Ship hydrodynamics

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==Ship hydrodynamics==

'''Eugenio  Oñate¹, Julio García¹ and Sergio R. Idelsohn²'''

{|
|-
|(¹)  International Center for Numerical Methods in Engineering (CIMNE)
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| Universitat Politécnica de Catalunya (UPC), Gran Capitán, s/n, 
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| 08034 Barcelona, Spain
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| (²)  CIMNE and International Center for Computational Methods in Engineering (CIMEC)
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| Universidad Nacional del Litoral and CONICET, Santa Fe, Argentina
|}
-->

==Abstract==

This chapter presents an overview of some computational methods for analysis of ship hydrodynamics problems. Attention is focused on the  description of a stabilized finite element formulation derived via a finite calculus procedure. Both arbitrary Lagrangian-Eulerian (ALE) and fully Lagrangian forms are presented. Details of the treatment of the free-surface waves and the interaction between the ship structure and the sea water are also given. Examples of application to a variety of ship hydrodynamics problems are shown.

'''Keywords''' Ship hydrodynamics, finite element method, finite calculus, fluid-structure interaction

==1 INTRODUCTION==

Accurate prediction of the hydrodynamic forces on a ship in motion is of paramount importance in ship design. The water resistance at a certain speed determines the required engine power and thereby the fuel consumption. Minimization of the hydrodynamic forces is therefore an important issue in ship hull design. Further, excitation of a wave pattern by ship motion not only induces wave resistance but may also limit the speed in the vicinity of the shore for environmental reasons, which must also be taken into account in ship design.

The usual simplification in ship hydrodynamics design is to separately consider the performance of the ship in still water and its behaviour in open sea. Hydrodynamic optimisation of a hull primarily requires the calculation of the resistance in a calm sea and the open sea effects  are generally taken into account as a wave added resistance.

The resistance of a ship in still water can be considered as the sum of several contributions: a viscous resistance associated with the generation of boundary layers, the wave resistance, the air resistance on the superstructure and the induced resistance related to the generation of lift forces.

Wave resistance in practical cases amounts to <math display="inline">10</math> to <math display="inline">60%</math> of the total resistance of a ship in still water (Raven, 1996 <span id="citeF-84"></span>[[#cite-84|[84]]]). It increases very rapidly at high speeds dominating the viscous component for high speed ships.  Furthermore, wave resistance is very sensitive to the hull form design and easily affected by small shape modifications. For all these reasons, the  possibility to predict and reduce the wave resistance is an important target.

The prediction of the wave pattern and the wave resistance of a ship has  challenged mathematicians and hydrodynamicists for over a century. The Boundary Element Method (BEM) is the basis of many computational algorithms developed in  past years. Here the flow problem is solved using a simple potential model.  BEM methods, termed  by  hydrodynamicists as Panel Methods may be classified into two categories. The first one uses the Kelvin wave source as the elementary singularity. The main advantage of such scheme is the automatic satisfaction of the radiation  condition. The theoretical background of this method was reviewed by Wehausen (1970) <span id="citeF-103"></span>[[#cite-103|[103]]], while computational aspects can be found in Soding (1996) <span id="citeF-87"></span>[[#cite-87|[87]]] and Jenson and Soding (1989) <span id="citeF-58"></span>[[#cite-58|[58]]]. The second class of BEM schemes uses the Rankine source as the elementary singularity. This procedure, first presented by Dawson (1977) <span id="citeF-29"></span>[[#cite-29|[29]]], has been widely applied in practice and many improvements have been addressed to account for the nonlinear wave effects. Among these, a succesful example is the Rankine Panel Method (Xia, 1986 <span id="citeF-106"></span>[[#cite-106|[106]]]; Jenson and Soding, 1989 <span id="citeF-58"></span>[[#cite-58|[58]]]; Nakos and Sclavounos, 1990 <span id="citeF-109"></span>[[#cite-109|[109]]]).

In addition to the important developments in potential flow panel methods for practical ship hydrodynamics analysis during the period 1960-1980, much research in the second half of the twentieth century was oriented towards the introduction of viscosity in the CFD analysis. In the 1960's the viscous flow research was mainly focused in 2D boundary layer theory and by the end of the decade several methods for arbitrary pressure gradients were available. This research continued to solve the 3D case during the following decade and an evaluation of the capability of the new methods to predict ship wave resistance was carried out at different workshops (Bai and McCarthy, 1979 <span id="citeF-7"></span>[[#cite-7|[7]]]; Larsson, 1981 <span id="citeF-61"></span>[[#cite-61|[61]]]; Noblesse and McCarthy, 1983 <span id="citeF-69"></span>[[#cite-69|[69]]]). Here application to some well specified test cases were reported and numerical and experimental results compared acceptable well for most part of the boundary layer along the hull, while wrong results were obtained near the stern. This prompted additional research and by the end of the 1980's a number of numerical procedures for solving the full viscous flow equation accounting for simple turbulence modes based on Reynolds averaged Navier-Stokes (RANS) equations were available. Considerable improvements for predicting the stern flow were reported in subsequent workshops organized in the 1990's (Kim and Lucas, 1990 <span id="citeF-59"></span>[[#cite-59|[59]]]; Reed ''et al.'' , 1990<span id="citeF-85"></span>[[#cite-85|[85]]]; Beck ''et al.'', 1993 <span id="citeF-8"></span>[[#cite-8|[8]]]; Raven, 1996 <span id="citeF-84"></span>[[#cite-84|[84]]]; Soding, 1996 <span id="citeF-87"></span>[[#cite-87|[87]]]; Janson and Larsson, 1996 <span id="citeF-57"></span>[[#cite-57|[57]]]; Alessandrini and Delhommeau, 1996 <span id="citeF-1"></span>[[#cite-1|[1]]]; Miyata, 1996 <span id="citeF-67"></span>[[#cite-67|[67]]], Löhner ''et al.'', 1998 <span id="citeF-64"></span>[[#cite-64|[64]]]). A good review of the status of CFD in ship hydrodynamics in the last part of the 20th century can be found in Larsson ''et al.'' (1998) <span id="citeF-62"></span>[[#cite-62|[62]]].

Independently of the flow equations used, the free-surface boundary condition has been solved in different manners. The exact free surface condition is nonlinear and several linearizations have been proposed (Baba and Takekuma, 1975 <span id="citeF-6"></span>[[#cite-6|[6]]]; Newmann, 1976 <span id="citeF-68"></span>[[#cite-68|[68]]]; Idelsohn ''et al.'', 1999 <span id="citeF-52"></span>[[#cite-52|[52]]]). Some of them use a fixed domain and others a moving one. An alternative is to solve the full nonlinear free surface equation on a reference surface which does not necessarily coincides with the free surface itself. In this way the updating of the surface mesh is minimized and sometimes is not even necessary.

The solution of the free-surface equation in a bounded domain brings in the necessity of a radiation condition to eliminate spurious waves. A way to introduce this condition was proposed by Dawson (1977) <span id="citeF-29"></span>[[#cite-29|[29]]] who used a finite difference (FD) formula based in four upwind points to evaluate the first derivatives appearing in the free-surface equation. This method became very popular and this is probably the main reason why a large majority of codes predicting the wave resistance of ships use FD methods on structured meshes (Larsson ''et al.'', 1998 <span id="citeF-62"></span>[[#cite-62|[62]]]).

Indeed the 1990's were a decade of considerable progress in CFD methods for ship hydrodynamics and the most important breakthrough was perhaps the coupled solution of the free-surface equation with the fluid flow equations. Here a number of viscous and inviscid solutions for the surface ship wave problem using finite element (FE) and finite volume (FV) methods with non structured grids were reported (Farmer ''et al.'', 1993 <span id="citeF-35"></span>[[#cite-35|[35]]], Hino ''et al.'', 1993 <span id="citeF-41"></span>[[#cite-41|[41]]];   Luo ''et al.'', 1995 <span id="citeF-66"></span>[[#cite-66|[66]]]; Storti ''et al.'', 1998a,b <span id="citeF-90"></span>[[#cite-90|[90]]-<span id="citeF-91"></span>[[#cite-91|91]]]; García ''et al.'', 1998 <span id="citeF-36"></span>[[#cite-36|[36]]]; García, 1999 <span id="citeF-37"></span>[[#cite-37|[37]]]; Alessandrini and Delhommeau, 1999 <span id="citeF-2"></span>[[#cite-2|[2]]]; Idelsohn ''et al.'', 1999 <span id="citeF-52"></span>[[#cite-52|[52]]]; Löhner ''et al.'', 1998 <span id="citeF-64"></span>[[#cite-64|[64]]], 1999 <span id="citeF-65"></span>[[#cite-65|[65]]]).

The current challenges in CFD research for ship hydrodynamics focus in the development of robust (stable) and computationaly efficient numerical methods able to capture the different scales involved in the analysis of practical ship hydrodynamics situations. Wave resistance coefficients for modern ship design are needed for a wide range of speeds and here the accurate prediction of the wave pattern and the hull pressure distribution at low speed (say below Froude number (<math display="inline">Fn = 0,2</math>) ) are still major challenges. Great difficulties also exist in the computation of the viscous resistance which requires very fine grids in the vicinity of the hull, resulting in overall meshes involving (at least) some <math display="inline">10^7-10^9</math> elements. Other relevant problems are the prediction of the wake details and the propeller-hull interaction. Fine meshing and advanced turbulence models are crucial for the realistic solution of these problems. Indeed the use of unstructured meshes is essential for problems involving complex shapes.

A different class of ship hydrodynamic problems require the modelling of breaking waves or the prediction of water inside the hull (green water) due to large amplitude waves typical of  sea keeping problems. Here Lagrangian flow methods where the motion of each flow particle is individually tracked using techniques developed for (incompressible) solid mechanics are a promising trend for solving a wide class of ship hydrodynamics problems.

The content of the chapter is structured as follows. In the next section the standard Navier-Stokes equations for an incompressible viscous flow are presented. The equations are formulated in an arbitrary Lagrangian-Eulerian (ALE) description allowing the independent motion of the mesh nodes from that of the fluid particles. Details of the problems posed  by the free-surface wave boundary condition are given. The difficulties encountered in the numerical solution of the fluid flow and the free-surface equations, namely the unstabilities induced by the convective terms and the limits in the approximation introduced by the incompressibility constraint are explained. A new procedure for deriving stabilized numerical methods for this type of problems based on the so called ''finite calculus'' (FIC) formulation is presented. The FIC method is based in redefining the standard governing equations in fluid mechanics by expressing the balance laws in a domain of ''finite size''. This introduces additional terms in the differential equations of the infinitessimal theory which are essential to derive stabilized numerical schemes (Oñate, 1998 <span id="citeF-70"></span>[[#cite-70|[70]]], 2000 <span id="citeF-71"></span>[[#cite-71|[71]]], 2004 <span id="citeF-72"></span>[[#cite-72|[72]]]). We present here a stabilized finite element method using equal-order linear interpolation for the velocity and the pressure variables. Both monolithic and fractional step time integration procedures are  described. A method for solving the coupled fluid-structure interaction problem induced by the motion of the ship due to the hydrodynamic forces is also presented.  A mesh moving algorithm  for updating the position of the free-surface nodes during the ship motion is given.

In the last part of the chapter the Lagrangian formulation for fluid flow analysis is presented as a particular case of the ALE form. As above mentioned the Lagrangian description has particular advantages for tracking the displacement of  the fluid particles in flows where large motions of the fluid surface occur. One of the advantages of the Lagrangian approach is that the convective terms dissapear in the governing equations of the fluid. In return, the updating of the mesh at almost every time step is now a necessity and efficient mesh generation algorithms must be used (Idelsohn ''et al.'', 2002 <span id="citeF-53"></span>[[#cite-53|[53]]], 2003a,b,c <span id="citeF-54"></span>[[#cite-54|[54]]-<span id="citeF-56"></span>[[#cite-56|56]]]).

The examples show the efficiency of the ALE and fully Lagrangian formulations to solve a variety of ship hydrodynamics problems.

==2 THE NAVIER-STOKES EQUATIONS FOR INCOMPRESSIBLE FLOWS. ALE FORMULATION==

===2.1 Momentum and mass conservation equations===

The  Navier-Stokes equations for an incompressible fluid in a domain <math display="inline">\Omega </math> can be written in an arbitrary Lagrangian-Eulerian form as

''Momentum''
<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\displaystyle{\rho \left({\partial u_i \over \partial t}+v_j {\partial u_i \over \partial x_j}+ u_i{\partial u_j \over \partial x_j} \right)+ {\partial p \over \partial x_i} - {\partial s_{ij} \over \partial x_j}-b_i =0 } \qquad \hbox{in }\Omega  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

''Mass conservation''
<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\displaystyle{{\partial u_i \over \partial x_i}=0} \qquad \hbox{in }\Omega  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}

In Eq.([[#eq-1|1]]) <math display="inline">u_i</math> is the velocity along the ith global reference axis, <math display="inline">u_i^m</math> is the velocity of  the moving mesh nodes, <math display="inline">v_i=u_i-u_i^m</math> is the relative velocity between the fluid and the moving mesh nodes, <math display="inline">\rho </math> is the (constant) density of the fluid, <math display="inline">b_i</math> are body forces, <math display="inline">t</math> is the time, <math display="inline">p</math> is the pressure and <math display="inline">s_{ij}</math> are the viscous stresses related to the viscosity <math display="inline">\mu </math> by the standard expression
<span id="eq-3"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>s_{ij}=2\mu \left(\varepsilon _{ij} - {1\over 3}\delta _{ij}{\partial u_k \over \partial x_k}\right) </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

where <math display="inline">\delta _{ij}</math> is the Kronecker delta and the strain rates <math display="inline">\varepsilon _{ij}</math> are
<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\varepsilon _{ij}= {1\over 2}\left({\partial u_i \over \partial x_j}+ {\partial u_j \over \partial x_i}\right) </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

The volumetric strain rate terms in the Eqs.([[#eq-1|1]]) and ([[#eq-3|3]]) can be neglected following Eq.([[#eq-2|2]]). We will however retain these terms as they are useful for the derivation of the stabilized formulation in Section [[#5 FIC EQUATIONS FOR VISCOUS INCOMPRESSIBLE FLOW. ALE FORMULATION|5]].

Eqs.([[#eq-1|1]]) and ([[#eq-3|3]]) are completed with the boundary conditions
<span id="eq-5a"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>n_j\sigma _{ij}-t_i =0 \quad \hbox{on }\Gamma _t</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5a)
|}
<span id="eq-5b"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>u_j - u_j^p =0 \quad \hbox{on }\Gamma _u</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5b)
|}

where <math display="inline">\sigma _{ij}=s_{ij}- p\delta _{ij}</math> are the total stresses, <math display="inline">n_j</math> are the component of the unit normal vector to the boundary and <math display="inline">t_i</math> and <math display="inline">u_j^p</math> are prescribed tractions and velocities on the Neumann and Dirichlet boundaries <math display="inline">\Gamma _t</math> and <math display="inline">\Gamma _u</math>, respectively where <math display="inline">\Gamma = \Gamma _t \cup \Gamma _u</math> is the boundary of the analysis domain <math display="inline">\Omega </math>. The boundary conditions are completed with the initial condition <math display="inline">u_j =u_j^0</math> for <math display="inline">t=t_0</math>.

In above equations <math display="inline">i,j=1,n_d</math> where <math display="inline">n_d</math> is the number of space dimension (i.e. <math display="inline">n_d =3</math> for 3D). Also, throughout the text the summation convention for repeated indexes is assumed unless specified otherwise.

===2.2 Free-surface boundary conditions===

The boundary conditions ([[#eq-5a|5a]]) on the surface tractions can be written in local normal and tangential axes as
<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\begin{array}{l}\sigma _n -t_n = 0\\ \sigma _{t_j}- t_{g_j}=0  \qquad j=1,2 \quad \hbox{for 3D} \end{array} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

where <math display="inline">\sigma _n</math> and <math display="inline">\sigma _{t_j}</math> are the normal and tangential stresses, respectively and <math display="inline">t_n</math> and <math display="inline">t_{g_j}</math> are the prescribed normal and tangential tractions, respectively.

