R E S E A R C H
Open Access
The Lagrange-Galerkin method for
fluid-structure interaction problems
Jorge San Martín
1, Jean-François Scheid
2and Loredana Smaranda
3**Correspondence:
[email protected] 3Department of Mathematics and
Computer Science, Faculty of Mathematics and Computer Science, University of Pite¸sti, Str. Târgu din Vale nr. 1, Pite¸sti, 110040, Romania
Full list of author information is available at the end of the article
Abstract
In this paper, we consider a Lagrange-Galerkin scheme to approximate a
two-dimensional fluid-structure interaction problem. The equations of the system are the Navier-Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the solid. We are interested in studying numerical schemes based on the use of the characteristics method for rigid and deformable solids. The schemes are based on a global weak formulation involving only terms defined on the whole fluid-solid domain. Convergence results are stated for both semi and fully discrete schemes. This article reviews known results for rigid solid along with some new results on deformable structure yet to be published.
1 Introduction
In this article, we present a modified characteristics method for the discretization of the equations modelling the motion of a solid immersed in a cavity filled by a viscous incom-pressible fluid. We are interested in rigid and deformable solids modelling some particu-late flows in the case of rigid solid and the swimming of slender, neutrally buoyant fish, for the deformable structure (see []). The presented methods are generalizations of the numerical scheme introduced in [], where the solid immersed in the fluid is rigid and has the same density with the fluid.
The fluid-structure interaction problem that we study is characterized by the strong coupling between the nonlinear equations of the fluid and those of the structure, as well as the fact that the equations of the fluid are written in a variable domain in time, which depends on the displacement of the structure. From the numerical point of view, in this kind of problems it is necessary to solve equations on moving domains. For this reason, in recent years various authors have proposed a number of different techniques [–].
For the numerical treatment of convection term in the Navier-Stokes equations, we dis-cretize the material derivative along trajectories (see []) combined with the Lagrange-Galerkin mixed finite element approximation of Navier-Stokes equations in a veloc-ity/pressure formulation studied in []. In [], the convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations is done.
The numerical analysis of some time decoupling algorithms in the case, where the de-formation of the structure induces an evolution in the fluid domain has been developed in [] (one-dimensional problem). For the ALE method applied to interaction problems, we may cite [] in the case of the unsteady Stokes equations in a time dependent domain
and [] for a two-dimensional problem describing the motion of a rigid body in a viscous fluid. In [, ], the authors have introduced a convergent numerical method based on finite elements with a fixed mesh for a two-dimensional fluid-rigid body problem, where the densities of the fluid and the solid are equal. In [, ], we have introduced crucial modifications on the characteristic function, and we have proposed a convergent numeri-cal scheme for a two dimensional fluid-rigid body problem where the densities of the fluid and the solid are different. In this paper, we go further, and we present a new characteris-tic function which gives us convergent algorithms for the simulation of aquacharacteris-tic organisms (for the existence and regularity of the solution in this kind of interactions, see []).
2 Setting of the problem 2.1 Notation and hypothesis
Let us now introduce some notation following paper [], where the existence and unique-ness for the solution of similar problem are treated. We denote bySthe domain occupied
by the solid in a reference configuration. We assume thatSis an open connected set with
C∞boundary, and we choose a system of coordinates with the origin at the mass center ofS. For the deformable solid, we suppose that the motion is given by a smooth mapping
X:S×[,∞)→R,
which satisfies
X(y,t) =ξ(t) +Rθ(t)X∗(y,t) ∀y∈S,t≥, ()
where for everyt≥,ξ(t) is the trajectory of the mass center, andθ(t) represents the angle giving the orientation of the solid.Rθdenotes the matrix associated to the rotation of angleθ. In (),X∗denotes an appropriate smooth mapping, representing the undulatory deformation of the creature. The rigid solid is obtained by considering the map
X∗= I,
where I denotes the identity map.
Throughout this paper, the deformable body will be calledthe creatureor sometimes justthe bodyin particular, when considering the rigid body case.
In the remaining part of this work, the functionsξ,θ are unknowns to be determined from the governing equations below, whereas the undulatory motionX∗will be supposed to be known and to satisfy several assumptions as in [], which will be recalled in the sequel.
