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Numerical Numerical modellingmodelling of competition rowing of competition rowing

boatsboats

Luca Formaggia

MOX, Department of Mathematics F. Brioschi

Politecnico di Milano, Italy

Other contributors : Edie Miglio, Nicola Parolini, Andrea Mola, Anna Scotti, Andrea

Paradiso, Lorenzo Tamellini

http://mox.polimi.it

IMA Workshop on Computing in Image Processing, Computer Graphics,

Virtual Surgery, and Sports

Minneapolis, 7-11 March 2011

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MOTIVATIONS

The study of the dynamics of a racing boat may help the

designer (shape optimization) as well as the trainers (crew

optimization)

Trimming

Sea keeping

Manouvering

Performance

Rower positioning

Rowing style

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An example of complex dynamics: accelerations

induced by the rowers action and movement

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Comparing different styles (horizontal accelerations)

Excellent rower Intermediate RowerImages taken from On board shell measurements of acceleration by K. Young and R. Muirhead, http://www.phys.washington.edu/users/jeff/courses/ken_young_webs/rowsci

Accurate simulation of the wave resistance caused by secondary motion may contribute to the optimization of the boat and of the athlete action.

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A multilevel approach

Boat material

and geometry

Rowers motion and

weight

Oarlock

forces

Dynamical system of rowing boat motion

FREE SURFACE DYNAMICS MODEL

RANS

(VOF)

Free surfaceHydrod.

3D Potential

Eqs.

Strip

Theory

B

O

A

T

M

O

T

I

O

N

Hull P

ositio

n

Hydrodynamic forces

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An example obtained with a RANS code

Costly computations! We wanted a simpler models for preliminary studies

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OUTLINE OF THE TALK

Modeling the dynamics of the boat and the rowers

A potential model for fast computations

A model based on free surface dynamics and

unilateral constraint

Conclusions and further work

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A typical rowing boat

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The dynamics of a rowing boat

sink

Horizontal acc.

Pitch

Movements in the (x,y) plane

These are the most important movements. A first model was developed accounting only for them

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Full 3D dynamics

R =

cos cos sin sin cos cos sin cos sin cos + sin sincos cos sin sin sin + cos cos cos sin sin sin cos sin sin cos cos cos

Boat reference system Absolute reference system

pitch angleyaw angle

roll angle

X = Gh +Rx

Gh

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Modelling the oar

Lwater

X

Y

boat hull

r

-F

F

-Fh

o

w

w

hr

The oar has been modeled as a simple lever

Fh = L rh

LFo

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Computation of the forces at footboards and seats

p=12

j

j=1,..n

h = hand

A dynamical model for the rower

This model together with the model for the oars allows us to write the forces at the seat Fs and at the footboards Ff as function of the force at the oarlocks Fo and the rower motion.

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The kinematics of rowing

The kinematics of the rowing is extracted

from a data base of measurements made

using video capturing techniques

In collaboration with C. Sforza of the Istituto di morfologia umana of Universit di

Milano, Italy

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The reconstructed kinematics

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Forces at the oarlocks

Oarlock forces may be reconstructed from experimental data. We need to separate the active phase from the recovery phase of the rowing action.

Fo,x =

{f1(t) 0 t < a

f2(t) a t < T

Fo,z = Fo,x

The vertical component is taken proportional to the horizontal

one, while f1 and f2 are approximated by a cubic and

quadratic polynomial in t, respectively

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The dynamical model of the boat-rowers system

Oarlock SeatFootboard

o oarlockMGh+ I1(, ;x

r) =2n

j=1 F oj + Mg + Fw

RIGR1 + RIGR

1 + I2(, ;xr) =

2nj=1

(Xolj G

h) F oj+

LrhL

2nj=1

(Xhlj G

h) F oj +M

w

rotation vector

L is the oar length and rh the part from the oarlock to the hand

Gh boat center of mass IG tensor of inertia

Fw

Mwand are forces and momenta due to the fluid.

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Reduced model for the hydrodynamic interaction

Computations with RANS codes are expensive in terms of

human and computer resources

In the preliminary design phase there is the need of a fast, yet

effective tools to compare different configurations

A fast tool can be used also by athletes and trainers to test

different rowing styles or boat arrangements

A first reduced model: we solve just ODEs (coupled with an off line Laplace equation)!