On the free boundary we have to ensure at all times that: 1) the pressure (which approximates the normal stres) equals the atmospheric pressure and the tangential tractions are zero unless specified otherwise, and 2) material particles of the fluid belong to the free-surface.

The condition on the pressure is simply especified as
<span id="eq-7"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>p=p_a </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}

where <math display="inline">p_a</math> is the atmospheric pressure (usually given a zero value).

The condition on the tangential tractions is satisfied by setting <math display="inline">t_{g_i} = 0</math> in Eq.([[#eq-6|6]]). This is automatically accounted for by the ''natural boundary conditions'' in the weak form of the momentum equations (Zienkiewicz and Taylor Vol. 3, 2000 <span id="citeF-107"></span>[[#cite-107|[107]]]).

The condition on the material particles is expressed (for steady state conditions) as
<span id="eq-8"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>u_in_i =0 \qquad \hbox{or } \qquad {\boldsymbol u}^T {\boldsymbol n}=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}

i.e. the velocity vector is tangent to the free-surface. An alternative form of Eq.([[#eq-8|8]]) can be found by noting that the normal vector has the following components
<span id="eq-9"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol n}=\left[-{\partial \beta  \over \partial x_1},1\right]^T\qquad \hbox{for 2D }\quad  \hbox{ and }\quad {\boldsymbol n}=\left[-{\partial \beta  \over \partial x_1}, -{\partial \beta  \over \partial x_2},1\right]^T \qquad \hbox{for 3D}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}

In Eq.([[#eq-9|9]]) <math display="inline">\beta = x_3 - x_3^{ref}</math> is the free-surface elevation measured in the direction of the vertical coordinate <math display="inline">x_3</math> relative to some previously known surface, which we shall refer to as the ''reference surface'' (Figure [[#img-2|2]]). This surface may  be horizontal (i.e. the undisturbed water surface) or may be simply a previously computed surface.

Introducing Eq.([[#eq-9|9]]) into ([[#eq-8|8]]) gives (for 3D)
<span id="eq-10"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>u_i {\partial \beta  \over \partial x_i}-u_3 =0 \quad , \quad i=1,2 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

Eq.([[#eq-10|10]]) is generalized for the transient 3D case as
<span id="eq-11"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\displaystyle{{\partial \beta  \over \partial t}+u_i {\partial \beta  \over \partial x_i}-u_3 =0}\quad , \quad i=1,2 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}<br/>

We observe that <math display="inline">\beta </math> obeys ''a pure convection'' equation with <math display="inline">u_3</math> playing the role of a (non linear) source term. The solution of Eq.([[#eq-11|11]]) with standard Galerkin FEM, or centred FD or FV methods will therefore suffer from numerical instabilities and some kind of stabilization is needed in order to obtain a physically meaningful solution. A  method to solve Eq.([[#eq-11|11]]), very popular in the context of the potential flow formulation, was introduced by Dawson (1977) <span id="citeF-29"></span>[[#cite-29|[29]]] using a four point FD upwind operator to evaluate the first derivatives of <math display="inline">\beta </math> on regular grids. Dawson's method has been extended by different authors to solve a many ship hydrodynamics problems (Raven, 1996 <span id="citeF-84"></span>[[#cite-84|[84]]]; Larsson ''et al.'', 1998 <span id="citeF-62"></span>[[#cite-62|[62]]]; Idelsohn ''et al.'', 1999 <span id="citeF-52"></span>[[#cite-52|[52]]]).

Solution of Eq.([[#eq-11|11]]) is strongly coupled with that of the fluid flow equations. The solution of the whole problem is highly non linear due to the pressence of the unknown velocities in Eq.([[#eq-11|11]]) and also to the fact that the free-surface position defining the new boundary conditions is also unknown. A number of iterative schemes have been developed for the solution of the non linear surface wave problem (Idelsohn ''et al.'', 1999 <span id="citeF-52"></span>[[#cite-52|[52]]]). They all basically involve solving Eq.([[#eq-11|11]]) for the new free-surface height <math display="inline">\beta </math>, for fixed values of the velocity field computed from the fluid solver in a previous iteration within each time increment. At this stage two procedures are possible, either the position of the free-surface is updated after each iteration and this becomes the new reference surface, or else an equivalent pressure of value <math display="inline">p=p_a + g (\beta -  \beta _{ref})</math>, where <math display="inline">g</math> is the gravity constant, is applied at the current reference surface as a boundary condition in the next flow iteration. The first option might require the regeneration of a new mesh, whereas the second one is less accurate but computationaly cheaper. Hence a compromise between the two alternatives is usually chosen in practice. The iterative process continues until a converged solution is found for the velocity, the pressure and the free-surface height at each time step. Details of the computational process are described in a next section.

An alternative method to treat the free-surface equation is based in the volume of fluid (VOF) technique (Hirt and Nichols, 1981 <span id="citeF-43"></span>[[#cite-43|[43]]]). In the VOF method the free-surface position is defined as the interface between two fluids interacting with each other, where the efect of one fluid on the other is very small (i.e. the water and the surrounding air). An interface function which takes the values 0 and 1 for each of the two fluids is transported with the fluid velocity using a time dependent advection equation. Early applications of the VOF in the context of the stabilized FEM were reported by Codina ''et al.'', 1994 <span id="citeF-26"></span>[[#cite-26|[26]]] and Tezduyar ''et al.'', 1998 <span id="citeF-102"></span>[[#cite-102|[102]]].  Examples of application of the VOF to ship hydrodynamics problems can be found in Tajima and Yabe, 1999 <span id="citeF-93"></span>[[#cite-93|[93]]] and Azcueta ''et al.'', 1999 <span id="citeF-5"></span>[[#cite-5|[5]]].

==3 ABOUT THE FINITE ELEMENT SOLUTION OF THE NAVIER-STOKES EQUATIONS==

The development of efficient and robust numerical methods for incompressible flow problems has been a subject of intensive research in last decades. Much effort has been spent in developing the so called ''stabilized'' numerical methods overcoming the two main sources of instability in incompressible flow analysis, namely those originated by the high values of the convective terms  and those induced by the difficulty in satisfying the incompressibility constraint.

The solution of above problems in the context of the finite element method (FEM) has been attempted in a number of ways. The underdiffusive character of the Galerkin FEM for high convection flows (which incidentaly also occurs for centred FD and FV methods) has been corrected by adding some kind of artificial viscosity terms to the standard Galerkin equations. A good review of such approach can be found in  Zienkiewicz and Taylor, Vol. 3 2000 <span id="citeF-107"></span>[[#cite-107|[107]]] and Donea and Huerta, 2003 <span id="citeF-31"></span>[[#cite-31|[31]]].

A popular way to overcome the problems with the incompressibility constraint is by introducing a pseudo-compressibility in the flow and using implicit and explicit algorithms developed for this kind of problems, such as artificial compressibility schemes (Chorin, 1967A <span id="citeF-14"></span>[[#cite-14|[14]]]) popular way to overcome the problems with the incompressibility constraint is by introducing a pseudo-compressibility in the flow and using implicit and explicit algorithms developed for this kind of problems, such as artificial compressibility schemes (Chorin, 1967 <span id="citeF-14"></span>[[#cite-14|[14]]]; Farmer ''et al.'', 1993 <span id="citeF-35"></span>[[#cite-35|[35]]]; Peraire ''et al.'', 1994 <span id="citeF-82"></span>[[#cite-82|[82]]]; Briley ''et al.'', 1995 <span id="citeF-10"></span>[[#cite-10|[10]]]; Sheng ''et al.'', 1996 <span id="citeF-86"></span>[[#cite-86|[86]]])  and preconditioning techniques (Idelsohn ''et al.'', 1995 <span id="citeF-51"></span>[[#cite-51|[51]]]). Other FEM schemes with good stabilization properties for the convective and incompressibility terms  are based in Petrov-Galerkin (PG) techniques. The background of PG methods are the non-centred (upwind) schemes for computing the first derivatives of the convective operator in FD and FV methods. More recently a general class of Galerkin FEM has been developed where the standard Galerkin variational form is extended with adequate residual-based terms in order to achieve a stabilized numerical scheme (Codina, 1998 <span id="citeF-16"></span>[[#cite-16|[16]]], 2000 <span id="citeF-17"></span>[[#cite-17|[17]]]). Among the many FEM of this kind  we can name the Streamline Upwind Petrov Galerkin (SUPG) method (Hughes and Brooks, 1979 <span id="citeF-45"></span>[[#cite-45|[45]]]; Brooks and Hughes, 1982 <span id="citeF-11"></span>[[#cite-11|[11]]]; Tezduyar and Hughes, 1983 <span id="citeF-97"></span>[[#cite-97|[97]]]; Hughes and Tezduyar, 1984 <span id="citeF-46"></span>[[#cite-46|[46]]]; Hughes and Mallet, 1986 <span id="citeF-48"></span>[[#cite-48|[48]]]; Idelsohn ''et al.'', 1995 <span id="citeF-51"></span>[[#cite-51|[51]]]; Storti ''et al.'', 1995 <span id="citeF-88"></span>[[#cite-88|[88]]], 1997 <span id="citeF-89"></span>[[#cite-89|[89]]]; Cruchaga and Oñate, 1997 <span id="citeF-27"></span>[[#cite-27|[27]]], 1999 <span id="citeF-28"></span>[[#cite-28|[28]]]), the Galerkin Least Square (GLS) method (Hughes ''et al.'', 1989 <span id="citeF-50"></span>[[#cite-50|[50]]]; Tezduyar, 1991 <span id="citeF-94"></span>[[#cite-94|[94]]]; Tezduyar ''et al.'', 1992a <span id="citeF-99"></span>[[#cite-99|[99]]]), the Taylor-Galerkin method (Donea, 1984 <span id="citeF-30"></span>[[#cite-30|[30]]]), the Characteristic Galerkin method (Douglas and Russell, 1982 <span id="citeF-32"></span>[[#cite-32|[32]]]; Pironneau, 1982 <span id="citeF-83"></span>[[#cite-83|[83]]]; Löhner ''et al.'', 1984 <span id="citeF-63"></span>[[#cite-63|[63]]]) and its variant the characteristic Based Split (CBS) method (Zienkiewicz and Codina, 1995 <span id="citeF-108"></span>[[#cite-108|[108]]]; Codina ''et al.'', 1998 <span id="citeF-23"></span>[[#cite-23|[23]]]; Codina and Zienkiewicz, 2002 <span id="citeF-25"></span>[[#cite-25|[25]]]), pressure gradient operator methods (Codina and Blasco, 1997 <span id="citeF-22"></span>[[#cite-22|[22]]], 2000 <span id="citeF-24"></span>[[#cite-24|[24]]])  and the Subgrid Scale (SGS) method (Hughes, 1995 <span id="citeF-44"></span>[[#cite-44|[44]]]; Brezzi ''et al.'', 1997 <span id="citeF-9"></span>[[#cite-9|[9]]]; Codina, 2000 <span id="citeF-17"></span>[[#cite-17|[17]]], 2002 <span id="citeF-21"></span>[[#cite-21|[21]]]).

In this work a stabilized FEM for incompressible flows is derived taking as the starting point the modified governing equations of the flow problem formulated via a finite calculus (FIC) approach. The FIC method is based in invoking the balance of fluxes in a domain of finite size. This introduces naturally additional terms in the classical differential equations of infinitessimal fluid mechanics which are a function of the balance domain dimensions. The new terms in the modified governing equations provide naturally the necessary stabilization to the standard Galerkin finite element method.

In the next section, the main concepts of the FIC approch are introduced via a simple 1D convection-diffusion model problem. Then the  FIC formulation of the fluid flow equations and the free-surface  wave equations using the finite element method  are presented.

==4 BASIC CONCEPTS OF THE FINITE CALCULUS (FIC) METHOD==

We will consider a convection-diffusion problem in a 1D domain <math display="inline">\Omega </math> of length <math display="inline">L</math>. The equation of balance of fluxes in a subdomain of size <math display="inline">d</math> belonging to <math display="inline">\Omega </math> (Figure [[#img-1|1]]) is written as
<span id="eq-12"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>q_A - q_B=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}

where <math display="inline">q_A</math> and <math display="inline">q_B</math> are the incoming and outgoing fluxes at points <math display="inline">A</math> and <math display="inline">B</math>, respectively. The flux <math display="inline">q</math> includes both convective and diffusive terms; i.e. <math display="inline">q=u\phi - k{d\phi \over dx}</math>, where <math display="inline">\phi </math> is the transported variable, <math display="inline">u</math> is the velocity and <math display="inline">k</math> is the diffusitivity of the material.

We express now the fluxes <math display="inline">q_A</math> and <math display="inline">q_B</math> in terms of the flux at an arbitrary point <math display="inline">C</math> within the balance domain (Figure  [[#img-1|1]]). Expanding <math display="inline">q_A</math> and <math display="inline">q_B</math> in Taylor series around point <math display="inline">C</math> up to second order terms gives
<span id="eq-13"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>q_A= q_C - d_1 \frac{dq}{d x}\vert _C+ \frac{d^2_1}{2}\frac{d^2q}{dx^2}\vert _C + O(d^3_1)\quad ,\quad  q_B= q_C + d_2 \frac{dq}{d x}\vert _C+\frac{d^2_2}{2}\frac{d^2q}{dx^2}\vert _C + O(d^3_2) </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

Substituting Eq.([[#eq-13|13]]) into Eq.([[#eq-12|12]]) gives after simplification
<span id="eq-14"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\frac{dq}{dx}-\underline{\frac{h}{2} \frac{d^2q}{dx^2}}=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

where <math display="inline">h=d_1-d_2</math> and all derivatives are computed at point <math display="inline">C</math>.

Standard calculus theory assumes  that the domain <math display="inline">d</math> is of infinitessimal size and the resulting balance equation is simply <math display="inline">{dq\over dx}=0</math>. We will relax this assumption and allow the balance domain to have a ''finite size''. The new balance equation ([[#eq-14|14]]) incorporates now the underlined term which introduces the ''characteristic length'' <math display="inline">h</math>. Obviously, accounting for higher order terms in Eq.([[#eq-13|13]]) would lead to new terms in Eq.([[#eq-14|14]]) involving higher powers of <math display="inline">h</math>.

Distance <math display="inline">h</math> in Eq.([[#eq-14|14]]) can be interpreted as a free parameter depending on the location of point <math display="inline">C</math> (note that <math display="inline">h=0</math> for <math display="inline">d_1=d_2</math>). However, the fact that Eq.([[#eq-14|14]]) is the exact balance equation (up to second order terms) for any 1D domain of finite size and that the position of point <math display="inline">C</math> is arbitrary, can be used to derive numerical schemes with enhanced properties, simply by computing the characteristic length parameter using an adequate “optimality” rule.

Consider, for instance, the modified equation ([[#eq-14|14]]) applied to the convection-diffusion problem. Neglecting third order derivatives of <math display="inline">\phi </math>, Eq.([[#eq-14|14]]) can be written as
<span id="eq-15"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>-u \frac{d\phi }{dx}+\left(k+\frac{u h}{2}\right)\frac{d^2\phi }{dx^2}=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|}

We see that the FIC procedure introduces ''naturally'' an additional diffusion term into the standard convection-diffusion equation. This is the basis of the popular “artificial diffusion” method (Hirsch, 1990 <span id="citeF-42"></span>[[#cite-42|[42]]]). The characteristic length <math display="inline">h</math> is typically expressed as a function of the cell or element dimensions. The optimal or critical value of <math display="inline">h</math> can be computed from numerical stability conditions such as obtaining a physically meaningful solution, or even obtaining “exact” nodal values (Zienkiewicz and Taylor, 2000 <span id="citeF-107"></span>[[#cite-107|[107]]]; Oñate and Manzan, 1999 <span id="citeF-76"></span>[[#cite-76|[76]]], 2000 <span id="citeF-77"></span>[[#cite-77|[77]]]; Oñate, 2004 <span id="citeF-72"></span>[[#cite-72|[72]]]).