(H) For everyt≥, the mappingy→X∗(y,t) is aC∞diffeomorphism fromSonto
S∗(t), whereS∗(t) =X∗(S
,t). Moreover, for everyy∈S, the mappingt→X∗(y,t) is of
classC∞andX∗(y, ) =y.
For everyt≥ we denote byY∗the inverse ofX∗,i.e., the diffeomorphism satisfying
X∗Y∗x∗,t,t=x∗, Y∗X∗(y,t),t=y
(H) The total volume of the creature is preserved,i.e.,
S∗(t)dx
∗= S
dy ∀t≥. ()
Denote byw∗the undulatory velocity of creature, written as a vector field onS∗(t),i.e.,
w∗x∗,t=∂X
∗
∂t (y,t)
y=Y∗(x∗,t)
∀x∗∈S∗(t),t≥. ()
LetρS,be the density field of the solid in the reference configurationS, and letρS∗(t) be
the density field ofS∗(t). The mass conservation principle applied to the whole body gives
S∗(t)ρ
∗ S
x∗,tdx∗=
S
ρS,(y)dy ∀t≥, ()
whereas the local form of the conservation of mass yields
ρS∗x∗,t= ρS,(Y
∗(x∗,t))
det(∇X∗)(Y∗(x∗,t)) ∀t≥,∀x
∗∈S∗(t), ()
where∇X∗stands for the Jacobian matrix ofX∗(·,t). (H)S∗(t)ρ∗S(x∗,t)w∗(x∗,t)dx∗= for allt≥.
(H)S∗(t)ρS∗(x∗,t)x∗⊥·w∗(x∗,t)dx∗= for allt≥, where we denote byx⊥the vector
x⊥=–xxforx=xx.
Conditions (H), (H) correspond to the so-calledself-propellingconditions which are natural requirements for understanding swimming viewed as a self-propelled phenomena. In particular, hypotheses (H) and (H) imply that the position of the center of mass of the creature is not affected by the undulatory motion, that is,
S
ρS,(y)X∗(y,t)dy= ∀t≥. ()
From (), it follows that the region occupied by the creature at timetis given by
Sξ(t),θ(t),t=Rθ(t)S∗(t) +ξ(t) ∀t≥. ()
Moreover, by differentiating equation () with respect tot, it follows that the Eulerian velocity field of the solid is given for everyt≥ by
uS(x,t) =ξ(t) +θ(t)
x–ξ(t)⊥+w(x,t) ∀x∈Sξ(t),θ(t),t, () where
w(x,t) =Rθ(t)w∗
R–θ(t)
x–ξ(t),t ∀x∈Sξ(t),θ(t),t. () The Eulerian density field of the body is given by
ρS(x,t) =ρS∗
R–θ(t)
withρS∗given by (). The massMof the body and its moment of inertia with respect to an axis orthogonal to the plane of the motion and passing by the mass center ofS(ξ(t),θ(t),t), are as usually given by
M=
S(ξ(t),θ(t),t)
ρS(x,t)dx. ()
I(t) =
S(ξ(t),θ(t),t)
ρS(x,t)x–ξ(t)
dx. ()
Let us notice that from (), () and (), we have that
M=
S
ρS,(y)dy.
Remark . In the caseX∗= I (rigid solid), all hypotheses (H)-(H) are satisfied, and the undulatory velocity fieldw∗is equal to zero.
2.2 Equations
Letbe an open bounded set inRrepresenting the domain occupied by the solid-fluid
system. Recalling thatS(ξ(t),θ(t),t) is the domain occupied by the solid at instantt, we have that the fluid fills, at instantt, the domainF(ξ(t),θ(t),t) =\S(ξ(t),θ(t),t).
