Numerical models for sailing yachts: from hull dynamics to wind/sails ...

Numerical models for sailing yachts:from hull dynamics to wind/sails FSI

Nicola Parolini, Matteo Lombardi

Dipartimento di MatematicaPolitecnico di Milano

HPC enabling of OpenFOAM for CFD applications

CINECA, Casalecchio di Reno - November 27, 2012

Sport Hydrodynamics: our contribution

America’s Cup Sailing Yachts

Appendage shape optimization

Free-surface hydrodynamics

Sink and trim boat dynamics

(with A. Quarteroni, D. Detomi, S. Piazza, M. Lombardi)

Olympic Rowing Boats

Boat/oar/rower system dynamics

Free-surface hydrodynamics

6-DOF dynamics and control

(with L. Formaggia, E. Miglio, A. Mola, M. Pischiutta)

Swimsuits

Drag reduction

Performance assessment

(with A. Veneziani, E. Foa, F. Biondi)

Collaboration with the Alinghi Design Team

31st America’s Cup

Auckland (NZ), February 2003

Defender: Team New Zealand (NZ)

Challenger: Alinghi (SUI)

32nd America’s Cup

Valencia (E), July 2007

Defender: Alinghi (SUI)

Challenger: Team New Zealand (TNZ)

33rd America’s Cup

Valencia (E), February 2010

Challenger: BMW Oracle Racing (USA)

33rd America’s Cup

Valencia (E), February 2010

Challenger: BMW Oracle Racing (USA)

Role of CFD in IACC yacht design

Towing TankWave drag on the

Wind TunnelGlobal forces on sails and

appendages

Potential FlowInviscid forces on

sails and appendages

appendages

RANS-based CFDViscous and pressureforces on sails and

appendages

For any given design configuration:Experimental tests and CFD simulations on a limited set of boat speed andattitude configurations F (Vi ,Aj );

Data regression on the range of parameters F (V ,A),(Vmin < V < Vmax,Amin < A < Amax);

Compute performance VEq,AEq with a Velocity Prediction Program (VPP).

appendagesVPP

appendages

For any given design configuration:Experimental tests and CFD simulations on a limited set of boat speed andattitude configurations F (Vi ,Aj );

Data regression on the range of parameters F (V ,A),(Vmin < V < Vmax,Amin < A < Amax);

Compute performance VEq,AEq with a Velocity Prediction Program (VPP).

Velocity Prediction Programs (VPP)

EquilibriumFor a given configuration, the VPP computes boat speed V and attitude A associated tothe force equilibrium state:

M ax = Ta(V ,A)− Dh(V ,A)

M ay = Sa(V ,A)− Sh(V ,A)

I ΩH = MH(V ,A)−MR(V ,A)

−→

Ta = Dh

Sa = Sh

MH = MR

−→

Ta = Dh

Sa = Sh

MH = MR

−→ VEq,AEq

Performance evaluation for yacht design

Objective: prediction of aero/hydrodynamic forces

Wave resistance and boat dynamicsevaluation

Laminar-to-turbulent transition regimes onappendages

Optimal flying shape of sails

Modeling approach:

Multiphase Reynolds-Averaged Navier-StokesEquations

SST k − ω turbulence model

Volume-of-Fluid method for interfacecapturing

Dynamical system for 6DOF boat motion

Fluid-structure interaction for sails

WIND/SAILS FSI

FREE-SURFACE HYDRODYNAMICS

APPENDAGE OPTIMIZATION

WIND/SAILS FSI

Wind/Sails Fluid-Structure interaction

Determination of the flying sail shape is crucial for per-formance evaluation

In Upwind sailing flow is mainly attached

Potential flow model can be adopted

FSI coupling between a sail structural model andpanel method

In Downwind sailing flow is stronglyseparated

Flow around gennaker/spinnaker needsviscous RANS solutions

Development of a RANS based FSIalgorithm required

Wind/Sails FSI: coupling algorithm

FSI problem as a fixed-point:

