1. AN ADVANCED FINITE ELEMENT METHOD FOR FLUID DYNAMIC ANALYSIS OF AMERICA’S CUP and IMS BOATS
J. García-Espinosa,
R. Luco-Salman, M. Salas2,
M. López-Rodríguez,
E. Oñate,
High Performance Yacht Design Conference 2002

2. Contents RANSE solver. Motivation. Free surface solver
Finite Increment Calculus (FIC) formulation
Implicit Fractional Step scheme
Monolithic Predictor-Corrector scheme
Free surface solver
Sink and Trim treatment
Numerical and Implementation Aspects
Examples
IMS AMC-CRC
Rioja de España
Conclusions

3. RANSE solver. Motivation.
Implicit scheme: In most of the cases of interest for the naval architecture the time step imposed by the stability criteria (smallest elements) may be orders of magnitude smaller than the time step required to obtain time-accurate results (physical time step), rendering explicit schemes impractical.
Pressure-Velocity segregation: Monolithic schemes treat advection term in an implicit manner, which avoids the mentioned disadvantages. Nevertheless, these methods are very expensive from a computational point of view (velocity and pressure discrete equations are coupled).
Finite Increment Calculus: Stability of the numerical algorithm is still today an open issue. There is a need for an accurate, pressure and advection stable algorithm, based on the physics of the problem.

4. Finite Increment Calculus (FIC) Foundations
We consider a convection-diffusion problem in a 1D domain of length L. The equation of balance of fluxes in a sub-domain of size d is A B C d1 d2 d x qA qB
where qA and qB are the incoming fluxes at points A and B. The flux q includes both convective and diffusive terms.
Let us express now the fluxes qA and qB in terms of the flux at an arbitrary point C within the balance domain. Expanding qA and qB in Taylor series around point C up to second order terms gives
Substituting above eqs. Into balance equation gives
with

5. Finite Increment Calculus (FIC) Formulation
We consider the motion around a body of a viscous incompressible fluid (RANSE-NS) including a free surface (FS). The stabilized finite calculus (FIC) form of the governing differential equations for the three dimensional (3D) problem can be written as:
These eqs. are the starting point for deriving a variety of stabilized numerical methods for solving the incompressible NS-RANSE equations. It can be shown, that a number of stabilized methods allowing equal order interpolations for velocity and pressure fields and stable and accurate advection terms integration can be derived from this formulation

6. Implicit Fractional Step Formulation
Discretization in time of the stabilized momentum equation using the trapezoidal rule (or θ method) as
An implicit fractional step method can be simply derived by splitting above equation as follows
This allow us to uncouple pressure and velocity calculations without loss of accuracy ( splitting errors of order 0(t2) )

7. Implicit Fractional Step formulation
Characteristics of the scheme:
Stable (convective terms - equal order velocity-pressure interpolations)
Implicit (higher time step)
Second order accuracy (theta scheme)
All the problems to be solved are scalar (less CPU and memory requirements)

8. Monolithic Predictor-Corrector scheme
All the arguments to derive Fractional Step scheme are still valid if we replace pressure term pn by any other pressure evaluation. In particular we may write (m iteration counter)
Characteristics of the scheme:
Stable (convective terms - equal order velocity-pressure interpolations)
Implicit and monolithic (higher time step)
Second order accuracy (theta scheme)
All the problems to be solved are scalar (less CPU and memory requirements)

9. Free surface solver
FIC method can be directly applied to the free surface equation.
Transpiration technique is used to couple free surface condition with RANSE solver. This technique is based on imposing pressure at free surface obtained by stress continuity as (neglecting tangential components):
Being g the surface tension coefficient and R the average curvature radius. Above condition is applied on a reference surface. FS eq. is also solved on this reference surface not necessary being the exact free surface. In order to improve accuracy of the solver a mesh updating procedure is applied.