With the notation above, the full system describing the self-propelled motion of the creature can be written as
ρF
∂u
∂t + (u· ∇)u
–μu+∇p=ρFf, x∈F
ξ(t),θ(t),t,t∈(,T), ()
divu= , x∈Fξ(t),θ(t),t,t∈(,T), ()
u= , x∈∂,t∈(,T), ()
u(x,t) =ξ(t) +θ(t)x–ξ(t)⊥+w(x,t), x∈∂Sξ(t),θ(t),t,t∈(,T), () Mξ(t) = –
∂S(ξ(t),θ(t),t)
σnd+
S(ξ(t),θ(t),t)
ρS(x,t)f(x,t)dx, t∈(,T), ()
Iθ(t) = –
∂S(ξ(t),θ(t),t)
x–ξ(t)⊥·σnd
+
S(ξ(t),θ(t),t) ρS(x,t)
x–ξ(t)⊥·f(x,t)dx, t∈(,T). ()
In the system above,ρF > andμ> stand for the density and the viscosity of the
fluid, which are supposed to be constant,uis the Eulerian velocity field of the fluid, and pdenotes the pressure field of the fluid. A prime stands for the derivation operator with respect to time. By using the classical notation
D(u) =
(∇u) + (∇u)T, () the stress tensor fieldσis defined by
where Id is the identity matrix inM(R). Moreover, fort∈[,T] andx∈∂S(ξ(t),θ(t),t)
we denote byn(x,t) the unit normal toS(ξ(t),θ(t),t) oriented towards the solid. Recall that the massMand the moment of inertiaI(t) of the solid at instanttare defined by () and ().
System ()-() is completed by the initial conditions
u(x, ) =u(x), x∈F
ξ(),θ(), , ()
ξ() =ξ, θ() =θ, ξ() =ξ, θ() =ω. ()
Remark . In the case of rigid solid, equation () becomes
u(x,t) =ξ(t) +θ(t)x–ξ(t)⊥, x∈∂Sξ(t),θ(t),t,t∈[,T], because the undulatory velocity fieldwis equal to zero.
2.3 Weak formulation
Letξ∈H((,T);R),θ∈H((,T);R) be two functions such thatS(ξ(t),θ(t),t)⊂for
allt∈[,T]. In the sequel, we defineF=F(ξ(),θ(), ) andS=S(ξ(),θ(), ). More-over, if no confusion is possible, we define
S(t) =Sξ(t),θ(t),t, F(t) =Fξ(t),θ(t),t=\S(t).
Let:R×[,T]→Rbe a mapping such that for everyt∈[,T], the function(·,t)
is aC∞-diffeomorphism fromFontoF(ξ(t),θ(t),t) and such that the derivatives
∂i+k+k
∂ti∂yk
∂y k
, i≤,k≥,k≥
exist and are continuous. The existence of such a function is due, in particular, to the fact thatdist(S(t),∂) > for allt(see []). We can now define the following functions spaces:
L,T;HF(t)= u|u∈L
,T;H(F),
H,T;LF(t)= u|u∈H
,T;L(F),
C[,T];HF(t)= u|u∈C
[,T];H(F),
L,T;HF(t)= p|p∈L
,T;H(F),
wherevdenotes the function defined byv(y,t) =v((y,t),t) for (y,t)∈F×(,T). In order to introduce the weak formulation, we first define some additional functions spaces. For everyt≥, let (ξ,θ) be an arbitrary position of the creature at timet, such thatS(ξ,θ,t)⊂. We denote
K(ξ,θ,t) = u∈H():D(u) = inS(ξ,θ,t), () M(ξ,θ,t) =
p∈L() :
p dx= andp= inS(ξ,θ,t)
, ()
Let (u,p,ξ,θ) be a solution of ()-(). The vector velocity fielduand the pressurep can be extended toby setting
u(x,t) =ξ(t) +θ(t)x–ξ(t)⊥+w(x,t) ifx∈Sξ(t),θ(t),t, () p(x,t) = ifx∈Sξ(t),θ(t),t. () The extended vector u(·,t) belongs toH
(). In the remaining part of this paper, the
solutionuandpof ()-() will be extended as above.
We also need to extend the density fieldρSof the creature (defined in ()) to the whole
domainby setting
ρ(x,t) =
⎧ ⎨ ⎩
ρF for allx∈F(ξ(t),θ(t),t), ρS(x,t) for allx∈S(ξ(t),θ(t),t),
t≥. ()
By a slight variation of the argument in Ladyzhenskaya [, p.], it can be shown that for everyδ> , there exists a continuous function (x∗,t)→∗(x∗,t) such that, for every t≥, the mapx∗→∗(x∗,t) isC∞onR\S∗(t) and such that the functiont→∗(x∗,t)
is of classC∞for everyx∗∈R\S∗(t) and
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
div∗= inR\S∗(t),t∈(,T),
∗(x∗,t) = ifdist(x∗,S∗(t))≥δ> ,t∈(,T),
∗(x∗,t) =w∗(x∗,t) ifx∗∈S∗(t),t∈(,T).
