Advances in Mathematical Physics

Volume 2019, Article ID 8083906, 14 pages

https://doi.org/10.1155/2019/8083906

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

^{1}Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal^{2}Concrete Dams Department, Laboratório Nacional de Engenharia Civil, Av. Brasil 101, 1700-066 Lisboa, Portugal^{3}CEMAT and Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Correspondence should be addressed to Ana L. Silvestre; tp.aobsilu.ocincet.htam@ertsevlis.ana

Received 22 August 2018; Accepted 5 November 2018; Published 2 January 2019

Academic Editor: Phuc Phung-Van

Copyright © 2019 Ricardo Faria et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 Section 2. 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 by , while the fluid domain (reservoir) is denoted by . Under the assumptions (i) small displacements and deformations in the structure, (ii) fluid displacement remains small; while interaction is substantial, the domains 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 and , we have the following: is the interface between and ; represents the ground or rock mass under the fluid; is the air-fluid interface; is an artificial ’wall’, a delimitation of the fluid domain’s extension; represents the rock mass under the structure; and finally, is constituted by the structure walls with contact with the air. The time interval domain during which the dynamic fluid-structure interaction occurs is , .

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 and initial conditions , , , and , find and such thatThe gravitational acceleration and the acceleration of the ground due to earthquake vibrations are included in the Navier equation, where is the structure density and is a positive constant representing a damping effect. The Cauchy stress tensor is given by , where and the parameters and are the Lamé constants. They relate to other elastic moduli constants, namely the Young’s modulus and the Poisson ratio , by: , The structure domain boundary is and is the outward unit normal vector at . 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, . The constant is the speed of sound in the fluid and is the acceleration due to gravity. The fluid domain boundary is . In the air-water interface , a wavelike motion is observed, and is permeable in the sense that the solution of (1b) in should be composed only of outgoing waves on , as no input from this boundary portion exists [3].

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

##### 3.1. Notation and Auxiliary Results

Let be a bounded domain and . We will use the classical notations and results for Lebesgue spaces , , , Sobolev spaces , , and their dual spaces . By and we denote the norms of and , respectively. We will use the notation for vector valued functions, and when and are tensor-valued functions, we will write , where represents contraction of tensors. Analogous meaning holds for .

Suppose that has nonzero -dimensional measure and let Recall that the Korn inequality is valid in : . Therefore is a Hilbert space with respect to the inner product and the associated norm is equivalent to the norm induced by .

If is a Banach space based on the domain , we will write for its dual space and for the evaluation of at . The Bochner spaces associated with will be denoted by , , will denote the space of weakly continuous functions with values in and will be a space of distributions.

We recall the method of mollifiers, which provides approximation by smooth functions. Here we will consider and mollification in the variable. Let the function satisfy the following properties: The mollifier , , is defined by (). For a Banach space , a smoothing operator can be defined by convolution with in the following way: given and , set for all , and let be defined by . Then Since convolution commutes with differentiation, we have and if , where and is a Banach space. Another property of mollification is + .

We will also use the following classical embedding result (see [19], pg. 392, Lemma 11.9).

Lemma 1. *Let be Hilbert spaces such that . Then *

##### 3.2. Variational Formulation

In order to derive the variational formulation of Problems (1a)-(1j), we consider the function spaces where is equipped with inner product and is equipped with inner product .

Suppose and are smooth solutions of Problem (1a)-(1j). Multiplying both sides of (1b) by and integrating by parts over yields where by (1d), (1e), (1f), and (1c)Therefore Analogously, dot multiplying both sides of (1a) by , integrating by parts over , and using yields Imposing the boundary conditions (1g) and (1h), we obtain

Hence the following variational problem is associated with (1a)-(1j): find and such thatand , , , and , 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]): In (12), and are the so-called mass operators, and are drag operators, and are stiffness operators, and and are interaction operators for the structure and the fluid, respectively. Note that , 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 of (1a)-(1j) exists and is sufficiently regular. If we try to obtain a (formal) energy estimate for such a solution by replacing in (11a) by and by in (11b), followed by time integration in , we end up with the identity

In order to “skew-symmetrize” the bilinear form associated with the equations on , so that the unbalanced terms are not present in the energy equation, it is convenient to introduce a* pressure potential* for (with respect to the variable): , .