Equation ([[#eq-13|13]]) can be extended to account for source and time effects. The full FIC equation for the transient convection-diffusion problem can be written in compact form as
<span id="eq-16"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>r- \underline{\frac{h}{2} \frac{dr}{dx}}- \underline{{\delta \over 2} {\partial r \over \partial t}}=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
|}
<span id="eq-17"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>r = -\left(\frac{d\phi }{dt}+u \frac{d\phi }{dx}\right)+ \frac{d}{dx}\left(k \frac{d\phi }{dx}\right)+ Q </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
|}

where <math>Q</math> is the external source and <math>\delta </math> is a time stabilization parameter (Oñate, 1998 <span id="citeF-70"></span>[[#cite-70|[70]]]; Oñate and Manzan, 1999 <span id="citeF-76"></span>[[#cite-76|[76]]]). For consistency a FIC form of the Neumann boundary condition should be used. This is obtained by invoking balance of fluxes in a domain of finite size next to the boundary <math>\Gamma _q</math> where the flux is prescribed to a value <math>\bar q</math>. The modified FIC boundary condition is
<span id="eq-18"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>k \frac{d\phi }{dx}+\bar q- \underline{\frac{h}{2}r}=0 \quad \hbox{at } \Gamma _q </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
|}

The definition of the problem is completed with the standard Dirichlet condition prescribing the value of <math display="inline">\phi </math> at the boundary <math display="inline">\Gamma _\phi </math> and the initial conditions.

==5 FIC EQUATIONS FOR VISCOUS INCOMPRESSIBLE FLOW. ALE FORMULATION==

The starting point are the FIC equations for a viscous incompressible fluid. For simplicity we will neglect the time stabilization term, as this is not relevant for the purposes of this work. The equations are written as (Oñate, 1998 <span id="citeF-70"></span>[[#cite-70|[70]]], 2000 <span id="citeF-71"></span>[[#cite-71|[71]]]; Oñate ''et al.'', 2002 <span id="citeF-79"></span>[[#cite-79|[79]]])

''Momentum''
<span id="eq-19"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>r_{m_i} - \underline{{1\over 2} h_j{\partial r_{m_i} \over \partial x_j}}=0  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
|}

''Mass balance''
<span id="eq-20"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\left({\partial u_k \over \partial x_k}\right)- \underline{{h_j\over 2} {\partial  \over \partial x_j} \left({\partial u_k \over \partial x_k}\right)}=0  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
|}

where
<span id="eq-21"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>r_{m_i} = \rho \left({\partial u_i \over \partial t}+v_j{\partial u_i \over \partial x_j}+u_i {\partial u_j \over \partial x_j}\right)+ {\partial p \over \partial x_i}- {\partial s_{ij} \over \partial x_j}-b_i \qquad i,j = 1, n_d </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
|}

and all the terms have been defined in Section [[#2.1 Momentum and mass conservation equations|2.1.]]

The Neumann boundary conditions for the FIC formulation are (Oñate, 1998 <span id="citeF-70"></span>[[#cite-70|[70]]], 2000 <span id="citeF-71"></span>[[#cite-71|[71]]])
<span id="eq-22"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>n_j \sigma _{ij} -t_i + \underline{{1\over 2} h_j n_j r_{m_i}}=0 \quad \hbox{on }\Gamma _t </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
|}

The Dirichlet and initial boundary conditions are the standard ones given in Section [[#2.1 Momentum and mass conservation equations|2.1.]]

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-22|22]]) these lengths define the domain where equilibrium of boundary tractions is established. The sign in front the <math display="inline">h_i</math> term in Eq.([[#eq-22|22]]) is consistent with the definition of <math display="inline">r_{m_i}</math> in Eq.([[#eq-21|21]]).

Eqs.([[#eq-19|19]])-([[#eq-22|22]]) are the starting  point for deriving stabilized FEM for solving the incompressible Navier-Stokes equations using equal-order interpolation for all variables.

===5.1 Stabilized integral forms===

From Eqs.([[#eq-19|19]]) and ([[#eq-3|3]]) it can be obtained
<span id="eq-23"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>{h_j\over 2} {\partial  \over \partial x_j} \left({\partial u_k \over \partial x_k}\right)\simeq   \sum \limits _{i=1}^{n_d}\tau _i     {\partial r_{m_i} \over \partial x_i} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (23)
|}

where
<span id="eq-24"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\tau _i = \left({8\mu \over 3h_i^2}+{2\rho u_i\over h_i}\right)^{-1} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (24)
|}

Substituting Eq.([[#eq-23|23]]) into Eq.([[#eq-20|20]]) leads to the following stabilized mass balance equation
<span id="eq-25></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>{\partial u_k \over \partial x_k}- \sum \limits _{i=1}^{n_d} \tau _i {\partial r_{m_i} \over \partial x_i}=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (25)
|}

The <math display="inline">\tau _i</math>'s in Eq.([[#eq-23|23]]) are ''intrinsic time parameters per unit mass''. Similar parameters  also appear in other stabilized formulations (Hughes and Mallet, 1986a <span id="citeF-48"></span>[[#cite-48|[48]]]; Tezduyar, 1991 <span id="citeF-94"></span>[[#cite-94|[94]]], 2001 <span id="citeF-95"></span>[[#cite-95|[95]]]; Codina, 2002 <span id="citeF-21"></span>[[#cite-21|[21]]]). Note that  the parameters of Eq.([[#eq-24|24]]) emerge naturally form the FIC formation and take the values of <math display="inline">\tau _i =\displaystyle{3h_i^2\over 8\mu }</math> and <math display="inline">\tau _i =\displaystyle{h_i\over 2\rho u_i}</math> for the viscous limit (Stokes flow) and the inviscid  limit (Euler flow), respectively.

The weighted residual form of the governing equations (Eqs.([[#eq-19|19]]), ([[#eq-22|22]]) and ([[#eq-25|25]])) is
<span id="eq-26"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\int _\Omega \delta u_i \left[r_{m_i} - {h_j\over 2} {\partial r_{m_i} \over \partial x_j}\right]+ \int _{\Gamma _t} \delta u_i (\sigma _{ij} n_j - t_i +{ h_j\over 2} n_j r_{m_i}) d\Gamma =0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (26)
|}
<span id="eq-27"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\int _\Omega q \left[{\partial u_k \over \partial x_k}- \sum \limits _{i=1}^{n_d} \tau _i {\partial r_{m_i} \over \partial x_i}\right]d\Omega =0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (27)
|}

where <math display="inline">\delta u_i</math> and <math display="inline">q</math> are arbitrary weighting functions representing virtual velocity and virtual pressure fields, respectively. Integrating by parts above equations leads to
<span id="eq-28"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\int _\Omega \left[\delta u_i\rho \left({\partial u_i \over \partial t}+v_j {\partial {u_i} \over \partial x_j}\right)+ \delta \varepsilon _{ij}(\tau _{ij}- \delta _{ij}p)\right]d\Omega -  \int _{\Omega } \delta u_i b_i d\Omega -\int _{\Gamma _t} \delta u_i t_id\Omega + </math>
|-
| style="text-align: center;" | <math> + \sum \limits _e \int _{\Omega ^e} {h_j\over 2}{\partial \delta u_i \over \partial x_j} r_{m_i} d\Omega =0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (28)
|}
<span id="eq-29"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\int _\Omega q r_d \,d\Omega + \int _\Omega \left[\sum \limits _{i=1}^{n_d} \tau _i {\partial q \over \partial x_i} r_{m_i}\right]d\Omega =0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
|}

The fourth integral in Eq.([[#eq-28|28]]) is computed as a sum of the element contributions to allow for discontinuities in the derivatives of <math display="inline">r_{m_i}</math> along the element interfaces. As usual <math display="inline">\delta \varepsilon _{ij} ={1\over 2} \left({\partial \delta u_i \over \partial x_j}+{\partial \delta u_j \over \partial x_i}\right)</math>. In Eq.([[#eq-28|28]]) we have neglected the volumetric strain rate term, whereas in the derivation of Eq.([[#eq-29|29]]) we have assumed that <math display="inline">r_{m_i}</math> is negligible on the boundaries.

===5.2 Convective and pressure gradient projections===

The convective and pressure gradient projections <math display="inline">c_i</math> and <math display="inline">\pi _i</math> are defined as
<span id="eq-30"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>c_i = r_{m_i} -\rho v_j {\partial u_i \over \partial x_j}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (30)
|}
<span id="eq-31"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math> \pi _i = r_{m_i} - {\partial p \over \partial x_i} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (31)
|}

We can now express <math display="inline">r_{m_i}</math> in Eqs.([[#eq-28|28]]) and ([[#eq-29|29]]) in terms of <math display="inline">c_i</math> and <math display="inline">\pi _i</math>, respectively which become additional variables. The system of integral equations is now augmented in the necessary number of equations by imposing that the residuals <math display="inline">r_{m_i}</math> vanish (in average sense) for both forms given by Eqs.([[#eq-30|30]]) and ([[#eq-31|31]]). The final system of integral equations is:
<span id="eq-32"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\int _\Omega \left[\delta u_i\rho \left({\partial u_i \over \partial t}+v_j {\partial {u_i} \over \partial x_j}\right)+ \delta \varepsilon _{ij}(\tau _{ij}- \delta _{ij}p)\right] d\Omega -  \int _{\Omega } \delta u_i b_i d\Omega - \int _{\Gamma _t} \delta u_i t_id\Omega +</math>
|-
| style="text-align: center;" | <math>  + \sum \limits _e \int _{\Omega ^e} {h_k\over 2}{\partial (\delta u_i) \over \partial x_k} \left(\rho v_j {\partial {u_i} \over \partial x_j} + c_i\right)d\Omega =0</math>
|-
| style="text-align: center;" | <math> \int _\Omega q \left({\partial u_k \over \partial x_k}\right) d\Omega +  \int _\Omega \sum \limits _{i=1}^{n_d} \tau _i {\partial q \over \partial x_i} \left({\partial p \over \partial x_i}+\pi _i\right)d\Omega =0</math>
|-
| style="text-align: center;" | <math> \int _\Omega \delta c_i \rho \left(\rho v_j {\partial {u_i} \over \partial x_j} + c_i\right)d\Omega =0 \qquad \hbox{ no sum in }i</math>
|-
| style="text-align: center;" | <math> \int _\Omega \delta \pi _i \tau _i \left({\partial p \over \partial x_i}+\pi _i\right)d\Omega =0\qquad \hbox{ no sum in }i </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (32)
|}

where <math display="inline">\delta c_i</math> and <math display="inline">\delta \pi _i</math> are appropriate weighting functions and the <math display="inline">\rho </math> and <math display="inline">\tau _i</math> terms are introduced in the last two equations for convenience. As usual <math display="inline">i,j,k=1,n_d</math>.

===5.3 Stabilized FIC equations for the free-surface wave condition===

We will derive next the FIC equation for the water wave surface.

Let us consider a 2D free-surface wave problem. Figure [[#img-3|3]] shows a typical free-surface segment line <math display="inline">AB</math>. The average vertical velocity for the segment <math display="inline">\bar u_2</math> is defined as
<span id="eq-33"></span>
<span id="eq-33a"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar u_2 ={u_2^A + u_2^B\over 2}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (33a)
|}

where <math display="inline">u_2^A</math> and <math display="inline">u_2^B</math> are the vertical velocities of the end points of the segment <math display="inline">A</math> and <math display="inline">B</math>.

The average vertical velocity <math display="inline">\bar u_2</math> can be computed from the wave heights at points <math display="inline">A</math> and <math display="inline">B</math> as
<span id="eq-33b"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar u_2 ={x_2^B -x_2^A\over \bar \delta }</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (33b)
|}

where <math display="inline">\bar \delta </math> is the time which a material particle takes to travel from point <math display="inline">A</math> to point <math display="inline">B</math> at the average velocity <math display="inline">\bar u_2</math> and <math display="inline">x_2 \equiv \beta </math> is the free-surface wave height. Equaling Eq.([[#eq-33a|33a]]) and ([[#eq-33b|33b]]) gives
<span id="eq-33c"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{u_2^A + u_2^B\over 2}= {x_2^B -x_2^A\over \bar \delta }</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (33c)
|}

We can now express the vertical velocities and the wave height at points <math display="inline">A</math> and <math display="inline">B</math> in terms of values at an arbitrary point <math display="inline">C</math> as
<span id="eq-34"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>u_2^A = u_2 (x_1^C -d_1, t^C-\delta _1) = u_2^C -d_1 {\partial \beta  \over \partial x_1}-\delta _1 {\partial \beta  \over \partial t}+ O (d_1^2,t_1^2) </math>
|-
| style="text-align: center;" | <math> u_2^B = u_2 (x_1^C +d_2, t^C +\delta _2) = u_2^C +d_2 {\partial \beta  \over \partial x_2}+\delta _2 {\partial \beta  \over \partial t}+ O (d_2^2, t_2^2)</math>
|-
| style="text-align: center;" | <math> x_2^A= x_2 (x_1^C -d_1, t^C-\delta _1) = x_2^C -d_1 {\partial \beta  \over \partial x_1}-\delta _1 {\partial \beta  \over \partial t}+{d_1^2\over 2} {\partial ^2 \beta \over \partial x_1^2}+ </math>
|-
| style="text-align: center;" | <math> +\, d_1\delta _1 {\partial ^2 \beta \over \partial x_1\partial t}+{\delta _1^2\over 2}{\partial ^2 \beta \over \partial t^2} + O (d_1^3, \delta _1^3)</math>
|-
| style="text-align: center;" | <math> x_2^B= x_2 (x_1^C +d_2, t^C +\delta _2) = x_2^C +d_2 {\partial \beta  \over \partial x_1}+\delta _2 {\partial \beta  \over \partial t}+{d_2^2\over 2} {\partial ^2 \beta \over \partial x_1^2}+ </math>
|-
| style="text-align: center;" | <math> +\, d_2\delta _2 {\partial ^2 \beta \over \partial x_1\partial t}+{\delta _2^2\over 2}{\partial ^2 \beta \over \partial t^2} + O (d_2^3, \delta _2^3) </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (34)
|}

where all the derivatives are computed at the arbitrary point <math display="inline">C</math>.

Substituting Eqs.([[#eq-34|34]]) into ([[#eq-33c|33c]]) and noting that <math display="inline">d\simeq u_1^C\bar \delta </math>, with <math display="inline">\bar \delta =(\delta _1+\delta _2)</math>, <math display="inline">d_1 \simeq u_1^C\delta _1</math> and <math display="inline">d_2 \simeq u_1^C\delta _2</math> (Figure [[#img-3|3]])  and that the position of point <math display="inline">C</math> is arbitrary, gives the FIC equation for  the free-surface height (neglecting high order terms) as
<span id="eq-35"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>r_\beta - \underline{{h\over 2}{\partial r_\beta  \over \partial x_1}} - \underline{{\delta \over 2}{\partial r_\beta  \over \partial t}}=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (35)
|}

with
<span id="eq-36"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>r_\beta = {\partial \beta  \over \partial t}+u_1 {\partial \beta  \over \partial x_1}-u_2 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (36)
|}

where <math display="inline">h=(d_1 -d_2)</math> and <math display="inline">\delta = (\delta _1 - \delta _2)</math> are space and time stabilization parameters. The standard infinitesimal form of the free-surface wave condition is obtained by making <math display="inline">h=\delta =0</math> in Eq.([[#eq-35|35]]) giving
<span id="eq-37"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\partial \beta  \over \partial t}+u_1 {\partial \beta  \over \partial x_1}-u_2=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (37)
|}

which coincides with equation ([[#eq-11|11]]) for the 2D case.

A simpler FIC expression can be derived from Eq.([[#eq-35|35]]) by retaining the second order space term only. This gives
<span id="eq-38"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\partial \beta  \over \partial t}+u_1 {\partial \beta  \over \partial x_1}-{u_1h\over 2} {\partial ^2\beta \over \partial x_1^2}-u_2 = 0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (38)
|}

This can be interpreted as the addition of an artificial diffusion term where <math display="inline">{u_1h\over 2}</math> plays the role of the new balancing diffusion coefficient.

