# A Fluid-Structure Interaction Model for Dam-Water Systems: Analytical Study and Application to Seismic Behavior

Authors: Ricardo Faria; Sérgio Oliveira; Ana L. Silvestre
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

We consider a dam-water system modeled as a fluid-structure interaction, specifically, a coupled hyperbolic second-order problem, formulated in terms of the displacement of the structure and the fluid pressure. Firstly, we investigate the well posedness of the corresponding variational formulation using Galerkin approximations, energy estimates, and mollification. Then, we apply the finite element method along with the state-space representation of the discrete problem in order to perform a 3D numerical simulation of Cabril arch dam (Zêzere river, Portugal). The numerical model is validated by comparison with available experimental data from a monitoring vibration system installed in Cabril dam.


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## Body

## 1. Introduction

The aim of this paper is the analysis and numerical simulation of the dynamics of a dam-water system, in particular, its response to earthquake actions.Since the core engineering problem lies on the safety and performance evaluation of the structure alone, many related problems have been mostly addressed from a solid mechanics perspective. A simple approach to incorporate the water effects in the dynamic behavior of the dam is to consider an extended Lagrangian model consisting merely of displacements variables in both the reservoir and the dam as in [1, 2] and references therein. However, to take into account the sensitivity of the dam body and foundations to water pressure, the techniques from solid mechanics alone may not be the most adequate. Fluid mechanics plays an important role as well, and appropriate equations describing the water’s behavior and its interaction with the structure should to be taken into account for a more realistic modeling of the system. For this purpose, a pressure-elastic displacement formulation is presented in [3], which is obtained by simplifying the equations of motion of the fluid, so that the velocity variable is eliminated. In [4–6], the authors perform a dynamic analysis of the two-dimensional problem. An advantage of the formulation based on the fluid pressure, a scalar unknown, is the reduction of the computational effort in the numerical simulations. Furthermore, the use of the pressure variable is advantageous due to the possibility of comparison of values from the numerical simulations of the mathematical model with real measurements of the water pressure alongside the dam-water wall. It is common that, in order to check if there exist cracks forming in the concrete and for general safety control [7–10], such structures are kept under constant monitoring, especially in seismic hazardous regions.To the best of our knowledge, the model we use in this paper for the simulation of dam-water systems, and used in similar engineering studies by other authors, e.g., [11–13], has not been analysed from the mathematical point of view. Therefore, our first aim is the well posedness of the fluid-structure interaction problem. The transmission condition on the fluid-structure interface causes new difficulties because the unbalanced order of the time derivatives of the pressure and the structure’s displacement do not allow a direct application of available mathematical results for linear hyperbolic systems. As a consequence, the regularity we will obtain for the two unknowns is not the same. If we had considered a fluid model formulated in terms of a velocity potential, then the problem would be a hyperbolic coupling similar to the problem treated in [14–17]. In these references, the fluid domain was unbounded and, in order to find the fluid velocity and the displacement field in an elastic body as a result of an incident acoustic wave, the authors used the Laplace transform with respect to the time variable, which is suitable for the application of discretization schemes based on the boundary element method and convolution quadrature method (see also [18]).In a second stage of our study, the discretization in space by the finite element method of the fluid-structure interaction model is followed by a state space representation and a diagonalization process that lead to a system of first-order linear differential equations. This approach is preferred over numerical time stepping schemes with less computational cost, as suggested in [3], because a modal analysis allows for a direct comparison with natural frequencies experimental data. In practice, such data may correlate eventual deterioration processes with changes in modal parameters over time [8]. Moreover, in the context of seismic responses, the modal analysis plays an important role, as it is crucial that a structure’s natural frequency does not match a frequency of an eventual earthquake; otherwise it may continue to resonate and experience severe structural damage.The plan of the paper is the following. The mathematical model is presented in Section2. In Section 3, we derive the variational formulation of the continuous problem in appropriate function spaces and investigate the a priori mathematical properties of solutions; for this, a pressure potential is introduced so that the classical techniques based on Galerkin approximations for second-order hyperbolic equations can be applied to the modified coupled system. The main results on the well posedness of the formulation in pressure-elastic displacement are stated in Theorem 3. We proceed, in Section 4, with the numerical solution of the three-dimensional problem: the discretization in space by the finite element method leads to a state-space representation of the problem (50), (47), and (48), followed by a suitable integration over time (58)-(59). The use of the modal superposition technique allows us to study, in Section 5, the main natural frequencies of the system and corresponding modal configurations. The comparison of these results with available experimental data from a monitoring vibration system of Cabril arch dam (Zêzere river, Portugal) will validate the numerical model. Finally, the seismic response of Cabril dam will be presented by means of displacement fields.


## 2. Analytical Modeling of the System

The region occupied by the structure (dam body and foundation) is denoted byΩs, while the fluid domain (reservoir) is denoted by Ωf. Under the assumptions (i) small displacements and deformations in the structure, (ii) fluid displacement remains small; while interaction is substantial, the domains Ωf,Ωs⊂R3 can be considered fixed, constant in time, and the Eulerian description can be adopted for both the fluid and the structure motions. For the boundaries of Ωs and Ωf, we have the following: Γ1 is the interface between Ωs and Ωf; Γ2 represents the ground or rock mass under the fluid; Γ3 is the air-fluid interface; Γ4 is an artificial ’wall’, a delimitation of the fluid domain’s extension; Γ5 represents the rock mass under the structure; and finally, Γ6 is constituted by the structure walls with contact with the air. The time interval domain during which the dynamic fluid-structure interaction occurs is [0,T], T>0.The formulation of the fluid-structure interaction problem in elastic displacement-pressure variables is the following (see [3], pgs.634-637, and [11]): given the seismic activity a→s and initial conditions u→0, v→0, p0, and q0, find u→=u→(x,t) and p=p(x,t) such that(1a)ϱs∂ttu→=∇x·σu→-ca∂tu→+ϱsg→+a→sinΩs×0,T(1b)∂ttp=c2ΔxpinΩf×0,T(1c)σu→n→s=pn→s,∂n→fp=-ϱf∂ttu→·n→sonΓ1×0,T(1d)∂nfp=0onΓ2×0,T(1e)∂ttp=-g∂nfponΓ3×0,T(1f)∂tp=-c∂nfponΓ4×0,T(1g)u→=0onΓ5×0,T(1h)σu→n→s=0onΓ6×0,T(1i)u→t=0=u→0,∂tu→t=0=v→0inΩs×0(1j)pt=0=p0,∂tpt=0=q0inΩf×0The gravitational acceleration g→ and the acceleration of the ground a→s due to earthquake vibrations are included in the Navier equation, where ϱs is the structure density and ca is a positive constant representing a damping effect. The Cauchy stress tensor σsu→ is given by σsu→≔λ∇x·u→I+2μεu→, where ε(u→)≔∇xu→+(∇xu→)⊤/2 and the parameters λ≥0 and μ>0 are the Lamé constants. They relate to other elastic moduli constants, namely the Young’s modulus E and the Poisson ratio ν, by: μ=E/2(1+ν), λ=νE/(1+ν)(1-2ν). The structure domain boundary is ∂Ωs=Γ1∪Γ5∪Γ6 and n→s=n→s(x) is the outward unit normal vector at ∂Ωs. The viscous effects in the fluid have been neglected and it was assumed that the fluid density varies very little, so that it may be considered constant, ϱf. The constant c is the speed of sound in the fluid and g≔g→ is the acceleration due to gravity. The fluid domain boundary is ∂Ωf≔Γ1∪Γ2∪Γ3∪Γ4. In the air-water interface Γ3, a wavelike motion is observed, and Γ4 is permeable in the sense that the solution of (1b) in Ωf should be composed only of outgoing waves on Γ4, as no input from this boundary portion exists [3].


## 3. Variational Formulation. Existence and Uniqueness of Weak Solutions

### 3.1. Notation and Auxiliary Results

LetΩ⊂R3 be a bounded domain and Γ⊆∂Ω. We will use the classical notations and results for Lebesgue spaces Lq(Ω), Lq(Γ), 1≤q≤∞, Sobolev spaces Hm(Ω):=Wm,2(Ω), m∈N, and their dual spaces H-m(Ω)≔H0mΩ′. By ·q,Ω and ·q,Γ we denote the norms of Lq(Ω) and Lq(Γ), respectively. We will use the notation (f,g)Ω≔∫Ωf·gdx for vector valued functions, and when f and g are tensor-valued functions, we will write (f,g)Ω≔∫Ωf:gdx, where f:g represents contraction of tensors. Analogous meaning holds for (f,g)Γ.Suppose thatΓD⊂∂Ω has nonzero 2-dimensional measure and let V0≔{v→∈H1(Ω)3:v→=0onΓD}. Recall that the Korn inequality is valid in V0: v→H1(Ω)3≤CK(Ω)ε(v→)2,Ω. Therefore V0 is a Hilbert space with respect to the inner product V0×V0∋(v→,w→)→∫Ωε(v→):ε(w→)dx and the associated norm V0∋v↦ε(v→)2,Ω is equivalent to the norm induced by H1(Ω)3.IfX is a Banach space based on the domain Ω, we will write X′ for its dual space and χ,uΩ for the evaluation of χ∈X′ at u∈X. The Bochner spaces associated with X will be denoted by Lr(a,b;X), 1≤r≤∞, Cw([a,b];X) will denote the space of weakly continuous functions with values in X and D′(0,T;X) will be a space of distributions.We recall the method of mollifiers, which provides approximation by smooth functions. Here we will considerw=w(t,x) and mollification in the t variable. Let the function J∈C0∞(R) satisfy the following properties: (2)suppJ⊆-1,1,0≤J≤1,∫-11Jτdτ=1,Jτ=J-τ∀τ∈R.The mollifier Jδ∈C0∞(R), δ>0, is defined by Jδ(τ)≔δ-1J(τ/δ) (τ∈R). For a Banach space X, a smoothing operator can be defined by convolution with Jδ in the following way: given w∈Lr(a,b;X) and 1≤r<∞, set w(t)=0 for all t∈[a,b], and let wδ=Jδ⋆w∈C∞(a,b;X) be defined by wδ(t)=(Jδ⋆w)(t)=∫RJδ(t-τ)w(τ)dτ. Then (3)wδ-wLra,b;X→0asδ→0.Since convolution commutes with differentiation, we have ∂twδ=(∂tw)δ and (4)∂twδ-∂twLsa,b;Y→0asδ→0,if ∂tw∈Ls(a,b;Y), where 1≤s<∞ and Y is a Banach space. Another property of mollification is suppwδ⊂suppJδ + suppw.We will also use the following classical embedding result (see [19], pg. 392, Lemma 11.9).Lemma 1.
LetX,Y be Hilbert spaces such that X↪dY. Then L∞(a,b;X)∩C([a,b];Y)⊂Cw([a,b];X).