Fluid(Struct(p)) = pFluid : fluid operator

Struct : structural operator

Given a pressure field pk , the fixed point iteration reads:

(Gk+1,Uk+1) = Struct(pk )

pk+1 = Fluid(Gk+1,Uk+1)

pk+1 = (1− θk )pk + θk pk+1

Steady algorithm

Interest only on converged steadysolution

Sail velocity (Uk+1) set to zero inflow solver

Less FSI iterations required

Steady solution may not be physical

Unsteady algorithm

Interest on transient solution

Unsteady flow solver required

Moving wall BC (Uk+1) in flowsolver

More FSI iterations required

Commercial software integration: the Virtual Wind Tunnel

Simulation of downwind sails: steady algorithm

Analysis of different sail shapesand trimmings

Changing sail trimming, the flowfield can dramatically change

Optimal trimming identification

-2 -1.5 -1 -0.5 0 0.5 1

Genn Sheet Trimming [m]

MainGennakerTotal Force

Gennaker Sheet Trimming GS=-1 m

Simulation of downwind sails: steady algorithm

Analysis of different sail shapesand trimmings

Changing sail trimming, the flowfield can dramatically change

Optimal trimming identification

-2 -1.5 -1 -0.5 0 0.5 1

Genn Sheet Trimming [m]

MainGennakerTotal Force

Gennaker Sheet Trimming GS=0.5 m

Wind/Sails FSI: open-source development

Fluid finite-volume model (OpenFOAM)

RANS equations with k − ω turbulence model

SIMPLE/PISO schemes for steady/transient solutions

Shell finite element model

SEDIS solverdeveloped at DIS, Politecnico diMilano (Prof. U. Perego)

MITC4 shell elements (Locking-free)

linear isotropic material

no need for wrinkle model

explicit in time

Fortran with OpenMP

Wind/Sails FSI: shell structure solver

Sail wrinkling detected loading the structure with a constant pressure field

Test case proposed in Fluid-structure interactions of anisotropic thin composite

materials for application to sail aerodynamics of a yacht in waves, Trimarchi, D.,Turnock, S.R., Chapelle, D. and Taunton, D. (2009).

Wind/Sails FSI: modelling

A COUPLED PROBLEM !

Wind/Sails FSI: possible coupling strategies

MONOLITHIC APPROACH

Better stability properties

MONOLITHIC APPROACH

Full matrix (fluid+structure)

MONOLITHIC APPROACH

Large memory requirements

MONOLITHIC APPROACH

PARTITIONED APPROACH

MONOLITHIC APPROACH

Modular approach

MONOLITHIC APPROACH

Modular approach

Reuse of existing software

MONOLITHIC APPROACH

Modular approach

Weakly Coupled

Efficient (1 flow and 1structure solution per timestep)

Can be unstable

MONOLITHIC APPROACH

Modular approach

Weakly Coupled

Efficient (1 flow and 1structure solution per timestep)

Can be unstable

Strongly Coupled

Subiterations andrelaxation

Better stability

Wind/Sails FSI: strongly coupled segregated scheme

Structural problem:

Mesh motion problem:

Flow problem:

Convergence test

Convergence test Number of FSI iterations

material properties

deformation magnitude

relaxation scheme

Wind/Sails FSI: implementation

Shell structural solver

• Explicit scheme =⇒ Many sub-time steps required to coverone fluid step

• Fortran code =⇒ OF is master, Fortran routine calledinside OF with fortran/c++ wrappers

• OpenMP =⇒ OF Master node calls structural solverwhile other CPUs are idle

⇓Whit more than one sail, structuralsolvers can be run in parallel

Non-conforming mesh (FV/FEM)

different grids

different collocations of DOF

interface interpolation (RBF)

Non-conforming mesh (FV/FEM)

different grids

different collocations of DOF

interface interpolation (RBF)

Mesh motion

moving 3D fluid grid (ALE)

large displacement

different possible approaches:

LaplacianRadial basis function (RBF)Inverse distance weighting (IDW)

