A finite element solution of an added mass formulation for

coupled fluid-solid vibrations

Alfredo Bermudez,∗ Rodolfo Rodrıguez† and Duarte Santamarina‡

Abstract

A finite element method to approximate the vibration modes of a structure incontact with an incompressible fluid is analyzed in this paper. The effect of the fluid istaken into account by means of an added mass formulation, which is one of the mostusual procedures in engineering practice. Gravity waves on the free surface of the liquidare also considered in the model.

Piecewise linear continuous elements are used to discretize the solid displacements,the variables to compute the added mass terms and the vertical displacement of thefree surface, yielding a non conforming method for the spectral coupled problem.Error estimates are settled for approximate eigenfunctions and eigenfrequencies.Implementation issues are discussed and numerical experiments are reported. Inparticular the method is compared with other numerical scheme, based on a puredisplacement formulation, which has been recently analyzed.

1 Introduction

In this paper we analyze a finite element method for the numerical solution of a spectralproblem arising in fluid-solid interactions. It concerns the numerical computation ofharmonic hydroelastic vibrations under gravity, i.e., harmonic vibrations of a coupledsystem consisting of an elastic solid in contact with an incompressible fluid; gravity effectsare included on the free surface of the fluid in order to allow for the so-called sloshing

modes.A first possibility to solve this problem is to consider a formulation in terms of

displacements in the solid and pressure in the fluid (see [13]). However, such approachleads to unsymmetric eigenvalue problems, which is an inconvenient from the numericalpoint of view.

Another approach has been recently introduced in [4] (see also [2] and [3]). It is basedon using displacements also in the fluid, discretized by lowest degree Raviart-Thomas

finite elements on a triangular mesh. Interface coupling between this discretization andclassical piecewise linear finite elements for the displacements in the solid is achieved in anonconforming way. Error estimates have been obtained and it has been proved that nospurious modes arise as it is typical in other discretizations of this formulation (see [10]).This method works in two or three dimensions. Furthermore, in the two-dimensional case

∗Departamento de Matematica Aplicada, Universidade de Santiago de Compostela, 15706 Santiago deCompostela, Spain.

†Departamento de Ingenierıa Matematica, Universidad de Concepcion, Casilla 4009, Concepcion, Chile.Partially supported by FONDECYT (Chile) through its grant No. 1.960.615 and Program A on NumericalAnalysis of FONDAP in Applied Mathematics.

‡Departamento de Matematica Aplicada, Universidade de Santiago de Compostela, 15706 Santiago deCompostela, Spain. Partially supported by Program C of FONDAP in Applied Mathematics.

1

2

it can be conveniently implemented by replacing the fluid displacement field by the curl of astream function. This strategy allows a significant saving of computational effort since onlya scalar magnitude (the stream function) needs to be discretized. However this techniquedoes not extend to three-dimensional problems.

Since the fluid is incompressible, one of the most usual procedures is to eliminate thefluid variables by using the so-called added mass formulation (see, for instance, [13] or[12]). This consists of taking into account the effect of the fluid by means of a Neumann-to-

Dirichlet operator (also called Steklov-Poincare operator) on the fluid-solid interface. Thisoperator can be furtherly discretized by means of boundary elements or finite elements.

In the present paper we propose and analyze a finite element method for the numericalsolution of this added mass formulation. We perform this analysis in the two-dimensionalcase, but the method can be readily extended to three-dimensional problems. In both casesit leads to convenient symmetric eigenvalue problems and does not involve vector variablesfor the fluid.

We start by giving a weak formulation of the spectral problem which allows us tocharacterize its solutions and to state their regularity properties. Error estimates areobtained both for eigenvectors and eigenvalues. An alternative formulation of the discretespectral problem, convenient from the computational point of view, is described and provedto be equivalent to the one theoretically analyzed. Finally numerical results for a testexample are presented and compared with those included in [4].

2 The model problem

We consider the problem of determining the vibration modes of a linear elastic structurecontaining an inviscid fluid. Our model problem consists of a 2D polygonal vessel filledwith a fluid with an open boundary as that in Figure 1.

·············································

·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ··

·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ···

-n

-ν

ΓO

ΩF

ΩS

ΓD

ΓN

ΓI

Fig. 1. Fluid and solid domains.

Let ΩF

and ΩS

be the domains occupied by the fluid and the solid, respectively, whichare not supposed to be either convex or simply connected; even interior angles of 2π areallowed. Let us denote by Γ

Fthe boundary of the fluid domain and by ν its unit normal

vector pointing outwards ΩF. This boundary is splitted into two parts: the interface

between solid and fluid ΓI

and the open boundary of the fluid ΓO. On the other hand,

the solid boundary is the union of three parts: the interface ΓI, Γ

Dand Γ

N; the structure

is supposed to be fixed along ΓD

(meas (ΓD) > 0) and free along Γ

N. Finally n denotes the

unit outward normal vector along ΓN.

Throughout this paper we use standard notation for Sobolev spaces, norms and

3

seminorms. We denote by C a generic constant not necessarily the same at each occurrence.We use the following notations for the physical magnitudes in the fluid:

u: the displacement vector,

p: the pressure,

ρF: the density,

and in the solid:v: the displacement vector,

ρS: the density,

λS

and µS: the Lame coefficients,

ε(v): the strain tensor defined by εij(v) := 12

(∂vi∂xj

+∂vj

∂xi

), i, j = 1, 2,

σ(v): the stress tensor, which we assume related to the strains by Hooke’s law:

σij(v) = λS

2∑

k=1

εkk(v)δij + 2µSεij(v), i, j = 1, 2.

We are interested in the small amplitude motions departing from the rest. The classicallinearization procedure yields the following eigenvalue problem for the vibration modes ofthe coupled system and their corresponding frequencies ω (see for instance [12]):

Find ω ≥ 0, u : ΩF−→ IR2, v : Ω

S−→ IR2 and p : Ω

F−→ IR, (u,v, p) 6= 0, such that:

∇p− ω2ρFu = 0, in Ω

F,(2.1)

div u = 0, in ΩF,(2.2)

div [σ(v)] + ω2ρSv = 0, in Ω

S,(2.3)

u · ν − v · ν = 0, on ΓI,(2.4)

σ(v)ν + pν = 0, on ΓI,(2.5)

ρFg u · ν − p = 0, on Γ

O,(2.6)

σ(v)n = 0, on ΓN,(2.7)

v = 0, on ΓD.(2.8)

The coupling between the fluid and the structure is taken into account by equations(2.4) and (2.5). The first one means that fluid and solid are in contact at the interface.The second one relates normal stresses of the solid on the interface with the pressure of thefluid.