### 3.2. Variational Formulation

In order to derive the variational formulation of Problems (1a)-(1j), we consider the function spaces (5)Hf≔q∈L2Ωf∪Γ3:∫Ωfqdx=0,Vf≔H1ΩfHs≔L2Ωs3,Vs≔v→∈H1Ωs3:v→=0onΓ5,where Hf is equipped with inner product (p,q)↦(p,q)Ωf+(p,q)Γ3 and Vs is equipped with inner product (v→,w→)↦(ε(v→),ε(w→))Ωs.Supposep and u→ are smooth solutions of Problem (1a)-(1j). Multiplying both sides of (1b) by ψ∈Vf and integrating by parts over Ωf yields (6)d2dt2∫Ωfpψdx=c2∫∂Ωf∂n→fpψds-∫Ωf∇xp·∇xψdx,where by (1d), (1e), (1f), and (1c)(7)∫∂Ωf∂n→fpψds=-ϱf∫Γ1∂ttu→·n→sψds-1g∫Γ3∂ttpψds-1c∫Γ4∂tpψds.Therefore (8)ϱfd2dt2∫Γ1u→·n→sψds+d2dt21c2∫Ωfpψdx+1g∫Γ3pψds+1cddt∫Γ4pψds+∫Ωf∇xp·∇xψdx=0,∀ψ∈Vf.Analogously, dot multiplying both sides of (1a) by φ→∈Vs, integrating by parts over Ωs, and using σu→:∇xφ→=λ(∇x·u→)(∇x·φ→)+2με(u→):ε(φ→) yields (9)ϱsd2dt2∫Ωsu→·φ→dx=∫∂Ωsσu→n→s·φ→ds-∫Ωsλ∇x·u→∇x·φ→+2μεu→:εφ→dx-caddt∫Ωsu→·φ→dx+ϱs∫Ωsg→+a→s·φ→dx.Imposing the boundary conditions (1g) and (1h), we obtain (10)ϱsd2dt2∫Ωsu→·φ→dx+caddt∫Ωsu→·φ→dx+∫Ωsλ∇x·u→∇x·φ→+2μεu→:εφ→dx-∫Γ1pn→s·φ→ds=ϱs∫Ωsg→+a→s·φ→dx,∀φ→∈Vs.Hence the following variational problem is associated with (1a)-(1j): find u→∈D′(0,T;Vs) and p∈D′(0,T;Vf) such that(11a)d2dt2Mu→,φ→+ddtCsu→,φ→+Ku→,φ→-Qsp,φ→=Fsφ→,φ→∈Vs(11b)d2dt2Sp,ψ+ddtCfp,ψ+Hp,ψ+d2dt2Qfu→,ψ=0,ψ∈Vfand u→(0)=u→0, ∂tu→(0)=v→0, p(0)=p0, and ∂tp(0)=q0, in some sense, to be specified in terms of continuity properties of the solution. In (11a)-(11b) we have used the following notation (see [3]): (12)Mu→,φ→≔ϱsu→,φ→Ωs,Sp,ψ≔1c2p,ψΩf+1gp,ψΓ3,Csu→,φ→≔cau→,φ→Ωs,Ku→,φ→≔2μεu→,εφ→Ωs+λ∇x·u→,∇x·φ→Ωs,Fsφ→≔ϱsg→+a→s,φ→Ωs,Cfp,ψ≔1cp,ψΓ4,Hp,ψ≔∇xp,∇xψΩf,Qsp,φ→≔pn→s,φ→Γ1,Qfu→,ψ≔ϱfu→,n→sψΓ1.In (12), M and S are the so-called mass operators, Cs and Cf are drag operators, K and H are stiffness operators, and Qs and Qf are interaction operators for the structure and the fluid, respectively. Note that Qf(u→,p)=ϱfQs(p,u→), but the interaction operators appear in (11a)-(11b) in an unbalanced way in terms of the order of the time derivatives, which makes it difficult to obtain good a priori estimates to use in the mathematical analysis of the model.


### 3.3. Some Considerations on the Regularity of the Pressure: Formal Energy Equation and a Priori Estimates

Let us assume that a solution(u→,p) of (1a)-(1j) exists and is sufficiently regular. If we try to obtain a (formal) energy estimate for such a solution by replacing φ→∈Vs in (11a) by ∂tu→ and ψ by ∂tp/ϱf in (11b), followed by time integration in [0,t], we end up with the identity (13)1ϱf∇xpt2,Ωf2+1ϱfc2∂tpt2,Ωf2+1ϱfg∂tpt2,Γ32+2ϱfc∫0t∂τpτ2,Γ42dτ+2μεu→t2,Ωs2+ϱs∂tu→t2,Ωs2+λ∇x·u→t2,Ωs2+2ca∫0t∂τu→τ2,Ωs2dτ=2ϱs∫0tg→+as→τ,∂τu→τΩsdτ+1ϱf∇p02,Ωf2+1ϱfc2q0Ωfdτ+1ϱfgq02,Γ32+2μεu→0t2,Ωs2+ϱsv→02,Ωs2+λ∇x·u→02,Ωs2+∫Γ1p∂tu→·n→spds-∫Γ1∂ttu→·n→s∂tpds.In order to “skew-symmetrize” the bilinear form associated with the equations onΓ1×]0,T], so that the unbalanced terms ∫Γ1p∂tu→·n→sds-∫Γ1∂ttu→·n→s∂tpds are not present in the energy equation, it is convenient to introduce apressure potential for p (with respect to the t variable): P∈D′(0,T;Vf), p=∂tP.Note that taking into account the relationϱf∂tv→=-∇xp, in Ωf×]0,T], between the fluid velocity and pressure in this model (see [3], pg. 634), the variable P can be seen as a velocity potential for the eliminated unknown v→ in the fluid equations. This is consistent with the assumption of irrotational flow. Now, replacing p=ϱf∂tP in (11a) yields (14)d2dt2Mu→,φ→+ddtCsu→,φ→+Ku→,φ→-ddtϱfQsP,φ→=Fsφ→,φ→∈Vs,while the same replacement in (11b) produces a higher order ordinary differential equation (15)d3dt3SP,ψ+d2dt2CfP,ψ+ddtHP,ψ+d2dt2Qfu→,ψ=0,ψ∈Vfon0,t.Then (16)d2dt2SP,ψ+ddtCfP,ψ+HP,ψ+ddtQfu→,ψ=C,ψ∈Vfwhere C is a constant (distribution) and, more precisely, C=S(q0,ψ)+Cf(p0,ψ)+Qfv→0,ψ. If u→, p, and P are regular functions with P(0)=0, then the chosen value for C is just the value of d2/dt2S(P,ψ)+d/dtCf(P,ψ)+H(P,ψ)+d/dtQfu→,ψ at t=0. Hence, we consider the new equation for the unknown P(17)d2dt2SP,ψ+ddtCfP,ψ+HP,ψ+ddtQfu→,ψ=Sq0,ψ+Cfp0,ψ+Qfv→0,ψ,ψ∈Vf,which is the result of formally integrating (11b) in [0,t]. The same procedure that leads to (13) allows us to derive a formal energy equation for (u→,P)(18)121ϱf∇xPt2,Ωf2+1ϱfc2∂tPt2,Ωf2+1ϱfg∂tPt2,Γ32+122μεu→t2,Ωs2+λ∇x·u→t2,Ωs2+ϱs∂tu→t2,Ωs2+1ϱfc∫0t∂τP2,Γ42dτ+ca∫0t∂τu→2,Ωs2dτ=ϱs∫0tg→+as→,∂τu→Ωsdτ+1ϱfc2∫0tq0,∂τPΩfdτ+1ϱfg∫0tq0,∂τPΓ3dτ+1ϱfc∫0tp0,∂τPΓ4dτ+v→0·n→s,PtΓ1+12ϱfc2p02,Ωf2+12ϱfgp02,Γ32+μεu→0t2,Ωs2+λ2∇x·u→02,Ωs2+ϱs2v→02,Ωs2and (18), in turn, yields the following basic a priori estimates for (u→,P):(19a)∇xu→L∞0,T;L2Ωs+∂tu→L∞0,T;L2Ωs≤C1,(19b)∇xPL∞0,T;L2Ωf+∂tPL∞0,T;L2Ωf≤C2(19c)∂tPL∞0,T;L2Γ3+∂tPL20,T,L2Γ4≤C3where the constants Ci, i=1,2,3, depend on the data. Relation (18) suggests that the pair (u→,P) satisfies P∈C([0,T];Vf), p=∂tP∈C([0,T];Hf) and u→∈C([0,T];Vs), and ∂tu→∈C([0,T];Hs). The above considerations also indicate that the time regularity we can expect of the pressure is less than that of the structure displacement, more precisely, (20)P=∫0·psds∈C0,T;H1Ωf,p∈C0,T;L2Ωf,pΓ3∈C0,T;L2Γ3,with p(0)=p0. Moreover, if p=∂tP and satisfies (21)1c2∂tp,ψΩf=-∇xP,∇xψΩf+1c2q0,ψΩf,∀ψ∈H01Ωf,and ∇xP(0)=0 in Ωf, it follows that ∂tp∈C([0,T];H-1(Ωf)) and ∂p(0)=q0inH-1(Ωf). Note that (21) is obtained from (17) taking test functions φ→∈H01(Ωf).Remark 2.
The variational formulations (14) and (17) are actually a formulation in displacement-velocity potential. From v→=∇xΨ and the relation ϱf∂tv→=-∇xp, in Ωf×]0,T], between the fluid velocity and the pressure in this model (see [3], pg. 634), we get p=-ϱf∂tΨ.


