## Introduction

The dam-reservoir system can be categorized as a coupled field system in which two physical domains of fluid and structure interact only at their interface. In such a problem, the presence of interaction implies that the time response of both subsystems must be evaluated simultaneously. Different approaches to the solution of the coupled field problem exist. Field elimination, simultaneous solution and partitioned solution are the three classes of solutions for the coupled field system. The advantages and disadvantages of each method were addressed by Felippa and Park (1980). The field elimination approach is not feasible in the case of nonlinear problems. The reduced system of equations has high order derivatives which cause some difficulties in applying the initial conditions. The simultaneous solution is time consuming and involves many operations, especially when a large number of elements is used. This method contains matrices with a large bandwidth and consequently requires a large amount of memory especially for the cases when the existing matrices are not symmetric. In addition, the main disadvantage of the first two classes of solution arises from the difficulties encountered in using available software while the partitioned solution has the capability of using existing software for each subsystem. Staggered solution was described by Felippa and Park (1980) as a partitioned solution procedure that can be organized in terms of sequential execution of single-field analyser.

Most of the physical systems are made of subsystems which interact with each other. These physical systems which are referred to as coupled systems, have been investigated by several researchers. Methods of solution vary depending on the governing differential equations of the subystems and may lead to different degrees of accuracy and stability of the solution (Park 1980). Coupled problems and their numerical solutions were addressed by Felippa and Park (1980); Park and Felippa (1984). Zienkiewicz and Chan (1989) proposed an unconditionally stable method for staggered solution of soil-pore fluid interaction problem. Huang (1995) proposed two unconditionally stable methods for the analysis of soil-pore fluid problem. The methods were named pressure correction method and displacement correction method. Zienkiewicz and Chan (1989) presented an unconditionally stable method for staggered solution procedure for the fluid-structure interaction problem. Their method was proved to be unconditionally stable when no damping term was included in the equations of the fluid and the structure. However, when the damping term is included in the equation of the subsystems, the proposed method may not be unconditionally stable. The problem of solutions instability when the damping term is included in the differential equation, was recognized by researchers. Most of the staggered solution applications in the field of fluid-structure interaction were conducted using a method which is not unconditionally stable.

There are two method of solution for dynamic analysis of a system. Time domain and frequency domain solutions are two methods of solutions and have different application depending on the nature of the system. For a linear system both method can be used while for a nonlinear system only time domain solution can be used to evaluate the dynamic response of the system.

The behaviour of a nonlinear system is different than the linear system. Lack of superposition property, multiplicity of equilibria and domains of attraction, local and global stability, frequency dependence of amplitude of free oscilation, the jump response, subharmonic and superharmonic generation are some of characteristics of a nonlinear system. In particular, when a linear system is subjected to harmonic excitation, the steady state response will be another harmonic with the same frequency and will be independent of the initial conditions. A nonlinear system may behave in a different manner. This character of nonlinear systems make them unable to be solved using frequency domain method.

In this Chapter, two methods of staggered solution procedure are applied to the dam-reservoir interaction problem. Both methods are shown to be unconditionally stable when the two differential equations of the fluid and structure include damping terms. The accuracy of the solution using both of the proposed methods, is investigated. Two different configurations of concrete gravity dams are analysed to illustrate the application of the proposed procedure and to compare the solution with available finite element solutions.

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