In the following we will use the 3D ALE form of Eq.([[#eq-35|35]]) neglecting the time stabilization term, given by
<span id="eq-39"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>r_\beta - \underline{{h_j\over 2}{\partial r_\beta  \over \partial x_j}}=0\qquad \hbox{with }\quad  r_\beta = {\partial \beta  \over \partial t}+ v_j {\partial \beta  \over \partial x_j}-v_3 \quad , \quad j=1,2 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (39)
|}

where, as usual, <math display="inline">v_i</math> are the relative velocities between the fluid and the moving mesh nodes.

==6 FINITE ELEMENT DISCRETIZATION==

===6.1 Discretization of the fluid flow equations===

We will choose <math display="inline">C^\circ </math> continuous linear interpolations of the velocities, the pressure, the convection projections <math display="inline">c_i</math> and the pressure gradient projections <math display="inline">\pi _i</math> over three node triangles (2D) and four node tetrahedra (3D). The interpolations are written as
<span id="eq-40"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\begin{array}{l}u_i = \displaystyle \sum \limits _{j=1}^n N_j \bar u_i^j \quad , \quad p = \sum \limits _{j=1}^n N_j \bar p^j\\ c_i = \displaystyle \sum \limits _{j=1}^n N_j \bar c_i^j \quad , \quad \pi _i = \sum \limits _{j=1}^n N_j \bar \pi _i^j \end{array} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (40)
|}

where <math display="inline">n=3</math> (4) for triangles (tetrahedra), <math display="inline">\bar {(\cdot )}</math> denotes nodal variables and <math display="inline">N_j</math> are the linear shape functions (Zienkiewicz and Taylor, Vol 1 2000 <span id="citeF-107"></span>[[#cite-107|[107]]]).

Substituting the approximations ([[#eq-40|40]]) into Eqs.([[#eq-32|32]]) and choosing the Galerking form with <math display="inline">\delta u_i =q=\delta c_i=\delta \pi _i =N_i</math> leads to following system of discretized equations
<span id="eq-41"></span>
<span id="eq-41a"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol M}\dot{\bar {\boldsymbol u}} + ({\boldsymbol A}+{\boldsymbol K}+\hat{\boldsymbol K}) \bar {\boldsymbol u} - {\boldsymbol G}\bar {\boldsymbol p}+{\boldsymbol C}\bar {\boldsymbol c}={\boldsymbol f}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (41a)
|}
<span id="eq-41b"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol G}^T \bar {\boldsymbol u} + {\boldsymbol L}\bar {\boldsymbol p}+{\boldsymbol Q}\bar {\boldsymbol \pi }={\boldsymbol 0}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (41b)
|}
<span id="eq-41c"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\qquad \qquad \hat{\boldsymbol C}\bar {\boldsymbol u}+ {\boldsymbol M}\bar {\boldsymbol c}={\boldsymbol 0} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (41c)
|}
<span id="eq-41d"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol Q}^T \bar {\boldsymbol p} + \hat{\boldsymbol M}\bar {\boldsymbol \pi }={\boldsymbol 0}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (41d)
|}

The element contributions are given by (for 2D problems)
<span id="eq-42"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\displaystyle {\boldsymbol M}_{ij}\!\!=\!\! \int _{\Omega ^e} \rho {\boldsymbol N}_i{\boldsymbol N}_j d\Omega \quad ,\quad {\boldsymbol A}_{ij}= \!\int _{\Omega ^e}\! {\boldsymbol N}_i \rho v_k {\partial {\boldsymbol N}_j \over \partial x_k} d\Omega \quad ,\quad \displaystyle {\boldsymbol K}_{ij}= \! \int _{\Omega ^e}\! {\boldsymbol B}_i^T {\boldsymbol D}{\boldsymbol B}_j d\Omega </math>
|-
| style="text-align: center;" | 
|-
| style="text-align: center;" | <math> \hat{\boldsymbol K}_{ij}\!\!= \!\! \int _{\Omega ^e} \rho {h_k v_l\over 2}\displaystyle {\partial {\boldsymbol N}_i \over \partial x_k} \displaystyle {\partial {\boldsymbol N}_j \over \partial x_l}d\Omega \!\!\quad ,\quad \!\! \displaystyle {\boldsymbol G}_{ij}= \! \int _{\Omega ^e}\! ({\boldsymbol \nabla } N_i)N_j d\Omega \!\!\quad ,\!\!\quad {\boldsymbol C}_{ij}= \! \int _{\Omega ^e}\! {h_k \over 2} \displaystyle {\partial {\boldsymbol N}_i \over \partial x_k}{\boldsymbol N}_j d\Omega \, d\Omega </math>
|-
| style="text-align: center;" | 
|-
| style="text-align: center;" | <math> \displaystyle L_{ij}\!\!=\!\!\int _{\Omega ^e} {\boldsymbol \nabla }^T N_i [\tau ] {\boldsymbol \nabla } N_j d\Omega ~,~ \hat {\boldsymbol C}_{ij}=\!\int _{\Omega ^e}\! {\boldsymbol N}_i \rho ^2 v_k \displaystyle {\partial {\boldsymbol N}_j \over \partial x_k}d\Omega ~,~ {\boldsymbol \nabla }= \! \left\{\begin{array}{c}\displaystyle {\partial ~ \over \partial x_1}\\ \displaystyle {\partial ~ \over \partial x_2}\end{array}\right\}~,~ [\tau ]= \!\left[\begin{array}{cc}\tau _1 &0\\ 0&\tau _2\end{array}\right]  </math>
|-
| style="text-align: center;" | 
|-
| style="text-align: center;" | <math> {\boldsymbol Q}\!\!=\!\! [{\boldsymbol Q}^1,{\boldsymbol Q}^2] \quad ,\quad  Q_{ij}^k = \int _{\Omega ^e}\tau _k {\partial N_i \over \partial x_k} N_jd\Omega \quad \hbox{ no sum in } k</math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (42)
|-
| style="text-align: center;" | 
|-
| style="text-align: center;" | <math> \hat{\boldsymbol M}\!\!=\!\! \left[\begin{array}{cc}\hat {\boldsymbol M}^1 &{\boldsymbol 0}\\  {\boldsymbol 0} & \hat {\boldsymbol M}^2\end{array}\right]\quad ,\quad  \hat{M}_{ij}^k = \int _{\Omega ^e} \tau _k N_i N_j d\Omega \quad ,\quad {\boldsymbol N}_i = N_i \left[\begin{array}{cc}1&0\\ 0&1 \end{array}\right]</math>
|-
| style="text-align: center;" | 
|-
| style="text-align: center;" | <math> {\boldsymbol f}_i\!\! =\!\! \int _{\Omega ^e} N_i {\boldsymbol b}d\Omega + \int _{\Gamma ^e}N_i {\boldsymbol t} d\Gamma \quad ,\quad  {\boldsymbol b}=[b_1,b_2]^T,\, {\boldsymbol t}=[t_1,t_2]^T </math>
|}
|}

with <math display="inline">i,j=1,n</math> and <math display="inline">k,l=1,n_d</math>.

In above <math display="inline">{\boldsymbol B}_i</math> is the standard strain rate matrix and <math display="inline">{\boldsymbol D}</math> the deviatoric constitutive matrix (for <math display="inline">\displaystyle {\partial u_i \over \partial x_i}=0</math>). For 2D problems
<span id="eq-43"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol B}_i =\left[\begin{array}{cc}{\partial N_i \over \partial x_1}&0\\ 0 & {\partial N_i \over \partial x_2}\\ {\partial N_i \over \partial x_1} & {\partial N_i \over \partial x_2}\end{array}\right]\quad ,\quad {\boldsymbol D} =\mu \left[\begin{array}{ccc}2&0&0\\ 0&2&0\\ 0&0&1\end{array}\right] </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (43)
|}

Note that the stabilization matrix <math display="inline">\hat{\boldsymbol K}</math> brings in an additional orthotropic diffusivity of value <math display="inline">\rho \displaystyle{h_k v_l\over 2}</math>. Matrices <math display="inline">{\boldsymbol A}, \hat {\boldsymbol K}</math> and <math display="inline">\hat{\boldsymbol C}</math> are dependent on the velocity field. The solution process can be advanced in time in a (quasi-nearly) implicit iterative manner using the following scheme.

<span id="step-1"></span>
''Step 1''
<span id="eq-44"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\boldsymbol u}^{n+1,i} = \bar{\boldsymbol u}^n - \Delta t {\boldsymbol M}^{-1} [({\boldsymbol A}^{n+\theta _1,i-1}+  {\boldsymbol K} + \hat {\boldsymbol K}^{n+\theta _1,i-1})\bar{\boldsymbol u}^{n+\theta _1,i-1}-{\boldsymbol G}{\boldsymbol p}^{n+\theta _2,i-1} + {\boldsymbol C} \bar {\boldsymbol c}^{n+\theta _3,i-1}-{\boldsymbol f}^{n+1}] </math>
|-
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (44)
|}
<span id="step-2"></span>
''Step 2''
<span id="eq-45"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol p}^{n+1,i}=- {\boldsymbol L}^{-1} [{\boldsymbol G}^T \bar {\boldsymbol u}^{n+1,i}+{\boldsymbol Q}\bar {\boldsymbol \pi }^{n+\theta _4,i-1}] </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (45)
|}
<span id="step-3"></span>
''Step 3''
<span id="eq-46"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar {\boldsymbol c}^{n+1,i}= - {\boldsymbol M}^{-1} \hat {\boldsymbol C}^{n+1,i}\bar {\boldsymbol u}^{n+1,i} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (46)
|}
<span id="step-4"></span>
''Step 4''
<span id="eq-47"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar {\boldsymbol \pi }^{n+1,i}= -\hat {\boldsymbol M}^{-1}{\boldsymbol Q}^T \bar {\boldsymbol p}^{n+1,i} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (47)
|}

In above <math display="inline">\theta _i</math> are time integration parameters with <math display="inline">0\le \theta _i\le 1</math> and <math display="inline">\bar{(\cdot )}^{n,i}</math> denotes nodal values at the <math display="inline">n</math>th time step and the ith iteration. <math display="inline">{\boldsymbol A}^{n+\theta _1,i-1}\equiv {\boldsymbol A}(\bar {\boldsymbol u}^{n+\theta _1,i-1})</math> etc. Also <math display="inline">(\cdot )^{n+\theta _i,0}\equiv (\cdot )^n</math> for the computations in step 1 at the onset of the iterations.

Steps [[#step-1|1]], [[#step-3|3]] and [[#step-4|4]] can be solved explicitely by choosing a ''lumped (diagonal) form'' of matrices  <math display="inline">M</math> and <math display="inline">\hat {\boldsymbol M}</math>. In this manner the main computational cost is the solution of step [[#step-2|2]] involving the inverse of a Laplacian matrix. This can be solved very effectively using an iterative method.

For <math display="inline">\theta _i\not =0</math> the iterative proces is unavoidable. The iterations follow until convergence is reached. This can be measured using an adequate error norm in terms of the velocity and pressure variables, or the residuals. Indeed some ot the <math display="inline">\theta _i</math>'s can be made equal to zero. Note that for <math display="inline">\theta _2=0</math> the algorithm is inconditionable unstable. A simple form is obtained making <math display="inline">\theta _1 = \theta _3=\theta _4=0</math>. This elliminates the non linear dependence with the velocity of matrices <math display="inline">{\boldsymbol A}</math> and <math display="inline">\hat{\boldsymbol K}</math>  during the iterative scheme.

Convergence of above solution scheme is however difficult for some problems. An enhanced version of the algorithm can be obtained by  adding the term <math display="inline">\hat {\boldsymbol  L} (\bar {\boldsymbol p}^{n+1,i} - \bar {\boldsymbol p}^{n+1,i-1})</math> where <math display="inline">\hat L_{ij} =\Delta t \int _{\Omega ^e} {\boldsymbol \nabla }^T N_i {\boldsymbol \nabla } N_j d\Omega </math> to the equation for the computation of the pressure in the second step. The new term acts as a preconditioner of the pressure equation given now by
<span id="eq-48"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar {\boldsymbol p}^{n+1,i}=-[{\boldsymbol L} + \hat {\boldsymbol L}]^{-1}[{\boldsymbol G}^T \bar {\boldsymbol u}^{n+1,i}+\hat {\boldsymbol L}\bar {\boldsymbol p}^{n+1,i-1} +{\boldsymbol Q}\bar {\boldsymbol \pi }^{n+\theta _4,i-1}] </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (48)
|}

Note that the added term vanishes for the converged solution (i.e. when <math display="inline">\bar {\boldsymbol p}^{n+1,i}= \bar {\boldsymbol p}^{n+1,i-1}</math>).

An alternative to above algorithm is to use the fractional step method described in the nex section.

===6.2 Fractional step method===

The pressure can  be split from the discretized momentum equations (see Eq.([[#eq-44|44]])) as
<span id="eq-49"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar {\boldsymbol u}^* = \bar {\boldsymbol u}^n -\Delta t {\boldsymbol M}^{-1}[({\boldsymbol A}^{n+\theta _1}+  {\boldsymbol K} + \hat {\boldsymbol K}^{n+\theta _1})\bar{\boldsymbol u}^{n+\theta _1}-\alpha {\boldsymbol G}\bar{\boldsymbol p}^{n} + {\boldsymbol C} \bar {\boldsymbol c}^{n+\theta _3}-{\boldsymbol f}^{n+1}] </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (49)
|}
<span id="eq-50"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\boldsymbol u}^{n+1}= \bar{\boldsymbol u}^*+\Delta t {\boldsymbol M}^{-1}{\boldsymbol G}\delta \bar{\boldsymbol p} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (50)
|}

In above equations <math display="inline">\alpha </math> is a variable taking values equal to zero or one. For <math display="inline">\alpha =0</math>, <math display="inline">\delta p \equiv p^{n+1}</math> (first order scheme) and for <math display="inline">\alpha =1</math>, <math display="inline">\delta p =\Delta p</math> (second order scheme) (Codina, 2001 <span id="citeF-19"></span>[[#cite-19|[19]]]). Note that in both cases  the sum of Eqs.([[#eq-49|49]]) and ([[#eq-50|50]]) gives the time discretization of the momentum equations with the pressures computed at <math display="inline">t^{n+1}</math>. The value of <math display="inline">\bar {\boldsymbol u}^{n+1}</math> from Eq.([[#eq-50|50]]) is substituted now into ([[#eq-41b|41b]]) to give
<span id="eq-51a"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol G}^T\bar {\boldsymbol u}^* + \Delta t {\boldsymbol G}^T {\boldsymbol M}^{-1}{\boldsymbol G}\delta \bar{\boldsymbol p} +  {\boldsymbol L} \bar{\boldsymbol p}^{n+1}+ {\boldsymbol Q}\bar {\boldsymbol \pi }^{n+\theta _4}=0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (51a)
|}

The product <math display="inline">{\boldsymbol G}^T {\boldsymbol M}^{-1}{\boldsymbol G}</math> can be approximated by a laplacian matrix, i.e.
<span id="eq-51b"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol G}^T {\boldsymbol M}^{-1}{\boldsymbol G}=\hat {\boldsymbol L} \quad \hbox{with } \hat {\boldsymbol L} \simeq \int _{\Omega ^e} {1\over \rho } ({\boldsymbol \nabla }^T N_i) {\boldsymbol \nabla } N_j \,d\Omega  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (51b)
|}

A semi-implicit algorithm can be derived as follows.<br/>

<span id="s1"></span>
''Step 1'' Compute the nodal fractional velocities <math display="inline">{\boldsymbol u}^*</math> explicitely from Eq.([[#eq-49|49]]) with <math display="inline">{\boldsymbol M}={\boldsymbol M}_d</math> where subscript <math display="inline">d</math> denotes a diagonal matrix.