### 3.4. Definition of Weak Solution and Well Posedness of the Mathematical Model

Based on the considerations made in the previous section, we shall say that aweak solution to Problems (1a)-(1j) is a pair (u→,p)∈D′(0,T;Vs)×D′(0,T;Vf) satisfying (11a)-(11b) and(22)u→∈C0,T;Vs∩C10,T;Hs,withu→0=u→0,∂tu→0=v→0,∫0tpsds∈C0,T;Vf,p∈C0,T;Hf,withp0=p0,∂tp∈C0,T;H-1Ωfwith∂tp0=q0inH-1Ωf.The main result concerning the well posedness of the mathematical models (1a)-(1j) is as follows.Theorem 3.
LetΩs,Ωf⊂R3 be locally Lipschitz bounded domains with the common boundary Γ1, as described in Section 2. Assume that a→s∈L1(0,T;L2(Ωs)3), p0∈Vf, q0∈Hf, and u→0,v→0∈Vs. Then there exists a unique solution (u→,p) to Problems (11a)-(11b).Theorem3 will be proved along several steps in the next subsections. The a priori estimates (19a)-(19c) for (u→,P) will be used to construct a weak solution for the problem, starting from a generalization of the classical Galerkin method for second-order hyperbolic problems (see [19, 20]). The method used for studying the transmission problem in [16] might also be adapted to the equations satisfied by (u→,P).


### 3.5. Galerkin Approximations

Let{ψk}k∈N and {φ→k}k∈N be basis of smooth functions for Vf and Vs, respectively. Let Vmf=span{ψ1,…,ψm} and Vms=span{φ→1,…,φ→m}. We seek approximating solutions in the form u→m(x,t)=∑l=1mcml(t)φ→l(x) and pm(x,t)=∑l=1mdml(t)ψl(x) with the coefficients cml(t) and dml(t) (0≤t≤T,l=1:m) obtained from(23a)M∂ttu→m,φ→k+Cs∂tu→m,φ→k-Qspm,φ→k+Ku→m,φ→k=Fsφ→k,(23b)S∂ttpm,ψk+Cf∂tpm,ψk+Qf∂ttu→m,ψk+Hpm,ψk=0,for k=1:m, and the initial conditions (24)u→m0=PVmsu→0,pm0=PVmfp0u→˙m0=PVmsv→0,pm˙0=PVmfq0,where PVms and PVmf are projection operators onto Vms and Vmf, respectively. This is a system of second-order ODEs, which has a unique solution. Indeed, (23a)-(23b) can be written in the form (25)M0ϱfQ⊤Scm¨dm¨+Cs00Cfcm˙dm˙+K-Q0Hcmdm=fs0,where we have used the notationy.=dy/dt and (26)Mij≔Mφ→i,φ→j,Cijs≔Cijsφ→i,φ→j,Kij≔Kφ→i,φ→jQij≔1ϱfQfφ→i,ψj,Sij≔Sψi,ψj,Cijf≔Cfψi,ψj,Hij≔Hψi,ψj,fjs≔Fsφ→j.The matrices M and S are symmetric, positive definite and since (27)det⁡M0ϱfQ⊤S=det⁡Mdet⁡S≠0the matrix M≔M0ϱfQ⊤S is nonsingular.Let(28)Dmlt≔∫0tdmlτdτ,Pmx,t≔∫0tpmx,τdτ,so that pm(x,t)=∂tPm(x,t) and Pm(x,0)=0. Then the approximating functions (u→m,Pm) satisfy(29a)M∂ttu→m,φ→k+Cs∂tu→m,φ→k-ϱfQs∂tPm,φ→k+Ku→m,φ→k=Fsφ→k,(29b)S∂ttPm,ψk+Cf∂tPm,ψk+Qf∂tu→m,ψk+HPm,ψk=SPVmfq0,ψk+CfPVmfp0,ψk+QfPVmsv→0,ψk.Now we multiply (29a) by c˙mk(t) and (29b) by D˙mk(t)/ϱf and sum from k=1 to k=m to conclude that Pm and u→m satisfy relations similar to (14) and (17) in the spaces Vms and Vmf and, consequently, the a priori estimates (19a)-(19c).Lemma 4.
For the Galerkin approximations(u→m,Pm) it holds that (30)∂tPmisboundedinL∞0,T;Hf,∂tu→misboundedinL∞0,T;Hs,PmisboundedinL∞0,T;Vf,u→misboundedinL∞0,T;Vs,∂tPmΓ4isboundedinL20,T;L2Γ4,The uniform bounds collected in Lemma4 allow us to obtain a pair (u→,P) as a weak limit of the Galerkin approximations (u→m,Pm) such that (31)P∈L∞0,T;Vf,∂tP∈L∞0,T;Hf,∂tPΓ4∈L20,T;L2Γ4,u→∈L∞0,T;Vs,∂tu→∈L∞0,T;Hs,together with the identities(32a)-∫0Tϱs∂tu→τ,φ→Ωs+cau→τ,φ→Ωs+Pτ,φ→·n→sΓ1ϕ˙τdτ+2μ∫0Tεu→τ,εφ→Ωs+λ∇x·u→τ,∇x·φ→Ωsϕτdτ=ϱsv→0,φ→Ωs+cau→0,φ→Ωsϕ0+ϱs∫0Tg→+as→τ,φ→Ωsϕτdτ,(32b)∫0t∇xPτ,∇xψΩfχτdτ-∫0t1c2∂tPτ,ψΩf+1g∂tPτ,ψΓ3χ˙τdτ-∫0t1cPτ,ψΓ4+ϱfu→τ·n→s,ψΓ1χ˙τdτ=1c2p0+q0,ψΩfχ0+1gp0+q0,ψΓ3+ϱfu→0+v→0·n→s,ψΓ1χ0,for all φ→∈Vs and ψ∈Vf and all ϕ,χ∈C1([0,T]) null in T. We now take φ→∈H01(Ωs)3 and ψ∈H01(Ωf) in (32a)-(32b) and obtain (33)ϱs∂ttu→,φ→Ωs=-ca∂tu→,φ→Ωs-μεu→,εφ→Ωs-λ∇x·u→,∇x·φ→Ωs+ϱsg→+as→,φ→Ωs,∀φ→∈H01Ωs3,1c2∂ttP,ψΩf=-∇xP,∇xψΩf+1c2q0,ψΩf,∀ψ∈H01Ωf.From these two relations, it follows ∂ttu→∈L∞(0,T;H-1(Ωs)3) and ∂ttP∈L∞(0,T;H-1(Ωf). By Lemma 1 and the well-known density result L2(Ω)↪dH-1(Ω), we obtain the following.Lemma 5.
u → ∈ C w ( [ 0 , T ] ; V s ) , ∂ t u → ∈ C w ( [ 0 , T ] ; H s ) and P∈Cw([0,T];Vf),∂tP∈Cw([0,T];Hf).