()
For every t≥, let (ξ,θ) be an arbitrary position of the creature at time t, such that S(ξ,θ,t)⊂. We then define(·,t;ξ,θ) by
(x,t;ξ,θ) =Rθ∗R–θ(x–ξ),t
∀x∈R. ()
Then the functionsatisfies
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
div= inR\S(ξ,θ,t),
= on∂,
=w inS(ξ,θ,t).
()
An important ingredient of the numerical method we use is given by the characteristic functions whose level lines are the integral curves of the velocity field. More precisely (see, for instance, [, ]) the characteristic functionψ: [,T]×→is defined as
the solution of the initial value problem
⎧ ⎨ ⎩ d
dtψ(t;s,x) =u(ψ(t;s,x),t) ∀t∈[,T],
ψ(s;s,x) =x. () It is well known that the material derivativeDtu=∂u/∂t+ (u· ∇)uof the velocity fieldu
at instanttsatisfies:
Dtu(x,t) =
d dt
uψ(t;t,x),t
Remark . By using a classical result of Liouville (see, for instance, [, p.]), if ξ∈H(,T), θ∈H(,T), u∈C[,T];H()
are such that for anyt∈[,T], we haveS(ξ(t),θ(t),t)⊂and
divu= inFξ(t),θ(t),t, then we get
detJψ(t,s,x) = ∀x∈F
ξ(t),θ(t),t, ()
where we have denoted by
Jψ=
∂ψi
∂yj
i,j
the Jacobian matrix of the transformationy→ψ(y).
In order to give the global weak formulation of our problem, we need to introduce the bilinear forms
a:H()×H()→R, b:H()×L()→R,
defined by
a(u,v) = μ
D(u) :D(v)dx ∀u,v∈H(),
b(u,q) = –
(divu)q dx ∀u∈H(),q∈L().
Proposition . Assume that
u∈L,T;HF(t)∩H,T;LF(t)∩C[,T];HF(t), p∈L,T;HF(t), ξ∈H(,T), θ∈H(,T),
and thatuand p are extended toas above.
Then (u,p,ξ,θ) is the solution of ()-() if and only if for all t∈ [,T], u(·,t) –
(·,t;ξ(t),θ(t))∈K(ξ(t),θ(t),t),p(·,t)∈M(ξ(t),θ(t),t),and(u,p)satisfies (ρDtu,v) +a(u,v) +b(v,p) =
ρf(t),v ∀v∈Kξ(t),θ(t),t, () b(u,q) = ∀q∈Mξ(t),θ(t),t, () for a.e.t∈(,T).
In the remainder of the paper, we suppose thatfandusatisfy
f∈C[,T];H(), u∈H(F), div(u) = inF,
u= on∂,
u(x) =ξ+ω(x–ξ)⊥+Rθw∗
R–θ(x–ξ),
forx∈∂S,
()
whereξ,ξ∈Randθ,ω∈Rare given as initial data in (). Let us also assume that
the corresponding solution (u,p,ξ,θ) of problem ()-() satisfies the following regularity properties:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
u∈C([,T];H(F(t)))∩H(,T;L(F(t))), D
tu∈L(,T;L(F(t))), u∈C([,T];C,()),
p∈C([,T];H(F(t))), ξ∈H(,T), θ∈H(,T).
()
Moreover, we suppose that there exists a nonempty open connected subsetofsuch
that for anyξ,ξ∈, we have
Sξ+λ(ξ–ξ),θ,t⊂ ∀λ∈[, ],∀θ∈[, π],∀t∈[,T]. ()
Using this notation, we assume that
ξ(t)∈, distS(t),∂> ∀t∈[,T]. ()
Remark . The hypotheses () and () imply the existence ofη> such that
distξ(t),∂> η, distS(t),∂> η ∀t∈[,T]. ()
3 Time discretization and first main result
In this section, based on a weak form of the governing equations, we describe a method for the time discretization of ()-().