RBF: an overview

Global map from a set of control points Cj :

f (x) =

γjφ(|x− xCj|) + q(x),

φ(·): radial basis (based on Euclidean distance)q: additional polynomial term

Imposing

exact mapping of control points f (xCj) = fCj

exact mapping of rigid body motions[BCC PC

RBF: interpolation

General RBF interpolation formula:

fI = [BIC PI ]︸︷︷︸

= [BIC PI ]

[BCC PC

︸︷︷︸

R−1CC

= HIC fC

Interpolation of displacement from structure (DS) to fluid (DF )

DF = [BFS PF ]︸︷︷︸

= [BFS PF ]

[BSS PS

︸︷︷︸

R−1SS

= HFSDS

Control points: structural nodesInterpolation points: fluid nodes

FSI: energy conservation

Exact energy transfer at the interface requires

WS(d) =

(σSdS) · d =

(σFdF ) · d = WF (d), ∀ d,

Numerically, the following stress transfer is obtained

ΣS = M−1S H

TFSMFΣF ,

For a finite-element structural solver, we need:

(σ · n)ψj =

(σ · n)iψiψj =⇒ MSΣS = HTFSMFΣF ,

RBF implementation improvements

Optimized libraries for LU factorization

Method OpenFOAM Boost ATLAS

NC = 1925 34.54 19.20 3.08NC = 2850 73.72 60.11 9.29

RBF implementation improvements

Optimized libraries for LU factorization

Method OpenFOAM Boost ATLAS

NC = 1925 34.54 19.20 3.08NC = 2850 73.72 60.11 9.29

... and matrix-vector multiplications (with ATLAS)

# proc. 2 4 8 16

Time (s.) 0.44 0.24 0.13 0.07

Each partition moves its own points and evaluates the local stress contribution:

MSΣS = HTFSMFΣF =

MFkΣFk

Mesh motion: Laplacian

Mesh motion based on the solution ofa Laplace problem

∇ · (γM∇d) = 0, in Ω,

d = d, on ∂ΩM ,

d = 0, on ∂ΩF ,

γM : variable artificial diffusion coefficient

inversely proportional to cell element size

inversely proportional to distance from moving patches

Mesh motion: RBF

Same RBF map defined for the interface interpolation

more interpolation points (all the 3d mesh points)

very large linear system to solve

to reduce the computational cost

some sampling of the control points (e.g. only one every N points)smooth cut-off function

Mesh motion: IDW

Inverse Distance Weighting (IDW) for multivariate interpolation.

Interpolation map: d(x) =∑NC

i=1|x−xi |

−pdi∑NCi=1

|x−xi |−p

, p = 2, 3, 4 usually

no linear system to solve

large matrix: # surface mesh points × # volume mesh points

local smooth cut-off functions to reduce computational cost

FSI: mesh motion

Laplacian: unsuccessful

RBF: too expensive

IDW: used in most simulations

FSI: benchmark test case

assessment of FSI algorithmsnot easy

comparison with availablebenchmark test cases

grid and time-step convergence

Wind/sail FSI: one sail setup

&%U%Boat%

155°%

Wind/sail FSI: one sail simulations

Intial transient due to non-equilibriuminitial configuration

Flow pushing on the edges of the sailbefore starting to stabilize

Wind/sail FSI: one-sail simulations - steady vs transient

Different possible approaches:1 Fully Transient

transient flow solution (PISO scheme)transient structure solutionmany FSI iterations for convergence (50-60)

2 Pseudo Transient

transient FSI solutionbut zero-velocity imposed on the sailfaster convergence (6-8 FSI iterations)smoother flow evolution

3 Steady

alternate steady flow solution (SIMPLE) and steady structuresolutiondisplacement from steady structural solver may be largedisplacement distributed over N sub-stepsmuch faster convergence, meaningful for steady physical solution