### 3.6. Strong Continuity, Energy Equality and Uniqueness

Let(u→,P) have the weak continuity properties established in the previous section and consider theenergy function defined by (34)Eu→,Pt≔12ϱf∇xPt2,Ωf2+12ϱfc2∂tPt2,Ωf2+12ϱfg∂tPt2,Γ32+μεu→t2,Ωs2+ϱs2∂tu→t2,Ωs2+λ2∇x·u→t2,Ωs2a.a.t∈0,T.We shall prove that E(u→,P)∈C([0,T]) and use this result to prove the strong continuity of (u→,P). To accomplish this, we will resort to the following intermediate result.Lemma 6.
If(u→,P) is the weak solution obtained in the previous section then (35)ddtEu→,P+1ϱfc∂tP2,Γ42+ca∂tu→2,Ωs2=ϱsg→+as→,∂tu→Ωs+1ϱf1c2q0,∂tPΩf+1gq0,∂tPΓ3+1cp0,∂tPΓ4+v→0·n→s,∂tPΓ1and E(u→,P):[0,T]→R is an absolutely continuous function.Proof.
Once (35) is proved, since (36)∂tP2,Γ42+∂tu→2,Ωs2∈L10,T,ϱsg→+as→,∂tu→Ωs+1ϱf1c2q0,∂tPΩf+1gq0,∂tPΓ3+1cp0,∂tPΓ4+v→0·n→s,∂tPΓ1∈L10,T,we immediately conclude that E(u→,P) is an absolutely continuous function.
To prove (35), we use the mollification method, as described in Section 3.1. Consider 0<t0<T∗<T, some fixed δ0>0 with δ0<t0, δ0<T-T∗, and let 0<δ<δ0. Inserting the special test functions ϕδ and χδ, with ϕ,χ∈C0∞(t0,T∗), in (32a)-(32b) and a simple calculation yields(37a)ϱs∂ttu→δ,φ→Ωs+ca∂tu→δ,φ→Ωs-∂tPδn→s,φ→Γ1+2μεu→δτ,εφ→Ωs+λ∇x·u→δ,∇x·φ→Ωs=ϱsg→δ+as→δ,φ→Ωs,∀φ→∈Vs,(37b)1c2∂ttPδ,ψΩf+1g∂ttPδ,ψΓ3+1c∂tPδ,ψΓ4+ϱf∂tu→δ·n→s,ψΓ1+∇xPδ,∇xψΩf=1c2q0δ,ψΩf+1gq0δ,ψΓ3+1cp0δ,ψΓ4+ϱfv→0·n→sδ,ψΓ1,∀ψ∈Vf.Now, since ∂tu→δ∈C0∞(t0,T∗;Vs) and ∂tPδ∈C0∞(t0,T∗;Vf), we can take φ→=∂tu→δ(t) and ψ=∂tPδ(t) in (37a)-(37b), thus obtaining (38)ddtEu→δ,Pδ+1ϱfc∂tPδ2,Γ42+ca∂tu→δ2,Ωs2=ϱsg→δ+as→δ,∂tu→δΩs+1ϱf1c2q0δ,∂tPδΩf+1gq0δ,∂tPδΓ3+1cp0δ,∂tPδΓ4+v→0·n→sδ,∂tPδΓ1.We let δ→0 and use the convergence properties (3)-(4) to obtain (35) in D′(t0,T∗). Since t0,T∗ are arbitrary, we conclude that (35) holds a.e. in (0,T).The result of Lemma5 can be improved toLemma 7.
The pair(u→,P) constructed in the previous subsection satisfies (39)P∈C0,T;Vf∩C10,T;Hf,PΓ3∈C10,T;L2Γ3,u→∈C0,T;Vs∩C10,T;Hsand the energy equality (18).Proof.
Using the weak continuity properties of(u→,P) and the continuity of E(u→δ,Pδ) from Lemma 6, we find that (40)12ϱf∇xPt-∇xPt02,Ωf2+12ϱfc2∂tPt-∂tPt02,Ωf2+12ϱfg∂tPt-∂tPt02,Γ32+ϱs2∂tu→t-∂tu→t02,Ωs2+μεu→t-εu→t02,Ωs2+λ2∇x·u→t-∇x·u→t02,Ωs2=Eu→,Pt+Eu→,Pt0-1ϱf∇xPt,∇xPt0Ωf+1ϱfc2∂tPt,∂tPt0Ωf+1ϱfg∂tPt,∂tPt0Γ3+ϱs∂tu→t,∂tu→t0Ωs+2μεu→t,εu→t0Ωs+λ∇x·u→t,∇x·u→t0Ωs→0ast→t0.Uniqueness for the constructed(u→,P) is consequence of the linearity of the problem (14) and (17) and the validity of the energy equality (18).Concerning uniqueness for the original(u→,p), it suffices to show that the only weak solution (in the sense of the definition that supports Theorem 3) with null data is (u→,p)=(0→,0). Again, the idea is to introduce a primitive P∈D′(0,T;Vf) for the original pressure p=∂tP. Since P=0 and another primitive of p will differ from P by a constant, it follows that p=0.


## 3.1. Notation and Auxiliary Results

LetΩ⊂R3 be a bounded domain and Γ⊆∂Ω. We will use the classical notations and results for Lebesgue spaces Lq(Ω), Lq(Γ), 1≤q≤∞, Sobolev spaces Hm(Ω):=Wm,2(Ω), m∈N, and their dual spaces H-m(Ω)≔H0mΩ′. By ·q,Ω and ·q,Γ we denote the norms of Lq(Ω) and Lq(Γ), respectively. We will use the notation (f,g)Ω≔∫Ωf·gdx for vector valued functions, and when f and g are tensor-valued functions, we will write (f,g)Ω≔∫Ωf:gdx, where f:g represents contraction of tensors. Analogous meaning holds for (f,g)Γ.Suppose thatΓD⊂∂Ω has nonzero 2-dimensional measure and let V0≔{v→∈H1(Ω)3:v→=0onΓD}. Recall that the Korn inequality is valid in V0: v→H1(Ω)3≤CK(Ω)ε(v→)2,Ω. Therefore V0 is a Hilbert space with respect to the inner product V0×V0∋(v→,w→)→∫Ωε(v→):ε(w→)dx and the associated norm V0∋v↦ε(v→)2,Ω is equivalent to the norm induced by H1(Ω)3.IfX is a Banach space based on the domain Ω, we will write X′ for its dual space and χ,uΩ for the evaluation of χ∈X′ at u∈X. The Bochner spaces associated with X will be denoted by Lr(a,b;X), 1≤r≤∞, Cw([a,b];X) will denote the space of weakly continuous functions with values in X and D′(0,T;X) will be a space of distributions.We recall the method of mollifiers, which provides approximation by smooth functions. Here we will considerw=w(t,x) and mollification in the t variable. Let the function J∈C0∞(R) satisfy the following properties: (2)suppJ⊆-1,1,0≤J≤1,∫-11Jτdτ=1,Jτ=J-τ∀τ∈R.The mollifier Jδ∈C0∞(R), δ>0, is defined by Jδ(τ)≔δ-1J(τ/δ) (τ∈R). For a Banach space X, a smoothing operator can be defined by convolution with Jδ in the following way: given w∈Lr(a,b;X) and 1≤r<∞, set w(t)=0 for all t∈[a,b], and let wδ=Jδ⋆w∈C∞(a,b;X) be defined by wδ(t)=(Jδ⋆w)(t)=∫RJδ(t-τ)w(τ)dτ. Then (3)wδ-wLra,b;X→0asδ→0.Since convolution commutes with differentiation, we have ∂twδ=(∂tw)δ and (4)∂twδ-∂twLsa,b;Y→0asδ→0,if ∂tw∈Ls(a,b;Y), where 1≤s<∞ and Y is a Banach space. Another property of mollification is suppwδ⊂suppJδ + suppw.We will also use the following classical embedding result (see [19], pg. 392, Lemma 11.9).Lemma 1.
LetX,Y be Hilbert spaces such that X↪dY. Then L∞(a,b;X)∩C([a,b];Y)⊂Cw([a,b];X).


## 3.2. Variational Formulation

In order to derive the variational formulation of Problems (1a)-(1j), we consider the function spaces (5)Hf≔q∈L2Ωf∪Γ3:∫Ωfqdx=0,Vf≔H1ΩfHs≔L2Ωs3,Vs≔v→∈H1Ωs3:v→=0onΓ5,where Hf is equipped with inner product (p,q)↦(p,q)Ωf+(p,q)Γ3 and Vs is equipped with inner product (v→,w→)↦(ε(v→),ε(w→))Ωs.Supposep and u→ are smooth solutions of Problem (1a)-(1j). Multiplying both sides of (1b) by ψ∈Vf and integrating by parts over Ωf yields (6)d2dt2∫Ωfpψdx=c2∫∂Ωf∂n→fpψds-∫Ωf∇xp·∇xψdx,where by (1d), (1e), (1f), and (1c)(7)∫∂Ωf∂n→fpψds=-ϱf∫Γ1∂ttu→·n→sψds-1g∫Γ3∂ttpψds-1c∫Γ4∂tpψds.Therefore (8)ϱfd2dt2∫Γ1u→·n→sψds+d2dt21c2∫Ωfpψdx+1g∫Γ3pψds+1cddt∫Γ4pψds+∫Ωf∇xp·∇xψdx=0,∀ψ∈Vf.Analogously, dot multiplying both sides of (1a) by φ→∈Vs, integrating by parts over Ωs, and using σu→:∇xφ→=λ(∇x·u→)(∇x·φ→)+2με(u→):ε(φ→) yields (9)ϱsd2dt2∫Ωsu→·φ→dx=∫∂Ωsσu→n→s·φ→ds-∫Ωsλ∇x·u→∇x·φ→+2μεu→:εφ→dx-caddt∫Ωsu→·φ→dx+ϱs∫Ωsg→+a→s·φ→dx.Imposing the boundary conditions (1g) and (1h), we obtain (10)ϱsd2dt2∫Ωsu→·φ→dx+caddt∫Ωsu→·φ→dx+∫Ωsλ∇x·u→∇x·φ→+2μεu→:εφ→dx-∫Γ1pn→s·φ→ds=ϱs∫Ωsg→+a→s·φ→dx,∀φ→∈Vs.Hence the following variational problem is associated with (1a)-(1j): find u→∈D′(0,T;Vs) and p∈D′(0,T;Vf) such that(11a)d2dt2Mu→,φ→+ddtCsu→,φ→+Ku→,φ→-Qsp,φ→=Fsφ→,φ→∈Vs(11b)d2dt2Sp,ψ+ddtCfp,ψ+Hp,ψ+d2dt2Qfu→,ψ=0,ψ∈Vfand u→(0)=u→0, ∂tu→(0)=v→0, p(0)=p0, and ∂tp(0)=q0, in some sense, to be specified in terms of continuity properties of the solution. In (11a)-(11b) we have used the following notation (see [3]): (12)Mu→,φ→≔ϱsu→,φ→Ωs,Sp,ψ≔1c2p,ψΩf+1gp,ψΓ3,Csu→,φ→≔cau→,φ→Ωs,Ku→,φ→≔2μεu→,εφ→Ωs+λ∇x·u→,∇x·φ→Ωs,Fsφ→≔ϱsg→+a→s,φ→Ωs,Cfp,ψ≔1cp,ψΓ4,Hp,ψ≔∇xp,∇xψΩf,Qsp,φ→≔pn→s,φ→Γ1,Qfu→,ψ≔ϱfu→,n→sψΓ1.In (12), M and S are the so-called mass operators, Cs and Cf are drag operators, K and H are stiffness operators, and Qs and Qf are interaction operators for the structure and the fluid, respectively. Note that Qf(u→,p)=ϱfQs(p,u→), but the interaction operators appear in (11a)-(11b) in an unbalanced way in terms of the order of the time derivatives, which makes it difficult to obtain good a priori estimates to use in the mathematical analysis of the model.