Note that taking into account the relation , in , between the fluid velocity and pressure in this model (see [3], pg. 634), the variable can be seen as a velocity potential for the eliminated unknown in the fluid equations. This is consistent with the assumption of irrotational flow. Now, replacing in (11a) yields while the same replacement in (11b) produces a higher order ordinary differential equation Then where is a constant (distribution) and, more precisely, . If , , and are regular functions with , then the chosen value for is just the value of at . Hence, we consider the new equation for the unknown which is the result of formally integrating (11b) in . The same procedure that leads to (13) allows us to derive a formal energy equation for and (18), in turn, yields the following basic a priori estimates for :where the constants , , depend on the data. Relation (18) suggests that the pair satisfies , and , and . 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, with . Moreover, if and satisfies and in , it follows that and . Note that (21) is obtained from (17) taking test functions .

*Remark 2. *The variational formulations (14) and (17) are actually a formulation in displacement-velocity potential. From and the relation , in , between the fluid velocity and the pressure in this model (see [3], pg. 634), we get .

##### 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 a* weak solution* to Problems (1a)-(1j) is a pair satisfying (11a)-(11b) andThe main result concerning the well posedness of the mathematical models (1a)-(1j) is as follows.

Theorem 3. *Let be locally Lipschitz bounded domains with the common boundary , as described in Section 2. Assume that , , , and . Then there exists a unique solution to Problems (11a)-(11b).*

Theorem 3 will be proved along several steps in the next subsections. The a priori estimates (19a)-(19c) for 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 .

##### 3.5. Galerkin Approximations

Let and be basis of smooth functions for and , respectively. Let and . We seek approximating solutions in the form and with the coefficients and () obtained fromfor , and the initial conditions where and are projection operators onto and , respectively. This is a system of second-order ODEs, which has a unique solution. Indeed, (23a)-(23b) can be written in the form

where we have used the notation and The matrices and are symmetric, positive definite and since the matrix is nonsingular.

Letso that and . Then the approximating functions satisfyNow we multiply (29a) by and (29b) by and sum from to to conclude that and satisfy relations similar to (14) and (17) in the spaces and and, consequently, the a priori estimates (19a)-(19c).

Lemma 4. *For the Galerkin approximations it holds that *

The uniform bounds collected in Lemma 4 allow us to obtain a pair as a weak limit of the Galerkin approximations such that together with the identitiesfor all and and all null in . We now take and in (32a)-(32b) and obtain From these two relations, it follows and . By Lemma 1 and the well-known density result , we obtain the following.

Lemma 5. * and .*

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

Let have the weak continuity properties established in the previous section and consider the* energy function* defined by We shall prove that and use this result to prove the strong continuity of . To accomplish this, we will resort to the following intermediate result.

Lemma 6. *If is the weak solution obtained in the previous section then and is an absolutely continuous function.*

*Proof. *Once (35) is proved, since we immediately conclude that is an absolutely continuous function.

To prove (35), we use the mollification method, as described in Section 3.1. Consider , some fixed with , , and let . Inserting the special test functions and , with , in (32a)-(32b) and a simple calculation yieldsNow, since and , we can take and in (37a)-(37b), thus obtaining We let and use the convergence properties (3)-(4) to obtain (35) in . Since are arbitrary, we conclude that (35) holds a.e. in .

The result of Lemma 5 can be improved to

Lemma 7. *The pair constructed in the previous subsection satisfies and the energy equality (18).*

*Proof. *Using the weak continuity properties of and the continuity of from Lemma 6, we find that