<br/>
<span id="s2"></span>
''Step 2'' Compute <math display="inline">\delta \bar {\boldsymbol p}</math> from Eq.([[#eq-51a|51a]]) (using Eq.([[#eq-51b|51b]])) as
<span id="eq-52"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\delta \bar {\boldsymbol p} =-({\boldsymbol L}+\Delta t \hat{\boldsymbol L})^{-1} [{\boldsymbol G}^T\bar{\boldsymbol u}^* +{\boldsymbol Q}\bar {\boldsymbol \pi }^{n+\theta _4}+{\boldsymbol L} \bar {\boldsymbol p}^n] </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (52)
|}
<span id="s3"></span>
''Step 3'' Compute the nodal velocities <math display="inline"> \bar{\boldsymbol u}^{n+1}</math> explicitely from Eq.([[#eq-50|50]]) with <math display="inline">{\boldsymbol M}={\boldsymbol M}_d</math>

<br/>
<span id="s4"></span>
''Step 4'' Compute <math display="inline"> \bar{\boldsymbol c}^{n+1}</math> explicitely from Eq.([[#eq-46|46]]) using <math display="inline">{\boldsymbol M}_d</math>.

<span id="s5"></span>
''Step 5'' Compute <math display="inline"> \bar{\boldsymbol \pi }^{n+1}</math> explicitely from Eq.([[#eq-47|47]]) with <math display="inline">\hat {\boldsymbol M} = \bar{\boldsymbol M}_d</math>.

This algorithm has an additional step than the iterative algorithm of Section [[#6.1 Discretization of the fluid flow equations|6.1.]] The advantage is that now Steps [[#s1|1]] and [[#s2|2]] can be fully linearized by choosing <math display="inline">\theta _1 = \theta _3=\theta _4=0</math>. Also the equation for the pressure variables in Step [[#s2|2]] has improved stabilization properties due to the additional laplacian matrix <math display="inline">\hat{\boldsymbol L}</math>.

Details on the stability properties of the FIC formulation for incompressible fluid flow problems can be found in Oñate ''et al.'' (2002) <span id="citeF-79"></span>[[#cite-79|[79]]].

===6.3 Discretization of free-surface wave equation===

The solution in time of Eq.([[#eq-39|39]]) can be written in terms of the nodal velocities computed from the flow solution, as
<span id="eq-53"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\beta ^{n+1} = \beta ^n -\Delta t \left[v_i^{n+1,i} {\partial \beta ^n \over \partial x_i}-v_3^{n+1,i}-{h_{\beta _j}\over 2} {\partial r_\beta ^n \over \partial x_j}\right] </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (53)
|}

Eq.([[#eq-53|53]]) can now be discretized in space using the standard Galerkin method and solved ''explicitely'' for the nodal wave heights at <math display="inline">t^{n+1}</math>. Typically the general algorithm will be as follows:

<ol>

<li>Solve for the nodal velocities <math display="inline">\bar{\boldsymbol u}^{n+1}</math> and the pressures <math display="inline">\bar{\boldsymbol p}^{n+1}</math> in the fluid domain using any of the algorithms of Sections [[#6.1 Discretization of the fluid flow equations|6.1]] and [[#6.2 Fractional step method|6.2.]] When solving for the pressure equation impose <math display="inline">p^{n+1}= p_a</math> at the free-surface <math display="inline">\Gamma _\beta </math>.  </li>

<li>Solve for the free-surface elevation <math display="inline">\beta ^{n+1}</math> (viz. Eq.([[#eq-53|53]])).  </li>

<li>Compute the new position of the mesh nodes in the fluid domain at time <math display="inline">t^{n+1}</math>. Alternatively, regenerate a new mesh. </li>

</ol>

The mesh updating process can also include the free-surface nodes, although this is not strictly necessary. An ''hydrostatic adjustement'' can be implemented once the new free-surface elevation is computed simply by imposing the pressure at the nodes on the reference surface as
<span id="eq-54"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>p^{n+1}=p_a + \rho g \Delta \beta \quad \hbox{with}\quad  \Delta \beta = \beta ^{n+1}- \beta ^{ref} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (54)
|}

where <math display="inline">g</math> is the gravity constant.

Eq.([[#eq-54|54]]) takes into account the changes in the free-surface without the need of updating the reference surface nodes. A higher accuracy in the flow solution can be obtained by updating these nodes after a number of time steps.

==7 FLUID-SHIP INTERACTION==

The algorithms of previous section can be extended to account for the ship motion due to the hydrodynamic forces. Here the ship hull structure can be modelled as a rigid solid defined by the three translations and the three rotations of its center of gravity. Alternatively, the full deformation of the ship structure can be computed by modelling the actual stiffness of the hull, the decks and the different structural members. Indeed, the first option usually suffices for the hull shape optimization.

In both cases the computation of the ship motion involves solving the dynamic equations of the ship structure written as
<span id="eq-55"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol M}_s \ddot {\boldsymbol d}+ {\boldsymbol K}_s {\boldsymbol d}={\boldsymbol f}_{ext} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (55)
|}

where <math display="inline">{\boldsymbol d}</math> and <math display="inline">\ddot {\boldsymbol d}</math> are the displacement and acceleration vectors of the nodes discretizing the ship structure, respectively, <math display="inline">{\boldsymbol M}_s</math> and <math display="inline">{\boldsymbol K}_s</math> are the mass and stiffness matrices of the structure and <math display="inline">{\boldsymbol f}_{ext}</math> is the vector of external nodal forces accounting for the fluid flow loads induced by the pressure and the viscous stresses. Clearly the main driving forces for the motion of the ship is the fluid pressure which acts in the form of a surface traction. Indeed Eq.([[#eq-55|55]]) can be augmented with an appropriate damping term. The form of all the relevant matrices and vectors can be found in standard books on FEM for structural analysis (Zienkiewicz and Taylor, Vol 2 2000 <span id="citeF-107"></span>[[#cite-107|[107]]]).

Solution of Eq.([[#eq-55|55]]) in time can be performed using implicit or fully explicit time integration algorithms. In both cases the values of the nodal displacements, the velocities and the accelerations at time <math display="inline">t^{n+1}</math> are found.

A simple coupled fluid-ship-structure solution in time using, for instance, the  fractional step method of Section [[#6.2 Fractional step method|6.2]] (for <math display="inline">\theta _1=\theta _2=\theta _3=\theta _4=0</math>)  involves the following steps.

<br/>
<span id="step1"></span>
'''Step 1''' Solve for the fractional velocities <math display="inline">\bar{\boldsymbol u}^*</math> using Eq.([[#eq-49|49]]). Here use of <math display="inline">\alpha =1</math> is recommended.

<br/>
<span id="step2"></span>
'''Step 2''' Compute <math display="inline">\delta \bar {\boldsymbol p}</math> from Eq.([[#eq-51a|51a]]) solving a simultaneous system of equations.

<br/>
<span id="step3"></span>
'''Step 3''' Compute explicitely the nodal velocities <math display="inline"> \bar{\boldsymbol u}^{n+1}</math> from Eq.([[#eq-50|50]]) with a diagonal mass matrix.

<br/>
<span id="step4"></span>
'''Step 4''' Compute explicitely the projected convective variables <math display="inline">\bar{\boldsymbol c}^{n+1}</math> from Eq.([[#eq-46|46]]) using <math display="inline">{\boldsymbol M}_d</math>.

<br/>
<span id="step5"></span>
'''Step 5''' Compute explicitely the projected pressure gradients <math display="inline">\bar {\boldsymbol \pi }^{n+1}</math> from Eq.([[#eq-47|47]]) using <math display="inline">\hat{\boldsymbol M}_d</math>.

<br/>
<span id="step6"></span>
'''Step 6''' Compute explicitely the new position of the free-surface elevation <math display="inline">\bar{\boldsymbol \beta }^{n+1}</math> from Eq.([[#eq-53|53]]).

<br/>
<span id="step7"></span>
'''Step 7''' Compute the movement of the ship by solving the dynamic equations of motion for the ship structure under the hydrodynamic forces induced by the pressures <math display="inline">\bar {\boldsymbol p}^{n+1}</math> and the viscous stresses <math display="inline">{\boldsymbol s}^{n+1}</math>.

<br/>
<span id="step8"></span>
'''Step 8''' Update the position of the mesh nodes in the fluid domain at <math display="inline">t^{n+1}</math> by using the mesh update algorithm described in the next section. The updating process can also include the free-surface nodes  although this is not strictly necessary  to be done at every time step and the hydrostatic adjustment of the pressure acting on the free-surface (Section [[#6.3 Discretization of free-surface wave equation|6.3]])  can be used as an alternative.<br/>

Obviously, the use of a value different from zero for the <math display="inline">\theta _1,\theta _2,\theta _3</math> or <math display="inline">\theta _4</math> parameters will require an iterative process between steps [[#step1|1]] and [[#step8|8]] until a converged solution for the variables describing the fluid and ship motions is found.

A common option is to update the position of the mesh nodes only when  the iterative process for computing the fluid and ship variables has converged. Clearly the regeneration of the mesh is unavoidable  when the distorsion of the elements exceed a certain limit.

==8 A SIMPLE ALGORITHM FOR UPDATING THE MESH NODES==

Different techniques have been proposed for dealing with mesh updating in fluid-structure interaction problems. The general aim of all methods is to prevent element distortion during mesh deformation (Tezduyar, 2001a <span id="citeF-95"></span>[[#cite-95|[95]]]; Tezduyar ''et al.'', 1992bc <span id="citeF-100"></span>[[#cite-100|[100]]-<span id="citeF-101"></span>[[#cite-101|101]]]).

Tezduyar ''et al.'' (1992c) <span id="citeF-101"></span>[[#cite-101|[101]]] and Chiandussi ''et al.'' (2000) <span id="citeF-13"></span>[[#cite-13|[13]]] have proposed simple method for  moving the  mesh nodes based on the iterative solution of a fictious linear elastic problem on the mesh domain. In the method introduced in Tezduyar ''et al.'' (1992c) <span id="citeF-101"></span>[[#cite-101|[101]]], the mesh deformation is handled selectively based on the element sizes and deformation modes, with the objective to increase stiffening of the smaller elements, which are typically located near solid surfaces. In Chiandusi ''et al.'' (2000) <span id="citeF-13"></span>[[#cite-13|[13]]] in order to minimize the mesh deformation the “elastic” properties of each mesh element are appropiately selected so that elements suffering greater movements are stiffer. A simple and effective procedure is to select the Poisson's ratio <math display="inline">\nu =0</math> and compute the “equivalent” Young modulus for each element by
<span id="eq-56"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>E = {\bar E \over 3 \bar  \varepsilon ^2 }(\varepsilon _1^2+\varepsilon _2^2 + \varepsilon _3^2) </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (56)
|}

where <math display="inline">\varepsilon _i</math> are the principal strains, <math display="inline">\bar E</math> is an arbitrary value of the Young modulus  and <math display="inline">\bar \varepsilon </math> is a prescribed uniform strain field. <math display="inline">\bar E</math> and <math display="inline">\bar  \varepsilon </math> are constant for all the elements in the  mesh.

In summary, the solution scheme proposed by Chiandusi ''et al.'' (2000) <span id="citeF-13"></span>[[#cite-13|[13]]] includes the following two steps.

''Step 1''. Consider the FE mesh as a linear elastic solid with homogeneous material properties characterized by a prescribed uniform strain field <math display="inline">\bar  \varepsilon </math>, an arbitrary Young modulus <math display="inline">\bar E</math> and <math display="inline">\nu =0</math>. Solve a linear elastic problem with imposed displacements at the mesh boundary defined by the actual movement of the boundary nodes. An approximate solution to this linear elastic problem, such as that given by the first iterations of a conjugate gradient solution scheme, suffices for practical purposes.

''Step 2''. Compute the principal strains  in each element. Repeat the (approximate) FE solution of the linear elastic problem with prescribed boundary displacements using the values of <math display="inline">E</math> of Eq.([[#eq-56|56]]).

The previous algorithm is able to treat the movement of the mesh due to changes in position of fully submerged and semi-submerged bodies such as ships. However if the floating body intersects the free-surface, the changes in the analysis domain  can be very important as emersion or inmersion of significant parts of the body can occur within a time step. A possible solution to this problem is to remesh the analysis domain. However, for most  problems a mapping of the moving surfaces linked to mesh updating algorithm described above can avoid remeshing. The surface mapping technique  used by the authors is based on transforming the 3D curved surfaces into reference planes (Figure [[#img-4|4]]). This makes it possible to compute within each plane the local (in-plane) coordinates of the nodes for the final surface mesh accordingly to the changes in the floating line. The final step is to transform back the local coordinates of the surface mesh in the reference plane to the final curved configuration which incorporates the new floating line (García, 1999 <span id="citeF-37"></span>[[#cite-37|[37]]]; Oñate and García, 2001 <span id="citeF-78"></span>[[#cite-78|[78]]]).

==9 MODELLING OF THE TRANSOM STERN FLOW==

The transom stern causes a discontinuity in the domain and the solution of the free-surface equation close to this region is inconsistent with the convective nature of the equation. This leads to instability of the wave height close to the transom region. This instability is found experimentally for low speeds. The flow at a sufficient high speed is physically  more stable although it still can not  be reproduced by  standard numerical techniques (Reed ''et al.'', 1990) <span id="citeF-85"></span>[[#cite-85|[85]]].

A solution to this problem is to apply adequate free-surface boundary conditions at the transom boundary. The obvious condition is to fix both the free surface elevation <math display="inline">\beta </math> and its derivative along the corresponding streamline to values given by the transom position and the surface gradient. However, prescribing those values can influence the transition between the transom flux and the lateral flux, resulting in unaccurate wave maps.

The method proposed in García and Oñate (2003) <span id="citeF-38"></span>[[#cite-38|[38]]] is to extend the free-surface below the ship. In this way the necessary Dirichlet boundary conditions imposed at the inflow  domain are enough to define a well posed problem. This method is valid both for the wetted and dry transom cases and it is also applicable  to ships with regular stern.

This scheme does not work for partially wetted transoms. This situation can occur for highly unsteady flows where wake vortex induces the free-surface deformation and the flow remains adhered to the transom. To favour the computation of the free-surface, an artificial viscosity term is added to the free-surface equation in the vicinity of the transom in these cases.

==10 LAGRANGIAN FLOW FORMULATION==

The Lagrangian formulation is an effective (and relatively simple) procedure for modelling the flow of fluid particles undergoing severe distorsions such as water jets, high amplitude waves, braking waves, water splashing, filling of cavities, etc. Indeed the Lagrangian formulation is very suitable for treating ship hydrodynamic  problems where the ship undergoes large motions. An obvious “a priori” advantage of the Lagrangian formulation is that both the ship and the fluid motion are defined in the same frame of reference.

The Lagrangian fluid flow equations are obtained by noting that the velocity of the mesh nodes and that of the fluid particles are the same. Hence the relative velocities <math display="inline">v_i</math> are zero in Eq.([[#eq-21|21]]) and the convective terms vanish in the momentum equations, while the rest of the fluid flow equations remain unchanged. Also, the solution of the free surface equation is not needed in the Lagrangian formulation, as the free surface motion is modelled naturally by computing the displacement of (all) the fluid particles at each time step. In any case,  the new position of the free surface nodes must be properly identified as described below.

The FEM algorithms for solving the Lagrangian flow equations are very similar to those for the ALE description presented earlier. We will focus in the  second order fractional step algorithm of Section [[#6.2 Fractional step method|6.2]] (for <math display="inline">\theta _1 = \theta _4=0</math> and <math display="inline">\alpha =1</math>) accounting also for fluid-ship interaction effects.

'''Step 1''' Compute explicitely a predicted value of the velocities <math display="inline">{\boldsymbol u}^*</math> as
<span id="eq-57"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\boldsymbol u}^{*} = \bar{\boldsymbol u}^n - \Delta t {\boldsymbol M}^{-1}_d [{\boldsymbol K} \bar{\boldsymbol u}^{n}- {\boldsymbol G}\bar {\boldsymbol p}^{n} -{\boldsymbol f}^{n+1}] </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (57)
|}

Note that matrices <math display="inlne">A</math> and <math display="inline">\bar{\boldsymbol K}</math> of Eq.([[#eq-48|48]]) emanating from the convective terms have been elliminated.