## 3.3. Some Considerations on the Regularity of the Pressure: Formal Energy Equation and a Priori Estimates

Let us assume that a solution(u→,p) of (1a)-(1j) exists and is sufficiently regular. If we try to obtain a (formal) energy estimate for such a solution by replacing φ→∈Vs in (11a) by ∂tu→ and ψ by ∂tp/ϱf in (11b), followed by time integration in [0,t], we end up with the identity (13)1ϱf∇xpt2,Ωf2+1ϱfc2∂tpt2,Ωf2+1ϱfg∂tpt2,Γ32+2ϱfc∫0t∂τpτ2,Γ42dτ+2μεu→t2,Ωs2+ϱs∂tu→t2,Ωs2+λ∇x·u→t2,Ωs2+2ca∫0t∂τu→τ2,Ωs2dτ=2ϱs∫0tg→+as→τ,∂τu→τΩsdτ+1ϱf∇p02,Ωf2+1ϱfc2q0Ωfdτ+1ϱfgq02,Γ32+2μεu→0t2,Ωs2+ϱsv→02,Ωs2+λ∇x·u→02,Ωs2+∫Γ1p∂tu→·n→spds-∫Γ1∂ttu→·n→s∂tpds.In order to “skew-symmetrize” the bilinear form associated with the equations onΓ1×]0,T], so that the unbalanced terms ∫Γ1p∂tu→·n→sds-∫Γ1∂ttu→·n→s∂tpds are not present in the energy equation, it is convenient to introduce apressure potential for p (with respect to the t variable): P∈D′(0,T;Vf), p=∂tP.Note that taking into account the relationϱf∂tv→=-∇xp, in Ωf×]0,T], between the fluid velocity and pressure in this model (see [3], pg. 634), the variable P can be seen as a velocity potential for the eliminated unknown v→ in the fluid equations. This is consistent with the assumption of irrotational flow. Now, replacing p=ϱf∂tP in (11a) yields (14)d2dt2Mu→,φ→+ddtCsu→,φ→+Ku→,φ→-ddtϱfQsP,φ→=Fsφ→,φ→∈Vs,while the same replacement in (11b) produces a higher order ordinary differential equation (15)d3dt3SP,ψ+d2dt2CfP,ψ+ddtHP,ψ+d2dt2Qfu→,ψ=0,ψ∈Vfon0,t.Then (16)d2dt2SP,ψ+ddtCfP,ψ+HP,ψ+ddtQfu→,ψ=C,ψ∈Vfwhere C is a constant (distribution) and, more precisely, C=S(q0,ψ)+Cf(p0,ψ)+Qfv→0,ψ. If u→, p, and P are regular functions with P(0)=0, then the chosen value for C is just the value of d2/dt2S(P,ψ)+d/dtCf(P,ψ)+H(P,ψ)+d/dtQfu→,ψ at t=0. Hence, we consider the new equation for the unknown P(17)d2dt2SP,ψ+ddtCfP,ψ+HP,ψ+ddtQfu→,ψ=Sq0,ψ+Cfp0,ψ+Qfv→0,ψ,ψ∈Vf,which is the result of formally integrating (11b) in [0,t]. The same procedure that leads to (13) allows us to derive a formal energy equation for (u→,P)(18)121ϱf∇xPt2,Ωf2+1ϱfc2∂tPt2,Ωf2+1ϱfg∂tPt2,Γ32+122μεu→t2,Ωs2+λ∇x·u→t2,Ωs2+ϱs∂tu→t2,Ωs2+1ϱfc∫0t∂τP2,Γ42dτ+ca∫0t∂τu→2,Ωs2dτ=ϱs∫0tg→+as→,∂τu→Ωsdτ+1ϱfc2∫0tq0,∂τPΩfdτ+1ϱfg∫0tq0,∂τPΓ3dτ+1ϱfc∫0tp0,∂τPΓ4dτ+v→0·n→s,PtΓ1+12ϱfc2p02,Ωf2+12ϱfgp02,Γ32+μεu→0t2,Ωs2+λ2∇x·u→02,Ωs2+ϱs2v→02,Ωs2and (18), in turn, yields the following basic a priori estimates for (u→,P):(19a)∇xu→L∞0,T;L2Ωs+∂tu→L∞0,T;L2Ωs≤C1,(19b)∇xPL∞0,T;L2Ωf+∂tPL∞0,T;L2Ωf≤C2(19c)∂tPL∞0,T;L2Γ3+∂tPL20,T,L2Γ4≤C3where the constants Ci, i=1,2,3, depend on the data. Relation (18) suggests that the pair (u→,P) satisfies P∈C([0,T];Vf), p=∂tP∈C([0,T];Hf) and u→∈C([0,T];Vs), and ∂tu→∈C([0,T];Hs). The above considerations also indicate that the time regularity we can expect of the pressure is less than that of the structure displacement, more precisely, (20)P=∫0·psds∈C0,T;H1Ωf,p∈C0,T;L2Ωf,pΓ3∈C0,T;L2Γ3,with p(0)=p0. Moreover, if p=∂tP and satisfies (21)1c2∂tp,ψΩf=-∇xP,∇xψΩf+1c2q0,ψΩf,∀ψ∈H01Ωf,and ∇xP(0)=0 in Ωf, it follows that ∂tp∈C([0,T];H-1(Ωf)) and ∂p(0)=q0inH-1(Ωf). Note that (21) is obtained from (17) taking test functions φ→∈H01(Ωf).Remark 2.
The variational formulations (14) and (17) are actually a formulation in displacement-velocity potential. From v→=∇xΨ and the relation ϱf∂tv→=-∇xp, in Ωf×]0,T], between the fluid velocity and the pressure in this model (see [3], pg. 634), we get p=-ϱf∂tΨ.


## 3.4. Definition of Weak Solution and Well Posedness of the Mathematical Model

Based on the considerations made in the previous section, we shall say that aweak solution to Problems (1a)-(1j) is a pair (u→,p)∈D′(0,T;Vs)×D′(0,T;Vf) satisfying (11a)-(11b) and(22)u→∈C0,T;Vs∩C10,T;Hs,withu→0=u→0,∂tu→0=v→0,∫0tpsds∈C0,T;Vf,p∈C0,T;Hf,withp0=p0,∂tp∈C0,T;H-1Ωfwith∂tp0=q0inH-1Ωf.The main result concerning the well posedness of the mathematical models (1a)-(1j) is as follows.Theorem 3.
LetΩs,Ωf⊂R3 be locally Lipschitz bounded domains with the common boundary Γ1, as described in Section 2. Assume that a→s∈L1(0,T;L2(Ωs)3), p0∈Vf, q0∈Hf, and u→0,v→0∈Vs. Then there exists a unique solution (u→,p) to Problems (11a)-(11b).Theorem3 will be proved along several steps in the next subsections. The a priori estimates (19a)-(19c) for (u→,P) will be used to construct a weak solution for the problem, starting from a generalization of the classical Galerkin method for second-order hyperbolic problems (see [19, 20]). The method used for studying the transmission problem in [16] might also be adapted to the equations satisfied by (u→,P).


## 3.5. Galerkin Approximations

Let{ψk}k∈N and {φ→k}k∈N be basis of smooth functions for Vf and Vs, respectively. Let Vmf=span{ψ1,…,ψm} and Vms=span{φ→1,…,φ→m}. We seek approximating solutions in the form u→m(x,t)=∑l=1mcml(t)φ→l(x) and pm(x,t)=∑l=1mdml(t)ψl(x) with the coefficients cml(t) and dml(t) (0≤t≤T,l=1:m) obtained from(23a)M∂ttu→m,φ→k+Cs∂tu→m,φ→k-Qspm,φ→k+Ku→m,φ→k=Fsφ→k,(23b)S∂ttpm,ψk+Cf∂tpm,ψk+Qf∂ttu→m,ψk+Hpm,ψk=0,for k=1:m, and the initial conditions (24)u→m0=PVmsu→0,pm0=PVmfp0u→˙m0=PVmsv→0,pm˙0=PVmfq0,where PVms and PVmf are projection operators onto Vms and Vmf, respectively. This is a system of second-order ODEs, which has a unique solution. Indeed, (23a)-(23b) can be written in the form (25)M0ϱfQ⊤Scm¨dm¨+Cs00Cfcm˙dm˙+K-Q0Hcmdm=fs0,where we have used the notationy.=dy/dt and (26)Mij≔Mφ→i,φ→j,Cijs≔Cijsφ→i,φ→j,Kij≔Kφ→i,φ→jQij≔1ϱfQfφ→i,ψj,Sij≔Sψi,ψj,Cijf≔Cfψi,ψj,Hij≔Hψi,ψj,fjs≔Fsφ→j.The matrices M and S are symmetric, positive definite and since (27)det⁡M0ϱfQ⊤S=det⁡Mdet⁡S≠0the matrix M≔M0ϱfQ⊤S is nonsingular.Let(28)Dmlt≔∫0tdmlτdτ,Pmx,t≔∫0tpmx,τdτ,so that pm(x,t)=∂tPm(x,t) and Pm(x,0)=0. Then the approximating functions (u→m,Pm) satisfy(29a)M∂ttu→m,φ→k+Cs∂tu→m,φ→k-ϱfQs∂tPm,φ→k+Ku→m,φ→k=Fsφ→k,(29b)S∂ttPm,ψk+Cf∂tPm,ψk+Qf∂tu→m,ψk+HPm,ψk=SPVmfq0,ψk+CfPVmfp0,ψk+QfPVmsv→0,ψk.Now we multiply (29a) by c˙mk(t) and (29b) by D˙mk(t)/ϱf and sum from k=1 to k=m to conclude that Pm and u→m satisfy relations similar to (14) and (17) in the spaces Vms and Vmf and, consequently, the a priori estimates (19a)-(19c).Lemma 4.
For the Galerkin approximations(u→m,Pm) it holds that (30)∂tPmisboundedinL∞0,T;Hf,∂tu→misboundedinL∞0,T;Hs,PmisboundedinL∞0,T;Vf,u→misboundedinL∞0,T;Vs,∂tPmΓ4isboundedinL20,T;L2Γ4,The uniform bounds collected in Lemma4 allow us to obtain a pair (u→,P) as a weak limit of the Galerkin approximations (u→m,Pm) such that (31)P∈L∞0,T;Vf,∂tP∈L∞0,T;Hf,∂tPΓ4∈L20,T;L2Γ4,u→∈L∞0,T;Vs,∂tu→∈L∞0,T;Hs,together with the identities(32a)-∫0Tϱs∂tu→τ,φ→Ωs+cau→τ,φ→Ωs+Pτ,φ→·n→sΓ1ϕ˙τdτ+2μ∫0Tεu→τ,εφ→Ωs+λ∇x·u→τ,∇x·φ→Ωsϕτdτ=ϱsv→0,φ→Ωs+cau→0,φ→Ωsϕ0+ϱs∫0Tg→+as→τ,φ→Ωsϕτdτ,(32b)∫0t∇xPτ,∇xψΩfχτdτ-∫0t1c2∂tPτ,ψΩf+1g∂tPτ,ψΓ3χ˙τdτ-∫0t1cPτ,ψΓ4+ϱfu→τ·n→s,ψΓ1χ˙τdτ=1c2p0+q0,ψΩfχ0+1gp0+q0,ψΓ3+ϱfu→0+v→0·n→s,ψΓ1χ0,for all φ→∈Vs and ψ∈Vf and all ϕ,χ∈C1([0,T]) null in T. We now take φ→∈H01(Ωs)3 and ψ∈H01(Ωf) in (32a)-(32b) and obtain (33)ϱs∂ttu→,φ→Ωs=-ca∂tu→,φ→Ωs-μεu→,εφ→Ωs-λ∇x·u→,∇x·φ→Ωs+ϱsg→+as→,φ→Ωs,∀φ→∈H01Ωs3,1c2∂ttP,ψΩf=-∇xP,∇xψΩf+1c2q0,ψΩf,∀ψ∈H01Ωf.From these two relations, it follows ∂ttu→∈L∞(0,T;H-1(Ωs)3) and ∂ttP∈L∞(0,T;H-1(Ωf). By Lemma 1 and the well-known density result L2(Ω)↪dH-1(Ω), we obtain the following.Lemma 5.
u → ∈ C w ( [ 0 , T ] ; V s ) , ∂ t u → ∈ C w ( [ 0 , T ] ; H s ) and P∈Cw([0,T];Vf),∂tP∈Cw([0,T];Hf).