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The main hypothesis

Main viscous phenomena are captured by static simulations at constant speed

Secondary motions dissipate energy mainly through wave radiation

The length of the waves generated by the secondary motions is comparable to that of the boat while the amplitude is small compared to the wave length

We can neglect non-linear phenomena when computing the effects of the secondary motions

Secondary motions are periodic with period T=2/ equal to the stroke period

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Decomposition of the hydraulic forces acting on the hull

Drag force

is estimated from off-line static computations

Bouyancy force

is the wet surface, dynamically computed

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Simulation of secondary motion effects

The secondary motion of the boat is considered as formed by elementary periodic movements

The induced velocity field is described by a potential in the complex plane

By linearising the free surface interface conditions and applying first order radiation condition at the artificial bounary we obtain a set of Laplace equations in the complex plane

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The potential problem

is a the generalised normal

(in the 3 d.o.f. case)

C.C. Mei, The applied dynamics of ocean surface waves, Wold Scientific, 1989

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The forces induced by the secondary motion

Added mass matrix

Damping matrix

Pitching movement Vertical movement

FD =M()v +

0

K()v(t )d

K(t) =1

Re

[

(M()M()) eitd

]

To account for more frequencies we need to solve a convolution integral

v = (Gh,)

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The control on sway and roll

The system is unstable in the roll degree of freedom and indifferently stable on the yaw degree of freedom. We have added a simple feedback control to simulate the action of the rowers

Fo,z =

{0 0 < t a

kroll a < t T

Fo,x =

{kY aw 0 < t a

0 a < t T

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The final system of equations

M(u(t), t)u(t) + K(u(t), t)u(t) = f(u(t), t) t > 0

y =

{u

u

}M(t,y)y = F (y, t).

Eventually, we have a system of non linear second order ODEs for the linear and angular displacements u

with given initial position and velocities. We reduce it to a first order system

solved with a standard RK45 scheme

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Example: single scull 80 Kg athlete

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Comparison: heavier vs ligher rower

Surge Heave Pitch

H

e

a

v

e

P

o

i

s

i

t

i

o

n

S

p

e

e

d

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The importance of the control

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Sensitivity Study

Multi-physics Model

VX

r

mr

FXmax

Oarlock Force

Rowing cadence

Mean surge velocity

Efficiency

Rower mass

Input PDF

Output PDF

(x, t, ) Pi=0

i(x, t)i()

Polynomial chaos expansion

1

T

T0

GhX(t)dt

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An example of sensitivity analysis

Male coxless four. Sensitivity study

Parameters: Horizontal oar force (4), rower weight (4), cadence

Input Param. VX sensit. sensit.FXmax 1 8.38 10

4 m

s/N 0.0014 1/N

FXmax 2 8.41 104 m

s/N 0.002 1/N

FXmax 3 8.34 104 m

s/N 0.0015 1/N

FXmax 4 8.37 104 m

s/N 0.0003 1/N

mr 1 0.0066m

s/Kg 0.0287 1/Kg

mr 2 0.0064m

s/Kg 0.0176 1/Kg

mr 3 0.0065m

s/Kg 0.0142 1/Kg

mr 4 0.0063m

s/Kg 0.0086 1/Kg

r 0.0343 ms

/( strokesmin

) 0.1589 1/( strokesmin

)

Sensitivities at point = 0

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The code in action

The model is currently used at Filippi Lido srl, a renown rowing boat manufacturer, for preliminary design and boat trimming.

www.filippiboats.it

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An alternative hydrodynamic model

At an intermediate level of complexity between full RANS

simulations and the potential model just described we have

hydrodynamic free surface models which describes the surface elevation explicitly.

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The problem setting

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Kinematical conditions

The velocity is indicated as

Free surface evolution

Incompressibility

Impermeable wall

Non slip condition at the bottom

Other conditions will be examined later on

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DAlambert-Lagrange principle

The flow motion satisfies at any time t the following relation

where P is any admissible virtual particle displacement

The corresponding differential equations are

Momentum equation

Boundary compatibility cond.Natural boundary conditions

`aaaa atmospheric pressure, W external surface forcing term

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Imposing of the presence of a boat by a inequality

constraint

Constraint

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Lagrange multiplier technique

and we add to the rhs of the variational formulation the term

which represents the virtual work done by leading to

We introduce a Lagrange multiplier

satisfying

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Intermpreting the augmented variational formulation

on ssssNote that

by which

The result is that the boundary condition on ssss becomes

It is convenient to rewrite the pressure term as

hydrodynamic correction

hydrostatic pressure

constraint reaction term

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The resulting surface Navier Stokes equations with

unilateral constraints

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An important note

The kinematic relation for together with the definition of N on ssssimplies that on we have the following slip condition on the velocity

that is no conditions are imposed on the tangential component

Indeed fixing the tangential component of the velocity on theboat surface is would be incompatible with the kinematicsof the free surface and the condition = on

Therefore the effect of friction on the boat may be only take into account empirically through the function W

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Specialising the model

Under the hypothesis that the basin is relatively shallow we

can make the following approximation

and neglect all horizontal components of the stress. This approximation leads to very convenient numerical schemes (but it is not crucial for the numerical algorithm for the imposition of the constraint)

Furthermore, by integrating along the vertical the kinematic condition for and using the continuity equation we have

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The final set of equations

CD is the Chezy coefficient. It accounts for friction at the bottom surface. We avoid resolving the boundary layer.

