Flexibility of dam structure affects the hydrodynamic pressure acting on the dam. Several approaches have been proposed to consider this effect. Most of these approaches are involved with an iterative scheme. Of course solving the total numerical model including the dam and the reservoir is the most accurate method, but it has certain deficiencies. Using the frontal solution method of total model, dam structure, and fluid domain and keeping the interface degrees of freedom in the front is proposed in the current study. Having the solution of the interface degrees of freedom, the structure and fluid may be analyzed separately. The main advantage of the method lies in the fact that the accuracy of the results is the same as analysis of the total model, no iteration is necessary, combination of Lagrangian and Eulerian formulations for solid and fluid may be used, and the unknown variables are of the same order. Performing the analysis in time domain extends the method to nonlinear analysis if required.

Although the dams are very stiff structures and experience small amount of deformations during earthquakes, these deformations are the source of drastic change in hydrodynamic pressure. Several approaches have been proposed to solve the coupled equations. One approach to include this interaction, uses iterative schemes, where first the dam geometry is assumed to be constant when the reservoir is solved. Then obtained pressure will be applied on the dam, and new configuration of dam is calculated. The iteration will be continued until the convergence is achieved. Fellippa and Park [

In the other approach, a complete model of the dam and reservoir is formed. Moreover, the two formulation methods of Eulerian (suitable for reservoir formulation) and Lagrangian (suitable for structure formulation) have to be combined or one of them has to be selected. Wilson and Khalvati [

Both of the abovementioned solution methods may be performed in frequency or time domain. A lot of work has been done by Chopra and coadjutor on dam-reservoir-foundation interaction in frequency domain covering several aspects of the problem [

Simulation method in time domain needs massive storage space because of solving all of equations including structure and fluid as well as far field variables for each time step. Furthermore, ill conditioning is dramaticaly conflictive and occurs during solving the equations of very different order variables.

Many analytical research studies have been carried out about dam-reservoir system. Tsai, Lee, and Ketter [

The proposed solution method in this study is a branch of complete model solution with the exception that only the selected degrees of freedom on the interface of dam and reservoir will be solved in time domain. The principal aim of this study is to employ frontal solution for solving the dam-reservoir system. The frontal method for solving the equations was presented by Irons [

The achievements can be explained as follows:

In the first step of the scheme, only the value of the pressure which acts on the interface is solved without losing any accuracy comparing with other solution methods which solve the entire system

In the second step of the scheme, Lagrangian or Eulerian approach can be used, which was better, for the solid domain and the fluid domain

This scheme is capable to model the nonlinear problems if time domain formulation is adopted

The dam-reservoir system is classified as a coupled problem where two fields each governed by a second-order differential equation, interact at their interface. The equation of motion for dam structure due to earthquake motion is given as

In above integration on the interface of the two domain, vector

In this study, Rayleigh damping is assumed, and so

The equation of motion for reservoir domain due to earthquake motion is given as [

Equations (

Equation (

By using step by step integration scheme of Newmark [

Multiplying the reservoir motion equation by

Different approaches have been proposed to solve the coupled equations. Most popular approach is iterative schemes [

An arbitrary pressure

Calculated

Calculated

Calculated

Check for convergence: if

This approach reduces coupled problem to subsystems. Therefore, only symmetric equations for each subsystem is calculated (Equations (

The direct numerical solution of dam and reservoir together in time domain may be considered as the best interaction solution for the following reasons:

Except the theoretical solution, it is the most accurate one

It does not have the geometrical and boundary condition limitations

It is capable of considering the nonlinear phenomenon

Although it has its own deficiencies,

the solution is extremely expensive,

it has to use uniform formulation; Lagrangian or Eulerian for both the dam and reservoir which Lagrangian and Eulerian formulation are suitable for the dam structure and the reservoir, respectively,

solving the equations of different order of variables has its own concerns, although it is solvable.

In common methods of matrix solution, the element’s information is gathered in element records and is kept on peripheral. Therefore, the element matrices, such as stiffness, are computed again any time needed. The advantage lies on the fact that almost all memory is free for assemblage of global stiffness.

Frontal solution of large matrices is based on special assembly. As soon as the coefficients of an equation are assembled from the contributions of all relevant elements, the corresponding variable can be eliminated [

In frontal method, the equations which are being formed at any given instant, their corresponding nodes, and degrees of freedom are named the front. The number of the variables in the front is named front width. The equations, nodes, and degrees of freedom (DOFs) belonging to the front are named active; those which have passed through the front and have been eliminated are named inactive. Active nodes can be deactivated after the last appearance. By eliminating variables as soon as their assembly have been completed, core storage is made available for variables yet to be assembled. The advantage lies on the fact that required memory for the solution will be reduced to the front.