Let us introduce a new variable, η := u · ν on ΓO, for the vertical displacement of

the free surface. Then from (2.1), (2.2), (2.4) and (2.6) the fluid displacements u can beeliminated and we obtain instead of these equations:

−∆p = 0, in ΩF,(2.9)

∂p

∂ν

= ω2ρFv · ν, on Γ

I,(2.10)

∂p

∂ν

= ω2ρFη, on Γ

O,(2.11)

p = ρFg η, on Γ

O.(2.12)

4

3 Variational formulation

Let X denote the product space defined by

X := H1Γ

D(Ω

S)2 × L2(Γ

O),

with H1Γ

D(Ω

S)2 := w ∈ H1(Ω

S)2 : w|Γ

D= 0, endowed with the usual product norm

‖(w, ξ)‖ :=[‖w‖2

H1(ΩS)2 + ‖ξ‖2

L2(ΓO

)

]1/2.

Let

W :=

ψ ∈ H1(ΩF) :

∫

ΓF

ψ dΓ = 0

.

To give a weak formulation of the spectral problem above, we first multiply (2.3) bya test function w ∈ H1

ΓD(Ω

S)2 and then integrate in Ω

S. By using a Green formula and

taking into account (2.5) and (2.7) we obtain

∫

ΩS

σ(v) : ε(w) −∫

ΓI

pw · ν dΓ = ω2∫

ΩS

ρSv · w.(3.1)

Similarly, by multiplying (2.9) by a test function ψ ∈ H1(ΩF), integrating in Ω

Fand

using a Green formula, (2.11) and (2.10), we obtain

∫

ΩF

∇p · ∇ψ = ω2

(∫

ΓI

ρFv · ν ψ dΓ +

∫

ΓO

ρFη ψ dΓ

)

.(3.2)

Now we are going to eliminate p in (3.1) by using (3.2). For this purpose let us denoteby L2

0(ΓF) the space

L20(ΓF

) :=

χ ∈ L2(ΓF) :

∫

ΓF

χ dΓ = 0

.

Since in the sequel we shall often use functions in L20(ΓF

) which are defined differently on ΓI

and ΓO, we will also denote them by means of their pairs of restrictions to Γ

Iand Γ

O(i.e.,

we identify L20(ΓF

) with (χ1, χ2) ∈ L2(ΓI) × L2(Γ

O) :

∫Γ

I

χ1 dΓ +∫Γ

O

χ2 dΓ = 0). For

instance, because of (2.2), (2.4) and the definition of η, the solution of Problem (2.1)-(2.8)satisfies the following “conservation of volume” constraint:

∫

ΓI

v · ν dΓ +

∫

ΓO

η dΓ = 0;

then we write (v · ν, η) ∈ L20(ΓF

).We denote by M the Neumann-to-Dirichlet operator for the Laplace equation on the

domain ΩF; i.e., given χ ∈ L2

0(ΓF), let φχ denote the unique solution in W of the problem

∫

ΩF

∇φχ · ∇ψ =

∫

ΓF

χψ dΓ, ∀ψ ∈W,(3.3)

and let M(χ) := φχ|ΓF. Then M is a bounded linear operator from L2

0(ΓF) into itself.

5

From the definition of M, (3.3) and (3.2), we deduce that there exists a constant c, tobe determined, such that p = ω2ρ

F[M(v · ν, η) + c] and hence (3.1) can be rewritten as

follows:∫

ΩS

σ(v) : ε(w) = ω2

∫

ΩS

ρSv · w +

∫

ΓI

ρF

[M(v · ν, η) + c] w · ν dΓ

.

Now we replace in (2.12) p by ρFω2 [M(v · ν, η) + c] then multiply by a test function

ξ ∈ L2(ΓO) and integrate to obtain:

∫

ΓO

ρFgηξ dΓ = ω2

∫

ΓO

ρF

[M(v · ν, η) + c] ξ dΓ.

We can summarize the weak formulation of our spectral problem as follows:

To find ω ≥ 0 and 0 6= (v, η, c) ∈ X × IR such that:

∫

ΩS

σ(v) : ε(w) +

∫

ΓO

ρFgηξ dΓ = ω2

∫

ΩS

ρSv · w(3.4)

+

∫

ΓI

ρF

[M(v · ν, η) + c] w · ν dΓ +

∫

ΓO

ρF

[M(v · ν, η) + c] ξ dΓ

, ∀(w, ξ) ∈ X,

∫

ΓI

v · ν dΓ +

∫

ΓO

η dΓ = 0.(3.5)

Constant c above can be considered as a Lagrange multiplier associated with theconstraint (3.5), which in its turn comes from the incompressibility of the fluid. Thisconstraint can be directly imposed in the space yielding another weak formulation whichwill be useful for the analysis to be done in the next section. Let V be the closed subspaceof X defined by

V :=

(w, ξ) ∈ X :

∫

ΓI

w · ν dΓ +

∫

ΓO

ξ dΓ = 0

.

Clearly any solution of (3.4)-(3.5) also solves the following spectral problem:

To find ω ≥ 0 and 0 6= (v, η) ∈ V such that:

∫

ΩS

σ(v) : ε(w) +

∫

ΓO

ρFgηξ dΓ = ω2

[∫

ΩS

ρSv · w(3.6)

+

∫

ΓI

ρFM(v · ν, η)w · ν dΓ +

∫

ΓO

ρFM(v · ν, η)ξ dΓ

]

, ∀(w, ξ) ∈ V.

In fact, for (w, ξ) ∈ V, the constant c disappears because∫Γ

I

w · ν dΓ +∫Γ

O

ξ dΓ = 0.

Actually both spectral problems are equivalent:

Proposition 3.1. ω ≥ 0 and 0 6= (v, η) ∈ V satisfy (3.6) if and only if there existsc ∈ IR such that (3.4) and (3.5) hold true. Moreover c is given by

c =1

ω2 meas (ΓO)

(∫

ΓO

gη dΓ − ω2∫

ΓO

M(v · ν, η) dΓ

)

.

6

Proof. Let ω ≥ 0 and 0 6= (v, η) ∈ V satisfing (3.6). Since (v, η) ∈ V, (3.5) holds. Onthe other hand, X = V ⊕ 〈(0, 1)〉, because

∫Γ

O

1 dΓ = meas (ΓO) 6= 0. For (w, ξ) ∈ V

(3.4) holds for any c ∈ IR. Thus we only need to prove that there is a constant c such that(3.4) is satisfied for (w, ξ) = (0, 1). By using this test function in (3.4) we see that this isachieved if and only if

∫

ΓO

ρFgη dΓ = ω2

∫

ΓO

ρF

[M(v · ν, η) + c] dΓ,

from which the result follows.The other implication was shown above.

4 Characterization of the spectrum and a priori estimates

Let us now consider the following continuous bilinear forms from V × V −→ IR:

a((v, η), (w, ξ)

):=

∫

ΩS

σ(v) : ε(w) +

∫

ΓO

ρFgηξ dΓ,

b((v, η), (w, ξ)

):=

∫

ΩS

ρSv · w +

∫

ΓI

ρFM(v · ν, η)w · ν dΓ +

∫

ΓO

ρFM(v · ν, η)ξ dΓ.