Flow equations

Free surface kinematics

Boundary Conditions

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Numerical treatment of the time derivative

where is obtained by solving

with a suitable time integration scheme

At each time step we set

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Time discrete equations

Linearization procedure

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Weak formulation setting

We define the following forms

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The b term explained

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The weak formulation

(a,b) denotes the L2() scalar product

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Interpratation of the weak formulation

The weak formulation may be reinterpreted as the KKT condition

for the following minmax problem

where

associated to the following minimization problem

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Numerical solution of the constrained problem

By the strong duality principle the problem is equivalently stated as where

which can be implemented as a succession of unconstrained problems.

In particular a projected steepest-descent type method applied to the problem in w leads to the well known Uzawa iterations

J. Cea, Lectures on Optimization Theory and Algorithms, Tata Institute, 1978

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Uzawa iterations

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Finite element space discretization

E. Miglio, A. Quarteroni and F. Saleri., CMAME, 1999

This choice allows to get an efficient and easy to parallelize scheme

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The algebraic setting

For every iteration of the Uzawa scheme we need to solve a system of the form

* Indicates the element-by-element vector

product

Horizontal velocity d.o.f.Elevation d.o.f.

Vertical velocity d.o.f.

Convergence tests

Lagrange multiplier d.o.f.

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A fractional step scheme

Hydrostatic iterations:

Intermediate vertical velocity

Hydrodynamic pressure computation

Hydrostatic correction

We can write it as an equation for only!

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A differential interpretation of the hydrostatic step

Let s. t. is the solution of

We can formally write the following equation for only

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Coupling the two dynamics

The force acting on the boat is computed as

where is the drag force, estimated by empirical formulae

is computed from a parametric description of the boat

displaced according to the computed rigid motion.

We use a simple explicit (staggered) scheme for the

interaction problem:

Fluid solution

Boat

dynamics

Fluid solution

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A Final Touch

Most recent simulations have been carried out rewriting the flow equation in a reference frame with the origin fixed in the xy plane: smaller computational domain. It suffices to add the term to the momentum equation

At the far field we have implemented a first order linearized

radiation condition to reduce unphysical reflections

Different numerical schemes of predictor/corrector type have

been implemented for the boat dynamics

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Example of grid on the xy plane an of a boat geometry

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First test case: wave produced by a wigley hull moving at

constant speed (hydrostatic approximation)

Froude number 0.316

Comparison with the

theoretical Mach line

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Sinking and pitching motion

Ellipsoid with the following

characteristics

3940 N

Sinking motion: return to equilibrium after a vertical

displacement

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Pitching motion

The boat returns to equilibrium after an angular displacement

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Wave hitting a boat

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Coupling with the full dynamics (hydrostatic)

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Hydrostatic vs hydrodynamic results

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Conclusions

We have presented two reduced models for the dynamic of a

rowing boat which are able to provide reasonable fast answer

to designers and trainers

Ongoing and future work

for both models

Validation with experiments in collaboration with Filippi

Lido and the University of Ferrara

for the inequality constraint model

Integration with real boat geometries from CAD data

Coupling with a model of boundary layer to account for

friction

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Acknowledgements

Filippi Lido s.r.l. for financial support and in particular Ing. Alessandro

Placido who introduced us to the wonderful world of rowing

Andrea Paradiso and Michele Altieri for the availability of some results

from their master thesis

Fausto Saleri for his important contribution to the original idea

The work has been partially supported by a PRIN07 project of the Italian

MIUR

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References

E. Miglio, A. Quarteroni and F. Saleri. Finite element approximation of quasi-3D

shallow water equations. Com. Meth. Appl. Mech. Engng. 174(3-4):335-369, 1999

P. Causin, E. Miglio and F. Saleri. Algebraic factorizations for 3D non hydrostatic free

surface flows, Comp. Vis. Sci. 5(2):85-92, 2002

A. Mola. Models for olympic rowing boats. PhD Thesis, Politecnico di Milano, 2009

A. Mola. Multiphysics and multilevel fidelity modelling and analysis of olympic rowing

boat dynamics. PhD Thesis, Virginia Tech, 2010

L. Formaggia, E. Miglio, A. Mola and A. Montano, A model for the dynamics of rowing

boats. Int. J. Numer. Meth. Fluids, 61(2):119-143, 2009

L. Formaggia, E. Miglio, A. Mola and A. Scotti. Numerical simulation of the dynamics

of boats by a variational inequality approach. In Variational analysis and aerospace

engineering. 213-227, Springer, 2009

L. Formaggia, A. Mola. N. Parolini and M. Pischiutta. A three-dimensional model for

the dynamics and hydrodynamics of rowing boats. Journal of Sport Engineering

and Technology, 224(1):51-61, 2010.

L. Tamellini, L. Formaggia, E.Miglio and A. Scotti. An Uzawa iterative scheme for the

simulation of floating boats. Submitted 2010.