## 3.6. Strong Continuity, Energy Equality and Uniqueness

Let(u→,P) have the weak continuity properties established in the previous section and consider theenergy function defined by (34)Eu→,Pt≔12ϱf∇xPt2,Ωf2+12ϱfc2∂tPt2,Ωf2+12ϱfg∂tPt2,Γ32+μεu→t2,Ωs2+ϱs2∂tu→t2,Ωs2+λ2∇x·u→t2,Ωs2a.a.t∈0,T.We shall prove that E(u→,P)∈C([0,T]) and use this result to prove the strong continuity of (u→,P). To accomplish this, we will resort to the following intermediate result.Lemma 6.
If(u→,P) is the weak solution obtained in the previous section then (35)ddtEu→,P+1ϱfc∂tP2,Γ42+ca∂tu→2,Ωs2=ϱsg→+as→,∂tu→Ωs+1ϱf1c2q0,∂tPΩf+1gq0,∂tPΓ3+1cp0,∂tPΓ4+v→0·n→s,∂tPΓ1and E(u→,P):[0,T]→R is an absolutely continuous function.Proof.
Once (35) is proved, since (36)∂tP2,Γ42+∂tu→2,Ωs2∈L10,T,ϱsg→+as→,∂tu→Ωs+1ϱf1c2q0,∂tPΩf+1gq0,∂tPΓ3+1cp0,∂tPΓ4+v→0·n→s,∂tPΓ1∈L10,T,we immediately conclude that E(u→,P) is an absolutely continuous function.
To prove (35), we use the mollification method, as described in Section 3.1. Consider 0<t0<T∗<T, some fixed δ0>0 with δ0<t0, δ0<T-T∗, and let 0<δ<δ0. Inserting the special test functions ϕδ and χδ, with ϕ,χ∈C0∞(t0,T∗), in (32a)-(32b) and a simple calculation yields(37a)ϱs∂ttu→δ,φ→Ωs+ca∂tu→δ,φ→Ωs-∂tPδn→s,φ→Γ1+2μεu→δτ,εφ→Ωs+λ∇x·u→δ,∇x·φ→Ωs=ϱsg→δ+as→δ,φ→Ωs,∀φ→∈Vs,(37b)1c2∂ttPδ,ψΩf+1g∂ttPδ,ψΓ3+1c∂tPδ,ψΓ4+ϱf∂tu→δ·n→s,ψΓ1+∇xPδ,∇xψΩf=1c2q0δ,ψΩf+1gq0δ,ψΓ3+1cp0δ,ψΓ4+ϱfv→0·n→sδ,ψΓ1,∀ψ∈Vf.Now, since ∂tu→δ∈C0∞(t0,T∗;Vs) and ∂tPδ∈C0∞(t0,T∗;Vf), we can take φ→=∂tu→δ(t) and ψ=∂tPδ(t) in (37a)-(37b), thus obtaining (38)ddtEu→δ,Pδ+1ϱfc∂tPδ2,Γ42+ca∂tu→δ2,Ωs2=ϱsg→δ+as→δ,∂tu→δΩs+1ϱf1c2q0δ,∂tPδΩf+1gq0δ,∂tPδΓ3+1cp0δ,∂tPδΓ4+v→0·n→sδ,∂tPδΓ1.We let δ→0 and use the convergence properties (3)-(4) to obtain (35) in D′(t0,T∗). Since t0,T∗ are arbitrary, we conclude that (35) holds a.e. in (0,T).The result of Lemma5 can be improved toLemma 7.
The pair(u→,P) constructed in the previous subsection satisfies (39)P∈C0,T;Vf∩C10,T;Hf,PΓ3∈C10,T;L2Γ3,u→∈C0,T;Vs∩C10,T;Hsand the energy equality (18).Proof.
Using the weak continuity properties of(u→,P) and the continuity of E(u→δ,Pδ) from Lemma 6, we find that (40)12ϱf∇xPt-∇xPt02,Ωf2+12ϱfc2∂tPt-∂tPt02,Ωf2+12ϱfg∂tPt-∂tPt02,Γ32+ϱs2∂tu→t-∂tu→t02,Ωs2+μεu→t-εu→t02,Ωs2+λ2∇x·u→t-∇x·u→t02,Ωs2=Eu→,Pt+Eu→,Pt0-1ϱf∇xPt,∇xPt0Ωf+1ϱfc2∂tPt,∂tPt0Ωf+1ϱfg∂tPt,∂tPt0Γ3+ϱs∂tu→t,∂tu→t0Ωs+2μεu→t,εu→t0Ωs+λ∇x·u→t,∇x·u→t0Ωs→0ast→t0.Uniqueness for the constructed(u→,P) is consequence of the linearity of the problem (14) and (17) and the validity of the energy equality (18).Concerning uniqueness for the original(u→,p), it suffices to show that the only weak solution (in the sense of the definition that supports Theorem 3) with null data is (u→,p)=(0→,0). Again, the idea is to introduce a primitive P∈D′(0,T;Vf) for the original pressure p=∂tP. Since P=0 and another primitive of p will differ from P by a constant, it follows that p=0.


## 4. Discretization of the Equations

### 4.1. Semidiscretization in Space

Once a mesh and the corresponding basis functions,ψi:Ωf→R and φ→i:Ωs→R3, have been generated, u→ and p are approximated by u→(x,t)≈u→∗(t,x):=∑i=13Nsui∗(t)φ→i(x), p(x,t)≈p∗(t,x)≔∑i=1Nfpi∗(t)ψi(x), and Problems (11a)-(11b) are approximated by (see (26)) (41)M0ϱfQ⊤Su∗¨p∗¨+Cs00Cfu∗˙p∗˙+K-Q0Hu∗p∗=f0.Matrices M, Cs, and K are 3Ns×3Ns dimensional, S, Cf, and H are Nf×Nf dimensional, Q is 3Ns×Nf dimensional, and f and s are, respectively, Ns- and Nf-dimensional vectors. The matrices M and S are symmetric, positive definite and the matrix (42)M≔M0ϱfQ⊤Sis nonsingular, as already seen in the previous section. It will be convenient to introduce additional notation(43)C≔Cs00Cf,K≔K-Q0H,y≔u∗p∗,f≔f0so that (41) can be written as (44)My¨+Cy˙+Ky=f.