The first one is symmetric and, by Korn’s inequality, coercive on X and hence on V; i.e.,there exists a positive constant α such that

a((v, η), (v, η)

)≥ α ‖(v, η)‖2, ∀(v, η) ∈ V.(4.1)

Regarding the second one we have the following result:

Proposition 4.1. The bilinear form b is symmetric and positive definite.Proof. The proposition is a direct consequence of the fact that the Neumann-to-Dirichletoperator M is self-adjoint and positive definite with respect to the L2(Γ

F) inner product.

In fact, for χ1, χ2 ∈ L20(ΓF

), let φχ1 , φχ2 ∈ W be the respective solutions of problem (3.3).Hence M(χi) = φχi |Γ

F, i = 1, 2, and

∫

ΓF

M(χ1)χ2 dΓ =

∫

ΓF

φχ1χ2 dΓ =

∫

ΩF

∇φχ1 · ∇φχ2 =

∫

ΓF

χ1φχ2 dΓ =

∫

ΓF

χ1M(χ2) dΓ.

Furthermore, for any χ ∈ L20(ΓF

)∫

ΓF

M(χ)χ dΓ =

∫

ΩF

|∇φχ|2 ≥ 0,(4.2)

and the equality holds only if φχ is constant which, together with∫Γ

F

φχ = 0, imply φχ = 0

and hence χ = 0.

The previous proposition shows that b(·, ·) is an inner product on V. Hence it definesa norm on this space that we denote | · |. Furthermore, because of (4.2), the followingcharacterization holds

|(v, η)|2 := b((v, η), (v, η)

)=

∫

ΩS

ρS|v|2 +

∫

ΩF

ρF

∣∣∣∇φ(v·ν,η)∣∣∣2,(4.3)

with φ(v·ν,η) ∈W denoting, as above, the solution of (3.3) for χ = (v · ν, η).

7

In order to analyze our spectral problem let us introduce the following bounded linearoperator:

T : V −→ V

(f , ζ) 7−→ (v, η)

with (v, η) being the solution of the elliptic problem

a((v, η), (w, ξ)

)= b

((f , ζ), (w, ξ)

), ∀(w, ξ) ∈ V.(4.4)

Then T is self-adjoint and positive definite with respect to a and b. Hence all of itseigenvalues are real and positive.

On the other hand, (λ, (v, η)) is an eigenpair of T if and only if ω = 1√λ

and (v, η) are

solution of (3.6). Therefore, the knowledge of the spectrum of T gives complete informationabout the solutions of our original problem.

In a similar way to Proposition 3.1 we can prove the following result:

Proposition 4.2. Let (v, η) = T(f , ζ). Then for

c =1

meas (ΓO)

(∫

ΓO

gη dΓ −∫

ΓO

M(f · ν, ζ) dΓ

)

,(4.5)

it holds:∫

ΩS

σ(v) : ε(w) +

∫

ΓO

ρFgηξ dΓ =

∫

ΩS

ρSf · w(4.6)

+

∫

ΓI

ρF

[M(f · ν, ζ) + c] w · ν dΓ +

∫

ΓO

ρF

[M(f · ν, ζ) + c] ξ dΓ, ∀(w, ξ) ∈ X.

We have the following a priori estimate for T(V):

Proposition 4.3. There exist constants t ∈ (0, 1] and C > 0 such that if (v, η) =T(f , ζ) with (f , ζ) ∈ V, then v ∈ H1+t(Ω

S)2, η ∈ H1/2(Γ

O) and

‖v‖H1+t(ΩS)2 + ‖η‖H1/2(Γ

O) ≤ C |(f , ζ)|.

Proof. Firstly, by taking (w, ξ) = (v, η) in (4.4), and using (4.1) and Cauchy-Schwartzinequality, we deduce

‖(v, η)‖ ≤1

α|(f , ζ)|.(4.7)

On the other hand, let φ(f ·ν,ζ) ∈W be the solution of (3.3) corresponding to χ = (f · ν, ζ);hence M(f · ν, ζ) = φ(f ·ν,ζ)|Γ

F∈ H1/2(Γ

F). Moreover, since in W the H1(Ω

F) norm is

equivalent to the norm in H1(ΩF)/IR, we have

‖M(f · ν, ζ)‖H1/2(ΓF

) ≤ C ‖φ(f ·ν,ζ)‖H1(ΩF

) ≤ C ‖∇φ(f ·ν,ζ)‖L2(ΩF

) ≤ C |(f , ζ)|,(4.8)

the latter because of the characterization (4.3) of this norm.Now, because of Proposition 4.2, we may test equation (4.6) separately with (w, 0) and

(0, ξ), for any w ∈ H1Γ

D(Ω

S)2 and ξ ∈ L2(Γ

O). So we obtain that v is solution (in the sense

of distributions) of the following elasticity problem:

−div [σ(v)] = ρSf , in Ω

S,

σ(v)ν = −ρF

[M(f · ν, ζ) + c]ν, on ΓI,

σ(v)n = 0, on ΓN,

v = 0, on ΓD,

8

and thatgη = M(f · ν, ζ) + c, on Γ

O,(4.9)

with c given in both by (4.5) and, hence, because of (4.7) and (4.8),

|c| ≤ C |(f , ζ)|.(4.10)

Therefore, from well known regularity results (see [9]), we have that there exist t ∈ (0, 1]such that v ∈ H1+t(Ω

S)2 with

‖v‖H1+t(ΩS)2 ≤ C

[‖f‖L2(Ω

S)2 + ‖ρ

F[M(f · ν, ζ) + c] ‖H1/2(Γ

I)

]≤ C |(f , ζ)|.

On the other hand, (4.9), (4.8), (4.10), shows that η ∈ H1/2(ΓO) with

‖η‖H1/2(ΓO

) ≤ C |(f , ζ)|,

therefore concluding the proof.

Now we are able to characterize the spectrum of the operator T.

Theorem 4.1. The spectrum of T consists of 0 and a sequence of strictly positive finitemultiplicity eigenvalues λn : n ∈ IN converging to 0.Proof. It is an immediate consequence of Proposition 4.3, the compact inclusion of[H1+t(Ω

S)2 × H1/2(Γ

I)] ∩ V into V (for t > 0) and the self-adjointness and positive

definiteness of T.

The following proposition states a priori estimates for the solution of problem (3.3)depending on the regularity of its data:

Proposition 4.4. For χ ∈ L20(ΓF

), let φχ be the solution of problem (3.3). There existconstants s ∈ [1/2, 1] and C > 0, such that φχ ∈ H1+s/2(Ω

F) and

‖φχ‖H1+s/2(ΩF

) ≤ C ‖χ‖L2(ΓF

).(4.11)

Furthermore, if χ ∈ H1/2(Γj), for Γj (j = 0, . . . , J) being the edges of ΓF, then φχ ∈

H1+s(ΩF) and

‖φχ‖H1+s(ΩF

) ≤ CJ∑

j=0

‖χ‖H1/2(Γj).(4.12)

Proof. We first prove (4.12). In this case φχ satisfies (strongly)

−∆φχ = 0, in ΩF,

∂φχ

∂ν= χ, on Γ

F.