10. Sink and Trim treatment
Dynamic sinkage and trim angle are calculated by
where z is a correction of the sinkage at the center of gravity,  is a trim angle correction, Fz and My are a net heave force and a trim moment. Awp is the water plane area, and Iy is the corresponding moment of inertia about the y axis. In order to take these changes into account a mesh updating process is carried out automatically several times during the calculation process. This process is based on a strategy of nodal displacement diffusion through the mesh. Nodal displacements are due to free surface deformation and sink and trim effects. However only sink and trim effects are taken into account close to the ship body.

11. Final geometry updating and mesh regeneration
Finally a new calculation is performed with the real geometry. This is done in four steps:
New free surface NURBS definition, taking the resulting deformation into account, is generated:
NURBS Cartesian support grid of MxN points is created.
Z coordinate of the points, representing the wave elevation, is interpolated into the grid.
Finally, the NURBS surface based on the support grid is generated.
Geometry of the vessel if moved according to calculated sinkage and trim angle.
New control volume and mesh are automatically generated
New calculation is carried out with fixed mesh

12. Numerical and Implementation Aspects
RANSE and FS equations are integrated by standard Finite Element Method (linear/quadratic tetrahedra, hexahedra, prisms, …)
RANSE and FS solvers have been optimized for working with unstructured meshed of linear tetrahedra/triangles
Implicit fractional step algorithm is used to go faster to steady state
Boundary conditions may be defined by analytical functions allowing to run different drifts angles with the same geometry/mesh

13. Numerical and Implementation Aspects
RANSE-FS solver has been implemented within the CFD system Tdyn
Tdyn includes a fully integrated pre/postprocessor based on GiD system, incorporating advanced CAD tools (NURBS importation, creation and edition)
Data insertion (control volume generation, physical properties, boundary conditions, etc) is guided by the use of wizard tools
Mesh can be automatically generated from the CAD information within the system. It also allows elements size assignment and quality check of the resulting mesh
System also includes a set of postprocessing options and tools for report generation

14. Application: IMS AMC-CRC
The first analyzed example is an IMS 37’ boat
Simulations have been carried out at full scale, using a two layer k turbulence model, in combination with an extended law of the wall
Results are compared to experimental data extrapolations performed at Ship Hydrodynamics Centre -Australian Maritime College (AMC) of the Maritime Engineering Cooperative Research Centre (CRC)

15. Application: IMS AMC-CRC
Acknowledgements: This work has partially funded by Universidad Austral de Chile.

16. Application: IMS AMC-CRC
Test
Heel angle
Drift angle
Rudder angle
E10D3
10º 3º
E20D5
20º 5º
Case E10D3 was towed at equivalent velocities of 5, 6, 7 and 8 knots
Case E20D5 was towed at equivalent velocities of 6, 7, 7.5 and 8 knots
Simulations were performed with fixed model. The position of the model in these calculations were obtained by static equilibrium of sails and hydrostatic forces.
Meshes have been generated with the same quality criteria (in terms of size transition and minimum angle), obtaining about tetrahedral elements in every case.

17. Application: IMS AMC-CRC
Pressure and velocity contours Streamlines around hull and appendages

18. Application: IMS AMC-CRC

19. Application: Rioja de España
The second analyzed example is the Spanish America’s Cup boat Rioja de España, participant in the edition of 1995
Simulations have been carried out at full scale, using a two layer k-e turbulence model, in combination with an extended law of the wall
Results are compared to experimental data extrapolations performed at El Pardo towing tank with a model at scale 1/3.5

20. Application: Rioja de España
Acknowledgements: Authors thank National Institute of Aerospace Technique (INTA) for permitting the publication of the towing tank tests of Rioja de España, and the model basin El Pardo (CEHIPAR) for sending the full documentation of the tests. Special gratefulness to IZAR shipbuilders for allowing the publication of the Rioja de España hull forms.

21. Application: Rioja de España
Test
Geometry
Heel angle
Drift angle
C0D0
Hull, no appendages 0º
E0D0
Hull, bulb and keel
E15D2
Hull, bulb, keel and rudder
15º 2º
E15D4 4º
E25D2
25º
Every case was towed at equivalent velocities of 10, 9, 8.5, 8.0, 7.5 and 7.0 knots

22. Application: Rioja de España
Test
Symmetry
# Elements
# Nodes
C0D0
Yes
75 000
E0D0
E15D2 No
E15D4
E25D2
All grids used have been generated with the same quality criteria (in terms of size transition and minimum angle) and using element sizes from 5mm to 2000 mm.