### 4.2. State-Space Representation of the Problem

We proceed with the so-calledstate-space representation of Problem (41) with subsequent diagonalization, so we end up with a system of independent first-order linear differential equations for the discretized problem. As already mentioned in the Introduction, this is preferred over a direct numerical time stepping scheme applied to (44) because it allows for a comparison with the measured natural frequencies for different values of water level.Starting from (44) and following [21, 22], a new variable z=y˙ is introduced so that (44) is equivalent to (45)Mz˙+Cy˙+Ky=fy˙=zwhich in turn can be written as (46)CM-I0y˙z˙+K00Iyz=f0.The matrix (47)B≔CM-I0=Cs0M00CfϱfQ⊤S-I0000-I00is nonsingular because det⁡(B)=det⁡(M)≠0. Defining (48)A≔K00I=K-Q000H0000I0000IF≔f0=f000and putting v≔u˙, q≔p˙, x≔[yz]⊤=[upvq]⊤, we can write (41) as (49)Bu˙p˙v˙q˙=Aupvq+FFrom now on, we consider the system of ODEs (50)Bx˙t=Axt+Ft,t∈0,Tx0=x0with the obvious identifications as concerns x0. In order to solve (50), the solution of the homogeneous equation Bx˙(t)=Ax(t),t∈]0,T] can be sought in the form x(t)=beλt, yielding the generalized eigenvalue problem Ab-λBb=0. Let m≔dim⁡A=dim⁡(B)=2(3Ns+Nf). If all the eigenvalues are distinct, the matrix pencil (A,B) is diagonalizable; that is, there exist m×m matrices W and V such that (51)W∗AV=ΩA≡diagα1,…,αm,W∗BV=ΩB≡diagβ1,…,βm.The matrices W and V contain the left and right eigenvectors of (A,B), that is αi=wi∗Avi and βi=wi∗Bvi. In practice, since the response of the mechanical system is typically dominated by a relatively small number of the lowest modes, it may not be necessary to compute the complete eigensystem of (A,B). Other approaches [12, 13], based on pseudo-symmetric techniques, can be used for computing the mode shapes and natural frequencies of this fluid-structure model.Using the transformation for modal coordinatesx=Vz and multiplying both sides of (50) by W∗, yields the modal system (52)ΩBz˙=ΩAz+Cin0,Tz0=z0with C=W∗F and z0 solution of the linear system Vz0=x0. From (51) we get z0=ΩA-1W∗Ax0 or z0=ΩB-1W∗Bx0 so that the calculation of z0 only requires direct matrix multiplications. Now we have a system of m independent first-order linear ordinary differential equations, which is easy to solve. Each equation of the system is of the form (53)βiz˙it=αizit+cit,t∈0,T,withzi0=zi,0where ci is the i-th component of the vector C, which in practice is only known in a set of discrete points.


### 4.3. Time Discretization of the Equations

For the discretization in time of the diagonalized system (52), we take the solution of each equation (53) in its integral form (54)zit=eλitzi0+eλit∫0te-λiτc~iτdτwhere λi≔αi/βi are the state eigenvalues and c~i=ci/βi, i=1:m. Each function ci is known in discrete instants of time 0<t1<⋯<tn-1<tn=T with Δt≔tk+1-tk, k=1:n-1, and therefore will approximate ci by a constant in each interval [tk,tk+1]: (55)cit≃citkt∈tk,tk+1.From the relation (56)zitk+1=eλiΔtzitk+eλitk+1∫tktk+1e-λiτc~iτdτand the approximation (55), we get (57)zitk+1≃eλiΔtzitk+eλitk+1c~tk∫tktk+1e-λiτdτ=eλiΔtzitk+ctkλieλiΔt-1.In vector form, the algorithm for computing z reads (58)ztk+1≃eΛΔtztk+Λ-1CtkeΛΔt-Iwith Λ=diag(λ1,…,λm). Once z is known, (59)xtk=Vztk,k=1:nwill provide the solution for the original variable.


## 4.1. Semidiscretization in Space

Once a mesh and the corresponding basis functions,ψi:Ωf→R and φ→i:Ωs→R3, have been generated, u→ and p are approximated by u→(x,t)≈u→∗(t,x):=∑i=13Nsui∗(t)φ→i(x), p(x,t)≈p∗(t,x)≔∑i=1Nfpi∗(t)ψi(x), and Problems (11a)-(11b) are approximated by (see (26)) (41)M0ϱfQ⊤Su∗¨p∗¨+Cs00Cfu∗˙p∗˙+K-Q0Hu∗p∗=f0.Matrices M, Cs, and K are 3Ns×3Ns dimensional, S, Cf, and H are Nf×Nf dimensional, Q is 3Ns×Nf dimensional, and f and s are, respectively, Ns- and Nf-dimensional vectors. The matrices M and S are symmetric, positive definite and the matrix (42)M≔M0ϱfQ⊤Sis nonsingular, as already seen in the previous section. It will be convenient to introduce additional notation(43)C≔Cs00Cf,K≔K-Q0H,y≔u∗p∗,f≔f0so that (41) can be written as (44)My¨+Cy˙+Ky=f.


## 4.2. State-Space Representation of the Problem

We proceed with the so-calledstate-space representation of Problem (41) with subsequent diagonalization, so we end up with a system of independent first-order linear differential equations for the discretized problem. As already mentioned in the Introduction, this is preferred over a direct numerical time stepping scheme applied to (44) because it allows for a comparison with the measured natural frequencies for different values of water level.Starting from (44) and following [21, 22], a new variable z=y˙ is introduced so that (44) is equivalent to (45)Mz˙+Cy˙+Ky=fy˙=zwhich in turn can be written as (46)CM-I0y˙z˙+K00Iyz=f0.The matrix (47)B≔CM-I0=Cs0M00CfϱfQ⊤S-I0000-I00is nonsingular because det⁡(B)=det⁡(M)≠0. Defining (48)A≔K00I=K-Q000H0000I0000IF≔f0=f000and putting v≔u˙, q≔p˙, x≔[yz]⊤=[upvq]⊤, we can write (41) as (49)Bu˙p˙v˙q˙=Aupvq+FFrom now on, we consider the system of ODEs (50)Bx˙t=Axt+Ft,t∈0,Tx0=x0with the obvious identifications as concerns x0. In order to solve (50), the solution of the homogeneous equation Bx˙(t)=Ax(t),t∈]0,T] can be sought in the form x(t)=beλt, yielding the generalized eigenvalue problem Ab-λBb=0. Let m≔dim⁡A=dim⁡(B)=2(3Ns+Nf). If all the eigenvalues are distinct, the matrix pencil (A,B) is diagonalizable; that is, there exist m×m matrices W and V such that (51)W∗AV=ΩA≡diagα1,…,αm,W∗BV=ΩB≡diagβ1,…,βm.The matrices W and V contain the left and right eigenvectors of (A,B), that is αi=wi∗Avi and βi=wi∗Bvi. In practice, since the response of the mechanical system is typically dominated by a relatively small number of the lowest modes, it may not be necessary to compute the complete eigensystem of (A,B). Other approaches [12, 13], based on pseudo-symmetric techniques, can be used for computing the mode shapes and natural frequencies of this fluid-structure model.Using the transformation for modal coordinatesx=Vz and multiplying both sides of (50) by W∗, yields the modal system (52)ΩBz˙=ΩAz+Cin0,Tz0=z0with C=W∗F and z0 solution of the linear system Vz0=x0. From (51) we get z0=ΩA-1W∗Ax0 or z0=ΩB-1W∗Bx0 so that the calculation of z0 only requires direct matrix multiplications. Now we have a system of m independent first-order linear ordinary differential equations, which is easy to solve. Each equation of the system is of the form (53)βiz˙it=αizit+cit,t∈0,T,withzi0=zi,0where ci is the i-th component of the vector C, which in practice is only known in a set of discrete points.


## 4.3. Time Discretization of the Equations

For the discretization in time of the diagonalized system (52), we take the solution of each equation (53) in its integral form (54)zit=eλitzi0+eλit∫0te-λiτc~iτdτwhere λi≔αi/βi are the state eigenvalues and c~i=ci/βi, i=1:m. Each function ci is known in discrete instants of time 0<t1<⋯<tn-1<tn=T with Δt≔tk+1-tk, k=1:n-1, and therefore will approximate ci by a constant in each interval [tk,tk+1]: (55)cit≃citkt∈tk,tk+1.From the relation (56)zitk+1=eλiΔtzitk+eλitk+1∫tktk+1e-λiτc~iτdτand the approximation (55), we get (57)zitk+1≃eλiΔtzitk+eλitk+1c~tk∫tktk+1e-λiτdτ=eλiΔtzitk+ctkλieλiΔt-1.In vector form, the algorithm for computing z reads (58)ztk+1≃eΛΔtztk+Λ-1CtkeΛΔt-Iwith Λ=diag(λ1,…,λm). Once z is known, (59)xtk=Vztk,k=1:nwill provide the solution for the original variable.


## 5. Numerical Simulations

The objective of this section is to illustrate how the displacement-pressure model can be used for studying the dynamic behavior of a dam. For this purpose, several numerical simulations of Cabril dam (shown in Figure1(a)) were performed, considering different water levels. This is a common procedure [23–25] as the natural frequencies of the system depend on the reservoir water level. Cabril dam, built in 1954, is the highest Portuguese arch dam. It presents some cracking near the crest which was considered in the computational geometry of the model (these are horizontal cracks, developed since the first filling of the reservoir due to a particular design shape of the dam crest). The numerical results will be compared with available experimental data from a permanent monitoring vibration system installed by LNEC in 2008. More precisely, the computed main natural frequencies (module of the state eigenvalues λi introduced in Section 4) and the corresponding modal configurations (obtained from the components ui of the state eigenvectors vi) will be compared with experimental data identified from acceleration records. A fast Fourier transform (FFT) technique was used in [26] to obtain the natural frequencies from these acceleration records.Figure 1
Cabril dam and computational domain.
(a)
Located at Zêzere river, 132 m high (b)
Finite-element meshes


### 5.1. Material Parameters and Finite Element Modeling of Cabril Arch Dam

A 3D finite element model of the dam-water system was developed, using meshes composed of serendipity isoparametric finite elements of cubic type, with 20 nodes in each element (see [27]) in the structure and fluid domains.Concerning the concrete and foundations parameters, Young’s modulus isE=32.5GPa, the Poisson’s ratio is ν=0.2, and the concrete specific mass is 2.45tn/m3. Although the damping effect on the structure is well defined through the drag operator Cs, the so-called Rayleigh’s hypothesis was used to model material damping. This assumes Cs as a linear combination of the mass and stiffness matrices, Cs=αM+βK, and we took, as in previous studies [26, 28], α=0.5 and β=0.0025.The finite element algorithm described in Section4, based on the geometry shown in Figure 5(b), was implemented in a Matlab program which allows to compute the modal characteristics of the whole system (static analysis) and the corresponding displacements and stress histories under prescribed forces (dynamic analysis), in particular, under seismic loads.


