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Self-Propelled Motions of Solids in a Fluid: Mathematical Analysis, Simulation and Control
Marius TUCSNAK Institut Elie Cartan de Nancy et INRIA Lorraine, projet CORIDA Collaborations with : Jorge SAN MARTIN (Santiago) Jean-François SCHEID (IECN) Takeo TAKAHASHI (IECN) 1
Motivation: understanding locomotion of aquatic organisms, in particular to design swimming-robots 2
Motion of solids in a fluid
• • • • Incompressible Navier-Stokes (Euler, Stokes) equations for the fluid ODE’s or PDE’s system for the motion of the solids Continuity of the velocity field at the interface Homogenous Dirichlet boundary conditions on the exterior boundary 3
Plan of the talk
• The model and the mathematical challenge • An existence and uniqueness theorem • Numerical method and simulations • Self-propelling at low Reynolds number 4
The model and the mathematical challenge
5
6
Kinematics of the creature (I)
S*(T)
7
Kinematics of the creature (II)
8
The governing equations
u t
div
u
u
u
0,
x
0,
x
( ),
t
[
0
,T]
, ( ),
t
[
0
,T]
,
u
h
x
h
w
,
x
( ),
t
[0,
T
],
Mh
S t
n d
,
t
[0,
T
], d d
t
J
x
h
n d
,
t
[0,
T
],
u
0)
u
0 (
x
) ,
h
(0)
h
0 ,
h i
'(0)
h
1
i
, (0 ) 0 9
Existence theorems
The Leray type mathematical theory has been initiated around year 2000. Early mathematical ref. for rigid-fluid interaction from a mathematical view-point : • • • • • D. Serre ( 1987) K. H. Hoffmann and V. Starovoitov ( 2000) B. Desjardins and M. Esteban ( 1999,2000) C. Conca, J. San Martin and M. Tucsnak (2001) J. San Martin, V. Starovoitov and M. Tucsnak (2002) Theorem (J. San Martin, J. –F. Scheid, T. Takahashi, M.T., ARMA (2008)).
If the given deformation is smooth enough than the system admits an unique strong solution. If no contact occurs in finite time then this solution is global. Control theoretic challenge: the input of the system is the geometry of the domain.
10
Numerical Method and Simulations
11
.
References
• • • • Carling, Williams and G.~Bowtell, (1998) Liu and Kawachi (1999) Leroyer and M.Visonneau, (2005) J. San Martin, J.-F. Scheid and M. Tucsnak ( SINUM 2005 , ARMA 2008) 12
Two bilinear forms
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Global weak formulation (J. San Martin, J.F –Scheid, T. Takahashi and M.T. in SINUM (2005) )
14
Semi-discretization with respect to time
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Choice of characteristics
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Finite element spaces
• • Fixed mesh Rigidity matrix is (partially) re-calculated at each time step
Convergence of solutions
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Straight-line swimming 19
Turning 20
Meeting
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Can they really touch?
The answer is no for rigid balls (San Martin, Starovoitov and Tucsnak (2002), Hesla (2006), Hillairet (2006) , but unknown in general.
22
Mickey’s Reconstruction
Kiss and go effect
Perspectives
25
• Coupling of the existing model to elastodynamics type models for the fish • Control problems • Infirming or confirming « Gray’s paradox » • Giving a rigorous proof of the existence of self-propelled motions 26
A Control Theoretic Approach to the Swimming of Cilia Micro-Organisms
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The model : One rigid ball in the whole space
u t
div
u
u
u
0,
x
0,
x
( ),
t
[
0
,T]
, ( ),
t
[
0
,T]
,
u
h
x
h
V R
,
x
( ),
t
[0,
T
],
Mh
B t
n d
,
t
[0,
T
], d d
t
(
J
)
x
h
n d
,
t
[0,
T
],
R
RS
( ).
)
R
3 \ ) 28
Change of coordinates :
y u t
div
u
u
u
u
0, y
F
,
t
[
0
,T]
, 0, y
F
,
t
[
0
,T]
,
u
h
M h V R
, y
S
,
t
[0,
T
],
B
n
d , d d
t
(
J
)
B x
n
d ,
t
[0,
T
],
R
RS
( )
B
B
,
F
3 \
B
29
A simplified finite dimensional model
div
u
p
0, y
F
,
t
[
0
,T]
, 0, y
F
,
t
[
0
,T]
,
u
h
0
B
( ) (
l
,
u
, y
B
,
t
[0,
T
],
n
d , 0
B
( )
x
(
l
,
h
l
,
R
u RS
( ),
n
d ,
t
[0,
T
],
u
0)
u
0 (
y
) ,
h
(0)
h
0 ,
h i
'(0)
h
1
i
, (0 ) 0 The control is:
i
6 1
i
i
30
The case of small deformations
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A controllability result
By « freezing » the deformation (Blake’s « layer model »), we obtain a simplified model which can be written as a dynamical system
Z
F
+
Bu
in
R
9 SO(3) .
• Proposition . (J. San Martin, T. Takahashi and M.T., QAM (2008)) • Generically (with respect to the shape functions), the above system • is controllable in any time. With “standard” choices of the shape functions • the controllabilty fails with less than 6 controls. Remark. For less symmetric shapes less than 6 controls suffice (Sigalotti and Vivalda, 2007) 32
Some perspectives
• Optimal control problems, in particular obtaining the motion of cilia by solving an optimization problems.
• Proving the existence of self-propelled motions for arbitrary Reynolds numbers • Control at higher Reynolds numbers.
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The case of small deformations