Let us first divide the time interval [,T] into subintervals [tk,tk+] withtk+–tk=t=
T
N, whereNis a positive integer andk∈ {, . . . ,N}. Let (uk,pk,ξ
k,θk) be the approximation
of the solution of ()-() at timet=tk(remark thatukandpkare functions defined on
the whole domain). We denote
Kk=Kξk,θk,tk, Mk=Mξk,θk,tk,
Sk=Sξk,θk,tk, Fk=Fξk,θk,tk, ()
and we consider the functions
k(x) =Rθk∗
R–θk
x–ξk,tk ∀x∈R,
ρk(x) =
⎧ ⎨ ⎩
ρS∗(R–θk(x–ξk),tk) ifx∈Sk, ρF ifx∈Fk.
Remark . Combining the regularity properties of∗andw∗, it follows that
k∈C,(). ()
Moreover, takingδ<ηin definition () of∗, whereηis defined in (), we have that
k+Kk⊆H().
Now, let us describe the numerical scheme for approximating the solutions of ()-(). This procedure is based on the weak form derived in Proposition ..
The first step of our scheme consists in computing the new position of the mass center and the new orientation of the creature by setting
ξk+=ξk+tukξk, ()
θk+=θk+ t I(tk)
Sk
ρkuk(x) –ukξk·x–ξk⊥dx. ()
The second step consists in computing the global velocity fielduk+and the global pres-sure fieldpk+. To this end, we look foruk+∈k++Kk+andpk+∈Mk+such that for
allv∈Kk+, and for allq∈Mk+, we have
ρk+u
k+–uk◦Xk uk t ,v
+auk+,v+bv,pk+=ρk+fk+,v, ()
buk+,q= , ()
wherefk+(x) =f(x,tk+) for anyx∈.
In the equations above, the approximate characteristic is given by
Xkuk(x) =χk
tk;tk+,x, ()
for allx∈, whereχkis the solution of the problem
⎧ ⎨ ⎩ d
dtχk(t;tk+,x) =uk(χk(t;tk+,x)), t∈[tk,tk+],
χk(tk+;tk+,x) =ξk+Rθ
k(tk;tk+,R–θk+(x–ξk+)),
()
with
uk(z) =uk(z) –ukξk–θk+–θk t
z–ξk⊥–k(z) ∀z∈R,
whereuk(z) is extended by zero outside of.
For alls∈[,T], andz∈R, function(·;s,z) corresponds to the characteristic function
of the extended undulatory velocity∗, defined by
⎧ ⎨ ⎩ d
dt(t;s,z) =∗((t;s,z),t), t∈[,T],
Remark . Let us note that for anyz∈S∗(tk+), equation () has the explicit solution
t;tk+,z=X∗Y∗z,tk+,t∈S∗(t). ()
Then for anyx∈S(ξk+,θk+,tk+), since
R–θk+
x–ξk+∈S∗tk+,
we obtain that the initial condition in () is
χktk+;tk+,x=ξk+RθkX∗
Y∗R–θk+
x–ξk+,tk+,tk.
Moreover, sincez=χk(tk+;tk+,x)∈S(ξk,θk,tk), we haveuk(z) = and
χkt;tk+,x=ξk+RθkX∗
Y∗R–θk+
x–ξk+,tk+,tk ∀t∈tk,tk+.
In particular, fort=tkwe have that
Xkuk(x) =ξk+RθkX∗
Y∗R–θk+
x–ξk+,tk+,tk ()
for allx∈S(ξk+,θk+,tk+).
It is easy to see that for anyk∈ {, . . . ,N}, equations ()-() represent a mixed formu-lation of a well-posed Navier-Stokes-type system, so that our scheme is well defined.
Let us now state our first main result concerning the convergence of the semi-discrete scheme ()-().
Theorem . Suppose thatis an open smooth bounded domain inR,fandusatisfy
(),and the exact solution(u,p,ξ,θ)of problem()-()satisfies hypotheses()-(). Then there exist a constant K> depending on T and a constantδ> independent of T such that for all <t≤δ,the solution(uk,pk,ξk,θk)
k∈{,...,N}of the time-discretization
problem()-()satisfies
sup ≤k≤N
utk–uk
L()+ξ
tk–ξk+θtk–θk≤Kt. ()
The complete proof of this result could be found in [] for the case of rigid solid and in the forthcoming paper [] for the deformable structure.