'''Step 2''' Compute <math display="inline">\delta \bar {\boldsymbol p}</math> from Eq.([[#eq-52|52]]).

'''Step 3''' Compute explicitely <math display="inline"> \bar{\boldsymbol u}^{n+1}</math> from Eq.([[#eq-50|50]]) with <math display="inline">{\boldsymbol M}={\boldsymbol M}_d</math>.

'''Step 4''' Compute <math display="inline">\bar {\boldsymbol \pi }^{n+1}</math> explicitely from Eq.([[#eq-46|46]]).

<span id="se-5"></span>
'''Step 5''' Solve for the motion of the ship structure by integrating Eq.([[#eq-55|55]]).

'''Step 6''' Update the mesh nodes in a Lagrangian manner as
<span id="eq-58"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol x}_i^{n+1} = {\boldsymbol x}_i^{n}+\bar {\boldsymbol u}_i \Delta t </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (58)
|}

'''Step 7''' Generate a new mesh and identify the new free surface nodes.

Further details of above algorithm can be found in Idelsohn ''et al.'' (2002 <span id="citeF-53"></span>[[#cite-53|[53]]], 2003a <span id="citeF-54"></span>[[#cite-54|[54]]]) and Oñate ''et al.'' (2003 <span id="citeF-80"></span>[[#cite-80|[80]]], 2004 <span id="citeF-81"></span>[[#cite-81|[81]]]).

The mesh regeneration process can be effectively performed using the extended Delaunay Tesselation described in Idelsohn ''et al.'' (2003b,c <span id="citeF-55"></span>[[#cite-55|[55]]-<span id="citeF-56"></span>[[#cite-56|56]]])). This method allows the fast generation of good quality meshes combining four  node tetrahedra (or three node triangles in 2D) with hexahedra and non standard polyhedra such as pentahedra (or quadrilaterals and pentagons in 2D) where linear shape functions are derived using non-Sibsonian interpolation rules. The mesh regeneration can take place at each time step (this has been the case for the examples presented in the paper), after a prescribed number of time steps, or when  the nodal displacements induce significant element distorsions.

The identification of the free-surface nodes in the Lagrangian analysis can be made using the Alpha Shape method. This is based on the search of all nodes which are on an empty Voronoi sphere with a radius greater than a specified distance defined in terms of the mesh size. For details see  Edelsbrunner and Mucke (1994) <span id="citeF-34"></span>[[#cite-34|[34]]] and  Idelsohn ''et al.'' (2002 <span id="citeF-53"></span>[[#cite-53|[53]]], 2003a,b,c <span id="citeF-54"></span>[[#cite-54|[54]]-<span id="citeF-56"></span>[[#cite-56|56]]]). A known value of the pressure (typically zero pressure or the atmospheric pressure) is prescribed at the free surface nodes.

The  conditions of prescribed velocities or pressures at the solid boundaries in the Lagrangian formulation are usually applied on a layer of nodes adjacent to the boundary. These nodes typically remain fixed during the solution process. Contact between water particles and the solid boundaries is accounted for by the incompressibility condition which naturally prevents the water nodes to penetrate into the solid boundaries. This simple way to treat the water-wall contact is another attractive feature of the Lagrangian flow formulation. For details see Idelsohn ''et al.'' (2002 <span id="citeF-53"></span>[[#cite-53|[53]]], 2003a <span id="citeF-54"></span>[[#cite-54|[54]]]) and Oñate ''et al.'' (2003 <span id="citeF-80"></span>[[#cite-80|[80]]], 2004 <span id="citeF-81"></span>[[#cite-81|[81]]]).

==11 MODELLING THE STRUCTURE AS A VISCOUS FLUID==

A simple and yet effective way to analyze the rigid motion of solid bodies in fluids with the Lagrangian flow description is to model the solid as a fluid with a viscosity much higher than that of the surrounding fluid. The fractional step scheme of Section [[#10 LAGRANGIAN FLOW FORMULATION|10]] can be readily applied skipping now step [[#se-5|5]] to solve for the motion of both fluid domains (the actual fluid and the fictitious fluid modelling the rigid body displacements of the solid).  An example of this type is presented in Sections [[#14.6.4 Floating wood piece|14.6.4]] and [[#14.6.7 Rigid cube falling in a recipient with water|14.6.7.]]

Indeed this approach can be further extended to account for the elastic deformation of the solid treated now as a visco-elastic fluid. This will however introduce some complexity in the formulation and the full coupled fluid-structure interaction scheme previously described is preferable.

==12 COMPUTATION OF THE CHARACTERISTIC LENGTHS==

The evaluation of the stabilization parameters is a crucial issue in stabilized methods. Most existing methods use expressions which are direct extensions of the values obtained for the simplest 1D case. It is also usual to accept the so called SUPG assumption, i.e. to admit that vector <math display="inline">{\boldsymbol h}</math> has the direction of the velocity field. This  restriction leads to instabilities when sharp layers transversal to the velocity direction are present. This deficiency is usually corrected by adding a shock capturing or crosswind stabilization term (Hughes and Mallet, 1986 <span id="citeF-48"></span>[[#cite-48|[48]]]; Tezduyar and Partk, 1986 <span id="citeF-98"></span>[[#cite-98|[98]]]; Codina, 1993 <span id="citeF-15"></span>[[#cite-15|[15]]]). Indeed, in the FIC formulation the components of <math display="inline">{\boldsymbol h}</math> introduce the necessary stabilization along both the streamline and transversal directions to the flow.

Excellent results have been obtained in all problems solved  with the ALE formulation using linear  tetrahedra with the value of the characteristic length vector  defined by
<span id="eq-59"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol h}=h_s {{\boldsymbol u}\over {u}}+h_{c} {{\boldsymbol \nabla } u\over \vert{\boldsymbol \nabla }u\vert } </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (59)
|}

where <math display="inline">u=\vert {\boldsymbol u}\vert </math> and <math display="inline">h_s</math> and <math display="inline">h_{c}</math> are the “streamline” and “cross wind” contributions given by
<span id="eq-60"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol h}_s=\max ({\boldsymbol l}^T_j {\boldsymbol u})/{u} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (60)
|}
<span id="eq-61"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> {\boldsymbol h}_{c}=\max ({\boldsymbol l}^T_j {\boldsymbol \nabla }u)/ \vert {\boldsymbol \nabla }u\vert \quad , \quad  j=1,n_s </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (61)
|}


where <math display="inline">{\boldsymbol l}_j</math> are the vectors defining the element sides (<math display="inline">n_s=6</math> for tetrahedra).

The form chosen for the characteristic  length parameters is similar to that proposed by other authors (see references of previous paragraph and also  Donea and Huerta, 2003 <span id="citeF-31"></span>[[#cite-31|[31]]] and Tezduyar, 2001b <span id="citeF-95"></span>[[#cite-95|[95]]]).

As for the free-surface equation the following value of the characteristic length vector <math display="inline">{\boldsymbol h}_\beta </math> has been taken
<span id="eq-62"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\boldsymbol h}_\beta =\bar h_s {{\boldsymbol u}\over {u}}+\bar h_c {{\boldsymbol \nabla }\beta \over \vert {\boldsymbol \nabla }\beta \vert }  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (62)
|}

The streamline parameter <math display="inline">\bar {\boldsymbol h}_s</math> has been obtained by Eq.([[#eq-60|60]]) using the value of the velocity vector <math display="inline">\boldsymbol u</math> over the 3 node triangles discretizing the free-surface and <math display="inline">n_s=3</math>.

The cross wind parameter <math display="inline">\bar h_c</math> has been computed by
<span id="eq-63"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar h_c = \max [{\boldsymbol l}_j^T {\boldsymbol \nabla }\beta ] {1\over \vert {\boldsymbol \nabla }\beta \vert } \quad ,\quad j=1,2,3 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (63)
|}

The cross-wind terms in eqs.([[#eq-59|59]]) and ([[#eq-62|62]]) take into account  the gradient of the solution in the stabilization parameters. This is a standard assumption in shock-capturing stabilization procedures.

A more consistent evaluation of <math display="inline">{\boldsymbol h}</math> based on a diminishing residual technique can be found in Oñate and García (2001) <span id="citeF-78"></span>[[#cite-78|[78]]].

==13 TURBULENCE MODELLING==

The detailed discussion on the treatment of turbulent effects in the flow equations falls outside the scope of this chapter. Indeed  any of the existing turbulence models is applicable.

In the examples presented next we have chosen a turbulence model based on the Reynolds averaged Navier-Stokes  equations where the deviatoric stresses are computed as sum of the standard viscous contributions and the so called Reynold stresses. Here we have chosen the Boussinesq assumption leading to a modification of the viscosity in the standard Navier-Stokes equations as sum of the “physical” viscosity <math display="inline">\mu </math> and a turbulent viscosity <math display="inline">\mu _T</math>.

One of the simplest and more effective choices for <math display="inline">\mu _T</math> is the Smagorinski LES model giving
<span id="eq-64"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mu _T=C_l h^e (2\varepsilon _{ij}\varepsilon _{ij})^{1/2}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (64)
|}

where <math display="inline">h^e</math> is the element size and <math display="inline">C</math> is a constant (<math display="inline">C\simeq 0.01</math>). In the examples analyzed <math display="inline">h^e</math> has been taken equal to <math display="inline">(\Omega ^e)^{1/2} </math> and <math display="inline">(V^e)^{1/3} </math> for 2D and 3D situations, respectively.

Many other options are possible such as the one and two equations turbulence models (i.e. the <math display="inline">k</math> model and the <math display="inline">k - \varepsilon </math> and <math display="inline">k - w</math> models) and the algebraic stress models. For further details the reader is refered to specialized publications (Celik ''el al.'', 1982 <span id="citeF-12"></span>[[#cite-12|[12]]]; Wilcox, 1994 <span id="citeF-105"></span>[[#cite-105|[105]]]).

==14 EXAMPLES==

The examples chosen show the applicability of the ALE and Lagrangian formulations presented to solve ship hydrodynamics problems. The fractional step algorithm of Section [[#6.2 Fractional step method|6.2]] with <math display="inline">\theta _1 = \theta _3=\theta _4=0</math> and <math display="inline">\alpha =1</math> has been used. The first ALE example is the flow past a submerged NACA 0012 profile.  The next ALE examples include the analysis of a Wigley hull, a scale model of a commercial ship and two American Cup racing sail boats. Numerical results obtained with linear tetrahedral meshes are compared with experimental data in all cases.

The last series of examples show applications of the Lagrangian formulation to simple  schematic 2D ship hydrodynamics situations.

===14.1 Submerged NACA 0012 profile===

A  submerged NACA0012 profile at <math display="inline">\alpha=5^\circ </math> angle  of attack is studied using a 3D finite element model. This configuration was tested experimentally by Duncan (1983) <span id="citeF-33"></span>[[#cite-33|[33]]] and modelled numerically using the  Euler equations  by several authors (Hino ''et al.'', 1993 <span id="citeF-41"></span>[[#cite-41|[41]]]; Löhner ''et al.'', 1998 <span id="citeF-64"></span>[[#cite-64|[64]]], 1999 <span id="citeF-65"></span>[[#cite-65|[65]]]; Idelsohn ''et al.'', 1999 <span id="citeF-52"></span>[[#cite-52|[52]]]). The submerged depth of the airfoil is equal to the chord length L. The Froude number for all the cases tested was set to <math display="inline"> Fr={{U}\over \sqrt{{g L}}}= 0.5672 </math> where <math display="inline">U</math> is the incoming flow velocity at infinity.

The stationary free  surface and the pressure distribution in the domain are shown in Figure [[#img-5|5]]. The non-dimensional wave heights compare well with the experimental results.

===14.2 Wigley hull===

We study here the well known  Wigley Hull, given by  the analytical formula <math display="inline">y=0.5B(1 - 4x^2)(1 - z^2/D^2)</math> where <math display="inline">B</math> and <math display="inline">D</math>  are the beam and the draft of the ship hull at still water.

The same configuration was tested experimentally (Noblesse and McCarthy, 1983 <span id="citeF-69"></span>[[#cite-69|[69]]]; Wigley, 1983 <span id="citeF-104"></span>[[#cite-104|[104]]] ) and modelled  numerically by several authors (Farmer ''et al.'', 1993 <span id="citeF-35"></span>[[#cite-35|[35]]]; Storti ''et al.'', 1998 <span id="citeF-91"></span>[[#cite-91|[91]]]; Idelsohn ''et al.'', 1999 <span id="citeF-52"></span>[[#cite-52|[52]]]; Löhner ''et al.'', 1999 <span id="citeF-65"></span>[[#cite-65|[65]]]). We use an unstructured 3D  finite  element mesh of <math display="inline">65434</math> linear tetrahedra,  with a reference surface of  <math display="inline">7800</math> triangles, partially  represented in Figure [[#img-6|6]]. A Smagorinsky turbulence model was chosen.

Three different viscous cases were studied for <math display="inline">L=6m, F_n=0.316, \mu=10^{-3} Kg/m.s</math>. In the first case the volume mesh was considered fixed, not allowing free-surface nor ship movements. Next, the volume mesh was updated due to free-surface movement, while keeping the model fixed. The third case corresponds to the analysis of a real free model including the mesh updating due to free-surface displacement  and ship movement (sinkage and trim).

Table [[#table1|1]] shows the obtained total resistance coefficient in the three cases studied compared with experimental data.

<span id="table1"></span>
{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
|+ style="font-size: 75%;" |Table. 1 Wigley Hull. Total resistance coefficient values
|- style="border-top: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  
| style="border-left: 2px solid;border-right: 2px solid;" | '''Experimental'''
| style="border-left: 2px solid;border-right: 2px solid;" | '''Numerical'''
|- style="border-top: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  Test 1 
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> 5.2\quad 10^{-3}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">4.9\quad 10^{-3}</math>
|- style="border-top: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |   Test 2 
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">5.2\quad 10^{-3}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">5.3\quad 10^{-3}</math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  Test 3 
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">4.9\quad 10^{-3}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" |  <math display="inline">5.1\quad 10^{-3}</math>

|}

Numerical values obtained for  sinkage and trim were <math display="inline">-0.1%</math> and <math display="inline">0.035</math>, respectively, while experiments gave <math display="inline">-0.15%</math> and <math display="inline">0.04</math>.

Figure [[#img-6|6a]] shows the pressure distribution obtained near the Wigley hull for the free model. Some streamlines have also been plotted. The mesh deformation in this case is shown in Figure [[#img-6|6a]]

Comparison of the obtained body wave profile with experimental data for the free and fixed models is shown in Figure [[#img-6|6b]].Significant differences are found close to stern for the fixed model.  The free-surface contours for the truly free ship motion are shown in Figure [[#img-6|6c]].

===14.3 KVLCC2 hull model===

The example is the analysis of the KVLCC2 benchmark model. Here a partially wetted tramsom stern is expected due to the low Froude number of the test. Figure [[#img-7|7]] shows the NURBS geometry obtained from the Hydrodynamic Performance Research team of Korea (KRISO). Numerical results are compared with the experimental data available from KRISO (2000).

The smallest element size used was <math display="inline">0.001</math> m and the largest <math display="inline">0.50</math> m.  The surface mesh chosen  is also shown in Figure [[#img-7|7]]. The 3D mesh included <math display="inline">550.000</math> tetrahedra.  The tramsom stern flow model described  was used.
<br />

<span id="test1"></span>
''Test 1.-'' ''Wave pattern calculation''.  The main characteristics of the analysis are listed below:

* Length: <math display="inline">5.52 m</math>, Beam (at water plane): <math display="inline">0.82 m</math>, Draught: <math display="inline">0.18 m</math>, Wetted Surface: <math display="inline">8.08 m^2</math>.
* Velocity: <math display="inline">1.05 m/seg</math>, Froude Number: <math display="inline">0.142</math>.
* Viscosity: <math display="inline">0.00126 Kg/m\cdot seg</math>, Density: <math display="inline">1000 Kg/m^3</math>, Reynolds number: <math display="inline">4.63\cdot 10^6</math>.

