"></span>[[#cite-31|[31]],<span id="citeF-33"></span>[[#cite-33|33]]]; Oñate and García (2001)<span id="citeF-34"></span>[[#cite-34|[34]]]; Oñate and Idelsohn (1988)<span id="citeF-31"></span>[[#cite-32|[32]]]].
+Eqs.([[#eq-7|7]])-([[#eq-10|10]]) are the starting  point for deriving stabilized finite element methods to solve the incompressible Navier-Stokes equations in a Lagrangian frame of reference using equal order interpolation for the velocity and pressure variables [Idelsohn ''et al.'' (2002; 2003a; 2003b; 2004) <span id="citeF-19"></span>[[#cite-19|[19]]-<span id="citeF-21"></span>[[#cite-21|21]],<span id="citeF-23"></span>[[#cite-23|23]]]]; Oñate ''et al.'' (2003)<span id="citeF-33"></span>[[#cite-33|[33]]]; Aubry ''et al.'' (2004)<span id="citeF-1"></span>[[#cite-1|[1]]]]. Application of the FIC formulation to finite element and meshless analysis of fluid flow problems can be found in [García and Oñate (2003)<span id="citeF-12"></span>[[#cite-12|[12]]]; Oñate (2000; 2004)<span id="citeF-28"></span>[[#cite-28|[28]],<span id="citeF-29"></span>[[#cite-29|29]]]; Oñate ''et al.'' (2000; 2004)<span id="citeF-31"></span>[[#cite-31|[31]],<span id="citeF-34"></span>[[#cite-34|34]]]; Oñate and García (2001)<span id="citeF-32"></span>[[#cite-32|[32]]]; Oñate and Idelsohn (1988)<span id="citeF-30"></span>[[#cite-30|[30]]]].
 ===3.1 Transformation of the mass balance equation. Integral governing equations===