4 Fully discrete formulation and second main result
In order to discretize problem ()-() with respect to the space variable, we introduce two families of finite element spaces which approximate spacesKk andMkdefined in
(), () and (). To this end, for any discretization parameterh∈(, ), we consider a quasi-uniform triangulationThof the domain. Suppose thatis a bounded convex
space associated withThfor the velocity field and byEh theP-finite elements space for
the pressure, that is,
Wh= v∈C():∀T∈Th,v|T∈P+ bubble(T)
,
Eh= q∈C() :∀T∈Th,q|T∈P
.
Then, we define the following finite elements spaces for a conform approximation of the fluid-solid system:
Kh(ξ,θ,t) =Wh∩K(ξ,θ,t) ∀ξ∈,∀θ∈[, π],∀t∈[,T],
Mh(ξ,θ,t) =Eh∩M(ξ,θ,t) ∀ξ∈,∀θ∈[, π],∀t∈[,T].
Let us recall an approximation property of the projection on Kh(ξ,θ,t)×Mh(ξ,θ,t)
(see []).
Lemma . Suppose thatV∈K(ξ,θ,t)and P∈M(ξ,θ,t).Then there exists a unique couple(Vh,Ph)inKh(ξ,θ,t)×Mh(ξ,θ,t)such that
⎧ ⎨ ⎩
a(V–Vh,v) +b(v,P–Ph) = ∀v∈Kh(ξ,θ,t),
b(V–Vh,q) = ∀q∈Mh(ξ,θ,t).
()
In addition,if we suppose thatV|\S(ξ,θ,t)∈H(\S(ξ,θ,t)) and P|\S(ξ,θ,t)∈H(\
S(ξ,θ,t)),then there exists a positive constant C,independent of h,such that
V–VhL()≤Ch.
In order to define the approximate characteristics, let us denote byFhtheP-finite
ele-ments space associated with the triangulationTh, and we introduce the space
Rh(ξ,θ,t) = ∇⊥φh:φh∈Fh,φh= on∂
∩K(ξ,θ,t),
∀ξ∈,∀θ∈[, π],∀t∈[,T], where∇⊥φh= –∂φh
∂y
∂φh ∂x
.
We denote byP(ξ,θ,t) the orthogonal projection fromL() ontoR
h(ξ,θ,t),i.e., for
anyu∈L(), then the projectionP(ξ,θ,t)u∈Rh(ξ,θ,t) is such that (u–P(ξ,θ,t)u,rh) =
for allrh∈Rh(ξ,θ,t).
LetNbe a positive integer. We denotet=T/Nandtk=ktfor allk∈ {, . . . ,N}. For
k= , we define
uh(·) =u(·, ), ξh=ξ and θh=θ, ()
where (u(·, ),p(·, ))∈Kh(ξ,θ, )×Mh(ξ,θ, ) is the projection of the initial
Assume that the approximate solution (ukh,pkh,ξhk,θhk) of ()-() at timet=tkis known.
We describe below the numerical scheme allowing to determinate the approximate solu-tion (uk+
h ,pk+h ,ξ k+
h ,θhk+) att=tk+. First, we computeξ k+
h ∈Randθhk+∈Rby
ξk+h =ξkh+uhkξkht, ()
θhk+=θhk+ t I(tk)
Sk
ρhkukh(x) –ukhξkh·x–ξkh⊥dx, ()
whereρk
h is defined by the identity () below.
We consider the approximated characteristic functionχk
hdefined as the solution of the
following ordinary differential equation:
⎧ ⎨ ⎩ d dtχ k
h(t;tk+,x) =P(ξkh,θhk,tk)ukh(χkh(t;tk+,x)), t∈[tk,tk+],
χk
h(tk+;tk+,x) =ξ k
h+Rθhk(tk;tk+,R–θhk+(x–ξ
k+ h )), () with uk h(z) =u
k h(z) –ukh
ξkh–θ
k+ h –θhk
t
z–ξkh⊥–hk(z) ∀z∈R,
whereukh(z) is extended by zero outside ofand
kh(x) =Rθk
h
∗R –θhk
x–ξkh,tk ∀x∈R. The characteristic functionis defined by ().