The turbulence model chosen was the <math display="inline">k</math> model. Figures [[#img-8|8]] and [[#img-9|9]] show the wave profiles on the hull and in  a cut at <math display="inline">y/L = 0.0964</math> obtained for Test [[#test1|1]], compared to experimental data.  The results are quantitatively good.
<br />

<span id="test2"></span>
''Test 2.-'' ''Wake analysis at different planes''. Several turbulence models were  used (''Smagorinsky'', <math display="inline">k</math> and <math display="inline">k-\epsilon </math> model) in order to verify the quality of the results. Here, only the results from the <math display="inline">k-\epsilon </math> model are shown. The velocity maps obtained even for the simplest <math display="inline">Smagorinsky</math> model were qualitatively good. The main characteristics of this analysis are:

* Length: <math display="inline">2.76 m</math>, Beam (at water plane): <math display="inline">0.41 m</math>, Draught: <math display="inline">0.09 m</math>, Wetted Surface: <math display="inline">2.02 m^2</math>.
* Velocity: <math display="inline">25 m/seg</math>.
* Viscosity: <math display="inline">3.05\cdot 10^{-5} Kg/m\cdot seg</math>, Density: <math display="inline">1.01 Kg/m^3</math>, Reynolds number: <math display="inline">4.63\cdot 10^6</math>.

Figures [[#img-10|10]] and [[#img-11|11]] present some results for Test [[#test2|2]].  Figure [[#img-10|10]] shows the contours of the axial (X) component of the velocity on a plane at <math display="inline">2.71 m</math> from the orthogonal aft. Figure [[#img-11|11]] shows the map of the kinetic energy at this plane. Experimental results are also plotted for comparison. Further results are reported in García and Oñate (2003)<span id="citeF-38"></span>[[#cite-38|[38]]].

===14.4 American Cup Rioja de España Model===

The Rioja de España boat representing Spain in the American Cup's edition of 1995 was analyzed. Figure [[#img-12|12]] shows the geometry of the boat based on standard NURBS surfaces. Numerical results are compared with an extrapolation of experimental data obtained in CEHIPAR basin (Spain) using a 1/3.5 scale model. Resistance tests were performed with the model free to sink and trim.  Experimental data  include drag, lift, sinkage, trim angles and wave profiles at <math display="inline">4.27m</math> (real scale) from symmetry plane.  Every model was towed at equivalent velocities of <math display="inline">10</math>, <math display="inline">9</math>, <math display="inline">8.5</math>, <math display="inline">8.0</math>, <math display="inline">7.5</math> and <math display="inline">7.0</math> knots.

Numerical analysis  were carried out at real scale. Characteristics of unstructured grids of four node linear tetrahedra used are shown in Table [[#table2|2]] together with the parameters of some of the  model studied.

<span id="table2"></span>
{|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
|+ style="font-size: 75%;" |Table. 2 Rioja de España. Analysis parameters.
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |   '''Test'''
| style="border-left: 2px solid;border-right: 2px solid;" | '''Geometry'''
| style="border-left: 2px solid;border-right: 2px solid;" | '''Heel''' 
| style="border-left: 2px solid;border-right: 2px solid;" | '''Drift'''
| style="text-align: center;border-left: 2px solid;border-right: 2px solid;" | '''Symmetry'''
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | '''Elements'''
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | '''nodes'''
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  E0D0 
| style="border-left: 2px solid;border-right: 2px solid;" |  Hull, bulb and keel 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0^\circ </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0^\circ </math>
| style="text-align: center;border-left: 2px solid;border-right: 2px solid;" | Yes 
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | 700 000 
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | 175 000
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  E15D2 
| style="border-left: 2px solid;border-right: 2px solid;" | Hull, bulb, keel and rudder 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>15^\circ </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>2^\circ </math>
| style="text-align: center;border-left: 2px solid;border-right: 2px solid;" | No 
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | 1 500 000 
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | 380 000
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  E15D4 
| style="border-left: 2px solid;border-right: 2px solid;" | Hull, bulb, keel and rudder 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>15^\circ </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>4^\circ </math>
| style="text-align: center;border-left: 2px solid;border-right: 2px solid;" | No 
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | 1 500 000 
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | 380 000
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  E25D2 
| style="border-left: 2px solid;border-right: 2px solid;" | Hull, bulb, keel and rudder 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>25^\circ </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>2^\circ </math>
| style="text-align: center;border-left: 2px solid;border-right: 2px solid;" | No 
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | 1 500 000 
| style="text-align: right;border-left: 2px solid;border-right: 2px solid;" | 380 000

|}

All grids used were generated with the same quality criteria and using element sizes from <math display="inline">5mm</math> to <math display="inline">2000 mm</math>. Some details of the grid used in the E0D0 case are shown in Figures [[#img-13|13]] and&nbsp;[[#img-14|14]]. A <math display="inline">k-\varepsilon </math> turbulence model in combination with an extended law of the wall was chosen for the analysis.

A time step of <math display="inline">0.1s</math> was used and this sufficed to achieve a steady state solution in all cases. Additional calculations were carried out  with <math display="inline">\Delta t  = 0.025s</math> and <math display="inline">0.01s</math>, in order to verify the influence of the time increment in the accuracy of the results. No significant influence was detected for the selected results.

Figure [[#img16|16]] shows a comparison of simulated and experimental wave profiles. The origin of the x axis has been taken at the fore perpendicular and the x+ sense is afore. In the non symmetric case, the measurements were performed at the opposite side of the heeled board. The ratio of maximum amplitudes of fluctuations (noise) and waves in the experimental measurements varies from <math display="inline">4%</math> to <math display="inline">12%</math>

Figures [[#img16|16]] and [[#img17|17]] show pressure contours on the bulb and keel for different cases, corresponding to a velocity of <math display="inline">8 kn</math>. Figures [[#img-18|18]] to [[#img-19|19]] show some of the wave patterns obtained  for a velocity of <math display="inline">9 kn</math>.  Figures [[#img-21|21]] and [[#img-22|22]] show some perspective views, including pressure and velocity contours, streamlines and cuts for some cases analyzed. Finally Figures [[#img-23|23]] and [[#img-24|24]] show resistance graphs where numerical results are compared with values ''extrapolated'' from experimental data. For further details see García ''et al.'' (2002) <span id="citeF-39"></span>[[#cite-39|[39]]].

===14.5 American Cup BRAVO ESPAÑA Model===

The finite element formulation presented was also applied to study the racing sail boat ''Bravo España'' participating in the 1999 edition of the American Cup. The  mesh of linear tetrahedra used is shown in Figure [[#img-25|25]]. Results presented in Figures&nbsp;[[#img-25|25]]&#8211;[[#img-27|27]] correspond to a non symmetrical case including appendages. Good comparison between experimental data and numerical results was again obtained Further details can be found in García and Oñate (2002) <span id="citeF-38"></span>[[#cite-38|[38]]].

===14.6 Lagrangian flow examples===

A number of simple problems have been solved in order to test the capabilities of the Lagrangian flow formulation to solve ship hydrodynamics problem. All examples have been analyzed with the algorithm described in Section [[#10 LAGRANGIAN FLOW FORMULATION|10]]. We note again that the regeneration of the mesh has been performed at each time step.

====14.6.1 Rigid ship hit by wave====

The first example is a very schematic representation of a ship when  is hit by a big wave produced by the collapse of a water recipient (Fig. [[#img-28|28]]). The ship can not move and initially the free-surface near the ship is horizontal. Fixed nodes represent the ship as well as the domain walls. The example tests the suitability of the Lagrangian flow formulation to solve water-wall contact situations even in the presence of curved walls. Note the breaking and splashing of the waves under the ship prow and the rebound of the incoming wave. It is also interesting to see the different water-wall contact situations at the internal and external ship surfaces and the moving free-surface at the back of the ship.

====14.6.2 Horizontal motion of a rigid ship in a reservoir====

In the next example (Fig. [[#img-29|29]]) the same ship of the previous example moves now horizontally at a fixed velocity in a water reservoir.   The free-surface, which was initially horizontal, takes the correct position at the ship prow and stern. Again, the complex water-wall contact problem is naturally solved in the curved prow region.

====14.6.3 Semi-submerged rotating water mill====

The  example shown in Figure [[#img-30|30]] is the analysis of a rotating water mill semi submerged in water. The blades of the mill are treated as a rigid body with an imposed rotating velocity, while the water is initially in a stationary  flat position. Fluid structure interactions with free-surfaces and water fragmentation are well reproduced in this example.

====14.6.4 Floating wood piece====

The next example shows an initially stationary recipient with a floating piece of wood where a wave is produced on the left side. The wood has been simulated by a liquid of higher viscosity as described in Section [[#11 MODELLING THE STRUCTURE AS A VISCOUS FLUID|11]]. The wave intercepts the wood piece producing a breaking wave and displacing the floating wood as shown in Figure [[#img-31|31]].

====14.6.5 Ships hit by an incoming wave====

In the example of Figure [[#img-32|32]] the motion of a fictitious rigid ship hit by an incoming wave is analyzed.  The dynamic motion of the ship is induced by the resultant of the pressure and the viscous forces acting on the ship boundaries.  We note that  the horizontal displacement of the gravity center of the ship was fixed to zero. In this way, the ship moves only vertically although it can freely rotate. The position of the ship boundary at each time step defines a moving boundary condition for the free surface particles in the fluid. 

Figure [[#img-32|32]] shows instants of the motion of the ship and the surrounding fluid. It is interesting to see the breaking of a wave on the ship prow as well as on the stern when the wave goes back. Note that some water drops slip over the ship deck.

Figure [[#img-33|33]] shows the analysis of a similar problem using precisely the same approach. The section of the ship analyzed corresponds now to that of a real container ship. Differently to the previous case the rigid ship is  free to move laterally due to the sea wave forces. The objective of the study was to asses the influence of the stabilizers in the ship roll. The figures show clearly how the lagrangian method predicts the ship and wave motions in a realistic manner.

====14.6.6 Tank-ship hit by a lateral wave====

Figure [[#img-34|34]] represents a transversal section of a tank-ship and a wave approaching to it. The tank-ship is modelled as a rigid body and it carries a liquid inside which moves freely within the ship domain.

Initially (<math display="inline">t = 0.00</math>) the ship is released from a fixed position and the equilibrium configuration found is consistent with Arquimides principle. During the following time steps the external wave hits the ship and both the internal and the external fluids interact with the ship boundaries. At times <math display="inline">t=6.15</math> and <math display="inline">8.55</math> breaking waves and some water drops slipping along the ship deck can be observed. Figure [[#img-35|35]] shows the pressure pattern at two time steps.

====14.6.7 Rigid cube falling in a recipient with water====

The last examples show the capacity of the Lagrangian formulation to analyze the motion of solids within water. In the first example a  solid cube is initially  free and falls down within a water recipient. In this example, the rigid solid is modeled first as a fictitious fluid with a higher viscosity,  similarly as for  the floating solid of Section [[#14.6.4 Floating wood piece|14.6.4.]] The results of this analysis are shown in Figure [[#img-36|36]]. Note that the method reproduces very well the interaction of the cube with the free surface as well as the overall sinking process. A small deformation of the cube is produced. This can be reduced by increasing further the fictitious viscosity of the cube particles.

The same problem is analyzed again considering now the cube as a  rigid solid subjected to  pressure and viscous forces acting in its boundaries. The resultant of the fluid forces and the weight of the cube are applied to the center of the cube. These forces govern the displacement of the cube which is computed by solving  the dynamic equations of motion as described in the fractional step algorithm of Section [[#10 LAGRANGIAN FLOW FORMULATION|10]], similarly as for the rigid ships of the three previous examples. Similarly as in those cases the moving cube contours define a boundary condition for the fluid particles at each time step.

Initially the solid falls down freely due to the gravity forces (Figure [[#img-37|37]]). Once in contact with the water surface, the alpha-shape method recognizes the different boundary contours which are shown with a thick line in the figure.  The pressure  and viscous forces are evaluated in all the domain and in particular on the cube contours. The fluid forces introduce a negative acceleration in the vertical motion until, once the cube is completely inside the water, the vertical  velocity  becomes zero. Then,  Arquimides  forces bring the cube up to the free-surface. It is interesting to observe that there is a rotation of the cube. The reason is that the center of the floating forces is higher in the rotated position than in the initial ones.

Figure [[#img-38|38]] shows a repetition of the same problem showing now all the finite elements in the mesh discretizing the fluid. We recall that in all the Lagrangian problems here described the mesh in the fluid domain ''is regenerated at each time step'' combining linear triangles and quadrilateral elements as described in Section [[#10 LAGRANGIAN FLOW FORMULATION|10]]. Note that some fluid particles separate from the fluid domain. These particles are treated as free boundary points with zero pressure and hence fall down due to gravity.

It is interesting to see that the final position of the cube is different from that of Figure [[#img-37|37]]. This is due to the unstable character of the cube motion. A small difference in the numerical computations (for instance in the mesh generation process) shifts the movement of the cube towards the right or the left. Note however that a final rotated equilibrium position is found in both cases.

Further examples of the Lagrangian flow formulation can be found in Oñate ''et al.'' (2003 <span id="citeF-80"></span>[[#cite-80|[80]]], 2004 <span id="citeF-81"></span>[[#cite-81|[81]]]) and Idelsohn ''et al.'' (2003a <span id="citeF-54"></span>[[#cite-54|[54]]], 2003 <span id="citeF-55"></span>[[#cite-55|[55]]]).

==15 CONCLUDING REMARKS==

An overview of different finite element schemes for solving ship hydrodynamics problems has been presented. The necessary stabilization in the numerical computation has been provided by the finite calculus formulation of the fluid flow equations accounting for surface wave effects.

Stabilized finite algorithms for solving a variety of ship hydrodynamics situations using ALE and fully Lagrangian descriptions have been presented. Both monolithic and fractional step algorithms have been derived. The interaction of the motion of the ship structure with the fluid equations can be accounted for in a straight forward manner within the general flow solution schemes. The ALE formulation is particularly adequate for ship hydrodynamics problems involving free-surface waves of moderate amplitude. The Lagrangian flow formulation allows to solve in a simple and  effective manner ship hydrodynamics problems involving large motions of the free-surface and complex fluid-structure interaction situations.

==ACKNOWLEDGEMENTS==



The authors thank Prof. R. Löhner, Prof. P. Bergan, Dr. C. Sacco and Dr. H. Sierra for many useful discussions.

Thanks are also given to Dr. F. del Pin and Mr. N. Calvo for their help and involvement in solving the problems with the Lagrangian formulation.

The authors are grateful to Copa America Desafio Español SA and CEHIPAR for providing the geometry and experimental data for the racing boats analyzed and to Det Norske Veritas for providing the geometry of the ship with stabilizers for the analysis shown in Figure [[#img-33|33]].

The Spanish company [http://www.compassis.com/ COMPASS] Ingeniería y Sistemas SA provided the computer code Tdyn for the numerical simulation of the examples solved with the ALE formulation (Tdyn, 2004 <span id="citeF-92"></span>[[#cite-92|[92]]]). This support is gratefully acknowledged.

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<div id="cite-1"></div>
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<div id="cite-2"></div>
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<div id="cite-3"></div>
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<div id="cite-5"></div>
[[#citeF-5|[5]]]  Azcueta R, Muzaferija S and Peric M. Computation of breaking bow waves for a very full hull ship. ''7th Int. Conf. on Num. Ship Hydro.'' 1999, Nantes, France, 6.2-1, 11.

<div id="cite-6"></div>
[[#citeF-6|[6]]] Baba E and Takekuma K. A study on free-surface flow around bow of slowly moving full forms. ''J. Soc. Naval Architects Japan'' 1975; 137.