Therefore, from classical regularity results (see [9]), we know that φχ ∈ H1+s(ΩF) and

(4.12) hold with s ∈ [1/2, 1] (in fact, s = 1 if ΩF

is convex and s = π/θ, with θ the largestreentrant angle, otherwise).

Now, for χ ∈ H−1/2(ΓF) such that 〈χ, 1〉H−1/2(Γ

F)×H1/2(Γ

F) = 0, problem (3.3) is well

defined and, in that case, φχ ∈ H1(ΩF) with ‖φχ‖H1(Ω

F) ≤ C ‖χ‖H−1/2(Γ

F). Therefore

(4.11) can be obtained by interpolation of Sobolev spaces (see for instance [5]).

9

The previous proposition allows us to prove further regularity for the eigenfunctions ofour problem, which will be used to obtain improved error estimates for the finite elementmethod to be given in the next section.

Proposition 4.5. Let (f , ζ) be an eigenfunction of T. Then f ∈ H1+t(ΩS)2, ζ ∈

H1/2+s(ΓO) and

‖f‖H1+t(ΩS)2 + ‖ζ‖H1/2+s(Γ

O) ≤ C |(f , ζ)|,(4.13)

with t and s as in Propositions 4.3 and 4.4, respectively.Proof. Let (v, η) = T(f , ζ) = λ(f , ζ). Since λ > 0, Proposition 4.3 implies

ζ =1

λη ∈ H1/2(Γ

O), with ‖ζ‖H1/2(Γ

O) ≤ C |(f , ζ)|

and

f =1

λv ∈ H1+t(Ω

S)2, with ‖f‖H1+t(Ω

S)2 ≤ C |(f , ζ)|.

Therefore, f |Γj · ν ∈ H1/2(Γj), for the edges Γj of ΓI(j = 1, . . . , J). So (4.12) holds for the

solution φ(f ·ν,ζ) of (3.3) corresponding to χ = (f · ν, ζ) and

‖φ(f ·ν,ζ)‖H1+s(ΩF

) ≤ C

J∑

j=1

‖f · ν‖H1/2(Γj)+ ‖ζ‖H1/2(Γ

O)

≤ C |(f , ζ)|.

Thus, (4.9) and (4.10) allow us to conclude the proof.

5 Finite element discretization

LetT F

h

and

T S

h

be two families of regular triangulations of Ω

Fand Ω

S, respectively.

The meshes do not need to have coincident nodes on their common interface ΓI, but we

assume that the end points of ΓD, Γ

N, Γ

Iand Γ

Ocoincide with nodes of the corresponding

triangulations.In order to compute the operator M we have to solve problem (3.3). For this we

use standard piecewise linear finite elements. Let Lh(ΩF) denote the space of continuous

piecewise linear functions on T Fh and let

Wh :=

ψh ∈ Lh(ΩF) :

∫

ΓF

ψh dΓ = 0

⊂W.

We define an approximate operator Mh : L20(ΓF

) −→ L20(ΓF

) by Mh(χ) = φχh|ΓF

, where φχh

is the unique solution in Wh of the discrete problem∫

ΩF

∇φχh · ∇ψh =

∫

ΓF

χψh dΓ, ∀ψh ∈Wh.(5.1)

Proposition 5.1. For χ ∈ L20(ΓF

), let φχ and φχh be the solutions of problems (3.3)

and (5.1), respectively. Then, for s ∈ [1/2, 1] as in Proposition 4.4,

‖φχ − φχh‖H1(Ω

F) ≤ C hs/2‖χ‖L2

0(Γ

F).(5.2)

10

Furthermore, if χ ∈ H1/2(Γj) for each edge Γj of ΓF

(j = 0, . . . , J), then

‖φχ − φχh‖H1(Ω

F) ≤ C hs

J∑

j=0

‖χ‖H1/2(Γj).(5.3)

Proof. It is a direct consequence of standard finite element error estimates and Proposition4.4.

Now we define an approximation of T. Let Lh(ΩS) be the space of continuous piecewise

linear functions on T Sh . Let Lh(Γ

O) be the space of restrictions to Γ

Oof functions in Lh(Ω

F).

Let Vh be the finite dimensional subspace of V defined by

Vh :=

(wh, ξh) ∈ Lh(ΩS)2 × Lh(Γ

O) : wh|Γ

D= 0 and

∫

ΓI

wh · ν dΓ +

∫

ΓO

ξh dΓ = 0

.

The following approximation property holds:

Proposition 5.2. There exists a linear operator Ih : V −→ Vh such that, ifv ∈ H1+t(Ω

S)2 and η ∈ Hq(Γ

O) for some t, q ∈ (0, 1], then

‖Ih(v, η) − (v, η)‖ ≤ C hr[‖v‖H1+t(Ω

S)2 + ‖η‖Hq(Γ

O)

],

with r = mint, q.Proof. Let vh be a Clement interpolant of v in Lh(Ω

S)2 such that vh|Γ

D= 0 (see [7]).

Let ηh be the L2(ΓO)-orthogonal projection of η onto the subspace of continuous piecewise

linear functions Lh(ΓO). Then standard error estimates yield

‖vh − v‖H1(ΩS)2 ≤ C ht‖v‖H1+t(Ω

S)2 ,(5.4)

‖ηh − η‖L2(ΓO

) ≤ C hq‖η‖Hq(ΓO

).(5.5)

On the other hand∫Γ

O

ηh dΓ =∫Γ

O

η dΓ = −∫Γ

I

v · ν dΓ but, in general, (vh, ηh) /∈ Vh.

Let ηh = ηh + ch with ch chosen such that

∫

ΓI

vh · ν dΓ +

∫

ΓO

ηh dΓ = 0.

Then (vh, ηh) ∈ Vh and furthermore

ch = −1

meas (ΓO)

(∫

ΓI

vh · ν dΓ +

∫

ΓO

ηh dΓ

)

= −1

meas (ΓO)

∫

ΓI

(vh − v) · ν dΓ.

Hence, by using (5.4), we have

|ch| ≤ C ‖vh − v‖H1(ΩS)2 ≤ C ht‖v‖H1+t(Ω

S)2 .(5.6)

Thus by defining Ih(v, η) := (vh, ηh), the proof follows from (5.4), (5.5) and (5.6).

Now let Th : V −→ V be the linear bounded operator given by Th(f , ζ) = (vh, ηh),with (vh, ηh) ∈ Vh being the solution of the discretized source problem

a((vh, ηh), (wh, ξh)

)= bh

((f , ζ), (wh, ξh)

), ∀(wh, ξh) ∈ Vh,(5.7)

11

where bh : V × V −→ IR is the bilinear form defined by

bh((v, η), (w, ξ)

):=

∫

ΩS

ρSv · w +

∫

ΓI

ρFMh(v · ν, η)w · ν dΓ +

∫

ΓO

ρFMh(v · ν, η)ξ dΓ.