Wind/sail FSI: one-sail simulations - steady

Pseudo-Transient vs. Steadymean forces and sail flying shapes veryclose

steady converges to a non oscillatingsolution

Wind/sail FSI: one-sail simulations - transient

Lower gennaker corner attachedto a sheeting rope (i.e. free tomove on a spherical path if undertension)

Pressure waves propagates fromthe corners (fixed and attachedto the sheet)

Wind/sail FSI: one-sail simulations - transient

Wind/sail FSI: two-sail simulations - steady

Different trimming of the gennaker (5m, 6m, 7m, 8m)

Wind/sail FSI: two-sail simulations - steady vs transient

Steady FSI (8 m) Transient FSI (8 m)

Wind/sail FSI: two-sail simulations - transient

1 initial large motion of sheet-attached vertex

2 when sail seems to collapse inward, the sheet gets under tension and sailrecovers

3 sail finally collapses inward due to the too open trimming

Wind/sail FSI: real-life instability

Wind/sail FSI: computational cost

Typical case size:

Fluid: 1.8M elements on 32 cores with MPI

Structure: 1800 nodes on 8 cores with OpenMP

Typical case size:

One FSI iteration timing:

Fluid solver =⇒ 8 s (44%)

Mesh motion =⇒ 3 s (17%)

Structural solver =⇒ 5 s (27%)

Other =⇒ 2 s (12%)

Typical case size:

Other =⇒ 2 s (12%)

Fluid/FSI cost comparison:

NS transient solver: 8 s per time step

FSI solver: 18 s × 30=540 s per time step (×70 !)

Typical case size:

Other =⇒ 2 s (12%)

Fluid/FSI cost comparison:

NS transient solver: 8 s per time step

FSI solver: 18 s × 30=540 s per time step (×70 !)

GPU acceleration:

GPU implementation of mesh motion

Hybrid GPU/CPU implementation of structural solver

(Andrea Bartezzaghi’s Master thesis)

Flow equations

Navier–Stokes equations

ρi∂tui + ρi (ui ·∇)ui −∇ · Ti (ui , pi ) = ρig, in Ωi

∇ · ui = 0,

with Ti (ui , pi ) = (µi + µt i )(∇ui +∇uiT )− pi I.

Flow equations

∇ · ui = 0,

Interface conditions

u1 = u2, on Γ,

T1 · n = T2 · n+ κσn on Γ.

Flow equations

∇ · ui = 0,

u1 = u2, on Γ,

T1 · n = T2 · n+ κσn on Γ.

One-fluid formulation

∂tρ+ u · ∇ρ = 0,

ρ∂tu+ ρ(u ·∇)u−∇ · T(u, p) = ρg + fΓ, in Ω

∇ · u = 0,

with T(u, p) = (µ+ µt)(∇u+∇uT )− pI.

Flow equations

∇ · ui = 0,

u1 = u2, on Γ,

T1 · n = T2 · n+ κσn on Γ.

∂tρ+ u · ∇ρ = 0,

∇ · u = 0,

ρ = ρ(x)

µ = µ(x)

Flow equations

∇ · ui = 0,

u1 = u2, on Γ,

T1 · n = T2 · n+ κσn on Γ.

∂tρ+ u · ∇ρ = 0,

∇ · u = 0,

ρ = ρ(x)

µ = µ(x)

fΓ = κσδΓn

Flow equations

∇ · ui = 0,

u1 = u2, on Γ,

T1 · n = T2 · n+ κσn on Γ.

∂tρ+ u · ∇ρ = 0,

∇ · u = 0,

ρ = ρ(x)

µ = µ(x)

fΓ = κσδΓn

Initial and boundaryconditions for u and ρ

Boat dynamics

Boat reference (Gc ; x, y, z) Global reference (O; X , Y , Z)

Rotation matrix

cos θ cosψ sinφ sin θ cosψ − cosφ sinψ cosφ sin θ cosψ + sinφ sinψcos θ cosψ sinφ sin θ sinψ + cosφ cosψ cosφ sin θ sinψ − sinφ cosψ− sin θ sinφ cos θ cosφ cos θ

