Onate et al 2004h - Revision history

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← Older revision		Revision as of 09:43, 29 May 2019
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−	Published in ''"Encyclopedia of Computational Mechanics'', Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes (Eds.), John Wiley & Sons Ltd, Vol. 3, Chapter 18, pp. 579 - 607, 2004<br />	+	Published in ''Encyclopedia of Computational Mechanics'', Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes (Eds.), John Wiley & Sons Ltd, Vol. 3, Chapter 18, pp. 579 - 607, 2004<br />
	DOI: 10.1002/9781119176817		DOI: 10.1002/9781119176817

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-<!-- metadata commented in wiki content
+Published in ''"Encyclopedia of Computational Mechanics'', Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes (Eds.), John Wiley & Sons Ltd, Vol. 3, Chapter 18, pp. 579 - 607, 2004<br />
-==Ship hydrodynamics==
+DOI: 10.1002/9781119176817
-−
-'''Eugenio  Oñate¹, Julio García¹ and Sergio R. Idelsohn²'''
-−
-{|
-|-
-|(¹)  International Center for Numerical Methods in Engineering (CIMNE)
-|-
-| Universitat Politécnica de Catalunya (UPC), Gran Capitán, s/n,
-|-
-| 08034 Barcelona, Spain
-|-
-| (²)  CIMNE and International Center for Computational Methods in Engineering (CIMEC)
-|-
-| Universidad Nacional del Litoral and CONICET, Santa Fe, Argentina
-|}
--->
 ==Abstract==

@@ Line 1,398: / Line 1,398: @@
 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]]]).
+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.'' (2002 <span id="citeF-53"></span>[[#cite-53|[53]]], 2003a <span id="citeF-54"></span>[[#cite-54|[54]]]).
 ==15 CONCLUDING REMARKS==

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 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>
+Figure [[#img-16|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]]].
+Figures [[#img-16|16]] and [[#img-17|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===

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 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]]]).
+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-96"></span>[[#cite-96|[96]]]).
 As for the free-surface equation the following value of the characteristic length vector <math display="inline">{\boldsymbol h}_\beta </math> has been taken

← Older revision	Revision as of 08:56, 10 July 2019
(No difference)

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 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-39"></span>[[#cite-39|[39]]] 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.
+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.

@@ Line 296: / Line 296: @@
 |}
-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, 1988, 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]]]).
+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

@@ Line 232: / Line 232: @@
 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]]]).
+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.

@@ Line 34: / Line 34: @@
 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-67"></span>[[#cite-67|[67]]]).
+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]]].