Notice that (5.7) is a nonconforming approximation of (4.4) because b has been replacedby bh.

The operator Th is self-adjoint and positive definite with respect to a and bh. Henceall of its eigenvalues are real and positive. The following proposition implies uniformconvergence of Th to T as h goes to 0, and will allow us to use the spectral approximationtheory in [1]:

Proposition 5.3. There exists a constant C > 0 such that, for (f , ζ) ∈ V, it holds

‖(T − Th)(f , ζ)‖

|(f , ζ)|≤ C hr,

with r = mins/2, t, for t ∈ (0, 1] and s ∈ [1/2, 1] as in Propositions 4.3 and 4.4,respectively.

Furthermore, if (f , ζ) is an eigenfunction of T, then the previous inequality holds forr = min3s/2, t.Proof. For (f , ζ) ∈ V, let (v, η) = T(f , ζ) and (vh, ηh) = Th(f , ζ). Then, for any(v, η) ∈ Vh, by using (4.1), (4.4) and (5.7), we have

α‖(v − vh, η − ηh)‖2 ≤ a((v − vh, η − ηh), (v − vh, η − ηh)

)(5.8)

= a((v − v, η − η), (v − vh, η − ηh)

)

+ a((v − vh, η − ηh), (v − vh, η − ηh)

)

= a((v − v, η − η), (v − vh, η − ηh)

)

+ b((f , ζ), (v − vh, η − ηh)

)− bh

((f , ζ), (v − vh, η − ηh)

).

Let φ(f ·ν,ζ) and φ(f ·ν,ζ)h be the solutions of (3.3) and (5.1), respectively, for χ =

(f · ν, ζ). Let φ((v−vh)·ν,η−ηh) and φ((v−vh)·ν,η−ηh)h be the corresponding solutions for

χ = ((v − vh) · ν, η − ηh). Then, by applying (3.3), (5.1), (5.2) and (4.8), we have

b((f , ζ), (v − vh, η − ηh)

)− bh

((f , ζ), (v − vh, η − ηh)

)(5.9)

=

∫

ΓI

ρF

(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)(v − vh) · ν dΓ

+

∫

ΓO

ρF

(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)(η − ηh) dΓ

=

∫

ΩF

ρF∇(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)· ∇φ((v−vh)·ν,η−ηh)

=

∫

ΩF

ρF∇(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)· ∇

(φ((v−vh)·ν,η−ηh) − φ

((v−vh)·ν,η−ηh)h

)

=

∫

ΩF

ρF∇φ(f ·ν,ζ) · ∇

(φ((v−vh)·ν,η−ηh) − φ

((v−vh)·ν,η−ηh)h

)

≤ C hs/2‖∇φ(f ·ν,ζ)‖L2(ΩF

)2

[‖(v − vh) · ν‖2

L2(ΓI) + ‖η − ηh‖

2L2(Γ

O)

]1/2

≤ C hs/2|(f , ζ)| ‖(v − vh, η − ηh)‖.

12

Now we use this estimate and the continuity of a in (5.8) to obtain

‖(v − vh, η − ηh)‖ ≤ C‖(v − v, η − η)‖ + C hs/2|(f , ζ)|, ∀(v, η) ∈ Vh.(5.10)

Hence, by using (v, η) = Ih(v, η) and Propositions 5.2 and 4.3 we have

‖(v − vh, η − ηh)‖ ≤ ‖(v − v, η − η)‖ + ‖(v − vh, η − ηh)‖(5.11)

≤ C‖(v − v, η − η)‖ + C hs/2|(f , ζ)|

≤ C hmin1/2,t[‖v‖H1+t(Ω

S)2 + ‖η‖H1/2(Γ

O)

]+ C hs/2|(f , ζ)|

≤ C hmins/2,t|(f , ζ)|,

which proves the first part of the Proposition.Assume now that (f , ζ) is an eigenfunction associated with an eigenvalue λ of T. Then

from Propositions 5.1 and 4.5 we obtain

‖φ(f ·ν,ζ) − φ(f ·ν,ζ)h ‖H1(Ω

F) ≤ C hs

J∑

j=0

‖f · ν‖H1/2(Γj))+ ‖ζ‖H1/2(Γ

O)

≤ C hs|(f , ζ)|.

Therefore, instead of (5.9) we have in this case

b((f , ζ), (v − vh, η − ηh)

)− bh

((f , ζ), (v − vh, η − ηh)

)

=

∫

ΩF

ρF∇(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)· ∇

(φ((v−vh)·ν,η−ηh) − φ

((v−vh)·ν,η−ηh)h

)

≤ C h3s/2|(f , ζ)| ‖(v − vh, η − ηh)‖, ∀(v, η) ∈ Vh,

and by repeating the arguments in (5.10) and (5.11) we obtain

‖(v − vh, η − ηh)‖ ≤ C hmin3s/2,t|(f , ζ)|.

As a consequence of the proposition above, since the bilinear form b defining the norm| · | is continuous on V, then Th converge to T in norm ‖ · ‖ (actually also in | · |) and henceisolated parts of the spectrum of T are approximated by isolated parts of the spectrumof Th (see [11]). More precisely, for any eigenvalue λ of T of finite multiplicity m, there

exist exactly m eigenvalues λ(1)h , . . . , λ

(m)h of Th (repeated according to their respective

multiplicities) converging to λ as h goes to zero. Furthermore, no spurious modes canarise as it is typical in some other discretizations of spectral problems in fluid-structureinteraction (see for instance [10]). We state this result in the following theorem, where σ(·)denotes the spectrum of an operator:

Theorem 5.1. Let I be a closed interval such that I ∩σ(T) = ∅. There exists a strictlypositive constant hI such that if h ≤ hI then I ∩ σ(Th) = ∅.

From now on and until the end of this section let λ be a positive fixed eigenvalue ofT of finite multiplicity m and let E be its associated eigenspace. For h small enough, let

λ(1)h , . . . , λ

(m)h be the m eigenvalues of Th converging to λ and let Eh be the direct sum of

the corresponding eigenspaces. Thus, by applying the spectral approximation theory forcompact operators as stated in [1] (Theorem 7.1) and by using Proposition 5.3, we obtainthe following error estimates:

Theorem 5.2. There exist a strictly positive constant C, independent of h, such that

13

i) for each (vh, ηh) ∈ Eh, dist((vh, ηh),E

)≤ C hr‖(vh, ηh)‖,

ii) for each (v, η) ∈ E, dist((v, η),Eh

)≤ C hr‖(v, η)‖,

where dist denotes the distance in norm ‖ · ‖ and r := min3s/2, t, with t ∈ (0, 1] and

s ∈[

12 , 1]

as in Propositions 4.3 and 4.4, respectively.