Boat tensor of inertia

Ixx Ixy IxzIyx Iyy IyzIzx Izy Izz

referred to the body-fixed reference system

Boat dynamics

Linear and angular momentum in the inertial reference system:

mXG = F

RIGR−1

ω + ω ×RIGR−1

ω = MG

Time integration on the system of first order ODE

mYG = F,

XG = YG ,

Adam-Bashforth scheme for the velocity

Yn+1 = Yn +∆t

2m(3Fn − Fn−1),

Crank-Nicolson scheme for the position of the center of mass

Xn+1 = Xn +∆t

2(Yn+1 + Yn).

Dynamical system coupled with the flow solver on a moving domain (in ALEframework).

Dynamics in wavy sea

Wave boundary condition at inlet

Seakeeping analysis in wavy seacan be performed

Dynamic response of the boat fordifferent wave lenghts andamplitudes

Maximum sink vs wave frequency

Free-surface simulation for AC32 and AC33

Free-surface solver in OpenFOAM

interFoam class solver

Validation on benchmark cases

Coupling with dynamic module

Implementation of external wavemodel

-0.015

-0.005

0 0.2 0.4 0.6 0.8 1

Wave e

levation [m

CoarseMedium

FineExp.

0 5 10 15 20 25 30D

]Time [s]

Wavy seaFlat sea

0 5 10 15 20 25 30

Time [s]

Wavy seaFlat sea

VOF model in interFoam

VOF uses a scalar indicator function to represent the phase of the fluid in each cell

α = α(x, t)

µ(x, t) = µwα+ µa(1− α)

ρ(x, t) = ρwα+ ρa(1− α)

density and viscosity (and, therefore α) are material properties of the fluids:

Dt= 0 →

∂t+ u · ∇α = 0 →

∂t+∇ · (uα) = 0

to keep the interface sharp, consider a modified governing equation

∂t+∇ · (uα) +∇ · (wα(1− α)) = 0

with w an artificial velocity field oriented normal to and towards the interface.

the relative magnitude of the artificial velocity can be changed (parameter cAlpha)

the α equation is solved using MULES (Multidimensional Universal Limiter withExplicit Solution) method

VOF model in interFoam

Ansys CFX (homogeneous) interFoam

CFX/OpenFOAM for AC33

Experimental vs. numerical drag prediction at different boat speeds

Automated Mesh Generation: snappyHexMesh

blockMesh+ STL geometry

castellatedMesh

snapEdge (in SHM since v2.0)

castellatedMesh

refineMesh

castellatedMesh

refineMesh

addLayers

snappyHexMesh: a more complex example

ad-hoc setup of snappyHexMesh

uniform refinement level over surfaces for layers

avoid non-conforming refinement in free-surface region

smooth refinement layer transition

Results on appendage design

Simulation campaign on all theappendage components at differentsailing regimes

Parametric studies on different designchoices

Investigation on radical new shapes

Results on appendage design

Simulation campaign on all theappendage components at differentsailing regimes

Parametric studies on different designchoices

Investigation on radical new shapes

Turbulent and transition models requireshighly refined block structured grids(Y+ ≈ 1)

Postprocessing for detection of local flowfeatures (separation and vortices)