23. Application: Rioja de España
E15D2 Initial mesh
Final mesh E0D0
All meshes have been generated with the same quality criteria (in terms of size transition and minimum angle) and using element sizes from 5mm to 2000 mm.
E15D2 keel-bulb union detail
E25D2 Final Mesh

24. Application: Rioja de España
E25D2 Streamlines and velocity modulus contours (V = 9 Kn)
Se presenta aquí un ejemplo de aplicación de las técnicas de simulación por ordenados y de experimentación. En concreto los análisis correspondientes al velero Rioja de España, participante en la edición de 1995 de la Copa América de vela. A la izquierda se muestran los resultados de la simulación con el programa Tdyn (ver en las condiciones de 9Kn de velocidad, 0º de deriva y 0º de escora, que corresponden al ensayo experimental con un modelo a escala 1:3.5, llevado a cabo en Cehipar que se muestra a la derecha.
Los ensayos experimentales y la simulación son herramientas que se usan complementariamente en diferentes fases del proyecto. La exactitud de las medidas y la gran experiencia acumulada frente a su dificultad de realización y gran inversión de recursos, provoca que los ensayos experimentales sean más adecuados para su realización en la última fase del proyecto, mientras que la facilidad de acceso y la posibilidad de ensayar diferentes configuraciones de manera prácticamente simultánea, hace que la simulación por ordenador sea la herramienta preferida en las fases intermedias del desarrollo del proyecto.
Por otra parte, hay multitud de casos en los que la experimentación es imposible de realizar en la práctica (grandes efectos de escala, falta de medios, limitaciones físicas, ...) por lo que la simulación por ordenador es, en estos casos, la única herramienta disponible.

25. Application: Rioja de España
Se presenta aquí un ejemplo de aplicación de las técnicas de simulación por ordenados y de experimentación. En concreto los análisis correspondientes al velero Rioja de España, participante en la edición de 1995 de la Copa América de vela. A la izquierda se muestran los resultados de la simulación con el programa Tdyn (ver en las condiciones de 9Kn de velocidad, 0º de deriva y 0º de escora, que corresponden al ensayo experimental con un modelo a escala 1:3.5, llevado a cabo en Cehipar que se muestra a la derecha.
Los ensayos experimentales y la simulación son herramientas que se usan complementariamente en diferentes fases del proyecto. La exactitud de las medidas y la gran experiencia acumulada frente a su dificultad de realización y gran inversión de recursos, provoca que los ensayos experimentales sean más adecuados para su realización en la última fase del proyecto, mientras que la facilidad de acceso y la posibilidad de ensayar diferentes configuraciones de manera prácticamente simultánea, hace que la simulación por ordenador sea la herramienta preferida en las fases intermedias del desarrollo del proyecto.
Por otra parte, hay multitud de casos en los que la experimentación es imposible de realizar en la práctica (grandes efectos de escala, falta de medios, limitaciones físicas, ...) por lo que la simulación por ordenador es, en estos casos, la única herramienta disponible.
E25D2 Pressure, velocity and eddy viscosity contours

26. Application: Rioja de España
E25D2 Streamlines on hull and appendages

27. Application: Rioja de España
E25D2 Wave profile 9Kn
Experimental Simulation
Se presenta aquí un ejemplo de aplicación de las técnicas de simulación por ordenados y de experimentación. En concreto los análisis correspondientes al velero Rioja de España, participante en la edición de 1995 de la Copa América de vela. A la izquierda se muestran los resultados de la simulación con el programa Tdyn (ver en las condiciones de 9Kn de velocidad, 0º de deriva y 0º de escora, que corresponden al ensayo experimental con un modelo a escala 1:3.5, llevado a cabo en Cehipar que se muestra a la derecha.
Los ensayos experimentales y la simulación son herramientas que se usan complementariamente en diferentes fases del proyecto. La exactitud de las medidas y la gran experiencia acumulada frente a su dificultad de realización y gran inversión de recursos, provoca que los ensayos experimentales sean más adecuados para su realización en la última fase del proyecto, mientras que la facilidad de acceso y la posibilidad de ensayar diferentes configuraciones de manera prácticamente simultánea, hace que la simulación por ordenador sea la herramienta preferida en las fases intermedias del desarrollo del proyecto.
Por otra parte, hay multitud de casos en los que la experimentación es imposible de realizar en la práctica (grandes efectos de escala, falta de medios, limitaciones físicas, ...) por lo que la simulación por ordenador es, en estos casos, la única herramienta disponible.