Finally, regarding the eigenvalues, we are going to prove a theorem providing animproved order of convergence. For this purpose we will exploit the fact that our spectralproblem is “variationally formulated” in the sense of [1]. However the results includedin this reference cannot be directly applied to our case since (5.7) is a nonconformingapproximation of (4.4).

Let us denote by H the Hilbert space obtained as the completion of space V withrespect to the norm | · |. Then (V, ‖ · ‖) is continuously and densely included in (H, | · |).Thus the operator T can be uniquely extended to H and this extension is also selfadjointwith respect to b. Similarly, Th can also be extended to H, however this extension is notselfadjoint with respect to b. Indeed, let us call T

∗h its adjoint with respect to this inner

product, i.e.,

b(T

∗h(v, η), (w, ξ)

)= b

((v, η), Th(w, ξ)

), ∀(v, η), (w, ξ) ∈ H.

Then we have

b(T

∗h(v, η), (w, ξ)

)= a

(T(v, η), Th(w, ξ)

), ∀(v, η), (w, ξ) ∈ H,

whereas

b(Th(v, η), (w, ξ)

)= a

(Th(v, η), T(w, ξ)

), ∀(v, η), (w, ξ) ∈ H,

which, in general, do not coincide. Nevertheless a larger order of convergence can be provedfor the approximation of the eigenvalues:

Theorem 5.3. There exist a strictly positive constant C, independent of h, such that∣∣∣λ− λ

(i)h

∣∣∣ ≤ C h2r, i = 1, . . . ,m,

where r := mins, t, with t ∈ (0, 1] and s ∈[

12 , 1]

as in Propositions 4.3 and 4.4,

respectively.Proof. By specializing Theorem 7.3 of [1] to our situation we have that

|λ− λ(i)h | ≤ C

sup

(f ,ζ)∈E

|(f ,ζ)|=1

sup(f ,ζ)∈E

|(f ,ζ)|=1

∣∣∣b((T − Th)(f , ζ), (f , ζ)

)∣∣∣+∣∣∣(T − Th)|E

∣∣∣∣∣∣(T − T

∗h)|E

∣∣∣

,

for i = 1, . . . ,m. Proposition 5.3 provides a bound of∣∣∣(T − Th)|E

∣∣∣, so we only need

to estimate the two remaining terms in the expression above. For the first one, let(f , ζ), (f , ζ) ∈ E and let

(v, η) := T(f , ζ) ∈ V, (v, η) := T(f , ζ) ∈ V,

(vh, ηh) := Th(f , ζ) ∈ Vh, (vh, ηh) := Th(f , ζ) ∈ Vh.

14

Then

b((T − Th)(f , ζ), (f , ζ)

)= a

((v, η) − (vh, ηh), (v, η)

)(5.12)

= a((v − vh, η − ηh), (v − vh, η − ηh)

)

+ a((v − vh, η − ηh), (vh, ηh)

).

The first term in the r.h.s of this expression can be easily bounded by using againProposition 5.3:

∣∣∣a((v − vh, η − ηh), (v − vh, η − ηh)

)∣∣∣ ≤ ‖(v − vh, η − ηh)‖ ‖(v − vh, η − ηh)‖(5.13)

≤ C h2min3s/2,t|(f , ζ)| |(f , ζ)|,

whereas the second one can be written in the following way:

a((v − vh, η − ηh), (vh, ηh)

)(5.14)

= b((f , ζ), (vh, ηh)

)− bh

((f , ζ), (vh, ηh)

)

=[b((f , ζ), (vh − v, ηh − η)

)− bh

((f , ζ), (vh − v, ηh − η)

)]

+[b((f , ζ), (v, η)

)− bh

((f , ζ), (v, η)

)].

Let φ(f ·ν,ζ) and φ(f ·ν,ζ)h be the solutions of (3.3) and (5.1), respectively, for χ = (f ·ν, ζ).

Then, for the first term in the r.h.s. of the expression above we have[b((f , ζ), (vh − v, ηh − η)

)− bh

((f , ζ), (vh − v, ηh − η)

)](5.15)

=

∫

ΓI

ρF

(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)(vh − v) · ν dΓ

+

∫

ΓO

ρF

(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)(ηh − η) dΓ

≤ C‖φ(f ·ν,ζ) − φ(f ·ν,ζ)h ‖L2(Γ

F)

[‖(vh − v) · ν‖2

L2(ΓI) + ‖ηh − η‖2

L2(ΓO

)

]1/2

≤ C‖φ(f ·ν,ζ) − φ(f ·ν,ζ)h ‖H1(Ω

F)‖(vh − v, ηh − η)‖

≤ C hmin5s/2,s+t|(f , ζ)| |(f , ζ)|,

where we have used (5.3), (4.13) and Proposition 5.3 for the last inequality.

To estimate the second term in the r.h.s. of (5.14), let φ(v·ν,η) and φ(v·ν,η)h be the

solutions of (3.3) and (5.1), respectively, for χ = (v · ν, η). It holds[b((f , ζ), (v, η)

)− bh

((f , ζ), (v, η)

)](5.16)

=

∫

ΓI

ρF

(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)v · ν dΓ +

∫

ΓO

ρF

(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)η dΓ

=

∫

ΩF

ρF∇(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)· ∇φ(v·ν,η)

=

∫

ΩF

ρF∇(φ(f ·ν,ζ) − φ

(f ·ν,ζ)h

)· ∇

(φ(v·ν,η) − φ

(v·ν,η)h

)

≤ C h2s|(f , ζ)| |(f , ζ)|,

15

where we have now used (5.3), (4.13) and Proposition 4.3 for the last inequality.Thus, as a consequence of (5.12)-(5.16), it holds

sup(f ,ζ)∈E

|(f ,ζ)|=1

sup(f ,ζ)∈E

|(f ,ζ)|=1

∣∣∣b((T − Th)(f , ζ), (f , ζ)

)∣∣∣ ≤ C hmin2s,2t.

On the other hand, regarding∣∣∣(T − T

∗h)|E

∣∣∣, since V is dense in H we have

|(T − T∗h)|E| = sup

(f ,ζ)∈E

|(f ,ζ)|=1

sup(f ,ζ)∈V

|(f ,ζ)|=1

b((T − T

∗h)(f , ζ), (f , ζ)

)

= sup(f ,ζ)∈E

|(f ,ζ)|=1

sup(f ,ζ)∈V

|(f ,ζ)|=1

b((T − Th)(f , ζ), (f , ζ)

).

Then b((T − Th)(f , ζ), (f , ζ)

)can be handled as above. However, now (f , ζ) /∈ E; so, by

repeating the arguments above, we only obtain∣∣∣b((T − Th)(f , ζ), (f , ζ)

)∣∣∣ ≤ C hmins,t|(f , ζ)| |(f , ζ)|.

Nevertheless, since Proposition 5.3 yields∣∣∣(T − Th)|E

∣∣∣ ≤ C hmin3s/2,t, this is enough to

allow us to conclude the theorem.