Shape optimization: possible strategies

Continuous adjoint approach in OpenFOAM

Drag Minimization

J = −

∫ΓBody

(2νσ(u)n − pn) · ufdγ

Gin Gout

Steady Navier-Stokes equations

(u · ∇)u − ∇ · (νσ(u)) + ∇p = 0 in Ω

∇ · u = 0 in Ω

u = uf on ΓIn

u = 0 on ΓBody

−2νσ(u)n + pn = 0 on ΓOut

Drag Minimization

J = −

∫ΓBody

Gin Gout

(u · ∇)u − ∇ · (νσ(u)) + ∇p = 0 in Ω

∇ · u = 0 in Ω

u = uf on ΓIn

u = 0 on ΓBody

Adjoint problem

(∇Tv)u − (u · ∇)v − ∇ · (2νσ(v)) + ∇q = 0 in Ω

∇ · v = 0 in Ω

v = −uf on ΓBody

v = 0 on ∂Ω \ ΓBody

Drag Minimization

J = −

∫ΓBody

Gin Gout

(u · ∇)u − ∇ · (νσ(u)) + ∇p = 0 in Ω

∇ · u = 0 in Ω

u = uf on ΓIn

u = 0 on ΓBody

Adjoint problem

(∇Tv)u − (u · ∇)v − ∇ · (2νσ(v)) + ∇q = 0 in Ω

∇ · v = 0 in Ω

v = −uf on ΓBody

Shape gradient

∇J = −(2νσ(u) : σ(v))n

Drag Minimization

J = −

∫ΓBody

Gin Gout

(u · ∇)u − ∇ · (νσ(u)) + ∇p = 0 in Ω

∇ · u = 0 in Ω

u = uf on ΓIn

u = 0 on ΓBody

Adjoint problem

(∇Tv)u − (u · ∇)v − ∇ · (2νσ(v)) + ∇q = 0 in Ω

∇ · v = 0 in Ω

v = −uf on ΓBody

Shape gradient

∇J = −(2νσ(u) : σ(v))n

Drag Minimization

J = −

∫ΓBody

Gin Gout

Volume constraint

augmented Lagrangianmethod

a-posteriori correction

Shape parametrization: FFD method

Sensitivity field V:

move the mesh points according to V

project V on a shape parametrization

Free-Form Deformation

psi invpsi

Omega Omegat

Map Ψ : (x1, x2) −→ (s, t) such that Ψ(D) = (0, 1)2, with Ω ⊂ D

psi invpsi

Omega Omegat

Control points: Pol,m(µl,m) = Pl,m + µl,m

psi invpsi

Omega Omegat

FFD map: T (x;µ) = Ψ−1(

∑Ll=0

∑Mm=0 b

L,Ml,m

(Ψ(x))Pol,m(µl,m)

psi invpsi

Omega Omegat

FFD map: T (x;µ) = Ψ−1(

∑Ll=0

∑Mm=0 b

L,Ml,m

(Ψ(x))Pol,m(µl,m)

Deformed domain: Ωo(µ) = T (Ω;µ), with T = T |Ω

psi invpsi

Omega Omegat

Shape optimization: 2D test case

Drag minimization of an airfoil

Initial shape NACA0030

Re=1000

0 10 20 30 40 50

Iteration

Drag coefficient

Velocity Velocity Adjoint Velocity

Initial geometry Final shape Final Shape

Shape optimization: 3D test case

only very preliminary results obtained

difficult extension to turbulent flows

Dakota/OpenFOAM integration

Dakota library (developed at Sandia Labs)

open-source multiobjctive optimization softwaredifferent optimization algorithms (gradient-based and not)tools for sensitivity analysis and robust design

Integration with external software (e.g. OpenFOAM) based on scripts

Simple example: bulb shape optimization

bulb drag minimization

fixed righting moment constraint

3 global design parameters controlling:

section aspect ratiomean camber linesection CG position

ReferenceconfigurationP1 = P2 = P3 = 0

Aspect RatioP1 = P1,min = −0.3

Aspect RatioP1 = P1,max = 0.3

Mean CamberP2 = P2,min = −0.4

Mean CamberP2 = P2,max = 0.4

Section CGP3 = P3,min = −0.4

Section CGP3 = P3,max = −0.4

Optimalconfigurationfor draft D=4 m

(Vittorio Bissaro’s Master thesis)

WIND/SAILS FSI

Integration of numerical tools for sailing yacht simulation

Full boat simulations

Integrate different models:

free-surface flow solver

rigid boat motion

wind/sail fluid-structure interaction

longitudinal motion (surge) treated withnon-inertial reference system

Different possible approaches:

one single domain

decoupling hydro and aero domains withsuitable domain interface conditions

rigid boat motion

one single domain

rigid boat motion

one single domain

rigid boat motion

one single domain

(Wibke Wriggers’s Master thesis)

Full boat simulation

Conclusions and perspectives

CFD in yacht design

Increased importance in design process

Acquired confidence from designer and sailors (thanks to proved accuracy)

New boat class (multi-hull) demanding new models (planing, cavitation, ...)