Finally, we define
Xkh(x) =χkhtk;tk+,x ∀x∈. () In the sequel, we shall split the mesh into the union of different types of triangle sub-sets. We first introduceAhas the union of all triangles intersecting the solidS(ξkh,θhk,tk),
i.e.,
Ah=
T∈Th
◦
T∩S◦(ξkh,θhk,tk)=∅
T.
We also denote byQh the union of all triangles such that all their vertices are contained
inAh. The triangles ofThare then split into the following four categories (see Figure ): • Fis the subset ofThformed by all trianglesT∈Thsuch thatT⊂S(ξkh,θhk,tk). • Fis the subset formed by all trianglesT∈Th\Fsuch thatT⊂Qh.
• Fis the subset formed by all trianglesT∈Thsuch thatT∩Qh=∅andT⊂Qh. • F=Th\(F∪F∪F).
We introduce the approximated density functionρk
has follows:
ρhk(x) =
⎧ ⎨ ⎩
ρ∗S(R–θk h(x–ξ
k
h),tk) ifx∈S(ξ k h,θhk,tk), ρF ifx∈\S(ξhk,θhk,tk).
Figure 1 In this figure, we see the splitting of the fixed triangulation related to the position of the solid at timet.
With these notations, we introduce the following mixed variational fully discrete for-mulation: Finduk+h ∈k+h +Kh(ξk+h ,θhk+,tk+),pk+h ∈Mh(ξk+h ,θhk+,tk+) such that
ρhk+u
k+
h –ukh◦X k h t ,v
+auk+h ,v+bv,pk+h =ρhk+fhk+,v ∀v∈Kh
ξk+h ,θhk+,tk+, ()
buk+h ,q= ∀q∈Mh
ξk+h ,θhk+,tk+, ()
wherefk+
h is theL()-projection offk+=f(tk+) on (Eh).
Let us now state the second main result of this paper, which asserts the convergence of the fully-discrete scheme ()-(). The complete proof of this result could be found in [] for the case of rigid body and in the forthcoming paper [] if the structure is deformable.
Theorem . Letbe a convex domain with a polygonal boundary.Suppose thatfand
usatisfy the conditions from(),and that(u,p,ξ,θ)is a solution of ()-()satisfying
regularity properties()-().Let C> and <α≤be two fixed constants.Then there
exist two positive constants K andτ∗,independent of h andt such that for all <t≤τ∗
and for all h≤Ct+α,we have
sup ≤k≤N
utk–ukh
L()+ξ
tk–ξkh+θtk–θhk≤Ktα.
Let us mention that in order to get an approximation of first order in time (i.e.,O(t) in Theorem .), we have to chooseα= . In this case, the corresponding condition onh becomesh≤Ctwhich is similar to the one obtained in [, Theorem .], where the
densities of the fluidρF and of the solidρSare equal.
under conditionh≤Ctand [], wherehandtare chosen such that h≤Ct≤
Chσ andσ> / (withhandtsmall enough). We also mention [] for an ALE scheme
applied to Stokes equations in a time-dependent domain, where the authors obtain an error estimate of orderO(t) under conditionh≤Ct/.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have contributed equally to this research work. All authors read and approved the final manuscript.
Author details
1Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile and Centro
de Modelamiento Matemático, UMR 2071 CNRS-UChile, Casilla 170/3-Correo 3, Santiago, Chile. 2Institut Elie Cartan de Nancy UMR 7502, Université de Lorraine, CNRS, INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France.3Department of Mathematics and Computer Science, Faculty of Mathematics and Computer Science, University of Pite¸sti, Str. Târgu din Vale nr. 1, Pite¸sti, 110040, Romania.
Acknowledgements
San Martín was partially supported by the Grant Fondecyt 1090239 and BASAL-CMM Project. Scheid gratefully acknowledges the Program ECOS-CONICYT (Scientific cooperation project between France and Chile) through the grant C07-E05. He was also partially supported by the ‘Agence Nationale de la Recherche’ (ANR), the Project CISIFS, the grant ANR-09-BLAN-0213-02. Smaranda was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-RU-TE-2011-3-0059.
Received: 30 June 2013 Accepted: 10 September 2013 Published:19 Nov 2013
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