### 5.2. Comparison between Numerical and Experimental Results of Cabril Dam. Natural Frequencies and Modal Configurations

We recall that the approximation process to simulate the behavior over time of the structure and the fluid ended up in the form(60)z˙it=λizit+citβi,t∈0,T,withzi0=zi,0,i=1:m,the original variable x being recovered from a linear combination of the right eigenvectors of the state-space pencil (A,B)(61)x=Vz=∑i=1mzivi,To eachvibration mode, i.e., each eigenvector vi, i∈{1,…,m}, there corresponds an eigenmode of the state-space system which is characterized by the way of responding to external forces, i.e., by vibrating with a certain frequency fi=λi, called anatural frequency of the system, and a certain relative damping coefficient ξi=-Re(λi)/fi.Mode shapes and natural frequencies for full reservoir are presented in Figure2. In the simulations, 202 serendipity finite elements with 20 nodes each were used at the dam body and foundations and 424 elements of the same type were defined at the fluid domain. Here only the displacement of the structure is shown for the first 8 modes. Concerning the graphical representation of the computed mode shapes, it should be noted that each mode i corresponds to an oscillatory motion at the frequency λi which is represented by waves at the nodal points (three waves at each point) whose amplitude and phase are calculated from the complex components of the eigenvector ui.Figure 2
Main natural frequencies and mode shapes, computed for full reservoir.In Figure3 we present a comparison between numerical and experimental values of the main natural frequencies of the dam-reservoir system for different values of the water level. The experimental values of the natural frequencies were obtained during the year 2014 by the analysis of several acceleration records from a permanent vibration monitoring system installed in Cabril dam [26]. The monitoring system includes 16 acceleration sensors (uniaxial sensors for measuring radial accelerations) located in the dam body, near the top: 9 at the crest gallery and 7 in a gallery under the cracked zone. Acceleration files were recorded every hour, during several months, in order to experimentally characterize the influence of the water level on the main natural frequencies of the system. The experimental natural frequencies were identified as abscises of spectral peaks obtained using a FFT technique to analyze the time series of measured accelerations.Figure 3
Variation of the main natural frequencies with the water level. Comparison between numerical and experimental values.The results presented in Figure3 show a good agreement between identified and computed natural frequencies for different values of water level, meaning that the implemented numerical model is reliable and well calibrated. Note that, for each mode, the natural frequency increases as the water level decreases, as usually calculated with classical approaches based on the simplified model of associated water masses proposed by Westergaard [23] for the simulation of hydrodynamic pressure on the upstream dam face.


### 5.3. Simulation of Seismic Behavior of Cabril Dam Using the Calibrated Model

We used the accelerogram of the 921 Earthquake, also known as Chichi Earthquake, as input data for ground motionsa→s. This accelerogram presents a peak ground acceleration of 0.1g (=1 m/s2), as shown in Figure 4. Such intensity corresponds to a strong perceived shaking when the earthquake occurs [29] and is in agreement with the characteristics of the seismic zone where the dam is located [30].Figure 4
Seismic accelerogram: magnitude ofa→s as a function of time.Figure 5
Computed seismic response of Cabril dam for a peak ground acceleration of 0.1g: displacement field.
(a)
Radial displacement history at the top (crest central point) (b)
Displacement field at the instant when the maximum displacement towards upstream occurs (approx. 15 mm)The corresponding time response was computed considering the diagonalization technique for the time integration described in Section4, which lead to algorithms (60)-(61). A reduced number of eigenvectors, more precisely 160 vibration modes, were fixed after several numerical tests have been made with a larger number of modes and response values with the same precision were obtained. Figure 5 shows the radial displacement response over time at the top of the central cantilever, with amaximum displacement at the top of the central section of about 15 mm.Themaximum stress values occur at the crest and are of about 1.6 MPa (tension and compression). In conclusion, Cabril dam withstands against the applied earthquake.


## 5.1. Material Parameters and Finite Element Modeling of Cabril Arch Dam

A 3D finite element model of the dam-water system was developed, using meshes composed of serendipity isoparametric finite elements of cubic type, with 20 nodes in each element (see [27]) in the structure and fluid domains.Concerning the concrete and foundations parameters, Young’s modulus isE=32.5GPa, the Poisson’s ratio is ν=0.2, and the concrete specific mass is 2.45tn/m3. Although the damping effect on the structure is well defined through the drag operator Cs, the so-called Rayleigh’s hypothesis was used to model material damping. This assumes Cs as a linear combination of the mass and stiffness matrices, Cs=αM+βK, and we took, as in previous studies [26, 28], α=0.5 and β=0.0025.The finite element algorithm described in Section4, based on the geometry shown in Figure 5(b), was implemented in a Matlab program which allows to compute the modal characteristics of the whole system (static analysis) and the corresponding displacements and stress histories under prescribed forces (dynamic analysis), in particular, under seismic loads.


## 5.2. Comparison between Numerical and Experimental Results of Cabril Dam. Natural Frequencies and Modal Configurations

We recall that the approximation process to simulate the behavior over time of the structure and the fluid ended up in the form(60)z˙it=λizit+citβi,t∈0,T,withzi0=zi,0,i=1:m,the original variable x being recovered from a linear combination of the right eigenvectors of the state-space pencil (A,B)(61)x=Vz=∑i=1mzivi,To eachvibration mode, i.e., each eigenvector vi, i∈{1,…,m}, there corresponds an eigenmode of the state-space system which is characterized by the way of responding to external forces, i.e., by vibrating with a certain frequency fi=λi, called anatural frequency of the system, and a certain relative damping coefficient ξi=-Re(λi)/fi.Mode shapes and natural frequencies for full reservoir are presented in Figure2. In the simulations, 202 serendipity finite elements with 20 nodes each were used at the dam body and foundations and 424 elements of the same type were defined at the fluid domain. Here only the displacement of the structure is shown for the first 8 modes. Concerning the graphical representation of the computed mode shapes, it should be noted that each mode i corresponds to an oscillatory motion at the frequency λi which is represented by waves at the nodal points (three waves at each point) whose amplitude and phase are calculated from the complex components of the eigenvector ui.Figure 2
Main natural frequencies and mode shapes, computed for full reservoir.In Figure3 we present a comparison between numerical and experimental values of the main natural frequencies of the dam-reservoir system for different values of the water level. The experimental values of the natural frequencies were obtained during the year 2014 by the analysis of several acceleration records from a permanent vibration monitoring system installed in Cabril dam [26]. The monitoring system includes 16 acceleration sensors (uniaxial sensors for measuring radial accelerations) located in the dam body, near the top: 9 at the crest gallery and 7 in a gallery under the cracked zone. Acceleration files were recorded every hour, during several months, in order to experimentally characterize the influence of the water level on the main natural frequencies of the system. The experimental natural frequencies were identified as abscises of spectral peaks obtained using a FFT technique to analyze the time series of measured accelerations.Figure 3
Variation of the main natural frequencies with the water level. Comparison between numerical and experimental values.The results presented in Figure3 show a good agreement between identified and computed natural frequencies for different values of water level, meaning that the implemented numerical model is reliable and well calibrated. Note that, for each mode, the natural frequency increases as the water level decreases, as usually calculated with classical approaches based on the simplified model of associated water masses proposed by Westergaard [23] for the simulation of hydrodynamic pressure on the upstream dam face.


## 5.3. Simulation of Seismic Behavior of Cabril Dam Using the Calibrated Model

We used the accelerogram of the 921 Earthquake, also known as Chichi Earthquake, as input data for ground motionsa→s. This accelerogram presents a peak ground acceleration of 0.1g (=1 m/s2), as shown in Figure 4. Such intensity corresponds to a strong perceived shaking when the earthquake occurs [29] and is in agreement with the characteristics of the seismic zone where the dam is located [30].Figure 4
Seismic accelerogram: magnitude ofa→s as a function of time.Figure 5
Computed seismic response of Cabril dam for a peak ground acceleration of 0.1g: displacement field.
(a)
Radial displacement history at the top (crest central point) (b)
Displacement field at the instant when the maximum displacement towards upstream occurs (approx. 15 mm)The corresponding time response was computed considering the diagonalization technique for the time integration described in Section4, which lead to algorithms (60)-(61). A reduced number of eigenvectors, more precisely 160 vibration modes, were fixed after several numerical tests have been made with a larger number of modes and response values with the same precision were obtained. Figure 5 shows the radial displacement response over time at the top of the central cantilever, with amaximum displacement at the top of the central section of about 15 mm.Themaximum stress values occur at the crest and are of about 1.6 MPa (tension and compression). In conclusion, Cabril dam withstands against the applied earthquake.


## 6. Conclusions

We studied a fluid-structure interaction model in pressure-displacement formulation which was derived from the Euler and Navier equations, as explained in [3].The difficulty in the theoretical study of the variational formulation of the coupled equations was caused by the unbalanced order of the time derivatives of the pressure and the structure’s displacement in the interface conditions. We introduced a pressure potential so that the classical techniques based on Galerkin approximations for second-order hyperbolic equations could be applied to investigate the well posedness of the fluid-structure interaction problem.Concerning the discretization of the problem, the application of the Finite Element Method (FEM) to the space variables was followed by a discretization in time associated with a state space approach appropriate for the dynamic analysis of arch dams. This numerical formulation was tested through a comparative study of the dynamic behavior of Cabril dam, namely, the simulation of dam’s natural frequencies and vibration mode shapes, for different reservoir water levels. The good agreement observed between modal identification outputs (from real measurements) and numerical results shows the reliability of the state space formulation implemented in the developed 3-D FEM program.



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*Source: 8083906-2019-01-02.xml*