Conclusions and perspectives

CFD in yacht design

Increased importance in design process

Acquired confidence from designer and sailors (thanks to proved accuracy)

New boat class (multi-hull) demanding new models (planing, cavitation, ...)

Development directions

Model integration

sail aerodynamicsfree-surface solver on appended hullFSI for sails and hull

CFD based VPP

Shape optimization and optimal control

References

N. P. and A. Quarteroni, Mathematical Models and Numerical Simulations for the America’s Cup. Comp.

Meth. Appl. Mech. Eng., 194, 1001–1026 (2005).

N. P. and A. Quarteroni. Modelling and numerical simulation for yacht design. In Proceedings of the 26th

Symposium on Naval Hydrodynamics Strategic Analysis, Inc., Arlington, VA, USA, 2007.

L. Formaggia, E. Miglio, A. Mola, and N. P. Fluid-structure interaction problems in free surface flows:application to boat dynamics. Int. J. Num. Meth. Fluids 56(8) 965–978 (2008)

D. Detomi, N. P. and A. Quarteroni, Mathematics in the Wind, in Monografias de La Real Academia de

Ciencias de Zaragoza 31, 35–56 (2009).

D. Detomi, N. P. and A. Quarteroni, Numerical Models and Simulations in Sailing Yacht Design. inComputational Fluid Dynamics for Sport Simulation, Lecture Notes in Computational Science andEngineering, 1–31, Springer, 2009.

M. Lombardi, N. P., A. Quarteroni and G. Rozza, Numerical simulation of sailing boats: dynamics, FSI, andshape optimization, in Variational Analysis and Aerospace Engineering: Mathematical Challenges forAerospace Design, G. Buttazzo and A. Frediani Eds., Optimization and its Applications series, Vol. 66,Springer, in press, 2012.

M. Lombardi, M. Cremonesi, A. Giampieri, N. P. and A. Quarteroni, A strongly coupled fluid-structureinteraction model for sail simulations, to appear in Proceedings of the 4th High Performance Yacht Design

Conference, Auckland, 2012.

M. Lombardi, N. P., A. Quarteroni, Radial basis functions for inter-grid interpolation and mesh motion in FSIproblems, MOX Report 40/2012.

thanks for your attention

This work has been partially supported by Regione Lombardia and CILEA througha LISA Initiative grant 2010/2012.

This work is not approved or endorsed by ESI Group, the producer of theOpenFOAM R© software and owner of the OpenFOAM R© trade mark.

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2017 CATALOGUE GREECE - Mariver Charter · 2017 SAILING HOLIDAYS 12 m Max 6 people + skipper SAILING YACHT SAILING YACHT SAILING YACHT The yachts have all the necessary equipment

Unforgettable sailing holidays at the best coasts of Turkey · 2020. 6. 24. · crewed luxury yachts Exclusive holidays with crewed luxury yachts CREWED LUXURY G SAILING ulets We

Viko Yachts (PL) Viko s22 - inautia.com · Viko Yachts (PL) Viko s22 Sailing cruiser (2020) Viko Yachts (PL) Viko s22 € 17.800 € Basic data Type: Sailing cruiser Year : 2020 Length:

beautiful people, beautiful yachts - Istion · 2016-08-31 · A wide range of sailing yachts, sailing catamarans and motor yachts up to 100 feet are offered to the finest destinations

SIMULATION-DRIVEN DESIGN OF SAILING YACHTS …€¦ · SIMULATION-DRIVEN DESIGN OF SAILING YACHTS AND MOTOR BOATS Bodo Hasubek, ... (SDD), Yacht Design, Computational Fluid Dynamics