28. Application: Rioja de España
E25D2 Wave profiles and pressure contours 9Kn
Se presenta aquí un ejemplo de aplicación de las técnicas de simulación por ordenados y de experimentación. En concreto los análisis correspondientes al velero Rioja de España, participante en la edición de 1995 de la Copa América de vela. A la izquierda se muestran los resultados de la simulación con el programa Tdyn (ver en las condiciones de 9Kn de velocidad, 0º de deriva y 0º de escora, que corresponden al ensayo experimental con un modelo a escala 1:3.5, llevado a cabo en Cehipar que se muestra a la derecha.
Los ensayos experimentales y la simulación son herramientas que se usan complementariamente en diferentes fases del proyecto. La exactitud de las medidas y la gran experiencia acumulada frente a su dificultad de realización y gran inversión de recursos, provoca que los ensayos experimentales sean más adecuados para su realización en la última fase del proyecto, mientras que la facilidad de acceso y la posibilidad de ensayar diferentes configuraciones de manera prácticamente simultánea, hace que la simulación por ordenador sea la herramienta preferida en las fases intermedias del desarrollo del proyecto.
Por otra parte, hay multitud de casos en los que la experimentación es imposible de realizar en la práctica (grandes efectos de escala, falta de medios, limitaciones físicas, ...) por lo que la simulación por ordenador es, en estos casos, la única herramienta disponible.

29. Application: Rioja de España
E0D0 Pressure contours 10Kn

30. Application: Rioja de España
E25D2 Pressure contours 9Kn

31. Application: Rioja de España
E0D0
E15D2
E25D2
E15D4

32. Conclusions
FIC technique applied to RANSE-FS equations allows to derive a number of stabilized schemes allowing equal-order velocity-pressure interpolation and adequate treatment of advection terms.
An implicit second-order accurate monolithic scheme, based on the FIC formulation has been derived to solve incompressible free surface flow problems. The final system of equations resulting from the time and space discretization is solved in each time step in an uncoupled manner.
The numerical experience indicates that the formulation is very efficient for free surfaces flows, when the critical time step of the problem is some orders of magnitude smaller than the time step required to obtain time-accurate results (physical time step), and that its time accuracy is excellent (i.e. 4 CPU h / 1.5Mtetras standard PC single processor, including s&t final calculation).
Sink and trim effects as well as free surface deformations are taking into account in the solution process by automatic mesh updating and a final geometry and mesh regeneration.
The solver has been optimized for using unstructured meshes of linear tetrahedra, allowing a simple and automatic mesh generation from complex NURBS based geometry, and the best mesh refinement control.

33. Conclusions
RANSE-FS solver has been integrated within the pre/postprocessing environment (GiD)
Graphical environment has been adapted to naval architecture needs by developing wizard tools
Numerical results obtained in the analysis of America’s Cup Rioja de España and IMS AMC-CRC boats indicate that the proposed method can be used with confidence for practical design purposes.
Evaluation of total resistance gives less that 5% difference with towing tank extrapolations in most part of the analysis range.
Evaluation of induced (lift) forces in non-symmetric cases gives even less differences.
Obtained wave profiles are also very close to those measured in towing tank.
Dynamic sinkage and trim angles show similar behavior in both experimental and numerical results. However, scale effects don’t allow further conclusions.
Qualitative results including: wave maps, streamlines, pressure and velocity contours and turbulence distribution, show also reasonable patterns

34. For further information ...