# A modiﬁed Lagrange Galerkin method for a ﬂuid rigid system

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A modiﬁed Lagrange-Galerkin method for a ﬂuid-rigid system with discontinuous density Jorge San Martín ? , Jean-François Scheid † , Loredana Smaranda ‡ Abstract In this paper, we propose a new characteristics method for the discretization of the two dimensional ﬂuid-rigid body problem in the case where the densities of the ﬂuid and the solid are di?erent. The method is based on a global weak formulation involving only terms deﬁned on the whole ﬂuid-rigid domain. To take into account the material derivative, we construct a special characteristic function which maps the approximate rigid body at the discrete time level tk+1 into the approximate rigid body at time tk. Convergence results are proved for both semi-discrete and fully-discrete schemes. 1 Introduction The aim of this paper is to present a modiﬁed characteristics method for the discretization of the equations modelling the motion of a rigid solid immersed into a viscous incompressible ﬂuid. Our method is a generalisation of the numerical scheme presented in San Martín, Scheid, Takahashi and Tucsnak [18] for the case where the ﬂuid and the solid have di?erent densities. The ﬂuid-rigid system occupies a bounded and regular domain O ? R2. The solid is assumed to be a ball of radius 1 whose center, at time t, is denoted by ?(t).

• rigid body

• discrete formulation

• has given

• stokes equations

• lagrange-galerkin method

• semi-discretization scheme

• characteristic function

• domain has

• ?0 ?

Sujets

##### Characteristic function

Informations

 Publié par Nombre de visites sur la page 34
Signaler un problème

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@t
u = 0; x2 ( t);t2 [0;T ];
u = 0; x2@O;t2 [0;T ];
0 ?
u = (t) +!(t)(x (t)) ; x2@B((t));t2 [0;T ];
Z Z
00
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dist(B((t));@O)> 0 8t2 [0;T ]:
> 0
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kk k(u ;p ; ) t = tkh h h
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> k: k k (t ;t ;x) =x u ( ) t:h k+1 k+1 h h
k k
X (x) = (t ;t ;x) 8x2O:k k+1h h

kk k kk kdiv P( )u ( (t;t ;)) P( )u ( ) = 0k+1h h h hh h
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detJ = 1:k
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k F T T2T TB( )1 h h h
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!
kk+1 ku u Xhk+1 h h k+1 k+1 ;’ +a(u ;’) +b(’;p )h h ht
k+1 k+1 k+1
= ( f ;’) 8’2K ( );h hh h
k+1 k+1b(u ;q) = 0 8q2M ( );hh h
k+1 2 2 k+1 2f L (O) f =f(t ) (E )k+1 hh
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u (u;p;;!)0
C > 0 0 < 10
C h t
1+0< t hC t0

k k sup j(t ) j +ku(t ) u k 2 2 Ct :k kh h L (O)
1kN
O( t)
2 = 1 h hC t0
=f s
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