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physics of rowinghydrodynamics
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1GEOMETRIC
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VALIDATION
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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
GEOMETRIC
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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
GEOMETRIC
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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.
2GEOMETRIC
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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
GEOMETRIC
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An example obtained with a RANS code
Costly computations! We wanted a simpler models for preliminary studies
GEOMETRIC
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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
GEOMETRIC
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A typical rowing boat
3GEOMETRIC
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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
GEOMETRIC
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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
GEOMETRIC
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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
GEOMETRIC
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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.
4GEOMETRIC
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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.
5GEOMETRIC
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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)!
GEOMETRIC
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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
6GEOMETRIC
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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,)
GEOMETRIC
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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
GEOMETRIC
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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
7GEOMETRIC
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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.
GEOMETRIC
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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
GEOMETRIC
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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
GEOMETRIC
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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
10
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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
GEOMETRIC
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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
GEOMETRIC
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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
GEOMETRIC
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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
GEOMETRIC
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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
GEOMETRIC
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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
GEOMETRIC
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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
GEOMETRIC
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Sinking and pitching motion
Ellipsoid with the following
characteristics
3940 N
Sinking motion: return to equilibrium after a vertical
displacement
GEOMETRIC
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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.