Taking advantage of frontal technique to keep the pressure DOFs on the interface of dam and reservoir in the front can highly help the solution. The procedure of the assembly for structure and reservoir by frontal method is illustrated in Figure

Frontal assembly procedure for two coupled domains. (a) The schematic model including fluid elements, structure elements, and nodes before beginning the assembly. (b) The first element of each media at the beginning of assembly. (c) The second element of each media for assembling. (d) Active nodes and deactivated nodes at the final step of assembly.

The solution basis for frontal method is Gaussian elimination. The interaction equations of dam and reservoir contain solid DOFs ({

{

Proposed solution procedure in this study is based on the frontal method; however, the order of elimination is somehow different. The solution procedure is illustrated in Figure _{s}} (sth equation in front). It must be noted that variable {_{s}} is _{s}} by using Gaussian elimination as follows:

Frontal solution procedure for dam-reservoir problem.

When assembly for all elements is finished, and all the variables are eliminated except the interface pressure variable. All the interface pressure DOFs remain in the front as shown in Figure

It was shown that in frontal solution, the pressure variables on interface can be kept in front and can be solved without continuing the solution for all equations. Then, the achieved results can be used as the boundary condition for solving the reservoir and as loading profile for solving the dam. In this way, no iteration is needed, there is no loss of accuracy, and each media can be solved by using its suitable formulation. Obviously, all deficiencies are removed without losing the accuracy.

A special computer code was generated to determine the dynamic response of structure-fluid systems by using frontal method in basis of abovementioned algorithm in two dimensional cases. The generated code uses the eight node plain strain elements for discretization of the structure domain and four node elements for discretization of the fluid domain. In this study, fluid is considered as compressible, inviscid, and irrotational, and concrete is assumed to be homogenous and isotropic.

As a first example, the dam reservoir system analyzed by Lee and Tsai [^{9} ton·m^{2} and weight per unit length of the structure = 36 ton/m. Lee and Tsai [

Ramp acceleration.

Geometrical detail of structure-reservoir system.

The material of the dam was assumed to be linearly elastic. For the concrete of the dam, Poisson’s ratio was taken as 0.2. For the water, unit weight and sound speed in the water were taken as 1000 kg/m^{3} and 1440 m/s, respectively.

In order to evaluate the accuracy and capability of the proposed solution, results are presented and compared with those presented by Lee and Tsai [

Crest displacement without interaction.

Crest displacement with interaction.

Hydrodynamic pressure at bottom of the reservoir.

For comparison purpose, the model was analyzed by using two different programs: first program solved the equations by frontal method and second program solved the equations by iterative scheme. The results are presented in Figure

Crest displacement with interaction.

Hydrodynamic pressure at bottom of the reservoir.

For second example, Pine Flat concrete gravity dam with 121.92 m height and reservoir with 116.12 m depth and 580 m length (Figure ^{3}, 0.2, and 22.75 GPa, respectively. For the water, unit weight and sound speed in the water were taken as 1000 kg/m^{3} and 1440 m/s, respectively.

Geometrical detail of dam-reservoir system.

The Taft Lincoln earthquake record (1952).

Crest displacement obtained from present study and response obtained by Fenves and Vargas-Loli [

Pine Flat crest displacement, peak acceleration scaled to 1.0 g.

As shown in Figure

In Figure

Pine Flat crest displacement, peak acceleration scaled to 1.0 g.

Hydrodynamic pressure at Pine Flat heel, peak acceleration scaled to 1.0 g.

As shown in the figures, excellent agreement between the results is achieved from proposed method and iterative solution.

Table

Execution time for two solution scheme.

Frontal method | Iterative method | |
---|---|---|

Example 1 | 20 | 27.36 |

Example 2 | 28.2 | 46.53 |

Frontal solution is proposed for solving the dynamic analysis of concrete gravity dams. Coupled equations of dam-reservoir interaction in time domain are obtained and solved by frontal method. A Fortran program is developed and its accuracy is verified by comparing results with published results. Two examples of dam-reservoir model were analyzed. The results of frontal solution were compared with the results of iterative method. The major advantages of the frontal solution are as follows:

The accuracy of the solution is the same as solution of the total equations in each time step.

Separate solution of the structure and reservoir causes the suitable formulation for each one to be used, namely, Lagrangian and Eulerian, respectively.

Solving very different order variables is eliminated.

The presented scheme takes less system memory and calculation time than iterative method. For two models examined in this paper, execution time was reduced to 37 and 65 percent for example 1 and example 2, respectively.

The presented scheme does not have the geometrical and boundary condition limitation.

The presented scheme is capable of considering the nonlinear phenomenon.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.