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==LIST OF FIGURES==

'''Figure '''[[#img-1|1]]. Equilibrium of fluxes in a  balance domain of finite size

'''Figure '''[[#img-2|2]]. Reference surface for the wave height <math display="inline">\beta </math>

'''Figure '''[[#img-3|3]]. Definition of free-surface parameters

'''Figure '''[[#img-4|4]]. Changes on the fluid interface in a floating body

'''Figure '''[[#img-5|5]]. Submerged NACA0012 profile. a) Detail of the mesh of 70000 linear tetrahedra chosen. b) Pressure contours. c) Stationary wave profile

'''Figure '''[[#img-6|6]]. Wigley hull. a) Pressure distribution and mesh deformation of the wigley hull (free model). b) Numerical and experimental body wave profiles.  c) free-surface contours for the truly free ship motion

'''Figure '''[[#img-7|7]]. KVLCC2 model. Geometrical definition based on NURBS surfaces and surface mesh used in the analysis

'''Figure '''[[#img-8|8]]. KVLCC2 model. Wave profile on the hull. Thick line shows numerical results

'''Figure '''[[#img-9|9]]. KVLCC2 model. Wave profile on a cut at y/L=0.0964. Thick line shows numerical results

'''Figure '''[[#img-10|10]]. KVLCC2 model. Map of the X component of the velocity on a plane at 2.71 m from the orthogonal aft. Experimental data shown in the right

'''Figure '''[[#img-11|11]]. KVLCC2 model. Map of the eddy kinetic energy (<math display="inline">K</math>) on a plane at 2.71 m from the orthogonal aft. Experimental data shown in the right

'''Figure '''[[#img-12|12]]. NURBS-based geometry used in the analysis of the Rioja de España America Cup boat

'''Figure '''[[#img-13|13]]. Detail of the final (boundary) mesh used in the E0D0 case. The mesh has been adapted taking into account sinkage, trim and free-surface deformation

'''Figure '''[[#img-14|14]]. Detail of the mesh used in the analysis of the E0D0 case, around keel-bulb union

'''Figure '''[[#img-15|15]]. Wave elevation profile for 10kn (left: E0D0, right: E15D2)

'''Figure '''[[#img-16|16]]. E0D0 8kn. Pressure contours on bulb

'''Figure '''[[#img-17|17]]. E25D2 8kn. Pressure contours on bulb

'''Figure '''[[#img-18|18]]. E15D2 9kn Wave map

'''Figure '''[[#img-19|19]]. E25D2 9kn Wave map

'''Figure '''[[#img-20|20]]. E15D2 7.5 kn. Pressure map on appendages and streamlines. Perspective view

'''Figure '''[[#img-21|21]]. E15D4 7.5 kn. Velocity modulus contours. Perspective view

'''Figure '''[[#img-22|22]]. E25D2 7.5 kn. Pressure contours on appendages and cuts. Perspective view

'''Figure '''[[#img-23|23]]. E0D0. Resistance graph. Comparison with results extrapolated from experimental data

'''Figure '''[[#img-24|24]]. E15D4. Resistance graph. Comparison with results extrapolated from experimental data

'''Figure '''[[#img-25|25]]. American Cup <math display="inline">Bravo ~Espa\tilde{n}a</math> racing sail boat. a) Mesh used in the analysis. b) Velocity contours

'''Figure '''[[#img-26|26]]. <math display="inline">Bravo ~Espa\tilde{n}a</math>. Streamlines

'''Figure '''[[#img-27|27]]. <math display="inline">Bravo~Espa\tilde{n}a</math>. Resistance test. Comparison of numerical results with experimental data

'''Figure '''[[#img-28|28]]. Fixed ship hit by incoming  wave

'''Figure '''[[#img-29|29]]. Moving ship with fixed velocity

'''Figure '''[[#img-30|30]]. Rotating water mill

'''Figure '''[[#img-31|31]]. Floating wood piece hit by a wave

'''Figure '''[[#img-32|32]]. Motion of a rigid ship hit by an income wave. The ship is modelled as a rigid solid restrained to move in the vertical direction

'''Figure '''[[#img-33|33]]. Ship with stabilizers hit by a lateral wave

'''Figure '''[[#img-34|34]]. Tank ship carrying an internal liquid hit by wave. Ship and fluids motion  at different time steps

'''Figure '''[[#img-35|35]]. Tank ship under lateral wave. Pressure distribution at two time steps.

'''Figure '''[[#img-36|36]]. Solid cube falling into a recipient with water. The cube is modelled as a very viscous fluid

'''Figure '''[[#img-37|37]]. Cube falling into a recipient with water. The cube is modelled as a rigid solid. Motion of the cube and free surface positions at different time steps

'''Figure '''[[#img-38|38]]. Cube falling in a water recipient. The cube is modeled as a  rigid solid. The finite element meshes generated at the selected instants are shown

<div id='img-1'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-majesus2.png|300px|Equilibrium of fluxes in a  balance domain of finite size]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Equilibrium of fluxes in a  balance domain of finite size
|}

<div id='img-2'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig2con216.png|350px|Reference surface for the wave height β]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Reference surface for the wave height <math>\beta </math>
|}

<div id='img-3'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig6con215.png|400px|Definition of free-surface parameters]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Definition of free-surface parameters
|}

<div id='img-4'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-fig2.png|340px|Changes on the fluid interface in a floating body]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Changes on the fluid interface in a floating body
|}

<div id='img-5'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig3_1.png|351px|]]
|[[Image:Draft_Samper_876641692-Fig3_2.png|251px|]]
|- style="text-align: center; font-size: 75%;"
|colspan="2"|a)
|-
|colspan="2"|[[Image:Draft_Samper_876641692-Fig3_3.png|351px|]]
|- style="text-align: center; font-size: 75%;"
|colspan="2"|b)
|-
|colspan="2"|[[Image:Draft_Samper_876641692-fig3b.png|351px|Submerged NACA0012 profile. a) Detail of the mesh of 70000 linear tetrahedra chosen. b) Pressure contours. c) Stationary wave profile]]
|- style="text-align: center; font-size: 75%;"
|colspan="2"|c)
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 5:''' Submerged NACA0012 profile. a) Detail of the mesh of 70000 linear tetrahedra chosen. b) Pressure contours. c) Stationary wave profile
|}

<div id='img-6'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig6Hull.png|400px|Wigley hull. a) Pressure distribution and mesh deformation of the wigley hull (free model). b) Numerical and experimental body wave profiles.  c) free-surface contours for the truly free ship motion ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 6:''' Wigley hull. a) Pressure distribution and mesh deformation of the wigley hull (free model). b) Numerical and experimental body wave profiles.  c) free-surface contours for the truly free ship motion 
|}

<div id='img-7'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-KCSgeo.png|300px|]]
|-
|[[Image:Draft_Samper_876641692-KCSmesh.png|300px|KVLCC2 model. Geometrical definition based on NURBS surfaces and surface mesh used in the analysis]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 7:''' KVLCC2 model. Geometrical definition based on NURBS surfaces and surface mesh used in the analysis
|}

<div id='img-8'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-KCSprof1.png|400px|KVLCC2 model. Wave profile on the hull. Thick line shows numerical results]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 8:''' KVLCC2 model. Wave profile on the hull. Thick line shows numerical results
|}

<div id='img-9'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-KCSprof2.png|400px|KVLCC2 model. Wave profile on a cut at y/L=0.0964. Thick line shows numerical results]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 9:''' KVLCC2 model. Wave profile on a cut at y/L=0.0964. Thick line shows numerical results
|}

<div id='img-10'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-KCSvelx1.png|400px|KVLCC2 model. Map of the X component of the velocity on a plane at 2.71 m from the orthogonal aft. Experimental data shown in the right]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 10:''' KVLCC2 model. Map of the X component of the velocity on a plane at 2.71 m from the orthogonal aft. Experimental data shown in the right
|}

<div id='img-11'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-KCSk.png|400px|KVLCC2 model. Map of the eddy kinetic energy (K) on a plane at 2.71 m from the orthogonal aft. Experimental data shown in the right]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 11:''' KVLCC2 model. Map of the eddy kinetic energy (<math>K</math>) on a plane at 2.71 m from the orthogonal aft. Experimental data shown in the right
|}

<div id='img-12'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig2Rioja.png|351px|NURBS-based geometry used in the analysis of the Rioja de España America Cup boat]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 12:''' NURBS-based geometry used in the analysis of the Rioja de España America Cup boat
|}

<div id='img-13'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig3Rioja.png|300px|Detail of the final (boundary) mesh used in the E0D0 case. The mesh has been adapted taking into account sinkage, trim and free-surface deformation]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 13:''' Detail of the final (boundary) mesh used in the E0D0 case. The mesh has been adapted taking into account sinkage, trim and free-surface deformation
|}

<div id='img-14'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig4Rioja.png|300px|Detail of the mesh used in the analysis of the E0D0 case, around keel-bulb union]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 14:''' Detail of the mesh used in the analysis of the E0D0 case, around keel-bulb union
|}

<div id='img-15'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig6Rioja.png|351px|]]
|[[Image:Draft_Samper_876641692-Fig7Rioja.png|351px|Wave elevation profile for 10kn (left: E0D0, right: E15D2) ]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 15:''' Wave elevation profile for 10kn (left: E0D0, right: E15D2) 
|}

<div id='img-16'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig10Rioja.png|300px|E0D0 8kn. Pressure contours on bulb]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 16:''' E0D0 8kn. Pressure contours on bulb
|}

<div id='img-17'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig13Rioja.png|300px|E25D2 8kn. Pressure contours on bulb]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 17:''' E25D2 8kn. Pressure contours on bulb
|}

<div id='img-18'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig15Rioja.png|351px|E15D2 9kn Wave map]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 18:''' E15D2 9kn Wave map
|}

<div id='img-19'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig17Rioja.png|351px|E25D2 9kn Wave map]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 19:''' E25D2 9kn Wave map
|}

<div id='img-20'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_876641692-Fig18Rioja.png|351px|E15D2 7.5 kn. Pressure map on appendages and streamlines. Perspective view]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 20:''' E15D2 7.5 kn. Pressure map on appendages and streamlines. Perspective view
|}

<div id='img-21'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_8440_21.JPG|400px|E15D4 7.5 kn. Velocity modulus contours. Perspective view]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 21:'''E15D4 7.5 kn. Velocity modulus contours. Perspective view
|}

<div id='img-22'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_7186_22.JPG|400px|E25D2 7.5 kn. Pressure contours on appendages and cuts. Perspective view]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 22:'''E25D2 7.5 kn. Pressure contours on appendages and cuts. Perspective view
|}

<div id='img-23'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_6639_23.JPG|400px|E0D0. Resistance graph. Comparison with results extrapolated from experimental data]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 23:'''E0D0. Resistance graph. Comparison with results extrapolated from experimental data
|}

<div id='img-24'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_4305_24.JPG|400px|E15D4. Resistance graph. Comparison with results extrapolated from experimental data]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 24:'''E15D4. Resistance graph. Comparison with results extrapolated from experimental data
|}

<div id='img-25'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_4979_25a.JPG|300px]]
|[[File:Draft_Samper_876641692_8356_25b.JPG|300px|American Cup Bravo Espa˜na racing sail boat. a) Mesh used in the analysis. b) Velocity contours]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" |a)
| colspan="1" |b)
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 25:'''American Cup ''Bravo España'' racing sail boat. a) Mesh used in the analysis. b) Velocity contours
|}

<div id='img-26'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_1371_26.JPG|400px|Bravo Espa˜na. Streamlines]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 26:'''''Bravo España''. Streamlines
|}

<div id='img-27'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_3722_27.JPG|600px|Bravo Espa˜na. Resistance test. Comparison of numerical results with experimental data]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 27:'''''Bravo España''. Resistance test. Comparison of numerical results with experimental data
|}

<div id='img-28'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_5176_28a.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_8566_28b.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_9682_28c.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_6767_28d.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_3969_28e.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_6912_28f.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_3203_28g.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_6414_28h.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_5208_28i.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_2893_28j.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_9590_28k.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_7368_28l.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_8712_28m.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_5839_28n.JPG|400px|Fixed ship hit by incoming wave]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 28:''' Fixed ship hit by incoming wave
|}

<div id='img-29'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_3563_29a.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_3230_29b.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_5022_29c.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_4964_29d.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_8265_29e.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_8929_29f.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_2933_29g.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_5829_29h.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_8757_29i.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_4267_29j.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_7646_29k.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_3001_29L.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_9513_29m.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_9661_29n.JPG|400px|Fixed ship hit by incoming wave]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 29:''' Moving ship with fixed velocity
|}

<div id='img-30'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_8039_30a.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_9067_30b.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_7260_30c.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_1982_30d.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_4468_30e.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_4181_30f.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_3489_30g.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_5484_30h.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_9476_30i.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_8131_30j.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_1923_30k.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_6250_30l.JPG|400px|Fixed ship hit by incoming wave]]
|- style="text-align: center; font-size: 75%;"
| colspan="3" | '''Figure 30:'''Rotating water mill.
|}

<div id='img-31'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_9238_31.JPG|400px|Floating wood piece hit by a wave]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 31:'''Floating wood piece hit by a wave
|}

<div id='img-32'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_8393_32.JPG|400px|Motion of a rigid ship hit by an incoming wave. The ship is modelled as a rigid solid restrained to move in the vertical direction.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 32:'''Motion of a rigid ship hit by an incoming wave. The ship is modelled as a rigid solid restrained to move in the vertical direction.
|}

<div id='img-33'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_6546_33.JPG|400px|Ship with stabilizers hit by a lateral wave]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 33:'''Ship with stabilizers hit by a lateral wave
|}

<div id='img-34'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|-
|[[File:Draft_Samper_876641692_3204_34a.JPG|400px|Tanks hip under lateral wave. Pressure distribution at two time steps]]
|-
|[[File:Draft_Samper_876641692_4060_34b.JPG|400px]]
|-
|[[File:Draft_Samper_876641692_9045_34c.JPG|400px|Tanks hip under lateral wave. Pressure distribution at two time steps]]
|-
|[[File:Draft_Samper_876641692_7873_34d.JPG|400px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 34:'''Tanks hip carrying an internal liquid hit by wave. Ship and fluids motion at different time steps
|}

<div id='img-34'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_4806_34cont_a.JPG|400px|Tanks hip under lateral wave. Pressure distribution at two time steps]]
|-
|[[File:Draft_Samper_876641692_1987_34cont_b.JPG|400px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 34:'''Cont.
|}

<div id='img-35'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_9026_35a.JPG|400px|Tanks hip under lateral wave. Pressure distribution at two time steps]]
|-
|[[File:Draft_Samper_876641692_7499_35b.JPG|400px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 35:'''Tanks hip under lateral wave. Pressure distribution at two time steps
|}

<div id='img-36'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_2874_36.JPG|500px|Solid cube falling into a recipient with water. The cube is modelled as a very viscous fluid.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 36:'''Solid cube falling into a recipient with water. The cube is modelled as a very viscous fluid.
|}

<div id='img-37'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_6617_37a.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_7210_37b.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_3674_37c.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_4304_37d.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_8199_37e.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_8063_37f.JPG|400px|Fixed ship hit by incoming wave]]
|-
|[[File:Draft_Samper_876641692_2444_37g.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_4396_37h.JPG|400px|Fixed ship hit by incoming wave]]
|[[File:Draft_Samper_876641692_9319_37i.JPG|400px|Fixed ship hit by incoming wave]]
|- style="text-align: center; font-size: 75%;"
| colspan="3" | '''Figure 37:'''Cube falling into a recipient with water. The cube is modelled as a rigid solid. Motion of the cube and free surface positions at different time steps.
|}

<div id='img-38'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_4347_38a.JPG|200px|Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.]]
|[[File:Draft_Samper_876641692_2484_38b.JPG|200px|Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.]]
|-
|[[File:Draft_Samper_876641692_9913_38c.JPG|200px|Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.]]
|[[File:Draft_Samper_876641692_6490_38d.JPG|200px|Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 38:'''Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.
|}

<div id='img-38'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_876641692_4499_cont_a.JPG|200px|Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.]]
|[[File:Draft_Samper_876641692_4134_cont_b.JPG|200px|Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.]]
|-
|[[File:Draft_Samper_876641692_4651_cont_c.JPG|200px|Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.]]
|[[File:Draft_Samper_876641692_8883_cont_d.JPG|200px|Cube falling in a water recipient. The cube is modeled as a rigid solid. The finite element meshes generated at the selected instants are shown.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 38:'''cont
|}