6 Numerical computations

It was proved in the previous section that the spectrum and eigenfunctions of Th convergeto those of T as h goes to zero. Equivalently, the solutions of the spectral problem (3.6)are approximated by those of the following discrete eigenvalue problem:

To find ωh ≥ 0 and 0 6= (vh, ηh) ∈ Vh such that:∫

ΩS

σ(vh) : ε(wh) +

∫

ΓO

ρFgηhξh dΓ = ω2

h

[∫

ΩS

ρSvh · wh(6.1)

+

∫

ΓI

ρFMh(vh · ν, ηh)wh · ν dΓ +

∫

ΓO

ρFMh(vh · ν, ηh)ξh dΓ

]

, ∀(wh, ξh) ∈ Vh.

Let Zh := wh ∈ Lh(ΩS)2 : wh = 0 on Γ

O and Xh := Zh ×Lh(Γ

O) the corresponding

discretization of the space X. In a similar way as for the continuous case, it can be provedan analogous of Proposition 3.1, showing the equivalence of the eigenvalue problem aboveand the following discrete version of (3.4)-(3.5):

To find ωh ≥ 0 and 0 6= (vh, ηh, ch) ∈ Xh × IR such that:∫

ΩS

σ(vh) : ε(wh) +

∫

ΓO

ρFgηhξh dΓ = ω2

h

∫

ΩS

ρSvh · wh(6.2)

+

∫

ΓI

ρF

[Mh(vh · ν, ηh) + ch] wh · ν dΓ +

∫

ΓO

ρF

[Mh(vh · ν, ηh) + ch] ξh dΓ

,

∀(wh, ξh) ∈ Xh,∫

ΓI

vh · ν dΓ +

∫

ΓO

ηh dΓ = 0.(6.3)

16

Let Yh := Zh × Lh(ΩF) × Lh(Γ

O). We show below that, except for ωh = 0, the

solutions of (6.2)-(6.3) are exactly those of the following problem, which is the one that wehave actually implemented:

To find ωh ≥ 0 and 0 6= (vh, φh, ηh) ∈ Yh such that:

∫

ΩS

σ(vh) : ε(wh) +

∫

ΓO

ρFgηhξh dΓ = ω2

h

[∫

ΩS

ρSvh · wh(6.4)

+

∫

ΓI

ρFvh · ν ψh dΓ +

∫

ΓO

ρFηhψh dΓ −

∫

ΩF

ρF∇φh · ∇ψh

+

∫

ΓI

ρFφh wh · ν dΓ +

∫

ΓO

ρFφhξh dΓ

]

, ∀(wh, ψh, ξh) ∈ Yh.

More precisely we have the following result:

Proposition 6.1. Let ωh 6= 0.

1. If ωh and (vh, ηh, ch) are solution of (6.2)-(6.3), then ωh and (vh, φ(vh·ν,ηh)h + ch, ηh)

are solution of (6.4), with φ(vh·ν,ηh)h being the solution of (5.1) for χ = (vh · ν, ηh).

2. If ωh and (vh, φh, ηh) are solution of (6.4), then ωh and (vh, ηh, ch) are solution of(6.2)-(6.3), with

ch =1

meas (ΓF)

∫

ΓF

φh dΓ.(6.5)

Proof. 1. From (6.3), (vh · ν, ηh) ∈ L20(ΓF

). Then, φ(vh·ν,ηh)h is well defined in Wh and

φh = φ(vh·ν,ηh)h + ch satisfies

∫

ΩF

∇φh · ∇ψh =

∫

ΓI

vh · ν ψh dΓ +

∫

ΓO

ηhψh dΓ, ∀ψh ∈Wh.

Furthermore, because of (6.3), the equation above is valid for all ψh ∈ Lh(ΩF). Now, this

fact and (6.2) imply (6.4) since, by definition, φh|ΓF

= Mh(vh · ν, ηh) + ch.2. By taking wh = 0 and ξh = 0 in (6.4) we deduce

∫

ΩF

∇φh · ∇ψh =

∫

ΓI

vh · ν ψh dΓ +

∫

ΓO

ηhψh dΓ, ∀ψh ∈ Lh(ΩF).

By taking ψh = 1 in the equation above, we obtain (6.3). Furthermore, for ch as in (6.5),φh − ch is the solution of problem (5.1) for χ = (vh · ν, ηh). Hence

φh|ΓF

= Mh(vh · ν, ηh) + ch.

Finally, by taking ψh = 0 in (6.4) and using the previous equation we obtain (6.2).

As it was said above we have implemented the generalized eigenvalue problem (6.4).Let us write it down in matrix form. Let Υ, Φ, Θ, Ω, Ψ and Ξ denote the vectors of nodalcomponents of vh, φh, ηh, wh, ψh and ξh, respectively. The matrices associated with the

17

bilinear forms in the variational formulation (6.4) are defined by

ΩtK

SΥ =

∫

ΩS

σ(vh) : ε(wh), ΩtM

SΥ =

∫

ΩS

ρSvh · wh,

ΨtFΦ =

∫

ΩF

ρF∇φh · ∇ψh, Ξ

tK

OΘ =

∫

ΓO

ρFgηhξh dΓ,

ΞtBΦ =

∫

ΓO

ρFφhξh dΓ, Ψ

tCΥ =

∫

ΓI

ρFvh · ν ψh dΓ.

KS

and MS

are the standard stiffness and mass matrices of the solid, respectively. F isthe mass matrix corresponding to the fluid. On the other hand C is the coupling matrixbetween solid and fluid variables, and K

Oand B are the matrices associated with the the

vertical displacement of the free surface and its coupling with the fluid variable.Problem (6.4) is written in terms of these matrices in the following way:

K

S0 0

0 0 0

0 0 KO

Υ

Φ

Θ

= ω2h

M

SC

t0

C −F Bt

0 B 0

Υ

Φ

Θ

.

This is a well posed generalized eigenvalue problem with symmetric indefinite bandedmatrices. Notice that ωh = 0 is an eigenvalue of this problem with eigenspace 0 ×Lh(Ω

F) × 0.

Appart from ω2h = 0 it has a finite number of eigenvalues, which are exactly those of

Problem (6.1). Since this last problem involves symmetric positive definite matrices, thenumber of non zero eigenvalues is equal to the dimension of Vh (and all of them are strictlypositive).

This implies that for any value σ, different from all those eigenvalues, the matrix

K

S0 0

0 0 0

0 0 KO

− σ

M

SC

t0

C −F Bt

0 B 0

=

K

S− σM

S−σCt

0

−σC σF −σBt

0 −σB KO

is non singular and banded. Consequently any “shift and inverse” method could beconveniently used to solve this eigenproblem.

This finite element method was applied to the computation of the vibration modes ofa 2D steel vessel partially filled with water as that shown in Figure 1. We have chosen thesame problem as in [4] in order to be able to compare both methods. In that reference thesame elements as in the present method were used for the structure, whereas the fluid wasdescribed by means of displacement variables discretized by lowest order Raviart-Thomaselements and coupled with those of the solid by a piecewise constant Lagrange multiplyer.

We have used the following physical parameters for the steel: density ρS

= 7.7 ×103 Kg/m3, Young modulus E = 1.44 × 1011 Pa and Poisson ratio ν

S= 0.35 (the Lame

coefficients being λS

=Eν

S

(1+νS)(1−2ν

S) and µ

S= E

1+νS

). The water has been idealized as

perfectly incompressible with density ρF

= 103 Kg/m3.Figure 2 shows the geometrical data of the fluid and solid domains and the coarsest

used mesh. To estimate the order of convergence of the method, the vibration modesof this coupled problem have been computed using several uniform refinements of thistriangulation. The refinement parameter N denotes the number of layers of triangles in thesolid (the mesh in Figure 2 corresponds to N = 1).

18

@@@@@@@@@@@@@@@@@@@@@@@@@@

@@ @@

@

@@

@@@

@@

@@

@@

@@

@@@

@@

@ @@

?6?

6

?

6

0.125 m

1.000 m

0.500 m

- --0.125 m 1.000 m 0.125 m

Fig. 2. Steel vessel filled with water: geometrical data and initial mesh.

Two kind of vibration modes have been computed by solving eigenvalue problem (6.4):low frecuency sloshing modes and hydroelastic vibration modes. The first ones corrrespondto the gravity waves on the surface of the liquid, whereas the second ones are the vibrationmodes of the elastic vessel modified by the interaction with the liquid.

We show in Table 1 the frecuencies ωSi of the first three sloshing modes (i.e., those

corresponding to the three lowest vibration frecuencies) computed with different succesivelyrefined meshes. Since no exact solution is known, we have used extrapolation to obtain anaproximation of the order of convergence of the method. We also include in this table theresults obtained in [4] with the same meshes.

Table 1

Computed eigenfrecuencies (in rad/s) of the first sloshing modes.

Added mass formulation Raviart-Thomas discretization [4]Mode N = 2 N = 3 N = 4 Order N = 2 N = 3 N = 4 OrderωS

15.3371 5.3242 5.3196 1.95 5.2946 5.3052 5.3090 1.93

ωS2

7.9811 7.8987 7.8697 1.98 7.7334 7.7877 7.8071 1.94ωS

310.0205 9.7937 9.7135 1.97 9.3491 9.4898 9.5414 1.87

Figures 3 to 5 show the corresponding computed fluid displacement and pressure fieldsfor the mesh corresponding to N = 3.

Fluid displacement field. Fluid pressure field.

Fig. 3. Fluid displacement and pressure fields of the ωS1

sloshing mode.

19

Fluid displacement field. Fluid pressure field.

Fig. 4. Fluid displacement and pressure fields of the ωS2

sloshing mode.

Fluid displacement field. Fluid pressure field.

Fig. 5. Fluid displacement and pressure fields of the ωS3

sloshing mode.

In Table 2 we show the computed frecuencies ωHi of the four lowest frequency

hydroelastic vibration modes. Once more the orders of convergence have been estimatedby extrapolation and the results in [4] have been included for comparison.

Table 2

Computed eigenfrecuencies (in rad/s) of the first hydroelastic vibration modes.

Added mass formulation Raviart-Thomas discretization [4]Modo N = 2 N = 3 N = 4 Order N = 2 N = 3 N = 4 OrderωH

1628.534 540.599 503.494 1.48 627.891 540.042 502.706 1.46

ωH2

2016.100 1766.640 1656.802 1.36 2014.433 1765.769 1656.178 1.36ωH

33499.512 3150.599 2981.838 1.09 3487.239 3145.029 2978.579 1.08

ωH4

3816.917 3323.111 3107.463 1.38 3803.743 3317.306 3104.094 1.37

Figures 6 to 9 show the deformed structure and the fluid displacement and pressurefields for these hydroelastic vibration modes.

The results obtained with the method discussed in this paper are almost identical tothose obtained in [4], in which lowest degree Raviart-Thomas element were used to discretizea pure displacement formulation of the same problem. However, notice that the latter arecomputationally more expensive (specially in three-dimensional domains), since a vectorialfield (displacements) is used to describe the motion of the fluid. Instead, the added massformulation is based on discretizing a scalar field in the fluid domain. Hence, for a fixedmesh, the computational cost is less expensive.

20

Deformed structure.

Fluid displacement field.

Fluid pressure field.

Fig. 6. Hydroelastic vibration mode of frequency ωH1

.

Deformed structure.

Fluid displacement field.

Fluid pressure field.

Fig. 7. Hydroelastic vibration mode of frequency ωH2

.

References

[1] I. Babuska and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, Vol.

21

Deformed structure.

Fluid displacement field.

Fluid pressure field.

Fig. 8. Hydroelastic vibration mode of frequency ωH3

.

Deformed structure.

Fluid displacement field.

Fluid pressure field.

Fig. 9. Hydroelastic vibration mode of frequency ωH4

.

II, P. G. Ciarlet and J. L. Lions, eds., North Holland, Amsterdam, 1991.[2] A. Bermudez, R. Duran and R. Rodrıguez, Finite element analisys of compressible and

incompressible fluid-solid systems. Math. Comp., 67 (1998) 111-136.

22

[3] A. Bermudez, R. Duran and R. Rodrıguez, Finite element solution of incompressiblefluid-structure vibration problems. Internat. J. Num. Meth. in Eng., 40 (1997) 1435-1448.

[4] A. Bermudez and R. Rodrıguez, Finite element analysis of sloshing and hydroelasticvibrations under gravity. M2AN (to appear).

[5] S. Brenner and L.R. Scott, Mixed and Hybrid Finite Element Methods, Springer-Verlag,New York, Berlin, Heidelberg, 1996.

[6] F. Brezzi and M. Fortin, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, Berlin, Heidelberg, 1991.

[7] P. Clement, Approximation by finite element functions using local regularization, RAIROAnal. Numer., 9 (1975) 77-84.

[8] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations,Springer-Verlag, Berlin, Heidelberg, New York, Tokio, 1986.

[9] P. Grisvard, Elliptic Problems for Non-smooth Domains, Pitman, Boston, 1985.[10] M. Hamdi, Y. Ousset and G. Verchery, A displacement method for the analysis of

vibrations of coupled fluid-structure systems, Internat. J. Numer. Methods Eng., 13 (1978)139-150.

[11] T. Kato, Perturbation theory for linear operators, Springer, Berlin, 1976.[12] H.J-P. Morand and R. Ohayon, Interactions Fluides-Structures, Recherches en Mathema-

tiques Appliquees 23, Masson, Paris, 1992.[13] O.C. Zienkiewicz and R.L. Taylor, The finite element method, Mc Graw Hill, London,

1989.